numericalanalysisconference.org.uk...Introduction Dear Participant, On behalf of the Strathclyde...

24th Biennial Conferenceon

Numerical Analysis

28 June - 1 July, 2011

Contents

Introduction 1Invited Speakers 3List of Participants 4Abstracts of Invited Talks 8Abstracts of Minisymposia 11Abstracts of Contributed Talks 28Timetable 48

Nothing in here

Introduction

Dear Participant,

On behalf of the Strathclyde Numerical Analysis Group, it is my pleasure to welcome you tothe 24th Biennial Numerical Analysis conference. This is the second time the meeting has beenheld at Strathclyde and continues the long series of “Dundee Numerical Analysis Conferences”.

The conference is rather unique in the sense that it seeks to encompass all areas of numericalanalysis, and the list of invited speakers reflects this aim. We have once again been extremelyfortunate in securing, what I hope you will agree is, an impressive line up of eminent plenaryspeakers.

The conference is funded almost entirely from the registration fees of the participants, aug-mented with contributions from the A.R. Mitchell Fund and the Gene Golub Fund. This year,we are grateful to the EPSRC and Scottish Funding Council funded centre for Numerical Algo-rithms and Intelligent Software (NAIS), for providing funding for some of the invited speakers.We are also indebted to the City of Glasgow for once again generously sponsoring a wine recep-tion at the City Chambers on Tuesday evening to which you are all invited.

We hope you will find the conference both stimulating and enjoyable, and look forward towelcoming you back to Glasgow again in 2013!

Mark AinsworthConference Chairman

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Information for participants

• General. There will be a registration desk in the foyer of the John Anderson building (building 16 on thecampus map, entry on Level 4 from Taylor Street as indicated). The conference office will be located in theCluster Manager’s office, room K4.12a of the John Anderson building. The organisers can be contactedthere during tea and coffee breaks.

• Accommodation. All rooms are in the Campus Village. Check-out time is 10:00 on day of departure.On Friday morning, luggage may be left in room K3.26.

• Meals. All meals (except lunch on Friday) will be served in the Lord Todd Dining Room (building 26 onthe campus map, entry as indicated). Breakfast is available from 07.30 until 09.00. The times of lunchesand dinners are as indicated in the conference programme. A buffet lunch will be served on Friday in thefoyer outside K3.25.

• Lecture rooms. These are mainly in the John Anderson building (building 16, enter on Level 4 fromTaylor Street). The main auditorium (K3.25) is down one floor from the main entrance, along with breakoutrooms K3.14, K3.17, K3.26 and K3.27. The additional breakout rooms are K4.12 (on the entrance level ofthe John Anderson building near the registration desk) and AR201 (in the Architecture building, no. 17on the campus map). These will be signposted.

• Chairing sessions. It is hoped that if you are listed as chairing a session, you will be willing to helpin this way. Minisymposium organisers should organise chairpeople for their own sessions (including anycontributed talks which follow) as appropriate. A break of 5 minutes has been allowed for moving betweenrooms. Please keep speakers to the timetable!

• Coffee and tea breaks. Coffee and tea will be provided at the advertised times in the foyer outsideK3.25.

• Bar. There is a bar in the Lord Todd building (building 26) next to the dining room.

• Reception. A reception for all participants hosted by Glasgow City Council will be held in the CityChambers on Tuesday 28th June from 20.00 to 22.00. The City Chambers is marked on the campus map:entry is from George Square.

• Conference dinner. The conference dinner will be held in the Lord Todd Dining Room on Thursday 30thJune at 19.00 for 19:30. The guest speaker will be Professor David Silvester, University of Manchester.

• Book displays. There will be books on display for the duration of the conference in room K3.26.

• Internet Access. Delegates will be provided with a username and password for internet access at reg-istration. Wireless access is available in all of the meeting rooms and in the Lord Todd bar/restaurant.Computer terminals will be available from 09:00-17:00 Tuesday-Thursday in room C3.09 of the ColvilleBuilding. This room can be accessed from the John Anderson building past lecture rooms K3.14 and K3.17and will be signposted.

• Sports facilities. Conference delegates can use the University sports facilities (building 3) by obtaininga card from the Student Village Office. The cost of the various facilities varies.

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Invited Speakers

Oscar Bruno Caltech [email protected]

Jack Dongarra University of Tennessee [email protected]

Chris Johnson University of Utah [email protected]

James Nagy Emory University [email protected]

Jorge Nocedal Northwestern University [email protected]

Djordje Peric Swansea University [email protected]

Alfio Quarteroni EPF de Lausanne [email protected]

Robert Schaback Universitat Gottingen [email protected]

Robert Skeel Purdue University [email protected]

Raul Tempone KAUST [email protected]

Nick Trefethen University of Oxford [email protected]

Barbara Wohlmuth TU Munchen [email protected]

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Participants

Shirley Abelman University of the Witwatersrand [email protected] Ainsworth University of Strathclyde [email protected] Akgun Middle East Technical University [email protected] Akman Institute of Applied Mathematics [email protected] Al-Baali Sultan Qaboos University [email protected] Allendes University of Strathclyde [email protected] Almarashi University of Strathclyde [email protected] Alrashidi University of Birmingham [email protected] Altintan Middle East Technical University [email protected] Andriamaro University of Strathclyde [email protected] Araujo University of Coimbra [email protected] Asner University of Oxford [email protected] Bandara University of Cambridge [email protected] Raul Barrenechea University of Strathclyde [email protected] Barth ETH Zurich [email protected] Berrut University of Fribourg [email protected] Bespalov University of Manchester [email protected] Betcke University College London [email protected] Bird University of Reading [email protected] Birkisson University of Oxford [email protected] Blanchard Grinnell College [email protected] Borsdorf University of Manchester [email protected] Brezinski University of Sciences and Technologies of Lille [email protected] Brito-Loeza Universidad Autonoma de Yucatan [email protected] Brown University of Canterbury [email protected] Brunner Memorial University of Newfoundland [email protected] Buckwar Heriot-Watt University [email protected] Burrage University of Oxford [email protected] Burrage Queensland University of Technology [email protected] Butler University College London [email protected] Cangiani University of Leicester [email protected] Chen University of Liverpool [email protected] Crofts Nottingham Trent University [email protected] Davies University of Strathclyde [email protected] Davydov University of Strathclyde [email protected] Deadman University of Manchester [email protected] Devolder Universite Catholique de Louvain [email protected] dos Reis Technical Universty Berlin [email protected] du Toit Smith Institute [email protected] Duncan Heriot-Watt University [email protected] England The Open University [email protected] Ergul University of Strathclyde [email protected] Fairag KFUPM [email protected] Fernandes Universidade do Minho [email protected] Fiala NAG Ltd [email protected] Fletcher University of Dundee [email protected] Gao China University of Petroleum [email protected] Garcia-Archilla Universidad de Sevilla [email protected] Georgoulis University of Leicester [email protected] Gonnet University of Oxford [email protected] Gould Science & Technology Facilities Council [email protected] Luis Gracia University of Zaragoza [email protected] Guettel University of Oxford [email protected] Hale University of Oxford [email protected] Hall University of Edinburgh [email protected] John Hart University of Edinburgh [email protected] Higham University of Strathclyde [email protected] Hogg STFC Rutherford Appleton Laboratory [email protected] Horvath Szechenyi Istvan University [email protected] Huangfu University of Edinburgh [email protected]

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Drahoslava Janovska Institute of Chemical Technology, Prague [email protected] Janovsky Charles University [email protected] Elizabeth Jenkins University of Bath [email protected] Kammogne University of Birmingham [email protected] Kang Communication University of China [email protected] Kecman Cambridge University [email protected] Rahim Khan King Fahd University of Petroleum and Minerals [email protected] Kirkland National University of Ireland [email protected] Klein University of Fribourg [email protected] Knight University of Strathclyde [email protected] Kocvara University of Birmingham [email protected] Kopteva University of Limerick [email protected] Kruse University Bielefeld [email protected] Lakkis University of Sussex [email protected] Lang Seminar fur Angewandte Mathematik [email protected] Langdon University of Reading [email protected] Lanzoni University College London [email protected] Lee University of Strathclyde [email protected] Lin University of Manchester [email protected] Linß Universitat Duisburg-Essen [email protected] Liu University of Strathclyde [email protected] Loghin University of Birmingham [email protected] Loisel Heriot-Watt University [email protected] Lotz The University of Edinburgh [email protected] Ludwig Technical University of Dresden [email protected] MacDonald University of Strathclyde [email protected] Mackenzie University of Strathclyde [email protected] Maischak Brunel University [email protected] V. Mantzaris University of Strathclyde [email protected] Martinsson University of Colarado [email protected] Matthews NAIS [email protected] McDonald University Of Strathclyde [email protected] Micheletti Politecnico di Milano [email protected] Miyajima Gifu University [email protected] Moroney Queensland University of Technology [email protected] Morton University of Oxford [email protected] Musolesi University of St Andrews [email protected] Nakatsukasa University of California, Davis [email protected] Newton Hewlett-Packard Laboratories [email protected] Nicaise University of Valenciennes [email protected] K Nichols University of Reading [email protected] Okayama Hitotsubashi University [email protected] Oseledets Russian Academy of Sciences [email protected] Pachon Credit-Suisse [email protected] Partridge University of Reading [email protected] Payne Heriot-Watt University [email protected] Pearson University of Oxford [email protected] Pekmen Atilim University [email protected] Petersen New York University [email protected] Phillips University College London [email protected] Phokaew University of Strathclyde [email protected] J.D. Powell University of Cambridge [email protected] Quinn Dublin City University [email protected]. Fabien Rabarison University of Strathclyde [email protected] Ramage University of Strathclyde [email protected] Rankin University of Strathclyde [email protected] Redivo-Zaglia University of Padova [email protected] Repin Steklov Institute of Mathematics at St. Petersburg [email protected] Riaz University of Strathclyde [email protected] Richardson University of Oxford [email protected] Richtarik University of Edinburgh [email protected]

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Martin Riedler Heriot-Watt University [email protected] Roj University of Strathclyde [email protected] Ruede University of Erlangen-Nuernberg [email protected] Rui University of Cambridge [email protected] Ruiz INPT - ENSEEIHT-Toulouse [email protected] Saeed University of Strathclyde [email protected] Savostyanov Russian Academy of Sciences [email protected] Scheichl University of Bath [email protected] Schieweck Institut fur Analysis und Numerik [email protected] Shi University of Bath [email protected] Silvester University of Manchester [email protected] Gerard Smith University of Strathclyde [email protected]ılia Sousa Coimbra University [email protected] Spence University of Bath [email protected] Stoyanov University of Reading [email protected] Stynes National University of Ireland, Cork [email protected] Sun Centre for Mathematical Imaging Techniques [email protected] Szpruch Univeristy of Oxford [email protected] Takac The University of Edinburgh [email protected] Tang Mamchester University [email protected] Taylor University of Manchester [email protected] Teckentrup University of Bath [email protected] Tezer-Sezgin Middle East Technical University [email protected] Thompson University of Edinburgh [email protected] Tisseur The University of Manchester [email protected] Townsend Oxford University [email protected] Tranah Cambridge University Press [email protected] Turk Middle East Technical University [email protected] Tyrtyshnikov Russian Academy of Sciences [email protected] Van lent University of the West of England [email protected] Varsakelis Universite catholique de Louvain [email protected] Verhille Universite des Sciences et Technologies de Lille [email protected] Vohralik University Paris 6 [email protected] von Schwerin King Abdullah University of Science and Technology [email protected] Wang University of Oxford [email protected] Wathen Oxford University [email protected] Watson University of Dundee [email protected] Yang Queensland University of Technology [email protected] Ping Yeo University of Strathclyde [email protected] Zhlobich University of Edinburgh [email protected] Zhu University of Oxford [email protected]

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Abstracts of Invited Talks

Spectral frequency- and time-domain PDE solversfor general domains

Oscar Bruno (Caltech)

We present fast frequency- and time-domain spectrally ac-curate solvers for Partial Differential Equations that ad-dress some of the main difficulties associated with simu-lation of realistic engineering systems. Based on a novelFourier-Continuation (FC) method for the resolution ofthe Gibbs phenomenon, associated surface-representationmethods and fast high-order methods for evaluation of in-tegral operators, these methodologies give rise to fast andhighly accurate frequency- and time-domain solvers forPDEs on general three-dimensional spatial domains. Ourfast integral algorithms can solve, with high-order accu-racy, problems of electromagnetic and acoustic scatteringfor complex three-dimensional geometries; our FC-baseddifferential solvers for time-dependent PDEs, in turn, giverise to essentially spectral time evolution, free of pollutionor dispersion errors. A variety of applications to linearand nonlinear PDE problems demonstrate the very signif-icant improvements the new algorithms provide over theaccuracy and speed resulting from other approaches.

On the Future of High Performance Computing:How to Think for Peta and Exascale Computing

Jack Dongarra (University of Tennessee)

In this talk we examine how high performance comput-ing has changed over the last 10-year and look towardthe future in terms of trends. These changes have hadand will continue to have a major impact on our software.Some of the software and algorithm challenges have al-ready been encountered, such as management of commu-nication and memory hierarchies through a combination ofcompile–time and run–time techniques, but the increasedscale of computation, depth of memory hierarchies, rangeof latencies, and increased run–time environment variabil-ity will make these problems much harder.

We will look at five areas of research that will havean importance impact in the development of software andalgorithms.

We will focus on following themes:

• Redesign of software to fit multicore and hybrid ar-chitectures

• Automatically tuned application software

• Exploiting mixed precision for performance

• The importance of fault tolerance

• Communication avoiding algorithms

Image-Based Biomedical Modeling, Simulation andVisualization

Chris Johnson (University of Utah)

Increasingly, biomedical researchers need to build func-tional computer models from images (MRI, CT, EM, etc.).The“pipeline” for building such computer models includesimage analysis (segmentation, registration, filtering), ge-ometric modeling (surface and volume mesh generation),large-scale simulation (parallel computing, GPUs), large-scale visualization and evaluation (uncertainty, error). Inmy presentation, I will present research challenges andsoftware tools for image-based biomedical modeling, sim-ulation and visualization and discuss their application forsolving important research and clinical problems in neuro-science, cardiology, and genetics.

Numerical Methods for Large Scale Inverse Prob-lems in Image Reconstruction

James Nagy (Emory University)

One of the most difficult challenges in numerical analysis isthe development of algorithms and software for large scaleill-posed inverse problems. Such problems are extremelysensitive to perturbations (e.g. noise) in the data. To com-pute a physically reliable approximation from given noisydata, it is necessary to incorporate appropriate regular-ization (i.e., stabilization) into the mathematical model.Numerical methods to solve the regularized problem re-quire effective numerical optimization strategies and effi-cient large scale matrix computations.

In this talk we describe how the challenges of solvinglinear inverse problems can be analyzed using the singularvalue decomposition, and how to efficiently implement theideas with iterative methods on realistic large scale prob-lems. We also discuss how the approaches can be adaptedfor use on nonlinear inverse problems. Applications frommedical imaging will be used for illustration.

Optimization Methods for Machine Learning

Jorge Nocedal (Northwestern University)

How good is today’s speech recognition software? How areessential features extracted from acoustic frames? Whatkind of numerical optimization techniques are capable oftraining a statistical model, using a very large data set,to make accurate predictions on such a difficult problem?I will begin this talk with a real time demonstration ofGoogle’s Voice Search to illustrate the progress that hasbeen made in speech recognition, as well as its limitations.This will lead us to the discussion of online vs batch op-timization methods, and to some interesting open ques-tions from the perspective of numerical optimization. Themain part of the talk will address a claim that is com-monly heard in the machine learning community, namelythat “the best optimization algorithms are not the bestlearning algorithms”. I will challenge this claim from botha practical and theoretical perspective. I will conclude

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the talk by presenting results of two new algorithms on aspeech recognition problem provided by Google.

On Computational Strategies for Fluid-StructureInteraction: Some Insights into Algorithmic Basiswith Applications

D. Peric & W. Dettmer (Swansea University)

This talk is concerned with algorithmic developments un-derpinning computational modelling of the interaction ofincompressible fluid flow with rigid bodies and flexiblestructures.

Fluid-structure interaction (FSI) represents a complexmultiphysics problem, characterised by a strong couplingbetween the fluid and solid domains along moving and of-ten highly deformable interfaces. Spatial and temporaldiscretisations of the FSI problem result in a coupled setof nonlinear algebraic equations [1], which is solved by avariety of different computational strategies, ranging fromweakly coupled partitioned schemes to strongly coupledmonolithic solvers.

This work discusses different options available to thedevelopers focusing on the so-called strongly coupled pro-cedures. Simple model problems are employed to illustratethe algorithmic properties of different methodologies, in-cluding a detailed convergence and accuracy analysis. Inparticular, we investigate the effects of different time inte-gration schemes in the sub-domains of a coupled problem[4]. Furthermore, careful analysis is performed of the cou-pling strategies and their effect on the convergence of theoverall procedure [3].

A comparative analysis between the exact Newton-Raphson methodology and its inexact version is performedon large scale simulations. In particular, we investigate:(i) the effect of the terms that couple the fluid flow withthe fluid mesh motion, and (ii) consequences of employ-ing different partitioned strategies on the efficiency andconvergence of the overall solution procedure [2].

Numerical examples will be presented throughout thetalk in order to illustrate the scope and benefits of thedeveloped strategies.

The examples include analysis of interaction of bothexternal and internal flows with rigid bodies and flexiblestructures relevant for different areas of engineering includ-ing civil, mechanical and bio-engineering.

Mathematical models for the cardiovascular sys-tem: analysis, numerical simulation, applications

A. Quarteroni (Ecole Polytechnique Federale de Lau-sanne and Politecnico di Milano )

The role of mathematics in understanding and simulat-ing fluid dynamics and biochemical processes in the phys-iological and pathological functioning of the human car-diovascular system is becoming more and more crucial.These phenomena are indeed correlated with the origin ofsome major cardiovascular pathologies, and influence theefficacy of the treatments to heal the arteries from theirdiseases.

Mathematical models allow the description of the com-plex fluid-structure interaction which govern the

artery wall deformation under the pressure pulse. More-over, appropriate reduction strategies can be devised to al-low for an effective description of the interaction betweenlarge, 3D components, and small 1D branches of the circu-latory system, as well as the transfer of drugs and chemi-cals between the arterial lumen and the vessel wall.

This presentation will address some of these issues anda few representative applications.

Kernel-Based Meshless Methods for Solving PDEs

Robert Schaback (Universitat Gottingen)

I shall focus on numerical methods using trial spaces spannedby functions K(·, xj) with a symmetric positive definitekernel K : Ω × Ω → R on a domain Ω ⊂ Rd, usingfinitely many scattered trial nodes xj ∈ Ω. This set-ting is independent of PDE solving and can be used forgeneralized Hermite–Birkhoff interpolation or for approx-imation, e.g. in Machine Learning.

When these trial spaces are used for PDE solving, theylead to a variety of numerical methods whose common fea-ture is being meshless, i.e. not using triangulations andexpressing everything “entirely in terms of nodes” (Be-lytschko et. al. 1996).

A first subtopic will beTrefftz–type methods, wherethe kernel–based trial space is adapted to the PDE tobe solved. For homogeneous problems, this comprises themethod of fundamental solutions and certain new tech-niques based on harmonic kernels. For inhomogeneousproblems, kernel–based trial spaces provide plenty of par-ticular solutions. Looking at the error of these methodswill lead to a non–roundoff version of a backward erroranalysis.

A second subtopic will focus on PDE–independenttrial spaces spanned by kernel translates, and provide aframework for error analysis of PDE solving using thesetrial spaces together with different independent forms ofstrong or weak testing. This applies to various numericaltechniques, including symmetric and unsymmetric mesh-less collocation and the Meshless Local Petrov–Galerkinmethod.

Numerical examples will be omitted, since there areplenty of them in the engineering literature. The focuswill be on mathematical analysis instead.

The best of fast N-body methods

Robert Skeel (Purdue University)

The problem of calculating a sum of pairwise interactionsfor a large set of particles is not only of great importancefor particle simulations but also of great value for con-tinuum simulations. The multilevel summation method(MSM), which does this calculation in linear time, em-bodies many if not most of the best features of both hier-archical clustering methods (tree methods, fast multipolemethods, hierarchical matrix methods) and FFT-based 2-level methods (particle–particle particle–mesh methods,

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particle–mesh Ewald methods). Like a hierarchical clus-tering method (HCM), the MSM parallelizes well and isspatially adaptable. Like FFT-based methods, the MSMis a simple and computes quantities that are continuous asa function of particle positions. The simplicity of MSMis demonstrated by a derivation that approximates theinteraction kernel as a linear combination of basis func-tions and represents this as a sparse matrix decomposition.This is followed by a detailed comparison with HCMs andFFT-based methods and concluded with a summary of re-cent work concerning performance on graphical processingunits and improved approximation schemes.

Error estimates and adaptive algorithms for singleand multievel Monte Carlo Simulation of Stochas-tic Differential Equations

Raul Tempone (KAUST)

This talk presents a generalization of the multilevel For-ward Euler Monte Carlo method introduced in [1] for theapproximation of expected values depending on the so-lution to an Ito stochastic differential equation [2,3]. Thework [1] proposed and analyzed a Forward Euler MultilevelMonte Carlo method based on a hierarchy of uniform timediscretizations and control variates to reduce the computa-tional effort required by a standard, single level, ForwardEuler Monte Carlo method. Here we introduce and analyzean adaptive hierarchy of non-uniform time discretizations,generated by single level adaptive algorithms previouslyintroduced in [4, 5]. These adaptive algorithms apply ei-ther deterministic time steps or stochastic time steps andare based on adjoint weighted a posteriori error expansionsfirst developed in [6]. Under sufficient regularity condi-tions, both our analysis and numerical results, which in-clude one case with singular drift and one with stopped dif-fusion, exhibit savings in the computational cost to achievean accuracy of O(TOL), from O(TOL−3) using the singlelevel adaptive algorithm to O((TOL−1 log (TOL))2); forthese test problems single level uniform Euler time step-ping has a complexity O(TOL−4).

[1] Giles, M. B., Multilevel Monte Carlo path simula-tion, Oper. Res., 56, no. 3, 607-617, (2008).

[2] Hoel, H., von Schwerin, E., Szepessy, A.; Tempone,R. Implementation and Analysis of an Adaptive MultiLevel Monte Carlo Algorithm, (In Review).

[3] Hoel, H., von Schwerin, E., Szepessy, A.; Tempone,R. Adaptive Multi Level Monte Carlo Simulation. To ap-pear in Lect. Notes Comput. Sci. Eng., Springer–Verlag,Berlin. (2011)

[4] Moon, K-S. ; von Schwerin, E. Szepessy, A.; andTempone, R., An adaptive algorithm for ordinary, stochas-tic and partial differential equations, Recent advances inadaptive computation, Contemp. Math., 383, 325-343,(2005).

[5] Moon, K-S.; Szepessy, A.; Tempone, R. and Zouraris,G. E., Convergence rates for adaptive weak approximationof stochastic differential equations, Stoch. Anal. Appl.,23, no. 3, 511-558, (2005).

[6] Szepessy, A.; Tempone, R. and Zouraris, G. E.:Adaptive weak approximation of stochastic differential equa-tions, Comm. Pure Appl. Math., 54, no. 10, 1169-1214,

(2001).

Six myths of polynomial interpolation and quadra-ture

Lloyd N. Trefethen (University of Oxford)

Computation with polynomials is a powerful tool for allkinds of numerical problems, but the subject has been heldback by longstanding confusion on a number of key points.In this talk I will attempt to sort things out as systemati-cally as possible by identifying six widespread misconcep-tions. In each case I will explain how the myth took hold— for all of them contain some truth — and then I’ll givetheorems and numerical demonstrations to explain why itis mostly false.

Finite element approximations for coupled prob-lems

Barbara Wohlmuth (TU Munchen)

Numerical simulation of coupled problems play an impor-tant role in many different fields. In a wide range of in-dustrial, environmental and medical applications, free-flowand porous media transport processes and elasto-acousticoccur. Although of high relevance, the mathematical mod-eling and the numerical simulation of complex coupledproblems remains challenging. This talk adresses two is-sues: firstly different areas are illustrated and numericalexamples are given. Characteristically for many appli-cations is that the coupling can be realized in terms ofbalance equations which result in saddle-point like struc-tures. A pair of variables has to guarantee the properinformation transfer across the interfaces. In addition tothis aspect many complex mathematical models for, e.g.,phase transition processes or plasticity involve additionalconstraints which result in a weak variational inequalitysetting. In the second part of the talk we review on stablediscretization schemes and on optimal a priori estimatesfor finite element approximations. Here special focus is onthe importance of uniform inf-sup stability and the role ofsurface or volume based Lagrange multipliers.

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Minisymposia abstracts

Minisymposium M1Recent developments in a-posteriori

error estimationOrganisers: Gabriel Barrenechea and

Richard Rankin

A posteriori error estimation and an adaptive schemefor the choice of the stabilization parameter in low-order finite element approximations of the Stokesproblem.

Alejandro Allendes, Mark Ainsworth, Gabriel R. Bar-renechea & Richard Rankin (University of Strathclyde).

The numerical solution of the Stokes problem can be facedusing two complementary approaches. One consists of us-ing classical inf-sup stable pairs of elements. The otherconsists of stabilizing an inf-sup unstable pair, by addinga mesh-dependent term controlling the unstable modes ofthe pressure to the formulation, where this term can de-pend on residuals of the equation at the element level, orit can just be based on compensating for the inf-sup defi-ciency of the pressure.

Now, most of the stabilized methods proposed so farrely on the presence of a positive constant multiplying theadded term. This constant is called the stabilization pa-rameter, for which a large amount of work has been de-voted to looking for the best choice of this parameter, asin some cases the results may heavily depend on it.

In the quest for the best possible parameter, which forus is the one that minimizes the error, we propose to min-imize an a posteriori error estimator which will be provedto be fully computable, a rigorous upper bound for theerror, and equivalent to the error, with a lower constantindependent of the stabilization parameter. Hence, hope-fully minimizing this estimator can give us a stabilizationparameter which also minimizes the error.

Following this idea, this estimator is built and anal-ysed by splitting the velocity error into a conforming anda nonconforming part, which play an important role in theperformance. As the finite element space for the velocityis continuous, the so called equilibrated boundary fluxes areused to build the estimator in conjunction with the recon-

struction of aH≈ (div,Ω) conforming lifting, for which an

explicit simple formula can be derived, which allows forsome minimization, making the estimation of the velocityerror very tight. Finally, the a posteriori error estimatorfor the pressure follows using the continuous inf-sup con-dition, and relies on the a priori knowledge of the inf-supconstant. Fortunately, tight estimates for this constant areavailable in the literature.

Finally, once the estimator is available, we propose theuse of a Trust-Region Derivative Free Optimization algo-rithm to approximate the minimiser of such an estimatorwith respect to the stabilization parameter.

A posteriori error estimates for linear wave prob-lems

Emmanuil H. Georgoulis, Omar Lakkis & Charalam-bos Makridakis (University of Leicester)

We address the error control of standard discretisationsof linear second order hyperbolic problems. More specifi-cally, we derive a posteriori error bounds in various normsfor two-step time-stepping methods (drawn from the fam-ily of 2nd order cosine schemes) and we discuss the exten-sions of these results to the fully discrete setting, wherebythe space discretisation is based on finite element meth-ods. The derivation of these bounds relies crucially on(classical) ideas for the a-priori error analysis of the samemethods due to Baker from the 1970s, as well as on carefulconstructions of suitable (space-)time-reconstruction func-tions.

A posteriori error estimates for classical and sin-gularly perturbed parabolic equations

Natalia Kopteva & Torsten Linß (University of Lim-erick)

For classical and singularly perturbed semilinear parabolicequations, we present maximum norm a posteriori error es-timates that, in the singulaly perturbed regime, hold uni-formly in the small perturbation parameter. The parabolicequations are discretized in time using the backward Eulermethod, the Crank-Nicolson method and the discontinu-ous Galerkin dG(1) method. Both semidiscrete (no spatialdiscretation) and fully discrete cases will be considered.The analysis invokes elliptic reconsturcions and elliptic aposteriori error estimates.

A finite element method for nonlinear elliptic equa-tions

Omar Lakkis & Tristan Pryer (University of Sussex)

Fully nonlinear equations are notoriously hard to solve nu-merically. Finite difference schemes have been known forsome of these equations, but the nonvariational structuremakes them harder nuts to crack with Galerkin methods.This almost precludes the use of adaptive methods (whichcome natural for finite element meshes) for the approx-imate solutions of such equations. In this research wederive a Newton’s method to solve fully nonlinear ellip-tic equations, as well as a fixed point iteration to solvequasilinear equations such as the ∞-Laplacian based ona nonvariational Galerkin finite element method. Adap-tive methods are derived based on residual-type aposteri-ori error estimates. Numerical experiments illustrate thederived methods in various situations.

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Error estimators for a partially clamped plate prob-lem with bem

Matthias Maischak (Brunel University)

The biharmonic equation models the bending of a thinelastic plate. Restricting the corresponding minimizationproblems on a convex subset of possible boundary condi-tions describing restrictions on the clamping of the plateboundary, we obtain a variational inequality. Using thefundamental solution we obtain a symmetric integral op-erator representation. The higher regularity requirementsof the biharmonic operator lead to the usage of C1 ba-sis functions, as well as to a C1 regular representation ofthe boundary. We will first present a high-order numericalquadrature scheme, suitable for a hp-method on curvedboundaries and second, we will derive a-posteriori errorestimates based on a hierarchical decomposition. Severalnumerical examples underline the theoretical results.

Anisotropic mesh adaptation via a posteriori errorestimation

S. Micheletti (Politecnico di Milano)

In this communication we present some results concerningthe employment of a posteriori error estimation to drivean anisotropic mesh adaptation.

The numerical simulation of advection-diffusion phe-nomena with strong advection, of the Navier-Stokes equa-tions at large Reynolds number, of ground-water flows inaquifers with varying hydraulic conductivity, where inter-nal and boundary layers may occur - or the unsteady mul-timaterial simulation of a water column collapsing undergravity, does benefit of the use of an adaptive technique soas to track the space heterogeneities and the time dynam-ics.

The automatism and reliability of the whole procedureis made possible by the introduction of an a posteriori es-timator, or indicator, of the discretization error. In par-ticular, we have developed a posteriori estimates for boththe energy norm and linear and nonlinear functionals ofthe error.

The applications tackled so far comprise standard second-order elliptic problems - diffusion equations and advec-tion dominated advection-diffusion-reaction systems, theirparabolic counterpart, as well as the Stokes and Navier-Stokes equations for uncompressible fluids. Recently, wehave also dealt with anisotropic mesh adaptation for PDE-constrained optimal control problems.

A residual-based a posteriori estimator for the A−φ magnetodynamic harmonic formulation of theMaxwell system

S. Nicaise, E. Creuse, Z. Tang, Y. Le Menach, N. Ne-mitz & F. Piriou (Universite de Valenciennes)

My talk will be devoted to the derivation of an a poste-riori residual-based error estimator for the A− φ magne-

todynamic harmonic formulation of the Maxwell system.The weak continuous and discrete formulations will be es-tablished, and the well-posedness of both of them will beaddressed. Some useful analytical tools will be derived;among them, an ad-hoc Helmholtz decomposition will beproven, which allows to pertinently split the error. Conse-quently, an a posteriori error estimator will be obtained,which will be proven to be reliable and locally efficient.Finally, some numerical tests that confirm the theoreticalresults will be presented.

A posteriori error estimation for finite element ap-proximations of linear elasticity problems

Richard Rankin&Mark Ainsworth (University of Strath-clyde)

Finite element approximation schemes can be used to ob-tain approximate solutions to partial differential equations.When using a numerical method to approximate the solu-tion to any problem it is important to be able to say howaccurate the approximation obtained is. We will discussthe a posteriori error estimators that we have obtained forbounding the energy norm of the error in finite element ap-proximations of linear elasticity problems in two and threedimensions.

A posteriori estimates for problems with lineargrowth energy functionals

Sergey Repin (Steklov Institute, St Petersburg)

Energy functionals having linear growth with respect tothe norm of the gradient (or strain) operator arise in thetheory of nonparametric minimal and capillary surfaces,perfect plasticity, and other problems. Original variationalformulations associated with these functionals are incor-rect and require a relaxation that leads to problems definedon spaces of functions whose derivatives are Radon mea-sures. The corresponding dual problems are well posed,but defined on rather complicated sets of admissible dualfunctions. For these reasons, getting fully reliable numer-ical approximations for such type problems is faced withserious difficulties. In the talk, it is shown a way to con-struct converging approximations with the help of stan-dard conventional finite element approximations and ex-plicitly control the error with the help of guaranteed aposteriori estimates that has been recently derived for thisclass of problems.

A posteriori error estimator for the Reissner-Mindlinproblem

Emmanuel Verhille, Emmanuel Creuse & Serge Nicaise(Laboratoire Paul Painleve)

We consider the Reissner-Mindlin problem with homoge-neous Dirichlet boundary conditions into a domain Ω ⊂ R2

defined by :

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−div Cε(ϕ) = γ in Ω,−div γ = g in Ω,γ = λt−2(∇ω − ϕ) in Ω,

(1)

where the unknowns are ω, a scalar function which repre-sents the transverse displacement of the median plane ofthe plate and ϕ, a vector-valued function which representsthe rotation of its normal under deformation. t representsthe plate thickness, gt3 the load acting on the plate, C isthe tensor of elastic moduli (depending of the Young mod-ulus and the Poisson coefficient determinated by the platenature), and λ a Lame coefficient.

The solution of the problem (1) is approximated by a P1-conforming finite element method : we look for (ωh, ϕh) ∈P1 × P2

1 ⊂ H10 (Ω) × H1

0 (Ω)2, that is the approximation

of the exact solution (ω, ϕ). The approximation error isdefined by

eroth := ∥ω − ωh∥2H10 (Ω) + ∥ϕ− ϕh∥2H1

0 (Ω)2+

λ−1t2∥γ−γh∥2L2(Ω)+λ−2t4∥rot(γ−γh)∥2L2(Ω)+∥γ−γh∥2H−1(Ω).

The aim of this work is to find an a posteriori estimatorη, robust on the plate thickness t, totally explicit, leadingto an upper-bound of type eroth ≤ η (reliability). We cannotice that the multiplicative constant is equal to one inthe upper-bound, which constitutes the originality of ourwork. Our technique is based on the derivation of an equi-librated estimator using two elements named x∗ and y∗,totally built from the approximations ωh and ϕh into adhoc functionnal spaces.

Additionally, we give a lower-bound of type η ≤ Ceroth

(efficiency) where the positive constant C depends onlyon the shape regularity of the mesh. Some numerical teststhat confirm the theoretical results will be presented.

A unified framework for a posteriori error estima-tion for the Stokes problem

Martin Vohralık, A. Hannukainen & R. Stenberg(Universite Paris)

A unified framework for a posteriori error estimation forthe Stokes problem is developed. It is based on [H1

0 (Ω)]d-

conforming velocity reconstruction and [H(div,Ω)]d- con-forming, locally conservative flux (stress) reconstruction.It gives guaranteed, fully computable global upper boundsas well as local lower bounds on the energy error. In orderto apply this framework to a given numerical method, twosimple conditions need to be checked. We show how to dothis for various conforming and conforming stabilized finiteelement methods, the discontinuous Galerkin method, theCrouzeix–Raviart nonconforming finite element method,the mixed finite element method, and a general class offinite volume methods. The tools developed and used in-clude a new simple equilibration on dual meshes and thesolution of local Poisson-type Neumann problems by themixed finite element method. Numerical experiments il-lustrate the theoretical developments.

Minisymposium M2Practical approximation withpolynomials and beyond

Organisers: Pedro Gonnet andNick Hale

Getting rid of spurious poles: a practical approach

Ricardo Pachon (Credit Suisse)

Rational approximation, in theory, is a powerful techniquethat can overcome deficiencies of polynomial approxima-tion. In practice, however, the use of rational functions isaffected by the appearance of spurious poles, also known as“Froissart doublets”, artificial pairs of zeros and poles thatare introduced in the computed approximation because ofrounding errors.

This talk will present work, developed in collaborationwith Gonnet, van Deun and Trefethen, for the removal ofspurious poles. We have established a connection betweenthem and the singular values of the linear system associ-ated to the rational interpolation problem. Moreover, wehave exploited this connection in an algorithm for robustand stable rational approximation, in which the spuriouspoles are removed and the conditions of the problem arechanged from interpolation to least-squares fitting. Wewill present this strategy in detail as well as many exam-ples.

Approximating functions with endpoint Singulari-ties

Mark Richardson (University of Oxford)

In this talk, we describe the ’Sincfun’ software system forcomputing with functions that may have an endpoint sin-gularity. Sincfun is modelled closely on Chebfun, but usessinc function expansions instead of Chebyshev series. Ourexperiment sheds light on the strengths and weaknesses ofsinc function techniques. It also serves as a review of someof the main features of sinc methods including construc-tion, evaluation, zerofinding, optimization, integration anddifferentiation.

Convergence theorems for polynomial interpola-tion in Chebyshev points

Lloyd N. Trefethen (University of Oxford)

We present two theorems that quantify the idea that thesmoother a function f is, the faster its degree n polynomialinterpolants in Chebyshev points converge as n→ ∞. Thefirst, going back in essence to Bernstein, asserts conver-gence at the rate O(Mρ−n) if f is analytic with |f(z)| ≤Min the ρ-ellipse around [−1, 1]. The second asserts conver-gence at the rate O(V n−ν) if f has a ν th derivative on[−1, 1] of bounded variation V . Curiously, the latter re-sult does not seem to be in the literature.

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Minisymposium M3

Novel algorithms for the solution ofboundary-value problems

Organisers: Pedro Gonnet andNick Hale

Automatic differentiation for automatic solution ofnonlinear BVPs

Asgeir Birkisson (University of Oxford)

A new solver for nonlinear boundary-value problems (BVPs)in Matlab is presented, based on the Chebfun softwaresystem for representing functions and operators automat-ically as numerical objects. The solver implements New-ton’s method in function space, where the derivatives in-volved are Frechet derivatives, computed using automaticdifferentiation (AD). We present our AD implementation,details of the Newton iteration and show various examplesof solving nonlinear BVPs with Chebfun.

Finite difference preconditioners for spectral meth-ods

Pedro Gonnet (University of Oxford)

Spectral collocation methods are a powerful tool for thenumerical solution of ODEs and systems of ODEs. Thesolution of a linear ODE in one dimension involves theconstruction of a dense discretized linear operator L of sizeN ×N , where N is the number of points used to representthe solution. The discretized solution u is then computedby solving Lu = f for some right-hand side f .

Although this solution is straight-forward and rela-tively easy to implement for single equation or systems ofequations, it involves solving for the dense matrix L, whichtypically requires O(N3) operations, making this methodprohibitively expensive for large N or large systems.

An alternative going back to Orszag (1980) and studiedextensively by Canuto and Quarteroni (1985) and Funaro(1987) is to solve the linear system of equations Lu = fiteratively using preconditioned gmres. As a precondi-tioner, a finite difference approximation of the operatorcan be used, which is sparse and easily invertible. Evalu-ating the operator does not require forming the matrix Land can be done in O(N logN) operations. This reducesthe total cost of solving the linear system of equations fromO(N3) to O(N logN).

In this talk, we discuss efficient ways of constructingand applying this preconditioner, some results on its per-formance and general applicability, as well as the imple-mentation of this solver as part of the Chebfun system.

“Rectangular pseudospectral differentiation matri-ces” or, “Why it’s not hip to be square”

Nick Hale & Toby Driscoll (University of Oxford)

The standard way of including boundary conditions inspectral collocation methods involves replacing certain rowsof the matrices with others that enforce the required condi-tions. For example, in a two-point boundary value problemwith Dirichlet conditions at each end, one replaces the firstand last rows with [1,zeros(1,N-1)] and [zeros(1,N-1),1]

respectively. However, a few things are troublesome withthis kind of approach:

• for more general boundary conditions the choice ofwhich rows to replace is often heuristic,

• the wrong choice can lead to singular matrices (par-ticularly in systems of equations),

• interpolation at just interior Chebyshev nodes canhave a much larger Lebesgue constant,

• products of differentiation and integration matricesdo not produce the identity matrix.

In this talk we discuss an alternative approach, wherebythe collocation matrix is projected onto a lower degreesubspace to make it rectangular and the boundary condi-tions used to ‘square it up’. We show how this approachcan be implemented by a simple pre-processing step, anddemonstrate that it overcomes the difficulties mentionedabove. We explore connections with other ideas in theliterature, such as the use of staggered grids and the ‘ficti-tious points’ advocated by Fornberg, and finally we showthe rectangular matrices in action on some boundary valueand eigenvalue problems.

Minisymposium M4

Stochastic computation

Organisers: Evelyn Buckwar andDes Higham

Multi-level Monte Carlo Finite Element methodsfor elliptic PDEs with stochastic coefficients

Andrea Barth (University of Zurich)

In Monte Carlo methods quadrupling the sample size halvesthe error. In simulations of stochastic partial differentialequations (SPDEs), the total work is the sample size timesthe solution cost of an instance of the partial differentialequation. A Multi-Level Monte Carlo method is intro-duced which allows, in certain cases, to reduce the overallwork to that of the discretization of one instance of the de-terministic PDE. The model problem is an elliptic equationwith stochastic coefficients. Multi-Level Monte Carlo er-rors and work estimates are given both for the mean of thesolutions and for higher moments. The overall complexityof computing mean fields as well as as k-point correlationsof the random solution is proved to be of log-linear com-plexity in the number of unknowns of a single Multi–Level

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solve of the deterministic elliptic problem. Numerical ex-amples complete the theoretical analysis.

Linear Stability Analysis of Numerical Methods forStochastic Differential Equations

Evelyn Buckwar, Conall Kelly & Thorsten Sickenberger(Heriot-Watt University)

Stochastic differential equations are increasingly used inseveral application areas to model random dynamics, theinfluence of physical noise or uncertainties in system pa-rameters. Numerical methods are a necessary tool to studythe mathematical models and thus in recent years stochas-tic numerics has made considerable progress. In this talkI will provide an overview of available methods and re-sults for a linear stability analysis of numerical methodsfor systems of stochastic differential equations.

From cells to tissue: coping with the heterogeneityvia fractional models

Kevin Burrage, Nicholas Hale and David Kay (Univer-sity of Oxford)

Fractional differential equations are becoming increasinglyused as a modelling tool for coping with anomalous diffus-ing processes and heterogeneous media. However, the pres-ence of a fractional differential operator causes memory(time fractional) or nonlocality (space fractional) issuesand this imposes a number of computational constraints.In this talk we discuss how fractional equations arise fromstochastic processes with heavy tails and then develop effi-cient, scalable techniques for solving fractional-in-space re-action diffusion equations using the finite element methodand robust techniques for computing the fractional powerof a matrix times a vector. Our approach is show-casedby solving the fractional Fisher and fractional Allen-Cahnreaction-diffusion equations in two spatial dimensions andanalysing the speed of the travelling wave and size of theinterface in terms of the fractional power of the underlyingLaplacian operator.

Using a modification of the SSA to simulate spatialaspects of chemical kinetics

Pamela Burrage, Tamas Szekely & Kevin Burrage (Queens-land University of Technology)

It is now well recognised that the dynamics of chemicalkinetics, when there are small numbers of interacting par-ticles, is best described by the Stochastic Simulation Al-gorithm (SSA). It is also recognised that in some cases,spatial variants are needed when there are spatial hetero-geneities or spatial confinements. However, such imple-mentations can be extremely computationally intensive.

In this talk we present a model of the colorectal cryptwhere there is a small niche of stem cells that differentiateinto other types of cells, which then migrate through the

crypt. Our approach is to introduce a probability densityfunction representing the time for a cell to be absorbed atthe ends of the crypt, and this allows us to modify the SSAby using this important spatial information without actu-ally performing a detailed Monte Carlo simulation. Weattempt to estimate the accuracy and efficiency of this ap-proach.

Numerics for quadratic FBSDE

Goncalo Dos Reis (TU Berlin)

In this presentation we discuss a class of stochastic dif-ferential equations (SDE) named Forward-Backward SDE(FBSDE) and give particular attention to the so-calledquadratic case. We explain how this class links to semi-linear Partial differential equations (PDE) with quadraticnon linearity in the gradient component. We cover some ofthe main results and difficulties concerning the numericsfor quadratic FBSDE.

Optimal Error Estimates of Galerkin Finite Ele-ment Methods for SPDEs with Multiplicative Noise

Raphael Kruse (University of Bielefeld)

In this talk we focus on Galerkin finite element meth-ods for semilinear stochastic partial differential equations(SPDEs) with multiplicative noise and Lipschitz continu-ous nonlinearities. We analyze the strong error of conver-gence for spatially semidiscrete approximations as well asa spatio-temporal discretization which is based on a lin-ear implicit Euler-Maruyama method. In both cases weobtain optimal error estimates. The results hold for differ-ent Galerkin methods such as the standard finite elementmethod or spectral Galerkin approximations.

If time permits we give a sketch of the proofs whichare based on sharp integral versions of well-known errorestimates for the corresponding deterministic linear homo-geneous equation together with optimal regularity resultsfor the mild solution of the SPDE.

Lax’s equivalence theorem for stochastic differen-tial equations

Annika Lang (University of Zurich)

Lax’s equivalence theorem is a well-known result in theapproximation theory of partial differential equations. Itstates: If an approximation is consistent, stability andconvergence are equivalent. In this talk I will generalizethis result to Hilbert space valued stochastic differentialequations. Furthermore, examples will illustrate consis-tency, stability, and convergence for some approximationsof Hilbert space valued stochastic dfferential equations.

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Euler–Maruyama-type Numerical Methods forStochastic Lotka-Volterra Model

Wei Liu & Xuerong Mao (University of Strathclyde)

Consider the one-dimensional stochastic Lotka-Volterra model

dx(t) = (bx(t)− ax2(t))dt+ σx(t)dB(t), (2)

with positive parameters a, b and σ. Many authors haveinvestigated this stochastic population model, from its ex-plicit solution through extinction to stationary distribu-tion. These theoretical results form an excellent base totest the ability of numerical methods for highly non-linearstochastic differential equations (SDEs).

In this talk, we are concerned with the Euler–Maruyama-type numerical methods. It is known that given a pos-itive initial value x(0) at t = 0, the stochastic Lotka-Volterra model has a unique global positive solution x(t)on t ∈ [0,∞). However this is not the case for the explicitEuler-Maruyama (EM) method

Xk+1 = Xk + (bXk − aX2k)∆t+ σXk∆Bk, (3)

due to the unboundedness of the normally distributed Brow-nian increment ∆Bk := B((k+1)∆)−B(k∆). To preservethe positivity of the EM method, we introduce the stoppedEM approximate solution using the technique of stoppingtime. The existing known results have only so far shownthat the stopped EM approximate solution converges tothe true solution in probability. One of our key contribu-tions here is to show that it converges to the true solutionin the strong sense (namely L2).

Moreover, the underlying SDE (2) is almost surely ex-ponentially stable if b − (1/2)σ2 < 0. We have not beenable to show that the stopped EM method is almost surelyexponentially stable under this same condition for suffi-ciently small step-size ∆t. However, we introduce a weakbackward EM method and successfully show that it repro-duces this stability property.

Numerical Simulation of Stochastic Reaction Net-works modelled by Piecewise Deterministic MarkovProcesses

Martin Riedler (Heriot-Watt University)

Hybrid systems, and Piecewise Deterministic Markov Pro-cesses in particular, are widely used to model and numer-ically study multiscale systems in biochemical reaction ki-netics and related areas where I put an emphasise on mod-els arising in mathematical neuroscience. I will presentan almost sure convergence analysis for numerical simula-tion algorithms for Piecewise Deterministic Markov Pro-cesses. These algorithm are built by appropriately dis-cretising constructive methods defining these processes.The stochastic problem of simulating the random, path-dependent jump times is reformulated as a hitting timeproblem for a system of ordinary differential equationswith random threshold which is solved using continuousapproximation methods. In particular show that the al-most sure asymptotic convergence rate of the stochastic

algorithm is identical to the order of the embedded deter-ministic method. Finally, we present an extension of ourresults to more general Piecewise Deterministic Processes.

Mean Exit Times and Multi-Level Monte CarloSimulations

Mikolaj Roj, Des Higham, Xuerong Mao, Qingshuo Song& George Yin (University of Strathclyde)

Many financial quantities such as path dependent options,volatility swaps and credit risk models are based on sim-ulations of stopping times. One has to use Monte Carlosimulations especially when a high number of dimensionsis involved. However, numerical methods for stochasticdifferential equations are relatively inefficient when usedto approximate mean exit times. In particular, althoughthe basic Euler–Maruyama method has weak order equalto one for approximating the expected value of the solu-tion, the order reduces to one half when it is used in astraightforward manner to approximate the mean valueof a (stopped) exit time. Consequently, the widely usedstandard approach of combining an Euler-Maruyama dis-cretization with a Monte Carlo simulation leads to a com-putationally expensive procedure. In this work, we showthat the multi-level approach developed by Giles (Oper-ations Research, 2008) can be adapted to the mean exittime context. In order to justify the algorithm, we analysethe strong error of the discretization method in terms of itsability to approximate the exit time. We then show thatthe resulting multi-level algorithm improves the expectedcomputational complexity by an order of magnitude, interms of the required accuracy. Computational results areprovided to illustrate the analysis.

Efficient Multilevel Monte Carlo simulations of non-linear financial SDEs without a need of simulatingLevy areas.

Lukasz Szpruch (University of Oxford)

In many financial engineering applications, one is inter-ested in the expected value of a financial derivative whosepayoff depends upon the solution of a stochastic differen-tial equation. Using a simple Monte Carlo method with anumerical discretization with first order weak convergence,to achieve a root-mean-square error of O(ϵ) would requireO(ϵ−2) independent paths, each with O(ϵ−1) time steps,giving a computational complexity which is O(ϵ−3). Re-cently, Giles [Giles, 2008] introduced a Multilevel MonteCarlo (MLMC) estimator, which enables a reduction ofthis computational cost to O(ϵ−2). In order to achievethis superior property of the MLMC estimator, the numer-ical discretization of a SDEs under consideration requirescertain convergence properties, namely that the numeri-cal approximation strongly converge to the solution of theSDEs with order 1. This carries some difficulties. First ofall, it is well known that it impossible to obtain an orderof convergence higher than 0.5 without a good approxima-tion of the Levy areas (which are very expensive to simu-late) [Cameron, 1980]. Second, convergence and stability

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of numerical methods are well understood for SDEs withLipschitz continuous coefficients, whereas most financialSDEs violates these conditions. It was demonstrated, thatfor SDEs with non-Lipschitz coefficients using the classi-cal methods, we may fail to obtain numerically computedpaths that are accurate for small step-sizes, or to obtainqualitative information about the behaviour of numericalmethods over long time intervals [Jentzen, 2011]. Ourwork addresses both of these issues, giving a customizedanalysis of the most widely used numerical methods. Inthis work we generalize the current theory of strong con-vergence rates for Euler-Maruyama-type schemes for somehighly non-linear multidimensional SDEs which appear infinance. We also construct a new MLMC estimator thatenables us to avoid simulation of Levy areas without af-fecting the required computational cost of order O(ϵ−2).We support our theoretical results with the simulations ofexpected values of options with various payoffs.

Adaptive stepsize control for the strong numericalsolution of SDEs

Phillip Taylor (University of Manchester)

The strong numerical solutions of stochastic differentialequations (SDEs) can be found using numerical meth-ods such as the Euler-Maruyama method and the Milsteinmethod. The use of fixed stepsizes for these methods hasbeen widely studied; however the theory for using adaptivestepsize control is far less developed, and there are manyissues to consider. Firstly, we must be careful since thereare methods which may look sensible but converge to thewrong solution. Once we have a method which convergesto the right solution we have, for example, many choices oferror estimates at each step, and different choices for howwe can change the stepsize and generate the Wiener path.Some methods may be suitable for solving SDEs generatedby a one-dimensional Wiener process but less suitable ifthe Wiener process is multi-dimensional. We investigatedifferent approaches to solving both Ito and StratonovichSDEs using adaptive stepsize control, and the issues in-volved.

Minisymposium M5

Iterative linear solvers for PDEs

Organiser: Andy Wathen

Fast Multigrid Algorithms and High OrderVariational Image Registration Model

Ke Chen (University of Liverpool)

Variational partial differential equations (PDEs) based meth-ods can be used for deformable image registration to de-rive reliable results for many cases Modersitzki [1]. Onechallenge inherent from other regularization modeling suchas denoising is that it is not trivial to design a single

model suitable for both smooth and non-smooth defor-mation problems. A model based on a curvature type reg-ularizer [2-3] appears to deliver excellent results for bothproblems and is proposed and studied in this paper. Weaddress the problem of developing a fast algorithm for thesystem of two coupled PDEs which is highly nonlinear andof fourth order.

In this paper, we first study several fixed-point typesmoothers, their local Fourier analysis and experiments toselect the most effective smoother before using a nonlinearmultigrid algorithm for the new model. We then gener-alize the work to multi-modal image registration model-ing based on combining intensity and geometric transfor-mations. Numerical tests showing advantages of our newmodels and algorithms are given.

[1] Jan Modersitzki, Numerical Methods for Image Regis-tration, OUP, 2004.[2] Noppadol Chumchob, Ke Chen and Carlos Brito, ”Afourth order variational image registration model and itsfast multigrid algorithm”, Multiscale Modeling and Simu-lation, Vol 9 (1), pp.89-128, 2011.[3] Carlos Brito and Ke Chen, “Multigrid Algorithm forHigh Order Denoising”, SIAM Journal on Imaging Sci-ences, Vol 3 (3), pp.363-389, 2010.

Interface Preconditioners for Flow Problems

Daniel Loghin (University of Birmingham)

Domain decomposition methods are established techniquesfor solving linear systems arising from the discretization ofPDE. A key feature of these methods is the solution of theinterface problem (or interface Schur complement) arisingfrom a non-overlapping decomposition of the domain. Forcertain scalar PDE (e.g., elliptic problems) the fast resolu-tion of this problem is possible through a range of robustalgorithms. However, the generalization of these methodsto the case of systems of PDE is not always straightfor-ward.

In this talk we examine a class of interface precondi-tioners for the Stokes and the Oseen and Navier-Stokesequations. The analysis considers the continuous inter-face problem and its discrete counterpart under variousstandard mixed finite element discretizations of the fullproblem. We show that the coercivity and continuity ofthe bilinear forms induced by the interface operator pro-vide mesh-independent preconditioners for both the Stokesand Oseen problems. For the Navier-Stokes system, we in-dicate how additional adaptive preconditioning techniquescan be used to enhance performance. Numerical resultsusing standard test problems are included to illustrate theprocedure and verify the optimality of the proposed solvertechnique.

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Sharp performance estimates for optimized domaindecomposition preconditioners

Sebastien Loisel& StephenW. Drury (Heriot-Watt Uni-versity)

The numerical solution of elliptic boundary value prob-lems is important in many fields of applications. Domaindecomposition methods can be used to solve such problemsin parallel on supercomputers. Optimized domain decom-position methods introduce a “Robin parameter” whichcan be tuned for higher performance. The analysis of suchmethods has been very challenging. In this talk, we willdiscuss symmetric and nonsymmetric optimized domaindecomposition methods and we will give new performanceestimates for the nonsymmetric methods used in combina-tion with the GMRES algorithm.

Preconditioned Iterative Methods for Convection-Diffusion Control Problems

John Pearson & Andrew J. Wathen (University of Ox-ford)

The development of iterative solvers for solving linearizedsystems of equations arising from finite element discretiza-tions of partial differential equations (PDEs) is a well-established field of numerical analysis. A more recent areaof interest is finding solvers for optimization problems withPDE constraints, so called PDE constrained optimizationproblems. We consider a particular class of problems ofthis type, namely distributed convection-diffusion controlproblems. Using an appropriate stabilization technique,the Local Projection Stabilization method, we develop pre-conditioned Minres and Bramble-Pasciak Conjugate Gra-dient solvers for such problems. To create these solvers,we draw on ideas from saddle point theory, the approxima-tion of finite element mass matrices, and multigrid tech-niques for the solution of convection-diffusion operators.We present supporting theory and numerical results todemonstrate the effectiveness and scope of our iterativemethods.

Saddle-point Problems in Liquid Crystal Modelling

Alison Ramage and Chris Newton (University of Strath-clyde)

The focus of this work is on the iterative solution of saddle-point problems which occur frequently, and in multipleways, in liquid crystal numerical modelling. For example,saddle-point problems arise whenever director models areimplemented, through the use of Lagrange multipliers forthe pointwise unit vector constraints (as opposed to usingangle representations). In addition, saddle-point systemsarise when an electric field is present that stems from aconstant voltage, irrespective of whether a director modelor a tensor model is used, or whether angle representationsfor directors or componentwise representation with point-wise unit vector constraints is employed. Furthermore,

the combination of these two situations (that is, a directormodel using components, associated constraints and La-grange multipliers, together with a coupled electric fieldinteraction) results in a novel double/inner saddle-pointstructure which presents a particular challenge in terms ofnumerical linear algebra.

In this talk we will present some examples of the saddle-point systems which arise in liquid crystal modelling anddiscuss their efficient solution using appropriate precondi-tioned iterative methods with the aim of giving an insightinto this new and exciting research area at the interfacebetween liquid crystal theory and numerical analysis.

Fast Iterative Solvers for Saddle Point Problems

David Silvester (University of Manchester)

We discuss the design of efficient numerical methods forsolving symmetric indefinite linear systems arising frommixed approximation of elliptic PDE problems with associ-ated constraints. The novel feature of our iterative solutionapproach is the incorporation of error control in the nat-ural energy norm in combination with an a posteriori es-timator for the PDE approximation error. This leads to arobust and optimally efficient stopping criterion: the itera-tion is terminated as soon as the algebraic error is insignif-icant compared to the approximation error. We describea proof of concept MATLAB implementation of this algo-rithm, which we call EST MINRES, and we illustrate itseffectiveness when integrated into our Incompressible FlowIterative Solution Software (IFISS) package (available on-line from http://www.manchester.ac.uk/ifiss/). This isjoint work with Valeria Simoncini.

Minisymposium M6

Complex Networks and MatrixComputations

Organisers: Des Higham andAlastair Spence

Googling the brain: Discovering hierarchical andasymmetric structure, with applications in neuro-science

Jonathan J. Crofts and Desmond J. Higham (Notting-ham Trent University)

Hierarchical organization is a common feature of many di-rected networks arising in nature and technology. As anexample, a well-defined message-passing framework basedon managerial status typically exists in a business organ-isation. However, in many real-world networks such pat-terns of hierarchy are unlikely to be quite so transparent.Due to the nature with which empirical data is collatedthe nodes will typically be ordered so as to obscure anyunderlying structure, in addition to this, the presence ofa small number of links violating the so-called “chain of

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command” makes the determination of such structures ex-tremely challenging. Here we address the issue of how toreorder a directed network in order to reveal this type ofhierarchy. Using ideas from the graph Laplacian litera-ture, we show that a relevant discrete optimization prob-lem leads to a natural hierarchical node ranking. In ad-dition, we present a generalisation of this node orderingalgorithm based upon the combinatorics of directed walksand show that there exists a natural link between the ideaspresented here and Google’s PageRank algorithm. We fin-ish by showing how the algorithms perform when appliedto both synthetic networks and a real-world network fromneuroscience where results may be validated biologically.

Bistability Through Triad Closure

Desmond J. Higham, Peter Grindrod & Mark C. Par-sons (University of Strathclyde)

The digital revolution is generating novel large scale exam-ples of connectivity patterns that change over time. Thisscenario may be formalized as a graph with a fixed setof nodes whose edges switch on and off. For example,we may have networks of interacting mobile phone users,emailers, on-line chat room participants, Facebookers orTweeters. To model and simulate the key properties ofsuch evolving networks, we can use a discrete time Markovchain setting, where edges appear and disappear accord-ing to appropriate probabilistic laws. I will describe a newmodel based on a natural ‘triad closure’ mechanism thatquantifies effects observed in social network analysis—newfriendships are more likely between individuals who sharecurrent friends. A mean field analysis for this type of evo-lution leads to a bistable difference equation—the systemcan evolve into one of two possible steady states. Predic-tions from the analysis will be illustrated with computa-tional simulations. I will also look at the issue of calibrat-ing the model to a real set of telecommunications data.

Bipartite Subgraphs and the Signless Laplacian Ma-trix

Steve Kirkland & Debdas Paul (National Universityof Ireland Maynooth)

Motivated by a problem arising in protein-protein inter-action networks, we consider the use of the so-called sign-less Laplacian matrix to help identify bipartite inducedsubgraphs of a given graph. Specifically, we use the eigen-vectors of the signless Laplacian matrix corresponding tosmall eigenvalues in order to help find bipartite inducedsubgraphs. We present some theoretical results that helpto justify the approach, and apply the technique to a par-ticular protein-protein interaction network.

Reordering multiple networks

Clare Lee, Desmond J. Higham, J. Keith Vass & DanielCrowther (University of Strathclyde)

Many large, complex networks contain hidden substruc-

tures that can be revealed using a range of post-processingalgorithms. In particular, reordering network nodes appro-priately may help to summarise key properties by exposingsignificant clusters, or more generally sets of neighbourswith similar features. Our results have been tested on realdata concerning the behaviour of genes and proteins incells. Microarray data produces large non square matricesof information recording the behaviour of a large numberof genes across a small number of samples. This data isby its very nature non-negative. One aim is to cluster ororder the genes/samples into groups where members be-have similarly to each other and differently to those inother groups. This allows us to find sets of genes whosebehaviour distinguishes different sample types. This workfocuses on the use of matrix factorisation methods to de-rive useful network reordering. These methods typicallyfactorise the original matrix into two smaller matrices witha pre-specified number of rows/columns. In particularwe look at non-negative matrix factorization techniques,which have the intuitive advantage of respecting the non-negative nature of the original data. We have also devel-oped a non-negative matrix factorisation technique thatreorders multiple data sets simultaneously. This has allow-ing us to incorporate multiple measurements of a similartype as well as other types of information.

Networks in Neuroscience: Using the GeneralisedSingular Value Decomposition to Compare Pairsof Networks

M McDonald, Des Higham, Neil Dawson & Judy Pratt(University of Strathclyde)

In this talk, we will look at a pair of graphs and discussa problem of comparison, with a view to identifying andexamining clustering, eventually also making account ofknown relationships between different groups of nodes. Wedescribe the use the Generalised Singular Value Decompo-sition (GSVD) - a close relative of Principal ComponentAnalysis (PCA) to re-order the data such that the cluster-ing is seen.

We will show that, by design, this compares both graphsat the same time, identifying structures that are mutuallyexclusive to each. Finally, an example where the algorithmis applied to neuroscience data is provided - here the thegraphs represent activity in the brain of control and drugdosed rats. We calculate the variance of sub-groups ofbrain regions within the re-ordered data and propose thatthis represents the expression of those groups of nodes. Us-ing this approach, we identified structures that are consis-tent with current biological understanding and suggestiveof other areas for examination.

Dynamic Communicators

Alex Mantzaris & Desmond J. Higham (University ofStrathclyde)

With new technologies and applications a plethora of datasets with novel properties have emerged. We look at datasets which are produced by snap shots of networks over

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time. Most methods of analysis have been done assuminga static network. To demonstrate and reinforce methodol-ogy for analysing temporal networks, we develop a modelwith a property observed in voice mail, email, and intu-itively plausible for organisations.Dynamic communicators have an in uence that is hiddenwhen an aggregate view of the time course of a networkis viewed. The model is able to create these dynamiccommunicators in the messages exchanged between themover the time course of the simulations. A key feature isthat there is a representation of hierarchy amongst nodeswhich affects the temporal downstream message propaga-tion. Computational tests on the synthetically produceddata reveal this feature.

Temporal Graphs and Robustness

Mirco Musolesi & Vito Latora (University of St An-drews)

The analysis of social and technological networks has at-tracted a considerable attention as social networking appli-cations and mobile sensing devices have given us a wealthof real data about people interactions. Classic studieslooked at analysing static or aggregated networks, i.e., net-works that do not change over time or built as the result ofaggregation of information over a certain period of time.Given the soaring number of collections of measurementsrelated to very large real network traces, researchers arequickly starting to realise that connections are inherentlyvarying over time and exhibit more dimensionality thanstatic analysis can capture.

We have recently proposed a novel theoretical frame-work to study this class of networks in order to quantifyfor example the speed/delay of information diffusion pro-cesses and structural properties, such as the presence ofconnected components. We have showed how these metricsare able to capture the temporal characteristics of time-varying graphs, such as delay, duration and time order ofinteractions, compared to the metrics used in the past onstatic graphs.

A key aspect is network robustness: the problem we arefacing in characterising it is related to the fact that most ofthe classic definitions of centrality, considering both staticand temporal graphs, are based on shortest paths. Byusing these metrics, alternative paths (or walks) are nottaken into consideration. However, among the existingdefinitions of centrality for static graphs, there are somemetrics that are calculated considering all existing walks;the most used are probably the so-called Bonacich andKatz centrality. Therefore, a possible solution is to derive ameasure of centrality based on multiple paths for temporalgraphs. An alternative is to try to “transform” a temporalgraph into a static graph and then apply the definitionsand use the theoretical results obtained for static graphs.In order to do so, we introduce the concept of reachabilitygraphs as a way of projecting temporal graphs into staticgraphs.

In this talk I will give a brief overview of the proposedconceptual framework and I will present some initial ideasabout the calculation of centrality metrics for characteris-ing network robustness by means of matrix computations

using the transformation of temporal graphs into reacha-bility graphs.

A graph model to explain the origin of early pro-teins

Chanpen Phokaew, D. Higham & E. Estrada (Univer-sity of Strathclyde)

We propose and analyse a mathematical model which de-scribes the evolutionary process of residue networks ina prebiotic condition. The model iteratively generatesthe connectivity structure. We have conducted compu-tational experiments to compare this model with real pro-tein residue networks based on various network properties,including average pathlength, degree distribution and as-sortativity coefficient. We illustrate that after a relaxationprocess, our artificial residue networks have network prop-erties that match real protein residue networks.

Communicability of networks

Zhivkho Stoyanov (University of Bath)

In this talk we shall discuss the notion of communicabil-ity of networks, which evolve with time. In particular, foreach agent in the network we shall consider two quanti-ties: the amount of information which passes from, andthrough, that agent. We shall show that these two quan-tities can be derived as a result of asymmetry in a matrixassociated with the network; this matrix will be called thecommunicability matrix of the network. A reason for theasymmetry in the communicability matrix is the order inwhich a sequence of events takes place on the network. Weshall conclude the talk by discussing a way of quantifyingthe significance of a given sequence of events.

Minisymposium M7

Numerical methods for singularlyperturbed problems

Organisers: Natalia Kopteva andMartin Stynes

Stabilization of convection-diffussion problems byShishkin mesh simulation

Bosco Garcia-Archilla (Univerisdad de Sevilla)

Standard numerical methods are liable to perform poorlyin convection-diffusion problems, particularly on convection-dominated situations, where spurious (unphysical oscilla-tions) are likely to pollute the numerical approximation.Several techniques have been developed in the past 30years to flatten the unwanted oscillations, although noclear winner has turned up yet. Among the most success-ful ones are the use of Shishkin meshes, their disadvantagebeing the difficulty of designing them on nontrivial geome-

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tries. We present a technique to simulate Shishkin mesheswithout the need to build them. The technique is applica-ble to irregular grids and on nontrivial geometries. The nu-merical approximation obtained present similar accuracyand lack of oscillations as real (not simulated) Shishkinmeshes.

Time and space–accurate numerical solution of one-dimensional parabolic singularly perturbed prob-lems of reaction–diffusion type

Jose Luis Gracia&C. Clavero (Universidad de Zaragoza)

In this talk we consider a parabolic singularly perturbedproblem given by

ut + Lx,εu = f(x, t), (x, t) ∈ Q = (0, 1)× (0, T ],

u(x, 0) = 0, x ∈ Ω, u(0, t) = u(1, t) = 0, t ∈ (0, T ],(4)

where the spatial differential operator Lx,ε is defined byLx,εu ≡ −εuxx + b(x, t)u. We assume that the singularperturbation parameter satisfies 0 < ε ≤ 1 and it can bearbitrarily small, and also that sufficient compatibility andregularity conditions hold among the data of the problem.It is well known that, in general, the solution has a multi-scale character, which exhibits strong gradients in narrowregions (boundary layers) close to both edges x = 0 andx = 1, of width O(

√ε| ln ε|).

The aim of this talk is twofold: to construct a robustscheme, with respect to the singular perturbation param-eter, and that the numerical method has high order con-vergence in the maximum norm. This norm is the mostappropriate in the context of singular perturbation prob-lems, since the maximum error occurs in the boundarylayer regions, but it makes more difficult the analysis ofthe convergence. In this talk the analysis of the conver-gence is based on two steps: first a semidiscretization intime and afterwards the resulting semidiscrete problemsare discretized in space. In this way the analysis of theconvergence of the fully discrete scheme is eluded.

In time we discretize by using the backward Eulermethod on a uniform mesh, and in space we consider ahybrid fourth order compact finite difference scheme ofpositive type, which is constructed on a special nonuni-form mesh condensing the grid points in the boundarylayer regions. The discrete maximum principle at eachtime level can be applied in the analysis based on the twostep discretization technique, which does not hold if thefully discrete scheme is considered instead of our approach.To deduce appropriate bounds of the error, we will need toknow the asymptotic behavior, with respect to ε, of the so-lution of the semidiscrete problems at each time level andits derivatives, showing the dependence on the singularperturbation parameter. These bounds can be obtainedusing an inductive argument. Then, a robust numericalmethod, having first order in time and almost four orderin space in the maximum norm, can be theoretical ana-lyzed.

On the other hand, to increase the order of uniformconvergence in time of the fully discrete method it is a dif-ficult task. So, in the second part of this talk, we will showthe theoretical difficulties appearing in the analysis of the

uniform convergence when the Crank–Nicolson method isused in the time discretization instead of the backwardEuler method. To finish, we propose the Richardson ex-trapolation as an alternative to increase the global order ofuniform convergence of the numerical scheme, showing inpractice the efficiency of this technique. This research waspartially supported by the project MEC/FEDER MTM2010-16917 and the Diputacion General de Aragon.

Collocation methods for singularly perturbed reaction-diffusion equations

Torsten Linß, Goran Radojev & Helena Zarin (Univer-sitat Duisburg-Essen)

A reaction-diffusion equation in one dimension is discre-tised by collocation using piecewise quadratic C1-splines.The method is shown to be robust with respect to the dif-fusion parameter on a modified Shishkin mesh. The rateof uniform convergence in the maximum norm is almosttwo. A posteriori error estimates are also derived. Theseallow the design of an adaptive mesh algorithm. Numeri-cal results illustrate the theoretical findings.This work is supported by Deutscher Akademischer Aus-tauschdienst; Grant 50740187.

Supercloseness of the SDFEM for convection-diffusionproblems with reduced regularity

Lars Ludwig (Technical University of Dresden)

The streamline-diffusion finite element method (SDFEM)is applied to a convection-diffusion problem posed onthe unit square using a rectangular mesh and bilinear el-ements. With u being the exact solution, the usual er-ror estimates for ∥ uI − uh ∥SD require a regularity of atleast u ∈ H3(Ω). ). In many cases, e.g. for polygonal do-mains or non-smooth data, this regularity cannot be takenfor granted. It will be shown that supercloseness resultsstill hold in some sense if the exact solution has reducedregularity. Finally, numerical experiments illustrate thetheoretical findings

A Linear Singularly Perturbed Problem with anInterior Turning Point

J. Quinn (Dublin City Univerity)

In this talk we examine the linear singularly perturbedproblem: Find yϵ such that

Lϵyϵ(x) := (ϵy′′ϵ − aϵy′ϵ − byϵ)(x) = f(x), x ∈ Ω := (0, 1),

(5a)

yϵ(0) = A > 0, yϵ(1) = B < 0, b(x) ≥ 0, x ∈ Ω, (5b)

d ∈ Ω : aϵ(x) ≥ 0, x ∈ [0, d), aϵ(d) = 0, (5c)

aϵ(x) ≤ 0, x ∈ (d, 1], (5d)

|aϵ(x)| ≥ |αϵ(x)|, x ∈ Ω, (5e)

αϵ(x) := θ tanh(rϵ(d− x)

), θ, r > 0. (5f)

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The numerical analysis for this problem differs from thestandard analysis for a problem with a non-zero discontin-uous convective coefficient of the form a > 0, x < d, a <0, x > d. Existence of a solution is first established acrossΩ, before the analysis for problem is split at the pointwhere the coefficient aϵ becomes zero. An upwind finitedifference operator is combined with a piecewise-uniformShishkin mesh centered around d to generate numericalapproximations. Numerical results are presented to illus-trate the theoretical parameter-uniform error bounds es-tablished. This research is funded by the Irish ResearchCouncil for Science, Engineering and Technology.

A balanced finite element method for singularlyperturbed reaction-diffusion problems

Martin Stynes & Runchang Lin (National University ofIreland, Cork)

Consider the singularly perturbed linear reaction-diffusionproblem −ε2∆u + bu = f in Ω ⊂ Rd, u = 0 on ∂Ω,where d ≥ 1, the domain Ω is bounded with (when d ≥ 2)Lipschitz-continuous boundary ∂Ω, and the parameter εsatisfies 0 < ε ≪ 1. It is argued that for this type ofproblem, the standard energy norm [ε2|u|21 + ∥u∥20]1/2 istoo weak a norm to measure adequately the errors in so-lutions computed by finite element methods because themultiplier ε2 gives an unbalanced norm whose differentcomponents have different orders of magnitude. A bal-anced and stronger norm is introduced, then for d ≥ 2 amixed finite element method is constructed whose solutionis quasi-optimal in this new norm. By a duality argumentit is shown that this solution attains a higher order of con-vergence in the L2 norm. Error bounds derived from theseanalyses are presented for the cases d = 2, 3. For a problemposed on the unit square in R2, an error bound that is uni-form in ε is proved when the new method is implementedon a Shishkin mesh. Numerical results are presented toshow the superiority of the new method over the standardmixed finite element method on the same mesh for thissingularly perturbed problem.

Minisymposium M8

Recent Advances in Numerical LinearAlgebra

Organiser: Francoise Tisseur

Two Two-Sided Procrustes Problems Arising inAtomic Chemistry

Rudiger Borsdorf (University of Manchester)

In atomic chemistry it is of interest to obtain a minimalset of localized atomic orbitals that describes the atom ofinterest and ensures certain properties of the atom. Theset is to be obtained from a density operator of a molec-ular system with large dimensions computed from resultsin quantum chemistry.

Let N ∈ Rn×n be symmetric and represent the den-sity operator. Let D ∈ Rm×m with m < n be given anddiagonal. The first problem that we are looking at is tominimize ∥Y TNY −D∥2F over Y ∈ Rn×m having orthonor-mal columns and with ∥ ·∥F denoting the Frobenius norm.This problem is equivalent to finding a minimal set de-scribing an atom with occupation numbers closest to theprescribed diagonal elements of D. In the second prob-lem only the number of electrons in the atom is desired tobe close to a prescribed value d ∈ Z, which is equivalentto minimizing the objective function (trace(Y TNY )−d)2.Both optimization problems involve the minimization overthe Stiefel manifold and are of the form of the two-sidedProcrustes problem.

Looking at the stationary points of the first problemon the manifold reveals the equivalence of this problem toan imbedding problem. This allows the application of atheorem by Fan and Pall from 1957 yielding the minimumfunction value. From this result we also obtain an optimalsolution. By means of this analysis we show the equiva-lence of the second problem to a convex quadratic problemwith box constraints. To solve this problem we propose touse the active-set method, as an optimal solution of the in-ner optimization problem can easily be obtained. Hence,this method yields a solution at low computational cost.

Blocked Methods for Computing the Square Rootof a Matrix

Edvin Deadman (University of Manchester)

The square root of a matrix can be computed via a Schurdecomposition. The square root of the resulting triangularmatrix can then be found either a column at a time or asuperdiagonal at a time. However this method does nottake advantage of level 3 BLAS operations. In this talkI will discuss implementations of the Schur method whichuse blocking and recursion to enable the square root to becomputed using level 3 BLAS. I will describe the speedupsobtained and will discuss applications to other triangularmatrix computations.

A Linear Algebraist’s Approach to Reforming TheUK Voting System

Philip A. Knight (University of Strathclyde)

Many modern democracies use forms of proportional rep-resentation (PR) as a means of allocating members of par-liament at both local and national level. The UK hasjoined in recently and a number of elections (e.g., Euro-pean, Scottish) use forms of PR; although there is strongresistance to bringing it into the Westminister election, asthe recent vote on AV indicates.

Biproportional apportionment is a novel method of ap-plying PR, used for the first time in 2006 in Switzerland.It works by applying a discrete form of matrix balancing.We investigate the viability of biproportional assignmentin UK elections; in particular how to overcome its limita-tions when applied to a system with over 600 constituen-cies all requiring a single representative. We show that

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a new multi-level apportionment process can be applied,but also look at alternative approach based on optimisingnetwork flow. We discuss pros and cons of such a system.

Along the way we prove a new convergence result formatrix balancing.

A method for the computation of Jordan blocks inparameter-dependent matrices

Alastair Spence (University of Bath)

Preconditioning for GMRES

Andy Wathen & Jen Pestana (University of Oxford)

For linear systems of equations with a real symmetric (orcomplex hermitian) coefficient matrix, there are standarditerative methods of choice: Conjugate Gradients for pos-itive definite problems and MINRES for indefinite prob-lems. For these methods there exist descriptive conver-gence theories based on eigenvalues, i.e. convergence boundsgiven in terms of polynomial approximation problems onthe eigenvalue spectrum which are often reasonably tight.A key consequence is that it is clear what one is tryingto achieve to get fast convergence via preconditioning inthese cases, namely well distributed eigenvalues or reducedspectral condition number.

By contrast, existing convergence bounds for Krylovsubspace methods such as GMRES for real nonsymmetriclinear systems give little mathematical guidance for thechoice of preconditioner. In this talk we will introduce adesirable mathematical property of a preconditioner whichguarantees that convergence of GMRES will essentially de-pend only on the eigenvalues of the preconditioned system,as is true in the symmetric case.

Minisymposium M9

High frequency and oscillatory problems

Organiser: Mark Ainsworth

Optimally blended finite element-spectral elementschemes for wave propagation

Mark Ainsworth & Hafiz Wajid (Strathclyde University)

In an influential article, Marfurt suggested that the ”best”scheme for computational wave propagation would involvean averaging of the consistent and lumped finite elementapproximations. Many authors have considered how thismight be accomplished for first order approximation, butthe case of higher orders remained unsolved. We describework on the dispersive and dissipative properties of a novelscheme for computational wave propagation obtained byaveraging the consistent (finite element) mass matrix andlumped (spectral element) mass matrix. The objective isto obtain a hybrid scheme whose dispersive accuracy is

superior to both of the schemes. We present the optimalvalue of the averaging constant for all orders of finite el-ements and prove that for this value, the scheme is twoorders more accurate compared with finite and spectralelement schemes and, in addition, the absolute accuracyis of this scheme is better than that of finite and spectralelement methods.

Recent Developments in nonpolynomial finite ele-ment methods for wave problems

T. Betcke & J. Phillips (University College London)

In recent years nonpolynomial finite element methods forthe solution of frequency domain wave problems such asthe Helmholtz equation have become of growing interest inMathematics and Engineering. Based on oscillatory basisfunctions instead of polynomials they have the potentialto lead to a significant decrease of the number of neededbasis functions for a satisfactory accuracy for solutions ofwave problems. In this talk we review some recent devel-opments for nonpolynomial finite element methods and inparticular focus on adaptivity and the behavior of thesemethods for inhomogeneous media problems.

Extremely Large Electromagnetics Problems: AreWe Going in the Right Direction ?

Ozgur Ergul & Levent Gurel (University of Strathclyde)

Recently, a popular approach for the solution of real-lifeelectromagnetics problems is to use boundary-element meth-ods with low-order discretizations. This strategy providesrobust and flexible implementations allowing for the fastand accurate analysis of three-dimensional complicated ob-jects with arbitrary geometries and electromagnetic exci-tations. Since accurate solutions via low-order elements re-quire dense discretizations, matrix equations derived fromreal-life applications often involve millions of unknowns. Inaddition, as opposed to those derived from finite-elementmethods, these matrices are not sparse. Hence, recentstudies in this area have mainly focussed on developingfast solvers, such as the multilevel fast multipole algo-rithm (MLFMA), and employing these low-complexity al-gorithms on parallel computers. In fact, the last decadehas seen marvelous advances in this area leading to a rapidincrease in the problem size from millions to more thanone billion. The developed full-wave implementations havebeen used to solve extremely large electromagnetics prob-lems involving complicated objects that are larger than1000λ, where λ is the wavelength, on moderate computerswith distributed-memory architectures.

Solutions of extremely large electromagnetics problems re-quire a well designed combination of diverse componentsfrom different areas, such as numerical methods, fast algo-rithms, iterative solutions, parallelization, and high per-formance computing. Involving diverse components, theresulting simulation environments contain various errorsources introduced at different stages. For example, par-allel MLFMA implementations have the following error

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sources, some of which are shared by other computationalmethods:

• Discretization of the three-dimensional model.

• Discretization of the integral-equation formulations.

• Numerical integrations on the discretization elementsto calculate the near-zone interactions.

• Factorization of the far-zone interactions and trun-cation of the number of harmonics.

• Interpolations in multilevel interactions.

All these error sources make contributions to the final out-put, e.g., computed scattered or radiated electromagneticfields. Recent works have focussed on increasing the prob-lem size and the number of unknowns, but less attentionhas been paid to the accuracy of the results, particularlycompared to alternative approximate techniques.

This study is on the accuracy of solutions of extremelylarge electromagnetics problems using a full wave solverbased on MLFMA. We discuss the aforementioned errorsources and their effects on the solutions. It is shown thatthe relaxation of some error sources would lead to more“efficient” solutions, as sometimes practiced in the litera-ture. Unfortunately, such relaxation techniques may sig-nificantly deteriorate the accuracy of the results such thatusing a full-wave solver becomes meaningless. To encour-age fair comparisons of the developed implementations forlarge-scale computations and emphasizing the accuracy ofthe results, we present a list of benchmark problems involv-ing canonical objects of various sizes and their accurate so-lutions using a sophisticated implementation of MLFMA.Problems include a sphere with a diameter of 680λ andthe NASA Almond that is larger than 1500λ. Both ob-jects are discretized with more than 500 million unknownsand analyzed with maximum 1% error.

A high frequency boundary element method forscattering by non-convex obstacles

Steve Langdon, Simon Chandler-Wilde, Dave Hewett &Ashley Twigger (University of Reading)

Standard finite or boundary element approaches for thenumerical solution of high frequency scattering problems,with piecewise polynomial approximation spaces, sufferfrom the limitation that the number of degrees of free-dom required to maintain accuracy must increase at leastlinearly with respect to frequency in order to maintainaccuracy. This can lead to excessively expensive compu-tations at high frequencies. To reduce the computationalcost, recent work has focused on the inclusion of oscilla-tory functions in the approximation space, so as to betterrepresent the oscillatory solution. In this talk we reviewthis approach as applied to some simple problems of scat-tering by convex obstacles, and we describe the extensionof these ideas to more complicated problems of scatteringby non-convex obstacles.

Minisymposium M10

Linear algebra in data-sparserepresentations

Organisers: Ivan Oseledets andPavel Zhlobich

Tensor trains and high-dimensional problems

Ivan Oseledets (Russian Academy of Sciences)

In this talk, I will introduce tensor train format, or sim-ply TT-format and explain, why it can be considered asone of the most promising low-parametric representationsfor high-dimensional problems in chemistry, physics andmathematics.

Rank-one QTT vectors with QTT rank-one andfull-rank Fourier images

Dmitry Savostyanov (Russian Academy of Sciences)

Recently, fast algorithm was proposed for the Fourier trans-form and related sine/cosine transforms, using QTT for-mat for data-sparse approximate representation of one–and multi–dimensional vectors (m-tensors). Although aFourier matrix itself does not possess a low-rank QTTrepresentation, it can be efficiently applied to m–tensorin the QTT format exploiting the multilevel structure ofthe Cooley-Tuckey algorithm. The m–dimensional Fouriertransform of an n×. . .×n vector with n = 2d has log2(size) =O(m2d2r3) complexity, where r is the maximum QTTrank of input, output and all intermediate vectors in theCooley-Tuckey scheme. The storage size in QTT formatis bounded by O(mdr2).

The proposed FFT-QTT algorithm outperforms theFFT algorithm for vectors with large n and m and mod-erate r. It is instructive to describe such vectors more ex-plicitely. In this talk we describe the class of QTT rank-one vectors with rank-one Fourier images. We also givean example of QTT rank-one vector that has the Fourierimages with full QTT ranks.

In numerical experiments we show that the use of QTTformat for such vectors relaxes the grid size constrains andallow high-resolution computations of Fourier images andconvolutions in high dimensions without “curse of dimen-sionality”.

Matrix computations with data in multi-index for-mats

E Tyrtyshnikov (Russian Academy of Sciences)

Vectors and matrices are usually considered as arrays withone or two indices. However, the same data can be eas-ily viewed as arrays with more (artificial) indices, e.g. avector of size n = 2d can be considered as an d-indexarray of size 2 × . . . × 2. When using recently proposedtensor-train decompositions one can design efficient algo-

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rithms for basic matrix operations of complexity linear ind. Thus, the well-known curse of dimensionality convertsinto the blessing of dimensionality. The new approach putsforth many wonderful perspectives and questions for fu-ture research. Applications include spectral problems ofcomputational chemistry, stochastic differential equations,general parametric problems, fast interpolation methodsfor multivariate data etc. The purpose of this talk is tooutline the new tensor representation techniques, relatedtheoretical findings and efficient algorithms reducing mul-tidimensional computations to matrix ones, applications,on-going works and perspectives.

Minisymposium M11

Optimization: complexity andapplications

Organiser: Peter Richtarik

The Fast Gradient Method as a Universal OptimalFirst-Order Method

Olivier Devolder, Francois Glineur and Yurii Nesterov(Universite Catholique de Louvain)

The goal of this talk is to show that the fast-gradientmethod, initially designed for smooth convex optimization(convex functions with Lipschitz-continuous gradient), canbe also applied to convex problems with weaker level ofsmoothness (including non-smooth problems with boundedsubgradients) and provides for these classes an optimalfirst-order method.

A new definition of inexact oracle was previously in-troduced in order to study the behavior of the first-ordermethods of smooth convex optimization (classical and fast-gradient methods) when only approximate first-order in-formation is available.

In this talk, we prove that an exact oracle of non-smooth convex optimization can be seen as an inexactoracle of smooth convex optimization. This observationgives us the possibility to apply the gradient and the fast-gradient methods to non-smooth functions, replacing gra-dients by subgradients and replacing the Lipschitz-constantof the gradient by a well-chosen quantity. We prove thatthese two adapted methods have a convergence rate ofO( 1√

k) on the non-smooth problems and are therefore op-

timal.

The same kind of results hold for weakly smooth func-tions (i.e. functions with Holder-continuous gradient). Weobtain that the fast-gradient method achieves the optimal

rate of convergence for this class: O

(1

k1+3ν

2

)where ν de-

notes the Holder parameter.

Finally, we prove that the fast-gradient method can beapplied to smooth and non-smooth strongly convex prob-

lems with here also an optimal complexity.

Solving topology optimization problems by domaindecomposition

Michal Kocvara (University of Birmingham)

We investigate two approaches to the solution of topologyoptimization problems using decomposition of the compu-tational domain. The first approach is based on a twostage algorithm. The basic method is the block Gauss-Seidel algorithm used in the same way as in domain de-composition methods for the solution of linear systems.Here, however, we apply the block Gauss-Seidel to a con-straint optimization problem. It is well known that sucha method, in general, does not converge to a critical pointof the optimization problem. To guarantee convergence,we follow one iteration of the block Gauss-Seidel methodby a few steps of a converging first-order method. Wewill show that the resulting two stage algorithm leads to asignificant improvement in the solution efficiency. The sec-ond approach uses reformulation of the original problemas a semidefinite optimization problem. This approach isparticularly suitable for problems with vibration or globalbuckling constraints. Standard semidefinite optimizationsolvers exercise high computational complexity when ap-plied to problems with many variables and a large semidefi-nite constraint. To avoid this unfavorable situation, we useresults from the graph theory that allow us to equivalentlyreplace the original large-scale matrix constraint by severalsmaller constraints associated with the subdomains. Thisleads to a significant improvement in efficiency, as will bedemonstrated by numerical examples. Joint work with M.Kojima (Tokyo Institute of Technology), D. Loghin and J.Turner (both University of Birmingham).

Efficiency of Randomized Coordinate Descent Meth-ods on Minimization Problems with a CompositeObjective Function

Martin Takac & Peter Richtarik (University of Edin-burgh)

We develop a randomized block-coordinate descent methodfor minimizing the sum of a smooth and a simple non-smooth block-separable convex function and prove thatit obtains an ϵ-accurate solution with probability at least1− ρ in at most O((2n/ϵ) log(1/ϵρ)) iterations, where n isthe dimension of the problem. This extends recent resultsof Nesterov (Efficiency of coordinate descent methods onhuge-scale optimization problems, 2010), which cover thesmooth case, to composite minimization, and improves thecomplexity by a factor of 4. In the smooth case we givea much simplified analysis. Finally, we demonstrate nu-merically that the algorithm is able to solve various ℓ1-regularized optimization problems with a billion variables.

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Minisymposium M12

Compressed Sensing: algorithms andtheory

Organiser: Peter Richtarik

GPU Accelerated Greedy Algorithms for SparseApproximation

Jeffrey Blanchard (Grinnell College)

Thresholding algorithms are efficient methods for deter-mining the best k-term approximation of a given signalwhen an incomplete set of linear measurements is known.These greedy algorithms have been extensively tested onunrealistically small problems. We show the efficacy ofGPU based greedy algorithms and observe speedups ona single Nvidia Tesla C2050 of 50 - 75 times a state ofthe art Intel Xeon 5650 CPU. These performance gainscomes from both the GPU and alterations to the algo-rithms which exploit the properties of the GPU, and it en-ables testing of more realistic sized signals of length 106.This is joint work with Jared Tanner.

Compressed Sensing Thresholds and Condition Num-bers in Optimization

Martin Lotz (Univeristy of Edinburgh)

We show how a precise error analysis of sparse recoveryproblems with noise can be obtained from the study ofgeneralizations of Renegar’s condition number for convexfeasibility problems. The probability distribution of thesecondition measures for random problem instances is wellstudied, leading in turn to results on robust compressedsensing.

A GPU accelerated coordinate descent method forlarge-scale L1-regularized convex minimization

Peter Richtarik (University of Edinburgh)

Compared with an efficient CPU code, a GPU implemen-tation of a coordinate descent method for L1-regularizedconvex optimization is capable of producing unprecedentedspeedup. Joint work with Martin Takac.

A new recovery analysis of Iterative Hard Thresh-olding for Compressed Sensing

Andrew Thompson (University of Edinburgh)

Compressed Sensing (CS) seeks to recover sparse or com-pressible signals from undersampled linear measurements -it asserts that the number of measurements should be pro-portional to the information content of the signal, ratherthan its dimension. One particular CS problem that hasreceived much attention recently is that of recovering a k-sparse signal x ∈ IRN from n linear measurements, where

k ≤ n ≤ N . Since the introduction of CS in 2004, manyalgorithms have been developed to solve this problem. Be-cause of the paradoxical nature of CS - exact reconstruc-tion from undersampled measurements - it is crucial forthe acceptance of an algorithm that rigorous worst-caseanalysis verifies the degree of undersampling the algorithmpermits. This aim can be accomplished by means of thephase transition framework in which we let (k, n,N) → ∞,while preserving the proportions δ = n/N and ρ = k/n.

We provide a new worst-case analysis for one of these re-covery algorithms, Iterative Hard Thresholding (IHT). Forthe specific case of Gaussian measurement matrices, com-parison with existing results by means of the phase transi-tion framework shows a substantial quantitative improve-ment. We derive two conditions for general measurementmatrices: Firstly, by analysing the fixed points of IHT weobtain a condition guaranteeing at most one fixed point(namely the original signal). Secondly, we give an im-proved condition guaranteeing convergence to some fixedpoint. If both conditions are satisfied, it follows that wehave guaranteed recovery of the original signal. A sta-tistical analysis of the fixed point condition for Gaussianmeasurement matrices allows us to derive a quantitativephase transition.

We also extend the consideration to variants of IHT withvariable step-size, including Normalized Iterative Hard Thresh-olding (NIHT). A similar analysis in this case yields a fur-ther significant improvement on the phase transition forGaussian measurement matrices.

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Abstracts of Contributed Talks

Reduction and solutions for unsteady flow of aSisko fluid for cylindrical geometry

S Abelman (University of the Witwatersrand)

In this study some reductions and solutions for unsteadyflow of a Sisko fluid are investigated. The governing nonlin-ear equation for unidirectional flow of a Sisko fluid is mod-elled in cylindrical polar coordinates. The exact steady-state solution for the nonlinear problem is obtained. Thereduction of the governing nonlinear problem is carriedout by using the similarity approach. Further, by timetranslation and scaling symmetries, the partial differentialequation is transformed into an ordinary differential equa-tion, which is integrated numerically taking into accountthe influence of the index n and the material parameter bof the Sisko fluid. A comparison between profiles of New-tonian and Sisko fluids is presented.

Bernstein-Bezier Finite Elements

Gaelle Andriamaro, Mark Ainsworth & Oleg Davydov(University of Strathclyde)

Algorithms with optimal complexity are presented for thecomputation of the element mass and stiffness matricesfor finite element spaces based on Bernstein-Bezier poly-nomials which, for a given simplex T ⊆ Rd, are definedby

Bnα =

(n

α

)λα, α ∈ In

d ,

where Ind is the set of non-negative integer (d + 1)-tuples

which sum up to n, and λ = (λ1, . . . , λd+1) is the set ofbarycentric coordinates with respect to T . Although theBernstein polynomials have several properties which makethem intensively used in computationally demanding ap-plications such as CAGD and visualisation, they have re-ceived little attention from the finite element community.The degrees of freedom corresponding to the obtained ba-sis are completely symmetrical, and the presented elementsare the first to achieve optimal complexity on simplicial el-ements.

The complexity needed to assemble the element matri-ces for the Bernstein polynomials is optimized using a re-markable feature of the Bernstein polynomials: Althoughnot based on a tensorial construction, the Bernstein-Bezierbasis possesses the key properties needed for the sum fac-torization techniques. In addition, the product of twoBernstein polynomials is again a Bernstein polynomial,which reduces the entries of the element matrices to com-ponents of the moment vector µν

α(f) =∫Tf(x)Bν

α(x)dx,α ∈ Iν

d , for appropriate ν, for example ν = 2n− 2 for thethe components of the stiffness matrix.

Discontinuous Galerkin methods for time-dependentadvection dominated optimal control problems

Tugba Akman, Hamdullah Yucel & Bulent Karasozen(Middle East Technical University)

We consider optimal control problems (OCPs) governedby time-dependent diffusion-advection-reaction equationswith distributed controls. For the numerical solution ofthe OCP, we use the discretize-then-optimize approach. Itis well known that the standard finite element discretiza-tions applied to convection dominated diffusion problemslead to strongly oscillations; the interior and boundary lay-ers can not be resolved properly. Several well-establishedstabilization techniques have been proposed and appliedin the literature. In contrast to the these, the discontin-uous Galerkin finite element method (DGFEM) providesaccurate solutions for convection dominated problems. Re-cently several methods were applied for stationary convec-tion dominated OCPs, but there exists few works for time-dependent problems. The state, adjoint and control vari-ables are discretized in space by the symmetric (SIPG), thenonsymmetric (NIPG) and the incomplete (IIPG) interiorpenalty Galerkin methods. For the temporal discretiza-tion, backward Euler, Crank-Nicolson and semi-implicitmethod are used. The resulting OCP has been solvedby the Newton-conjugate gradient, limited BFGS and thesteepest descent method. The numerically obtained con-verge orders for the cost function confirm the a priori errorestimates.

DRBEM and DQM Solutions of Natural Convec-tion Flow in a Cavity Under a Magnetic Field

Nagehan Akgun & Munevver Tezer Sezgin (Middle EastTechnical University)

In this paper, the dual reciprocity boundary element method(DRBEM) and the differential quadrature method (DQM)are applied to solve the two-dimensional, unsteady natu-ral convection flow in a square cavity under an externallyapplied magnetic field. The governing equations are givenin terms of stream function (ψ), vorticity (w) and temper-ature (T) as

∇2ψ = −w

Pr∇2w =∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+Ha2Pr

∂v

∂x−RaPr

∂T

∂x

∇2T =∂T

∂t+ u

∂T

∂x+ v

∂T

∂y(6)

where u =∂ψ

∂y, v = −∂ψ

∂xare velocity components, Pr,

Ha and Ra are the Prandtl number, Hartmann numberand Rayleigh number, respectively. Vorticity transportand energy equations are transformed to the form of mod-ified Helmholtz equations by utilizing forward differencewith relaxation parameters for the time derivatives, andapproximating also Laplacian terms at two consecutive

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time levels. In the DRBEM solution the resulting mod-ified Helmholtz equations are solved by using the funda-mental solution 1

2πK0(x) of the modified Helmholtz equa-

tion whereas in the stream function Poisson’s equation12π

ln(x) (the fundamental solution of Laplace equation)is made use of. In the DQM solution also these modi-fied Helmholtz equation are used keeping all the previoustime level terms as inhomogeneity. In both DRBEM andDQM procedures since the equations are solved in termsof modified Helmholtz equations containing the time ad-vancement in it, there is no need to use another time inte-gration scheme, and one has the advantage of using largetime increments. In the DQM solution procedure Gauss-Chebyshev-Lobatto space points are used which are clus-tered through the end points. The comparison between theDRBEM and DQM solutions are made in terms of com-putational cost and accuracy. DQM has the advantageof using very small number of discretization points, andgiving accurate and efficient solutions. Both DQM andDRBEM are applied on the natural convection flow un-der a magnetic field in a square cavity heated and cooledon the vertical walls, and adiabatic conditions imposed onthe horizontal walls. Solutions are obtained for Rayleighnumber values between 103 and 106, and Hartmann num-bers up to Ha = 300. Although, DRBEM and DQM givealmost the same accuracy, DQM is desirable since it usesconsiderably small number of grid points resulting withless computational work.

Convergence Analysis of a Family of Damped Quasi-Newton Methods for Nonlinear Optimization

Mehiddin Al-Baali (Sultan Qaboos University)

In this talk we will extend the technique in the dampedBFGS method of Powell (Algorithms for nonlinear con-straints that use Lagrange functions, Mathematical Pro-gramming, 14: 224–248, 1978) to a family of quasi-Newtonmethods with applications to unconstrained optimizationproblems. An appropriate way for defining this damped-technique will be considered to enforce safely the positivedefiniteness property for both positive and indefinite quasi-Newton updates. It will be shown that this techniquemaintains the global and superlinear convergence propertyof a restricted class of quasi-Newton methods for convexfunctions. This property will be also enforced for an inter-val of divergent quasi-Newton methods. Numerical expe-riences will be described to show that the proposed tech-nique improves the performance of quasi-Newton methodssubstantially and significantly in certain robust and ineffi-cient cases (eg, the BFGS and DFP methods), respectively.

How to solve boundary value problems by usingVariational Iteration Method

Derya Altıntan & Omur Ugur (Middle East TechnicalUniversity)

Many problems of science are represented by a system oflinear or nonlinear differential equations. It is in generaldifficult to solve such type of systems and, hence, find-

ing approximate or, if possible, closed-form solutions hasbeen the subject of many researchers. The VariationalIteration Method (VIM) is an alternative to numericallysolving those equations and has proven its applicability.The method constructs a correction functional by usingLagrange multipliers which will be identified by using cal-culus of variations and restricted variations. We studythe method for solving boundary value problems, and ob-tain a relation between the Lagrange multipliers and thecorresponding adjoint differential equation. Furthermore,we obtain the generalisation of the method to the matrix-valued Lagrange multipliers, and hence, enjoy the usage ofno restricted variations in the linear part of the splittingoperator. To illustrate the method we apply method tovarious systems.

Stability and convergence of finite difference schemesfor complex diffusion processes

Aderito Araujo, Sılvia Barbeiro & Pedro Serranho (Uni-versity of Coimbra)

Diffusion processes are commonly used in image process-ing in order to remove noise. The main idea is that ifone pixel is affected by noise, than the noise should bediffused among the neighboring pixels in order to smooththe region. In this way proper diffusion partial differen-tial equations have been considered to achieve this end.Taking the heat equation

∂u(x, t)

∂t= ∇ · (D(x, t, u)∇u(x, t))

where u(x, t) represents the denoised image at time t withthe initial noisy image u(x, 0), the choice of the diffusionparameter D plays a very important role for the purposeof denoising. Roughly speaking, one wants D to allow dif-fusion on homogeneous areas affected only by noise and toforbid diffusion on edges to preserve features of the origi-nal denoised image. In this way, several expressions for Dhave been suggested.

The first approaches indicated that D should dependon the gradient of u with an inverse proportion. However,this approach had a few handicaps. For instance, within aramp edge the diffusion coefficient is similar along all theedge delaying the diffusion process, not distinguishing be-tween the end points and interior points of the ramp edgewhere diffusion should differ. Therefore, the use of thelaplacian was suggested as being more appropriate since ithas a higher amplitude near the end points and low mag-nitude elsewhere. The drawback is that the computationof a higher order derivative is needed leading to higherill-posedness in the first steps while the image is stronglyaffected by noise. To overcome this problem, some authorssuggested to use complex diffusion. The use of a complexdiffusion coefficient turns the partial differential equationinto some sort of combination between the heat and theSchrodinger equation.

In recent years, nonlinear complex diffusion proved tobe a numerically well conditioned technique that have beensuccessfully applied in medical imaging despeckling anddenoising. The main purpose of this work is to obtaina stability condition and establish to prove the conver-

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gence for a class on finite difference schemes for a nonlin-ear complex diffusion equation. Some numerical results,obtained in the context of a collaboration with the Insti-tute of Biomedical Research in Light and Image (IBILI),a research institution of the Faculty of Medicine of theUniversity of Coimbra, are also presented.

A posteriori error estimation for coupled problems

Liya Asner, Simon Tavener & David Kay (University ofOxford)

The coupling of multi-physics systems is essential for avariety of problems arising in engineering and applied sci-ence. Given their scale and complexity, such problemsare usually solved numerically, often using finite elementmethods. It is important to verify the accuracy of the ob-tained approximate solutions, for instance if the computa-tional model in question aids decision-making in a clinicalsetting. A posteriori methods provide an efficient prac-tical tool for error estimation. They can also be used todrive mesh refinement algorithms, which minimize the er-ror given limited computational resources.We present an a posteriori error representation formula fora time-dependent coupled problem. The representation re-lies on the solution of a dual problem. Both systems aresolved using a finite element method with Lagrange mul-tipliers, so that the system remains fully coupled and themethod stability is not compromised. The accuracy of therepresentation formula is verified for regular quantities ofinterest. A space-time mesh refinement strategy based onlocal error estimates is proposed. The effectivity of therepresentation formula and the performance of the meshrefinement algorithm are tested numerically in a numberof two-dimensional test problems. In particular, we con-sider a travelling solution, and show that its non-trivialbehaviour is captured. The meshes produced using thedeveloped strategy require a significantly lower number ofdegrees of freedom than the uniform meshes, which achievesimilar accuracy.

A low order local projection stabilized finite ele-ment method for the Oseen equation

Gabriel R. Barrenechea & Frederic Valentin (Univer-sity of Strathclyde)

This talk is devoted to describe a new local projectionstabilized finite element method (LPS) for the Oseen prob-lem. The method adds to the Galerkin formulation newfluctuation terms which are symmetric and easily com-putable at the element level. Proposed for the pair P1/Pl,l = 0, 1, when the pressure is continuously or discontinu-ously approximated, well-posedeness and error optimalityare proved. In addition, we introduce a cheap strategyto recover an element-wise mass conservative velocity fieldin the discontinuous pressure case, a property usually ne-glected in the stabilized finite element context. Numericsvalidate the theoretical results and show that the presentmethod improves accuracy to represent boundary layerswhen compared with alternative approaches.

A formula for the influence of jumps on finite sincinterpolants

Jean–Paul Berrut (Universite de Fribourg)

In an article that appeared in “Numerical Algorithms” in2007, we have discovered that a sinc interpolant on a finiteinterval essentially is the product of a harmonic functionby the difference of two quadrature formulae for a Cauchyprincipal value integral. In a second paper in the samejournal, that appeared recently, we have proved an Euler–Maclaurin formula for the error of such quadratures in thepresence of jumps of the integrand.

The present work combines the two results to obtain aformula that quantifies the impact of an arbitrary numberjumps on finite sinc interpolants. Numerical results showthat the application of the formula, together with a quoti-enting method against the Gibbs phenomenon, allows fora spectacular damping, when not total elimination, of thisimpact.

Differential complexes and interpolation operatorsin the context of high-order numerical methods forelectromagnetic problems

Alex Bespalov & N Heuer (University of Manchester)

It is now well known that differential complexes (or, exactsequences of vector spaces and differential operators), e.g.,

H1 ∇−→ H(curl)curl−→ L2,

are important in the analysis of time-harmonic problemsof electromagnetics.

When discretising these problems, it is equally impor-tant to use the approximation spaces forming the corre-sponding discrete sub-complexes. This leads to stable dis-cretisations enjoying some local conservation properties.For instance, a discrete counterpart of the above exact se-quence is

X1N

∇−→ XNedN

curl−→ X0N ,

where X1N is the space of continuous piecewise polynomi-

als, XNedN is the H(curl)-conforming finite element space

based on the first Nedelec family, and X0N is the space of

piecewise (possibly discontinuous) polynomials (here, N isa generic discretisation parameter).

The continuous and discrete complexes linked by in-terpolation operators (projectors) form de Rham diagram,which commutes if the discrete spaces and interpolationoperators are chosen properly. The commuting diagramproperty, and the corresponding error estimates for in-terpolation operators, have immediate applications to thenumerical analysis of time-harmonic electromagnetic prob-lems. In particular, these results are critical to prove thediscrete compactness property, which in turn implies theconvergence of approximations, as well as for the erroranalysis.

In this talk, we demonstrate how the above construc-tions are applicable to numerical analysis of the electricfield integral equation in three dimensions. We focus onthe boundary element discretisations employing high-order

28

polynomials (i.e., p- and hp-versions of the boundary el-ement method). Inspired by the work of Demkowicz andBabuska on high-order finite element approximations ofMaxwell’s equations, we introduce and analyse a new pro-jection based interpolation operator which satisfies the com-muting diagram property involving appropriate trace spaces.We also establish an estimate for the interpolation errorin the norm of the trace space H−1/2(÷), which is closelyrelated to the energy spaces for boundary integral formu-lations of time-harmonic problems of electromagnetics inthree dimensions.

Nonlinear functional equations satisfied by orthog-onal polynomials

C. Brezinski, (Universite des Sciences et Technologiesde Lille)

Let c be a linear functional defined by its moments c(xi) =ci for i = 0, 1, . . .. We proved that the nonlinear func-tional equations P (t) = c(P (x)P (αx + t)) and P (t) =c(P (x)P (xt)) admit polynomial solutions which are thepolynomials belonging to the family of formal orthogonalpolynomials with respect to a linear functional related toc. This equation relates the polynomials of the familywith those of the scaled and shifted family. Other types ofnonlinear functional equations whose solutions are formalorthogonal polynomials are also presented. Applicationsto Legendre and Chebyshev polynomials are given. Then,orthogonality with respect to a definite inner product isstudied. When c is an integral functional with respectto a weight function, the preceding functional equationsare nonlinear integral equations, and these results lead tonew characterizations of orthogonal polynomials on thereal line, on the unit circle, and, more generally, on analgebraic curve.

On numerical algorithms for level set and curva-ture based models for surface fairing

Carlos Brito-Loeza (Universidad Autonoma de Yucatanand University of Liverpool)

The problem of surface fairing has been successfully ad-dressed using variational models. To this end, differentmodels involving the minimization of different functionals,some using the Gaussian curvature of the surface, othersthe mean curvature and the rest using variations or combi-nations of the principal curvatures of the surface has beenproposed and tested by different authors. Either way us-ing an implicit representation of the surface by level sets oran explicit representation by triangulation, a usually stiffand high order partial differential equation arising fromthe minimization problem has to be solved. So far theexplicit representation is more popular particularly withinthe computer graphics community since apparently it de-livers a faster numerical solution than the level set method.For the latter and up to our knowledge only a explicit Eulermethod has been reported in the literature. In this talk wewill present our success in using the stabilized fixed pointmethod proposed in [C. Brito-Loeza and K. Chen, Multi-

grid algorithm for high order denoising, SIAM J. ImagingSci., 3 (2010), pp. 363-389.] to rapidly find the solutionof the model

∫Ωh2|∇Φ|dΩ where h = ∇ · ∇Φ

|∇Φ| stands forthe mean curvature of the surface represented by the zerolevel set of Φ. We will also discuss our experiences inapplying the same algorithm to the model presented in[Matthew Elsey and Selim Esedoglu, Analogue of the to-tal variation denoising model in the context of geometryprocessing, SIAM Multiscale Model. Simul., 7 (2009), pp.1549-1573]

Using reduced-order models and waveform relax-ation to solve cerebral bloodflow autoregulationDEs on a large binary tree network

Richard Brown& Tim David (University of Canterbury)

Under certain assumptions the network of cerebral bloodvessels can be modelled by a purely resistive binary treewhere each vessel is characterised by a single resistance,computed as a function of its length and radius. Eachvessel has the ability to vasoconstrict or vasodilate, hencechanging their resistance, in response to local variablessuch as cellular Ca2+ concentration within the vessel, andcollectively they autoregulate the cerebral bloodflow. Thesedynamics are highly nonlinear and can be described phe-nomenologically by a system of ODEs for each vessel.

Mathematically, the overall system is a large networkof coupled ODEs, which can comprise up to millions ofvariables. The overall system is highly coupled, and canexhibit stiffness, so solving by traditional sequential ODEalgorithms can be very expensive. The tree structure sug-gests an obvious parallelisation of the problem into sub-trees, where the coupling between subtrees can be cap-tured by a single variable: the blood pressure pi(t) at thejunction where the trees are coupled. The system can besolved iteratively by a waveform relaxation variant, how-ever convergence can be very slow. To speed up conver-gence we investigate using a small linear reduced ordermodel obtained from a linearisation of the overall systemto get a close initial approximation to start the waveformrelaxation procedure.

Mimetic Finite Difference methods for elliptic prob-lems

Andrea Cangiani (University of Leicester)

In this talk we present recent developments in MimeticFinite Difference (MFD) methods for elliptic problems.MFDmethods, at least in their low order, original, versionsare techniques to extend classical finite element methodsto general polygonal/polyhedral meshes. General meshesarise in the treatment of complex solution domains andheterogeneous materials (e.g. reservoir models) and areparticularly indicated in moving meshes techniques as wellas adaptive mesh refinement/de-refinement.

A more well known technique to extend finite elementsto polygonal meshes is given by harmonic finite elements.We will present a novel interpretation of MFDs showingthat these combine the accuracy and ease of implemen-

29

tation of standard, polynomial finite elements, with themeshing flexibility of harmonic elements. We shall alsodescribe MFDs for elliptic eigenvalue problems as well asconvection-diffusion problems. Stabilization techniques inthe convection-dominated regimes will also be presented.

Convolution quadrature revisited for Volterra in-tegral equations

Penny J Davies & Dugald B Duncan (University ofStrathclyde)

Convolution quadrature (CQ) methods were introducedby Lubich in 1988 to approximate convolution integrals.More recently they have been used for the temporal ap-proximation of time–dependent boundary integral equa-tions together with a finite element or collocation approxi-mation in space, because they typically give a more stablenumerical algorithm than standard time-stepping meth-ods. The drawback is that the support of the associatedCQ basis functions is infinite, which gives rise to densesystem matrices and makes the method inefficient. Weshow how using a spline-based implementation of the CQmethodology can circumvent this.

RBF-FD Methods for Elliptic Equations

Oleg Davydov & Dang Thi Oanh (University of Strath-clyde)

We investigate low order RBF-based generalised finite dif-ference methods (RBF-FD) on irregular centres. This ap-proach is genuinely meshless and can compete in accuracywith the linear finite element method that requires solvinglinear systems of comparable bandwidth/density. Algo-rithms and numerical results for stencil support selection,adaptive refinement and optimal shape parameter will bediscussed.

Postprocessing Reservoir Simulations to Obtain Ac-curate Well Pressures

Dugald B Duncan & Nneoma Ogbonna (Heriot-WattUniversity)

We discuss a new post-processing technique for standardreservoir simulators to efficiently determine detailed andaccurate information about the time-dependent pressureat a wellbore. This information is important in monitor-ing and assessing the properties and capacity of reservoirsof various types. The wellbore is a relatively tiny hole ina large reservoir. Standard simulators use point or linesource well models and do not resolve down to the tinyscale of the well radius, which is where the required infor-mation actually is. Our method involves just a local solveof the time dependent pressure equation in the vicinityof the wellbore, using as boundary data information fromthe main reservoir simulation. We describe the connec-tion with the widely used Peaceman wellbore index andits variants, discuss the accuracy (both from analysis andexperiments) and show results from a number of test cases.

Comparing two iterated projection approximationsfor integral operators in terms of convergence andcomputational cost

Rosario Fernandes, Mario Ahues & Filomena d’Almeida(Universidade do Minho)

We will consider the Fredholm integral equation of thesecond kind

(T − zI)φ = f, (7)

where T is a compact linear operator defined in a Hilbertspace X by

(Tφ)(s) =

∫ b

a

k(s, t)φ(t)dt, s ∈ [a, b], φ ∈ X,

I denotes the identity operator on X, z belongs to theresolvent set of T and f belongs to X.

An approximate solution of the equation (1) can beobtained as a solution of an approximate problem

(Tn − zI)φn = f, (8)

where Tn is one element of a sequence (Tn)n≥1 which isν−convergent to T .

This sequence can be obtained by projection meth-ods, such as the Classical Galerkin, the Kantorovich orthe Sloan methods. Given πn a projection from X onto afinite dimentional subspace Xn, those approximations maybe defined as TG

n = πnTπn, TKn = πnT and TS

n = Tπn,respectively.

Recently, R. Kulkarni has proposed a new projectionmethod defined by TR

n = TKn + TS

n − TGn and proved that,

although the size of the linear system to be solved is thesame as in the Galerkin method, the rate of convergenceof the method proposed is higher than the order of theClassical Galerkin and the Sloan methods.

We will compare this method with the iterated Kan-torovich approximation, both from the convergence pointof view and their computational cost. When we mentioniterated Kantorovich method we mean that we solve theapproximate problem (11) thus obtaining φK

n and then werefine it using (10) that yields φIK

n = 1z(TφK

n − f).

From all standard projection approximations of a boundedlinear operator in a Hilbert space, iterated projection andKulkarni’s discretization exhibit a global superconvergenterror bound.

Limited memory spectral gradient methods

Roger Fletcher (University of Dundee)

Spectral Gradient methods such as the Barzilai-Borweinmethod have given new impetus to the capability of solv-ing large scale unconstrained optimization methods. Thetalk will described how significant improvements can beobtained by making use of a few extra ‘long vectors’ ofmemory. The possible application of these ideas to non-linearly constrained optimization will also be explored.

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Energy Conservation Laws of Maxwell’s Equationsand Their Application in Numerical Analysis ofFDTD

Liping Gao (China University of Petroleum)

In this talk, I would like to introduce some new energy con-servation laws of the 3D Maxwell’s equations in time do-main in the case of the perfectly electric conducting bound-ary conditions. Based on these new continuous forms ofidentities for the Maxwell’s equations, discrete forms ofidentities for FDTD are derived, which show that FDTDis energy conserved in terms of the discrete H1 norms.Conditional stability of FDTD in the discrete H1 norms isthen proved. By the energy-conserved identities of FDTD,we also prove that FDTD are second order accurate bothin time and space in the discrete H1 norms when a condi-tion that is stronger than the CFL condition is satisfied.This means that FDTD is super convergent in the dis-crete H1 norms. Numerical experiments are carried outand computational results to test the energy conservation,stability and super convergence are presented.

Puiseux-based extrapolation for large-scale degen-erate quadratic programming

Nick Gould, Dominique Orban & Daniel Robinson (STFC-Rutherford Appleton Laboratory)

We describe the design of a new software package CQP forlarge-scale convex quadratic programming. The methodis based on high-order Taylor and Puiseux approximationto a variety of infeasible arcs connecting the current iter-ate to a better target point. Possible arcs include thoseby Zhang and by Zhou and Sun based on work by Stoer,Mizuno, Potra and others. The resulting algorithm is prov-ably both globally and polynomially convergent at an ul-timately high-order (depending on the series used) in bothnon-degenerate and degenerate cases. We will illustrate anumber of algorithmic options on the CUTEr QP test set.CQP is available as part of the fortran 90 library GALA-HAD.

A parallel integrator for linear initial-value prob-lems

Stefan Guttel & Martin J. Gander (University of Ox-ford)

We propose a novel parallel method for the integrationof linear initial-value problems

u′(t) = Au(t) + f(t), u(0) = u0, (9)

where A is a large sparse or structured matrix, and u, fare vectors depending on time. As opposed to most exist-ing parallel methods which exploit the linear structure of(9), our method does not require direct or inverse Laplacetransforms of f . We utilize an overlapping time decompo-sition of (9) into decoupled inhomogeneous and homoge-neous subproblems, and a near-optimal Krylov method forthe fast exponential propagation of the homogeneous sub-

problems. The solution of the original problem is then ob-tained by the superposition of subproblem solutions. Theparallel speedup depends on the ratio by which homoge-neous problems can be solved faster than inhomogeneousproblems, and hence becomes large when the inhomogene-ity f is sufficiently stiff. The efficiency of this approach isdemonstrated at some model problems.

Matrix-free IPM with GPU acceleration

J.A.J. Hall, J. Gondzio & E. Smith (University of Edin-burgh)

Interior point methods (IPM) with direct solution of theunderlying linear systems of equations have been used suc-cessfully to solve very large scale linear programming (LP)and quadratic programming (QP) problems. However, thelimitations of direct methods for some classes of problemshave led to iterative techniques being considered. Thematrix-free method is one such approach and is so namedsince the iterative solution procedure requires only the re-sults of operations Ax and ATy, where A is the matrix ofconstraint coefficients. Thus, in principle, it may be ap-plied to problems where A is not known and only an oracleis available for computing Ax and ATy. Since the com-putational cost of these operations may well dominate thetotal solution time for the problem, it is important thatthe techniques used to perform them are efficient.

This talk will outline the matrix-free interior pointmethod and, for several classes of LP problems, demon-strate its overwhelmingly superior performance relative tothe simplex method and IPM with equations solved di-rectly. The dominant cost of the operations Ax and ATywill then be used to motivate their implementation on aGPU to yield further performance gains. Different com-putational schemes for these sparse matrix-vector productswill be discussed. A comparison of the speed-up achievedusing a many-core GPU implementation with that for amulti-core CPU implementation indicates the former hasbetter potential.

Experience of linear solvers in an nonlinear inte-rior point method

Jonathan Hogg (STFC Rutherford Appleton Laboratory)

We present the results of some extensive testing of varioussymmetric indefinite factorization codes inside the well-known nonlinear interior point solver IPOPT. We describevarious experiments to improve performance and attemptsto classify the problems on which each solver performsbest.

In addition to the solvers that are available as standardwe also give results on the latest HSL solvers aimed atmulticore platforms.

A fast matrix assembling for interior penalty dis-continuous Galerkin method

Tamas L. Horvath (Szechenyi University)

31

We investigate the matrix assembling for the IPDG so-lution of

−∇ · (A∇u) +B · ∇u = f on Ω,

u|∂Ω = uD,

with A and B piecewise constant. Using IPDG methodwe have to solve the following: find w ∈ VDG with a(w, v)+b(w, v) = φ(v), ∀v ∈ VDG where:

a(w, v) =∑E∈τh

∫E

A∇w · ∇v −∑e∈Γ0

∫e

A∇w · ηe [[v]] +

ϵ∑e∈Γ0

∫e

A∇v · ηe [[w]] +∑e∈Γ0

σ0

|e|

∫e

[[v]] [[w]] ,

b(w, v) = −∑E∈τh

∫E

wB · ∇v +∑e∈Γ0

∫e

B · ηewup [[v]] ,

φ(v) =

∫Ω

fv +∑e∈Γ

∫e

(ϵA∇v · ηe +

σ0

|e|v +B · ηev)uD,

where [[u]] is the jump of u, u is the average of uon the interelement faces. τh denotes a triangular tessel-lation of Ω, Γ = ∂Ω, Γ0 =

∪E∈τh

∂E, σ0 is the penaltyparameter, ε ∈ −1, 0, 1 and VDG contains polynomialsof a given order on each element.

To set up the linear equations we have to computea(Φi,Φj) for the basis functions of VDG. The computationof the corresponding integrals element-by-element is quiteexpensive especially if we are using higher order elements.

Standard reference domain based computations are wellknown for matrix assembling in comforming finite elementprocedures for constant diffusion matrices. These requireonly an affine linear mapping between the reference do-main and the actual one and the integrals of the basisfunctions on the reference domain. For interior penaltythe main difficulties emerge when calculating the integralsof jumps and averages on interelement faces.

These terms can be rewritten in a more convenientform without jumps and averages, see e.g. the book byRiviere. To calculate these integrals one should invest hugecomputational work for computing the quadrature pointsand evaluate the basis functions here.

To overcome these difficulties we develop a referencedomain based algorithm, where we could calculate inte-grals on a reference domain and on its neighbours, and weonly have to use the matrices containing these informa-tion. The advantage of our approach becomes clear whenwe are using higher order elements. The running times wepresent prove the efficiency of the method both in two andthree dimensions.

A high performance dual simplex solver

Q. Huangfu & J.A.J. Hall (University of Edinburgh)

The dual simplex method is frequently the preferred ap-proach when solving large scale linear programming (LP)

problems. Algorithmic enhancements of the dual sim-plex method that are important for its efficient serial im-plementation are the dual steepest-edge framework andbound flipping ratio test. As the improved performanceof modern computer architectures is now being achievedvia parallelism, it is important that this be exploited bythe dual simplex method. Suboptimization is a relativelyunknown algorithmic variant of the dual simplex methodthat significantly increases the scope for exploiting paral-lelism. This talk will discuss how these three algorithmicenhancements may be combined within a parallel imple-mentation by means of novel computational techniques.Results will be given that demonstrate the competitive-ness of this implementation relative to an efficient serialdual simplex solver.

Zeros of matrix polynomials: A case study

Drahoslava Janovska&Gerhard Opfer (Institute of Chem-ical Technology, Prague)

Let us consider a matrix polynomial p,

p(X) =

N∑j=0

AjXjBj , where

X, Aj , Bj ∈ K2×2, j = 0, . . . , N, A0B0, AN , BN = 0,

K stands for the field of real or complex numbers, thematrices Aj , Bj , j = 0, 1, . . . , N, N ≥ 1, are givenmatrices of order 2. If the matrix X has property thatp(X) = 0, we callX a zero of p. In general, the polynomialcan consist of multiple terms of the same degree.

Let X ∈ K2×2 and let χX(z) = z2 − tr(X) z + det(X)be its characteristic polynomial. Then there exist numbersαj , βj , j = 0, 1, . . . , such that j−power of the matrix Xcan be written as

Xj = αjX+βjI for all j = 0, 1, . . . , where α0 := 0, β0 := 1,

αj+1 := tr(X)αj + βj , βj+1 := −αjdet(X), j ≥ 0,

and the polynom p has the form

p(X) =

N∑j=0

Aj(αjX+ βjI)Bj

= A0B0 +

N∑j=1

αjAjXBj +

N∑j=2

βjAjBj .

In this particular case, i.e. for X, Aj , Bj ∈ K2×2, we givea classification of all zeros of the polynomial in terms of therank of the corresponding system. We have also developedan algorithm for the computation of these zeros.

In two papers [1], [2], quaternionic polynomials, bothsimple and two-sided, are investigated. Due to the factthat there is a class of 2 × 2 complex matrices and aclass of 4×4 real matrices, which isomorphically representquaternions, the quaternionic polynomials are implicitlycontained in the investigation of matrix polynomials. In[1] we have shown that for one-sided quaternionic polyno-mials there are two classes of zeros: isolated and spherical.Two-sideded quaternionic polynomials have, in addition,

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three more classes of zeros defined by the rank of a certainreal 4× 4 matrix, [2].

These two special examples of matrix polynomials showthat the task to classify and compute zeros of the matrixpolynomials is quite complicated even when the coefficientsand unknown matrices are only of order two.

The research was supported by DFG grant OP 33/19-1 and by the research project MSM 6046137306 of TheMinistry of Education, Youth and Sports, Czech Republic.

References

[1] D. Janovska, G. Opfer: A note on the computation ofall zeros of simple quaternionic polynomials, SIAM J. Nu-mer. Anal. 48, No.244 (2010), 244-256.

[2] D. Janovska, G. Opfer: The classification and the com-putation of the zeros of quaternionic, two-sided polyno-mials, Numerische Mathematik, Volume 115, No.1 (2010),81-100.

On periodic patterns in the traffic flow

Vladimır Janovsky & Lubor Buric (Charles University)

We investigate microscopic models of the road traffic, seee.g. [1]. In particular, we consider a car-following modelfor a single-line traffic flow on a circular road, [2]. Themodel is characterized by a choice of a particular optimalvelocity function (OV) which is related to each individualdriver. The aim is to investigate traffic jams. Mathemat-ically, the jams are periodic solutions of the model. Theyform smooth rotating waves. In [3], it was shown that pe-riodic solutions (cycles) of the model are due to the Hopfbifurcation of steady state solutions. Nevertheless, onecan check that many cycles become non physical for largeparameter regions.

The analysis shows that a trajectory becomes non phys-ical since an event which can be interpreted as a collisionwith the foregoing car. The natural action of a driverat that moment would be to overtake the slower car. In[4], we proposed a model of an overtaking. It implicitlydefines a maneuver consisting of deceleration/accelerationjust shortly before/after the event. The maneuver is fullydefined by the OV of the driver. The resulting model is anevent-driven model. It is no longer smooth. In [4], we alsohinted at a large variety of oscillatory solutions (oscillatorypatterns) of this new model.

In our presentation, we

• formulate our model as a particular Filippov system,see [5],

• define selected oscillatory patterns as invariant ob-jects of this Filippov system,

• use the standard software (AUTO97) to continuethese patterns with respect to a parameter,

• consider bifurcations of the patterns.

REFERENCES

[1] D. Helbing, “Traffic and related self-driven many-partical systems”, Rev. Modern Phys., Vol. 73,pp. 1067–1141, (2001).

[2] M. Bando, K. Hasebe, A. Nakayama, A. Shibata,Y. Sugiyama, “Dynamical model of traffic con-gestion and numerical simulation”, Phys. Rev. E,Vol. 51, pp. 1035–1042, (1995).

[3] I. Gasser, G. Sirito, B. Werner, “Bifurcation anal-ysis of a class of ’car following’ traffic models”,Physica D, Vol. 197, pp. 222–241, (2004).

[4] L. Buric, V. Janovsky, “On pattern formation ina class of traffic models”, Physica D, Vol. 237, pp.28–49, (2008).

[5] L. Buric, V. Janovsky, “The traffic problem:Modeling of the overtaking”, Mathematical Meth-ods in the Applied Sciences, Vol. 32, pp. 2319–2338, (2009).

The Effects of Dispersive and Dissipative Proper-ties of Numerical Methods on the Linear Advec-tion Equation during Data Assimilation

Sian Jenkins, Chris Budd, Melina Freitag & NathanSmith (University of Bath)

Data assimilation is a method often used in forecastingwhere the results from a numerical model simulating aphysical system are compared to past observations of thetrue physical system. The aim of the comparison is to iden-tify parameters for the numerical model which will providean improved forecast of future observations of the physicalsystem. There are many different methods for data as-similation. In this instance 4D-Variational (4D-Var) dataassimilation is considered.

4D-Var data assimilation involves using an approxima-tion for the initial condition, called the analysis vector, ofthe model in order to form results for comparison withthe observations. The analysis vector is found such thatit produces the least squares solution between the resultsfrom the simulation and the observations taken over theassimilation window. The simulation is continued for aperiod of time longer than the assimilation window in or-der to generate a forecast for the physical system. This ishow operational weather forecast centers generate weatherforecasts.

However errors can affect the results of 4D-Var dataassimilation and come from many different sources. Forexample, model error is introduced through the equationsof the model inaccurately simulating the true physical sys-tem, and numerical error during the solution of the equa-tions. In this case we are interested in the numerical errorand its effects during 4D-Var data assimilation. In order tounderstand the effects of numerical model error, the linearadvection equation is investigated, u : R× R → R,

ut(x, t) + ux(x, t) = 0.

Numerical model error is introduced when simulatingthe linear advection equation through the approximationof the derivatives using finite difference methods. The aimis to understand how the application of different finite dif-

33

ference methods for the equation affects the accuracy ofthe outcome from the inverse problem.

Many finite difference schemes have been designed tosolve the linear advection equation, such as the Lax-Wendroffand Box Scheme. The dispersive and dissipative effects ofthese methods become apparent over time through the af-fect of the numerical methods on the different frequencycomponents of the wave. In this presentation we discussthe effects of the dispersive and dissipative properties ofseveral methods in relation to 4D-Variational data assim-ilation. Here we consider a shock profile with periodicboundary conditions and consider the accuracy of the re-sultant analysis vector and its forecast over the assimila-tion window.

Domain Decomposition for Reaction-Diffusion Sys-tems

Rodrigue Kammogne & Daniel Loghin (University ofBirmingham)

DomainDecomposition (DD) methods have been success-fully used to solve elliptic problems , as they are dealingwith the problem in a more elegant and efficient way by di-viding the domain into subdomains and then obtaining thesolution by solving smaller problems on these subdomains.Furthermore DD−techniques can incorporate in their im-plementation not only the physics of the different phenom-ena associated with the modeling , but also the enhance-ment of parallel computing. They can be divided into twomajor categories: with and without overlapping. The mostimportant factor in either case is the ability to solve theinterface problem referred as the Steklov-Poincare. Forsolving the interface problem there exist two approaches.The first approach,which consists of approximating the in-terface problem by a sequence of subdomains problem andthe second approach, which aims to tackle the interfaceproblem directly. As far as we know none of these havebeen applied on reaction diffusion systems.In the past few years reaction diffusion systems have re-ceived a strong attention as the have been widely usedin biological , chemical and recently in financial model-ing. Obtaining a numerical solution for reaction diffusionsystems remains a challenging task.In our presentation we consider the following generic sys-tem :

ut −∆u = f(u) in ΩBu = g on ∂Ω

where u = (u1, u2) and f ∈ R2 .

We will describe how DD−methods can be used to tacklereaction diffusion systems problems in the non-overlappingcase. Furthermore by addressing the problem directly onthe interface we will present and analyze different DD-Preconditioners of the Schur-complement for solving thearising linear system , as this lead to a solution techniqueindependent to the mesh parameter. More precisely , wewill exploit the fact that the Steklov-Poincare operatorsarising in a non-overlapping DD−algorithm are coerciveand continuous with respect to Sobolev norms of index

1/2. Finally we will validate the theoretical results onnumerical experiments drawn from biological applications.

Improved T − ψ finite element schemes for eddycurrent problems

Tong Kang & Tao Chen (Communication University ofChina)

The aim of this paper is to propose improved T − ψ finiteelement schemes for eddy current problems in the three-dimensional bounded domain with a simply-connected con-ductor. In order to utilize nodal nite elements in spacediscretization, we decompose the magnetic eld into sum-mation of a vector potential and the gradient of a scalarpotential in the conductor; while in the nonconducting do-main, we only deal with the gradient of the scalar scalar.As distinguished from the traditional coupled scheme withboth vector and scalar potentials solved in a discretizingequation system, the proposed decoupled scheme is pre-sented to solve them at two separate equation systems,which avoids solving a saddle-point equation system likethe coupled scheme and leads to an important saving incomputational eort. The simulation results and the datacomparison of TEAM Workshop Benchmark Problem 7between the traditional coupled scheme and our decoupledscheme show the validity and eciency of the decoupled one.

Convergence of an implicit algorithm for a finitefamily of asymptotically quasi-nonexpansive maps

Abdul Rahim Khan, Hafiz Fukhar-ud-din & Muham-mad Aqeel Ahmad Khan (King Fahd University of Petroleumand Minerals)

The proof of the celebrated Banach contraction principlehinges on ”Picard iteration”. The principle, on the onehand, itself can be implemented on a computer to approx-imate fixed point of a contractive map to any required ac-curacy and on the other hand, it is applicable to a varietyof subjects such as partial differential equations, integralequations and image processing. This signifies prime im-portance of the iterative approximation of fixed points intheoretical numerical analysis.

In this paper, we study weak and strong convergenceof a new two-step implicit algorithm for a finite familyof asymptotically quasi-nonexpansive maps in a uniformlyconvex Banach space. The results are proved for a moregeneral implicit iterative scheme under weaker assump-tions on the control sequences of parameters. Our resultsare generalization of several well-known results from thecurrent literature.

Linear rational finite differences and applications

Georges Klein & Jean–Paul Berrut (University of Fri-bourg)

The barycentric rational interpolants introduced by Floaterand Hormann in 2007 are “blends” of polynomial inter-polants of fixed degree d. In some cases these rational

34

functions achieve approximations of much higher qualitythan the classical polynomial interpolants, which, e.g., areill-conditioned and lead to Runge’s phenomenon if theinterpolation nodes are equispaced. Besides being goodinterpolants for such a distribution of nodes, the ratio-nal functions are suited for applications as well. We willpresent linear rational finite difference methods and someof their applications.

Norm Growth and Pseudospectra for PML Dis-cretisations of Open Systems

Jacopo Lanzoni & Timo Betcke (University College ofLondon)

A standard tool for the solution of acoustic scatteringproblems in open systems is the use of perfectly matchedlayers to cut-off the computational domain. Althoughthe corresponding operator has a bounded inverse for realwavenumbers, it can nevertheless exhibit large norm-growthon the real axis in the neighborhood of complex reso-nances. In this talk we investigate this norm growth in theneighborhood of eigenvalues of PML discretisations. Thesuitable tool to do this is to compute generalised pseu-dospectra of the discretised problem. Since PML discreti-sations lead to large, sparse matrices we will use reductiontechniques to make the pseudospectra computations feasi-ble. We apply this approach to several interesting prob-lems, where resonances are close to the real axis and alsostudy the influence of the PML parameters on the pseu-dospectra of the discretised problems.

Fractional Matrix Powers

Lijing Lin & Nicholas J. Higham (University of Manch-ester)

The aim of this work is to devise a reliable algorithm forcomputing Ap for A ∈ Cn×n and arbitrary p ∈ R. Theneed to compute fractional powers Ap arises in a varietyof applications, including Markov chain models in financeand healthcare, fractional differential equations, discreterepresentations of norms corresponding to finite elementdiscretizations of fractional Sobolev spaces, and the com-putation of geodesic-midpoints in neural networks. Here,p is an arbitrary real number, not necessarily rational. Of-ten, p is the reciprocal of a positive integer q, in which caseX = Ap = A1/q is a qth root of A. Various methods areavailable for the qth root problem. However, none of thesemethods is applicable for arbitrary real p. In this work,a new algorithm is developed for computing arbitrary realpowers Ap. The algorithm starts with a Schur decomposi-tion, takes k square roots of the triangular factor T , evalu-

ates an [m/m] Pade approximant of (1−x)p at I −T 1/2k ,and squares the result k times. The parameters k and mare chosen to minimize the cost subject to achieving doubleprecision accuracy in the evaluation of the Pade approxi-mant, making use of a result that bounds the error in thematrix Pade approximant by the error in the scalar Padeapproximant with argument the norm of the matrix. ThePade approximant is evaluated from the continued fraction

representation in bottom-up fashion, which is shown to benumerically stable. In the squaring phase the diagonaland first superdiagonal are computed from explicit formu-

lae for T p/2j , yielding increased accuracy. Since the basicalgorithm is designed for p ∈ (−1, 1), a criterion for reduc-ing an arbitrary real p to this range is developed, makinguse of bounds for the condition number of the Ap problem.How best to compute Ak for a negative integer k is alsoinvestigated. In numerical experiments the new algorithmis found to be superior in accuracy and stability to severalalternatives, including the use of an eigendecompositionand approaches based on the formula Ap = exp(p log(A)).

Robust Solution of a BVP arising in Liquid Crys-tal Modelling

John Mackenzie, Craig Macdonald, Chris Newton & Al-ison Ramage (University of Strathclyde)

In this talk I will discuss the use of an adaptive finiteelement method to solve a non-linear singularly perturbedboundary value problem which arises from a one-dimensionalQ-tensor model of liquid crystals. The adaptive non-uniformmesh is generated by equidistribution of a strictly positivemonitor function which is a linear combination of a con-stant floor and a power of the second derivative of the nu-merical solution. By an appropriate selection of the moni-tor function parameters, we show that the accuracy of thecomputed numerical solution is robust to the size of thesingular perturbation parameter and achieves an optimalrate of convergence with respect to the mesh density.

Fast Direct Solvers for Elliptic PDEs

Per-Gunnar Martinsson (University of Colorado at Boul-der)

That the linear systems of algebraic equations arising uponthe discretization of elliptic PDEs can be solved very rapidlyis well-known, and many successful iterative solvers withlinear complexity have been constructed (multigrid, Krylovmethods, etc). More recently, it has been demonstratedthat it is also often possible to directly compute an ap-proximate inverse (or LU/Cholesky factorization) to thecoefficient matrix in linear or close to linear time. Theinverse is computed in a data-sparse format that exploitsinternal matrix structure such as rank-deficiencies in theoff-diagonal blocks.The talk will focus on methods relying on the Hierarchi-cally Semi-Separable (HSS) matrix format to efficientlyrepresent the solution operator to the PDE. This formatis less versatile than the more popular H and H2 matrixformats, but typically results in very high performance interms of speed and accuracy when it can be made to work.For problems on 1D domains such as a boundary integralequation (BIE) on a domain in the plane, the adaptationof the HSS format is straight-forward, and problems inhigher dimensions can be handled via recursive domaindecomposition techniques that reduce the dimensionalityof the domain on which the compressed operator acts.The talk will describe numerical examples in both two and

35

three dimensions. Variable coefficient problems are han-dled via accelerated nested dissection methods, while con-stant coefficient problems are solved via the correspondingboundary integral equation formulations.

The relation between two algorithms for enclosingmatrix eigenvalues

Shinya Miyajima (Gifu University)

In this talk, we are concerned with accuracy of computedeigenvalues for

Ax = λx, A ∈ Cn×n, λ ∈ C x ∈ Cn, (1)

where λ is an eigenvalue and x is an eigenvector corre-sponding to λ.

There are several algorithms for enclosing eigenvaluesin (1), e.g., [2, 3]. Especially we consider the algorithmsfor enclosing all eigenvalues which is applicable even whenA is not Hermitian. Such an algorithm has been proposedin [2, Algorithm 2], which is based on the following theo-rem:

Theorem 1 (Oishi [2]) Assume as a result of numericalcomputation, we have an n × n complex diagonal matrixD and an n × n complex matrix X such that AX ≈ XDfollows approximately. Let λi, i = 1, . . . , n and ∥ · ∥ be the(i, i) element of D and a norm satisfying ∥ FG ∥≤∥ F ∥∥ Gfor F,G ∈ Cn×n, respectively. Denote the n × n identitymatrix by I. For an arbitrary n × n complex matrix Y ,define n × n complex matrices R and S as R := Y AX −D, S := Y X − I. Then it holds that min≤i≤n |λ− λi| ≤ϵo, ϵo :=∥ R ∥ + ∥ A ∥∥ S ∥ .

On the other hand, a theorem has been presented in [1]for enclosing all eigenvalues in the generalized eigenvalueproblem

Ax = λBx, A,B ∈ Cn×n, λ ∈ C x ∈ Cn (2)

This theorem is also applicable even when A is not Hermi-tian. We can utilize this theorem for enclosing all eigenval-ues in the standard eigenvalue problem (1) by substitutingB = I into (2). Then we obtain the following theorem:

Theorem 2 (Miyajima [1]) LetD, X, λi, ∥ · ∥, I, Y andS be defined as in Theorem 1. Define an n × n complexmatrix T as T := Y (AX − XD. If ∥ S ∥< 1, then X andY are nonsingular, and it follows that min1≤i≤n |λ− λi| ≤

ϵm, ϵm :=∥ T ∥

1− ∥ S ∥ .

The purpose of this talk is to present a theorem show-ing that ϵo ≥ ϵm holds if max1≤i≤n |λi| + ϵo ≤∥ A ∥ and∥ S ∥< 1 hold. We discuss the validity of these assump-tions, and report some numerical results for confirmingthat the presented theorem is true and showing that theseassumptions follow in many cases.

References

1 Miyajima, S.: Fast enclosure for all eigenvalues ingeneralized eigenvalue problems. J. Comput. Appl.Math. 233, 2994–3004 (2010)

2 Oishi, S.: Fast enclosure of matrix eigenvalues andsingular values via rounding mode controlled com-putation. Linear Algebra Appl. 324, 133–146 (2001)

3 Rump, S.M.: Computational error bounds for mul-tiple or nearly multiple eigenvalues. Linear AlgebraAppl. 324, 209–226 (2001)

An efficient finite volume method applied to flowin heterogeneous porous media

Tim Moroney, Ben Cumming, Kevin Burrage & IanTurner (Queensland University of Technology)

We present a finite volume method for efficiently solvingRichards’ equation in heterogeneous porous media. Thespatial discretisation of the governing partial differentialequation naturally handles heterogeneous media on un-structured meshes by allowing discontinuities in conservedquantities, such as moisture content, at the interface be-tween media. At each node, a pair of algebraic and differ-ential equations is produced, and the resulting system issolved efficiently using backward differentiation formulae.The inclusion of the algebraic equations results in excel-lent mass conservation in the computed solution, and anappropriately tailored preconditioner largely mitigates theadditional cost.

This formulation makes extensive use of basic denseand sparse linear algebra operations, making it an excel-lent candidate for graphics processing unit (GPU) accel-eration. We show how C++ templates allow for the samehigh-level code to be run on either CPUs or GPUs, withtechnologies such as CUDA, MPI and openMP used toextract maximum performance from the machine.

Results are exhibited that show how this formulationproduces accurate and efficient solutions to problems inheterogeneous media. Timings for multicore CPU andGPU runs demonstrate that this formulation is amenableto effective GPU acceleration.

Arithmetic and communication minimizing algo-rithms for the symmetric eigenvalue problem andthe singular value decomposition

Yuji Nakatsukasa (University of California, Davis)

The total cost of an algorithm is a combination of its arith-metic cost and communication cost. Recently there hasbeen considerable progress in the development of linear al-gebra algorithms that minimize communication (e.g., [?]).However, the reduction in communication cost sometimescomes at the expense of significantly more arithmetic. Thisincludes the recently-proposed communication-minimizingalgorithms [?] for computing the symmetric eigenvalue de-composition and the singular value decomposition (SVD).These algorithms also suffer from potential instability.

In this talk I will describe algorithms for these twodecompositions that minimize both communication andarithmetic, up to a constant factor smaller than 3. Bothalgorithms are based on a QR-based algorithm [?] for com-puting the polar decomposition. The essential cost for eachof these algorithms is in performing QR decompositions

36

of the form

[XI

]= QR, of which we require no more

than 6 for the symmetric eigendecomposition, and 12 forthe SVD. I will establish backward stability of these al-gorithms under mild assumptions. Preliminary numericalexperiments demonstrate their efficiency.

Optimal state and parameter estimation using ahybrid data assimilation scheme

N K Nichols P J Smith and S L Dance (University ofReading)

A numerical model can never completely describe the com-plex physical processes underlying the behaviour of a realworld dynamical system. State of the art computationalmodels are becoming increasingly sophisticated, but inpractice these models suffer from uncertainty in their ini-tial conditions and parameters. Mathematical techniquesfor combining observational data with prior model predic-tions, known as data assimilation techniques, can be usedto construct accurate estimates of the initial conditionsand parameters in the model and hence improve the abil-ity of the model to predict the true system state. Undercertain statistical assumptions on the errors in the prior es-timates, the solution to the assimilation problem yields theoptimal (maximum a posteriori likelihood) estimate of thetrue state of the system. A key diffculty in the constructionof a data assimilation scheme is specification of the priorerror covariances. These covariances play an importantrole in the filtering and spreading of observational data;however, propagating the background error covariance ma-trix is computationally expensive. Here we combine ideasfrom 3D variational assimilation methods and extendedKalman filter techniques to produce a new hybrid as-similation scheme that provides a ow dependent approx-imation to the state-parameter cross covariances withoutexplicitly propagating the full system covariance matrix.This method has been tested in several linear and non-linear dynamical models, successfully recovering the theinitial states and true parameter values to good accuracy,even with noisy observations. The new technique has alsobeen applied to a simplified model of coastal and estuar-ine sediment transport to deliver estimates of uncertainmorphodynamic parameters with excellent results. Such asystem can be used to improve ood forecasting, with clearsocial and economic bene

ts. This work was funded by a NERC Flood Risk FromExtreme Events (FREE) Case Award with the Environ-ment Agency. References P.J. Smith, S.L. Dance and N.K.Nichols, A hybrid data assimilation scheme for model pa-rameter estimation: application to morphodynamic mod-elling, Computers and Fluids, 46, 2011, 436 - 441. Smith,P.J, Baines, M.J, Dance, S.L, Nichols, N.K, Scott, T.R.,Variational data assimilationj for parameter estimation:application to a simple morphodynamic model. Ocean Dy-namics, 59, 2009, pp 697-708.

Theoretical analysis of Sinc-collocation methods forweakly singular Volterra integral equations of thesecond kind

Tomoaki Okayama, Takayasu Matsuo & Masaaki Sugi-hara (Hitotsubashi University)

We are concerned with Volterra integral equations of thesecond kind of the form

u(t)−∫ t

a

k(t, s)

(t− s)pu(s) ds = g(t), a ≤ t ≤ b, (1)

where g and k are given smooth functions, p is a constantwith 0 < p < 1, and u is a solution to be determined. Theequations have a weakly singular kernel (t − s)−p, calledthe Abel kernel, which diverges at s = t. In addition, dueto the singularity of the kernel, the solution u may have aderivative singularity at the endpoint t = a [1]. Because ofthese two singularities, it is not an easy task to constructa high order numerical scheme.

Sinc-collocation methods, which have been developedby Riley [2] and Mori et al. [3], seem successful for theequation (1) in the sense that the resulting schemes canachieve exponential convergence. Riley [2] has estimatedthe error of his numerical solution uN as

maxa≤t≤b

|u(t)−uN (t)| ≤ C∥A−1N ∥∞

√N logN e−

√πd(1−p)N (2)

where AN denotes a coefficient matrix of the resulting lin-ear equation, and d is a parameter that indicates the regu-larity of the solution u. It has been noted that ∥A−1

N ∥ ≤ 4for N in a “practical range” if the interval b − a is “suffi-ciently small”[2]. Afterward, Mori et al. [3] have improvedthe scheme, and estimated the error of the numerical so-lution as

maxa≤t≤b

|u(t)−uN (t)| ≤ C∥B−1N ∥∞ logN e−πdN/ log(2dN/p)(3)

where BN is also a coefficient matrix. This estimate alsosuggests that the scheme can achieve exponential conver-gence, if ∥B−1

N ∥∞ do not increase rapidly. This has alsobeen confirmed by some numerical experiments.

One of the purposes of this study is to prove exponen-tial convergence of the two schemes in a rigorous sense.This is done by showing

∥A−1N ∥∞ ≤ C1 logN, ∥B−1

N ∥∞ ≤ C2 logN, (4)

for all sufficiently large N , and for any length of the inter-val b − a. The existence of the inverse matrices A−1

N andB−1

N is also proved. Another purpose of this study is toestimate the parameter d, which is indispensable for im-plementing the schemes and affects the convergence rate asestimated above. This is done by applying the techniqueestablished for Fredholm integral equations [4].

References:

[1] Ch. Lubich: Runge–Kutta theory for Volterra andAbel integral equations of the second kind, Math.Comp., 41 (1983), 87–102.

[2] B. V. Riley: The numerical solution of Volterra in-tegral equations with nonsmooth solutions based onsinc approximation, Appl. Numer. Math., 9 (1992),pp. 249–257.

[3] M. Mori, A. Nurmuhammad, and T. Murai: Nu-merical solution of Volterra integral equations withweakly singular kernel based on the DE-sinc method,Japan J. Indust. Appl. Math., 25 2008, 165–183.

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[4] T. Okayama, T. Matsuo, and M. Sugihara: Sinc-collocation methods for weakly singular Fredholmintegral equations of the second kind, J. Comput.Appl. Math., 234 (2010), 1211–1227.

Using data assimilation within a moving mesh ap-proach to ice sheet modelling

Dale Partridge, Mike Baines & Nancy Nichols (Univer-sity of Reading)

Computational studies of glaciers are particularly chal-lenging to the modeller. Although ice sheet models arewell-established, prediction of profiles are generally infeasi-ble analytically and difficult to achieve numerically. Manyfixed grid methods lose large amounts of information dueto the sparse nature of grid points, necessitated by thelarge spatial scales ice sheets can cover. A moving meshapproach to numerical modelling of glaciers has been pro-posed, based on the shallow ice approximation equations,where the mesh is driven by relative mass conservation.This approach is being tested against benchmark experi-ments set up to compare different models.

We apply the techniques of data assimilation, combiningthe model with observations to improve our estimate ofthe state of the system in order to correct the model andovercome some of the deficiancies that arise under movingmesh methods. Due to the nature of the methods, themesh points themselves are variables in the model whichcan be manipulated bu correctly applied assimilation.

Dual Lumping of Isogeometric Finite Element Schemesusing Petrov-Galerkin Techniques

Mark Payne & Dugald Duncan (Heriot-Watt University)

The isogeometric finite element method, introduced byHughes et al. 2005, uses non-uniform rational B-splines(NURBS) as a basis for test and trial functions. Thestrengths of using NURBS is the fact that they have excel-lent approximation properties and can easily be producedvia the Cox-de Boor recursion formula. NURBS however,except for crude row sum mass lumping techniques, can-not easily produce diagonal mass matrices for hyperbolicproblems. In particular, the standard spectral element ap-proach cannot be used due to their non-negative nature.One idea is to use a Petrov-Galerkin approach, producingan orthogonal dual basis which is then used as the testfunctions as discussed in Cottrell et al. 2009. Common ex-amples of dual basis are discontinuous and therefore inte-gration by parts cannot be used. We adapt the technique,using a smooth dual basis that is orthogonal to the NURBSbasis only in respect to an approximate quadrature. Thisallows us to intergrate by parts with the additional ad-vantage to choose the test functions suitably to optimizeconvergence and accuracy. We apply this approach to theacoustic wave equation and compare it with the spectralelement method and the row sum mass lumping method.

DQM Time-DQM Space Solution of Hyperbolic

Telegraph Equation

Bengisen Pekmen & Munevver Tezer-Sezgin(Atilim Universit)

Differential Quadrature Method (DQM) is proposed bothin space and time directions for the numerical solutions ofone- and two-dimensional hyperbolic telegraph equations.The equation in two-space dimension is utt+2αut+β

2u−uxx − uyy = f(x, y, t), 0 ≤ x, y,≤ 1, t ≥ 0 where α ≥ β >0, and it is subjected to appropriate initial and boundaryconditions. The DQM formulation of this equation givesa main system [B] u = knowns of which B is the co-efficient matrix consisting of the known DQM weigthingcoefficients, and the known vector contains f(xi, yj , tl) asentries. Both the initial condition u(x, 0) and Dirichletboundary conditions are inserted to the system directly,reducing the size and the computational cost. The initialcondition ut(x, 0) and Neumann boundary conditions arediscretized by using DQM and added to the main systemresulting an overdetermined system.

For the solution, the least squares method or QR fac-torization is used. The solvability of the DQM result-ing overdetermined system of equations is related withthe column rank of the coefficient matrix. When the ini-tial/or Neumann type boundary conditions are discretizedusing DQM, and added to the system, the row size iscertainly greater than the column size which makes thesystem overdetermined. The small number of grid pointsespecially in time direction provides a full column rankcoefficient matrix. This is one of the main advantage ofDQM method in using for the time direction which allowsus to be able to use large time increment ∆t.

Both polynomial based differential quadrature (PDQ)and Fourier based differential quadrature (FDQ) are usedin space directions whereas PDQ is made use of in timedirection. Numerical solution is obtained by using Gauss-Chebyshev-Lobatto points in space intervals and equallyspaced points for the time interval. DQM in time direc-tion gives the solution directly at a required time level orsteady-state without the need of iteration, and it is stableeven large ∆t is used. The computations are carried forseveral one- and two-dimensional telegraph equations withDirichlet and/or Neumann boundary conditions. DQMalso has the advantage of giving very good accuracy withconsiderably small number of discretization points.

Finite elements on pyramids

Joel Phillips (University College London)

The differing strengths of tetrahedral and hexahedral fi-nite elements mean that it is sometimes desirable to com-bine both within the same method. A mesh that con-tains both tetrahedral and hexahedra will, in general, needquadrilateral-based pyramids to act as “glue” between them;so, if we are to build conforming finite element methodsusing such a mesh, we need to be able to construct pyra-midal finite elements.

In this talk, I will give a brief history of attempts toconstruct pyramidal finite elements and present our con-struction of a family of pyramidal analogues to Nedelec’s

38

elements. These are arbitrarily high order elements foreach of the spaces of the de Rham complex which are com-patible with tetrahedral and hexahedral elements throughtheir triangular and quadrilateral faces. They satisfy thecommuting diagram property necessary to build stable ap-proximations to mixed problems.

I will also prove the surprising result that the approxi-mation space for a pyramidal element cannot be composedonly of polynomials. We will see that this means that thestandard analysis of the effects of numerical integrationon finite elements is inapplicable to pyramidal elements.I will give some insight into how this problem is resolvedand then present some numerical examples.

The augmented Lagrangian method with boundson the dual variables

M.J.D. Powell (University of Cambridge)

Let some or all of the constraints of an optimization cal-culation be treated by an augmented Lagrangian method,the Lagrangian being minimized by the primal variablesfor each choice of the dual variables (Lagrange multiplierestimates). Then usually the dual variables are optimalwhen the resultant least value of the Lagrangian takes itsmaximum value, but, if the given constraints are incon-sistent, then this maximum value and some correspondingvalues of the dual variables may become infinite. Failuresof this kind can be avoided by imposing prescribed boundson the dual variables. Let these bounds be active at theend of the calculation. Then we find, in several situations,that the final values of the primal variables solve a prob-lem where some of the original constraints are replacedby L1 penalty terms, as in work of P.E. Gill and D.P.Robinson. An interesting application of this technique isdescribed, developed by D.W. Wood, A.A. Groenwold andL.F.P. Etman, for Sequential Approximate OptimizationAlgorithms that have huge numbers of variables.

Algorithm and error bounds for piecewise linearapproximation on anisotropic triangulations

A.F. Rabarison & O. Davydov (University of Strath-clyde)

Given a square domain Ω ⊂ R2 and a convex function f ∈C2(Ω), we propose an algorithm for generating a sequenceof anisotropic triangulations TN : #(TN ) ≤ N,N ≥ N0of Ω. For any p ∈ [1,∞), the resulting piecewise linearinterpolant fN to f over the triangulation TN satisfies theasymptotic estimations

lim supN→∞

N∥f − fN∥Lp(Ω) ≤(∫

Ω

Kf,p(z)qdz

)1/q

, (10)

lim supN→∞

N12 |f − fN |W1

p (Ω) .(∫Ω

Kf,p(z)qDz

) 12(∫

Ω

| detHf (z)|q4 ∥Hf (z)∥

p22 Dz

) 1p

(11)

where 1/q = 1+1/p, Hf (z) denotes the Hessian matrix of

f at z ∈ Ω, whereas

Kf,p(z) := inf|T |=1

∥πz − ITπz∥Lp(T )

πz(x, y) =1

2

∂2f(z)

∂x2x2 +

∂2f(z)

∂x∂yxy +

1

2

∂2f(z)

∂y2y2,

and ITπ is the linear interpolation polynomial of π over thetriangle T . It is known [2] that the estimation (10) cannotbe improved on any sequence of triangulations TN suchthat supT∈TN

diam(T ) . N−1/2. Previous constructionsdo not satisfy (11), with |f−fN |W1

p (Ω) possibly unboundedas N → ∞.

Hirota’s bilinear method and the ε–algorithm

M Redivo-Zaglia & C. Brezinski, (University of Padova)

Hirota’s bilinear method [2] can be quite useful in the solu-tion of nonlinear differential and difference equations. Inthis talk, we show how this method can lead to a proofthat the ε–algorithm of Wynn [5] implements the Shanks’sequence transformation [3, 4] and, reciprocally, that thequantities it computes are expressed as ratios of Hankeldeterminants as given by Shanks. New identities betweenthe quantities involved in Hirota’s method are obtained.Then, the same bunch of results is showed to hold also forthe confluent form of the ε–algorithm. Its relation to theLotka–Volterra equation allows to express the solution ofthis equation under a closed form. The same techniquecan be also used for obtaining a multistep extension of theε–algorithm [1].

References

[1] C. Brezinski, Y. He, X.-B. Hu, M. Redivo-Zaglia,J.-Q. Sun, Multistep ε-algorithm, Shanks’ transformation,and Lotka-Volterra system by Hirota’s method, Math. Com-put. in press.

[2] R. Hirota, The Direct Method in Soliton Theory,Cambridge University Press, Cambridge, 1992.

[3] D. Shanks, An analogy between transient and math-ematical sequences and some nonlinear sequence–to–sequencetransforms suggested by it, Part I, Memorandum 9994,Naval Ordnance Laboratory, White Oak, July 1949.

[4] D. Shanks, Non linear transformations of diver-gent and slowly convergent sequences, J. Math. Phys.,34 (1955) 1–42.

[5] P. Wynn, On a device for computing the em(Sn)transformation, MTAC, 10 (1956) 91–96.

Towards Exascale Computing: Multilevel Methodsand Flow Solvers for Millions of Cores

Ulrich Ruede (University of Erlangen-Nurnberg)

This presentation will report on experiences implementingPDE solvers on Peta-Scale computers, such as the 290000core IBM Blue Gene system in the Julich Supercomput-ing Center. The talk will have two parts, the first onepresenting the Hierarchical Hybrid Grid (HHG) method.HHG is a prototype parallel multigrid algorithm that has

39

been used to solve FE systems with up to a trillion (1012)degrees of freedom arising form a tetrahedral FE mesh.Excellent scalability and a good absolute efficiency areachieved by a using a matrix-free data structure and acarefully optimized implementation. The second part ofthe talk will present a brief overview of our work on simu-lating complex flow phenomena. This will include work ona parallel fluid-structure-interaction technique with manymoving rigid objects that are embedded in the flow. Fullyresolved geometric representations of each particle are usedin a massively parallel simulation of particle-ladden flows.The talk will conclude with some remarks on the chal-lenges that the developers of numerical algorithms will befacing on the path to exascale in the coming years.

Numerical Solution of Fully Nonlinear Elliptic PDEswith a Modified Argyris Element

Abid Saeed & Oleg Davydov (University of Strathclyde)

We discuss an implementation of Bohmer’s finite elementmethod for fully nonlinear elliptic partial differential equa-tions. In this method the nonlinear problem is solved it-eratively by a Newton scheme yielding a sequence of lin-ear elliptic equations for good initial guess. Discretisa-tion is done by modified Argyris finite elements that ad-mit a stable splitting. Implementation is done using theBernstein-Bezier techniques for modified Argyris elements.We consider the Dirichlet problems for several fully nonlin-ear equations including Monge-Ampere equation over con-vex polygonal domains. The numerical results for severaltest problems illustrate the expected rate of convergenceof the scheme.

Energy Minimizing Coarse Spaces with FunctionalConstraints

Robert Scheichl, Panayot Vassilevski & Ludmil Zikatanov(University of Bath)

We will report on the construction of energy minimiz-ing coarse spaces for the robust approximation of PDEsolutions in the case of strongly varying and high con-trast coefficients. The spaces are built by patching to-gether solutions to appropriate local saddle point prob-lems or eigenproblems. They can be used either in two-level Schwarz methods or directly to approximate the PDEsolution on meshes that do not resolve the coefficient vari-ation. We first set an abstract framework for such con-structions, and then we give two examples of construct-ing the coarse space and a stable interpolation operatorfor the two level Schwarz method. The stability and ap-proximation bounds of the constructed interpolant are in aweighted norm and are independent of the variations in thecoefficients. Part of this work was done jointly with Vic-torita Dolean (University of Nice Sophia–Antipolis), Fred-eric Nataf and Nicole Spillane (both University of ParisVI).

Higher order variational time discretizations fornonlinear systems of ordinary differential equations

Friedhelm Schieweck & Gunar Matthies (University ofMagdeburg)

We discuss different time discretizations of variational typeapplied to a nonlinear system of ordinary differential equa-tions which is typically generated by a semi-discretizationin space of a given nonlinear parabolic partial differen-tial equation like, for instance, the non-stationary Burgersequation.

Among these methods we compare the known con-tinuous Galerkin-Petrov and the discontinuous Galerkinmethod with time polynomial ansatz functions of orderk (cGP(k)- and dG(k)-method) with respect to accuracy,stability and computational costs. Moreover, we proposetwo new extended methods (cGP-C1(k+1)- and dG-C0(k+1)-method) which have on the one hand a one higher degreeof ansatz functions and accuracy, the same stability prop-erties and on the other hand the same computational costsas the original methods. We prove A-stability for the cGP-methods and strong A-stability for the dG-methods.

We present optimal error estimates and the close rela-tionship between the original and extended methods whichprove as a byproduct the super-convergence of the originalmethods cGP(2) and dG(1) in the endpoints of all discretetime intervals. Finally, we present first numerical resultsfor the non-stationary Burgers equation in one space di-mension.

A Lax-Wendroff type numerical method for a frac-tional advection diffusion equation

Ercılia Sousa (Coimbra University)

Recently, many transport problems, involving diffusion,have been formulated on fractional differential equations.Fractional derivatives are non-local opposed to the localbehaviour of the integer derivatives and they are used tomodel a phenomenon called anomalous diffusion, whichcan be subdiffusion or supersiffusion. In this work, we areinterested in the superdiffusion.

A one dimensional fractional advection diffusion modelis considered, where the usual second-order derivative givesplace to a fractional derivative of order α, with 1 < α ≤2. The fractional derivative is defined by the Riemann-Liouville operator, which is an integral operator. We pro-pose an explicit difference method which is second orderaccurate. Consistency and stability of the method are ex-amined and numerical tests are presented.

A new splitting method for mean curvature-basedvariational image denoising

Li Sun & Ke Chen (University of Liverpool)

Denoising perhaps is one of the most fundamental tasksin image processing. It has been deeply investigated formany years and many excellent results have been obtainedby using Total Variation (TV) based L1-model. While themodel can preserve important features such as sharp edgesand contours, for smooth images however, the TV model

40

produces undesirable staircasing effect. In order to rem-edy this drawback, many researchers have turned to higherorder models. The particular model using mean curvatureas a regulariser, shown to be effective for general denoisingby Zhu and Chan 2006 (W. Zhu and T. F. Chan, “ImageDenoising Using Mean Curvature”, preprint,http://www.math.nyu.edu/ wzhu/, 2006); Brito and Chen2010 (Carlos Brito and Ke Chen, “Multigrid Algorithmfor High Order Denoising”, SIAM Journal on Imaging Sci-ences, Vol 3 (3), pp.363-389, 2010), is studied in this workand we have proposed an efficient and robust iterativemethod for its numerical solution.Due to strong nonlinearity, the resulting Euler-Lagrangepartial differential equation is of fourth order and conver-gence is extremely slow with gradient descents [Zhu andChan 2006]. The stabilisation method adopted by CarlosBrito and Ke Chen 2010 was a major improvement butit has to severely regularise the nonlinearity. After firstapplying the 2-level splitting method of Tai 2011 (Xue-Cheng Tai, J. Hahn, and G. J. Chung, “A Fast Algorithmfor Euler’s Elastica Model Using Augmented LagrangianMethod”, SIAM Journal on Imaging Sciences, Vol 4, pp.313-344 , 2011), we found that the restored quality is notas good as the original curvature model although the re-sulting numerical scheme is indeed fast. To reduce the levelof approximations, we consider one-level splitting methodin this work. We derived and implemented the preciseboundary conditions (BCs) using staggered grids and lo-cally coupled iterative solvers instead of using the usualNeumann BCs which are found not suitable. It turns outthat both the Newton method and a Fixed point methodare convergent, and they are suitable for a nonlinear multi-grid method. Numerical results can show that our modelcan deliver better quality of restoration than the previousfast methods, by using less regularisation of the nonlinear-ity than Brito and Chen 2010.

iGen: a program for the automated generation ofreduced-dimension models

Dan F. Tang (Manchester University)

In this presentation I will describe a novel approach tothe generation of reduced dimension models from high-dimensional models. The approach involves expressing thehigh-dimensional model as a computer program written inC++. The source code of this program is then fed intoa newly developed computer program called iGen. iGenanalyses the structure of the source code, represents sec-tions of code as multivariate polynomials with constanterror bounds and approximates these polynomials withlower-order polynomials. From the analysis, iGen derivesa reduced-dimension model that closely approximates theoriginal, while reporting bounds on the error introduced byany approximations. I will describe the application of iGento a number of increasingly complex physical systems andshow that iGen has the ability to generate reduced modelsthat execute typically orders of magnitude faster than theunderlying, high-dimensional models from which they arederived.

Multilevel Monte Carlo for highly heterogeneousmedia

Aretha Teckentrup, J. Charrier, K.A. Cliffe, M.B. Gilesand R. Scheichl (University of Bath)

The quantification of uncertainty in groundwater flow playsa central role in the safety assessment of radioactive wastedisposal and of CO2 capture and storage underground.Stochastic modelling of data uncertainties in the rock per-meabilities lead to elliptic PDEs with random coefficients.Because of the typically large variances and short corre-lation lengths in groundwater flow applications, methodsbased on truncated Karhunen-Loeve expansions are onlyof limited use and Monte Carlo type methods are stillmost commonly used in practice. To overcome the noto-riously slow convergence of conventional Monte Carlo, weformulate and implement a novel variance reduction tech-nique based on hierarchies of spatial grids/models. Thismultilevel Monte Carlo method was first introduced forhigh-dimensional quadrature by Heinrich (2001) and forstochastic ODEs in mathematical finance by Giles (2007).We will demonstrate theoretically and practically on a typ-ical model problem the significant gains with respect toconventional Monte Carlo that are possible with this newapproach, leading (in the best case) to an asymptotic com-putational cost that is proportional to the cost of solvingone deterministic PDE to the same accuracy.

MHD flow and heat transfer between parallel plateswith Navier-slip wall condition

Munevver Tezer-Sezgin & Onder Turk (Middle EastTechnical University )

The magnetohydrodynamic (MHD) flow of a dusty fluidwith heat transfer between parallel plates, in which thefluid has temperature dependent thermal conductivity andviscosity, is solved by using Chebyshev spectral collocationmethod. An external uniform magnetic field is appliedperpendicular to the plates, and the fluid is driven by aconstant pressure gradient G. The plates are electricallyinsulated and kept at constant temperatures. The dustparticles are assumed to be spherical in shape and uni-formly distributed throughout the fluid. The governingequations are given in terms of fluid velocity and temper-ature u, T , and particles velocity and temperature up, Tp,respectively, as

Re∂u

∂t= Re G+

∂

∂y

[µ(T )

∂u

∂y

]−Ha2u−R(u− up),

∂up

∂t=

1

Re g(u− up),

Re∂T

∂t=

1

Pr

∂

∂y

(κ(T )

∂T

∂y

)+ Ec µ(T )

(∂u

∂y

)2

+Ec Ha2 u2 + 2R3Pr

(Tp − T ),∂Tp

∂t= −L(Tp − T ),

where 0 ≤ y ≤ 1, t > 0, and µ(T ) = e−aT , κ(T ) = ebT .

Navier-slip condition is imposed for both the fluid and par-

41

ticle velocities on the plates which constitutes the propor-tion of velocities to the tangential viscous stresses with aparameter. The Chebyshev spectral method allows one tobe able to use considerably small number of Chebyshev-Gauss-Lobatto space points clustered through ends. Im-plicit backward difference is used for the temperature deriva-tives which is unconditionally stable and uses quite largetime steps. This makes the whole solution procedure com-putationally cheap and efficient. The effect of the wallslip parameter, viscosity µ(T ) and thermal conductivityκ(T ) variations, and the uniform magnetic field for boththe fluid and the dust particles is studied. It is found thatthe velocities of both fluid and dust particles decrease asHartmann number Ha is increased, when Reynolds num-ber Re, Prandtl number Pr, Eckert number Ec, particleconcentration parameter R are kept fixed. An increasein the Navier-slip parameter causes increases in the mag-nitude of velocities of dust particles and the fluid. Theeffect of viscosity parameter variation is more stressed onthe velocity profiles than the temperature. Variations (es-pecially increase) in the thermal conductivity parameterresult in temperature variations in non-linear manner, butthe velocities are not much affected.

Multiscale analysis in Sobolev Spaces on boundeddomains with restricted interpolation points

Alex Townsend (University of Oxford)

A multiscale scheme is studied for the approximation ofSobolev functions on bounded domains with restricted datasites. Our method employs compactly supported radial ba-sis functions with centers at scattered data sites restrictedat each level to ensure the support of the interpolant is con-tained within the domain. The multiscale approximationis constructed by successive residual corrections with eachlevel using a support radii appropriate to capture dfferentscales. Important steps in the proof of the convergencetheorem for the scheme will be shown with an improvedrate if the target function and its derivatives vanishes onthe boundary. We will also present some numerical ex-amples of the scheme which reinforce the theory. Thisanalysis is hoped to be a first step towards understandingdomain boundary effects for radial basis function inter-polation and to improve the numerical solution of partialdifferential equations on bounded domains.

Chebyshev spectral collocation method for unsteadyMHD flow and heat transfer between parallel plates

Onder Turk (Middle East Technical University)

The magnetohydrodynamic (MHD) flow with heat transferof a Newtonian, incompressible fluid which has tempera-ture dependent viscosity is solved between parallel plates.An external uniform magnetic field is applied perpendic-ular to the plates, and the fluid is driven by a constantpressure gradient G. The movement of the upper platewith a constant velocity Ru, and the convection actionin terms of inflow/outflow through plates are also consid-ered. The plates are electrically insulated. The governing

equations are written in non-dimensional form in terms ofvelocity u, temperature T

∂u

∂t+Rv

∂u

∂y= G+

∂

∂y

[µ(T )

∂u

∂y

]−Ha2u,

∂T

∂t+Rv

∂T

∂y=

1

Pr

∂2T

∂y2+ Ec µ(T )

(∂u

∂y

)2

+ Ec Ha2 u2,

(12)where −1 ≤ y ≤ 1, t > 0 and µ(T ) = e−aT is the variableviscosity.

These coupled nonlinear partial differential equationstogether with convenient boundary conditions are solvednumerically. The Chebyshev spectral collocation methodis used for space derivatives whereas implicit backward dif-ference is made use of for temporal derivatives which is un-conditionally stable and does not need too small time step.The Chebyshev spectral method allows one to be able touse considerably small number of clustered Chebyshev-Gauss-Lobatto points in space direction. Also, it is easyand practical to obtain higher order derivative approxi-mations of the solution by using only multiplications ofChebyshev differentiation matrices. This makes the wholeprocedure computationally cheap and efficient. Numericalresults are visualized in terms of velocity and temperatureof the fluid for several values of Hartmann number (Ha)and viscosity parameter (a) for depicting the influences onthe flow and temperature. Results for the velocity andtemperature field are obtained for several values of Hart-mann number, viscosity parameter, upper plate velocity(Ru), inflow/outflow parameters (Rv) when Prandtl num-ber Pr = 1, Eckert number Ec = 0.2 and G = 5 aretaken. The effects of variable (temperature dependent)viscosity and applied magnetic field are deeply examined.The fluid starts with the interaction of pressure gradientand the externally applied magnetic field. An increase inviscosity parameter a causes increase in the magnitudesof both velocity and temperature. But as Ha increases,flow is decreased sharply, temperature also drops but set-tles down for Ha ≥ 10. As Ha is increased boundarylayers are formed near the insulating plates which is theexpected behavior of MHD flow. As time progresses bothvelocity and temperature increase and reach steady-statebefore t = 5. Drop in the magnitudes of velocity andtemperature is also observed in the presence of convectionterms (inflow/outflow through plates). With the move-ment of the upper plate the temperature is not affectedas much as the velocity. The use of Chebyshev spectralmethod when it is combined with the unconditionally sta-ble backward difference time integration, enables one toobtain very good accuracy with considerably small num-ber of collocation points, and quite large time steps. Thismakes the solution procedure computationally efficient.

Distributing Points in a Triangle using OptimalTransport Techniques

Jan Van lent & Chris Budd (University of the West ofEngland)

Generating point distributions and meshes for a given do-

42

main is an important problem in numerical analysis. Theone-dimensional problem of generating points in an inter-val that are suitable for interpolation and quadrature isfairly well understood. Chebyshev points, for example,perform very well. However, already for a simple two-dimensional domain such as a triangle, generating goodpoint distributions is still an active research topic. Inthis talk I will discuss techniques for generating pointsand meshes based on numerical methods for the Monge-Kantorovich optimal transport problem. I will illustratehow these techniques can be used for triangular domains.

Numerical analysis of two-phase granular flows atlow Mach numbers

C. Varsakelis&M. V. Papalexandris (Universite Catholiquede Louvain)

In this talk we present an algorithm for the numericaltreatment of a continuum two-velocity, two-pressure modelfor two-phase granular flows. The proposed method be-longs to the class of fractional-step methods and employs ageneralized projection scheme for the momentum equationof each phase. It is further able to deal with strong spatialgradients and moving interfaces; both common character-istics of the flows of interest. The talk is concluded withthe presentation of numerical simulations which illustratethe accuracy and robustness of the proposed algorithm.

A Posteriori Error Analysis for a QuasicontinuumMethod

Hao Wang & Christoph Ortner (University of Oxford)

The Quasicontinuum (QC) Method is a method for mod-eling an atomistic system by coupling an atomistic modeland a continuum approximation together. The atomisticmodel is used in the region where large deformation suchas fracture happens so that the full atomistic detail is re-tained while a continuum model is used in the rest of thematerial so that the computational work is significantlyreduced. In this work, we have given the a posteriorierror analysis for Consistent Atomistic/Continuum Cou-pling QC method, which was developed recently and is anconsistent and efficient coupling method in one and twodimensional atomistic system with two body interactionpotentials. This method is also promising in the exten-sion to three dimensional systems and general interactionpotentials. There are only a few researches on the a poste-riori error analysis of QC methods. In our work, we use thenegative-norm residual analysis to derive an efficient esti-mator of the global error of the deformation in a Sobolevnorm. We also give the estimator of the error of the en-ergy, i.e, the difference between the energy under the com-puted deformation and that under the real deformation.An adaptive mesh refinement strategy is developed basedon these error estimators and is illustrated in some nu-merical experiments. A comparison is also made betweenthe efficiency of this refinement strategy and that of ana priori quasi-optimal mesh generation strategy developedin another literature.

Novel numerical methods for nonlinear time-spacefractional diffusion equations in two dimensions

Qianqian Yang, TimMoroney, Kevin Burrage, Ian Turner& Fawang Liu (Queensland University of Technology)

In this talk, nonlinear time-space fractional diffusion equa-tions in two dimensions (ntsfde-2d) are considered. Thentsfde-2d is obtained from the standard diffusion equa-tion by replacing the first-order time derivative with theCaputo fractional derivative, and the second order spacederivatives with the fractional Laplacian. Using the ma-trix transfer technique proposed by Ilic, Liu, Turner, andAnh (Fractional Calculus and Applied Analysis, 9, 333-349, 2006), the ntsfde-2d is transformed into a time frac-tional differential system, where A is the approximate ma-trix representation of the standard Laplacian. Traditionalapproximation of the fractional Laplacian requires diag-onalisation of A, which is very time-consuming for largematrices. The novelty of our proposed numerical schemeis that, using the finite difference method to approximatethe Caputo time fractional derivative, the solution of thentsfde-2d is written in terms of a matrix function vec-tor product at each time step. We use the finite volumemethod over unstructured meshes to generate the matrixA, and the Lanczos method to approximate the matrixfunction vector product. Numerical results are presentedto verify the accuracy and efficiency of the proposed nu-merical solution strategies.

C1 Piecewise Quadratic Hierarchical Bases

Wee Ping Yeo & Oleg Davydov (University of Strath-clyde)

On general triangulations of arbitrary polygonal domainsΩ ⊂ R2, we construct C1 continuous piecewise quadratichierarchical bases which generate Riesz bases for the Sobolevspaces Hs(Ω), with s ∈ (1, 5

2). The nested approxima-

tion spaces used in our construction are the spaces of C1

piecewise quadratic polynomials on combination of Powell-Sabin-6 and Powell-Sabin-12 triangulations, the same splinespaces used by Jia and Liu (Adv. Comput. Math.., 77(2007), 287–312) in their construction of wavelet baseswhich generate Riesz bases for Hs(Ω), with s ∈ (1.618, 5

2).

The idea of such bases is to construct them recursivelyby adding to the basis from the previous approximationspace a set of locally supported functions spanning a com-plement space such that the union is a basis of the currentapproximation space. In particular, we will use hierarchi-cal bases meaning that the functions that are added arejust a subset of a basis for the space on the current level.The starting point for our construction is the Lagrangebasis constructed by Nurnberger and Zeilfelder (Proceed-ings Multivariate Approximation. Birhauser, Basel, 2001)for C1 piecewise quadratic polynomials on Powell-Sabin-6 triangulations of arbitrary polygonal domains. Withslight modification to the construction of NZ01 on theinitial level and another algorithm developed for succes-sive refinement levels we construct nested Lagrange inter-polation sets, which is the key step in the constructionof hierarchical bases. Compared to the Riesz bases con-

43

structed by Jia and Liu, our Riesz bases have a biggerrange of stability, and to the earlier hierarchical Riesz basesfor Hs(Ω), s ∈ (1, 5

2) of also Lagrange type constructed

by Davydov and Stevenson on checkerboard quadrangu-lations, our construction is an advantage over (O. Davy-dov and R. Stevenson, Hierarchical Riesz bases for Hs(Ω),1 < s < 5

2, Constr. Approx., 22 (2005), 365–394), as it ap-

plies to general triangulations of arbitrary polygonal do-mains. Since the range of stability of our Riesz bases forHs(Ω) also includes s = 2 like the other two construc-tions, we plan to explore the use of the corresponding hi-erarchical basis in preconditioning the fourth order ellipticproblems such that the stiffness matrices will be uniformlywell-conditioned.

The numerical linear algebra of approximation in-volving Radial Basis Functions

Shengxin Zhu & Andy Wathen (University of Oxford)

Radial Basis Functions have attracted a lot of attentionsin recent years as an elegant approximation scheme formultidimensional data and an attractive way to solve par-tial differential equations. These methods share many ad-vantages, being meshfree/meshless in any dimension, andhaving good approximation properties, however they leadto significant problems of linear algebra—the related lin-ear systems of equations are always highly ill-conditioned.Thus constructing effective fast solvers plays an impor-tant role in practical use of these methods and will shapetheir future utility for large scale problems. We are go-ing talk about issues related to coping with the severelyill-conditoned systems as well as bringing some open prob-lems.

44

Tuesday 28th June

Chair: K3.25: Ainsworth

9:00-9:05 Opening Remarks

9:05-10:05 R SchabackKernel-Based Meshless Methods for Solving PDEs

10:05-11:05 R SkeelThe best of fast N-body methods

11:05–11:30 COFFEE/TEA

Chair: K3.25: M1 K3.14: M2 K3.17: M4

11:30-11:55 S Repin M1 L N Trefethen M2 P Burrage M4A posteriori estimates for prob-lems with linear growth energyfunctionals

Convergence theorems for poly-nomial interpolation in Chebyshevpoints

Using a modification of the SSAto simulate spatial aspects ofchemical kinetics

11:55-12:20 S Micheletti M1 M Richardson M2 A Lang M4Anisotropic mesh adaptation viaa posteriori error estimation

Approximating functions withendpoint Singularities

Lax’s equivalence theorem forstochastic differential equations

12:20-12:45 A Allendes M1 R Pachon M2 K Burrage M4A posteriori error estimation andan adaptive scheme for the choiceof the stabilization parameter inlow-order finite element approxi-mations of the Stokes problem

Getting rid of spurious poles: apractical approach

From cells to tissue: coping withheterogeneity via fractional mod-els

12:45-14:00 LUNCH-Lord Todd

Chair: K3.25: Knight

14:00-15:00 L N TrefethenSix myths of polynomial interpolation and quadrature

Chair: K3.25: M1 K3.14: M3 K3.17: M4

15:00-15:25 M Maischak M1 A Birkisson M3 M Riedler M4

Error estimators for a partiallyclamped plate problem with bem

Automatic differentiation for au-tomatic solution of nonlinearBVPs

Numerical Simulation of Stochas-tic Reaction Networks mod-elled by Piecewise DeterministicMarkov Processes

15:25-15:50 O Lakkis M1 N Hale M3 A Barth M4

A finite element method for non-linear elliptic equations

“Rectangular pseudospectral dif-ferentiation matrices” or, “Whyit’s not hip to be square”

Multilevel Monte Carlo Finite El-ement Methods for Elliptic PDEswith Stochastic Coefficients

15:50-16:15 N Kopteva M1 P Gonnet M3 L Szpruch M4

A posteriori error estimates forclassical and singularly perturbedparabolic equations

Finite difference preconditionersfor spectral methods

Efficient Multilevel Monte Carlosimulations of non-linear financialSDEs without a need of simulat-ing Levy areas

16:15-16:45 COFFEE/TEA

Chair: K3.25: Watson

16:45-17:55 A Quarteroni (A.R. Mitchell Lecture)Mathematical models for the cardiovascular system: analysis, numerical simulation, applications

18:15-19:15 DINNER-Lord Todd

20:00-22:00 RECEPTION-Glasgow City Chambers

M1 A-posteriori error estimation, M2 Approximation with polynomials, M3 Novel algorithms for BVPs, M4 Stochastic computation

45

Tuesday 28th June


Chair: K4.12: M5 K3.27: Abelman

11:30-11:55 S Loisel M5 E SousaSharp performance estimates for optimizeddomain decomposition preconditioners

A Lax-Wendroff type numerical method fora fractional advection diffusion equation

11:55-12:20 K Chen M5 R Brown

Fast Multigrid Algorithms and High OrderVariational Image Registration Model

Using reduced-order models and waveformrelaxation to solve cerebral bloodflow au-toregulation DEs on a large binary tree net-work

12:20-12:45 D Loghin M5 C Brezinski

Interface preconditioners for flow problemsNonlinear functional equations satisfied byorthogonal polynomials


Chair: K4.12: M5 K3.27: Davis

15:00-15:25 D Silvester M5 M Redivo-ZagliaFast Iterative Solvers for Saddle Point Prob-lems

Hirota’s bilinear method and the ε–algorithm

15:25-15:50 A Ramage M5 F Schieweck

Saddle-point Problems in Liquid CrystalModelling

Higher order variational time discretizationsfor nonlinear systems of ordinary differentialequations

15:50-16:15 J Pearson M5 S GuttelPreconditioned Iterative Methods forConvection-Diffusion Control Problems

A parallel integrator for linear initial-valueproblems


M5 Iterative linear solvers for PDEs

46

Wednesday 29th JuneChair: K3.25: Higham

9:00-10:00 B WohlmuthFinite element approximations for coupled problems

10:00-11:00 J NocedalOptimization Methods for Machine Learning


Chair: K3.25: M1 K3.14: M4 K3.17: M6

11:30-11:55 V M Vohralik M1 M Roj M4 S Kirkland M6A unified framework for a posteri-ori error estimation for the Stokesproblem

Mean Exit Times and Multi-LevelMonte Carlo Simulations

Bipartite Subgraphs and the Sign-less Laplacian Matrix

11:55-12:20 R Rankin M1 R Kruse M4 M Musolesi M6

A posteriori error estimation forfinite element approximations oflinear elasticity problems

Optimal Error Estimates ofGalerkin Finite Element Methodsfor SPDEs with MultiplicativeNoise

Temporal Graphs and Robustness

12:20-12:45 E Verhille M1 E Buckwar M4 M McDonald M6

A posteriori error estimator forthe Reissner-Mindlin problem

Linear Stability Analysis of Nu-merical Methods for StochasticDifferential Equations

Networks in Neuroscience: Usingthe Generalised Singular ValueDecomposition to Compare Pairsof Networks


Chair: K3.25: Mackenzie

14:00-15:00 C JohnsonImage-Based Biomedical Modeling, Simulation and Visualization

Chair: K3.25: M1 K3.14: M4 K3.17: M6

15:00-15:25 E Georgoulis M1 W Liu M4 C Lee M6

A posteriori error estimates forlinear wave problems

Euler–Maruyama-type NumericalMethods for Stochastic Lotka-Volterra Model

Reordering multiple networks

15:25-15:50 S Nicaise M1 G dos Reis M4 J Crofts M6A residual-based a posteriori es-timator for the A − φ magneto-dynamic harmonic formulation ofthe Maxwell system

Numerics for Quadratic FBSDE

Googling the brain: Discover-ing hierarchical and asymmetricstructure, with applications toneuroscience

15:50-16:15 L Asner P Taylor M4 D Higham M6A posteriori error estimation forcoupled problems

Adaptive stepsize control for thestrong numerical solution of SDEs

Bistability Through Triad Closure


Chair: K3.25: Brunner K3.14: Langdon K3.17: M6

16:45-17:10 P J Davies A Teckentrup A Mantzaris M6Convolution quadrature revisitedfor Volterra integral equations

Multilevel Monte Carlo for highlyheterogeneous media

Dynamic Communicators

17:10-17:35 T Okayama T Moroney C Phokaew M6Theoretical analysis of Sinc-collocation methods for weaklysingular Volterra integral equa-tions of the second kind

An efficient finite volume methodapplied to flow in heterogeneousporous media

A graph model to explain the ori-gin of early proteins

17:35-18:00 R Fernandes D Duncan Z Stoyanov M6Comparing two iterated projec-tion approximations for integraloperators in terms of convergenceand computational cost

Postprocessing Reservoir Simula-tions to Obtain Accurate WellPressures

Communicability of networks


M1 A-posteriori error estimation, M4 Stochastic computation, M6 Complex Networks and Matrix Computations

47

Wednesday 29th June


Chair: K4.12: Garcia-Archilla K3.27: Hall

11:30-11:55 JP Berrut M J D PowellA formula for the influence of jumps on finitesinc interpolants

The augmented Lagrangian method withbounds on the dual variables

11:55-12:20 G Klein R FletcherLinear rational finite differences and applica-tions

Limited memory spectral gradient methods

12:20-12:45 A F Rabarison N GouldAlgorithm and error bounds for piecewise lin-ear approximation on anisotropic triangula-tions

Puiseux-based extrapolation for large-scaledegenerate quadratic programming


Chair: K4.12: Stynes K3.27: Fletcher

15:00-15:25 A Townsend Q HuangfuMultiscale analysis in Sobolev Spaces onbounded domains with restricted interpolationpoints

A high performance dual simplex solver

15:25-15:50 O Davydov J HallRBF-FD Methods for Elliptic Equations Matrix-free IPM with GPU acceleration

15:50-16:15 G Andriamaro J Hogg

Bernstein-Bezier Finite ElementsExperience of linear solvers in an nonlinearinterior point method


Chair: K4.12: Scheichl K3.27: Ergul

16:45-17:10 O Turk N K NicholsChebyshev spectral collocation method forunsteady MHD flow and heat transfer be-tween parallel plates

Optimal state and parameter estimation us-ing a hybrid data assimilation scheme

17:10-17:35 C Varsakelis S Jenkins

Numerical analysis of two-phase granularflows at low Mach numbers

The Effects of Dispersive and DissipativeProperties of Numerical Methods on the Lin-ear Advection Equation during Data Assimi-lation

17:35-18:00 S Abelman D PartridgeReduction and solutions for unsteady flow ofa Sisko fluid for cylindrical geometry

Using data assimilation within a movingmesh approach to ice sheet modelling


48

Thursday 30th JuneChair: K3.25: Barrenechea

9:00-10:00 O BrunoSpectral frequency- and time-domain PDE solvers for general domains

10:00-11:00 R TemponeError estimates and adaptive algorithms for single and multievel Monte Carlo Simulation of Stochastic Differential Equations


Chair: K3.25: M7 K3.14: M8 K3.17: M9

11:30-11:55 B Garcia-Archilla M7 R Borsdorf M8 M Ainsworth M9Stabilization of convection-diffusion problems by Shishkinmesh simulation

Two Two-Sided Procrustes Prob-lems Arising in Atomic Chemistry

Optimally blended finite element-spectral element schemes forwave propagation

11:55-12:20 J L Gracia M7 E Deadman M8 T Betcke M9Time and space-accurate numer-ical solution of one-dimensionalparabolic singularly perturbedproblems of reaction-diffusiontype

Blocked Methods for Computingthe Square Root of a Matrix

Recent Developments in nonpoly-nomial finite element methods forwave problems

12:20-12:45 T Linß M7 P Knight M8 O Ergul M9Collocation methods for singu-larly perturbed reaction-diffusionequations

A Linear Algebraist’s Approach toReforming The UK Voting Sys-tem

Extremely Large Electromagnet-ics Problems: Are We Going inthe Right Direction ?


Chair: K3.25: Ramage

14:00-15:00 J DongarraOn the Future of High Performance Computing: How to Think for Peta and Exascale Computing

Chair: K3.25: M7 K3.14: M8 K3.17: M9

15:00-15:25 L Ludwig M7 A Spence M8 S Langdon M9Supercloseness of the SDFEMfor convection-diffusion problemswith reduced regularity

A method for the computationof Jordan blocks in parameter-dependent matrices

A high frequency boundary ele-ment method for scattering bynon-convex obstacles

15:25-15:50 J Quinn M7 A Wathen M8 L Gao

A Linear Singularly PerturbedProblem with an Interior TurningPoint

Preconditioning for GMRES

Energy Conservation Laws ofMaxwell’s Equations and TheirApplication in Numerical Analysisof FDTD

15:50-16:15 M Stynes M7 F Tisseur A Bespalov

A balanced finite element methodfor singularly perturbed reaction-diffusion problems

Structure Preserving Transforma-tions for Quadratic Matrix Poly-nomials

Differential complexes and inter-polation operators in the contextof high-order numerical methodsfor electromagnetic problems


Chair: K3.25: Linß K3.14: Deadman K3.17: Duncan

16:45-17:10 J Mackenzie L Lin B PekmenRobust Solution of a BVP arisingin Liquid Crystal Modelling

Fractional Matrix PowersDQM Time-DQM Space Solutionof Hyperbolic Telegraph Equation

17:10-17:35 J Van lent S Miyajima N AkgunDistributing Points in a Triangleusing Optimal Transport Tech-niques

The relation between two algo-rithms for enclosing matrix eigen-values

DRBEM and DQM Solutions ofNatural Convection Flow in aCavity Under a Magnetic Field

17:35-18:00 A Araujo Y Nakatsukasa M Tezer-Sezgin

Stability and convergence of fi-nite difference schemes for com-plex diffusion processes

Arithmetic and communicationminimizing algorithms for thesymmetric eigendecompositionand the singular value decompo-sition

MHD flow and heat transfer be-tween parallel plates with Navier-slip wall condition

19:00 for 19:30 DRINKS RECEPTION and CONFERENCE DINNER-Lord Todd

M7 Singularly perturbed problems, M8 Numerical Linear Algebra, M9 High frequency and oscillatory problems

49

Thursday 30th June


Chair: K4.12: Silvester K3.27: Berrut AR201: K Burrage

11:30-11:55 M Payne C Brito-Loeza Q YangDual Lumping of IsogeometricFinite Element Schemes usingPetrov-Galerkin Techniques

On numerical algorithms for levelset and curvature based modelsfor surface fairing

Novel numerical methods for non-linear time-space fractional diffu-sion equations in two dimensions

11:55-12:20 R Scheichl L Sun A Khan

Energy Minimizing Coarse Spaceswith Functional Constraints

A new splitting method for meancurvature-based variational imagedenoising

Convergence of an implicit algo-rithm for a finite family of asymp-totically quasi-nonexpansivemaps

12:20-12:45 J Phillips W Yeo D Altintan

Finite elements on pyramidsC1 Piecewise Quadratic Hierar-chical Bases

How to solve boundary valueproblems by using Variational It-eration Method


Chair: K4.12: Georgoulis K3.27: Loisel AR201: Nichols

15:00-15:25 G Barrenechea D Janovska U RuedeA low order local projection sta-bilized finite element method forthe Oseen equation

Zeros of matrix polynomials: Acase study

Towards Exascale Computing:Multilevel Methods and FlowSolvers for Millions of Cores

15:25-15:50 A Cangiani R Kammogne J Lanzoni

Mimetic Finite Difference meth-ods for elliptic problems

Domain Decomposition forReaction-Diffusion Systems

Norm growth and pseudospectrafor PML discretisations of opensystems

15:50-16:15 A Saeed PG Martinsson D TangNumerical Solution of Fully Non-linear Elliptic PDEs with a Modi-fied Argyris Element

Fast Direct Solvers for EllipticPDEs

iGen: a program for the au-tomated generation of reduced-dimension models


Chair: K4.12: Cangiani K3.27: Gould

16:45-17:10 T Kang S Zhu

T − ψ finite element schemes foreddy current problems

The numerical linear algebra ofapproximation involving RadialBasis Functions

17:10-17:35 T Horvath H WangA fast matrix assembley forinterior penalty discontinuousGalerkin method

A Posteriori Error Analysis for aQuasicontinuum Method

17:35-18:00 V Janovsky M Al-Baali

On periodic patterns in the trafficflow

Convergence analysis of a familyof damped quasi-Newton meth-ods for nonlinear optimization

19:00 for 19:30 DRINKS RECEPTION and CONFERENCE DINNER-Lord Todd

50

Friday 1st JulyChair: K3.25: Davydov

9:00-10:00 D PericOn Computational Strategies for Fluid-Structure Interaction: Some Insights into Algorithmic Basis with Applications

10:00-11:00 J NagyNumerical Methods for Large Scale Inverse Problems in Image Reconstruction


Chair: K3.25: M10 K3.14: M11 K3.27: M12

11:30-11:55 E Tyrtyshnikov M10 O Devolder M11 J Blanchard M12

Matrix computations withdata in multi-index formats

The Fast Gradient Methodas a Universal Optimal First-Order Method

GPU Accelerated Greedy Al-gorithms for Sparse Approxi-mation

11:55-12:20 D Savostyanov M10 M Takac M11 P Richtarik M12

Rank-one QTT vectors withQTT rank-one and full-rankFourier images

Efficiency of Randomized Co-ordinate Descent Methods onMinimization Problems witha Composite Objective Func-tion

A GPU accelerated coor-dinate descent method forlarge-scale L1-regularisedconvex minimization

12:20-12:45 P Zhlobich M10 M Kocvara M11 M Lotz M12Stability of QR-based systemsolvers for a subclass of Qua-siseperable Order One matri-ces

Solving topology optimizationproblems by domain decom-position

Compressed sensing thresh-olds and condition numbers inoptimization

12:45-13:10 A Thompson M12A new recovery analysis of It-erative Hard Thresholding forCompressed Sensing

13:15-14:30 LUNCH-Foyer outside K3.25

END OF CONFERENCE

M10 Linear algebra in data-sparse representations, M11 Optimization: complexity and applications, M12 Compressed Sensing: algorithmsand theory

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