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Doctoral Thesis

Sparse perturbation algorithms for elliptic PDE's with stochasticdata

Author(s): Todor, Radu Alexandru

Publication Date: 2005

Permanent Link: https://doi.org/10.3929/ethz-a-005151392

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Diss. ETH No. 16192

Sparse Perturbation Algorithmsfor Elliptic PDE's with Stochastic Data

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the degree of

Doctor of Mathematics

presented by

RADU ALEXANDRU TODOR

Dipl. Math. University of Bucharest

born November 12, 1973

citizen of Romania

accepted on the recommendation of

Prof. Dr. Christoph Schwab, examiner

Prof. Dr. Ralf Hiptmair, co-examiner

Prof. Dr. Reinhold Schneider, co-examiner

2005

Dank

Die vorliegende Arbeit ist unter der Leitung von Herrn Prof. Christoph Schwab entstanden.

Ich danke ihm ganz herzlich für seine stete Unterstützung, seine hilfreichen und motivieren¬

den Ratschläge, und seine immer sehr engagierte Betreuung meiner Doktorarbeit.

Herrn Prof. Ralf Hiptmair und Herrn Prof. Reinhold Schneider danke ich für ihre kon¬

struktiven Bemerkungen und die Übernahme des Koreferats.

Ferner gilt mein Dank meiner Familie, die mich immer liebevoll unterstützt hat, sowie

meinen Kolleginnen und Kollegen aus dem Seminar für Angewandte Mathematik, die mir

den Aufenthalt an der ETH angenehm gemacht haben.

Financial support by the European Union (contract number HPRN-CT-2002-00286) and

the Swiss Federal Office for Science and Education (grant BBW 02.0418), within the IHP

network 'Breaking Complexity', is gratefully acknowledged.

i

Seite Leer /

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Abstract

In this work we investigate elliptic partial differential equations with stochastic data. We

develop fully deterministic algorithms using statistics of the data (expectation and/or

higher order moments) as input, for the computation of similar statistics of the stochastic

solution. We introduce a sparsified higher order moment representation which, in the con¬

text of a classical perturbation method enables us to formulate efficient algorithms for the

computation of solution moments, using sparse grids and wavelet techniques. Assuming

analytic spatial regularity of the data fluctuation, we prove almost linear complexity for

these algorithms.

Choosing stationary diffusion in a bounded domain as a model problem, we discuss in

Chapter 1 the setting, basic notations and definitions, plus the main results.

We devote Chapter 2 to the case of a stochastic source term and deterministic diffusion

coefficient. We derive deterministic moment equations and use sparse grids discretization

to preserve almost optimal convergence rates. We propose a new solution algorithm for

the resulting linear system given by a well-conditioned (due to the use of a wavelet basis),

yet fully populated matrix with a tensor product block structure, to achieve log-linear

complexity (in the number of degrees of freedom) despite the high dimensionality of the

moment problem.

In Chapter 3 we address the more general case of a stochastic diffusion coefficient and

stochastic source term. We show that the expectation of the stochastic solution can be

computed starting from the higher order moments of the data. We introduce approximate

sparsified representations of these moments and show that using them as input in the com¬

putation of the solution expectation results in an algorithm of almost linear complexity,

under the analyticity assumption mentioned above. We finally combine this algorithm with

the one developed in Chapter 2 to obtain similar, efficient algorithms for the computation

of higher order moments of the stochastic solution.

We conclude by discussing in Chapter 4 possible further theoretical developments and

extensions of the results presented in this thesis.

m

Seite Leer /

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Kurzfassung

Gegenstand dieser Arbeit ist die Untersuchung elliptischer partieller Differentialgleichungenmit stochastischen Daten. Basierend auf Datenstatistiken (Erwartungswert, höhere Mo¬

mente) werden rein deterministische Algorithmen zur Berechnung von ähnlichen Statistiken

der stochastischen Lösung entwickelt. Effiziente Darstellungen der höheren Momente eines

Zufallsfeldes werden eingeführt, und zusammen mit dünnen Gittern und wavelet Techniken

im Rahmen einer Störungsmethode verwendet, um schnelle Verfahren zur Berechnung der

Lösungsmomente zu erhalten. Im Falle der im Ort analytischen Datenfluktuationen wird

fast lineare Gesamtkomplexität gezeigt.

In Kapitel 1 wird das Modelproblems (stationäre Diffusion im beschränkten physikalischen

Gebiet), die grundlegenden Konzepte und Definitionen eingeführt, sowie die Hauptresul¬tate erläutert.

Der Fall einer stochastischen Quelle mit deterministischem Diffusionskoeffizient wird in

Kapitel 2 behandelt. Hoch dimensionale, deterministische Momentgleichungen werden

hergeleitet, mit Hilfe von dünnen Gittern und Waveletbasen diskretisiert, was zu gut kon¬

ditionierten, allerdings voll besetzten linearen Systemen führt. Ein Lösungsverfahren wird

entwickelt, das die Tensorproduktstruktur des ursprünglichen Momentproblems ausnutzt,

um log-lineare Gesamtkomplexität zu erreichen.

Für das vollständig stochastische Diffusionsproblem (sowohl Quelle als auch Diffusionsko¬

effizient werden als Zufallsfelder modelliert) wird in Kapitel 3 ein Algorithmus zur Berech¬

nung dos Erwartungswertes entwickelt. Höhere Momente der stochastischen Daten wer¬

den algorithmisch in eine Approximation des Erwartungswertes der stochastischen Lösung

umgesetzt. Im Falle einer im Ort regulären Fluktuation der Daten und unter Verwendungder effizienten Darstellungen der Momente als Eingabe wird erneut fast lineare Gesamtkom¬

plexität bewiesen. Ferner wird dieser Algorithmus mit dem in Kapitel 2 beschriebenen Ver¬

fahren kombiniert, was auch eine effiziente Berechnung der höheren Momente ermöglicht.

Im letzten Kapitel werden Möglichkeiten zur Verallgemeinerung der zuvor entwickelten

Theorie aufgezeigt.

v

Contents

Dank i

Abstract iü

Kurzfassung v

1 Introduction 1

1.1 Problem Formulation 1

1.2 Main Results 4

1.3 Notations and Definitions 8

2 Stochastic Source Term 11

2.1 Generalities 11

2.2 The Moment Equation 13

2.3 Discretization with Sparse FEM 14

2.4 Iterative Solution and Complexity 17

3 Stochastic Coefficient 27

3.1 Random Fields 27

3.2 Discretization 30

3.3 Neumann Series Decompsition 31

3.4 Discrete Mixed Moment Problem 33

3.4.1 Mixed Moment Algorithm 36

3.4.2 Sparsification 41

3.4.3 Complexity 43

3.5 Mean Field Computation 44

3.6 Computation of Higher Order Moments 45

3.6.1 Discretization 46

3.6.2 Algorithms 48

4 Conclusions and Open Issues 57

vii

5 Appendix 59

5.1 A Trace Lemma 59

5.2 A Fast Decaying Sequence 61

Bibliography 63

Curriculum Vitae 67

viii

Chapter 1

Introduction

1.1 Problem Formulation

The rapid development of (adaptive) algorithms, hardware and software in recent years

has made the accurate numerical solution of elliptic partial differential equations a routine

matter in many (but not all) engineering applications. Despite this, results of accurate finite

element (FE) computations often deviate significantly from the responses of the physicalsystem under consideration. Having eliminated the discretization error, and assumingthat the modelling error inherent in the selected partial differential equations (pde's) is

negligible (i.e. that the adopted pde's precisely describe the physics of the system under

consideration), the gap between simulation and observation must be due to uncertainty in

the input data.

One task for applied mathematics and scientific computing is therefore (see e.g. [GS91],[ER03]) to develop tools for uncertainty processing, i.e. the systematic and quantitativenumerical representation of uncertainty in input data and its propagation to the outputof a finite element simulation. This requires new developments in several areas of appliedmathematics and engineering: input parameters are replaced by random variables resp.

by random fields with known or estimated statistics, and traditional deterministic finite

element methods are reformulated to allow for randomness in input data and solution.

In this work, we focus on this latter aspect, i.e. the formulation, design and analysis of

deterministic finite element solutions of elliptic pde's with stochastic data.

We consider as a model problem diffusion in a bounded domain D C Rd, where the data

(source term / and diffusion coefficient a) and therefore the solution u too, are allowed to

depend randomly on a new parameter w £ Q,

-div(a(-,u;)Vu(-,aO) = /(-,£*;) in H~l{D) P-a.e. w e ft. (1.1)

Here we assume the sample set Ü to be equipped with a probability measure P quantifyingthe uncertainty and giving the probability of occurance for each event a in a set E C 2U

of admissible events (that is, (ft, £, P) probability space).

1

2 CHAPTER 1. INTRODUCTION

Identifying the data a, f as well as the solution u with random vector fields we rewrite (1.1)as

Affl(w)«(w) = /M in H~l{D) P-a.e. eu G ft. (1.2)

Both (1,1) and (1.2) are to be understood here and throughout this work variationally in

H£{D), that is,

/ a(x, uj)Vu(x) Vu(x) da; = (/(•, w), u(-)>H-i(o),Hä(D) Vu H^(D) P-a.e. weft.

D

Under the uniform ellipticity assumption (A denotes here the Lebesgue measure)

a_ < a(x,uj) < a+ Ax P-a.e. (x,u) e D x ft,

(1.1) is well-posed in L°(Ü,II^(D)) for data / L°(Ü, H-^D)) and a G L°(D x ft) (hereL° denotes measurability). For equation (1.1) we then formulate the following

Problem. Given statistical information w.r.t. uj about the data (a, /), compute statistical

information w.r.t. u) about u solution to (1.1) (statistical information comprises first and

higher order moments, variance, probability distribution function etc.).

Since for a problem with stochastic data knowing all joint probability densities (i.e. com¬

plete statistical information) is in practice hardly the case, reasonable assumptions can be

made only on some data statistics. Defining for any ieN+ the moment of order k of a

measurable u : D x ft — R by

Mk(u) :Dk :=DxDx---xD-+R

Mk(u)(x1,x2,..., xfc) := / u(xi,u)u(x2,ui) u(xk,u) dP(uj)n

whenever the integral exists, the moment version of the problem above reads,

Moment Problem. Given moments w.r.t. uj of the data (a,/), compute moments w.r.t.

üj of u solution to (1.1) (most relevant are first and second order moments).

Several approaches have been proposed in the literature to treat these two problems numer¬

ically, of which we mention and briefly discuss here the Monte Carlo (MC), the perturbation

and the stochastic Galerkin (SG) methods.

The MC simulation is probably the simplest and most natural approach, based on samplingthe coefficients and solving deterministic problems by a standard (e.g. Galerkin) finite el¬

ement method (see, e.g. [BTZ04]). The computation is fully parallelizable (therefore easy

to code), but requires knowledge of data distribution everywhere in the physical domain,

1.1. PROBLEM FORMULATION 3

a very strong assumption. Besides, the method is in general considered to converge slowlydue to the statistical error generated by sampling, although the overall complexity can be

easily shown to behave polynomially in inverse accuracy (see Definition 1.1).

The perturbation approach is widely used in engineering applications (sec [KH92] and ref¬

erences therein). There are several versions, of which we mention here only the Neumann

expansion (see [Kel64]), the first order second moment (FOSM) (see [DW81]), and the

Wiener/generalized polynomial chaos (W/gPC) (see [Wie38], [SchOO], [XK02]) methods.

In one of its mathematically rigorous forms the perturbation method consists in invert¬

ing the operator with random coefficients as a Neumann series around its deterministic

mean and solving then sequentially a large number of deterministic problems with differ¬

ent source terms. Its good performance has been demonstrated in practice (at least for

small fluctuations), but only superalgebraic complexity estimates have been obtained in

theory (see e.g. [BC02]). In this work we also adopt the Neumann perturbation approach

and, after carrying out a careful error analysis, we propose an improved version, involvingalso sparse grids and wavelet techniques, and leading, under additional but reasonable reg¬

ularity assumption on the fluctuation, to moment computation algorithms of subalgebraic,

nearly optimal complexity (see Definition 1.1 below). Note that the fluctuation regularity

assumption allows the control of the data truncation error and acts here therefore as a

substitute for the standard moment closure hypothesis.

The parametrization of uncertainty is one of the key points in the numerical treatment of

problems with stochastic data. The number of parameters used to describe the random

fields has in general a huge impact on the algorithm complexity. A random field expansion

of Karhunen-Löeve type separating the deterministic and stochastic variables at an opti¬

mal approximation rate (see e.g. [Loè77], [Loè78]) has become in recent years a standard

procedure to transform the original stochastic problem into a parametric deterministic one,

where the parameter involved belongs to a high dimensional space.

The SG method interprets then the transformed problem variationally in both the param¬

eter and the physical variable and uses a standard (Galerkin) FEM to produce a numerical

solution (note the need for numerical integration schemes in high dimensional domains).Backward substitution gives the numerical solution to the original stochastic problem and

postprocessing is required to obtain the desired statistical information. Just as in the case

of a MC simulation, very detailed information on the data distribution is needed, and the

finite elements in use today to treat the transformed problem can only ensure superalge¬

braic complexity raten (see e.g. [BTZ04], [FST05], [MKÛ5]).An interesting and promising alternative to the SG method would be to essentially ap¬

ply the same complexity reduction method used here (see section 3.4.2) in the context of

reconstruction techniques (collocation, interpolation) for the solution of the transformed


problem, by taking into account its analytic parameter dependence. This leads to results

comparable with those presented in this work - nearly optimal complexity under qualita¬

tively similar assumptions.

Definition 1.1. We say that an algorithm for the computation of M.k{u) with u solution

to (1.1) has nearly optimal complexity if, assuming that the deterministic diffusion problem

in D with data given by the expectations of a and / can be solved with an accuracy e using

0(e~/3) operations and memory (for some ß > 0, asymptotically ase-> 0), the algorithm

requires at most 0{e~^~°^>) operations and memory for an ^-accurate computation of

Mk(u).

In other words, an algorithm for the computation of Mk(u) (where u solves a stochastic

model problem) is nearly optimal if the computational effort required is equivalent, asymp¬

totically as the prescribed accuracy e goes to 0, to the effort needed to approximately solve

one deterministic model problem. For example, if the deterministic model problem can be

solved in linear complexity (/? — 1), e.g. by a multigrid method, then a nearly optimal

algorithm for Mk{u) has (nearly) linear complexity, too.

Example 1.2. An algorithm for the computation of M}(u) with u solution to (1.1), which

amounts to solving e~°(1) deterministic diffusion problems in D for an accuracy of e — 0

and with data as regular as the expectations of a, f, has nearly optimal complexity.

We mention that the mean field computation algorithm we are going to present is of the

type mentioned in the example above and will be developed around some very classical

ideas of the perturbation approach. Besides, it is almost fully parallelizable and its imple¬

mentation can be based on any préexistent FE solver of the deterministic model (diffusion)

problem in D (to be called with many different r.h.s.'s).

1.2 Main Results

We give in the following a summary of the main results contained in this work. We empha¬

sise that the main novelty consists in improved perturbation algorithms of nearly optimal

complexity (thus more efficient than MC, standard perturbation or SG methods, under

similar assumptions).

The case of a deterministic diffusion coefficient a in (1.1) (that is, depending only on the

physical variable x, a(x,u) = a(x) \fx G D) with a stochastic source term / (that is,

depending on both the physical and the stochastic variables x and ui respectively) is a key

step in the development of solution algorithms for (1.1) using a perturbation approach.

1.2. MAIN RESULTS 5

This analysis has been carried out in [ST03a], [ST03b]. We briefly review next the main

results obtained in [ST03b].

Defining the solution operator to the model problem, Aa :— —div(aV) : Hq(D) —> H~1(D),and

Aa ® Aa ® • • ® Aa : Hl(Dk) -> H~\Dk)

where

Hs{Dk) := HS(D)®HS(D)®---®HS(D) VsG[-l,oo[

Ht(Dk) := Hl{D)®Hl{D)®---®Hl{D)

are the anisotropic Sobolev spaces of mixed highest derivatives (e.g. H1(D2) ~ {w(x,y) G

L2(D x L>) | Vxv, Vyv, VxVyv E £2}), a deterministic equation for Mk(u) in terms of

Mk(f) is

(Aa ® Aa ® • • • ® Aa)A**(u) - Mk{f) in H~1{Dk).

This deterministic moment problem for A^fc(w) in terms of Mk{f) shows that we can trade

randomness in the original pde at the expense of high dimension in the problem domain.

Employing a sparse tensor product discretization and a wavelet preconditioning procedure,

the corresponding discrete moment problem is shown to lead to a well-conditioned linear

system. Further, a solution algorithm for the resulting system (given in general by a fully

populated matrix, but having a special blockwise tensor product structure) is developed

to ensure log-linear (in N, number of dof's in D) number of operations and memory re¬

quirements. The high dimensional fc-th moment problem is therefore solvable numerically

in work essentially proportional to that of the mean field equation, i.e. for k — 1, up to

logarithmic terms. In other words, nearly optimal complexity is achieved by the algorithm

presented in [ST03b] in the case of a deterministic diffusion coefficient.

In chapter 3 we take up the case of a stochastic coefficient. The main difficulty is now that,

in contrast to the case of a deterministic coefficient, there is no obvious pde with Mk(u) as

a solution. We start with the simplest such problem, that of computing Ml{u), the mean

field (or expectation) of u solution to (1.1). The idea is, as in the widely used perturbation

approach, to use a representation (decomposition)

a(x, lü) — e(x) + r(x, uj) — 'deterministic expectation' + 'random fluctuation'

where the fluctuation is assumed to be smaller than the expectation in the sense that

\\r(x> ')\\L°°(m0 < sup

" v

/",(U)

< 1.

XD e(x)

Choosing e appropriately, it is easily seen that this condition is (almost) equivalent to the

ellipticity of Aa.

We further assume the fluctuation r to be analytic in the physical domain D C [— 1, l]d,but emphasize that this condition can be relaxed to only piecewise finite Sobolev regularity,

r<=A([-l,l]d,L°°(Sl)).

The analyticity assumption, which is satisfied e.g. if the covariance of a is Gaussian,

ensures the existence of a fluctuation expansion as a fast (quasi-exponential) convergent

scries separating the deterministic and stochastic variables,

r(x,u)=J2(x)xm(u)> \\<t>®Xm\\L~(Dxfl)<e-°(ml/dî VmGN+. (1.3)

Such a representation is clearly not unique - one can choose e.g. the Karhuneri-Loeve

(KL) expansion of r, or (4>m)me^+ to be the Legendre polynomials in [— 1, l]d. In both

cases the deterministic part (4>m)meN+ of the expansion (1.3) is available (by an additional

eigenvalue computation for the integral operator associated to the kernel M2(r), in the

KL case) and statistical information on the stochastic part (Xm)mepj+ follows by testing r

against (0m)m6N+)

Xm(u)) = / r(x, u)c/>m(x) dP(uj) Vm G N+.

n

Choosing a truncation order M for the fluctuation series,

M

rjif := 53 4>m{x)Xm(uj)m=l

and a FE discretization level L (corresponding to the FE space VL of dimension NL), we

define recursively, with Ae — —div(eV), the sequence (vj^jm by

Aev0Jj(-,uj) = f(-,u)

in VI P-a.e. we!],

Aei>j,i(-,o;) = div(rM(',a;)^_i,L(-,w)) Vj G N+

The Neumann series Y1jçnvj,l *s then easily shown to converge in Hq(D), uniformly in

to, like a geometric series. Truncating it a level n G N and assuming the data e, f to be

sufficiently regular in D as to ensure a FE convergence rate $(Nl) uniformly in lo G ft, we

have

H^V) - E-**W)lku» £ e-°(Mi/d) + *(Nl) + e~°M. (1.4)3=0

In order to balance the contributions of the three discretization steps taken so far (trun¬cation of fluctuation series, FE in D, truncation of Neumann series) and to achieve an

accuracy of order s > 0 we choose

M~|log£|d, L s.t. $(NL) < e, n~\\oge\. (1.5)

1.2. MAIN RESULTS 7

However, the computation of Ml(v]iL) for 0 < j < n in (1.4) is not an obvious task. We

show that Ail(vjtL) can be computed exactly using any deterministic solver in D at FE

discretization level L, starting from the mixed moment

Mb'l){rM,f)(K,x) ~ I rM(xi,Lj)rM(x2,uj) • -rM(xj,uj)f(x,Lü) dP(uj) xG D3,x G D

a

via the following composition of j + 1 bounded operators defined in terms of the div, V

operators in D, the trace operator on the diagonal set in D x D and the FE solution

operator A~^ of the deterministic diffusion problem in D with coefficient e, at FE level L,

mA(D*,H-\D)) ld^®A<\ A^,VL) - A{ÎP-\VL) - •••

MM(TM,f)L~iD3) e,L> M^(rM,v0Jj) - M^(rM,Vl,L)

... _> Vl

- M\v3,L)

The alternative representation of the (j, 1) mixed moment of (rM,f),

M^\rM,f)= J2 | / Xmi (u)--- Xmj (u)f(x, u) dP(uj) ) 0mi ® • ® 0mj. (1.6)

^»a 02v

1/ £L°°(D3)G P"1^)

is processed by the diagram (f) and transformed into M1(vjtL). The mean field computa¬

tion algorithm consists therefore in running (f) for all 0 < j < n and with (1.6) as input.

The total cost is shown to be essentially as high as the computational effort needed to

run the first step in (f) for j — n (definition of the following j operators in (j) thus omit¬

ted). Taking into account the mixed moment symmetry w.r.t. permutations of the index

set (mi,m2,... ,m3), we deduce that the algorithm requires solving as many deterministic

diffusion problems in D at FE discretization level L as there are terms in the alternative

representation (1.6) of the mixed moment of (r^, /), that is at least

Mnjn\ ~ £-°(1)|log£|°(((*-1)|log£|) deterministic problems in D with accuracy e.

This superalgebraic (for d > 1) overall complexity (which renders the algorithm rather

inefficient, compared to a standard Monte Carlo simulation of quadratic complexity, see

discussion below) can be substantially reduced by noting that, due to (1.4), (1.5), not exact

but only approximate, £-accurate knowledge of M^iy^i) for 0 < j < n is required. Based

on the fast decay (1.3) of the fluctuation series, we show that many terms in the alternative

representation (1.6) of the mixed moment of (rM, f) can be discarded without affecting the

0(e) computation accuracy. Using the resulting sparsified mixed moment representation

as input in (f), the algorithm complexity is shown to become nearly optimal, consisting in

solving

~ | log£|°(|log£l+1 ) < e~°W deterministic problems in D with accuracy e. (1.7)

We conclude the presentation of the mean field computation algorithm by a comparison

with the Monte Carlo method, in terms of overall complexity. Note first that sampling

a, f over a finite sample set fto requires knowledge of the data distribution everywhere in

D. Averaging the discrete (FE level L) solutions (ui(cj))ueQ0 of the sampled diffusion

problems we obtain the accuracy estimate

It follows that the sampling can not distinguish between a rough and a smooth (in the

physical variable) fluctuation, and that the Monte Carlo method amounts in this case to

solving

~ e~2 deterministic problems in D with accuracy e, (1.8)

which is the desired quadratic complexity estimate.

Summarizing, in the first part of chapter 3 we propose a perturbation mean field compu¬

tation algorithm and prove that additional information on the data (smoothness in the

physical variable) allows a nearly optimal complexity estimate, asymptotically as e —> 0.

The algorithm is thus shown to be less expensive than the Monte Carlo simulation ((1.7)vs. (1.8)) and all other known algorithms (SG, W/gPC, FOSM etc.).In the second part of chapter 3 we combine the perturbation algorithm presented above

for the mean field computation with the sparse grid solution algorithm for the higher or¬

der moment problem discussed in chapter 2. The result is a new algorithm also of nearly

optimal complexity for the computation of Mk(u) for k > 2.

1.3 Notations and Definitions

We conclude this introductory part by setting up basic notations and definitions that will

be constantly used throughout this work. We consider a bounded open set D C Md (the

physical domain) with Lipschitz boundary F = dD, we denote by A the Lebesgue com¬

pletion of the Borelian er-algebra on D, and use the standard notation A for the Lebesgue

measure on D. We model the uncertainty in the data through a probability space which

we denote by (ft, £,P), Equipped with the product a-algebra A x S, the space D x ft

becomes a a-algebra.

Definition 1.3. If n G N+, a measurable function g : D x ft — W1 is called a random

vector field on D. If n — 1, g is simply a random field on D.

1.3. NOTATIONS AND DEFINITIONS 9

In the following, mathematical objects (sets, functions, spaces, etc.) defined in terms of

the physical domain D will be referred to as deterministic, whereas those depending on ft

will be termed stochastic.

To formulate a minimal regularity requirement on the random fields we define the Bochner

spaces of vector-valued measurable functions.

Definition 1.4. For an arbitrary Banach space (P, || • \\B), we define the space Lgtep(ft, P)of all B-valued E-measurable step-functions on ft by

£steP(^> B) '= {9 Œ -> B I 3 (fti)ieN+ C S partition of ft, (k)ieN+ C B s.t. g = ^ XnM,

where xn* denotes for i G N+ the indicator function of the set ft» C ft.

The Bochner space of B-valued ^-measurable functions on ft is then given by

L°(ft, B) := {g : ft - B | 3 (gn)n^+ C P°ep(ft, P) s.t. gn > g P-a.e. in ft}.n—too

Clearly then, if (P, || • ||) is a space of measurable (vector valued) functions on D, the

corresponding Bochner space consists of random (vector) fields.

The notion of positivity for a stochastic diffusion coefficient reads

Definition 1.5. A random field a G L°(ft, L°°(D)) is called strictly positive if a(-) is P-a.e.

strictly positive, i.e. there exists AT G S such that P(N) = 0 and

Vw G ft \ N 3a_iW, a+fLJ G R+ s.t. a_,w < a(cu) < a+iU1 A-a.e. in D. (1.9)

The existence of a unique stochastic solution to an elliptic problem with stochastic data

follows then from the well-posedness of the corresponding deterministic problem (see also

[Bab61]). In other words, the well-posedness is preserved by tensoring with measurable

functions.

Proposition 1.6. If a e P0(ft,L°°(r;)) is strictly positive and f G L°(ft, H~1(D)), then

there exists a unique u G L°(Ü,H^(D)) such that

-div(o(w)V«(w)) =/(w) inir1(D) P-a.e. in ft. (1.10)

Proof. Since the uniqueness part follows immediately from the well-posedness of the de¬

terministic diffusion problem ((1.10) pointwise in ft), we only check here the existence of

a (measurable) solution to (1.10). To this end, we consider two sequences (an)n>i) (fn)n>i

as in Definition 1.4, as well as the corresponding representations an — X)»eN Xnnian,i

and fn — 5jjeN+Xn»,</n,i respectively (w.l.o.g. we assume them subordinate to the same

partition of ft). With the notations in (1.9) we set for any n G N+

In '•— {i G N+ I a_ u/2 < an^i < 2a+ilil A-a.e. in D for some u G ftn,i}. (1.11)

From (1.9) and the P-a.e. convergence of the sequence (an)„e^+ to a we deduce that 2n / 0

for n large enough. Clearly, for any i G In the diffusion problem in D with data (on>i, fnti)has a unique solution un^ Pi\(D). We then set

un~ E X<W

• un,i + Xn\uieInnnii• 0 G Ls°tep(ft, H^D)) (1.12)

and prove next that (un)n£^+ converges P-a.e. in Hq(D). The limit, denoted by u, is in

view of Definition 1.4 an element of L°(ft, H^(D)) and will be seen to satisfy (1.10).

Let A' G S be a null set such that the pointwise convergence of the sequences (an)n>!,

(fn)n>\ as well as (1.9) hold on ft \ N. Fixing an arbitrary to G ft \ N, it follows that there

exists n^ > 1 such that for any n > nw there exists i^ G Xn with the property u G ft^,^.

This implies for any n > n^ that un(oj) — Un^ solves the deterministic diffusion problem

in D with data (aniiul, fn,^)- The ellipticity condition (1.11), uniform in n > n^, of all these

problems ensures via a standard Strang estimate the Hq(D) convergence of un(u) to u(ui)

solution to the deterministic diffusion problem in D with data (a(ui), f (uj)), as n — oo,

which concludes the proof. D

Remark 1.7. Under the assumptions in Proposition 1.6, the Doob-Dynkin Lemma (see

e.g. [Fed69]) ensures the existence of a Borel measurable mapping

h : Raïï^L°°(D) x H~1(D) - P0X(P>) (1.13)

such that

u(uj) = h(a(u),f(uj)) P-a.e. in ft. (1.14)

Troughout this work we assume that problem (1.10) satisfies the ellipticity condition (1.9)

uniformly in ft, that is it holds

Assumption 1.8. There exist a_,a+ G M+ such that

0 < a_ < a(u) < a+ A-a.e. in D, P-a.e. in ft. (1-15)

Note that for numerous engineering applications assumption (1.15) is satisfied (see e.g.

[GS91]). However, (1.15) is clearly less general than (1.9) and does not cover e.g. the

case of a log-normal diffusion coefficient, a — exp(F) with Y a gaussian random field

(for details and applications of log-normally distributed random fields in groundwater and

polution monitoring problems see [Gil87]).

Chapter 2

Stochastic Source Term

Throughout this section the diffusion coefficient is assumed to be deterministic, that is, a

depends only on the physical variable x G D. (1.15) becomes in this case

0 < a_ < a(x) < a+ < co A-a.e. in D. (2.1)

The source term / of (1.10) is a random field with finite first (or higher) order moments

fLk(n,H-\D)) &GN+. (2.2)

Under these assumptions we study the existence and computability of second and higher

order moments of u solution to (1.10), if the corresponding moments of the data / are

known.

2.1 Generalities

We begin with an introduction of the appropriate functional spaces which will help us

define the moments associated to a random field. The standard reference for the material

we briefly review in the following is [Yos95].

Definition 2.1. If k > 1 is an integer and H is an arbitrary separable Hilbcrt space, we

define the space of Lk, H-valued Bochner integrable functions (see e.g. [Yos95]) on ft by

Lk(Q; H) := i f : ft - H | /measurable, f \\f(cü)\\kH dP(w) < oo > / ~

ii/iiW) := j \\m\\kH dp(u),

where we use the same notation for a P-a.e. equivalence (denoted by ~) class and one of

its members.

A useful characterisation of these spaces follows from

11

12 CHAPTER 2. STOCHASTIC SOURCE TERM

Theorem 2.2 (Bochner). / G Lk(Û; II) if and only if there exists a sequence of H-valued

step functions (fj)jeN such that

fj-*fP-a.e.onSl and / \\fj - f\\kH -» 0, as j - oo. (2.3)

n

The case k = 1 is particularly important, since the elements of L1(ft; II) can be integrated.

Theorem 2.3. If f Ll(Çt; H), then the vector-valued integral

l f(uj)dP(u)) GP (2.4)

o

is well-defined by means of a sequence of II-valued step functions (fj)jen satisfying (2.3)

for k — 1, by

[ f(uj) dP(u) := lim / fJu) dP(u) G H. (2.5)J j-HX Jü Ü

The role of the abstract Hilbert space H will be played in the context of the moments

associated to random fields by the anisotropic Sobolev spaces.

Definition 2.4. If k G N* and v = (vi,v2,..., Vk) G Rk, we denote by

k

Pv(P)fc):=(g)P^(P) (2.6)3=1

the anisotropic Sobolev space of order v on Dk. Similarly we define the homogeneous

anisotropic Sobolev space H%(Dk) (Convention: for any s G R, s :— (s, s,..., s) G Rk).

Definition 2.5. If k > 1 is an integer, s G R and the random field f on D satisfies

/ePfc(ft,Ps(D)),then

/() ® /(•) <g> • • • ® /() G L\Q, Hs(Dk)), (2.7)s

v/

k times

so that the k-th order moment M.k(f) of the random field f (sometimes called k-point

correlation) is defined via Theorem 2.3 by

Mk(f) := J f{u) <g> /H ® • • • ® /(w) dP(cu) G Hs(Dk). (2.8)

n

Remark 2.6. Under the assumptions in Definition 2.5 it is easy to see that if s > 0, then

the A;-th order moment of / given by (2.8) satisfies

Mk(f)(xx,... ,xk) = J f(Xl, w) f(x2, Lu)--- f(xk, lu) dP(u), (2.9)

for any (xi, x2,..., xk) G Dk.

2.2. THE MOMENT EQUATION 13

2.2 The Moment Equation

Here we derive a deterministic PDE for the k-th order moment Mk(u), establish its well-

posedness in anisotropic Sobolev scale in Dk, and study the solution regularity.

Theorem 2.7. If f e Pfc(ft, P"1^)), then u solution to (1.10) satisfies

ueLk(Çl,H*(D)), (2.10)

so that Mk(u) G H$(Dk).

Proof. The conclusion follows at once from Definition 2.5 and the well-posedness of the

deterministic diffusion problem in D,

IMw)Hffi(D) <ca\\f(uj)\\h-hd) P-a.e. w G ft, (2.11)

where the constant ca > 0 depends only on the diffusion coefficient a. G

We prove next that M.k(u) satisfies the deterministic equation in Dk obtained by tensorising

(1.10) with itself k times. To this end we introduce the following notations,

k

a®,k ._ Ç§a£L°"(Dk)i=i

k

V®-fc := (g)V GBftf1 (I?*), ®J=1£2(£>)**)3=1

where we denote by B(X, Y) the space of bounded linear operators between the Hilbert

spaces X and Y, with B(X) := B(X,X).

Theorem 2.8. Mk(u) is the unique solution in Hl(ük) of the variational problem

(a®>kV®>kMk(u),V®'kM)LHD)M = {Mk(f),M)H-i{Dk)^{Dk) \JM G Hl(Dk). (2.12)

Proof. The existence and uniqueness of a solution to (2.12) are easily proved using the Lax-

Milgram Lemma in appropriate anisotropic Sobolev spaces, noting that tensor products

of bounded positive homeomorphisms between Hilbert spaces induce homeomorphisms

between corresponding tensor products spaces.

Now, since / G Lk(ü, H~l(D)), there exists a sequence (/„)„6n, (hn)nçN of P-valued step

functions on ft satisfying (2.3) with H :— H~1(D). By dominated convergence and (2.5),

Urn Ma(fn) - Ma(f) in H~1(Dk). (2.13)71—K»

We write /„ — J2qejn fn^-nq,„ where ln,,n denotes the indicator function of ft9i„ G S and,

f% G H~l(D) for any q,n (for each n the family (ftgiTs)(7ejn is a partition of ft). To the


deterministic data /* we associate the solution u^ G H^(D) of the corresponding diffusion

problem,

(aVulVv)LHD)ä = (K,v)H-HD)iHiiD) Vt; G H^D), (2.14)

and set un :~ J2qejn un^q,n- The continuous dependence (2.11) on the data of the solution

of the deterministic problem and (2.3) for / imply

lim un — u P-a.e. on ft, lim \\\u(u) — un(u))\\k„i(TT.dP(u]) = Q. (2.15)n—>oo n—*oo J -"ov^

From Definition 2.5, (2.15) and (2.5) we deduce that

lim Mh(un) = Mk(u) in P1^). (2.16)71—t-OO

Choosing in (2.14) & different deterministic test functions v\,v%,..., vk, taking the product

of the resulting k equations and summing over q with weights P(ft9>n), we obtain that

Mk(un) solves the deterministic problem

(a®'kV^kMk(un), V®'kM)LHD)ak = {Ma(fn),M)H-1(Dk)Mèm WW G H^(Dk) (2.17)

(use here that tensor products of total sets in Hilbert spaces are total in product spaces).The desired equation for Mk(u) follows then from (2.13) and (2.16) by letting n — oo in

(2.17). D

The regularity of Mk(u) follows naturally from that of the data Ma(f), and the result,

as well as its proof, is analogous to the one presented in [ST03a] for k — 2 (bounded

invertibility of operators acting in Hilbert spaces is preserved by tensorization). We only

state it, as follows. Recall that a deterministic diffusion problem in D is said to satisfy the

shift theorem at order s > 0 if H~1+s regularity of the data implies H1+s regularity of the

solution.

Theorem 2.9. // the deterministic diffusion problem in D with coefficient a satisfies the

shift theorem at order s, then (2.12) satisfies a shift theorem at order s too, in the sense

that ifMk(f) G H~1+s(Dk) implies Mk(u) G H1+s(Dk).

Remark 2.10. In the case of a polygon or polyhedron D, a shift theorem at order s >

0 holds in weighted spaces Hlß+8'2(D) (see [BG88]). The proof of Theorem 2.9 can be

correspondingly adapted to deduce then a shift theorem for the moment equation (2.12)in an anisotropic weighted Sobolev scale.

2.3 Discretization with Sparse FEM

We investigate now the numerical approximation of M,k(u), using the Finite Element

Method for the deterministic elliptic problem (2.12). We start by defining general FE

spaces, so let

V0 C Vi C ... C VL C ... C Hl{D) (2.18)

2.3. DISCRETIZATION WITH SPARSE FEM 15

be a dense hierarchical sequence of finite dimensional subspaces of Pq(P), with Nl :=

dim(Vi) < oo for all L.

Assumption 2.11. The following approximation property holds,

min ||W - v\\HuD) < $(NL, s)\\u\\H.+i(D), Vu G Hs+l(D) n Hl0(D), (2.19)

where Q(N, s) —» 0 for s > 0 as N —» oo denotes the convergence rate of the given FE

space sequence.

For regular solutions the usual FE spaces based on quasiuniform, shape regular meshes are

suitable.

Example 2.12. If T — {7i}LeN is a nested sequence of regular triangulations of D of

meshwidth hi = h^-i/l, we choose Vl to be the space of all continuous piecewise polyno¬

mials of degree p on 72 vanishing on dD. Then Assumption 2.11 holds with

NL~cT2d-L, $(N,s) = cTN-s, 6:=mm{p,s}/d. (2.20)

Since the A;-th order moment M.k(u) of u solves the elliptic problem (2.12) on Dk, we

construct FE spaces in Dk starting from the FE spaces {VlJ^o in D. Full tensor product

spaces,k

Vi:=(g)Vi3=1

are natural candidates, but, due to efficiency reasons, we use here the sparse tensor product

spaces that are defined by (see [Zen91], [BG99])

VL := Span I (g) Vi} | 0 < h + i2 + ... + ik < L I. (2.21)

Another description of the sparse tensor space (2.21) can be given in terms of the hierarchic

excess Wi at level L > 0 of the scale {Vl}l>q, which is chosen to be an arbitrary algebraic

complement of Vl-i in Vl (set V_i :— {0}). Since then

Vl = © Wu (2.22)0<i<L

it is easily seen that VL admits the direct algebraic (not necessarily L2 or H1 orthogonal!)

decomposition

VL:= 0 <g)WtjC 0 <g)W,=(g)VL. (2.23)0<u+i2+...+ik<L j'=l Q<h,i2,...,ik<L j=l j=l

The discretized version of (2.12) using the sparse FE space VL reads

{a®'kV®>kMkL(u\ V®'kML)L2{D)M = (Mk(f)ML)w^),Him VML G VL, (2.24)

where we denote by MkL(u) G Vl the discrete solution of (2.12). The approximation

property (2.19) allows an estimate of the discretization error in terms of the functional <E>,

as follows.

Proposition 2.13. If M.L(u) is the solution to (2.24), L > k — 1 and the approximation

property (2.19) holds, then

k

\\Mk(u)-MkL(u)\\2H1(Dk)<ca,k-J2c^^- E ll-^HIlWw (2-25)3=1 JCO, ,fc}

Card(J)^j

where ej G {0, l}k,ej(j) = 1 iff j G J and

3-i

cO'.*) = E E (HNh,s).^(Nl2,s)--^(Nlm,s).HN0,s))2+m=l li+...+lm=L~m+l

£ $(Nll,s)2.Q>(Nh,s)2---<S>(Nh,s)2 (2.26)h+...+l3=L-j+l

Besides, the constant caik depends exponentially on k, that is, ca,k ~ ct-

Proof. The estimate (2.25) follows using the quasioptimality of the FE solution to (2.24),

the approximation property (2.19) and the description (2.23) of the sparse tensor space with

Wl defined as the orthogonal complement of Vl-\ in Vj, w.r.t the standard scalar product

(•,) in Hq(D). More precisely, we consider the following orthogonal decomposition in

P^o(P'fc) equipped with the tensor product Hilbert structure induced by <S)j=i(v) (hereand in the following orthogonality in H^(Dk) is to be understood w.r.t. this Hilbert

structure).k

Mk(u) - PÈL(M\u)) = Y. ®PLMk(u) (2.27)

at>0,l<t<k

where P^ denotes the orthogonal projection on Wa w.r.t. (-,-), acting in the i-th dimension

of Dk. As the notation suggests, PyL denotes the Po(Pfc)-orthogonal projection on Vl-

Further we consider Q%a to be the projection on Va acting in the i-th direction of Dk and

note that the sum in the decomposition (2.27) is Po(£>fe)-orthogonal, since the excesses

(M/l)lsn are pairwise Po(P)-orthogonal. We rewrite the r.h.s. of (2.27), pointing out

those dimensions 1 < j < k for which ct3 — 0 (coarsest approximation),

E (®^W(«) = E E E (®^,®^W(«),£L«,>i+i \*=i / p=i jc{i,2 ,k} £]eja]>L+i \jeJ 3%j j

a,>0CardJ=p aj>!

\ /

2.4. ITERATIVE SOLUTION AND COMPLEXITY 17

and we cast the first inner sum of projections for J — {ji,j2, ,jp} in the form

E fé)^:<8)po ) =(id-«(g)(id-prt(g)Poja„>l

+ E p£®(Id - <ft-j <8> (Id - pon) (8) po"l<£ n>3 jéj

2

Ql'a2^1

+ • + E ^^-«-ET-iJ®^ (2-28)

Here the /-th sum in the decomposition (2.28) consists of those terms in the l.h.s. corre¬

sponding to indices ai,a2,...ap > 1 with £)n=i ctn > L + 1 and for which / L + 1.

The conclusion follows from (2.28) using the trivial estimate ||P„|| < ||Id —QL-iH (operatornorm in Pß(Pfe))and the approximation property (2.19). D

Specializing Proposition 2.13 by choosing the FE spaces from Example 2.12 we obtain

Corollary 2.14. For the sparse tensor product FE space based on the construction given

in Example 2.12 and for any L > k — 1 it holds

\\Mk(u)-MkL(u)\\H,{Dk) < c^AlogNLf-^NL'-WM^W^^= caAT,u(^ëNLYk-^2N£s (2.29)

and

dim Vh < CkrQo&Ntf^NL, (2.30)

where s — (s,s,..., s), Ô — minjj), s}/d, and all the constants depend exponentially on k.

Note that the full tensor product space would require 0(NL\) degrees of freedom for a

relative tolerance 0(N£5).

2.4 Iterative Solution and Complexity

Sparse FE spaces discussed in section 2.3 allow an important reduction of the number of

degrees of freedom needed to compute a discrete solution to the moment problem (2.12) up

to a prescribed accuracy. After choosing a basis in the sparse FE space Vl and computing

the corresponding stiffness matrix SL, finding the discrete solution of (2.24) amounts to

solving the linear system

8LMk(u) = Mk(f). (2.31)

In order to be able to solve (2.31) efficiently using an iterative method (e.g. the conjugate

gradient (CG) method), we must ensure well-conditioning and sparsity of the involved

matrix SL. While the first property can be ensured by a wavelet preconditioning procedure,

the latter (which does not hold, in fact) will be replaced by a proper use of the anistropic

structure of the problem, which will show that computation and storage of SL is not

necessary.

Concerning notations, T will be in the following a family of double indices running in

W x Nd.

Assumption 2.15. There exist a family * — (ipj,i)(j,i)eT C Hq(D) and constants cu c2 > 0

such that any u G 111(D) can be expanded as a convergent series,

« = E ci^j,i G I$(D) (2.32)

and the following stability condition is fulfilled

ci>* E icî2-

il E ciîMii(u) ^ c2,* E icî2- (2-33)

Constructions of families \[> satisfying Assumption 2.15 are known for intervals (D =]0,1[),

hypercubes (D —]0, l[d) or polygonal domains (see e.g. [DS99]).

Example 2.16 (hierarchical basis on interval). For D —]0,1[, let <fi be the hat function

on E, piecewise linear, taking values 0,1,0 at 0,1/2,1 and vanishing outside ]0,1[. Setting

? '•= {0'> *) I 0 < j, l<i< 2j} and

î(x) := 2'j/24>(2jx -Ï + 1) Vx G]0,1[,

the family \& — (ipj,i)(j,%)eF satisfies Assumption 2.15.

Example 2.17 (prewavelets on interval). With D, T and </> as above, we define on R

the function ijj, piecewise linear, taking values (1, -6,10, -6,1) at (1/2,1,3/2, 2,, 5/2) and

vanishing outside ]0,3[. Similarly, ijj1 take (9, —6,1) at (1/2,1,3/2) and V>r assumes values

(1,-6,9) at (1/2,1,3/2). Further, we define the scaling function by ipo,i :— <j> and the

boundary wavelets,

^,i(x) := 2~3^l(23x)

*/)j#(x) - 2~j/2ipT(2jx - 2j + 1) Vx G]0,1[, Vj > 1.

The interior wavelets are given by

ipj,i(x) := 2-j/2ip(2jx -z + 2) Vx G]0,1[, V2 < 2 < 2j - 1J > 2. (2.34)

The family \I> = (Tpj.ufjtier satisfies then Assumption 2.15.

For further examples we refer the reader to [Coh03], [Dah97] and the references therein.

Example 2.18 (wavelets on hypercube). In D =]0, l[d choose T := {(j, i) G Nd x Nd | 0 1 U„U)(.xq) Vl = Wi<«<Jei) (2-35)9=i

we obtain (after rescaling in PX(P>)) a family ^ = (ipj,i)(j,i)eF which satisfies Assumption

2.15 (see e.g. [G095]).

Formally, an increasing FE space sequence in D C Rd can be defined in terms of the family

VL := Span{V^| 0 < [j^ < L} (2.36)

(j is in general a multi-index, so that \j\oo :— maxi<g<d jq).We consider the algebraic complement WL of VL~i in VL given by

WL := Span{^ | tfU = L}, (2.37)

and obtain, via (2.23), the following representation of the sparse tensor space Vl,

k k

VL = Span{^,i := <g) VjM,iM I E U(^)loo < ^}, (2-38)f=l v=\

where \(v) denotes the z^-th line of the k x d matrix j and similarly for i.

The algebraic excess Wl of the sparse tensor scale (Vl)l>o is then given by

k k

WL = Span{V>j,i := <g) ^M)iH | £ |j(i/)|«, = P}, (2.39)

and can be further decomposed as

Wl=@Wi_ with WL = Spaiityj,, | |j(^)U = /„}, (2.40)

|I|=L

where

|/| := I, + ;2 + ... + lk V/ G Nfc.

For further use we collect, for P > 0, the basis functions in the definition (2.37) of WL in

a vector denoted ^l- Similarly, for / G Nfc we consider \E^ to be the vector containing all

basis functions of Wi_ as defined in (2.40).

Concerning the properties of the stiffness matrix SL that are of interest for solving (2.31)

(well-conditioning and sparsity), it holds

Proposition 2.19. The matrices (Sl)l>o have uniformly bounded condition number

K(SL)<ca!T,*,k VP>0, (2.41)

where the upper bound cati^tk depends exponentially on k.

Proof. From (2.33) it follows that the basis (ipj,i)(j,i)eT gives a homeomorphism of Hilbert

spaces between £2 and Hq(D), or that

« = E °i.^ — \< - E M2 (2-42)

defines an equivalent norm on Hq(D). The same holds then for the basis ijj^\ introduced in

(2.38). It follows that for M := (-Mj,i)j,i G M^ with NL := dim t>L, X := Ej,i^J,i^J,iis an element of Vl and

(SLM,M)^L = (a®>kV®>kM,V®'kM)LHD)<* ~ \\M\\2Hèm ~ J] l^wl2 ^ H^llk-

D

Proposition 2.20. For Examples 2.16, 2.17, 2.18 above as well as for similar wavelet

constructions the matrix SL is not sparse, in the sense that (compare (2.30)j

nnz(SL) > O(Nl) VP > 0. (2.43)

Proof. It is easily seen that the entries of SL corresponding to the indices i,j,i',j' with

j(l)=j'(2)= (P, P,..., L) are in general nonzero, implying the desired lower bound. D

The nonsparsity makes the storage and use of SL rather costly, and shows that ideas be¬

yond multilevel preconditioning are needed for an efficient solution of (2.31). The key

observation is the special structure of the discrete operator (or, equivalently, of SL), which

inherits (blockwise) the tensor product structure of the continuous operator (see (2.12)).A matrix-vector multiplication algorithm of log-linear complexity will be derived in the

following, exploiting the special tensor product structure of the problem. It will be shown

that, in terms of memory, only (blockwise) storage of the matrix SL corresponding to the

case k = 1 is necessary. Relating SL to SL will show that one step of CG algorithm can

be performed by manipulating only sparse blocks of SL. Moreover, an algorithm of log-

linear complexity in Ni for performing the matrix-vector multiplication required by the

CG method will be derived.

Storage of the load vector is necessary too, but, due to (2.30), the corresponding memory

requirement grows only log-linearly in NL, as P — oo.

We start with the derivation of the relation between SL and SL. To this end we denote by

(•, •)„, the scalar product associated with the norm (2.42). (•, -)w is obviously equivalent to


the standard scalar product in Hl(D) and (^j,i){j,i)^r is an orthonormal basis of Hq(D)

equipped with (•, -)w. Let PL and QL be the orthogonal projections in Pq(P) w.r.t. (-, •)„„on VL and WL given by (2.36) and (2.37) respectively. Clearly then,

i=o

Correspondingly, we denote by PL and QL the orthogonal projections on VL and WL (see

(2.38), (2.39)) w.r.t. the scalar product on Hl(Dk) obtained by tensorization of (•, -)w.On account of (2.38), (2.39) we obtain the multilevel decomposition

L

Pl = Y,Ql, (2.44)1=0

as well as the detail decomposition

Ql = E Qb (2-45)

wherek

Ql-=®Qi„ (2-46)j/=i

is the projection on the space Wi_ introduced in (2.40).We further denote by Qk the bilinear form of the moment problem (2.12),

Qk:=Q® Q®...®Q, (2.47)

where

Q(u,v):=(AVu,Vv)L2(D) Vu,v G H^(D). (2.48)

The discrete problem in Vl is then given by the bilinear form

QkL(u,v) := Qk(pLU,PLv) \/u,ve VL C Ht(Uk), (2.49)

or, inserting (2.44) and (2.45) in (2.49), by

L

QkL(u,v)=J2 E Qk(QLu,Qnv) Vu,v G VL. (2.50)U'=0 üeN*

|i|»i,IÜ=i'

Recalling that #/ is the vector containing the basis functions of W\ given in (2.40), we

write

Qi_u = u[ */, (2.51)

with real vector coefficients ui and similarly for v.

Using (2.51) in (2.50), we obtain

L

Ql(«,v)=£ E ul-êLli-v^ (2-52)l,l'=0 i,i^nk

\L\=i,\lL\=i'

where the matrix Sfo, is given by evaluating the bilinear form on the basis functions,

But, in view of (2.47) and (2.39), we have

k k

§& = Qk(Vi, *r) = ® e(*z„, H) = ® SU> (2-53)

where

S\'y ~ Q&t, *J') V0<l,l'<L (2.54)

are the blocks of the stiffness matrix SL corresponding to the deterministic diffusion prob¬

lem in D with coefficient a (or, equivalently, to the case A; = 1).The representation formulae (2.52) and (2.53) show that

Q£(«,«)=E E «M®SWtW. (2-55)1,1'=0 i,i£eN* \f=l /

111=1,11:1=!'

that is the stiffness matrix SL of the k-th moment problem computed w.r.t. the basis

(2.38) of the FE space Vl has a block structure

^ = \^l,l') LlleN* )"~

\l\=l<LIÜ=('<L

and each block is a tensor product of certain blocks of the stiffness matrix SL.

Moreover, SL is almost sparse, once for the basis (^,i)(j,t)e.F the following local support

assumption holds true. We remark that Examples 2.16, 2.17, 2.18 as well as similar wavelet-

type constructions are in this category.

Assumption 2.21. There exists p G N+ such that for all 1 '')values of i!.

Remark 2.22. From Assumption 2.21 it follows by a simple counting argument that

nnz(Sj,) ^ Pd • (min(*>l') + i)*"1 • 2*max^-''> V0 <l,l' < L. (2.56)

To formulate the matrix-vector multiplication algorithm we consider also, for each pair

I — (lv)ï=vlL ~ (0*=i> a reordering (permutation) gijj_ of {1,2,..., k} such that

Q k f k k }

E l"(») + E U») ^ max i E ^ E ï» \ v 1 ^ i ^k- (2-57)

The existence of such a permutation a is easy to prove, by choosing x„ — l„, yu — l'v for

any 1 < v < k in the following result.

Lemma 2.23. If (xv)\<u<k and (yv)i<v<k are two families of positive real numbers, then

there exists a permutation a of the set {1,2,..., k} such that

q k ( k k \

E x»w+ E y<v) -

maxi Exv' E y» \ v * ^ <? < fc- (2-58)

v=l u-q+l \v=l v=l )

Proof. We use induction on k. Since for k = 1 the claim is trivial, we assume that it holds

also for some k > 1. Consider (xv)i<.v<k+\ and (yu)i<u<k+i two families of positive real

numbers and define zv :— xv for 1 < v < k — 1 and Zk :— Xk + Xk+\, as well as tv :— yv for

1 < v < k — 1 and tk :— yk + yk+i- The induction assumption ensures the existence of a

permutation r of {1, 2,..., k} such that

q k ( k k }

E Zr(y)+ E M") ^ max ] EXi" EVv \ 'v : -q -k- (2-59)

We set then a(u) := t(v) for all i/ < r~1(k) and a(z^) :— t(v—1) for all ^ > r-1(A;)+l. Now,

if 2/jt + Xfe+i < Xk + 2/&+1 holds, we set <7(t_1(ä;)) := k, a(T_1(k) + 1) :— k + 1. Otherwise,

that is if ji/fc + xfc+i > xk + yk+l, we define <7(t_1(ä;)) := k + 1 and a(r~1(k) + 1) :~ A;. With

this choice for a one can easily check the inequalities (2.58). D

In order to simplify the exposition of the algorithm we introduce, for any 1 < q < k, an

arbitrary index pair (/, J/_) and a permutation a = <7y/ associated to it in the sense explained

above, the following tensor product matrices

k

Tt^-®U^ (2-6°)i>=i

where

v e{<7(l),<7(2),...,<r(</-l)}

U»:=< SL^ v =<& (2-61)

v £{a(q+l),a(q + 2),...,a(k)}

and Id/,/ denotes for / > 0 the identity matrix of size dimW/. With these notations, each

block in (2.55) can be expressed as a product of matrices of the type introduced in (2.60),

k

OC Siv,vv = Tijs,k ' ^iz,k-i ' ' ' ^i£,v (2.62)i>=i

For later use we remark that (2.56) implies the following sparsity estimate for Ttj, Q.

Remark 2.24. It holds

5-1 k

m<TULq) ^ C?W n&W + l)d • (min(^(9), C(9)) + I)""1 • II (W) + 1)"v—L u~q+l

. 2d-Œï=î/"(")+max^(«)'C(9)}+Ef=<I+i^(1,))i (2.63)

Proof. The estimate (2.63) follows from the obvious equality

/i-l k

nnz(Thu) = TTdimW/, >

" nnz(S/' ,, ) • TT dimWr ,

Q=l 5=/i+l

the asymptotic estimate dimW/ < cT(l + l)d • 2d' and (2.56). D

Based on the factorization formula (2.62), we derive now the multiplication algorithm of

the matrix S by a vector x.

input : blockwise storage of SL — (ä'/lf)o</,/'<l (sparse), x — (xJh+h+-h<L

output: SLx

l for / satisfying X^„=1 lv < L do

2 initialize (SLx)i :— 0

3 for U_ satisfying £^t=i /[, < P do

4 compute 2// := T^k T^h-i • • ?&i •

^

5 update (SLx)L := (5Lx)^ + j«

6 end

7 end

Algorithm 1: Sparse tensor matrix-vector multiplication

Remark 2.25. The order in the product on line 5: of Algorithm 1 is essential for the

efficiency of the multiplication.

Remark 2.26. Due to (2.60), (2.61), the multiplication of 7}jj, by a vector x is done by

multiplying the blocks of x by Sf v .

Wc estimate next the complexity of Algorithm 1.

Theorem 2.27. Algorithm 1 performs the matrix-vector multiplication x — SLx using at

most 0((log NL)kd+2k~2Nl) floating point operations, and it requires only storage of the

stiffness matrix SL corresponding to the case k = 1 (mean field problem) and of x.

Proof. Due to (2.55), (2.62) we write

k

(SLx)L = £ (g) Stu a* = E T&j, 2&fc-i • • T^ •

xL.V_ v=\ a

U1\<L \H\<L

The multiplication under the summation can be performed using at most

k

#/,,:= £nnz(?^) (2.64)9=1

floating point operations. From (2.63) we obtain that

k

#UL < CTA* E(k(D + V* ' * (^-A) + ^ (min(W C(,)) + I)'"1'?=i

• (Cfa+i) + l)d- ' • (C(fc) + !)d • 2*(Sî'*<">+{'*t«)'W+£î=«+i£<.o).

Using the defining property (2.57) of er = ery/ we deduce that for P > 1,

#U' < or,*.* (max (HI, |f I})"*"1 . 2*"<W ^.

The block (SLx)i can be computed using £^, #/(// operations. Finally, the number of

operations needed to perform x — SLx (collect all blocks (8Lx)i for all V) admits the

upper estimate

I H\L\<L \i±\<L

(I+ k — 1\

J < (/ + l)fc_1 solutions

l G Nfe, we conclude

#flops(x -, SLx) < cw £(/ + l)dk+2k~2 2dl < cw(\ogNL)kd+2k-2NL.

D

Due to Proposition 2.19, the number of steps required by the CG algorithm to compute

the discrete solution up to a prescribed accuracy is bounded once we use the solution at

level P — 1 as initial guess of the solution at level P. Thus it holds

Theorem 2.28. The deterministic problem (2.12) for the k-th order moment Mk(u) G

Hs+1(Dk) n H^(Dk) of the random solution u to (1.10) is solvable by a sparse tensor

product FEM with NL dof's in D and for any L > k — 1 at a cost of

cr,kA^gNL)kd+2k-2NL (2.65)

floating point operations, with a memory

CT,k(l°&NL)k-lNL, (2.66)

for a relative accuracy of

ca,u,TdlogNLyk-lV2NZs, (2.67)

where ö — min{p, s}/d.

Note that, up to logarithmic terms, the estimates (2.65), (2.66), (2.67) are asymptotically

as L —> oo similar to those of the case k = 1 (mean field problem). Further, all constants

involved depend exponentially on k.

Chapter 3

Stochastic Coefficient

Under the uniform positivity Assumption 1.8 we investigate equation (1.10),

Aa(w)u(w) = f(uj) in H'1(D) P-a.e. u G Q

using a perturbation argument. We describe the stochastic diffusion coefficient a as a

random fluctuation r around a deterministic expectation (mean field) e and separate the

deterministic and stochastic parts in r by a tensorial representation. Under regularity

assumptions on r(x,u>) in the physical variable x (but not on the size of r, apart from

the standard one ensuring the well-posedness of (1.10)) we prove that arbitrary (weighted)moments of the solution u to (1.10) can be computed in essentially the same complexity

as the solution of one deterministic diffusion problem in physical domain P. Besides, the

resulting algorithm requires only a deterministic solver for diffusion problems in D and it

is partially parallelizable.

3.1 Random Fields

We consider a splitting of the diffusion coefficient into a deterministic expectation e and a

random fluctuation r,

a(x,Lü) = e(x) +r(x,u) V(x,w) G D x Ü (3.1)

with a positive e G L°°(D) (not necessarily equal to / a(.,Lü)dP(uj)),

a

0 < e_ < e(x) < e+ < oo Vx G P. (3.2)

It follows that the random fluctuation r G L°°(D x Ü) too, and we require that the

fluctuation be pointwise smaller than the expectation.

Assumption 3.1. For the representation (3.1) it holds

0 < „ := esssup^11^"'-/11.^^ < 1. (3.3)

27

28 CHAPTER 3. STOCHASTIC COEFFICIENT

Remark 3.2. The constant expectation choice

e(x) := (o_ + a+)/2 Vx G D (3.4)

satisfies Assumption 3.1 with v < (a+ — a_)/(a+ + a_) < 1.

The more natural (from a statistical point of view) choice

e(x):= I a(x,u))dP(uj) Vx G D (3.5)

n

satisfies (3.3) if the density function of r(x, •) is even for any x G D (positive and negative

fluctuations occur with equal probabilities).

Concerning the fluctuation term r we formulate also a modelling assumption as well as a

condition of regularity in the physical variable.

Assumption 3.3. The random fluctuation r can be represented in Lœ(D x SI) as a con¬

vergent seriesoo

r = E t ® Xm (3-6)m—1

with known <pm G C°°(D) (deterministic), Xm G P°°(S1) (stochastic).

The representation formula (3.6) describes the tensor product nature of the random field

r, satisfying therefore the separation ansatz in the deterministic and stochastic variables

x G D and wë!] respectively. A standard representation of this type is the Karhunen-

Loève expansion (see e.g. [Loè77], [Loè78]), and a further example is given below (see

Example 3.6).

The regularity of the random field r is quantified by the convergence rate of the series

(3.6).

Assumption 3.4. The random fluctuation r admits a representation (3.6) for which there

exist constants cr, Ci>r, a > 0 such that

\\4>m ® A"m||LOO(SxS2) < c,. exp(-chrma) Vm G N+. (3.7)

The regularity Assumption 3.4 holds for instance for random fields r which are analytic

in the physical variable, with values in L°°(Q). This follows from standard approximation

theory of analytic functions (see e.g. [Dav63]).

Definition 3.5. For an arbitrary Banach space P and an open set U Ç Cn we denote by

A(U, B) the space of P-valued complex analytic functions on U which are also continuous

on the closure of U. Note that A(U, B) equipped with

MU(t/,i?) := sup ||w(x)||ß V22 G A(U,B)xeu

becomes a Banach space.

3.1. RANDOM FIELDS 29

Example 3.6. If D C [-1, l]d and r G A([-l, l]d, P°°(f2)), then a representation (3.6)exists with (0m)meN+ the Legendre polynomials in [—1, l]^ (tensor products of standard

Legendre polynomials in [—1,1], scaled to have P2-norm equal to 1) and

Xm(uj) := / r(x, Lü)4>m(x) dx P-a.e. lü G Ü, Vm G N+. (3.8)

H,x]d

Moreover, (3.7) holds with a — \jd and c^T depending on the size of the analyticity domain

of r in a complex neighbourhood of [—1, l]d.

Remark 3.7. By knowing Xm for m G N+ we understand that any mixed moment of any

finite subset of the random variable family (ATm)m6N+ is (computationally) available. For

instance, for the construction presented in Example 3.6, the analyticity assumption on r

ensures that the integral on the r.h.s. of (3.8) can be estimated using a quadrature rule

(xq,wq)qQ in [-1, l]d. It follows

Xm(-) ^ Er(X9' ')<t)rn(Xq)wq. (3.9)geQ

Given the realisations (a(-,a;))i„eno) an arbitrary k-th order mixed moment of the family

(Xm)mçN+ is then estimated by the mean

fxmi(u;)Xm2(uj)...Xmk(uj)dP(u;) ~ -L J^ fl (E7^'^"»»« ) . (3.10)

Remark 3.8. The following arguments, proofs and algorithms carry over to the case of a

random fluctuation r satisfying Assumption 3.3 only locally on D (i.e. if a representation

formula (3.6) holds locally on a finite covering of D). Moreover, a simple scaling argument

shows that we can assume w.l.o.g. from now on that D C [—1, l]d.

Since computations can handle only finite data sets, we truncate the infinite expansion

(3.6) of r and introduce for any M G NU {oo} (convention: r0 — 0) the M-term truncated

fluctuationM

Tm :=E^"® Xm> aM -=e + rM. (3.11)

The error introduced thereby in the solution of (1.10) is then controlled due to the Strang

Lemma.

Proposition 3.9. 7/ Assumption 3.4 holds, then there exists MT G N such that for any

M > Mr the problem of finding uM G L°(Q, H^(D)) such that

AaMMwMH-/(w) inH~l(D) P-a.e. LueÜ (3.12)

is well-posed and

\\u(lo) - um(u)\\hI(d) < ce,rex.p(-chTMa)\\f(cu)\\H-y(D) P-a.e. u G Ü. (3.13)

Remark 3.10. If (Xrn)mG^+ is an independent family of random variables with vanishing

mean (e.g. for the (3.5) choice of e), then Mr can be chosen equal to 0, due to the fact that

problem (3.12) becomes elliptic uniformly in M G N. This can be easily seen by taking

the conditional expectation of (1.15) w.r.t. the <r-algebra generated by X\, X2,..., Xm-

Throughout this section we will constantly make use of the mixed moments of vector valued

random fields, which are defined as follows.

Definition 3.11. If H is a separable Hilbert space, j G N+ U {oo}, p = (pi,p2, ,Pj) G

Lk(ü, H)3 and n — (nj, n2,..., n3) G W with ||n||i :— 72i + n2 H \-n3 — k, then

Mn(p) := pi(Lü)®--.®p1(u)®...®pj(uj)®---®pj(u;) dP(uj) G II ® ® II (3.14)J "

v' *>

vS

"V

'

nni times n3 times k times

is called the mixed moment of p of order n.

Further notations will be 13 := (1,1,..., 1) and, for m G N^_, Xm :— (Xmi,Xm2,..., Xmj).

j times

3.2 Discretization

Considering the hierarchical FE space sequence (Vl)lêN given by (2.18) and satisfying As¬

sumption 2.11, we formulate for any M G NU {oo} and any FE discretization level P G N

the discrete counterpart of (3.12),

VL : Find uMiL G L°(Ü, VL) such that

ABM(w)UW,L(w) =/(w) in VI P-a.e. ue Ü. (3.15)

From Assumption 3.4, PL is well-posed for M > Mr.

Assuming that the conditions ensuring a shift theorem at level 5 for the diffusion problem in

D with coefficient a(-, lo) and data f(-,oj) are satisfied uniformly in lo G Ü, the discretization

error satisfies the standard estimate.

Assumption 3.12. There exists s > 0 such that a G L°°(Q,CS(D)) (following e.g. from

e G CS(D) and r G A(D, L°°(U))), and dD G Cs+1, so that the deterministic diffusion

problem in D with coefficient o satisfies a shift theorem at level s > 0 (see, e.g. [GT01]).

Proposition 3.13. /// G P0(^,P"1+S(P)) and Assumptions 2.11, 3.4, 3.12 hold, then

\\um(lü) - uMiL(uj)\\hi(D) < caJ\f(uj)\\H-i+siD)$(NL, s) P-a.e. u G Ü, (3.16)

and for any M > Mr, P G N. Here the functional $ quantifies the approximation property

of the scale (VL)LeN (see also Example 2.12).

3.3. NEUMANN SERIES DECOMPSITION 31

3.3 Neumann Series Decompsition

Considering / G P°(fi, H~1(D)) and the hierarchical FE space sequence (2.18), we formu¬

late for any M G N U {oo}, M > Mr and any FE discretization level P G N the following

sequence of well-posed problems,

VL : Find um,l e P°(fi, VL) such that

&aM(u)UM,L(u) = f(u) in VI P-a.e. lo G fi, (3.17)

V0il Find v0tL G P°(fi, VL) such that

AeVo,LM = /H in VI P-a.e. w G fi, (3.18)

and for jf G N+

^,l : Find v3tL G P°(fi, KL) such that

Aevj<L(uj) = diY(rM(oj)Vv3-ltL(uj)) in Vt P-a.e. u G fi. (3.19)

Using basic P1 estimates for the diffusion problem in D we show next the exponential

decay of vjiL, the solution to (3.19), in the energy norm as j —> oo, and the exponential

convergence of the series with general term v3iL to um,l solution to (3.17). To this end, we

introduce, for any e G C,

aw.« := e + erm G P°°(P> x fi) (3.20)

and study the diffusion problem with stochastic coefficient üm,<:-

Proposition 3.14. Under Assumptions 3.1, 3.4, there exists 8U > 0 such that the problem

of finding uM,L,t G P°(fi, Vl ® C) with

~Km^)Um,l,z(u) = /H m^ P-a. e.wGfi (3.21)

is well-posed uniformly in M > Mr, P G N and e G S where S denotes the complex strip

S:=[-l-<5„,l + <S,,]xzRcC. (3.22)

Proof. For notational ease we drop u from all notations. The real and imaginary parts

^um,l,(. G Vl and ^Sum,l, Vl should solve P-a.e. a; G fi the system equivalent to (3.21),

f &maMßuM,L* ~ A$aMlßUM,L,e = f m V£ .

23.

\ AaaM£SRuM,£,£ + AsROM)eöuM,L,£ = 0 in V£

Choosing an ONB in Vl (equipped with the Hq(D) norm), (3.23) becomes a linear system

whose matrix A has the form

with Ai G RNlxNl the real-valued symmetric matrix associated to the bilinear form of

AsßaMt

in Vl, and A2 G RNlxNl the real-valued symmetric matrix associated to the bilinear

form of AoaMe in Vl- Noting that

5RaM,e = e + (3fc)rM (3.25)

is bounded away from zero for e in 5 given by (3.22) with 5V small enough we deduce

that Ai has (positive) spectrum, bounded away from 0 uniformly in lo G fi, P G N and

M >Mr. Since

\ ( °ß ) ' ( °ß ) ) = {Aia,a)RNL + ^l/3'^ (3.26)

we obtain that the spectrum of A is bounded away from 0 uniformly in lo G fi, P G N and

M >Mr. D

Proposition 3.15. Under Assumptions 3.1, 3.4 ,the mapping

S 3 e ^ %,v(w) G H^(D) (3.27)

imï/i Mm,i,£ solution to (3.21) is holomorphic P-a.e. in ft for any M > Mr, P G N, with Mr

as in Proposition 3.9.

Proof. Fixing e0 G S, we show that UM,L,e is holomorphic as a function of e in a neighbour¬

hood of eo by explicitly constructing the solution of (3.21) as a Taylor expansion around

e0. To this end we consider the following sequence of well-posed problems

AaMieoMuM,L,eoH = /(<*>) in VI P-a.e. u) G ft, (3.28)

and for j G N+

A«Af,.0M^,L,eo(w) = div(rM(w)Vvi-i,L,eo(w)) in V£ P-a.e. well, (3.29)

where by convention v0,l^0 :~ uM,L,e0- From (3.29) with v3tL,e as test function we deduce

that, P-a.e. in fi it holds,

||V^,e»||L2(Z)) < ^^^llVvj.^MllLHo) VjGN+. (3.30)

Note that the existence of c£0 > 0 is ensured by Proposition 3.14. From (3.30) we obtain

the exponential decay of |p|,7'||wj),L,eo(u;)llfl'1(z>) with j —> oo, uniformly P-a.e. in fi for p in

a small complex neighbourhood of 0 depending on e0 and ||r||x,»(£,xn). In particular the

series

E^^'oM (3-31)

converges in E\(D), uniformly P-a.e. in fi for p in a small complex neighbourhood of 0.

Moreover, the sum solves the diffusion problem with stochastic coefficient a^tü+p, as one

can easily see by summing (3.28) and (3.29) for j G N+. The value of (3.31) is therefore

equal to UM.L.eo+p^) -P-a.e. in fi, which concludes the proof. D

3.4. DISCRETE MIXED MOMENT PROBLEM 33

The domain of analyticity of the mapping (3.27) contains therefore the closed disc of center

0 and radius 1 in the complex plane, which ensures that the distance between eo — 0 and

the singularities of (3.27) is strictly larger than 1. As a consequence, the Taylor expansion

of (3.27) around eo = 0 converges exponentially in p — 1.

Proposition 3.16. If f L°(ft,H~1+s(D)) and Assumptions 3.1, 3.4 hold, there exists

cli0 > 0 such that

\\vj<L(uj)\\Hi{D)<ca\\f(uj)\\H-i(D)exp(-clAj) P-a.e. lu G fi, (3.32)

and

n

\\um,l(^) - E'^L^Hffo(ß) - cJ/Mlliï-H-D) exp(-CiiQn) P-a.e. lu G fi, (3.33)3=0

and for any M > Mr, j,n,L G N.

Proof. Taylor expansion of (3.27) around eo = 0 gives,

um,l,p(uj) = ^p'v3;l,o(uj) (3.34)

so that

v3,l,o(uj) = j$um,l,z(u)\c=0. (3.35)

Noting that VjtL,o — 'v3\l for any j G N+ and um,l,i = um,l, all we have to check is the

exponential convergence in Pq(P>) of the series on the l.h.s. of (3.34) for p = 1. But this is

equivalent to the exponential decay of Vjtii0 as j —» oo in the Hq(D) norm, which in turn

follows from (3.35) and the Cauchy representation formula

diuM,LMl=» = ^ï] z3+ldz V? G N

r

applied on an arbitrary circle F of center 0 and radius strictly larger than 1, contained in

the strip S. D

3.4 Discrete Mixed Moment Problem

Since our primary aim is the development and the analysis of an algorithm based on the

Neumann expansion (3.18), (3.19) for the computation of the simplest (first/second order)

statistics of it solution to (1.10), let us collect now in the next result the errors introduced by

the discretization steps of the previous sections (modelling, FE discretization and Neumann

truncation errors, see (3.13), (3.16) and (3.33)).

Proposition 3.17. /// G ^(ft^^+^D)) and Assumptions 3.1, 3.4, 3.12 hold, then

M") ~^2vjiL(uj)\\Hi{D) <

3=0

< ca,s\\f(uj)\\H-i+s(D)(^NL,s) + exp(-chTMa) + exp(-chan)) P-a.e. u G fi, (3.36)

for any M > Mr, n, L G N.

Note that, after averaging (3.36) over lj G fi and balancing the truncation orders M,n

and the discretization level P, the computation of the mean fields M.1(v3,l) for 0 < j < n

produces also an approximation of M.l(u). However, computing the moments M1(vj!l) for

0 < j < n is not an obvious task. Under additional regularity assumptions on the random

fluctuation r, we develop next an algorithm which makes this computation possible in nearly

optimal complexity (compare Definition 1.1). It turns out that this computation has to be

based on knowledge of higher order mixed moments of the pair (tm , /) • Assuming these

are available, we derive a general algorithm for the computation of the mixed moments

associated to the pair (rjh, u) (and not only of A^1(m)). This algorithm will be later used

in the context of second (and higher) order moment computation.

Assumption 3.18. There exists an open complex neighbourhood U Ç Cd of D such that

Assumption 3.4 is satisfied and the decay estimate (3.7) holds in L°°(U x fi) (in particular,

4>m G A(U) for all m G N+).

Remark 3.19. Assumption 3.18 implies r G A(U, P°°(fi)) and the convergence of the

series (3.6) in A(U, L°°(ft)). Reciprocally, if r G A(U', P°°(fi)) with V c U' Q Cd (which

holds e.g. for a diffusion coefficient a with Gaussian covariance), then Assumption 3.18 is

satisfied with (3.6) the Legendre/Karhunen-Loève expansion of r (compare Example 3.6).

Definition 3.20. If A" is a bounded linear operator between two Banach spaces B\ and

P2, and U Ç Cn is an open set, let A(U, Pj) be the Banach space of P^-valucd functions

which are continuous on U and analytic in U (i = 1,2). We define Idu <S> X to be the

bounded linear operator given by

Idu ® X : A(U, Pi) -> A(U, B2), ((ldu®X)h)(z)--X(h(z)) Vz g V.

Remark 3.21. The direct sum of d copies of the trace operator S defined in Lemma 5.1

induces by restriction to functional spaces on U a bounded linear operator denoted for

notational ease again by S,

S : A(U, L2(D)d) - L2(D)d, (ß(u)){x) - u(x, x) A-a.e. x G D. (3.37)

We formulate next for j G N+ arbitrary the problem to which the mixed moment of

(rMi'Vj,L) of order (j, 1) is solution. To this end we introduce the notation

A^ : H~\D) -> Vl (3.38)


for the bounded FE solution operator of the homogeneous diffusion problem with coefficient

e corresponding to the FE space Vl equipped with the Hq(D) norm.

Further, for any U Ç Cd and any j G N+ we define the product domain

U3:=UxUx-..xUc C3d.s

v'

j times

Definition 3.22. For any j G N+ and L G N we define the operator T3j4 as the following

composition of linear operators,

A(U^ Vl) ^22îZ A(W, L\D)d) Id"~10S> A(W-\ I?(D)d) Id^-l8div1

Id--10diV> A(U3~\ H-\D)) Id"-l8Aä, A{U3-^ yLy (3.39)

Similarly, for any j G N and P G N we define the operator Sj,l given by

Id,,., ® A~L

A(U3, H~\D)) — ^ A(U3, VL). (3.40)

It then holds

Proposition 3.23. The operator T^l defined by (3.39) is bounded uniformly in j, L and

maps M^'^^m^Vj^l) into M^~l,l\rM, Vj'+i.-l) for any f G N.

Similarly, the operator S3tL defined by (3.40) is bounded uniformly in j, P and maps M^(rM, f)

into M^l\rM,VQiL).

Proof. Note first that the mixed moments have been introduced in Definition 3.11. Observe

next that the second operator in (3.39) is bounded uniformly in j due to Lemma 5.1 (see

Appendix) and Definition 3.20, while the others are trivially continuous, uniformly in P.

The mapping property follows by taking the tensor product of Pj'+i,l (see (3.19)) with

I'm(w) ® • • ® rju(w) (j — 1 times) and averaging the resulting equation over lo G fi-

The mapping property of S3tL follows then analogously, by taking the tensor product of V0,l

(see (3.18)) with rM(u) ® ® rM(Lo) (j — 1 times) and averaging the resulting equation

over a; G fi.

Proposition 3.23 shows in particular that the first order moment (expectation) A41(v3>l) =

M^^(rM-,Vjti) follows by successive composition of operators of type (3.39) from the

higher order moment M^'^(rM,v0,L)- This in turn is obtained by mapping M(3'^(rM,f)via SjtL- The discrete problem solved by Jv^^Uj^l) is therefore given by

Corollary 3.24. For any j,f G N it holds

M{3'>l\rM, vJtL) = (Tr+1,L o Tr+2,L o • • • o T3,+3,l o Sr+3tL)(M^+^(rM, /)). (3.41)

In particular, for j' = 0,

M\vjtL) = (TltL oT2iLo..-o Tj,L o S3j)(M^(rM, /)). (3.42)

Note that (3.41) can be viewed as a recursion of parametric deterministic diffusion problems

in D where after each recursion step taking the trace via S reduces the dimension of the

parameter space by d.

3.4.1 Mixed Moment Algorithm

From Corollary 3.24 and Proposition 3.17 it follows that the computation of the exact

mean field Ml(u) requires knowledge of the mixed moments jM^'^rM, /) f°r j G N. The

more such moments are available, the higher the truncation order of the Neumann series

and the better the expected approximation of Ml(u).

Concerning the mixed moments M.^3'^(rM, f) we have the following representation formula,

and estimates following from Assumption 3.18.

Proposition 3.25. If j G N and r satisfies Assumption 3.18, then for

M^l)(rM, f) = JrM(uj)®...®rM(Lo)®f(Lü) dP(uj) G A(W, H~l(D)) (3.43)

ü j times

it holds

M^(rMJ)= J2 M^^X^f)®^, (3.44)men',

where for any index m = (mi, m2,..., m3) G N+, UniH^ :— max^^ \mk\,

</>m := <t>my <8> (pm2 <S> • • ® mj G A(U3), (3.45)

and (in view of Definition 3.11)

M^(XmJ) = JXmi(Lo)Xm2(to) Xrnj(Lo)f(uj)dP(uj) G H-\D).n

Further, the general term of the expansion (3.44) satisfies

||A^lj+1(^m,/)®0m|U(w,H-»(D)) < cjexp(-ci,r||m||S)||/|Ui(n,H-i(D)) (3-46)

with ||m||a :— (m" + rn2 + • + rap1/".

Equation (3.44) and the definitions of all operators involved in (3.42) enable us to derive

the following algorithms for the computation of the mixed moments of the pairs (tm,v3^l)and (rM,UM,L) respectively, once a solver for the deterministic diffusion problem in D is

available.

input : • M, j,f, L

• approximation of M.^'+3'^(rM, f) of the form

M%£j*\rM,f)= J2 h^^ with hm G H-\D)

% e.g. hm := M1i'+>+1(Xm, f) for exact representation of M^'^'^rM, /)• solver of diffusion problem in D with deterministic coefficient e and

FE space VL

output: • approximation of M^'^^m,^^) of the form

M%£)(rM,VjiL) = Yl Srnfa

l|m|l»<M

with gm G Vl

1 compute for all m G N3++3, Hm^ < M

gm- ^Lhm G VL

2 for k — j' + j : — 1 : f + 1 do

3 compute for all m G N+ x, HmH^ < M

9m := A"i I div ^ </>mV^(m,m) 1 G Vl

M

m=l

4 end

5 compute and return

T

Algorithm 2: M%pX)(rM,v3tL) = Algorithm 2 MJ,f,L,M%£j,1)(rM,f)

input : • M, n,f, L

• approximations of M^'^'^^m, f) for all 0 < j < n, of the form

Mïp^W/) = E ^- with hmeH~1(D)

% e.g. hm :— M1''+'+1(Xm,f) for exact representation of M{-3'+3,1\rM, f)

• solver of diffusion problem in D with deterministic coefficient e and

FE space Vl

output: • approximation of M^3 '^(r^, um,l) of the form

M%£l(rM, uMiL) = Y 9ui(S>r.

||m||00<M

with gm G Vl

l compute for all j — 0 : 1 : n

M^W^l) = Algorithm 2 [m, j,/,!^^1'^,/)


M^^M, UMtL) := J2M%p](rM, HU3=0

Algorithm 3: M%pl(rM,uM,L) = Algorithm 3 M, n,f, L, (Xip^J,1)(rM, f))0<3<

Proposition 3.26. For Algorithm 2 it holds

M%V(rM, v3tL) = (Ty+ltL o Ty+2,L o • • o Ty+3tL o S3,+3,L)(M%;3>lXrM, /)), (3.47)

so that

||A4(/,1W vhL) ~ M%vl)(rM,v3,L)\\A{w',ul{D)) <

< Ca\\M«+3»(rM, f) - M%^\rM, f)\\A{U3'+J,u-m). (3.48)

Proof. Estimate (3.48) is a direct consequence of (3.47) and Proposition 3.23, so that it

suffices to prove (3.47). To this end, we simply note that, in view of (3.40), applying S3>+3,l

to A4^ppJ,1(rM, /) corresponds to step 1: of the algorithm, whereas (3.39) ensures that step

3: consists of successive applications of the operators T3>+]jl, 2y+:?_li/J,..., Ty+])/-,. D

Proposition 3.27. For Algorithm 3 it holds

II.M^VM.UM.i) - MU£l(rM,uMtL)\\A{uriHèm <

<ca||/||Li(n,/f-i(D))lkM||i00(i7xn)exp(-ci,an)+n

+ EàLWMWWHm, f) ~ M%^l\rM, f)\\A{ui'+itH-HD)). (3.49)3=0

Proof. The l.h.s. of (3.49) can be estimated from above using the triangle inequality by

n n

\\MW(rM,uM,L - J>,l)|| + E \\M{j''l)(rM,v3jj) - M%P(rMivi%L)\\ (3.50)3=0 3=0

(all norms in A(U3 , Pq(P))). The conclusion follows then from the error estimates in

Propositions 3.16, 3.26.

One important property of the mixed moments which can be immediately used to reduce

the complexity of Algorithms 2, 3 is their symmetry w.r.t. m. To describe it, we define on

N+ the equivalence relation

m ~ m' <é=^ 3cj- permutation of {1,2,..., j} s.t. m'k = ma(k) VI < k < j, (3.51)

so that for any m G N^_ the orbit 0(m) of m is given by

ö(m) :— {m! G NJ+ | 3a permutation of {1, 2,... ,j} s.t. m'k — m^k) VI < k < j}.

Due to Definition 3.11 it then clearly holds

Remark 3.28. The mixed moment Mlj+1(Xm, f) is symmetric in m, that is,

m~m' = M1^(Xm,f) = M1^(Xmr,f)eH~1(D). (3.52)

Remark 3.28 shows that reasonable approximations of the mixed moments of the pair

(rM, f) used as input for Algorithm 2 should also satisfy the symmetry condition w.r.t. m,

m - m' =* hm = h^ G H'l(D). (3.53)

A simple backward induction argument on k shows then that this symmetry is preserved

by step 1: and the recursion in step 3: of Algorithm 2.

Proposition 3.29. If the input data (hm) m6NJ'+^ is symmetric w.r.t. m, then gm givenIMI»<M

by steps 1: and 3: of Algorithm 2 is also symmetric in m,

m~m' =» 9r* = 9m>ZVL<zHl{D). (3.54)

Remark 3.30. The complexity of the mean field computation using Algorithm 3 with

f — 0 and exact input data hm — M.lj+1(Xm, f) is in general superalgebraic. This follows

by a simple comparison error estimate (3.36) vs. total number of deterministic problems

which are to be solved in steps 1: and 3: of Algorithm 2. After choosing

M ~ P1A\ n ~ L (3.55)

in order to balance the contributions of the discretization steps to the total error estimate

(3.36), Algorithm 3 requires solving (even after taking into account the symmetry of the

t/m's) at least

^OKl/a-DIog^+l) (356)Mn

deterministic diffusion problems in D at level P (already in step 1:, Algorithm 2 with

j = n) for an accuracy of 0(N£S). Recalling that a = 1/d for analytic regularity of

the perturbation r (see also Example 3.6), (3.56) shows the superalgebraic complexity of

Algorithm 3 in dimension d > 2. In view of the algebraic (quadratic) complexity of the

Monte Carlo method, Algorithm 3 becomes then too expensive to use. Note however that

a subexponential complexity upper bound can be easily shown for Algorithm 2 by counting

arguments similar to those loading to (3.56).

The high complexity of Algorithm 3 can be significantly reduced by taking into account

the analyticity assumption on r, through the estimate (3.46). We show that representation

(3.44) can be sparsified, that is, many terms can be neglected (due to their small size as

estimated by (3.46)), without affecting the total discretization error estimate (3.36). We

consider thus for any j G N an index set (the explicit construction will follow in section

3.4.2)

T3 C {m G W+ | llmlloo < M} (3.57)

(convention: N^ — {0},T0 :— {0}) and apply Algorithms 2, 3 using as input the approxi¬

mation of MS1' +3'^(rM, f) given by

M<£*1KruJ):=M%$l\ru,f)= £ hm ® <f>m, (3.58)

which amounts to simply setting the input coefficients hm :— 0 for m ^ Ty+J-.

Defining

M%^(rM, v3,L) := (7>+i,l o 7>+2,L o •. o 7>+,,L o Sr+^)(M%^1](rM, /)), (3.59)

we rewrite Algorithms 2, 3 corresponding to the truncation index sets (Tj)j£n (sec Algo¬

rithms 4, 5).

Concerning notations, we introduce for any j, j' G N with j > f the projection operator

onto the first f coordinates, U3i3> : N3+ — N3+, and for any index set r C N3+ we put

U.jy(T) := |J njy(m) = {m' G W+ \ 3m G T s.t. m'k = mk VI < k < j'} Ç Nj.mer

Remark 3.31. The approximation error estimates for Algorithms 4, 5 are the same as

those given in Propositions 3.26 and 3.27, with approximate moments MaVP replaced by

sparsified approximate moments Ma.pP}r-

By sparsifying the moment representation using a convenient index family (Tj)je^ we gain

a substantial complexity reduction of Algorithm 3 (number of deterministic problems to

be solved in step 1:, see also Remark 3.30), without deteriorating the error estimate (3.49).

This will be shown in section 3.5, after the explicit construction of the index sets (T3)3£n-Here we only observe that if the index sets (Y3)jem are compatible with the equivalence

relation ~, that is,

meïj^ 0(m) Ç T3 \fj G N, (3.60)

and with the projection operators IT, that is,

n.yOf,-)^ Vj,j'N,j>f, (3.61)

then the following complexity estimate for Algorithm 4 is a consequence of Proposition

3.29, which applies also in the case of sparsified input data.

Proposition 3.32. If the index family (T3)je^ satisfies the compatibility conditions (3.60),

(3.61); then Algorithm 4 requires solving \T3'+3/ ~ | + \T3i+j_i/ ~ | + |Tj/+J_2/ ~ | H (-

IT?'/ ~ I deterministic diffusion problems in D at FE discretization level L.

3.4.2 Sparsification

For a fixed M G N and any j, q G N we define the sparsification index sets

Tjjq :— {m G N+ | ||m||oo < M and m has less than q different entries }. (3.62)

Proposition 3.33. Under Assumption 3.18 there exists c2<r > 0 such that for any j, q G N

it holds

|| J2 M^(Xm, f) ® Kh^WM-HD)) < 4eM-C2,rq1+a)\\f\\mn,H-HD). (3.63)

l|m|loo<M

Proof. In view of (3.46) it suffices to estimate (convention: empty sum equals 0)

Sj,q-=4 E exP(-<Vllmlla)- (3-64)

l|m|!oo<M

To each m G N3+ with HmH^ < M we associate the vector n— (rii,n2,... ,Um) G NM

defined by

VI < m < M nm:= p <=$> m has p entries equal to m. (3.65)

Then ||n||i := ni + n2 + • + Um = j and for any n with ||n||i — j there are exactlyfi

indices m G N+ with ||m||oo < M to which n is associated via (3.65).72i!n2!- .-riM

Moreover, we have

m G Tjj9 <=$> nm > 1 for less than q different values of m taken from {1,..., M}.

(3.66)With this notation, the sum S3tq in (3.64) can be written

M

SJ,Q = 4 E ,/' rexp(-ci,r Y]nmma),

neN-M

llnlli^j.lsuppfn)!^«

77,1!n2!

• -nM!nW« L ^

m=l

or, parametrizing the indices n through their support and the values they assume there,

min{j,M}.,

i

5M=4 E E E \3..n i^kE^O' (3-67)î=9 l<mi<-<m,<M neN^,||n||i=j

mi '

'

*"*' fc=l

supp(n)={m1, ,«!,}

The general term of the sum (3.67) can be estimated from above by

(nmi-l)!.">mt-l)!eXp(-Cl'-E(^ " ^TJ^xp^g^). (3.68)

Using (3.68) in (3.67) and performing the innermost sum via the multinomial formula gives

Sj,q < 4^ E (Eexp(~Cl>rmfcM P »!expKrE<)i>q l<mi<--<m,<M \k=l ) ^ ' k=\

* ^E^_i(i)<! E «?(-<*,, £>g). (3.69)«>9 l<mi<-"<m2<M k—\

Due to m/t > k for any 1 < k < i we absorb the factorial in the exponential under the

inner summation and obtain (with any 0 < c2<v < ci,r)

min{j,M} ,

.^s

s3« < 4 E *\M E cxP(-C2,rx;ot=g

^ 'l<mi<---<m,<M fc=l

< c3. max V) exp(-c2,rVm£). (3.70)l<mi<-<m,<M A;=l

The conclusion follows then from Lemma 5.2 (see Appendix). D

Proposition 3.33 shows that the sparsification error in the mixed moment representation

(3.44) with the index set TJ>g (3.62) is just as large as the largest neglected term, up to

the constants involved.

A further obvious property of the index sets (Tm)j£n is their consistency with the equiv¬

alence relation ~ and with the projection operators II in the sense of (3.60), (3.61). The

number of equivalence classes in TJt9 w.r.t. ~ can be bounded using a simple counting

argument.

Proposition 3.34. The number of equivalence classes in Tji9 satisfies the bound

rw~i<(M+i)«o'+i)'. (3-71)

Proof. Since every equivalence class contains exactly one element with non-decreasing entry

sequence, we need to count (via (3.65)) the distinct solutions n = (rii, n2,..., Um) G NM

of ri\ + 77-2 + ... + riM — j with nm > 1 for less than q values of 1 < m < M. We obtain

iw-i-E(T)(2-î)-(M+iro'+irD

3.4.3 Complexity

The computational effort required by Algorithm 4 with sparsification index sets given by

(3.62),

r3-T3,q V;GN, (3.72)

and with q G N to be chosen later follows directly from Propositions 3.32 and 3.34.

Proposition 3.35. Algorithm 4 consists in solving at most (j + j' + 1)9+1(M + l)9 deter¬

ministic diffusion problems at FE discretization level L.

Combining Propositions 3.27, 3.33, 3.35 we obtain the following error and complexity

estimate for Algorithm 5 below with exact input data,

hm := .MV-m+1 (Xm, /) Vm G r3,+j, VO < j < n. (3.73)

and sparsification index sets given by (3.72). We denote in the following the output of

Algorithm 5 below with exact input data (3.73) by

M{4'V(rM,uM,L). (3.74)

Proposition 3.36. /// G P1^, P-1(P0) and Assumptions 3.1, 3.18 hold, then for the

solution um,l G Ll(ft, Vl) to (3.17),

^aM{uj)UM,L(u) ~ /(w) in Vl P-a-e. lo £ ft (3.75)

it holds (w.l.o.g. ca > 1)

\\Mb'>l)(rM,UM,L) -M^^M^M^hiW^iD)) <

< cJ/||Li(n,H-i(D))(lkM|li'oo(c/x0) exp(-c1)a72) + c{+nexp(-c2trq1+a)). (3.76)

Moreover, M4'n (rM,UM,L) can be computed starting from M^J(rM, f) for all 0 <j<

n via Algorithm 5 below at the cost of solving at most (j' + 72 + l)q+2(M +l)q deterministic

diffusion problems in D at FE level L.

3.5 Mean Field Computation

From Propositions 3.17 and 3.36 with f — 0 we obtain

Proposition 3.37. /// G P1^, H~1(D)) and Assumptions 3.1, 3.18 hold, then

\\M}(uMiL) - M^^Um^WhUd) < Ca||/IUi(n,H-i(D))(exp(-Ci,an) + c^exp(-c2,ro1+a)).

(3.77)

//, in addition, f G LS(ft, H~1+S(D)) and Assumption 3.12 holds too, then

\\MHu) ~ M^n(uMiL)\\HèiD) <

< c0,a||/||Li(n,H-1+'(D))(exp(-c1,rMa) + $(NL, s) + exp(-ci,„n) + c>xp(-c2,rg1+Q))

(3.78)

for any M > MT, n,L,qe N.

Moreover, M\^(um,l) can be computed starting from M^ (^m,/) for all 0 < j < n

via Algorithm 5 at the cost of solving at most (n + l)q+2(M + l)q deterministic diffusion

problems in D at FE level P.

Note that the four terms on the r.h.s. of (3.78) correspond to the four discretization steps

(modelling assumption, FEM in D, truncation of Neumann series, and mixed moment

sparsification for the pair (r^f,/) respectively). Balancing the discretization errors in

(3.78) by the (unique up to multiplicative constants) appropriate choice of parameters

M~P1/a, n~L, <7~P1/(1+a) (3.79)

and comparing with the computational effort estimate obtained in Proposition 3.37 we

deduce (assuming also exact integration for all r.h.s.'s in step 3: of Algorithm 4),

Theorem 3.38. Under Assumptions 3.1 (well-posedness), 3.12 (uniform data regular¬

ity), 3.18 (analytic perturbation), the mean field of u random solution to (1.10) with

f G Ll(ft, H~l+S(D)) is numerically computable using Algorithm 5 at nearly the same

cost as one deterministic diffusion problem in D.

If the available deterministic solver in D needs 0(e~@) floating point operations and mem¬

ory to reach an accuracy e for one deterministic diffusion problem with coefficient e and

Hs regular source term (usually ß = l/«5 + o(l) with Ô = min(s,p)jd), then Algorithm 5 re¬

quires at most 0(e~@~°^) operations and memory to achieve the same accuracy £ —» 0 for

M1^) in the case of stochastic data. Algorithm 5 has therefore nearly optimal complexity

in the sense of Definition 1.1.

Proof. Choosing the discretization parameters M,n,q as in (3.79), the number of deter¬

ministic diffusion problems in D to be solved at FE level P (and given in Proposition 3.37)

admits then (up to a multiplicative constant) the upper bound

Lo(LVU+*>) < exp(0((logiVL)1/(1+^ loglogNL)) < N°L(1) as P -> oo, (3.80)

3.6. COMPUTATION OF HIGHER ORDER MOMENTS 45

which concludes the proof. D

Algorithms 4, 5 turn out to be also amenable to (almost full) parallelization.

Remark 3.39. The computation in step 1: of Algorithm 4 is fully parallelizable w.r.t.

m G Tj'+j. The computation in step 3: of Algorithm 4 is also fully parallelizable w.r.t.

m G Ilj'+3tk_i(Yji+j), but sequential w.r.t. k = / + j : — 1 : f + 1 (that is, j serial steps

are to be taken). Note that, due to (3.79), j < n ~ P ~ — loge, where e is the prescribed

accuracy to be reached in the mean field/higher order moment computation.

Further, the computation in step 1: of Algorithm 5 is also fully parallelizable w.r.t. j —

0 : 1 : n.

Remark 3.40. All results and algorithms derived for the mean field computation can

be easily generalized to the case of arbitrary first order statistics (weighted expectations,

obtained by replacing the probability measure P by a new one Q such that dQ/dP exists

in P°°(f2) and it is known).

3.6 Computation of Higher Order Moments

Based on the results obtained in Chapter 2 (stochastic source term), on the mean field

computation Algorithm 5 and under the regularity Assumption 3.18 we show that also

higher order moments of the solution u to (1.10) can be computed in almost the same

complexity as the solution of one deterministic diffusion problem in D.

We formulate first a continuous moment problem which we subsequently discretize and

decompose as direct sum of several mean field problems. These are in turn solved using

Algorithm 5.

Remark 3.41. In the following we discuss (for simplicity and since correlation and variance

kernels are often more relevant than higher order moments) only the computation of the

second order moment (k — 2). Corresponding results hold for all higher order moments

and can be derived in a similar manner.

For / G L°(ft, H~X(D)) we recall that the diffusion problem (1.10) with stochastic data

consists in finding u G L°(ft, Hq(D)) such that

Aa(lJ)«(w) - f(oo) in H~\D) P-a.e. toZft. (3.81)

After truncating the fluctuation expansion (3.6) at level M (see also (3.11)) we are left

with the problem (3.12) of finding uM G L°(ft,H^(D)) such that

Km{.)Um(^) = /M in H~l(D) P-a.e. a; G £1 (3.82)

From Proposition 3.9 we immediately obtain

Proposition 3.42. If Assumption 3.4 holds, then for any M > MT

\\u(lo)®u(lo)-um(lo)®um(uj)\\h^d^) < Ce,r,a-ex-P(-ci,rMa)\\f(Lo)\\2H-HD) P-a.e. ueft,

(3.83)

so that

\\M2(u) - .M2(22M)IUw ^ c*,r,a- exp(-c1<rMa)\\f\\h{n<H-HD)y (3.84)

We consider also the diffusion problem with constant coefficient 1 and stochastic right hand

side / G L°(ft, H~l(D)), whose solution we denote by / G L°(ft, H£(D)),

Af(u) = f(u) in P-'(P) P-a.e. lo G ft. (3.85)

We formulate next a continuous variational problem, to which um(-)®um(-) G P°(^, H^(D2))

is the unique solution. This problem will be in turn fully discretized. By Id#i(£)) we denote

in the following the identity operator acting in Hq(D).

Proposition 3.43. P-a.e. in ft it holds,

(A®A)(f(Lo)®f(uj)) = f(Lo)®f(uj) inH-\D)®H~l(D), (3.86)

(AaM(w) <g> ldm(D))(uM(u>) ® f(uj)) = (A ® IdHm)(f(Lo) ® J(lo)) in II~l(D) ® H^D),

(3.87)

(IdHi(D)®AaM(ij))(uM(Lo)®uM(üo)) = (ïdHi{D)®A)(uM(iu)®f(Lv)) in H^(D)®H~\D).

(3.88)

Proof. All three equations follow directly from (3.82) and (3.85) by tensorization. D

3.6.1 Discretization

We introduce the sparse tensor product spaces in D x D by

Vl := Span {Vt®Vv | 0 </ + /'< L} c H^(D2). (3.89)

Under Assumption 2.11, the approximation property of the scale (Vl)lsn reads (see e.g.

[Zen91], [ST03b])

min ||u - v\\HHD2} < $(NL, a)||«|U-i+.(Da) Vu G H~1+S(D2) n I$(D2), (3.90)

where the functional l> can be explicitly written in terms of <£. For instance we have

ê(NL) := cT(\ogNL)1/2N£s with 5 := min{p, $}/d for the FE spaces discussed in Example

2.12.

We further assume the existence of a wavelet family in the physical domain D, consistent

with the FE space scale (Vl)l£N-

Assumption 3.44. There exist a family * = (ipki) im C Hq(D) and two constantsfcëiAj

£%*, c-2,® > 0 such that any u Hq(D) can be expanded as a convergent series,

u=5>k,^M ^Ho(D) (3.91)

fceA;

and the following three conditions are fulfilled

1. stability,

ci,* ^2 lCfc-'|2-

Il S cuM\hï(d) ^ c2,* X] IcmI2- (3-92)lew (N leN

fceA; feeA; fceA;

2. local support,

|A;| ~ 2di and for any /, /', the set supp (ipk,i) H supp (V>fc',Z') has nonempty interior for at

most ~(l + l' + l)^-1 • 2max^-r> values of (k, k') G A, x Ar.

3. consistency,

V"L - span{V>M | 0 < / < L, k G Vi}.

Constructions of families \P satisfying Assumption 3.44 are known for intervals (D =]0,1[),

hypercubes (D =]0, l[d) or polygonal domains (see e.g. [DS99]). Note that the stability

condition ensures the existence of a scalar product (•, •)* in H^(D) which is equivalent to

the standard one and has the property that the wavelet family * :— (ipk,i) ie« is an ONB

in (IP0(D), (.,-)*).

We consider the discrete counterparts of (3.86), (3.87), (3.88) associated to the sparse

FE space hierarchy (Vljlcn, and consisting in finding (//)l, (W)m,l, (uu)m,l G L°(ft, VL)

such that

(A®A)((ff)L(Lo)) = f(u;)®f(u3) in V£, (3.93)

(AffiMM ® ldHm)((uf)MiL(u)) = (A ® IdHi(D))((//)L(a;)) in V£, (3.94)

(Idffi(D) ® AaM(Lj))((uu)M,L(uj)) = (ldHi{D) <» A)((u/)MiL(o;)) in 0L*. (3.95)

More precisely, by (3.94) we understand the variational formulation which in the second

variable is associated to the scalar product (bilinear form) (•,•)*, and similarly for (3.95).

Proposition 3.45. Under Assumption 3.12, it holds, P-a.e. in ft and for all M > Mr

(here Id := Idjjx^)),

\\UM(UJ) ® UM(W) - (uu)jlf,LM||ffi(D3) < Ca,*(||(Id ~ ^X/M ® /(^)) IU01(^) +

+ ||(Id - PyL)(uM(uo) ® /»)||Wol(D=) + ||(Id - PyL)(uM(uj) ® «mM)||hi(^)), (3-96)

so t/iai z/ also Assumption 3.12 is satisfied, then P-a.e. lo G ft

\\um(uj) ® %(w) - (u«)M,L(w)||iii(Da) < c^,**(iVL)||/(w)||^-i+.(£)) (3.97)

and

\\M2(uM) - Ml((uu)MtL)\\HHDi) < ca,*e(NL)\\f\\h(n,H-i+sm. (3.98)

Recall that $(NL) :— cT(log NlY^N^5 with 5 :— min{p, s}/d for the FE spaces discussed

in Example 2.12.

Proof. The P-a.e. uniform ellipticity of the bilinear forms giving (the l.h.s.'s of) (3.86),

(3.87), (3.88) for M > Mr ensures the quasi-optimality of the corresponding FE solutions

in (3.93), (3.94), (3.95). We thus have, P-a.e. in ft,

||«m(w) ®«Af(w) - (MU)M,LM||jfi(£)2) < Caty(\\(ld - PyL)(uM(u) ® Um(uj))\\H1(D>)+

+ \\um(u) ® f(co) - (uJ)m,l(u)\\hUd*)) (3-99)

\\UM(UJ) ® /(w) - K/)m,lM||tf0i(L>2) < Ca,*(II(Id - PvL){uM{w) ® /H)||fl-i(£,a)+

+ ll/>) «8 /» " (/7)lMII*oW) (3-100)

||/» ® /(W) - (//)L(a;)||ffi(D3) < ||(Id - P^X/M ® /("))||*oW. (3-101)

Inserting (3.101) in (3.100) and then the resulting estimate (3.100) in (3.99) we obtain

(3.96). Further, (3.97) follows from (3.96) and the approximation property (3.90) of the

sparse tensor product spaces. Integrating (3.97) over ft we obtain (3.98).

3.6.2 Algorithms

We start by decomposing equations (3.94), (3.95) into simpler, mean field problems using

the wavelet family * := (ipk,i) '^ Since (ff)L, (uJ)ml, (uu)m,l £ L°(ft, VL), we can write

(/7)l = J2 ^,i®^k,L-i (3-102)0<1<L

fceAL_;

(uf)u,L = J2 ßk,i®^k,L-i (3-103)

= J2 ^L-i®-yk,i (3-104)0<1<L

keAL_t

(uu)M;L = Yl V'fc.L-Z®^ (3.105)0<1<L

*eAt_;

where ak,i, ßk,i, lk,u h,i G L°(ft, V) for all 0 < / < L, k G AL-i-

For future use we note here that the family (7^/) o<i<l can be obtained from the familyfceAL_,

(ßki) o<i<l by a rearrangement of its coefficients in the Vl basis (^/)o<kz,. The proof'

fcAL„,'

fceA;

follows by simple manipulations of (3.103) and (3.104).

Proposition 3.46. It holds

7W(üj) = ^2 (ßk\L-i'(u),ipk,L-i)vipk>,i> P-a.e. lü eft. (3.106)0<l'<l

fc'eAj,

Using the decompositions (3.102), (3.103), (3.104), (3.105) in the variational formulation

of (3.94), (3.95) we obtain

Proposition 3.47. Equation (3.94) is equivalent to

AaM{Lj)ßktl(Lo) = Aakil(uj) in Vf V0 < I < P, k G A^ P-a.e. oo G ft. (3.107)

Similarly, equation (3.95) is equivalent to

AaM(^ök,i(Lo) = A-ïk,i(u) in VC V0 < I < L, k G AL-i P-a.e. u G ft. (3.108)

Keeping in mind the results of section 3.4, let us explain how the correlation computation

algorithm based on the discretization step given by (3.93), (3.94), (3.95) should work.

The correlation kernel A/(1((wu)m,l) can be obtained using (3.105) from the mean fields

•M1^,;), which in turn follow in view of (3.108) from the mixed moments of (rM,7fc,;)-

By rearrangement via (3.106), the mixed moments of (rM,lk,i) are expressed in terms of

the mixed moments of (r^, ßk,i), which follow then from the mixed moments of (rM, ctk,i)-

Finally, the mixed moments of (rM,ctk,i) can be obtained, due to (3.102), from the mixed

moments of (rM, (//)l) (or of (rM, f ® /)). The algorithm will therefore have to perform

all these steps in reverse order, starting from the available information, the mixed moments

of the pair (rM,f®f)-

In the following we use the notation

(A®A)^ : H~\D) ® H~\D) - VL (3.109)

for the discrete solution operator associated to the bilinear form of A <S> A (coercive on

Hq(D2)) and the sparse FE space Vl- Further, we denote by

iW.HÏM^C, i*,i,(0 :=(Ç,il>)v VÇeHl(D) (3.110)

the contraction in the Hilbert space (Hq(D), (, •)$) with vector ip G Hq(D).

With these notations, simple algebra shows the connection between the mixed moments of

the pair (rM, f ® f) and those of (rM, ak<t).

Proposition 3.48. In A(Uj, H^(D)) it holds

M$'1\rM,0[k,l)= Y2 9m,akzl ® <f>m (3.111)meTj

where

0m,aW = (IdHi(D)®z*^,L_!)(A®A)Z1X^«(Xm,/®;E/) G Vt (3.112)

satisfy the symmetry condition w.r.t. m (invariance under entry permutations).

Proof. Using (3.102) and the linearity of the mixed moments of order (j, 1) in the last

argument we write

M%'1)(rM,(ff)L)= J2 M^1](rM,ctk,i)®^k,L-i. (3.113)

fceAL_,

But (3.93) shows that

M^\rM, (ff)L) = Y, (A^A)^1^^, / ®x f) ® <j>m, (3.114)

so that the conclusion follows by applying \dHi^ ® z*,^fcL_, ® Id^«^) to (3.114) and

(3.113). D

Noting that, due to (3.93) and (3.102), / G L2(ft,II^(D)) ensures ak,i G L\ft, H^(D))

boundedly, we apply Proposition 3.36 to the equation (3.107).

Proposition 3.49. Under Assumptions 3.1, 3.18, M^^'(rM,ßkti) can be computed star-

rting from My3'

(rM,ctkj) for all 0 < j < n via Algorithm 5 at the cost of solving at

most (j' + n + l)q+2(M + l)q deterministic diffusion problems in D at FE level I. The

error estimate reads

<

< Ca||aw||Li(Q^i(Z3))(||rM||{00(i7xn)cxp(-clian) + c{+"exp(-c2]7.<71+a)). (3.115)

The next step of the algorithm, passing from the mixed moments of the pair (rM,ßkj) to

the mixed moments of (rM, 7k,i), can be analyzed using Proposition 3.46 and the linearity

of the approximate mixed moment M^1' in the last argument (see (3.59)).

Proposition 3.50. //

M(4'»(rM, ßk,i) = J2 SmA, ® &n, (3-116)

mT3,

with #m,Ä,i e Vi, then

M^\rM,lk,i) = J2 0m,7fcll ® ^m (3.U7)

meTj,

where #m,7M G VJ is given by

9m,yk,i = /] (9m,ßk,iL_l,,lI>k,L-l)*ll1k',l'- (3.118)

fc'eA,,

Moreover, the following error estimate holds,

IIA^W.Tw) - M^(rM,lkMA{WM1(D))<

< Ca,*||/||22(n>jy-ip))(lkM|lioc([/xn)exp(-Cl,an) + c3a+nexp(-c2ir(? +a)). (3.119)

Proof. From (3.106) we obtain

M{4£\'rM,lk,l) = Y M?,n\rM,(ßk',L-l'(-),lJ1k,L~l)'lpk>,l')0<l'<l

fc'eA,,

= J^ (IdL-(^') ® **,^.l-i ® IdHHD))M%t£\rM, ßk',L-l>) ® ^fc',J'

0<l'<l meïjifc'eA,,

from which (3.118) follows by interchanging the summations. To check (3.119) we first

note that jk,i solves, in view of (3.106),

AttM(w)7fc.iM = Y (ßk<,L-i'W),i>k,L-i)ykaM{w)'il)ki,i> h\VL* P-a.e. uj ft. (3.120)0</'</

fc'A"J,

Next we observe that the ^(ft, H~1(D)) norm of the r.h.s. in (3.120) can be estimated

from above by

Ca>* /( XI W^'.L-l'^Wjji^Y^dP^) < C,*||(u/)jlf,L||Li(n,/rçp»)) < Ca,*|l/llï,2(n,H-i(£>))»o

0<i'<i"

fc'eA,,

so that (3.119) follows from Proposition 3.36 applied to the equation (3.120). O

Finally we investigate the computation of ök,i solution to (3.108). We apply the T version

of Proposition 3.27 (see also Remark 3.31) to equation (3.108) using as input data for

Algorithm 5 the approximate moments

M%p%(rM,Alk,i) := Mî\rM, A7*,i) (3-121)

and obtain

Proposition 3.51. Under Assumptions 3.1, 3.18, an approximate mean field

Ml^Âî) := Algorithm 5 [m, n', 0, (M^\rM, A7w))0<,-,<n,] (3.122)

can be computed at the cost of solving at most (n' + \)q+2(M + l)q deterministic diffusion

problems in D at FE level I and for an accuracy (w.l.o.g. ca, \\rM\\L°°(Uxü) > lj

\\MH6u) - MlnwVMmD) £

< ca\\f\\2mn,H-i(D))(eM-Ci,an') + ||rM ||2»([,xn)exp(-cli0n) + <'+nexp(-c2,^1+a)).(3.123)

Due to the orthogonal decomposition (3.105) we then have, with NL := diml4>

Corollary 3.52. Under Assumptions 3.1, 3.18 it holds

\\M1((uu)M,l) ~ Yl ^k,L-l®Mlpp,ytntnl(SkMHZ{D*)<0<1<L

fceAj,.,

<ca\\f\\2LHnM-imÛl/2(exp(-chan')^rM\\^(3.124)

We summarize more formally in Algorithm 6 the second moment computation we have

developed and analyzed in this section. The total error estimate follows by collecting all

discretization errors, as given by Propositions 3.42, 3.45 and Corollary 3.52.

Proposition 3.53. It holds

II^'H" Y, V'fc,L-/®Xapp,T,n,n'(^)ll//01(Û2)^0<1<L

fceAL_,

< Ca,r||/||ia(n,H-ip))(exp(-clirMa) + *(NL) + A>i/3(exp(-Cl,an')+

+ lkM||2Vx«)exp(-Cl,an) + c;?'+"exp(-c2,7y+Q))). (3.125)

The complexity of Algorithm 6 can be estimated using Propositions 3.49 and 3.51.

Proposition 3.54. Step 1 of Algorithm 6 consists in solving |Tn/+„/ ~ | + |T„<+n_i/ ~

\-\ 1-1 To/ ~ | second moment diffusion problems in D x D with constant coefficient and

in the sparse FE space Vl.

Steps 2 and 4 of Algorithm 6 consist in solving at most 2|A/-/_z|(n' + n + \)q+2(M + l)q

deterministic diffusion problems in D with coefficient e and in the FE space Vi for all

0 < / < P.

Step 3 of Algorithm 6 consists only of a sorting procedure applied to the coefficient sets of

(9m,ßki) °s'<l in the wavelet basis ^ (compare (3.118)/

Proof. The complexity estimate of step 1 follows from the moment symmetry w.r.t. m,

whereas those of steps 2 and 4 are direct consequences of Proposition 3.36.

Note that a more explicit complexity estimate of step 1: in Algorithm 6 follows from the

main result in Chapter 2, that we recall next.

Theorem 3.55. Under Assumption 2.15 and if f G L2(ft, H~1+S(D)), each problem in

step 1: of Algorithm 6 is numerically solvable for any L > 1 at a cost of

cr,*(logiVL)2d+2AL (3.126)

floating point operations, with

CT(logNL)NL (3.127)

memory, for an accuracy of

ca,u,r(logNL)1/2NZô, (3.128)

where Ô = min{p, s}/d.

Balancing the discretization errors in (3.125) by the choice of parameters

M-P1/", n'~L, n~P, q ~ Lll{l+a) (3.129)

and comparing with the computational effort estimate obtained in Proposition 3.54 we

deduce (assuming also exact integration for the r.h.s. of all deterministic problems to be

solved in Algorithm 6),

Theorem 3.56. Under Assumptions 3.1 (well-posedness), 3.12 (uniform data regularity),

3.18 (analytic perturbation) and 2.15 (existence of Hq(D)-stable wavelet family), the sec¬

ond order moment (correlation) of u random solution to (1.10) with f G L'2(ft,H~1+s(D))

is computable using Algorithm 6 at nearly the same cost as one deterministic diffusion

problem in D.

More precisely, if the available wavelet based solver in D needs 0(e~ß) floating point op¬

erations and memory to reach an accuracy £ for one deterministic diffusion problem with

coefficient e and Hs regular source term (usually ß — I/o + o(l) with ö = min(s,p)/d),

then Algorithm 6 requires at most 0(e~@~0^) operations and memory to achieve the same

accuracy e —> 0 for M2(u) in the case of stochastic data. Algorithm 6 has therefore nearly

optimal complexity in the sense of Definition 1.1.

Proof Due to Propositions 3.34, 3.54 and Theorem 2.28, the computational effort of step

1 consists in at most NL+ operations as P —> oo, by the same argument used in (3.80).

This argument shows also that the cost of steps 2 and 4 admits (up to a multiplicative

constant) the upper bound

Y\AL-i\N°LWNl+0{1) < NL+0{1) asP-oo, (3.130)

for an accuracy of &(Nl) ~ N£ (compare Theorem 3.55). D

Remark 3.57. All solution operations contained in steps 1:, 2:, 3:, 4: of Algorithm 6 are

fully parallelizable w.r.t. the indices l,j,j' and k G Al-i-

input : • M, j,f, L

• approximation of M^ +3,1\rM, f) of the form

M%^1}(rM,f) = Y h^ with hmeH~1(D)

% e.g. hm :— M1y+i+1(Xm, f) for exact sparsified representation of

M{4'+jA)(rMJ) (see (3.73))• solver of diffusion problem in D with deterministic coefficient e and

FE space VL

output: • approximation of M^'^^tmjVj^) of the form

,0M)M.%£t(rM,vjjL)= Y 9m^

tn&n3'+3,j'(Tj'+3^

with gm G VL

1 compute for all m g Yy+3

2 for k = f + j : -1 : j' + 1 do

3

gm := A~^m G Vl

compute for all m G Uj'+j^-iC^f+j)

(

9m -- A-L div y ^m'fcVp,, G Vi

4 end

5 return

M%p^(rM,v3,L) := 5Z #m<^

men/+j,3'(T/+3)

Algorithm 4: ^^(rM,^) = Algorithm 4 MJ,/, P, A<VM,/)<(j'-H,l)/

3.6. COMPUTATION OP HIGHER ORDER MOMENTS 55

input : • M, n,f, L

• approximations of M^'+3'l\rM, f) f°r all 0 < j < n, of the form

M%^l)(rM,f)= Y h<t> with hmeH~1(D)meTj'+l

% e.g. hm := M1^'+^1(Xm, f) for exact sparsified representation of

M%'+lA\rMJ) (see (3.73))

• solver of diffusion problem in D with deterministic coefficient e and

FE space VL

output: • approximation of M^'^^m^m^) of the form

Mp'Ä,n(rM, um,l) = Y 9m<^m

with gm G Vl

l compute for j — 0 : 1 : n

<0M)Mi;/r(rM)vJtL):= Algorithm 4

M,j,j',L,M^'(rM,f)Aj'+3,y)t


M%p\n(rM,uM,L) :=YMZ&(rM,v3,L)j=0

,0',»Algorithm 5:

M^^n(rM, uM,L) = Algorithm 5 M, n,j>, L, (M^'^m, f))0<3<nA3'+3,1) I

input : • M, n, n', L

• approximations of M^'+3'1\tm, f ®x f) for all 0 < j < n, 0 < / < n'

of the form

M%+jA\rM,f<8>f)= Y h^-lW+3,1),

;i• m,j w j )=

with hm := A^V+J+i(Xm, / ®x /) H-\D2)

• solver of deterministic diffusion problem in D with coefficient e

on FE space scale (V/)o<kl

• ^-wavelet based deterministic solver of second moment diffusion problem

in D x D with sparse FE space Vl

output: • approximation Miw^tnn,((uu)M,L) £ VL of M2(u)

1 compute for all m G T^+j-, 0 < j < n, 0 < f < n' and all 0 < / < P, k G AL-i

9m,akil := (Idffi(I3) ® i*,^iL_,)(A®A)£1/iin G Vi

2 compute for all 0 < f < n! and all 0 (rM, Aaw))0<,<

3 compute for all 0 < f < n' and all 0 < / < P, A; G Al_/ via (3.118),

M$;1){TM,'1kÙ= Y 3m,lkil ® (pm

meYji

4 compute for all 0 ((UU)M,L) •= Y ^-' ® ^W.T.n.n'OW)

fceAL_,

Algorithm 6: (M\w T n n,(£M)) °<<<^ = Algorithm 6[M, n, n', P, (A4?'+j,1)(rM, / ®

/)) 0<j<n ]0<j'<Ji'

Chapter 4

Conclusions and Open Issues

Classical perturbation algorithms for the moment computation of the solution to an ellip¬

tic problem with stochastic data have been further developed and investigated in terms of

complexity. Statistical information about the data, needed as algorithm input, has been

explicitly determined. It has been shown that, under essentially only one additional as¬

sumption, the spatial regularity of the fluctuation, the algorithms are, asymptotically in

work vs. accuracy, more efficient than the standard MC/SG/WPC methods, and a priori

nearly optimal (in the sense of Definition 1.1) complexity bounds have been proven. As in

the case of a standard MC simulation, the computation is almost fully parallelizable and

the implementation requires essentially only one routine to call any available, préexistent

FE code for the corresponding deterministic elliptic problem and run it successively with

different source terms.

We conclude by enumerating several possible extensions of the results presented in this

thesis, along with some further open problems for the solution of which the techniques and

ideas used here might be, in our opinion, relevant.

1. Relaxation of the fluctuation regularity condition r G A(D, L°°(ft)) to only finite

Sobolev regularity. We further conjecture that, in order that perturbation algorithms

retain nearly optimal complexity, the stochastic fluctuation must be at least piecewise

continuous in the physical domain D.

More generally, it could be of interest the construction of a regularity space scale

(Xs)o<s<2 interpolating between smooth and rough data corresponding to 6 — 0 and

5 = 2 respectively, such that r G Xs ensures the complexity rate e~5+of-^ for the

first/higher order moment algorithms, or for some combination of perturbation and

Monte Carlo algorithms (quasi-Monte Carlo).

2. Development of parallel results for the stochastic Galerkin (SG) method which is

known to deliver, under stronger assumptions (known joint distributions of the ran¬

dom variables (Xm)me^+ in the fluctuation expansion), complete statistical informa-

57

58 CHAPTER 4. CONCLUSIONS AND OPEN ISSUES

tion (i.e. the probability distribution function) on the stochastic solution u.

3. Study of the perturbation algorithm scalability w.r.t. the correlation length of the

diffusion coefficient a and the case of a small correlation length, equivalent to a slow

(still exponential) convergence of the fluctuation expansion due to unfavourable val¬

ues of the constants involved in the decay estimates, or to a large pre-asymptotic

domain for all results presented in this thesis. This pre-asymptotic analysis should

also give some insight into the case of a rough stochastic fluctuation, including white

noise, and might require new random field representation techniques, in order to

parsimoniously parametrize the data uncertainty, plus an investigation of the appli¬

cability of perturbation algorithms in this new context.

4. Generalization and adaptation of perturbation algorithms to the case of unbounded

stochastic diffusion coefficient with bounded samples (lower and upper bounds are

uniform in the physical domain, but are sample dependent, e.g. log-normally dis¬

tributed a).

5. Extension to general second order elliptic/parabolic/hyperbolic problems with stochas¬

tic data (advection-diffusion, linear elasticity, heat, wave equations, Stokes problems

etc.).

Chapter 5

Appendix

5.1 A Trace Lemma

Lemma 5.1. If s,t,u are real numbers such that

t>s>0, u>s>0 and t + u~s>d/2, (5.1)

then the mapping (the trace on the diagonal set in Rd x Rd)

S(R2d) 3 0 -=* Z(<p)(x) = (f>(x,x) G S(Rd) (5.2)

has a unique linear and continuous extension from P'(IRd) ® Hu(Rd) to Ps(Rd).

Proof. Denoting by Td (and by hat) the Fourier transformation (and transform) in Rd, we

compute the Fourier transform of the mapping S (denote by dx the normalized Lebesgue

measure in Rd, dx = (2ir)-d/2d\),

S$(p) := (ft o H o f£(4>))(p) = J e~^(E o F£$)){x) dx

= f e~^x f eix<t+r>^(Ç,r])dÇdridx

= J e-irxe0î>{tO-Ç)dïMdx = j fa,p-i)d£. (5.3)

R3rf

Since the uniqueness of an extension follows by a density argument, we only show that

S : <S(M2d) — <S(Md) computed in (5.3) has a linear and continuous extension from Kt(Md)®

Ku(Rd) to Ks(Rd) if s,t,u satisfy (5.1) and

Kr(Rd) := L2(Rd, (x)2rdx) with (x)r := (1 + |x|2)r/'2 Vr > 0. (5.4)

59

60 CHAPTER 5. APPENDIX

We estimate therefore

\m\\h = J(p)2s-(jktP-odA dp

2

(pY

(^(p~0u(0*(p-0"-^p-0^ dp

<

RdRd NR"* / \r<* /

r (pY3- il y /g\2t/p _ ^

^lU"^,^) • iMk*®*«- (5.5)

All we have to check is that (5.1) ensures the finiteness of the P°° norm in (5.5). To this

end, we note first that it is enough to prove the uniform boundedness of the integral in

(5.5) for p outside a compact set in Rd, e.g. for \p\ > 1. Let 1/^/2 < c < 1 be an arbitrary

real number. We split the integral in two parts, coresponding to the integration domains

D\ :— {£ | (p — 0 > c- (p)} and D2 :— {£ | (p — £) < c- (p)} and estimate the corresponding

integrals as follows

f (p)2s ! f 1

J (02t(p~t)2udli -PJ (02t(p-02{u-s)d^ (5'6)

But it is easy to see that (p — £) > c • (p) with l/y/2 < c < 1 and |p| > 1 implies

(p — 0 > c' • (£) for any 0 < d < y/2c2 — 1. Inserting in (5.6) we obtain (w > 5)

f (P)2'dC<

1 /• 1g (57)

y (£)2*(p _ ^2u"< -

C2S . cJ2(u-s) J ß)2(t+u-s)"Ç ^' >

-Dl Krf

and the r.h.s. of (5.7) is finite, due to t + u — s > d/2, and does not depend on p.

To estimate the integral over D2, we note first that D2 is a ball of center p and radius

y/c2 • (p)2 — 1, so that (£) > c' (p) with some positve d > 0 depending only on c < 1. It

follows(^)2S

„ ^

! / (P)2(s-t]f (P)2Sd: <

J_/

J (02t(p-02uç -

d*'J {p-02uD2 Di

di

r^j

< <p)2^- j (r)"2"^"1 dr

0

(p)2(S-t)-2u+d if u < rf/2

(p)2(s-*) • log(p) if u^d/2 (5.8)

(p)2(s-t) if u> d/2

with constants independent of p. The uniform boundedness follows in all cases from the

assumptions t > s and t + u — s > d/2.

5.2. A FAST DECAYING SEQUENCE 61

5.2 A Fast Decaying Sequence

Here we prove that, if y,a > 0 and j G N+, the sum of the series with general term

exp(—y 2~Di=i m%) indexed over 1 < mi < ... < m3 < oo is, qualitatively and uniformly in

j G N+, just as large as the leading term, corresponding to mk — k for all 1 < k < j. More

precisely, it holds

Lemma 5.2. If a > 0, and x > y > z > 0, then there exist ca]X>2/, caiytZ > 0 such that

1J

1

c^expf-x--—jl+a) < Y T\eM-yrnk)<ca,y,zexp(-z-^j1+a) (5'9)l<mi<...<mj<oo k—1

VjGN+.

Proof. For y > 0 and j, J G N+ U {oo} with j < J we set

s*- e n««pHK)- (5-iq)l<mi<...<raj<J k=l

We check first the lower bound in (5.9). To this end we note that the sum in (5.10) contains

the term corresponding to mk — k for all 1 < k < j, therefore we have

S3<J>exp(--yYka). (5-11)fc=i

The conclusion follows by noting that

} 1

Yk° ^ a + !)1+a / x<X(ix = 0' + i)1^!^- (5-12)

It remains to prove the upper bound of the sum in (5.9). From the definition (5.10) it

follows that the sequence (Sj>3)3^+ is rapidly decaying, that is

Sj>3 < ca,y,0ß3' VjGN+,V/3>0. (5.13)

From (5.10) we derive the recursive formula

j-i

Sj,J+1 = S3,j + exV(~y(J+ir) Y Y[eM-y<)l<mi<...<mj_i<J fc=l

= Sjj + expi-viJ + mSj-uj (5-14)

From (5.14) it follows immediately by induction on j that

Sjtj < S3,oo = Jim Sj,j < oo Vj G N+) (5.15)

62 CHAPTER 5. APPENDIX

and that

Sjt00 < Sjj + Y eM-yk°) ^-1>00. (5.16)fc=J+l

Now, for an arbitrary 7 G (0,1) we have, for j large (j > j0, with j0 depending on y, a, 7),

00

Y exp(-yfc*)<7, (5-17)fc=j+i

which ensures

S/,00 < Sj-j + 7-SVi.oo, Vi > jo- (5-18)

From (5.13) and (5.18) we deduce that

Sjt00 < ca,vß((7 + ß)j + 7J-J0+15i0-i,oo), (5.19)

which shows that Sjj0O — 0 as j —> 00, by choosing ß such that 7 + /? < 1. The sequence

(Âj,0O)jeN+ is in particular bounded, that is

S;,«, < Cc,tf VjGN+. (5.20)

Since this inequality holds for any y > 0, the conclusion follows then from (5.20) upon

replacing y by y — z. d

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«a*«».

Curriculum Vitae

Name Radu Alexandru Todor

Date of Birth November 12, 1973

Place of Birth Brasov, Romania

Nationality Romanian

Education

1980-1992

1988, '89, '90

1991

1992

1992-1997

1997-1999

1999

2000-present

Primary School/Secondary School/College

"Andrei Saguna" Brasov, Romania

First Prize, Romanian National Mathematical Olympiad

Bronze Medal, International Mathematical Olympiad, Sigtuna, Sweden

Gold Medal, Jury Prize, Balkan Mathematical Olympiad, Athens, Greece

Silver Medal, International Mathematical Olympiad, Moscow, Russia

Study of Mathematics, Bucharest University

B. Sc. Diploma Thesis: Morse Inequalities on Covering Manifolds

Advisor: Prof. V. Iftimie

Post Graduate Studies, Bucharest University

Master Program: Nonlinear Equations and Modelling

Master Thesis: Spectral Theory for Uniformly Propagative Systems

Advisor: Prof. V. Iftimie

(April-June) Research Stay at Department of Mathematics,

Université Cergy-Pontoise, Paris, France

Ph. D. Student and Research Assistant, ETH Zurich

Advisor: Prof. Ch. Schwab

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