Inverse Multiscale Problems Martin Burger Institut für Numerische und Angewandte Mathematik...

Inverse Multiscale Problems

Martin Burger

Institut für Numerische und Angewandte MathematikEuropean Institute for Molecular Imaging (EIMI)

Center for Nonlinear Science (CeNoS)Westfälische Willhelms-Universität Münster

5.9.2007 BICS Workshop, Bath

Various processes in the natural, life, and social sciences involve multiple scales in time and space.

An accurate description can be be obtained at the smallest (micro) scale, but the arising microscopic models are usually not tractable for simulations.

In most cases one would even like to solve inverse problems for these processes (identification from data, optimal design, …), which results in much higher computational effort.

Introduction


In order to obtain sufficiently accurate models that can be solved numerically with reasonable effort there is a need for multiscale modelling.

Multiscale models are obtained by coarse-graining of the microscopic description. The ideal result is a macroscopic model based on differential equations, but some ingredients in these models often remain to be computed from microscopic models.

Introduction


In many models some parameters (function of space, time, nonlinearities) are not accesible directly, but have to be identified from indirect measurements.

For most processes one would like to infer improved behaviour with respect to some aspect – optimal design / optimal control

For such identification and design tasks, a similar inverse multiscale modelling is needed.

Introduction


Transport of charged particles arises in many applications, e.g. semiconductor devices, ion channels, or nanopores

The particles are transported along (against) the electrical field with additional diffusion. Self-consistent coupling with electrical field via Poisson equation. Possible further interaction of the particles at different scales: recombination, ionization, precipitation, size exclusion

Electron / Ion Transport

Ion ChannelCourtesy Bob Eisenberg

MOSFETS, from www.st.com


Microscopic models from statistical physics (MD, Langevin, Boltzmann) or quantum mechanics (Schrödinger), coupled to Poisson (self-consistency)

Coarse-graining to macroscopic PDE-Models classical research topic in applied math. Long hierarchy of models, well understood for semiconductors, not yet so well for channels and nanopores (due to crowding effects)


Sketch of l-type CaChannel

Sketch of geometry of a MESFET

Mock 84, Markowich 86, Markowich-Ringhofer-Schmeiser 90, Jüngel 2002Eisenberg et al 01-06


Other end of the hierarchy are Poisson-Drift-Diffusion / Poisson-Nernst-Planck equations (zero-th and first moment of Boltzmann-Poisson with respect to velocity)


Poisson-Nernst-Planck

Poisson-Drift-Diffusion


Size exclusion in ion channels significantly increases

computational effort(e.g. nonlocal functionals in DFT)


Densities and Potential in an L-type Ca channel (PNP-DFT)

Densities in a nanopore

Ca2+

Na+

Cl-


The main characteristics of the function of a device are current-voltage (I-V) curves (think of ion channels as a biological device).

These curves are also the possible measurements (at different operating conditions, e.g. at different ion concentrations in channels)

For semiconductor devices one can also measure capacitance-voltage (C-V) curves



Inverse Problem 1: identify structure of the device (doping profiles, contact resistivity, relaxation times / structure of the protein, effective forces) from measurements of I-V Curves (and possibly C-V curves)

Inverse Problem 2: improve performance (increased drive current at low leakage current, time-optimal behaviour / selectivity) by optimal design of the device (sizes, shape, doping profiles / proteins, nanopore geometry)



Many herding models can be derived from micro-scopic individual-agents-models, using similar paradigms as statistical physics. Examples are

- Crowding effects in molecular biology (ion channels, chemotaxis)

- Swarming / Herding / Schooling / Flocking of animals, humans (birds, fish, insect colonies, human crowds in evacuation and panic)

- Traffic flow- Opinion formation- Volatility clustering, price herding in financial

markets

Emergent Behaviour


Microscopic models can be derived in terms of SDEs, like Langevin equations for particle position / state

Interaction kernels not determined by physics / natural laws in such applications

A lot of macroscopic data collected

Emergent Behaviour


Coarse-graining to PDE-models similar to statistical physics (Boltzmann /Vlasov-type, Mean-Field Fokker Planck), but N smaller

New effects yield also new types of interaction and advanced issues in PDE-models (general nonlocal interaction, scaling limits to nonlinear diffusion, ..)

Emergent Behaviour


Natural inverse problems in emergent behaviour (mostly future work):

- Identification of interaction potentials from observation Bianchi et el 06,

- Identification of dynamic parameters (effective diffusions, mobilities) McCarthy et al 07

- Optimal control (boundary or via external potentials) Lebdiez-Maurer 04, McCarthy et al 05

- Optimal shape / topology design (e.g. evacuation routes, traffic flow)

Emergent Behaviour


In some models blow-up is undesirable (e.g. chemotaxis and swarming due to finite size of individuals), in others it is wanted.

E.g in opinion formation, the blow-up (as a concentration to delta-distributions) can explain the formation of extremist opinions (in stubborn societes)

Blow-up is an enormous challenge with respect to the construction of stable numerical schemes and for inverse problems

Emergent Behaviour

Porfiri, Stilwell, Bollt 2006


Swarming

mb-Capasso-Morale 05mb-DiFrancesco 06

Example:

swarming models

without

repulsive

force

(blowup)


Swarming

mb-Capasso-Morale 05mb-DiFrancesco 06

Example:

swarming models

with local

repulsive

force

(small

nonlinear

diffusion)


Chemotaxis

mb-Dolak-DiFrancesco, SIAP 07

Example:

Chemotaxis

models with

Quorum

sensing,

Formation

of clusters /

coarsening

mburg_01


Inverse Problem 1: identify interaction or external potentials (or dynamic parameters like mobilities) from observations [mostly future work]

Inverse Problem 2: design or control system to optimal behaviour [some results, a lot of future work]

Emergent behavious


Molecular Imaging (PET, SPECT, …) techniques are usually based on some tracer that attaches to specific molecules

Radioactive decay at some time, emitted photons (two directions) recorded outside the body

Decay rate proportional to density inside the body, hence identification of right-hand side in transport equations

HR Molecular Imaging


Future clinical applications need increase of spatial resolution

Current test setup – small animal PET

Small scale effects become important: - Inaccuracy of decay position- Inaccuracy of emission axis- Scattering events

HR Molecular Imaging


Small animal pet: mouse heart (courtesy SFB 656 MoBiL,reconstruction by

Coronal Sagittal Thomas Kösters)

Transverse

HD Molecular Imaging


Typical characteristics of the inverse problems are

- huge amounts of data - low sensitivities of identification / design variables

with respect to data nonetheless- simulation of data requires many solutions of

forward model, high computational effort / memory- can be formulated as optimization problems (least-

squares or optimal design) with model as constraints

- sophisticated optimization models difficult to apply (even accurate computation of first-order variations might be impossible)

Summary of Issues


Inverse problems techniques usually formulate a forward map F between the unknowns x and the data y

Evaluating the map F(x) amounts to simulate the forward model for specific (given) x and compute macroscopic observables

The inverse problem is formulated as

or the associated least-squares problem / maximum likelihood estimation problem

Towards a General Theory

F (x) = y


In the multiscale case, we might think of a scale-dependent problem

and a coarse-grained problem (maybe not completely scale-independent) related to


F²(x²) = y

F0(x0) = y² ! 0


How can we benefit from the coarse grained model ?

- Replace -problem by reduced problem- Preconditioning of reconstruction

algorithms by coarse-grained solvers- Multiscale computations directly for the

inversion- Interplay with regularization, appropriate

coarse-graining for the unknown



Setup

with u solving

Multiscale scheme available

Multiscale Methods for the Inversion

F²(p) = Bu

E ²(u;p) = 0

E ²;H (uH ;pH ) = 0


Variational regularization of the inverse problem

with constraint


E ²(u;p) = 0

D(u;y) +®R(p) ! minu;p


Schemes based on KKT-System

Multiscale computation for last equation available, how to construct computation for first two ?


@uD(u;y) +@uE ²(u;p)¤w = 0

®R(p) +@pE ²(u;p)¤w = 0

E ²(u;p) = 0


Example: source estimation in

with macroscopic observation

Homogenization techniques / multiscale FEM can be used


Bu = k¤u

E ²(u;p) = r ¢(a(¢)¢²)r u) +p


Adjoint problem

Optimality

Can again be solved with the same methodError estimates carry over


@pE ²(u;p)¤w= w

@uE ²(u;p)¤w= r ¢(a(¢;¢²)r w)


Multiple Time Scales: highly-oscillatory ODE (similar schemes for SDES, see Kevrekidis)

HMM for multiscale ODEs (Engquist-Tsai 05-07):

- Macro time grid Tk and micro time grids Tk = tk,0 < tk,1 < .. < tk,n

- Effective force estimation by local ODE integration on micro grid

- Extrapolate to macro time step


E ²(u;p) = @tu f ²(t;u;p)¡


Adjoint problem

with final value w(T) = 0, backward in time

Could try to apply same HMM scheme:- Macro grid Tk - Micro grid Tk = k,0 > k,1 > .. > k,n


¡@uE ²(u;p) = ¡ @tw @uf ²(t;u;p)w


HMM scheme for adjoint:- Needs to evaluate derivative of f at new

micro time steps (values of u there !)- Corresponding solution is not the adjoint of

the HMM scheme for the forward ODE- No convergence / error estimates for the

regularized problem guaranteed- Not even existence of solution clear (even

with regularization !)



Alternative: use discrete adjoint to approximate adjoint equation

- Backward integration on same micro/macro grid, no other values of u needed

- Force correction rather than force estimation

- Discrete optimality system, existence guaranteed, energy descent for time stepping

- Estimation for approximation of dual variable w carries over



Numerical experiment for linear model

Coefficient a one-periodic

Micro time step 0.00001

Macro time step 0.005


f ²(t;u;p) =1²a(

t²)(u ¡ p)


u v



Multiscale Least-squaresReconstruction functional



Error estimate for the (regularized) inverse problem and its multiscale approximation can be derived


kp pM Sk · C®¡ º (¿+²¿)¡


In molecular imaging, the main quantity of interest is the main variation of the density

Small oscillations appear, but it is not realistic to find them from macroscopic data

Look for structures with small total variation

¸2kAu ¡ f k2 +

12kLuk2 ! min

u

Interplay with regularization


Modelling via prior in the log-likelihood functional (Poisson distribution of noise)

¸2kAu ¡ f k2 +

12kLuk2 ! min

u


X

j

yj logyj

(F u)j+®

Zjr uj


Two-step Algorithm for Minimization (EM-TV)

uk+1 as minimizer of

Implemented by Alex Sawatzky

¸2kAu ¡ f k2 +

12kLuk2 ! min

u


uk+1=2 = ukF ¤(yj

(F u)j)

12

Z(u ¡ uk+1=2)2

uk+®

Zjr uj


106 events

standard EM EM-TV

Inversion and Cartooning

¸2kAu ¡ f k2 +

12kLuk2 ! min

u


250.000 events

standard EM EM-TV


¸2kAu ¡ f k2 +

12kLuk2 ! min

u


50.000 events

standard EM EM-TV


¸2kAu ¡ f k2 +

12kLuk2 ! min

u


Problem of highest technological importance is the identification of doping profiles (non-destructive device testing for quality control)

In order to determine the doping profile many current measurements at different operating conditions are needed.

Inverse problem is of the form (k=1,…,N)

Fk(doping profile) = Current Measurementk

Evaluating each Fk means to solve the model once

Identification of Doping Profiles

mb-Engl-Markowich-Pietra 01mb-Engl-Markowich 01, mb-Engl-Leitao-Markowich 04


Identification of Doping Profiles

Sketch of a two-dimensional pn-diode Identification of a doping profile of

a pn-diode by a nonlinear Kaczmarz-method


Analogous problem in ion channels: identify permament charge of the channel

More realistic: identify external potential (forces caused by the channel structure) acting on the permament charge distribution

More data and higher sensitivity than for semiconductors, since concentrations can be varied

Higher computational effort for the inverse problem

Identification of Channel Structures

mb-Eisenberg-Engl, SIAP 07


Less operating conditions are of interest for optimal design problems (usually only on- and off-state), at most two different boundary conditions

Possible non-uniqueness from primary design goal

Secondary design goal: stay close to reference state (currently built design)

Sophisticated optimization tools possible for Poisson-Drift-Diffusion models

Optimal Design of Doping Profiles

Hinze-Pinnau 02 / 06mb-Pinnau 03


Fast optimal design by simple trick

Instead of C, define new design variable as the total charge Q = -q(n-p-C)

Partial decoupling, simpler optimality system

Globally convergent Gummel method for design

Optimal Design of Doping Profiles

mb-Pinnau 03 / 07


Fast optimal design technique, optimal design with computational effort compareable to 2-3 forward simulations.

Optimal Design of Diodes and MESFET

Optimized Doping Profiles for a pn-diode

Optimized Doping Profiles for a npn-diode and IV-curve

mb-Pinnau, SIAP 03

Optimized MESFET Doping Profile.Current increased by 50% relative to reference state


Papers and talks at

www.math.uni-muenster.de/u/burger

Email

[email protected]

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