(Inverse) Multiscale Modelling Martin Burger Institut für Numerische und Angewandte Mathematik Westfälische Willhelms-Universität Münster

(Inverse) Multiscale Modelling

Martin Burger

Institut für Numerische und Angewandte Mathematik

Westfälische Willhelms-Universität Münster

3.4.2007 CeNoS Kolloquium

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 behaviour – optimal design / optimal control

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

Introduction


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- 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)

Introduction


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

The inverse problem is formulated as the equation

F(x) = y

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

Introduction


Resulting problem is regularized by iterative methods with early termination or by adding regularizing energies to resulting optimization problems (and subsequent application of iterative methods)

Iterative methods for nonlinear problems usually require the computation of sensitivities (derivatives of F with respect to unknown)

Derivative of least-squares functional

F‘(x)* (F(x) – y)

requires computation of adjoint of F‘

Introduction


„Adjoint methods“ do not compute F‘(x)* as a linear map (matrix), but only the evaluation F‘(x)*

This requires the implementation of an adjoint model, which might not always be possible (e.g. if the simulation tool for the forward model is black-box) or if the computational effort is too large

In such cases it can be benefitial to consider surrogate models for (locally) replacing the complicated F by a simpler model and thus simpler map

Introduction


The computational issues in the solution of inverse problems raises the need for deriving reduced macroscopic or multiscale models

This talk will give several examples from various application fields.

Introduction


Joint work with

Mary Wolfram (WWU)

Heinz Engl, K.Arning (Linz)

Peter Markowich (Wien)

Bob Eisenberg (Rush Medical University Chicago)

Rene Pinnau (Kaiserslautern)

Michael Hinze (Hamburg)

Paola Pietra (Pavia)

Antonio Leitao (Florianopolis)

Electron / Ion Transport


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

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 (recombination, size exclusion)


Ion ChannelCourtesy Bob Eisenberg

MOSFETS, from www.st.com


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)



In order to get structure from function, we first need a model predicting function from given structure.

Microscopic models either from statistical physics (Langevin, Boltzmann) or quantum mechanics (Schrödinger), coupled to Poisson

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 (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)

Voltage enters as boundary value for the electric potential, current is computed from boundary flux of the electrons / ions

Some effects (energy transport, quantum effects, … ) need improved models (higher moments, QDD, ..)


Poisson-Nernst-Planck

Poisson-Drift-Diff.


Numerical simulation is a strong challenge due to occurence of internal layers in the densities and due to nonlinear coupling with Poisson.


Electric Potential in a MESFET

Electric Potential in a synthetic channel(computed by M.Wolfram)

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

Densities in a synthethic channel

Ca2+

Na+

Cl-


Problem of highest technological importance is the identification of doping profiles (non-destructive 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

Fj(doping profile) = Current Measurementj

Evaluating each Fj 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


Less operating conditions are of interest for optimal design problems (usually only on- and off-state)

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 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


Joint work with

Marco Di Francesco (L‘Aquila)

Daniela Morale (Milano)

Enzo Capasso (Milano)

Yasmin Dolak-Struss (Wien)

Christian Schmeiser (Wien)

Herding / Aggregation Models


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

- Formation of galaxies- Crowding effects in molecular biology (ion

channels, chemotaxis)- Swarming of animals, humans (birds swarms, fish

populations, insect colonies, motion of human crowds, evacuation and panic)

- Volatility clustering and price herding in financial markets



Ctd.

- Traffic flow - Swarming of animals, humans (birds swarms, fish

population, insect colonies, evacuation and panic)- Opinion formation - …



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

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

Since interactions are not based on fundamental physical laws, the interaction potentials (also external potentials) are not known exactly



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]



Classical herding models with (long-range) attractive force lead to blow up (sometimes in finite time !)

Modelling Issues in Herding Models


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 !


Porfiri, Stilwell, Bollt 2006


Global interaction



Finite range interaction



To understand overcrowding and counteracting mechanisms, go back to microscopic models

- Diffusion / Brownian Motion leading to macroscopically linear diffusionPossibly not enough to prevent blow-up !

- Quorum sensing (particles not allowed to go to occupied locations) leading to decreasing effective aggregation for high densities

- Local repulsive forces leading to macroscopically nonlinear diffusion

Prevention of Overcrowding

Hillen-Painter 02

Raschev-Rüschendorf 97

Capasso-Morale-Oelschläger 04


Example: chemotaxis models with quorum sensing,

Formation of plateaus

Chemotaxis

mb-Dolak-DiFrancesco, SIAP 07


Inverse Problem 1: identify mobility from dynamic observations

Inverse Problem 2: control system to achieve a certain pattern of cells in finite time

Chemotaxis

McCarthy et al 07

Lebdiez-Maurer 04,McCarthy et al 05


Example:

swarming models

with local

repulsive force (small nonlinear diffusion)

Swarming

mb-Capasso-Morale 05mb-DiFrancesco 06


Joint work with

Enzo Capasso (Milano)

Alessandra Micheletti (Milano)

Livio Pizzochero (Milano)

Gerhard Eder (Linz)

Heinz Engl (Linz)

Bo Su (Iowa State)

Montell SpA (Ferrara), now Basell Polyolefins

Solidification of Polymers


Polymeric materials solidify over a large temperature range, between glass transition point and melting point by a process called crystallization (in analogy to the growth of crystalline structures)

Crystallization consists of nucleation of grains (randomly with a probability dependent on local temperature) and subsequent growth (with normal velocity dependent on local temperature)

Due to phase change latent heat is released, which influences the temperature evolution


Isotactic Poly-hydroxybutynate.Courtesy G.Eder, Phys. Chemistry, JKU Linz


Microscopic model taking into account lamellar structure of the material. By far too fine, can be coarse-grained to a „macroscopic model“ for the nucleation and growth of (almost spherical) grains




Phase-change model: PDE for temperature as a function of space and time, coupled to front growth model for grains and stochastic nucleation model (heterogeneous Poisson process)

Solved by finite differencing, level set method for grain growth


mb, Free Boundary Problems Proc. 2002


Large-Scale Simulation of Volume-Filling (i-pp)



mb-MichelettiJ.Math.Chem 2002


Large-Scale Simulation, Boundary Cooling (i-pp)



mb-MichelettiJ.Math.Chem 2002


Since there are 106 – 1018 grains in typical processes, even this „macroscopic“ phase change model can hardly be used for real-life predictions.

Further coarse-graining starting from phase change as „microscopic model“. Identify „mesoscopic“ size between the macroscopic size L and microscopic size l (size of single grains)




Mesoscale averaging: compute average volume fraction of solidified material in balls of radius Does not yield good reduced models.

Stochastic averaging: compute expected value of local phase function (=1 if point is inside the solidified region, 0 else). Reduced model can be obtained if nucleation is modeled as a Poisson process : generalization of Schneider rate equations. Variance of phase function is not small.

Mesoscale averaging of stochastic average reduces variance and yields computable models.



mb-Capasso-Pizzochero 06, mb-Capasso 01mb-Capasso-Eder 01, mb-Capasso-Salani 01


Macroscale model consisting of PDEs for temperature, mean volume fraction and auxiliary variables. Efficient simulation possible.



mb-Capasso 01Götz-Struckmeier 05


Inverse Problem: Identify nucleation rate (= rate of heterogeneous Poisson process) as a function of temperature.

Classical Technique: make single experiment for each temperature (sudden quench to respective temperature), count (high number) of grains at the end. Ratajski, Janeschitz-Kriegl 1996

Inverse Problem Technique: make single experiment with continuous decrease of temperature, identify rate from accessible temperature measurements at the boundary (by DSC).


mb-Capasso-Engl 99, mb 01


Nonlinear Inverse Problem: determine map F: nucleation rate → boundary temperature. Solve nonlinear equation

F(nucleation rate) = measured temperature

Map F is given only implicitely by solving the model for given nucleation rate.

Inverse problem is ill-posed (small data errors can lead to completely different solutions). We need to use sophisticated regularization techniques


mb-Capasso-Engl 99, mb 01


Synthetic data, 1% noise: reconstructed (primitive of) nucleation rate vs. exact one



Optimization problem: optimal control of the boundary heating to obtain „good structure“ at the end. Best mechanical properties for small grains of uniform size.

Define objective functional based on macroscopic models for total number of grains (maximized), local mean volume fraction (close to one), and local mean contact interface density (homogeneous)



Optimal switching of cooling temperature for 2d rectangular sample


Götz-Pinnau-Struckmeyer 06


Joint work with

Frank Hausser (CAESAR Bonn)

Christina Stöcker (CAESAR Bonn)

Axel Voigt (Dresden)

Growth of Nanostructures


Inverse Problem 1: identify parameters in the model from observed patterns (diffusion coefficients, kinetic coefficients)

Inverse Problem 2: obtain organization to ordered structure of islands of certain size on the thin film, by controlling temperature, deposition rate, prepatterning, applying electric field


Bauer et al 99


SiGe Nanostructures (and similar system) grow by a surface diffusion mechanism. Effective energy is influenced by elastic relaxation effects in the bulk (Si and Ge have different atomic lattices)

Microscopic model atomistic, KMC simulation. Can nowadays be upscaled to reasonable sizes for nanoscale system. But computation of elastic effects still causes too high computational effort.

Coarse-graining to semicontinuous BCF models (discrete in vertical directions) or directly continuum models of surface diffusion.


Bauer et al 99


Effective energies for vicinal nanosurfaces with elastic effects can be computed in continuum description (as functions of the slope). Possibly non-convex for compressive strain


Bauer et al 99

Shenoy 2004


Non-convexity of the energy causes faceting (only preferred slopes), can also cause backward diffusion effects in the surface evolution (theoretical and computational problems)

Regularization of the surface energy is needed in order to overcome the ill-posedness

Natural regularization is obtained by considering more than nearest neighbour interaction in the microscopic energy. This translates to curvature-dependent terms in the macroscopic energy


° =°0(n)+®· 2

° = ~°0(µ) +

®j@Sµj2


Curvature regularized surface energy yields a two-scale model: faceted surfaces (large scale) with rounded corners and edges (small scale)

Numerical simulation by surface representation as a graph or by level set method. Yields systems of stiff differential equations - efficient solution by variational discretization that preserves energy dissipation (large time steps possible).



Coarsening behaviour for random perturbations of flat surface


Bauer et al 99

mb, JCP 2005mb, Hausser, Stöcker, Voigt, JCP 2006


Papers and talks at

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

Email

[email protected]

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(Inverse) Multiscale Modelling Martin Burger Institut für Numerische und Angewandte Mathematik Westfälische Willhelms-Universität Münster