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

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

CeNoS Kolloquium 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

CeNoS Kolloquium 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

CeNoS Kolloquium 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

CeNoS Kolloquium 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

CeNoS Kolloquium 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

CeNoS Kolloquium „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

CeNoS Kolloquium 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

CeNoS Kolloquium 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

CeNoS Kolloquium 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) Electron / Ion Transport Ion Channel Courtesy Bob Eisenberg MOSFETS, from

CeNoS Kolloquium 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 Electron / Ion Transport

CeNoS Kolloquium 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) Electron / Ion Transport

CeNoS Kolloquium 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) Electron / Ion Transport Sketch of l-type CaChannel Sketch of geometry of a MESFET Mock 84, Markowich 86, Markowich- Ringhofer-Schmeiser 90, Jüngel 2002 Eisenberg et al 01-06

CeNoS Kolloquium 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,..) Electron / Ion Transport Poisson-Nernst- Planck Poisson-Drift-Diff.

CeNoS Kolloquium Numerical simulation is a strong challenge due to occurence of internal layers in the densities and due to nonlinear coupling with Poisson. Electron / Ion Transport 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 Ca 2+ Na + Cl -

CeNoS Kolloquium 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 F j (doping profile) = Current Measurement j Evaluating each F j means to solve the model once Identification of Doping Profiles mb-Engl-Markowich-Pietra 01 mb-Engl-Markowich 01, mb-Engl-Leitao-Markowich 04

CeNoS Kolloquium 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

CeNoS Kolloquium 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 / 06 mb-Pinnau 03

CeNoS Kolloquium 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

CeNoS Kolloquium Joint work with Marco Di Francesco (L‘Aquila) Daniela Morale (Milano) Enzo Capasso (Milano) Yasmin Dolak-Struss (Wien) Christian Schmeiser (Wien) Herding / Aggregation Models

CeNoS Kolloquium 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 Herding / Aggregation Models

CeNoS Kolloquium Ctd. -Traffic flow -Swarming of animals, humans (birds swarms, fish population, insect colonies, evacuation and panic) -Opinion formation -… Herding / Aggregation Models

CeNoS Kolloquium 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 Herding / Aggregation Models

CeNoS Kolloquium 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] Herding / Aggregation Models

CeNoS Kolloquium Classical herding models with (long-range) attractive force lead to blow up (sometimes in finite time !) Modelling Issues in Herding Models

CeNoS Kolloquium 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 ! Modelling Issues in Herding Models Porfiri, Stilwell, Bollt 2006

CeNoS Kolloquium Global interaction Modelling Issues in Herding Models

CeNoS Kolloquium Finite range interaction Modelling Issues in Herding Models

CeNoS Kolloquium To understand overcrowding and counteracting mechanisms, go back to microscopic models -Diffusion / Brownian Motion leading to macroscopically linear diffusion Possibly 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

CeNoS Kolloquium Example: chemotaxis models with quorum sensing, Formation of plateaus Chemotaxis mb-Dolak-DiFrancesco, SIAP 07

CeNoS Kolloquium 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

CeNoS Kolloquium Example: swarming models with local repulsive force (small nonlinear diffusion) Swarming mb-Capasso-Morale 05 mb-DiFrancesco 06

CeNoS Kolloquium 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

CeNoS Kolloquium 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 Solidification of Polymers Isotactic Poly- hydroxybutynate. Courtesy G.Eder, Phys. Chemistry, JKU Linz

CeNoS Kolloquium 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 Solidification of Polymers Isotactic Poly- hydroxybutynate. Courtesy G.Eder, Phys. Chemistry, JKU Linz

CeNoS Kolloquium 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 Solidification of Polymers mb, Free Boundary Problems Proc. 2002

CeNoS Kolloquium Large-Scale Simulation of Volume-Filling (i-pp) Solidification of Polymers Isotactic Poly- hydroxybutynate. Courtesy G.Eder, Phys. Chemistry, JKU Linz mb-Micheletti J.Math.Chem 2002

CeNoS Kolloquium Large-Scale Simulation, Boundary Cooling (i-pp) Solidification of Polymers Isotactic Poly- hydroxybutynate. Courtesy G.Eder, Phys. Chemistry, JKU Linz mb-Micheletti J.Math.Chem 2002

CeNoS Kolloquium Since there are 10 6 – 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) Solidification of Polymers Isotactic Poly- hydroxybutynate. Courtesy G.Eder, Phys. Chemistry, JKU Linz

CeNoS Kolloquium 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. Solidification of Polymers Isotactic Poly- hydroxybutynate. Courtesy G.Eder, Phys. Chemistry, JKU Linz mb-Capasso-Pizzochero 06, mb-Capasso 01 mb-Capasso-Eder 01, mb-Capasso-Salani 01

CeNoS Kolloquium Macroscale model consisting of PDEs for temperature, mean volume fraction and auxiliary variables. Efficient simulation possible. Solidification of Polymers Isotactic Poly- hydroxybutynate. Courtesy G.Eder, Phys. Chemistry, JKU Linz mb-Capasso 01 Götz-Struckmeier 05

CeNoS Kolloquium 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). Solidification of Polymers mb-Capasso-Engl 99, mb 01

CeNoS Kolloquium 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 Solidification of Polymers mb-Capasso-Engl 99, mb 01

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

CeNoS Kolloquium 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) Solidification of Polymers

CeNoS Kolloquium Optimal switching of cooling temperature for 2d rectangular sample Solidification of Polymers Götz-Pinnau-Struckmeyer 06

CeNoS Kolloquium Joint work with Frank Hausser (CAESAR Bonn) Christina Stöcker (CAESAR Bonn) Axel Voigt (Dresden) Growth of Nanostructures

CeNoS Kolloquium 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 Growth of Nanostructures Bauer et al 99

CeNoS Kolloquium 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. Growth of Nanostructures Bauer et al 99

CeNoS Kolloquium 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 Growth of Nanostructures Bauer et al 99 Shenoy 2004

CeNoS Kolloquium 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 Growth of Nanostructures ° = ° 0 ( n ) + ®· 2 ° = ~ ° 0 ( µ ) + ® S µ j 2

CeNoS Kolloquium 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). Growth of Nanostructures

CeNoS Kolloquium Coarsening behaviour for random perturbations of flat surface Growth of Nanostructures Bauer et al 99 mb, JCP 2005 mb, Hausser, Stöcker, Voigt, JCP 2006

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