Inverse Multiscale Modelling Martin Burger Institut für Numerische und Angewandte Mathematik European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische Willhelms-Universität Münster

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen „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

Seminar, AICES, RWTH Aachen The computational issues in the solution of inverse problems raises the need for deriving reduced macroscopic or multiscale models, and efficient numerical solution This talk will give several examples from various application fields Introduction

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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) Electron / Ion Transport Poisson-Nernst- Planck Poisson-Drift- Diffusion

Seminar, AICES, RWTH Aachen 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)

Seminar, AICES, RWTH Aachen Size exclusion in ion channels significantly increases computational effort (nonlocal functionals in DFT) Electron / Ion Transport Densities and Potential in an L- type Ca channel (PNP-DFT) Densities in a synthethic channel Ca 2+ Na + Cl -

Seminar, AICES, RWTH Aachen Voltage enters as boundary value for the electric potential, current is computed from boundary flux of the electrons / ions Boundary concentrations satisfy charge neutrality, uniquely determined in semiconductors, varied quite arbitrarily in ion channels Some effects (energy transport, quantum effects, … ) need improved models (higher moments, QDD,..) Size exclusion in in ion channels needs additional treatment (MC, Boltzmann, DFT) Electron / Ion Transport Poisson-Nernst- Planck Poisson-Drift-Diff.

Seminar, AICES, RWTH Aachen 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) F k (doping profile) = Current Measurement k Evaluating each F k means to solve the model once Identification of Doping Profiles mb-Engl-Markowich-Pietra 01 mb-Engl-Markowich 01, mb-Engl-Leitao-Markowich 04

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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 / 06 mb-Pinnau 03

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen Joint work with Marco Di Francesco (L‘Aquila) Daniela Morale (Milano) Enzo Capasso (Milano) Yasmin Dolak-Struss (Wien) Christian Schmeiser (Wien) Herding / Aggregation Models

Seminar, AICES, RWTH Aachen 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

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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen Classical herding models with (long-range) attractive force lead to blow up (sometimes in finite time !) Modelling Issues in Herding Models

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen Key issue for stable schemes is a metric gradient flow formulation on a manifold of probability measures equipped with the 2-Wasserstein metric Energy can be nonconvex depending on size of G. For A=0, energy minimizers are only concentrated states (Dirac-deltas) Modelling Issues in Herding Models E [ ½ ] = Z A ( ½ ) d x Z ½ ( G ¤ ½ ) d x Ambrosio, Gigli, Savare 05, mb-DiFrancesco 07

Seminar, AICES, RWTH Aachen Natural (implicit) time discretization of a metric gradient flow with metric d and energy E, concept of minimizing movements Discretization to Stable Numerical Schemes

Seminar, AICES, RWTH Aachen Discrete manifold e.g. by finite elements, discrete approximations of metric and energy e.g. by quadratic Taylor-expansion around last time step with flux Discretization in H div, e.g. with Raviart-Thomas Stable Numerical Schemes

Seminar, AICES, RWTH Aachen Porous Medium Equation, m=2 Entropy

Seminar, AICES, RWTH Aachen Porous Medium Equation, m=2 Error (L 2 )

Seminar, AICES, RWTH Aachen Porous Medium Equation, m=2 Porous medium in multi-D (implementation by M.Wolfram in NGSolve)

Seminar, AICES, RWTH Aachen Porous Medium Equation, m=3 Entropy

Seminar, AICES, RWTH Aachen Thin Film equation Entropy Additional work due to natural choice of FE basis for metric gradient flow: nonconforming approximation of the entropy

Seminar, AICES, RWTH Aachen Thin Film equation Entropy

Seminar, AICES, RWTH Aachen Thin Film equation Entropy

Seminar, AICES, RWTH Aachen Global interaction, no diffusion Modelling Issues in Herding Models

Seminar, AICES, RWTH Aachen Finite range interaction, no diffusion Modelling Issues in Herding Models

Seminar, AICES, RWTH Aachen Swarming mb-Capasso-Morale 05 mb-DiFrancesco 06 Example: swarming models with local repulsive force (small nonlinear diffusion)

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen Example: chemotaxis models with quorum sensing Chemotaxis mb-Dolak-DiFrancesco, SIAP 07 mb-DiFrancesco- Dolak 06

Seminar, AICES, RWTH Aachen Example: chemotaxis models with quorum sensing, Formation of plateaus Chemotaxis mb-Dolak-DiFrancesco, SIAP 07

Seminar, AICES, RWTH Aachen Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity: Plateau motion

Seminar, AICES, RWTH Aachen Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity Asymptotics at hyperbolic time-scale

Seminar, AICES, RWTH Aachen Chemotaxis Limit is a nonlinear, nonlocal conservation law: we need entropy solutions Entropy inequality

Seminar, AICES, RWTH Aachen Chemotaxis Stationary solutions These are entropy solutions iff

Seminar, AICES, RWTH Aachen Chemotaxis Asymptotics for large time by time rescaling Look for limiting solutions

Seminar, AICES, RWTH Aachen Chemotaxis Asymptotic expansion in interfacial layer (as for Cahn-Hilliard) Note: entropy condition

Seminar, AICES, RWTH Aachen Chemotaxis We obtain a surface diffusion law with diffusivity and chemical potential Corresponding energy functional D = ¡ n S ¹ = ¡ S 2 = ¡ S [ ­ ] 2

Seminar, AICES, RWTH Aachen Chemotaxis Flow is volume conserving Flow has energy dissipation property

Seminar, AICES, RWTH Aachen Chemotaxis Stability of stationary solutions can be studied based on second (shape) variations on the energy functional Stability condition for normal perturbation Instability without entropy condition ! Otherwise high-frequency stability, possible low- frequency instability

Seminar, AICES, RWTH Aachen Chemotaxis Instability

Seminar, AICES, RWTH Aachen Chemotaxis Instability

Seminar, AICES, RWTH Aachen Chemotaxis Instability

Seminar, AICES, RWTH Aachen Chemotaxis Surface Diffusion

Seminar, AICES, RWTH Aachen Chemotaxis Surface Diffusion

Seminar, AICES, RWTH Aachen Chemotaxis Surface Diffusion, 3D

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen Synthetic data, 1% noise: reconstructed (primitive of) nucleation rate vs. exact one Solidification of Polymers

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen Optimal switching of cooling temperature for 2d rectangular sample Solidification of Polymers Götz-Pinnau-Struckmeyer 06

Seminar, AICES, RWTH Aachen Joint work with Frank Hausser (CAESAR Bonn) Christina Stöcker (CAESAR Bonn) Axel Voigt (Dresden) Growth of Nanostructures

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen 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

Seminar, AICES, RWTH Aachen Surface diffusion appears in many important applications - in particular in material and nano science Growth of a surface  with velocity Surface Diffusion

Seminar, AICES, RWTH Aachen F... Deposition flux D s.. Diffusion coefficient ... Atomic volume ... Surface density k... Boltzmann constant T... Temperature n... Unit outer normal ... Chemical potential = energy variation Surface Diffusion

Seminar, AICES, RWTH Aachen Level Set / Graph Formulation Level set function or graph parametrization u of surface determined from - (graph) (level t u = ¡ d i v ( Pr · ) · = d i v ( r u Q ) P = Q ( I r u Q ­ r u Q ) Q = p 1 + j r u j 2 Q = j r u j

Seminar, AICES, RWTH Aachen Level Set Formulation We have to deal with fourth-order equation, no maximum principle No global level set formulation Efficient computations and proofs still widely open (One of the „major mathematical challenges in materials science“, Jean Taylor, AMS, 2002 / Robert Kohn, SIAM, 2002)

Seminar, AICES, RWTH Aachen SD can be obtained as the limit (  →0) of minimization subject to Minimizing Movement: SD

Seminar, AICES, RWTH Aachen Level set version: subject to Minimizing Movement: SD

Seminar, AICES, RWTH Aachen Basic idea: Semi-implicit time discretization + Splitting into two / three second-order equations + Finite element discretization in space Natural variables for splitting: Height u, Mean Curvature , Chemical potential  (denoted u, v, w in the following) Numerical Solution

Seminar, AICES, RWTH Aachen Based on variational principle, minimizing movement subject to Time Discretization

Seminar, AICES, RWTH Aachen Quadratic approximation of the convex terms in the energy, linear approximation of the non- convex terms around u(t) Rewrite local variational problem as minimization over u, v, and w With constraints defining v and w KKT condition yields indefinite linear system, Lagrangian variables are multiples of v and w Time Discretization

Seminar, AICES, RWTH Aachen Discretization of the variational problem in space by piecewise linear finite elements and P(u) are piecewise constant on the triangularization, all integrals needed for stiffness matrix and right-hand side can be computed exactly Spatial Discretization

Seminar, AICES, RWTH Aachen After few manipulations we obtain indefinite linear system for the nodal values A stiffness matrix from diffusion coefficient 1/Q B stiffness matrix from diffusion coefficient P/Q M mass matrix for identity, C mass matrix for 1/Q Iterative solution by multigrid-precond. GMRES Discrete Problem

Seminar, AICES, RWTH Aachen SD  = 3.5,  = 0.02,  

Seminar, AICES, RWTH Aachen SD  = 1.5,  = 0.02,  

Seminar, AICES, RWTH Aachen Faceting Graph Simulation: mb JCP 04, Level Set Simulation: mb-Hausser-Stöcker-Voigt 06 Adaptive FE grid around zero level set

Seminar, AICES, RWTH Aachen Faceting Anisotropic mean curvature flow

Seminar, AICES, RWTH Aachen Faceting of Thin Films Anisotropic Mean Curvature Anisotropic Surface Diffusion mb 04, mb-Hausser- Stöcker-Voigt-05

Seminar, AICES, RWTH Aachen Faceting of Bulk Crystals Anisotropic surface diffusion

Seminar, AICES, RWTH Aachen 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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