Shape Optimization, Homogenization and Optimal Control: DFG-AIMS workshop held at the AIMS Center Senegal, March 13-16, 2017 (International Series of Numerical Mathematics, 169)

The contributions in this volume give an insight into current research activities in Shape Optimization, Homogenization and Optimal Control performed in Africa, Germany and internationally. Seeds for collaboration can be found in the first four papers in the field of homogenization. Modelling and optimal control in partial differential equations is the topic of the next six papers, again mixed from Africa and Germany. Finally, new results in the field of shape optimization are discussed in the final international three papers. This workshop, held at the AIMS Center Senegal, March 13-16, 2017, has been supported by the Deutsche Forschungsgemeinschaft (DFG) and by the African Institute for Mathematical Sciences (AIMS) in Senegal, which is one of six centres of a pan-African network of centres of excellence for postgraduate education, research and outreach in mathematical sciences.-- Provided by publisher Preface 6 Contents 7 Homogenization in Magnetic-Shape-Memory Polymer Composites 9 1 Introduction 10 2 The Model 11 3 Analytical Homogenization 13 4 Numerical Study of the Cell Problem 20 5 Conclusions and Outlook 23 References 24 Homogenization of Stochastic Parabolic Equationsin Varying Domains 26 1 Introduction 26 2 Homogenization of Semilinear Stochastic Parabolic Equations in a Domain with Fine Grained Boundaries 29 2.1 Our Notion of Solution and Its Existence 31 2.2 Probability Measures Generated by the Sequence ( W( s) ,u( s) ) and Their Tightness 34 2.3 Cell Problem 36 2.4 Main Result and Ideas of Its Proof 38 2.4.1 Ideas of the Proof of Theorem 8 39 3 Homogenization of Stochastic Heat Equation in Perforated Domains: The Cioranescu-Murat Approach 42 4 Homogenization of Stochastic Heat Equation in Varying Domains 45 References 45 Two-Scale Convergence: Obviousness of the Choice of Test Functions? Not Always 48 1 Introduction 48 2 Fundamentals of Homogenization 51 3 An Appropriate Variational Formulation and Test Functions 52 3.1 Preliminaries 52 3.2 A Suitable Variational Equation 54 3.3 Building of Appropriate Test Functions 55 4 The Homogenization Result for fC1([0,T];L2(Ω;R)), f(0)=ω 56 References 62 Corrector Problem and Homogenization of Nonlinear Elliptic Monotone PDE 63 1 Introduction 63 2 Proofs of Theorems 1.1 and 1.3 69 3 Proof of Theorem 1.4 76 References 79 Instantaneous Optimal Control of Friction Dominated Flow in a Gas-Network 80 1 Introduction 80 1.1 Modeling of Gas Flow in a Single Pipe 80 1.2 Network Modeling 82 1.2.1 Optimal Control Problems and Outline 83 1.3 Time Discretization 84 2 Well-Posedness 86 3 Optimal Control 89 References 93 Multilevel Iterations for the Combined Moving Horizon Estimation and Nonlinear Model Predictive Control for PDE Models 94 1 Introduction 95 2 Problem Formulation: NMPC and MHE 96 3 Decomposition of the MHE Problem 98 4 Decomposition of the NMPC Problem 104 5 Multilevel Iterations 108 5.1 Establishing the Reduced Problems 108 5.2 Iteration on the Level A 110 5.3 Iteration on the Level B 111 5.4 Iteration on the Level C 112 6 Numerical Results 113 7 Conclusions 116 References 116 Some Recent Developments in Optimal Control of Multiphase Flows 118 1 Introduction 118 1.1 Notation 122 2 Existence of Solutions of the Cahn–Hilliard–Navier–Stokes System 122 2.1 The Semi-Discrete CHNS System 124 2.2 Existence of Solutions 127 3 The Optimal Control Problem 128 4 Stationarity Conditions 129 4.1 Boundary Control 135 5 Numerical Solution Methods 136 5.1 Adaptive Mesh Refinement Techniques 136 5.2 Numerical Results 139 6 Future Directions 143 References 144 Localized Model Reduction in PDE Constrained Optimization 148 1 Introduction 148 2 Weak Formulation for the Parameterized Multiscale Problem and Non-conforming Reference Approximation 150 3 A General Non-conforming Weak Formulation for Numerical Multiscale Methods 151 4 A General Framework for Localized Model Reduction Based on Numerical Multiscale Methods with Oversampling 153 5 The Reduced Basis Heterogeneous Multiscale Method (RB-HMM) 154 5.1 A Posteriori Error Estimation 155 5.2 Optimization and Computation of Derivative Information 156 5.3 Numerical Experiments 157 6 The Localized Reduced Basis Multiscale Method (LRBMS) with Online Enrichment 160 6.1 A Posteriori Error Estimation 161 6.2 Optimization and Online Enrichment 162 6.3 Numerical Experiment 163 References 166 Numerical Simulation for a Dimensionless Coupled System of Shallow Water Equation with Long Term Dynamic of Sand Dunes Equation 169 1 Introduction and Model 169 2 Finite Volume Scheme for the Dimensionless Coupled System (SW-LTDD) 172 2.1 Finite Volume Scheme for the Dimensionless Shallow Water Equation (SWE) 173 2.2 Finite Volume Scheme for the Dimensionless Long Term Dynamic of Dunes of sand (LTDD) 176 2.3 Stability of the Scheme 178 2.3.1 Stability of the LTDD Scheme 179 3 Numerical Simulation 182 3.1 First Test 183 3.2 Second Test 184 3.3 Third Test 187 3.4 Fourth Test 190 4 Conclusion and Perspectives 193 References 193 Coupling the Navier-Stokes Equations with a Short Term Dynamic of Sand Dunes 195 1 Introduction and Modeling 196 2 Scaling of the Reference Model and Parametrized Model 197 3 On the Incompressible Navier-Stokes Equation 200 4 Asymptotics Analysis of the Navier-Stokes Equation 205 References 214 Branching Structures in Elastic Shape Optimization 216 1 Introduction 217 2 Composite Structures 218 3 Elastic State Equation and Compliance Optimization 220 4 Discretization 221 5 Shape Optimization 224 6 Results 225 References 228 Shape and Topological Derivatives via One Sided Differentiation of the Minimax of Lagrangian Functionals 229 1 Introduction 230 2 Illustrative Examples from PDE Control 231 3 Shape Derivative via the Velocity Method 233 3.1 Simple Illustrative Example 233 3.2 Generic Construction of Micheletti: Metric Group, Courant Metric, and Its Tangent Space 234 3.3 Examples of and Continuity with Respect to the Courant Metric 235 3.4 Hadamard Semidifferentiability 236 4 State Constrained Objective Functions: Averaged Adjoint 237 4.1 Abstract Framework 237 4.1.1 Preliminaries 237 4.1.2 Minimax and Averaged Adjoint 238 4.2 Original Condition of Sturm 240 4.3 Sturm's Theorem Revisited and Extended: A New Condition 241 4.4 A New Condition with the Standard Adjoint 243 4.5 Another Form of Hypothesis (H3') 243 4.6 Back to the Simple Illustrative Example from PDE Control 244 5 Example of a Topological Derivative: Non-zero Extra Term 246 5.1 Topological Derivative 246 5.2 One-Dimensional Example: Objective Function and State Equation 246 5.3 Perturbed Problems via Dilation of the Point a 247 5.4 Non-convex Lagrangian and Standard Adjoint Equation 249 5.5 Computation of the Derivative dtG(t,φ,ψ) 250 5.6 Averaged Adjoint State Equation 250 5.7 Material Derivative 252 6 Multivalued Case: Two Theorems without and with the Extra Term 252 6.1 First Theorem: Mild Generalization 253 6.2 Second Theorem: General Case 253 6.3 Second Theorem: A Non-convex Example Where X(0) is Not a Singleton 254 References 258 On Optimization Transfer Operators in Shape Spaces 260 1 Introduction 260 2 Overview of Shape Spaces 261 3 Optimization Methods on the Space of Smooth Shapes 262 4 Optimization Methods on the Space of H1/2-Shapes 267 References 274 Front Matter ....Pages i-viii Homogenization in Magnetic-Shape-Memory Polymer Composites (Sergio Conti, Martin Lenz, Matthäus Pawelczyk, Martin Rumpf)....Pages 1-17 Homogenization of Stochastic Parabolic Equations in Varying Domains (Mogtaba A. Y. Mohammed, Mamadou Sango)....Pages 19-40 Two-Scale Convergence: Obviousness of the Choice of Test Functions? Not Always (Hubert Nnang)....Pages 41-55 Corrector Problem and Homogenization of Nonlinear Elliptic Monotone PDE (Jean Louis Woukeng)....Pages 57-73 Instantaneous Optimal Control of Friction Dominated Flow in a Gas-Network (Günter Leugering, Gisèle Mophou)....Pages 75-88 Multilevel Iterations for the Combined Moving Horizon Estimation and Nonlinear Model Predictive Control for PDE Models (Ekaterina Kostina, Gregor Kriwet)....Pages 89-112 Some Recent Developments in Optimal Control of Multiphase Flows (Michael Hintermüller, Tobias Keil)....Pages 113-142 Localized Model Reduction in PDE Constrained Optimization (Mario Ohlberger, Michael Schaefer, Felix Schindler)....Pages 143-163 Numerical Simulation for a Dimensionless Coupled System of Shallow Water Equation with Long Term Dynamic of Sand Dunes Equation (Mouhamadou A. M. T. Baldé, Diaraf Seck)....Pages 165-190 Coupling the Navier-Stokes Equations with a Short Term Dynamic of Sand Dunes (Ibrahima Faye, Mariama Ndiaye, Diaraf Seck)....Pages 191-211 Branching Structures in Elastic Shape Optimization (Nora Lüthen, Martin Rumpf, Sascha Tölkes, Orestis Vantzos)....Pages 213-225 Shape and Topological Derivatives via One Sided Differentiation of the Minimax of Lagrangian Functionals (Michel C. Delfour)....Pages 227-257 On Optimization Transfer Operators in Shape Spaces (Volker H. Schulz, Kathrin Welker)....Pages 259-275