Optimization with PDE Constraints

Michael Hinze, Rene Pinnau, Michael Ulbrich, Stefan Ulbrich1402088388, 9781402088384

This book presents a modern introduction of pde constrained optimization. It provides a precise functional analytic treatment via optimality conditions and a state-of-the-art, non-smooth algorithmical framework. Furthermore, new structure-exploiting discrete concepts and large scale, practically relevant applications are presented. The main focus is on the algorithmical and numerical treatment of pde constrained optimization problems on the infinite dimensional level. A particular emphasis is on simple constraints, such as pointwise bounds on controls and states. For these practically important situations, tailored Newton- and SQP-type solution algorithms are proposed and a general convergence framework is developed. This is complemented with the numerical analysis of structure-preserving Galerkin schemes for optimization problems with elliptic and parabolic equations. Finally, along with the optimization of semiconductor devices and the optimization of glass cooling processes, two challenging applications of pde constrained optimization are presented. They demonstrate the scope of this emerging research field for future engineering applications.

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Table of contents :
Preface……Page 5
Acknowledgements……Page 6
Contents……Page 8
Introduction……Page 11
Examples for Optimization Problems with PDEs……Page 14
Boundary Control……Page 15
Boundary Control with Radiation Boundary……Page 16
Optimization of an Unsteady Heating Processes……Page 17
Optimal Design……Page 18
Linear Functional Analysis and Sobolev Spaces……Page 19
Basic Definitions……Page 20
Linear Operators and Dual Space……Page 21
Lebesgue Measurable Functions and Lebesgue Integral……Page 23
Definition of Lebesgue Spaces……Page 25
Density Results and Convergence Theorems……Page 27
Regular Domains and Integration by Parts……Page 28
Sobolev Spaces……Page 29
Poincaré’s Inequality……Page 31
Sobolev Imbedding Theorem……Page 32
The Dual Space H-1 of H01……Page 33
Weak Convergence……Page 34
Dirichlet Boundary Conditions……Page 36
Boundary Conditions of Robin Type……Page 40
Weak Solutions of Uniformly Elliptic Equations……Page 42
An Existence and Uniqueness Result for Semilinear Elliptic Equations……Page 43
Interior Regularity……Page 44
Boundary Regularity……Page 45
Weak Solutions of Parabolic PDEs……Page 46
Bochner Spaces……Page 47
Weak Solutions……Page 50
Abstract Parabolic Evolution Problem……Page 53
Energy Estimate and Uniqueness Result……Page 54
Existence Result by Galerkin Approximation……Page 55
Operator Formulation……Page 57
Regularity Results……Page 58
An Existence and Uniqueness Result for Semilinear Parabolic Equations……Page 59
Basic Definitions……Page 60
Existence Result for a General Linear-Quadratic Problem……Page 62
Existence Results for Nonlinear Problems……Page 64
Distributed Control of Elliptic Equations……Page 66
Reduced Problem, Sensitivities and Adjoints……Page 67
Sensitivity Approach……Page 68
Adjoint Approach……Page 69
Application to a Linear-Quadratic Optimal Control Problem……Page 70
A Lagrangian-Based View of the Adjoint Approach……Page 73
Second Derivatives……Page 74
Optimality Conditions for Simply Constrained Problems……Page 75
Optimality Conditions for Control-Constrained Problems……Page 80
A General First Order Optimality Condition……Page 81
Necessary First Order Optimality Conditions……Page 82
General Linear-Quadratic Problem……Page 83
Distributed Control of Elliptic Equations……Page 84
Distributed Control of Semilinear Elliptic Equations……Page 86
Boundary Control of Parabolic Equations……Page 88
Optimality Conditions for Problems with General Constraints……Page 90
A Basic First Order Optimality Condition……Page 91
Constraint Qualification and Robinson’s Regularity Condition……Page 92
Karush-Kuhn-Tucker Conditions……Page 93
Application to PDE-Constrained Optimization……Page 94
Elliptic Problem with State Constraints……Page 96
Optimal Control of Instationary Incompressible Navier-Stokes Flow……Page 98
Functional Analytic Setting……Page 99
Analysis of the Flow Control Problem……Page 101
Reduced Optimal Control Problem……Page 104
Synopsis……Page 106
Unconstrained Optimization……Page 108
Armijo Rule……Page 110
Optimization on Closed Convex Sets……Page 113
Projected Armijo Rule……Page 116
Unconstrained Problems-Newton’s Method……Page 118
Nonsmooth Reformulation Approach and Generalized Newton Methods……Page 119
SQP Methods……Page 121
General Inequality Constraints……Page 122
Nonsmooth Reformulation Approach and Generalized Newton Methods……Page 123
Motivation: Application to Optimal Control……Page 124
A General Superlinear Convergence Result……Page 125
The Classical Newton’s Method……Page 128
Generalized Differential and Semismoothness……Page 129
Semismooth Newton Methods……Page 132
Semismooth Newton Method for Finite Dimensional KKT Systems……Page 133
Pointwise Bound Constraints in L2……Page 134
Semismoothness of Superposition Operators……Page 135
Pointwise Bound Constraints in L2 Revisited……Page 138
Application to Optimal Control……Page 139
General Optimization Problems with Inequality Constraints in L2……Page 141
Distributed Control……Page 142
Neumann Boundary Control……Page 144
Optimal Control of the Incompressible Navier-Stokes Equations……Page 146
Lagrange-Newton Methods for Equality Constrained Problems……Page 149
Generalized Equation……Page 153
SQP Methods for Inequality Constrained Problems……Page 157
SQP Subproblem……Page 159
State-Constrained Problems……Page 160
Semismooth Newton Methods……Page 161
Moreau-Yosida Regularization……Page 162
Lavrentiev Regularization……Page 163
Mesh Independence……Page 164
Other Methods……Page 165
Introduction……Page 166
Stationary Model Problem……Page 167
First Discretize, Then Optimize……Page 169
First Optimize, Then Discretize……Page 170
Discussion and Implications……Page 172
The Variational Discretization Concept……Page 173
Error Estimates……Page 176
Uniform Estimates……Page 179
Numerical Examples for Distributed Control……Page 180
Boundary Control……Page 186
Neumann and Robin-Type Boundary Control……Page 187
Numerical Examples for Robin-Type Boundary Control……Page 192
Dirichlet Boundary Control……Page 198
Numerical Example for Dirichlet Boundary Control……Page 202
Some Literature Related to Control Constraints……Page 205
Constraints on the State……Page 206
Pointwise Bounds on the State……Page 207
Finite Element Discretization……Page 208
Error Analysis……Page 212
Piecewise Constant Controls……Page 216
Numerical Examples for Pointwise Constraints on the State……Page 221
Some Literature for (Control and) State Constraints……Page 227
Pointwise Bounds on the Gradient of the State……Page 228
Finite Element Discretization……Page 230
A Numerical Experiment with Pointwise Constraints on the Gradient……Page 235
Mathematical Model, State Equation……Page 236
Discretization……Page 238
Further Literature on Control of Time-Dependent Problems……Page 240
Optimal Semiconductor Design……Page 242
Semiconductor Device Physics……Page 243
Charge Transport……Page 244
The Potential Equation……Page 245
The Continuity Equations……Page 246
The Current Densities……Page 247
Scaling……Page 248
The Optimization Problem……Page 249
The First-Order Optimality System……Page 253
Steepest Descent……Page 255
The Reduced Newton Method……Page 257
Optimal Control of Glass Cooling……Page 259
Radiation……Page 260
SPN-approximations……Page 262
Optimal Boundary Control……Page 263
Derivatives……Page 266
Newton’s Method……Page 267
Numerical Results……Page 269
References……Page 273