Hinze / Pinnau / Ulbrich Optimization with PDE Constraints

1. Auflage 2008
ISBN: 978-1-4020-8839-1
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 270 Seiten

Solving optimization problems subject to constraints given in terms of partial d- ferential equations (PDEs) with additional constraints on the controls and/or states is one of the most challenging problems in the context of industrial, medical and economical applications, where the transition from model-based numerical si- lations to model-based design and optimal control is crucial. For the treatment of such optimization problems the interaction of optimization techniques and num- ical simulation plays a central role. After proper discretization, the number of op- 3 10 timization variables varies between 10 and 10 . It is only very recently that the enormous advances in computing power have made it possible to attack problems of this size. However, in order to accomplish this task it is crucial to utilize and f- ther explore the speci?c mathematical structure of optimization problems with PDE constraints, and to develop new mathematical approaches concerning mathematical analysis, structure exploiting algorithms, and discretization, with a special focus on prototype applications. The present book provides a modern introduction to the rapidly developing ma- ematical ?eld of optimization with PDE constraints. The ?rst chapter introduces to the analytical background and optimality theory for optimization problems with PDEs. Optimization problems with PDE-constraints are posed in in?nite dim- sional spaces. Therefore, functional analytic techniques, function space theory, as well as existence- and uniqueness results for the underlying PDE are essential to study the existence of optimal solutions and to derive optimality conditions.

Autoren/Hrsg.

Hinze, Michael

Pinnau, Rene

Ulbrich, Michael

Ulbrich, Stefan

Inhaltsverzeichnis

1;Preface;6
1.1;Acknowledgements;7
2;Contents;9
3;Analytical Background and Optimality Theory;12
3.1;Introduction and Examples;12
3.1.1;Introduction;12
3.1.2;Examples for Optimization Problems with PDEs;15
3.1.3;Optimization of a Stationary Heating Process;16
3.1.3.1;Boundary Control;16
3.1.3.2;Boundary Control with Radiation Boundary;17
3.1.3.3;Distributed Control;18
3.1.3.4;Problems with State Constraints;18
3.1.4;Optimization of an Unsteady Heating Processes;18
3.1.4.1;Boundary Control;19
3.1.5;Optimal Design;19
3.2;Linear Functional Analysis and Sobolev Spaces;20
3.2.1;Banach and Hilbert Spaces;21
3.2.1.1;Basic Definitions;21
3.2.1.2;Linear Operators and Dual Space;22
3.2.2;Sobolev Spaces;24
3.2.2.1;Lebesgue Spaces;24
3.2.2.2;Lebesgue Measurable Functions and Lebesgue Integral;24
3.2.2.3;Definition of Lebesgue Spaces;26
3.2.2.4;Density Results and Convergence Theorems;28
3.2.2.5;Weak Derivatives;29
3.2.2.6;Regular Domains and Integration by Parts;29
3.2.2.7;Sobolev Spaces;30
3.2.2.8;Poincaré's Inequality;32
3.2.2.9;Sobolev Imbedding Theorem;33
3.2.2.10;The Dual Space H-1 of H01;34
3.2.3;Weak Convergence;35
3.3;Weak Solutions of Elliptic and Parabolic PDEs;37
3.3.1;Weak Solutions of Elliptic PDEs;37
3.3.1.1;Weak solutions of the Poisson equation;37
3.3.1.2;Dirichlet Boundary Conditions;37
3.3.1.3;Boundary Conditions of Robin Type;41
3.3.1.4;Weak Solutions of Uniformly Elliptic Equations;43
3.3.1.5;An Existence and Uniqueness Result for Semilinear Elliptic Equations;44
3.3.1.6;Regularity Results;45
3.3.1.7;Interior Regularity;45
3.3.1.8;Boundary Regularity;46
3.3.2;Weak Solutions of Parabolic PDEs;47
3.3.2.1;Uniformly Parabolic Equations;48
3.3.2.2;Bochner Spaces;48
3.3.2.3;Weak Solutions of Uniformly Parabolic Equations;51
3.3.2.4;Weak Solutions;51
3.3.2.5;Existence and Uniqueness of Weak Solutions;54
3.3.2.6;Abstract Parabolic Evolution Problem;54
3.3.2.7;Abstract Parabolic Evolution Problem, Equivalent Form;55
3.3.2.8;Energy Estimate and Uniqueness Result;55
3.3.2.9;Existence Result by Galerkin Approximation;56
3.3.2.10;Operator Formulation;58
3.3.2.11;Regularity Results;59
3.3.2.12;An Existence and Uniqueness Result for Semilinear Parabolic Equations;60
3.4;Gâteaux- and Fréchet Differentiability;61
3.4.1;Basic Definitions;61
3.4.2;Implicit Function Theorem;63
3.5;Existence of Optimal Controls;63
3.5.1;Existence Result for a General Linear-Quadratic Problem;63
3.5.2;Existence Results for Nonlinear Problems;65
3.5.3;Applications;67
3.5.3.1;Distributed Control of Elliptic Equations;67
3.5.3.2;Boundary Control of Semilinear Elliptic Equations;68
3.6;Reduced Problem, Sensitivities and Adjoints;68
3.6.1;Sensitivity Approach;69
3.6.2;Adjoint Approach;70
3.6.3;Application to a Linear-Quadratic Optimal Control Problem;71
3.6.4;A Lagrangian-Based View of the Adjoint Approach;74
3.6.5;Second Derivatives;75
3.7;Optimality Conditions;76
3.7.1;Optimality Conditions for Simply Constrained Problems;76
3.7.2;Optimality Conditions for Control-Constrained Problems;81
3.7.2.1;A General First Order Optimality Condition;82
3.7.2.2;Necessary First Order Optimality Conditions;83
3.7.2.3;Applications;84
3.7.2.4;General Linear-Quadratic Problem;84
3.7.2.5;Distributed Control of Elliptic Equations;85
3.7.2.6;Distributed Control of Semilinear Elliptic Equations;87
3.7.2.7;Boundary Control of Parabolic Equations;89
3.7.3;Optimality Conditions for Problems with General Constraints;91
3.7.3.1;A Basic First Order Optimality Condition;92
3.7.3.2;Constraint Qualification and Robinson's Regularity Condition;93
3.7.3.3;Karush-Kuhn-Tucker Conditions;94
3.7.3.4;Application to PDE-Constrained Optimization;95
3.7.3.5;Applications;97
3.7.3.6;Elliptic Problem with State Constraints;97
3.8;Optimal Control of Instationary Incompressible Navier-Stokes Flow;99
3.8.1;Functional Analytic Setting;100
3.8.2;Analysis of the Flow Control Problem;102
3.8.3;Reduced Optimal Control Problem;105
4;Optimization Methods in Banach Spaces;107
4.1;Synopsis;107
4.2;Globally Convergent Methods in Banach Spaces;109
4.2.1;Unconstrained Optimization;109
4.2.1.1;Armijo Rule;111
4.2.2;Optimization on Closed Convex Sets;114
4.2.2.1;Projected Armijo Rule;117
4.2.3;General Optimization Problems;119
4.3;Newton-Based Methods-A Preview;119
4.3.1;Unconstrained Problems-Newton's Method;119
4.3.2;Simple Constraints;120
4.3.2.1;Nonsmooth Reformulation Approach and Generalized Newton Methods;120
4.3.2.2;SQP Methods;122
4.3.3;General Inequality Constraints;123
4.3.3.1;Nonsmooth Reformulation Approach and Generalized Newton Methods;124
4.3.3.2;SQP Methods;125
4.4;Generalized Newton Methods;125
4.4.1;Motivation: Application to Optimal Control;125
4.4.2;A General Superlinear Convergence Result;126
4.4.3;The Classical Newton's Method;129
4.4.4;Generalized Differential and Semismoothness;130
4.4.5;Semismooth Newton Methods;133
4.4.5.1;Semismooth Newton Method for Finite Dimensional KKT Systems;134
4.4.5.2;Discussion;135
4.5;Semismooth Newton Methods in Function Spaces;135
4.5.1;Pointwise Bound Constraints in L2;135
4.5.2;Semismoothness of Superposition Operators;136
4.5.3;Pointwise Bound Constraints in L2 Revisited;139
4.5.4;Application to Optimal Control;140
4.5.5;General Optimization Problems with Inequality Constraints in L2;142
4.5.6;Application to Elliptic Optimal Control Problems;143
4.5.6.1;Distributed Control;143
4.5.6.2;Neumann Boundary Control;145
4.5.7;Optimal Control of the Incompressible Navier-Stokes Equations;147
4.6;Sequential Quadratic Programming;150
4.6.1;Lagrange-Newton Methods for Equality Constrained Problems;150
4.6.2;The Josephy-Newton Method;154
4.6.2.1;Generalized Equation;154
4.6.3;SQP Methods for Inequality Constrained Problems;158
4.6.3.1;SQP Subproblem;160
4.6.3.2;Application to Optimal Control;161
4.7;State-Constrained Problems;161
4.7.1;SQP Methods;162
4.7.2;Semismooth Newton Methods;162
4.7.2.1;Moreau-Yosida Regularization;163
4.7.2.2;Lavrentiev Regularization;164
4.8;Further Aspects;165
4.8.1;Mesh Independence;165
4.8.2;Application of Fast Solvers;166
4.8.3;Other Methods;166
5;Discrete Concepts in PDE Constrained Optimization;167
5.1;Introduction;167
5.2;Control Constraints;168
5.2.1;Stationary Model Problem;168
5.2.2;First Discretize, Then Optimize;170
5.2.3;First Optimize, Then Discretize;171
5.2.4;Discussion and Implications;173
5.2.5;The Variational Discretization Concept;174
5.2.6;Error Estimates;177
5.2.6.1;Uniform Estimates;180
5.2.6.2;Numerical Examples for Distributed Control;181
5.2.7;Boundary Control;187
5.2.7.1;Neumann and Robin-Type Boundary Control;188
5.2.7.2;Numerical Examples for Robin-Type Boundary Control;193
5.2.7.3;Dirichlet Boundary Control;199
5.2.7.4;Numerical Example for Dirichlet Boundary Control;203
5.2.8;Some Literature Related to Control Constraints;206
5.3;Constraints on the State;207
5.3.1;Pointwise Bounds on the State;208
5.3.1.1;Finite Element Discretization;209
5.3.1.2;Error Analysis;213
5.3.1.3;Piecewise Constant Controls;217
5.3.1.4;Numerical Examples for Pointwise Constraints on the State;222
5.3.1.5;Some Literature for (Control and) State Constraints;228
5.3.2;Pointwise Bounds on the Gradient of the State;229
5.3.2.1;Finite Element Discretization;231
5.3.2.2;A Numerical Experiment with Pointwise Constraints on the Gradient;236
5.4;Time Dependent Problem;237
5.4.1;Mathematical Model, State Equation;237
5.4.2;Optimization Problem;239
5.4.3;Discretization;239
5.4.4;Further Literature on Control of Time-Dependent Problems;241
6;Applications;243
6.1;Optimal Semiconductor Design;243
6.1.1;Semiconductor Device Physics;244
6.1.1.1;Charge Transport;245
6.1.1.2;The Potential Equation;246
6.1.1.3;The Continuity Equations;247
6.1.1.4;The Current Densities;248
6.1.1.5;Scaling;249
6.1.2;The Optimization Problem;250
6.1.2.1;The First-Order Optimality System;254
6.1.3;Numerical Results;256
6.1.3.1;Steepest Descent;256
6.1.3.2;The Reduced Newton Method;258
6.2;Optimal Control of Glass Cooling;260
6.2.1;Modeling;261
6.2.1.1;Radiation;261
6.2.1.2;SPN-approximations;263
6.2.2;Optimal Boundary Control;264
6.2.2.1;Derivatives;267
6.2.2.2;Newton's Method;268
6.2.3;Numerical Results;270
7;References;274

Leseproben

Abstract The following chapter is devoted to the study of two industrial applications, in which optimization with partial differential equations plays a crucial role. It shall provide a survey of the different mathematical settings which can be handled with the general optimal control calculus presented in the previous chapters. We focus on large scale optimal control problems involving two well-known types of partial differential equations, namely elliptic and parabolic ones. Since real world applications lead generally to mathematically quite involved problems, we study in particular nonlinear systems of equations. The examples are chosen in such a way that they are up-to-date and modern mathematical tools are used for their specific solution. The industrial fields we cover are modern semiconductor design and glass production. We start each section with a modeling part to introduce the underlying physics and mathematical models, which are then followed by the analytical and numerical study of the related optimal control problems.

4.1 Optimal Semiconductor Design

Each student learns in the first lecture on numerical mathematics that the enormous speed-up of numerical simulations during the last 30 years is rooted in two facts, namely the significant improvement of algorithms and the ongoing miniaturization in electronics which allows for faster computing times. In the previous chapters we already learned in which way fast numerical algorithms can be developed. Now we study the impact of mathematical optimization on advanced semiconductor design.

There are several stages at which optimization and control are used in semiconductor industry, e.g., in circuit design, thermal control of the circuit board or, on a smaller level, the design of the semiconductor device itself. Even the control of the whole production process itself is under mathematical investigation. The most popular semiconductor device is indeed the so-called MOSFET (metal oxide silicium field effect transistor), which is employed in many applications (see Fig. 4.1) [129].

In the design cycle one changes the geometry of the device (miniaturization!) and the so-called doping profile, which describes the density of charged background ions. This doping profile defines the type of the semiconductor device under consideration. In the conventional design cycle simulation tools are employed to compute the so called current-voltage characteristics of the device, from which the electrical engineer can deduce many performance characteristics of the device. This is done for a certain set of design parameters, and then these parameters are adjusted empirically. Thus, the total design time depends crucially on the knowledge and experience of the engineer.

In standard applications a working point, i.e., a certain voltage-current pair, for the device is fixed. In particular for MOSFET devices in portable systems it is most important to have on the one hand a low leakage current (in the off-state), which maximizes the battery lifetime, and on the other hand one wants to maximize the drive current (in the on-state) [128]. Now, we want to study how one can apply the previously introduced techniques to optimize such a device and we pose the following design question [77]: Is it possible to gain an amplified current at the working point only by a slight change of the doping profile?