Optimization with PDE Constraints

Transcription

1 Optimization with PDE Constraints

2 MATHEMATICAL MODELLING: Theory and Applications VOLUME 23 This series is aimed at publishing work dealing with the definition, development and application of fundamental theory and methodology, computational and algorithmic implementations and comprehensive empirical studies in mathematical modelling. Work on new mathematics inspired by the construction of mathematical models, combining theory and experiment and furthering the understanding of the systems being modelled are particularly welcomed. Manuscripts to be considered for publication lie within the following, nonexhaustive list of areas: mathematical modelling in engineering, industrial mathematics, control theory, operations research, decision theory, economic modelling, mathematical programming, mathematical system theory, geophysical sciences, climate modelling, environmental processes, mathematical modelling in psychology, political science, sociology and behavioural sciences, mathematical biology, mathematical ecology, image processing, computer vision, artificial intelligence, fuzzy systems, and approximate reasoning, genetic algorithms, neural networks, expert systems, pattern recognition, clustering, chaos and fractals. Original monographs, comprehensive surveys as well as edited collections will be considered for publication. Managing Editor: R. Lowen (Antwerp, Belgium) Series Editors: R. Laubenbacher (Virginia Bioinformatics Institute, Virginia Tech, USA) A. Stevens (University of Heidelberg, ) For other titles published in this series, go to

3 Optimization with PDE Constraints M. Hinze Universität Hamburg R. Pinnau Technische Universität Kaiserslautern M. Ulbrich Technische Universität München and S. Ulbrich Technische Universität Darmstadt

4 Michael Hinze Dept. Mathematik Universität Hamburg Bundesstr Hamburg Rene Pinnau FB Mathematik TU Kaiserslautern Erwin-Schrödinger-Str Kaiserslautern Gebäude 48 Michael Ulbrich Technische Universität München Fakultät für Mathematik Lehrstuhl für Mathematische Optimierung Boltzmannstr Garching mulbrich@ma.tum.de Stefan Ulbrich Technische Universität Darmstadt Fachbereich Mathematik, AG10 Schloßgartenstr Darmstadt ulbrich@mathematik.tu-darmstadt.de ISBN e-isbn Library of Congress Control Number: Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper springer.com

5 Preface Solving optimization problems subject to constraints given in terms of partial differential 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 simulations to model-based design and optimal control is crucial. For the treatment of such optimization problems the interaction of optimization techniques and numerical simulation plays a central role. After proper discretization, the number of optimization variables varies between 10 3 and 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 further explore the specific 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 mathematical field of optimization with PDE constraints. The first chapter introduces to the analytical background and optimality theory for optimization problems with PDEs. Optimization problems with PDE-constraints are posed in infinite dimensional 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. These results form the foundation of efficient optimization methods in function space, their adequate numerical realization, mesh independence results and error estimators. The chapter starts with an introduction to the necessary background in functional analysis, Sobolev spaces and the theory of weak solutions for elliptic and parabolic PDEs. These ingredients are then applied to study PDE-constrained optimization problems. Existence results for optimal controls, derivative computations by the sensitivity and adjoint approaches and optimality conditions for problems with control-, state- and general constraints are considered. All concepts are illustrated by elliptic and parabolic optimal control problems. Finally, the optimal control of instationary incompressible Navier-Stokes flow is considered. The second chapter presents a selection of important algorithms for optimization problems with partial differential equations. The development and analysis of these methods is carried out in a Banach space setting. This chapter starts with introducing a general framework for achieving global convergence. Then, several variants of generalized Newton methods are derived and analyzed. In particular, necessary and sufficient conditions for fast local convergence are derived. Based on this, the concept of semismooth Newton methods for operator equations is introduced. It is shown how complementarity conditions, variational inequalities, and optimality systems can be reformulated as semismooth operator equations. Applications to constrained optimal control problems are discussed, in particular for elliptic

6 vi Preface partial differential equations and for flow control problems governed by the incompressible instationary Navier-Stokes equations. As a further important concept, the formulation of optimality systems as generalized equations is addressed and the Josephy-Newton method for generalized equations is analyzed. This provides an elegant basis for the motivation and analysis of sequential quadratic programming (SQP) algorithms. The second chapter concludes with a short outline of recent algorithmic advances for state constrained problems and a brief discussion of several further aspects. The third chapter gives an introduction to discrete concepts for optimization problems with PDE constraints. As models for the state elliptic and parabolic PDEs are considered which are well understood from the analytical point of view. This allows to focus on structural aspects in discretization. The approaches First discretize, then optimize and First optimize, then discretize are compared and discussed, and a variational discrete concept is introduced which avoids explicit discretization of the controls. Special focus is taken on the treatment of constraints. This includes general constraints on the control, and also pointwise bounds on the state, and on the gradient of the state. The chapter presents the error analysis for the variational discrete concept and accomplishes the analytical findings with numerical examples which confirm the analytical results. Finally, the fourth chapter is devoted to the study of two industrial applications, in which optimization with partial differential equations plays a crucial role. It provides a survey of the different mathematical settings which can be handled with the general optimal control calculus presented in the previous chapters. The chapter focuses 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, in particular nonlinear systems of equations are studied. 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 covered are modern semiconductor design and glass production. Each section starts 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. Acknowledgements This Book is based on lecture notes of the autumn school Modellierung und Optimierung mit Partiellen Differentialgleichungen which was held in September 2005 at the Universität Hamburg. It was supported by the Collaborative Research Center 609, located at the Technische Universität Dresden, and by the Priority Programme 1253, both sponsored by the Deutsche Forschungsgemeinschaft, as well as by the Schwerpunkt Optimierung und Approximation at the Department Mathematik of the Universität Hamburg. All support is gratefully acknowledged. Finally we would like to thank a number of colleagues whose collaboration and support influenced the material presented in this book. These include Günter Bärwolff, Martin Burger, Klaus Deckelnick, John Dennis, Michael Ferris, Andreas

7 Preface vii Günther, Matthias Heinkenschloss, Michael Herty, Michael Hintermüller, Axel Klar, Karl Kunisch, Günter Leugering, Ulrich Matthes, Christian Meyer, Danny Ralph, Ekkehard Sachs, Anton Schiela, Alexander Schulze, Mohammed Seaid, Norbert Siedow, Guido Thömmes, Philippe Toint, Fredi Tröltzsch, Andreas Unterreiter, Luís Vicente, and Morten Vierling.

8 Contents Preface... v 1 Analytical Background and Optimality Theory... 1 StefanUlbrich Introduction and Examples Introduction ExamplesforOptimizationProblemswithPDEs Optimization of a Stationary Heating Process Optimization of an Unsteady Heating Processes OptimalDesign Linear Functional Analysis and Sobolev Spaces Banach and Hilbert Spaces Sobolev Spaces Weak Convergence Weak Solutions of Elliptic and Parabolic PDEs Weak Solutions of Elliptic PDEs WeakSolutionsofParabolicPDEs Gâteaux- and Fréchet Differentiability BasicDefinitions Implicit Function Theorem Existence of Optimal Controls Existence Result for a General Linear-Quadratic Problem Existence Results for Nonlinear Problems Applications Reduced Problem, Sensitivities and Adjoints Sensitivity Approach Adjoint Approach Application to a Linear-Quadratic Optimal Control Problem A Lagrangian-Based View of the Adjoint Approach ix

9 x Contents Second Derivatives Optimality Conditions Optimality Conditions for Simply Constrained Problems Optimality Conditions for Control-Constrained Problems Optimality Conditions for Problems with General Constraints Optimal Control of Instationary Incompressible Navier-Stokes Flow Functional Analytic Setting AnalysisoftheFlowControlProblem Reduced Optimal Control Problem Optimization Methods in Banach Spaces Michael Ulbrich Synopsis Globally Convergent Methods in Banach Spaces Unconstrained Optimization OptimizationonClosedConvexSets General Optimization Problems Newton-Based Methods A Preview Unconstrained Problems Newton s Method SimpleConstraints General Inequality Constraints Generalized Newton Methods Motivation:ApplicationtoOptimalControl A General Superlinear Convergence Result TheClassicalNewton smethod Generalized Differential and Semismoothness Semismooth Newton Methods Semismooth Newton Methods in Function Spaces Pointwise Bound Constraints in L Semismoothness of Superposition Operators Pointwise Bound Constraints in L 2 Revisited ApplicationtoOptimalControl General Optimization Problems with Inequality Constraints in L Application to Elliptic Optimal Control Problems Optimal Control of the Incompressible Navier-Stokes Equations Sequential Quadratic Programming Lagrange-Newton Methods for Equality Constrained Problems TheJosephy-NewtonMethod SQP Methods for Inequality Constrained Problems...148

10 Contents xi 2.7 State-ConstrainedProblems SQP Methods Semismooth Newton Methods Further Aspects Mesh Independence ApplicationofFastSolvers Other Methods Discrete Concepts in PDE Constrained Optimization Michael Hinze Introduction ControlConstraints Stationary Model Problem FirstDiscretize,ThenOptimize FirstOptimize,ThenDiscretize DiscussionandImplications The Variational Discretization Concept ErrorEstimates Boundary Control SomeLiteratureRelatedtoControlConstraints ConstraintsontheState Pointwise Bounds on the State Pointwise Bounds on the Gradient of the State Time Dependent Problem Mathematical Model, State Equation OptimizationProblem Discretization Further Literature on Control of Time-Dependent Problems Applications René Pinnau Optimal Semiconductor Design Semiconductor Device Physics TheOptimizationProblem NumericalResults OptimalControlofGlassCooling Modeling Optimal Boundary Control NumericalResults References...265