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This special volume focuses on optimization and control of
processes governed by partial differential equations. The
contributors are mostly participants of the DFG-priority program
1253: Optimization with PDE-constraints which is active since 2006.
The book is organized in sections which cover almost the entire
spectrum of modern research in this emerging field. Indeed, even
though the field of optimal control and optimization for
PDE-constrained problems has undergone a dramatic increase of
interest during the last four decades, a full theory for nonlinear
problems is still lacking. The contributions of this volume, some
of which have the character of survey articles, therefore, aim at
creating and developing further new ideas for optimization, control
and corresponding numerical simulations of systems of possibly
coupled nonlinear partial differential equations. The research
conducted within this unique network of groups in more than fifteen
German universities focuses on novel methods of optimization,
control and identification for problems in infinite-dimensional
spaces, shape and topology problems, model reduction and
adaptivity, discretization concepts and important applications.
Besides the theoretical interest, the most prominent question is
about the effectiveness of model-based numerical optimization
methods for PDEs versus a black-box approach that uses existing
codes, often heuristic-based, for optimization.
This special volume focuses on optimization and control of
processes governed by partial differential equations. The
contributors are mostly participants of the DFG-priority program
1253: Optimization with PDE-constraints which is active since 2006.
The book is organized in sections which cover almost the entire
spectrum of modern research in this emerging field. Indeed, even
though the field of optimal control and optimization for
PDE-constrained problems has undergone a dramatic increase of
interest during the last four decades, a full theory for nonlinear
problems is still lacking. The contributions of this volume, some
of which have the character of survey articles, therefore, aim at
creating and developing further new ideas for optimization, control
and corresponding numerical simulations of systems of possibly
coupled nonlinear partial differential equations. The research
conducted within this unique network of groups in more than fifteen
German universities focuses on novel methods of optimization,
control and identification for problems in infinite-dimensional
spaces, shape and topology problems, model reduction and
adaptivity, discretization concepts and important applications.
Besides the theoretical interest, the most prominent question is
about the effectiveness of model-based numerical optimization
methods for PDEs versus a black-box approach that uses existing
codes, often heuristic-based, for optimization.
Many of the most challenging problems in the applied sciences
involve non-differentiable structures as well as partial
differential operators, thus leading to non-smooth distributed
parameter systems. This edited volume aims to establish a
theoretical and numerical foundation and develop new algorithmic
paradigms for the treatment of non-smooth phenomena and associated
parameter influences. Other goals include the realization and
further advancement of these concepts in the context of robust and
hierarchical optimization, partial differential games, and
nonlinear partial differential complementarity problems, as well as
their validation in the context of complex applications. Areas for
which applications are considered include optimal control of
multiphase fluids and of superconductors, image processing,
thermoforming, and the formation of rivers and networks. Chapters
are written by leading researchers and present results obtained in
the first funding phase of the DFG Special Priority Program on
Nonsmooth and Complementarity Based Distributed Parameter Systems:
Simulation and Hierarchical Optimization that ran from 2016 to
2019.
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.
Das Buch gibt eine Einfuhrung in zentrale Konzepte und Methoden der
Nichtlinearen Optimierung. Es ist aus Vorlesungen der Autoren an
der TU Munchen, der TU Darmstadt und der Universitat Hamburg
entstanden. Der Inhalt des Buches wurde insbesondere auf
mathematische Bachelorstudiengange zugeschnitten und hat sich als
Basis entsprechender Vorlesungen sowie fur eine anschlieende
Vertiefung im Bereich der Optimierung bewahrt. Der Umfang
entspricht zwei zweistundigen oder einer vierstundigen Vorlesung,
wobei etwa in gleichem Umfang sowohl unrestringierte
Optimierungsprobleme als auch Optimierungsprobleme mit
Nebenbedingungen behandelt werden. Im Teil uber die unrestringierte
Optimierung werden sowohl Trust-Region- als auch
Liniensuch-Methoden zur Globalisierung behandelt. Fur letztere wird
ein ebenso leistungsfahiges wie intuitives Konzept der zulassigen
Suchrichtungen und Schrittweiten entwickelt. Die schnelle lokale
Konvergenz Newton-artiger Verfahren und ihre Globalisierung sind
weitere wichtige Themengebiete. Das Kapitel uber restringierte
Optimierung entwickelt notwendige und hinreichende
Optimalitatsbedingungen und geht auf wichtige numerische Verfahren,
insbesondere Sequential Quadratic Programming, Penalty- und
Barriereverfahren ein. Der Bezug von Barriereverfahren zu den
aktuell intensiv untersuchten Innere-Punkte-Verfahren wird
ebenfalls hergestellt.
Semismooth Newton methods are a modern class of remarkably powerful
and versatile algorithms for solving constrained optimization
problems with partial differential equations (PDEs), variational
inequalities, and related problems. This book provides a
comprehensive presentation of these methods in function spaces,
striking a balance between thoroughly developed theory and
numerical applications. Although largely self-contained, the book
also covers recent developments in the field, such as
state-constrained problems and offers new material on topics such
as improved mesh independence results. The theory and methods are
applied to a range of practically important problems, including
optimal control of semilinear elliptic differential equations,
obstacle problems, and flow control of instationary Navier-Stokes
fluids. In addition, the author covers adjoint-based derivative
computation and the efficient solution of Newton systems by
multigrid and preconditioned iterative methods.
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