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Domain Decomposition Methods in Optimal Control of Partial Differential Equations

'International Series of Numerical Mathematics (ISNM)'. 2004. Auflage. Book. Sprache: Englisch.
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While domain decomposition methods have a long history dating back well over one hundred years, it is only during the last decade that they have become a major tool in numerical analysis of partial differential equations. This monograph emphasizes do... weiterlesen
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Titel: Domain Decomposition Methods in Optimal Control of Partial Differential Equations
Autor/en: John Lagnese, Günter Leugering

ISBN: 3764321946
EAN: 9783764321949
'International Series of Numerical Mathematics (ISNM)'.
2004. Auflage.
Book.
Sprache: Englisch.
Birkhäuser Basel

27. September 2004 - gebunden - 464 Seiten

Beschreibung

While domain decomposition methods have a long history dating back well over one hundred years, it is only during the last decade that they have become a major tool in numerical analysis of partial differential equations. This monograph emphasizes domain decomposition methods in the context of so-called virtual optimal control problems and treats optimal control problems for partial differential equations and their decompositions using an all-at-once approach.

Inhaltsverzeichnis

1 Introduction.
2 Background Material on Domain Decomposition.
2.1 Introduction.
2.2 Domain Decomposition for 1-d Problems.
2.2.1 Unbounded Domains.
2.2.2 Bounded Domains.
2.2.3 Semi-discretization.
2.3 Domain Decomposition Methods for Elliptic Problems.
2.3.1 Review of Basic Methods.
2.3.2 Virtual Controls.
2.3.3 The Basic Algorithm of P.-L. Lions.
2.3.4 An Augmented Lagrangian Formulation.
2.3.5 General Elliptic Problems and More General Splittings.
2.3.6 An a Posteriori Error Estimate.
2.3.7 Interpretation as a Damped Richardson iteration.
2.3.8 A Serial One-Dimensional Problem.
3 Partial Differential Equations on Graphs.
3.1 Introduction.
3.2 Partial Differential Operators on Graphs.
3.3 Elliptic Problems on Graphs.
3.3.2 Domain Decomposition.
3.3.3 Convergence.
3.3.4 Interpretation as a Richardson Iteration.
3.4 Hyperbolic Problems on Graphs.
3.4.1 The Model.
3.4.2 The Domain Decomposition Procedure.
4 Optimal Control of Elliptic Problems.
4.1 Introduction.
4.2 Distributed Controls.
4.2.2 Domain Decomposition.
4.2.3 A Complex Helmholtz Problem and its Decomposition.
4.2.4 Convergence.
4.2.5 Methods for Elliptic Optimal Control Problems.
4.2.6 An A Posteriori Error Estimate.
4.3 Boundary Controls.
4.3.2 Domain Decomposition.
4.3.3 Convergence.
4.3.4 An A Posteriori Error Estimate.
5 Control of Partial Differential Equations on Graphs.
5.1 Introduction.
5.2 Elliptic Problems.
5.2.1 The Global Optimal Control Problem on a Graph.
5.2.2 Domain Decomposition.
5.2.3 Distributed Controls.
5.2.4 Boundary Controls.
5.3 Hyperbolic Problems.
5.3.1 The Global Optimal Control Problem on a Graph.
5.3.2 The Domain Decomposition Procedure.
6 Control of Dissipative Wave Equations.
6.1 Introduction.
6.2 Optimal Dissipative Boundary Control.
6.2.1 Setting the Problem.
6.2.2 Existence and Regularity of Solutions.
6.2.3 The Global Optimality System.
6.3 Time Domain Decomposition.
6.3.1 Description of the Algorithm.
6.3.2 Convergence of the Iterates.
6.3.3 A Posteriori Error Estimates.
6.3.4 Extension to General Dissipative Control Systems.
6.4 Decomposition of the Spatial Domain.
6.4.1 Description of the Algorithm.
6.4.2 Convergence of the Iterates.
6.4.3 A Posteriori Error Estimates.
6.5 Space and Time Domain Decomposition.
6.5.1 Sequential Space-Time Domain Decomposition.
6.5.2 Sequential Time-Space Domain Decomposition.
7 Boundary Control of Maxwell's System.
7.1 Introduction.
7.2 Optimal Dissipative Boundary Control.
7.2.1 Setting the Problem.
7.2.2 Existence and Uniqueness of Solution.
7.2.3 The Global Optimality System.
7.3 Time Domain Decomposition.
7.3.1 Description of the Algorithm.
7.3.2 Convergence of the Iterates.
7.3.3 A Posteriori Error Estimates.
7.4 Decomposition of the Spatial Domain.
7.4.1 Description of the Algorithm.
7.4.2 Convergence of the Iterates.
7.4.3 A Posteriori Error Estimates.
7.5 Time and Space Domain Decomposition.
7.5.1 Sequential Space-Time Domain Decomposition.
7.5.2 Sequential Time-Space Domain Decomposition.
8 Control of Conservative Wave Equations.
8.1 Introduction.
8.2 Optimal Boundary Control.
8.2.1 Setting the Problem.
8.2.2 Existence and Regularity of Solutions.
8.2.3 The Global Optimality System.
8.3 Time Domain Decomposition.
8.3.1 Description of the Algorithm.
8.3.2 Convergence of the Iterates.
8.3.3 A Posteriori Error Estimates.
8.3.4 Extension to General Conservative Control Systems.
8.4 Decomposition of the Spatial Domain.
8.4.1 The Local Optimality Systems.
8.4.2 The Domain Decomposition Algorithm.
8.4.3 Convergence of the Iterates.
8.5 The Exact Reachability Problem.
8.5.1 The Global Optimality System.
8.5.2 The Limit of the Local Optimality Systems.
8.5.3 Application to Domain Decomposition.
8.5.4 Convergence to the Global Optimality System.
9 Domain Decomposition for 2-D Networks.
9.1 Elliptic Systems on 2-D Networks.
9.1.2 Examples.
9.1.3 Existence and Uniqueness of Solutions.
9.1.4 Domain Decomposition.
9.1.5 Convergence of the Algorithm.
9.2 Optimal Control on 2-D Networks.
9.2.1 Optimal Final Value Control.
9.2.2 Existence and Regularity of Solutions.
9.3 Decomposition of the Spatial Domain.
9.3.2 The Decomposition Algorithm.
9.3.3 Convergence of the Algorithm.

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