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.
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.
