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:: Table of Contents |
Chapter 1. Introduction
1.1 Motivation
1.2 Gauss and Jacobi
1.3 Laplace's equation as a typical example
1.4 An advection-reaction-diffusion problem as a typical non-
symmetric example
1.5 Problems
Chapter 2. Stationary Iterative Methods
2.1 Error, residual, and difference of iterates
2.2 Convergence analysis
2.3 Convergence factor and convergence rate
2.4 Regular splittings and M-matrices
2.5 Jacobi
2.6 Gauss-Seidel
2.7 Successive over-relaxation: SOR
2.8 Richardson
2.9 Problems
Chapter 3. Krylov Methods
3.1 Steepest Descent
3.2 The conjugate gradient method
3.3 The Arnoldi iteration
3.4 The Lanczos algorithm
3.5 Generalized minimal residual: GMRES
3.6 Two families of Krylov methods
3.7 Problems
Chapter 4. Preconditioning
4.1 Stationary iterative methods and preconditioning
4.2 Left and right preconditioning
4.3 Preconditioning in practice
4.4 Flexible GMRES: FGMRES
4.5 Algebraic preconditioning methods
4.6 Schwarz domain decomposition methods
4.7 Dirichlet-Neumann domain decomposition method
4.8 Neumann-Neumann domain decomposition method
4.9 Comparison of Schwarz, Dirichlet-Neumann and
Neumann-Neumann
4.10 Multigrid methods
4.11 Problems
Chapter 5. Optimal Control
5.1 Optimal control of the Laplace equation
5.1.1 Existence and uniqueness of a minimizer
5.1.2 Optimality system and adjoint equation
5.2 Reduced approach
5.3 All-at-once approach
5.3.1 Optimized Schwarz methods
5.3.2 Block-diagonal preconditioners
5.3.3 Schur-complement-based preconditioners
5.3.4 Collective smoothing
5.3.5 Multigrid methods
5.4 Further preconditioners for optimal control problems
5.5 Problems
Chapter 6. Appendix
6.1 Existence, uniqueness and well-posedness of Schwarz iterates
6.2 Some polynomial identities
6.3 Sobolev embedding theorems
6.4 Lax-Milgram Theorem
6.5 Weak compactness
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