Iterative Methods and Preconditioners for Systems of Linear Equations

Gabriele Ciaramella, Martin J. Gander

Published by SIAM

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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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