Matrix preconditioning techniques and applications.

*(English)*Zbl 1079.65057
Cambridge Monographs on Applied and Computational Mathematics 19. Cambridge: Cambridge University Press (ISBN 0-521-83828-2/hbk). xxiii, 568 p. (2005).

Matrix computing arises in many problems from image restoration to the numerical solution of partial differential equations. This book is mainly about recent preconditioning techniques and algorithms and their applications in different interesting fields. This is a very well-written and complete book which presents a lot of recent and original algorithms of upmost importance. Special attention is directed towards the algorithmic point of view, showing that the author is a specialist of this area of scientific computing. Everyone interested in preconditioning methods, from students to high level researchers, should be interested in this original book.

Topics in preconditioning techniques covered by this book are: matrix splitting preconditioners (in the finite element method (FEM) setting), approximate inverse preconditioners (FEM setting), multilevel (approximate inverse) preconditioners (FEM setting), recursive Schur complements preconditioner (FEM setting), matrix splitting and approximate inverses (wavelet setting), recursive Schur complements preconditioner (wavelet setting) and implicit wavelet preconditioner (FEM setting).

The different selected applications are: the iterative solution of boundary integral equations arising in acoustic scattering problems, coupled matrix problems like fluid structure interaction, EHL equations for modelling isothermal and thermal cases, Oseen problems, nonlinear total variation equation for image restoration, optimization, level set application to interfaces tracking, image segmentation, bifurcation problems in voltage stability in electrical power transmission systems. A chapter is also devoted to basics in parallel computing. Some Matlab codes are provided showing that the author always treats a problem from the mathematical formulation to the numerical implementation.

It is difficult to detail every point that you can meet in such a valuable book (domain decomposition, fast Fourier transform and fast wavelet algorithms, multigrid algorithms, etc.) but be sure that this is a good investment if you are interested in these topics or if you want to have a good overview.

Topics in preconditioning techniques covered by this book are: matrix splitting preconditioners (in the finite element method (FEM) setting), approximate inverse preconditioners (FEM setting), multilevel (approximate inverse) preconditioners (FEM setting), recursive Schur complements preconditioner (FEM setting), matrix splitting and approximate inverses (wavelet setting), recursive Schur complements preconditioner (wavelet setting) and implicit wavelet preconditioner (FEM setting).

The different selected applications are: the iterative solution of boundary integral equations arising in acoustic scattering problems, coupled matrix problems like fluid structure interaction, EHL equations for modelling isothermal and thermal cases, Oseen problems, nonlinear total variation equation for image restoration, optimization, level set application to interfaces tracking, image segmentation, bifurcation problems in voltage stability in electrical power transmission systems. A chapter is also devoted to basics in parallel computing. Some Matlab codes are provided showing that the author always treats a problem from the mathematical formulation to the numerical implementation.

It is difficult to detail every point that you can meet in such a valuable book (domain decomposition, fast Fourier transform and fast wavelet algorithms, multigrid algorithms, etc.) but be sure that this is a good investment if you are interested in these topics or if you want to have a good overview.

Reviewer: Xavier Antoine (Vandœuvre-lès-Nancy)

##### MSC:

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65F10 | Iterative numerical methods for linear systems |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65Nxx | Numerical methods for partial differential equations, boundary value problems |

65Z05 | Applications to the sciences |

65T60 | Numerical methods for wavelets |

68W30 | Symbolic computation and algebraic computation |