Matrix preconditioning techniques and applications /

Matrix preconditioning techniques and applications /

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Bibliographic Details
Author / Creator:Chen, Ke, 1962-
Imprint:Cambridge ; New York : Cambridge University Press, 2005.
Description:xxiii, 568 p., [8] p. of plates : ill. (some col.) ; 24 cm.
Language:English
Series:Cambridge monographs on applied and computational mathematics ; 19.
Subject:
Sparse matrices -- Data processing.
Differential equations -- Numerical solutions -- Data processing.
Iterative methods (Mathematics) -- Data processing.
Integral equations -- Numerical solutions -- Data processing.
Differential equations -- Numerical solutions -- Data processing.
Integral equations -- Numerical solutions -- Data processing.
Iterative methods (Mathematics) -- Data processing.
Sparse matrices -- Data processing.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/5700666
Hidden Bibliographic Details
ISBN:9780521838283
0521838282
Notes:Includes bibliographical references (p. [530]-555) and indexes.
Table of Contents:
  • Preface
  • Nomenclature
  • 1. Introduction
  • 1.1. Direct and iterative solvers, types of preconditioning
  • 1.2. Norms and condition number
  • 1.3. Perturbation theories for linear systems and eigenvalues
  • 1.4. The Arnoldi iterations and decomposition
  • 1.5. Clustering characterization, field of values and [set membership]-pseudospectrum
  • 1.6. Fast Fourier transforms and fast wavelet transforms
  • 1.7. Numerical solution techniques for practical equations
  • 1.8. Common theories on preconditioned systems
  • 1.9. Guide to software development and the supplied Mfiles
  • 2. Direct methods
  • 2.1. The LU decomposition and variants
  • 2.2. The Newton-Schulz-Hotelling method
  • 2.3. The Gauss-Jordan decomposition and variants
  • 2.4. The QR decomposition
  • 2.5. Special matrices and their direct inversion
  • 2.6. Ordering algorithms for better sparsity
  • 2.7. Discussion of software and the supplied Mfiles
  • 3. Iterative methods
  • 3.1. Solution complexity and expectations
  • 3.2. Introduction to residual correction
  • 3.3. Classical iterative methods
  • 3.4. The conjugate gradient method: the SPD case
  • 3.5. The conjugate gradient normal method: the unsymmetric case
  • 3.6. The generalized minimal residual method: GMRES
  • 3.7. The GMRES algorithm in complex arithmetic
  • 3.8. Matrix free iterative solvers: the fast multipole methods
  • 3.9. Discussion of software and the supplied Mfiles
  • 4. Matrix splitting preconditioners [T1]: direct approximation of A[subscript n x n]
  • 4.1. Banded preconditioner
  • 4.2. Banded arrow preconditioner
  • 4.3. Block arrow preconditioner from DDM ordering
  • 4.4. Triangular preconditioners
  • 4.5. ILU preconditioners
  • 4.6. Fast circulant preconditioners
  • 4.7. Singular operator splitting preconditioners
  • 4.8. Preconditioning the fast multipole method
  • 4.9. Numerical experiments
  • 4.10. Discussion of software and the supplied Mfiles
  • 5. Approximate inverse preconditioners [T2]: direct approximation of [characters not reproducible]
  • 5.1. How to characterize A[superscript -1] in terms of A
  • 5.2. Banded preconditioner
  • 5.3. Polynomial preconditioner p[subscript k](A)
  • 5.4. General and adaptive sparse approximate inverses
  • 5.5. AINV type preconditioner
  • 5.6. Multi-stage preconditioners
  • 5.7. The dual tolerance self-preconditioning method
  • 5.8. Near neighbour splitting for singular integral equations
  • 5.9. Numerical experiments
  • 5.10. Discussion of software and the supplied Mfiles
  • 6. Multilevel methods and preconditioners [T3]: coarse grid approximation
  • 6.1. Multigrid method for linear PDEs
  • 6.2. Multigrid method for nonlinear PDEs
  • 6.3. Multigrid method for linear integral equations
  • 6.4. Algebraic multigrid methods
  • 6.5. Multilevel domain decomposition preconditioners for GMRES
  • 6.6. Discussion of software and the supplied Mfiles
  • 7. Multilevel recursive Schur complements preconditioners [T4]
  • 7.1. Multilevel functional partition: AMLI approximated Schur
  • 7.2. Multilevel geometrical partition: exact Schur
  • 7.3. Multilevel algebraic partition: permutation-based Schur
  • 7.4. Appendix: the FEM hierarchical basis
  • 7.5. Discussion of software and the supplied Mfiles
  • 8. Sparse wavelet preconditioners [T5]: approximation of [characters not reproducible] and [characters not reproducible]
  • 8.1. Introduction to multiresolution and orthogonal wavelets
  • 8.2. Operator compression by wavelets and sparsity patterns
  • 8.3. Band WSPAI preconditioner
  • 8.4. New centering WSPAI preconditioner
  • 8.5. Optimal implementations and wavelet quadratures
  • 8.6. Numerical results
  • 8.7. Discussion of software and the supplied Mfiles
  • 9. Wavelet Schur preconditioners [T6]
  • 9.1. Introduction
  • 9.2. Wavelets telescopic splitting of an operator
  • 9.3. An exact Schur preconditioner with level-by-level wavelets
  • 9.4. An approximate preconditioner with level-by-level wavelets
  • 9.5. Some analysis and numerical experiments
  • 9.6. Discussion of the accompanied Mfiles
  • 10. Implicit wavelet preconditioners [T7]
  • 10.1. Introduction
  • 10.2. Wavelet-based sparse approximate inverse
  • 10.3. An implicit wavelet sparse approximate inverse preconditioner
  • 10.4. Implementation details
  • 10.5. Dense problems
  • 10.6. Some theoretical results
  • 10.7. Combination with a level-one preconditioner
  • 10.8. Numerical results
  • 10.9. Discussion of the supplied Mfile
  • 11. Application I: Acoustic scattering modelling
  • 11.1. The boundary integral equations for the Helmholtz equation in R[superscript 3] and iterative solution
  • 11.2. The low wavenumber case of a Helmholtz equation
  • 11.3. The high wavenumber case of a Helmholtz equation
  • 11.4. Discussion of software
  • 12. Application II: Coupled matrix problems
  • 12.1. Generalized saddle point problems
  • 12.2. The Oseen and Stokes saddle point problems
  • 12.3. The mixed finite element method
  • 12.4. Coupled systems from fluid structure interaction
  • 12.5. Elasto-hydrodynamic lubrication modelling
  • 12.6. Discussion of software and a supplied Mfile
  • 13. Application III: Image restoration and inverse problems
  • 13.1. Image restoration models and discretizations
  • 13.2. Fixed point iteration method
  • 13.3. Explicit time marching schemes
  • 13.4. The Primal-dual method
  • 13.5. Nonlinear multigrids for optimization
  • 13.6. The level set method and other image problems
  • 13.7. Numerical experiments
  • 13.8. Guide to software and the supplied Mfiles
  • 14. Application IV: Voltage stability in electrical power systems
  • 14.1. The model equations
  • 14.2. Fold bifurcation and arc-length continuation
  • 14.3. Hopf bifurcation and solutions
  • 14.4. Preconditioning issues
  • 14.5. Discussion of software and the supplied Mfiles
  • 15. Parallel computing by examples
  • 15.1. A brief introduction to parallel computing and MPI
  • 15.2. Some commonly used MPI routines
  • 15.3. Example 1 of a parallel series summation
  • 15.4. Example 2 of a parallel power method
  • 15.5. Example 3 of a parallel direct method
  • 15.6. Discussion of software and the supplied MPI Fortran files
  • Appendix A. A brief guide to linear algebra
  • A.1. Linear independence
  • A.2. Range and null spaces
  • A.3. Orthogonal vectors and matrices
  • A.4. Eigenvalues, symmetric matrices and diagonalization
  • A.5. Determinants and Cramer's rule
  • A.6. The Jordan decomposition
  • A.7. The Schur and related decompositions
  • Appendix B. The Harwell-Boeing (HB) data format
  • Appendix C. A brief guide to MATLAB
  • C.1. Vectors and matrices
  • C.2. Visualization of functions
  • C.3. Visualization of sparse matrices
  • C.4. The functional Mfile and string evaluations
  • C.5. Interfacing MATLAB with Fortran or C
  • C.6. Debugging a Mfile
  • C.7. Running a MATLAB script as a batch job
  • C.8. Symbolic computing
  • Appendix D. List of supplied M-files and programs
  • Appendix E. List of selected scientific resources on Internet
  • E.1. Freely available software and data
  • E.2. Other software sources
  • E.3. Useful software associated with books
  • E.4. Specialized subjects, sites and interest groups
  • References
  • Author Index
  • Subject Index

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