Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from t...

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GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems

: This manual describes the next generation of the modular three-dimensional transport model, MT3D, with significantly expanded capabilities, including the addition of (a) a third-order total-variation-diminishing (TVD) scheme for solving the advection term that is mass conservative but does not introduce excessive numerical dispersion and artificial oscillation, (b) an efficient iterative solver based on generalized conjugate gradient methods and the Lanczos/ORTHOMIN acceleration scheme to remove stability constraints on the transport time-step size, (c) options for accommodating nonequilibrium sorption and dual-domain advection-diffusion mass transport, and (d) a multicomponent program structure that can accommodate add-on reaction packages for modeling general biological and geochemical reactions MT3DMS can be used to simulate changes in concentrations of miscible contaminants in groundwater considering advection, dispersion, diffusion, and some basic chemical reactions, with various types of boundary conditions and external sources or sinks The basic chemical reactions included in the model are equilibrium-controlled or rate-limited linear or nonlinear sorption and first-order irreversible or reversible kinetic reactions MT3DMS can accommodate very general spatial discretization schemes and transport boundary conditions, including: (a) confined, unconfined, or variably confined/unconfined aquifer layers, (b)inclined model layers and variable cell thickness within the same layer, (c) specified concentration or mass flux boundaries, and (d) the solute transport effects of external hydraulic sources and sinks such as wells, drains, rivers, areal recharge, and evapotranspiration

/pdf/mt3dms-a-modular-three-dimensional-multispecies-transport-35iuw2lll4.pdf

MT3DMS: A Modular Three-Dimensional Multispecies Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems; Documentation and User's Guide

We consider a class of iterative algorithms for solving systems of linear equations where the coefficient matrix is nonsymmetric with positive-definite symmetric part. The algorithms are modelled after the conjugate gradient method, and are well suited for large sparse systems. They do not make use of any associated symmetric problems. Convergence results and error bounds are presented.

Variational Iterative Methods for Nonsymmetric Systems of Linear Equations

http://scgroup.hpclab.ceid.upatras.gr/class/SCII/Various/SaadvdVorst2000.pdf

Iterative solution of linear systems in the 20th century

Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods

On the simplification of generalized conjugate-gradient methods for nonsymmetrizable linear systems

Self-gravitational force calculation of infinitesimally thin gaseous disks

The classical form of the conjugate gradient method (CG method), developed by Hestenes and Stiefel, for solving the linear system Au = b is applicable when the coefficient matrix A is symmetric and positive definite (SPD). In this paper we consider various alternative forms of the CG method as well as generalizations to cases where A is not necessarily SPD. This analysis includes the ''preconditioned conjugate gradient method'' which is equivalent to conjugate gradient acceleration of a basic iterative method corresponding to a preconditioned system. Both the symmetrizable case and the nonsymmetrizable case are considered. For the nonsymmetrizable case there are very few useful theoretical results available. A package of programs, known as ITPACK, has been developed as a tool for carrying out experimental studies on various algorithms. Preliminary conclusions based on experimental results are given. 42 refs.

Kang C. Jea

Papers

Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods

On the simplification of generalized conjugate-gradient methods for nonsymmetrizable linear systems

Self-gravitational force calculation of infinitesimally thin gaseous disks

Preconditioned conjugate gradient algorithms and software for solving large sparse linear systems

On the effectiveness of adaptive Chebyshev acceleration for solving systems of linear equations