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

In this book, which focuses on the use of iterative methods for solving large sparse systems of linear equations, templates are introduced to meet the needs of both the traditional user and the high-performance specialist Templates, a description of a general algorithm rather than the executable object or source code more commonly found in a conventional software library, offer whatever degree of customization the user may desire

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Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods

Preface How to Get the Software Part I. Linear Equations. 1. Basic Concepts and Stationary Iterative Methods 2. Conjugate Gradient Iteration 3. GMRES Iteration Part II. Nonlinear Equations. 4. Basic Concepts and Fixed Point Iteration 5. Newton's Method 6. Inexact Newton Methods 7. Broyden's Method 8. Global Convergence Bibliography Index.

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Iterative Methods for Linear and Nonlinear Equations

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

This book presents a carefully selected group of methods for unconstrained and bound constrained optimization problems and analyzes them in depth both theoretically and algorithmically. It focuses on clarity in algorithmic description and analysis rather than generality, and while it provides pointers to the literature for the most general theoretical results and robust software, the author thinks it is more important that readers have a complete understanding of special cases that convey essential ideas. A companion to Kelley's book, Iterative Methods for Linear and Nonlinear Equations (SIAM, 1995), this book contains many exercises and examples and can be used as a text, a tutorial for self-study, or a reference. Iterative Methods for Optimization does more than cover traditional gradient-based optimization: it is the first book to treat sampling methods, including the Hooke& Jeeves, implicit filtering, MDS, and Nelder& Mead schemes in a unified way.

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Iterative Methods for Optimization

Estimates above and below are obtained for the height of the equilibrium free surface of a liquid when the liquid partially fills a cylindrical container whose cross section contains a corner with interior angle 2α. The surface is characterized by the condition that its mean curvature be proportional to its height above a reference plane (or, in the case of zero gravity, that the mean curvature be constant), and by the requirement that it meet the container wall with prescribed contact angle γ. It turns out that the qualitative behavior of such a surface near the vertex changes markedly, according as α + γ < ½π, or α + γ ≥ ½π. In the former case, the surface is either unbounded or fails to exist, while in the latter case every such surface is bounded. Some experimental comparisons are indicated, and an application to the problem of describing the mechanism of water rise in trees is discussed. 

The above results describe a limiting case among corresponding properties that hold for surfaces defined over domains with smooth boundaries. This extension is indicated, as well as a formal extension to n-dimensional surfaces; here the interest centers on the fact that it is the mean curvature of an (n-1)-dimensional boundary element that controls the local behavior of the n-dimensional solution surface.

https://www.pnas.org/content/pnas/63/2/292.full.pdf

On the behavior of a capillary surface in a wedge

We consider a generalized conjugate gradient method for solving sparse, symmetric, positive-definite systems of linear equations, principally those arising from the discretization of boundary value problems for elliptic partial differential equations. The method is based on splitting off from the original coefficient matrix a symmetric, positive-definite one that corresponds to a more easily solvable system of equations, and then accelerating the associated iteration using conjugate gradients. Optimality and convergence properties are presented, and the relation to other methods is discussed. Several splittings for which the method seems particularly effective are also discussed, and for some, numerical examples are given.

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A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations

Block preconditionings for the conjugate gradient method are investigated for solving positive definite block tridiagonal systems of linear equations arising from discretization of boundary value problems for elliptic partial differential equations. The preconditionings rest on the use of sparse approximate matrix inverses to generate incomplete block Cholesky factorizations. Carrying out of the factorizations can be guaranteed under suitable conditions. Numerical experiments on test problems for two dimensions indicate that a particularly attractive preconditioning, which uses special properties of tridiagonal matrix inverses, can be computationally more efficient for the same computer storage than other preconditionings, including the popular point incomplete Cholesky factorization.

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Block Preconditioning for the Conjugate Gradient Method

for a scalar funct ion u(x) ~ over a region ~ in n -d imens iona l Euc l idean space, n>~2. W e assume the b o u n d a r y Z of ~ to sa t i s fy smoothness hypotheses , which v a r y wi th the cont ex t . F o r some purposes i t suffices t h a t Y, E C (1) in local pa ramete rs . However , a t t imes , we shall have to refer, a t leas t locally, to a mean cu rva tu re a t po in ts on Y~. A l though our resul ts could be s t a t ed in t e rms of general ized b o u n d a r y curva tu res in such cases, i t is preferable for the purpose of these papers to assume t h a t Y~ E C (~ a t these points . F ina l ly , cer ta in resul ts dea l wi th boundar ies on which r egu la r i ty fails a t a set of points , Small enough to pe rmi t a res t r i c ted form of the d ivergence theo rem to hold. The t y p e of s ingular set t h a t is admiss ible in th is sense will be clarif ied la ter . O u r pr incipal concern is the case of (1) of special phys ica l in teres t ,

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On capillary free surfaces in the absence of gravity

We consider a generalized conjugate gradient method for solving systems of linear equations having nonsymmetric coefficient matrices with positive-definite symmetric part. The method is based on splitting the matrix into its symmetric and skew-symmetric parts, and then accelerating the associated iteration using conjugate gradients, which simplifies in this case, as only one of the two usual parameters is required. The method is most effective for cases in which the symmetric part of the matrix corresponds to an easily solvable system of equations. Convergence properties are discussed, as well as an application to the numerical solution of elliptic partial differential equations.

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Paul Concus

Papers

On the behavior of a capillary surface in a wedge

A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations

Block Preconditioning for the Conjugate Gradient Method

On capillary free surfaces in the absence of gravity

A generalized conjugate gradient method for nonsymmetric systems of linear equations