Iterative accelerating algorithms with Krylov subspaces for the solution to large-scale nonlinear problems
type: Item
instance of: scholarly article
title: Iterative accelerating algorithms with Krylov subspaces for the solution to large-scale nonlinear problems
zbMATH Open document ID: 0951.65047
DOI: 10.1023/A:1019187614377
author: Franck Risler
author: Christian Rey
published in: Numerical Algorithms
publication date: 2000-06-07T00:00:00Z
review text: Numerical methods for solving nonlinear equations typically require the solution of a sequence of linear systems (1) \(A^k x^k = b^k\), \(k = 1,2,\ldots\). In continuation of earlier work [see e.g. ibid. 17, No. 3-4 (1998; Zbl 0908.65034)], the authors consider the case when the \(A^k\) in (1) are different large, sparse, symmetric, positive definite matrices, and each system (1) is solved iteratively by a conjugate gradient method. Several new solvers are introduced that rely on a reuse of the Krylov subspaces associated with (1) for earlier indices. The principal idea is to split the search for the required minimum into two subspaces that are \(A^k\)-conjugate where the first subspace enables some filtering of information from the previous Krylov subspaces related to the large eigenvalues of the current \(A^k\). For the numerical assessment of the approach a class of problems is considered concerning large deformations of heterogeneous hyperelastic bodies. Certain computational aspects of memory management and complexity, as well as the use of parallel computer systems, are also addressed.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65Y05
Mathematics Subject Classification ID: 74B20
zbMATH DE Number: 1456932
zbMATH Keywords: nonlinear equation
zbMATH Keywords: iteration
zbMATH Keywords: conjugate gradients
zbMATH Keywords: Krylov subspaces
zbMATH Keywords: parallel computing
zbMATH Keywords: hyperelastic bodies
reviewed by: Werner C. Rheinboldt
MaRDI profile type: Publication
full work available at URL: a:1019187614377
OpenAlex ID: W1637050567
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description: scientific article; zbMATH DE number 1456932