Iterative accelerating algorithms with Krylov subspaces for the solution to large-scale nonlinear problems
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.
scientific article; zbMATH DE number 1456932