Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing

Abstract

Abstract. We provide a general framework for the understanding of inexact Krylov subspace methods for the solution of symmetric and nonsymmetric linear systems of equations, as well as for certain eigenvalue calculations. This framework allows us to explain the empirical results reported in a series of CERFACS technical reports by Bouras, Frayssé, and Giraud in 2000. Furthermore, assuming exact arithmetic, our analysis can be used to produce computable criteria to bound the inexactness of the matrix-vector multiplication in such a way as to maintain the convergence of the Krylov subspace method. The theory developed is applied to several problems including the solution of Schur complement systems, linear systems which depend on a parameter, and eigenvalue problems. Numerical experiments for some of these scientific applications are reported.

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

“…This finally shows the main relations (28) and (29) of Theorem 1 that are satisfied at the end of the i + 1-th cycle. …”

“…We analyze the performances of various flexible methods used with four iterations of unpreconditioned GMRES as a preconditioner. This polynomial preconditioner is a variable nonlinear function which thus requires a flexible Krylov subspace method as an outer method [28]. Table 3 collects the number of matrix-vector products of some flexible methods minimizing over a subspace of same dimension i.e.…”

Cooperative Application/OS DRAM Fault Recovery

“…Inexact Krylov methods generalize flexible methods by allowing both the preconditioner and the matrix to change in every iteration [19,20]. However, inexact Krylov methods require error bounds, and thus cannot be used to provide tolerance against arbitrary data and computational faults when applying the matrix A.…”

Comparing Some Methods and Preconditioners for Hydropower Plant Flow Simulations

“…The applications of these preconditioners generate inner-outer iterations schemes [23][24][25], and generate two linear system that ought to be solved for the right-hand side residual.…”