Daniel B. Szyld
Temple University, Philadelphia
Krylov subspace methods such as GMRES are extensively used for the solution of (preconditioned) linear equations, especially those arising from discretization of differential equations. Often, because of the size of the problem, one cannot use the full (orthogonal) basis of the Krylov subspace. It is common to use restarted methods, and these work well sometimes, but not other times. We present two alternatives: inexact and truncated versions. In inexact Krylov subspace methods, the matrix vector product Av is replaced by (A+E)v, where E is some error matrix. Using a computable criterion, this error can be monitored, relaxing the accuracy of the matrix-vector product as the method progresses. One particular application is dynamically adapted inexact additive Schwarz preconditioners, where the convergence tolerance of the local solves are adapted as the overall procedure advances. Large computational savings can be obtained with this approach. Alternatively, one can contain the cost of the methods by orthogonalizing only with respect to a fixed number of previous vectors. Our analysis confirms the observed fact that the orthogonality of the basis is not important, only the need to maintain linear independence is. Numerical examples illustrate our theoretical results. (This is a joint work with V.Simoncini, University of Bologna)