Nested Krylov Methods Based on GCR

摘要

Recently the GMRESR method for the solution of linear systems of equations hasbeen introduced. GMRESR involves an outer and an inner method. The outer method is GCR, which is used to compute the optimal approximation over a given set of direction vectors in the sense that the residual is minimized. The inner method is GMRES, which at each step computes a new direction vector by approximately solving the residual equation. This direction vector is then used by the outer algorithm to compute a new approximation. We propose to preserve the orthogonality relations of GCR in the inner GMRES algorithm. This gives optimal correction with an 'improved' operator, which should also lead to faster convergence. However, this involves using Krylov methods with a singular, non-symmetric operator. We will discuss some important properties of this. We will show by experiments, that in terms of matrix vector multiplications this modification (almost) always leads to better convergence.