GMRESDR

GMRESDR, short for GMRES with deflated restarting, is an iterative method for solving large sparse linear systems, particularly nonsymmetric ones. It extends the restarted GMRES framework by incorporating deflation of troublesome eigencomponents to accelerate convergence. The approach targets cases where the coefficient matrix has small or clustered eigenvalues that impede standard restart strategies.

In GMRESDR, each solve proceeds in cycles. During a cycle, GMRES constructs a Krylov subspace of a fixed size m using the Arnoldi process. From this subspace, approximate eigenpairs (Ritz values and vectors) are computed. A subset of these approximate eigenvectors, typically k, is retained to form a deflation subspace. In subsequent cycles, these deflation vectors are used to augment or modify the search space so that the residual is reduced more effectively by deflating the influence of the troublesome eigencomponents. This augmentation can be viewed as projecting out the directions associated with the deflated eigenvectors, thereby improving convergence after restarts.

GMRESDR is particularly advantageous when solving sequences of linear systems with the same matrix but different right-hand sides, or when A has near-singular behavior that slows convergence of standard GMRES. It reduces iteration counts and can lower overall computational effort at the cost of additional work to compute Ritz vectors and manage the deflation subspace. Typical variants are denoted GMRES-DR(m,k), where m is the restart length and k is the number of deflated vectors kept.

Overall, GMRESDR offers a practical approach to mitigating slow convergence in restarted GMRES by deflating problematic spectral components, making it a useful tool for challenging sparse linear systems arising in PDE discretizations and related applications.