Preconditioning Low Rank Generalized Minimal Residual Method (GMRES) for Implicit Discretizations of Matrix Differential Equations

This work proposes a new class of preconditioners for the low rank Generalized Minimal Residual Method (GMRES) for multiterm matrix equations arising from implicit timestepping of linear matrix differential equations. We are interested in computing low rank solutions to matrix equations, e.g. arising from spatial discretization of stiff partial differential equations (PDEs). The low rank GMRES method is a particular class of Krylov subspace method where the iteration is performed on the low rank factors of the solution. Such methods can exploit the low rank property of the solution to save on computational and storage cost. Of critical importance for the efficiency and applicability of the low rank GMRES method is the availability of an effective low rank preconditioner that operates directly on the low rank factors of the solution and that can limit the iteration count and the maximal Krylov rank. The preconditioner we propose here is based on the basis update and Galerkin (BUG) method, resulting from the dynamic low rank approximation. It is a nonlinear preconditioner for the low rank GMRES scheme that naturally operates on the low rank factors. Extensive numerical tests show that this new preconditioner is highly efficient in limiting iteration count and maximal Krylov rank. We show that the preconditioner performs well for general diffusion equations including highly challenging problems, e.g. high contrast, anisotropic equations. Further, it compares favorably with the state of the art exponential sum preconditioner. We also propose a hybrid BUG - exponential sum preconditioner based on alternating between the two preconditioners.