Accelerating the iterative solution of convection-diffusion problems using singular value decomposition
For researchers solving convection-diffusion problems with iterative solvers, this offers an alternative recycling method that may improve convergence, though the gains are incremental.
The paper proposes a new recycling strategy for Krylov subspace methods that uses singular value decomposition to select previous solutions, accelerating the iterative solution of convection-diffusion problems. Numerical tests show promising results for high-order methods.
The discretization of convection-diffusion equations by implicit or semi-implicit methods leads to a sequence of linear systems usually solved by iterative linear solvers such as GMRES. Many techniques bearing the name of \emph{recycling Krylov space methods} have been proposed to speed up the convergence rate after restarting, usually based on the selection and retention of some Arnoldi vectors. After providing a unified framework for the description of a broad class of recycling methods and preconditioners, we propose an alternative recycling strategy based on a singular value decomposition selection of previous solutions, and exploit this information in classical and new augmentation and deflation methods. The numerical tests in scalar non-linear convection-diffusion problems are promising for high-order methods.