On adaptive strategy for overcome stagnation in LGMRES(m, l)

Juan Carlos Cabral Figueredo¹; Christian E. Schaerer¹

¹ Polytechnical School, National University of Asuncion.

doi:10.20906/CPS/CILAMCE2017-0970

Resumo

The solution of sparse linear systems is the most time-consuming step in running reservoir simulations. Iterative solution based on projection onto Krylov subspaces is typically used. The most popular choice used when linear systems are sparse and nonsymmetric is the Generalized Minimal Residual algo- rithm (GMRES). This method and its variants are often refered to as an optimal method because it finds the approximate solution in the Krylov subspace that minimizes the 2-norm of the residual. In particular, we will focus on the method LGMRES (m.l), which in addition to using a Krylov subspace of dimension m, uses the last d error approximations. These parameters remain constant in each step. This methodology allows improve the convergence, but can also be stalled if the parameters are not selected correctly. The present work proposes an efficient way to exploit the Krylov subspace information and modify the restart parameters adaptively in order to avoid the stagnation. Numerical results, solving problems of reservoir simulation, show improvements over the traditional implementations, especially, where the restarted parameters are not varied.

Palavras-chave: Iterative method; Krylov subspace; Adaptive GMRES(m); Reservoir simulation