Improving GMRES(m) using an adaptive switching controller

Abstract

Summary The restarted generalized minimal residual (denoted as GMRES(m)) normally used for solving a linear system of equations of the form Ax=b has the drawback of eventually presenting a stagnation or a slowdown in its rate of convergence at certain restarting cycles. In this article, a switching controller is introduced to modify the structure of the GMRES(m) when a stagnation is detected, enlarging and enriching the subspace. In addition, an adaptive control law is introduced to update the restarting parameter to modify the dimension of the Krylov subspace. This combination of strategies is competitive from the point of view of helping to avoid the stagnation and accelerating the convergence with respect to the number of iterations and the computational time. Computational experiments corroborate the theoretical results.

Experiment and Simulation of One Deep Sewage Discharge Type: Transport of Low‐Salinity Sediment‐Laden Flow Discharged

“…The convection terms were discretized in the non‐conservation form, combining the discretization method of the central differencing scheme and the upwind scheme. The Generalized Minimal Residual Algorithm (GMRES) method was used to solve the velocity‐pressure terms in the discrete equations (Cabral et al., 2020). The volume of fluid (VOF) method was utilized to track the free surface (Park & Park, 2021).…”