Convergence analysis for nonlinear GMRES

Abstract In this work we revisit nonlinear generalized minimal residual method (NGMRES) applied to nonlinear problems. NGMRES is used to accelerate the convergence of fixed-point iterations, which can substantially improve the performance of the underlying fixed-point iterations. We consider NGMRES with a finite window size m, denoted as NGMRES (m). However, there is no convergence analysis for NGMRES (m) applied to nonlinear systems. We prove that for general m0 the residuals of NGMRES (m) converge r-linearly under some conditions. For m=0, we prove that the residuals of NGMRES (0) converge q-linearly.