Convergence analysis for nonlinear GMRES

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 $m>0$ 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.