Speaker
Description
Many applications require the solution of large-scale linear systems that have nonlinear parameter dependence. The Infinite GMRES algorithm, developed by Jarlebring and Correnty in 2022, converts these systems into infinite-dimensional systems with linear parameter dependence. This transformation involves a linearization process that results in a system with a special companion-like structure. Due to the shift invariance of the Krylov subspaces, all choices of parameter give the same Krylov subspaces. Thus performing the Arnoldi process for one parameter enables the efficient construction of the GMRES iterates for all parameter values. The special structure of the right-hand-side vector of the infinite-dimensional system allows this Arnoldi process to be executed with finite-dimensional vectors, where the dimension grows with every iteration. As such, Infinite GMRES uses finite-length Krylov basis vectors to solve an infinite-dimensional system. The special structure of the infinite-dimensional system invites further investigation into the convergence behavior of the Infinite GMRES residual norm. We aim to understand how various components of this system, including the size and nature of the block matrices, the number of blocks, and the choice of parameter value influence convergence behavior. In this presentation, we will provide rigorous convergence results using various tools of analysis, including the numerical range.