Monday 6th February 2023 – LJLL
Nicole Spillane CNRS – Polytechnique

Title : Some remarks on the convergence of GMRES with weighted norms, preconditioning and deflation

Summary :

GMRES is a very well established iterative solver for general linear systems. At iteration k, it computes the approximate solution x_k to the linear system Ax = b that minimizes the norm of the residual in a certain space called the Krylov subspace. It is also well known that convergence can be accelerated by means of:
– preconditioning, i.e., providing the solver with an (easier to compute) approximate of the inverse of A,
– deflating, i.e., pre-solving a projected version of the problem,
– weighting, i.e., changing the norm that is minimized at each iteration.

I will show how convergence of GMRES can be analyzed theoretically. I will focus particularly on problems whose symmetric part (A+A*)/2 is positive definite and on symmetric positive definite preconditioners.

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