Preconditioned Infinite GMRES for Parameterized Linear Systems

We are interested in obtaining solutions to parameterized linear systems of the form for many values of the parameter . Here is large, sparse, and nonsingular with a nonlinear, analytic dependence on . Our approach approximates the solution to a linearized system in a flexible GMRES setting [Y. Saad, SIAM J. Sci. Comput., 14 (1993), pp. 461–469], where the linearization is based on a companion matrix similar to the operator in the infinite Arnoldi method [E. Jarlebring, W. Michiels, and K. Meerbergen, Numer. Math., 122 (2012), pp. 169–195]. This novel approach applies the action of a preconditioner inexactly, providing performance improvement over the method infinite GMRES [Jarlebring and Correnty, SIAM J. Matrix Anal. Appl., 43 (2022), pp. 1382–1405] without a loss of accuracy in general. The method returns a function which is cheap to evaluate for different . We show that the error of our method is estimated based on the magnitude of the parameter , the inexactness of the preconditioning, and the spectrum of the companion matrix. Numerical examples from a finite element discretization of a Helmholtz equation with a parameterized material coefficient illustrate the competitiveness of our approach. The software used in the simulations is publicly available online, and all the experiments are reproducible.