Compact Rational Krylov for Parametrized Systems with Application to BEM Frequency Sweeping

Abstract

In parametrized linear systems $\mathsf{P}(\mu )\mathsf{x}=\mathsf{b}$ the system matrix $\mathsf{P}$ depends nonlinearly on a parameter $\mu$ and solutions are sought for many values of this parameter. We show that the compact rational Krylov (CORK) framework, originally introduced to solve nonlinear eigenvalue problems, can be used to efficiently produce approximate solutions to such a system for many values of the parameter at once. In this approach, the parametrized system is first linearized, resulting in a large shifted linear system $(\mathsf{A}-\mu \mathsf{B})\mathsf{y}=\mathsf{d}$. We formulate a left- and right-preconditioned rational Krylov GMRES method for shifted linear systems. In the setting of parametrized linear systems, these can exploit the structure in the linearization, and in combination with the CORK framework, computational and memory complexity mainly depend on the problem size, less the degree of the linearization. Additionally, we show how to incorporate a right-hand side $\mathsf{b}(\mu )$ that also depends on the parameter, how to choose the shifts to steer convergence and how to allow for inexact solves at these shifts throughout the iterations. As an application we consider the 'frequency sweeping' of Helmholtz scattering problems through the Boundary Element Method (BEM), enabled via an efficient representation of the dense but data-sparse wavenumber-dependent system matrix.