Siobhan Correnty (KTH Stockholm)
We consider parameterized linear systems of the form $A(\mu) x(\mu) = b$ for many different $\mu$, where $A(\mu)$ is large, sparse and nonsingular, with a nonlinear dependence on $\mu$. In this work we propose two methods to compute a partial parameterization $\tilde{x} \approx x(\mu)$, where $\tilde{x}(\mu)$ is cheap to evaluate for many values of $\mu$. Both methods utilize a companion linearization, inspired by [Effenberger, Kressner, 2012], based on an accurate Chebyshev interpolation of $A(\mu)$ on the interval $[-a,a]$, $a \in \mathbb{R}$. We approximate the solution to the linearization in a preconditioned multishift BiCG setting, as proposed in [Ahmad, Szyld, van Gijzen, 2017]. This process leads to two short-term recurrence methods, where one execution of either algorithm produces the approximation $\tilde{x}(\mu)$ for many different $\mu \in [-a,a]$ simultaneously. The first proposed method considers an exact application of a shift-and-invert preconditioner on each iteration, and the second applies an approximation to the same preconditioner in an inexact, flexible setting. The competitiveness of our methods are illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with parameterized material coefficient.
