#### Volume 58, pp. 629-656, 2023.

## Preconditioned Chebyshev BiCG method for parameterized linear systems

Siobhán Correnty, Elias Jarlebring, and Daniel B. Szyld

### Abstract

We consider the problem of approximating the solution to $A(\mu) x(\mu) = b$ for many different values of the parameter $\mu$. Here, $A(\mu)$ is large, sparse, and nonsingular with a nonlinear dependence on $\mu$. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of $A(\mu)$ on the interval $[-a,a]$, $a \in \mathbb{R}_+$, inspired by Effenberger and Kressner [BIT, 52 (2012), pp. 933–951]. The solution to the linearization is approximated in a preconditioned BiCG setting for shifted systems, as proposed in Ahmad et al. [SIAM J. Matrix Anal. Appl., 38 (2017), pp. 401–424], where the Krylov basis matrix is formed once. This process leads to a short-term recurrence method, where one execution of the algorithm produces the approximation of $x(\mu)$ for many different values of the parameter $\mu \in [-a,a]$ simultaneously. In particular, this work proposes one algorithm which applies a shift-and-invert preconditioner exactly as well as an algorithm which applies the preconditioner inexactly based on the work by Vogel [Appl. Math. Comput., 188 (2007), pp. 226–233]. The competitiveness of the algorithms is illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with a parameterized material coefficient. The software used in the simulations is publicly available online, and thus all our experiments are reproducible.

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### Key words

parameterized linear systems, short-term recurrence methods, Chebyshev interpolation, inexact preconditioning, Krylov subspace methods, companion linearization, shifted linear systems, parameterized Helmholtz equation, time-delay systems

### AMS subject classifications

15A06, 65F08, 65F10, 65F50, 65N22, 65P99

### Links to the cited ETNA articles

[38] | Vol. 37 (2010), pp. 123-146 Yvan Notay: An aggregation-based algebraic multigrid method |

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