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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 629-656, 2023/12/07
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.
Keywords: 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