Open this publication in new window or tab >>2023 (English)In: Electronic Transactions on Numerical Analysis, E-ISSN 1068-9613, Vol. 58, p. 629-656Article in journal (Refereed) Published
Abstract [en]
We consider the problem of approximating the solution to A(μ)x(μ) = b for many different values of the parameter μ. Here, A(μ) is large, sparse, and nonsingular with a nonlinear dependence on μ. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of A(μ) on the interval [-a; a], a 2 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(μ) for many different values of the parameter μ ∈ [-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.
Place, publisher, year, edition, pages
Osterreichische Akademie der Wissenschaften, Verlag, 2023
Keywords
Chebyshev interpolation, companion linearization, inexact preconditioning, Krylov subspace methods, parameterized Helmholtz equation, parameterized linear systems, shifted linear systems, short-term recurrence methods, time-delay systems
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:kth:diva-341929 (URN)10.1553/etna_vol58s629 (DOI)2-s2.0-85180534256 (Scopus ID)
Note
QC 20240108
2024-01-082024-01-082024-02-27Bibliographically approved