{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/preconditioned-chebyshev-bicg-for","title":"Preconditioned Chebyshev BiCG for parameterized linear systems","arxiv_id":"2212.04295","date":"2022-12-08","proceeding":null,"authors":["Siobhán Correnty","Elias Jarlebring","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 we assume $A(\\mu)$ 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 $[-a,a]$, $a \\in \\mathbb{R}$. The solution to the linearization is approximated in a preconditioned BiCG setting for shifted systems, 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 to $x(\\mu)$ 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. The competitiveness of the algorithms are illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with parameterized material coefficient. The software used in the simulations is publicly available online, and thus all our experiments are reproducible.","url_abs":"https://arxiv.org/abs/2212.04295v3","url_pdf":"https://arxiv.org/pdf/2212.04295v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"preconditioned-chebyshev-bicg-for","repo_url":"https://github.com/siobhanie/chebyshevbicg","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}