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Efficient Solution of Stochastic Galerkin Matrix Equations via Reduced Basis and Tensor Train Approximation
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SYSNO ASEP 0586684 Document Type C - Proceedings Paper (int. conf.) R&D Document Type Conference Paper Title Efficient Solution of Stochastic Galerkin Matrix Equations via Reduced Basis and Tensor Train Approximation Author(s) Béreš, Michal (UGN-S) ORCID, RID, SAI Number of authors 1 Source Title Large-Scale Scientific Computations, Lecture Notes in Computer Science, 13952. - Cham : Springer Nature Switzerland AG, 2024 / Lirkov I. ; Margenov S. - ISSN 0302-9743 - ISBN 978-3-031-56207-5 Pages s. 205-214 Number of pages 10 s. Publication form Online - E Action LSSC 2023: International Conference on Large-Scale Scientific Computations /14./ Event date 05.06.2023 - 09.06.2023 VEvent location Sozopol Country BG - Bulgaria Event type WRD Language eng - English Country CH - Switzerland Keywords stochastic Galerkin method ; reduced basis ; tensor train approximation Subject RIV BA - General Mathematics OECD category Applied mathematics Institutional support UGN-S - RVO:68145535 UT WOS 001279202200021 EID SCOPUS 85195469853 DOI https://doi.org/10.1007/978-3-031-56208-2_20 Annotation This contribution focuses on the development of a computational method to efficiently solve matrix equations arising from stochastic Galerkin (SG) discretization of steady Darcy flow problems with uncertain and separable permeability fields. The proposed method consists of a two-step solution process. Firstly, we construct a reduced basis for the finite element portion of the discretization using the Monte Carlo (MC) method. We consider various sampling techniques for the MC method. Secondly, we use a tensor polynomial basis to handle the stochastic aspect of the problem and employ a tensor-train (TT) approximation to approximate the overall solution of the reduced SG system. To enhance the convergence of the TT approximation, we use an implicitly preconditioned system with a Kronecker-type preconditioner. Moreover, we also develop low-cost error indicators to assess the accuracy of both thereduced basis and the final solution of the reduced system. Workplace Institute of Geonics Contact Lucie Gurková, lucie.gurkova@ugn.cas.cz, Tel.: 596 979 354 Year of Publishing 2025 Electronic address https://link.springer.com/book/10.1007/978-3-031-56208-2
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