0586684 - ÚGN 2025 RIV CH eng C - Konferenční příspěvek (zahraniční konf.)
Béreš, Michal
Efficient Solution of Stochastic Galerkin Matrix Equations via Reduced Basis and Tensor Train Approximation.
Large-Scale Scientific Computations. Vol. 13952. Cham: Springer Nature Switzerland AG, 2024 - (Lirkov, I.; Margenov, S.), s. 205-214. ISBN 978-3-031-56207-5. ISSN 0302-9743. E-ISSN 1611-3349.
[LSSC 2023: International Conference on Large-Scale Scientific Computations /14./. Sozopol (BG), 05.06.2023-09.06.2023]
GRANT EU: European Commission(XE) 847593 - EURAD
Institucionální podpora: RVO:68145535
Klíčová slova: stochastic Galerkin method * reduced basis * tensor train approximation
Obor OECD: Applied mathematics
Web výsledku:
https://link.springer.com/book/10.1007/978-3-031-56208-2DOI: https://doi.org/10.1007/978-3-031-56208-2_20

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.
Trvalý link: https://hdl.handle.net/11104/0354116