Szczegóły publikacji
Opis bibliograficzny
Augmenting Petrov-Galerkin method with optimal test functions by DNN learning the inverse of the Gram matrix / Tomasz SŁUŻALEC // W: Computational Science – ICCS 2025 Workshops : 25th international conference : Singapore, Singapore, July 7–9, 2025 : proceedings, Pt. 1 / eds. Maciej Paszyński, Amanda S. Barnard, Yongjie Jessica Zhang. — Cham : Springer Nature Switzerland, cop. 2025. — (Lecture Notes in Computer Science ; ISSN 0302-9743 ; LNCS 15907). — ISBN: 978-3-031-97553-0; e-ISBN: 978-3-031-97554-7. — S. 112–125. — Bibliogr., Abstr. — Publikacja dostępna online od: 2025-07-07
Autor
Dane bibliometryczne
| ID BaDAP | 161023 |
|---|---|
| Data dodania do BaDAP | 2025-07-10 |
| DOI | 10.1007/978-3-031-97554-7_9 |
| Rok publikacji | 2025 |
| Typ publikacji | materiały konferencyjne (aut.) |
| Otwarty dostęp |  |
| Wydawca | Springer |
| Konferencja | International Conference on Computational Science 2025 |
| Czasopismo/seria | Lecture Notes in Computer Science |
Abstract
The Petrov-Galerkin (PG) method is a robust alternative to the Galerkin method for finite element simulations of challenging partial differential equations (PDEs). The solution of the Galerkin method is obtained from the linear system Bx = F resulting from discretization of trial and test spaces. Although the Galerkin method enforces the equality of trial and test spaces (Uh = Vh) and relies on the inf-sup stability condition to ensure solution accuracy, it often fails for difficult problems where the discrete inf-sup constant αh significantly deviates from the abstract inf-sup constant α. This discrepancy leads to numerical instability and incorrect solutions. The PG method addresses this by allowing distinct trial and test spaces (Uh≠Vh), enabling the selection of test functions that improve the discrete inf-sup constant αh. Of particular interest is the Petrov-Galerkin method with optimal test functions (PGO), where the test functions are computed to maximize αh, ensuring stable solutions even for ill-conditioned problems. The PGO method modifies the discrete test space to approximate the abstract stability properties as closely as possible. The computation of optimal test functions involves solving GW = B, where B is the Galerkin matrix, and G is the Gram matrix of the test space’s inner product. Solving BTWx = WTF then yields a stable solution. However, the added computational cost of inverting G-1 poses a significant overhead. In this work, we propose a novel approach that leverages deep neural networks (DNNs) to approximate the inverse of the Gram matrix for a class of advection-diffusion problems with variable diffusion coefficients. By training the DNN to predict G-1, we eliminate the computational overhead of matrix inversion, enabling efficient and stable solutions of PDEs. Our results demonstrate the effectiveness of the DNN-enhanced PGO method in maintaining stability and accuracy, even for difficult computational problems where the standard Galerkin method fails. This approach represents a significant advancement in the practical applicability of the Petrov-Galerkin framework for solving complex PDEs.
