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Deep learning accelerated algebraic multigrid methods for polytopal discretizations of second-order differential problems.

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Title: Deep learning accelerated algebraic multigrid methods for polytopal discretizations of second-order differential problems.
Authors: Antonietti, Paola F.1 (AUTHOR) paola.antonietti@polimi.it, Caldana, Matteo1 (AUTHOR) matteo.caldana@polimi.it, Gentile, Lorenzo Bruno Donato1 (AUTHOR) lorenzo3.gentile@mail.polimi.it, Verani, Marco1 (AUTHOR) marco.verani@polimi.it
Source: Mathematical Models & Methods in Applied Sciences. Jul2026, Vol. 36 Issue 8, p1649-1677. 29p.
Subjects: Algebraic multigrid methods, Deep learning, Numerical solutions to partial differential equations, Mathematical optimization, Partial differential equations, Galerkin methods
Abstract: Algebraic Multigrid (AMG) methods are state-of-the-art algebraic solvers for Partial Differential Equations. Still, their efficiency depends heavily on the choice of suitable parameters and/or ingredients. Paradigmatic examples include the so-called strong threshold parameter, which controls the algebraic coarse-grid hierarchy, as well as the smoother, i.e. the relaxation methods used on the fine grid to damp out high-frequency components of the error. In AMG, since the coarse grids are constructed algebraically (without geometric intuition), the smoother's performance is even more critical. For the linear systems stemming from polytopal discretizations, such as Polytopal Discontinuous Galerkin (PolyDG) and Virtual Element (VEM) methods, AMG sensitivity to such choices is even more critical due to the significant variability of the underlying meshes, which results in algebraic systems with different sparsity patterns. In this paper, we focus on the linear systems of equations stemming from polytopal discretizations of second-order elliptic problems. We propose a novel deep learning approach that automatically tunes the strong threshold parameter and the smoother choice in AMG solvers, thereby maximizing AMG performance. We test various differential problems in both two- and three-dimensional settings, with heterogeneous coefficients and polygonal/polyhedral meshes, and demonstrate that the proposed approach generalizes well. In practice, we demonstrate that we can reduce AMG solver time by up to 2 7 % with minimal changes to existing PolyDG and VEM software libraries. [ABSTRACT FROM AUTHOR]
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Abstract:Algebraic Multigrid (AMG) methods are state-of-the-art algebraic solvers for Partial Differential Equations. Still, their efficiency depends heavily on the choice of suitable parameters and/or ingredients. Paradigmatic examples include the so-called strong threshold parameter, which controls the algebraic coarse-grid hierarchy, as well as the smoother, i.e. the relaxation methods used on the fine grid to damp out high-frequency components of the error. In AMG, since the coarse grids are constructed algebraically (without geometric intuition), the smoother's performance is even more critical. For the linear systems stemming from polytopal discretizations, such as Polytopal Discontinuous Galerkin (PolyDG) and Virtual Element (VEM) methods, AMG sensitivity to such choices is even more critical due to the significant variability of the underlying meshes, which results in algebraic systems with different sparsity patterns. In this paper, we focus on the linear systems of equations stemming from polytopal discretizations of second-order elliptic problems. We propose a novel deep learning approach that automatically tunes the strong threshold parameter and the smoother choice in AMG solvers, thereby maximizing AMG performance. We test various differential problems in both two- and three-dimensional settings, with heterogeneous coefficients and polygonal/polyhedral meshes, and demonstrate that the proposed approach generalizes well. In practice, we demonstrate that we can reduce AMG solver time by up to 2 7 % with minimal changes to existing PolyDG and VEM software libraries. [ABSTRACT FROM AUTHOR]
ISSN:02182025
DOI:10.1142/S0218202526420029