A Multilevel Schwarz Preconditioner Based on a Hierarchy of Robust Coarse Spaces

Daas, Hussam Al; Grigori, Laura; Jolivet, Pierre; Tournier, Pierre-Henri

doi:10.1137/19m1266964

Cited by 20 publications

(27 citation statements)

References 32 publications

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“…In the last two decades, there has been a great advance in the development of spectral coarse spaces that yield efficient preconditioners. Spectral coarse spaces were initially proposed in the multigrid community for elliptic PDEs with self-adjoint operators [15,19,24,34], and similar ideas were later picked up by the DD community for the same kind of problems [2,3,4,7,33,32,40,48,49]. The past three years have seen several approaches to tackle symmetric indefinite systems and non-self-adjoint problems.…”

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confidence: 99%

“…Such preconditioners have been used to solve a large class of systems arising from a range of engineering applications (see, for example, [3,5,29,35,38,47,51] and references therein). We denote by…”

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confidence: 99%

“…First presented and analyzed in [2], local HPSD splitting matrices provide a framework to construct robust two-level Schwarz preconditioners for sparse HPD matrices. Recently, this has led to the introduction of robust multilevel Schwarz preconditioners for finite element SPD matrices [3], sparse normal equations matrices [5], and sparse general SPD matrices [4].…”

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confidence: 99%

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Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices

Daas¹,

Jolivet²,

Rees³

2022

Preprint

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Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against stateof-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.

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Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices

Daas¹,

Jolivet²,

Rees³

2022

Preprint

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“…Thus the GenEO method would not scale to the full Hawk machine. This estimate could be improved by employing parallel direct solvers, but the scalability of these solvers is limited [21,26,3].…”

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confidence: 99%

“…While robustness with respect to coefficient variations is proven and demonstrated numerically, the problem sizes are rather small. A multilevel version of GenEO was proposed in [3] based on the SPSD splitting introduced in [2].…”

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Multilevel Spectral Domain Decomposition

Bastian¹,

Scheichl²,

Seelinger³

et al. 2021

Preprint

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Highly heterogeneous, anisotropic coefficients, e.g. in the simulation of carbon-fibre composite components, can lead to extremely challenging finite element systems. Direct solvers for the resulting large and sparse linear systems suffer from severe memory requirements and limited parallel scalability, while iterative solvers in general lack robustness. Two-level spectral domain decomposition methods can provide such robustness for symmetric positive definite linear systems, by using coarse spaces based on independent generalized eigenproblems in the subdomains. Rigorous condition number bounds are independent of mesh size, number of subdomains, as well as coefficient contrast. However, their parallel scalability is still limited by the fact that (in order to guarantee robustness) the coarse problem is solved via a direct method. In this paper, we introduce a multilevel variant in the context of subspace correction methods and provide a general convergence theory for its robust convergence for abstract, elliptic variational problems. Assumptions of the theory are verified for conforming, as well as for discontinuous Galerkin methods applied to a scalar diffusion problem. Numerical results illustrate the performance of the method for two-and three-dimensional problems and for various discretization schemes, in the context of scalar diffusion and linear elasticity.

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AI‐enhanced iterative solvers for accelerating the solution of large‐scale parametrized systems

Nikolopoulos,

Kalogeris,

Stavroulakis

et al. 2023

Numerical Meth Engineering

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SummaryRecent advances in the field of machine learning open a new era in high performance computing for challenging computational science and engineering applications. In this framework, the use of advanced machine learning algorithms for the development of accurate and cost‐efficient surrogate models of complex physical processes has already attracted major attention from scientists. However, despite their powerful approximation capabilities, surrogate model predictions are still far from being near to the ‘exact’ solution of the problem. To address this issue, the present work proposes the use of up‐to‐date machine learning tools in order to equip a new generation of iterative solvers of linear equation systems, capable of very efficiently solving large‐scale parametrized problems at any desired level of accuracy. The proposed approach consists of the following two steps. At first, a reduced set of model evaluations is performed using a standard finite element methodology and the corresponding solutions are used to establish an approximate mapping from the problem's parametric space to its solution space using a combination of deep feedforward neural networks and convolutional autoencoders. This mapping serves as a means of obtaining very accurate initial predictions of the system's response to new query points at negligible computational cost. Subsequently, an iterative solver inspired by the Algebraic Multigrid method in combination with Proper Orthogonal Decomposition, termed POD‐2G, is developed that successively refines the initial predictions of the surrogate model towards the exact solution. The application of POD‐2G as a standalone solver or as preconditioner in the context of preconditioned conjugate gradient methods is demonstrated on several numerical examples of large scale systems, with the results indicating its strong superiority over conventional iterative solution schemes.

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A Multilevel Schwarz Preconditioner Based on a Hierarchy of Robust Coarse Spaces

Cited by 20 publications

References 32 publications

Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices

Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices

Multilevel Spectral Domain Decomposition

AI‐enhanced iterative solvers for accelerating the solution of large‐scale parametrized systems

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