Adaptive GDSW Coarse Spaces for Overlapping Schwarz Methods in Three Dimensions

Heinlein, Alexander; Klawonn, Axel; Knepper, Jascha; Rheinbach, Oliver

doi:10.1137/18m1220613

Cited by 34 publications

(30 citation statements)

References 44 publications

(80 reference statements)

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“…Let us remark that the basic procedure is the same for all adaptive DDMs which rely on the solution of localized eigenvalue problems on edges or, in three dimensions, on edges and faces. Hence, we describe the approach independently of the specific DDM and show numerical results for two very different examples, that is, adaptive FETI‐DP and adaptive generalized Dryja‐Smith‐Widlund (GDSW) [26]. Let us remark that we combine an overlapping adaptive DDM with our machine learning approach for the first time in this article and thus also show the generality of our approach.…”

Section: Machine Learning In Adaptive Domain Decompositionmentioning

confidence: 94%

Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review

et al. 2021

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Scientific machine learning (SciML), an area of research where techniques from machine learning and scientific computing are combined, has become of increasing importance and receives growing attention. Here, our focus is on a very specific area within SciML given by the combination of domain decomposition methods (DDMs) with machine learning techniques for the solution of partial differential equations. The aim of the present work is to make an attempt of providing a review of existing and also new approaches within this field as well as to present some known results in a unified framework; no claim of completeness is made. As a concrete example of machine learning enhanced DDMs, an approach is presented which uses neural networks to reduce the computational effort in adaptive DDMs while retaining their robustness. More precisely, deep neural networks are used to predict the geometric location of constraints which are needed to define a robust coarse space. Additionally, two recently published deep domain decomposition approaches are presented in a unified framework. Both approaches use physics‐constrained neural networks to replace the discretization and solution of the subdomain problems of a given decomposition of the computational domain. Finally, a brief overview is given of several further approaches which combine machine learning with ideas from DDMs to either increase the performance of already existing algorithms or to create completely new methods.

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Section: Machine Learning In Adaptive Domain Decompositionmentioning

confidence: 94%

Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review

et al. 2021

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“…However, for many realistic applications, where unstructured grids and domain decompositions are used, a coarse triangulation is typically not available and, in addition, di cult to obtain. On the other hand, heterogeneous problems might require an additional treatment of the heterogeneities by the coarse space; see, e.g., [1,23,3] for multiscale coarse spaces and, e.g., [21,49,16,22,28,18,26,27] for adaptive coarse spaces.…”

Section: 2mentioning

confidence: 99%

“…Several coarse spaces for linear Schwarz methods have been proposed which can be constructed based on unstructured domain decompositions, without the need for an additional coarse triangulation; see, e.g., [14,11,10,12,13,48,3,16,22,28,18,26,27,24,25]. Most of those approaches make use of energy-minimizing extensions based on the di↵erential operator of the PDE.…”

Section: 2mentioning

confidence: 99%

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Additive and Hybrid Nonlinear Two-Level Schwarz Methods and Energy Minimizing Coarse Spaces for Unstructured Grids

Heinlein¹,

Lanser²

2020

SIAM J. Sci. Comput.

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Nonlinear domain decomposition (DD) methods, such as, e.g., ASPIN (Additive Schwarz Preconditioned Inexact Newton), RASPEN (Restricted Additive Schwarz Preconditioned Inexact Newton), Nonlinear-FETI-DP, or Nonlinear-BDDC methods, can be reasonable alternatives to classical Newton-Krylov-DD methods for the solution of sparse nonlinear systems of equations, e.g., arising from a discretization of a nonlinear partial di↵erential equation. These nonlinear DD approaches are often able to e↵ectively tackle unevenly distributed nonlinearities and outperform Newton's method with respect to convergence speed as well as global convergence behavior. Furthermore, they often improve parallel scalability due to a superior ratio of local to global work. Nonetheless, as for linear DD methods, it is often necessary to incorporate an appropriate coarse space in a second level to obtain numerical scalability for increasing numbers of subdomains. In addition to that, an appropriate coarse space can also improve the nonlinear convergence of nonlinear DD methods. In this paper, four variants how to integrate coarse spaces in nonlinear Schwarz methods in an additive or multiplicative way using Galerkin projections are introduced. These new variants can be interpreted as natural nonlinear equivalents to well-known linear additive and hybrid twolevel Schwarz preconditioners. Furthermore, they facilitate the use of various coarse spaces, e.g., coarse spaces based on energy-minimizing extensions, which can easily be used for irregular domain decompositions, as, e.g., obtained by graph partitioners. In particular, Multiscale Finite Element Method (MsFEM) type coarse spaces are considered, and it is shown that they outperform classical approaches for certain heterogeneous nonlinear problems. The new approaches are then compared with classical Newton-Krylov-DD and nonlinear onelevel Schwarz approaches for di↵erent homogeneous and heterogeneous model problems based on the p-Laplace operator.

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“…interface between subdomains or phenomena as incompressibility and plastification in problems from solid mechanics, the classical condition number bounds do not hold anymore and the convergence of the classical domain decomposition approaches deteriorates. In recent years, several adaptive coarse spaces techniques have been developed to cope with these issues [5,35,34,33,57,56,4,6,51,52,41,40,31,16,17,13,11,60,61,24,23,22,18]. In these approaches, local eigenvalue problems on parts of the interface, e.g., edges or faces, are solved in advance and eigenvectors belonging to certain eigenvalues are used to automatically design a coarse space.…”

mentioning

confidence: 99%

Machine Learning in Adaptive Domain Decomposition Methods---Predicting the Geometric Location of Constraints

Heinlein¹,

Klawonn²,

Lanser³

et al. 2019

SIAM J. Sci. Comput.

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Domain decomposition methods are robust and parallel scalable, preconditioned iterative algorithms for the solution of the large linear systems arising in the discretization of elliptic partial di↵erential equations by finite elements. The convergence rate of these methods is generally determined by the eigenvalues of the preconditioned system. For second-order elliptic partial di↵erential equations, coe cient discontinuities with a large contrast can lead to a deterioration of the convergence rate. A remedy can be obtained by enhancing the coarse space with elements, which are often called constraints, that are computed by solving small eigenvalue problems on portions of the interface of the domain decomposition, i.e., edges in two dimensions or faces and edges in three dimensions. In the present work, without restriction of generality, the focus is on two dimensions. In general, it is di cult to predict where these constraints have to be added and therefore the corresponding local eigenvalue problems have to be computed, i.e., on which edges. Here, a machine learning based strategy using neural networks is suggested to predict the geometric location of these edges in a preprocessing step. This reduces the number of eigenvalue problems that have to be solved before the iteration. Numerical experiments for model problems and realistic microsections using regular decompositions as well as decompositions from graph partitioners are provided, showing very promising results.

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Adaptive GDSW Coarse Spaces for Overlapping Schwarz Methods in Three Dimensions

Cited by 34 publications

References 44 publications

Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review

Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review

Additive and Hybrid Nonlinear Two-Level Schwarz Methods and Energy Minimizing Coarse Spaces for Unstructured Grids

Machine Learning in Adaptive Domain Decomposition Methods---Predicting the Geometric Location of Constraints

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