Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems

Klawonn, Axel; Kühn, Martin Joachim; Rheinbach, Oliver

doi:10.1553/etna_vol49s1

Cited by 17 publications

(34 citation statements)

References 40 publications

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“…It also generalizes these results to the approximate case. The idea of building a coarse space by solving local eigenproblems in each subdomain was introduced in [15,29]; it was successfully applied for other DDM such as FETI-DP [12] or BDDC [9] in [22,23,24].…”

Section: And Thementioning

confidence: 99%

Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

Agullo¹,

Giraud²,

Poirel³

2019

SIAM J. Matrix Anal. Appl.

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The solution of large sparse linear systems is one of the most time consuming kernels in many numerical simulations. The domain decomposition community has developed many efficient and robust methods in the last decades. While many of these solvers fall into the abstract Schwarz (aS) framework, their robustness has originally been demonstrated on a case-by-case basis. In this paper, we propose a bound for the condition number of all deflated aS methods provided that the coarse grid consists of the assembly of local components that contain the kernel of some local operators. We show that classical results from the literature on particular instances of aS methods can be retrieved from this bound. We then show that such a coarse grid correction can be explicitly obtained algebraically via generalized eigenproblems, leading to a condition number independent of the number of domains. This result can be readily applied to retrieve or improve the bounds previously obtained via generalized eigenproblems in the particular cases of Neumann-Neumann (NN), Additive Schwarz (AS) and optimized Robin but also generalizes them when applied with approximate local solvers. Interestingly, the proposed methodology turns out to be a comparison of the considered particular aS method with generalized versions of both NN and AS for tackling the lower and upper part of the spectrum, respectively. We furthermore show that the application of the considered grid corrections in an additive fashion is robust in the AS case although it is not robust for aS methods in general. In particular, the proposed framework allows for ensuring the robustness of the AS method applied on the Schur complement (AS/S), either with deflation or additively, and with the freedom of relying on an approximate local Schur complement. Numerical experiments illustrate these statements.

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Section: And Thementioning

confidence: 99%

Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

Agullo¹,

Giraud²,

Poirel³

2019

SIAM J. Matrix Anal. Appl.

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“…Therefore, it is beneficial to reduce the number of necessary eigenvalue problems to a necessary minimum, e.g., by filtering out eigenvalue problems which do not add any new constraints for a given tolerance T OL. A heuristic approach based on the coe cient jumps on the considered faces and edges as well as the residual after one step of the FETI-DP or BDDC method is already considered in [34,35]. Our approach is to train a neural network to make this decision automatically.…”

Section: Computational E↵ortmentioning

confidence: 99%

“…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.…”

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confidence: 99%

“…Unfortunately, it is not known in advance which of the eigenvalue problems have to be considered. To decide which eigenvalue problems can be omitted, a heuristic approach based on the coe cient jumps as well as the residual after one step of the FETI-DP or BDDC method are already considered in [34,35]; see also [40] for a first preliminary result in this direction. A heuristic approach for the detection of necessary eigenvalue problems and the construction of adaptive constraints without the solution of eigenvalue problems for overlapping Schwarz methods is proposed in [21,24].…”

mentioning

confidence: 99%

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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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“…In recent years, different approaches for the construction of adaptive coarse spaces for domain decomposition methods have been studied extensively; see, e.g., [6,7,61,76,23,24,65,66,13,62,87,78,80,81,82,11,79,28,47,49,50,69,27,39,40,89,25,31,1,4,67,9,90,68,38,14,8,44,59,10].…”

Section: Introductionmentioning

confidence: 99%

Parallel adaptive FETI‐DP using lightweight asynchronous dynamic load balancing

Klawonn

Kühn

Rheinbach

2019

Numerical Meth Engineering

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A parallel FETI-DP domain decomposition method using an adaptive coarse space is presented. The implementation builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses small, local eigenvalue problems for faces and, additionally, for a small number of edges. The condition number of the preconditioned operator then satisfies a bound which is independent of coefficient heterogeneities in the problem. The computational cost of the local eigenvalue problems is not negligible, and also a significant load imbalance can be introduced. As a remedy, certain eigenvalue problems are discarded by a theory-guided heuristic strategy, based on the diagonal entries of the stiffness matrices. Additionally, a lightweight pairwise dynamic load balancing strategy is implemented for the eigenvalue problems. The load balancing is supervised by an orchestrating rank using asynchronous point-to-point communication. The resulting method shows good weak and strong scalability up to thousands of cores while fast convergence is obtained even for heterogeneous problems.

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Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems

Cited by 17 publications

References 40 publications

Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

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

Parallel adaptive FETI‐DP using lightweight asynchronous dynamic load balancing

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