Universität zu Köln

Heinlein, Alexander, Klawonn, Axel ORCID: 0000-0003-4765-7387, Lanser, Martin and Weber, Janine (2019). MACHINE LEARNING IN ADAPTIVE DOMAIN DECOMPOSITION METHODS-PREDICTING THE GEOMETRIC LOCATION OF CONSTRAINTS. SIAM J. Sci. Comput., 41 (6). S. A3887 - 26. PHILADELPHIA: SIAM PUBLICATIONS. ISSN 1095-7197

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Abstract

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 differential 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 differential equations, coefficient 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 difficult 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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Heinlein, Alexander UNSPECIFIED UNSPECIFIED UNSPECIFIED Klawonn, Axel UNSPECIFIED orcid.org/0000-0003-4765-7387 UNSPECIFIED Lanser, Martin UNSPECIFIED UNSPECIFIED UNSPECIFIED Weber, Janine UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-159695
DOI:	10.1137/18M1205364
Journal or Publication Title:	SIAM J. Sci. Comput.
Volume:	41
Number:	6
Page Range:	S. A3887 - 26
Date:	2019
Publisher:	SIAM PUBLICATIONS
Place of Publication:	PHILADELPHIA
ISSN:	1095-7197
Language:	English
Faculty:	Central Institutions / Interdisciplinary Research Centers
Divisions:	Weitere Institute, Arbeits- und Forschungsgruppen > Center for Data and Simulation Science (CDS)
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language OVERLAPPING SCHWARZ METHODS; PRIMAL FETI METHODS; COARSE SPACES; BDDC METHODS; SUBSTRUCTURING METHODS; UNIFIED FRAMEWORK; ELLIPTIC PROBLEMS; MULTISCALE FLOWS; DP METHODS; PRECONDITIONERS Multiple languages Mathematics, Applied Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/15969