Alexander Heinlein


08 May 2019


12:45 - 13:30 (Lunch available from 12:35)


RA 1501 (Ravelijn)


Alexander Heinlein

Title: Designing robust and efficient domain decomposition methods for highly heterogeneous problems using local spectral information and machine learning techniques


Large linear systems arising from discretizations of partial differential equations are typically solved using iterative solvers in combination with preconditioners. Domain decomposition methods are a class of preconditioners that are scalable for a wide range of problems, e.g., from solid or fluid mechanics. However, the presence of large variations or jumps within material properties, resulting in large variations in the spectrum of the system, can significantly affect the convergence of these methods. Variations in the material properties appear in many applications, for instance, in the simulation of composite materials or porous media. In order to retain robustness of domain decomposition methods for such problems, the coarse space can be enriched by additional coarse basis functions, which are computed from the solutions of local generalized eigenvalue problems. In the domain decomposition community, coarse spaces of this type are typically called adaptive coarse spaces.

In this talk, reasons for the deterioration of the convergence rate of classical domain decomposition methods for highly heterogeneous problems are discussed, and adaptive coarse spaces that can recover robustness of these methods are introduced. As a drawback, the solution of the local generalized eigenvalue problems typically takes up a significant part of the total solution time. However, for many realistic problems, already the coarse space constructed from a part of all local eigenvalue problems is robust, whereas the solution of many eigenvalue problems could be avoided. Therefore, the second part of the talk deals with the application of neural networks  to identify the set of relevant eigenvalue problems, which is unknown a priori. Thereby, the number of eigenvalue problems that are solved is reduced.