Axel Klawonn, Professor, Universität zu Köln

Jizu Huang, Associate Professor

Learning Coarse Spaces in Domain Decomposition Methods

15:00-16:00 September 16(Tuesday), N226

Domain decomposition methods (DDMs) are highly parallel scalable, iterative solvers for the solution of large systems of linear equations, for example, arising from the discretization of elliptic partial differential equations (PDEs). The convergence rate of classic DDMs in general deteriorates severely for coefficient distributions with large contrasts in the coefficient function. To retain the robustness for such problems, the coarse space of the DDM can be enriched by additional coarse basis functions, often obtained by solving local generalized eigenvalue problems. However, the set-up and the solution of these eigenvalue problems typically takes up a significant part of the total time to solution. Additionally, for many realistic model problems, only the solution of a small number of the eigenvalue problems is necessary to design a robust algorithm. In general, it is difficult to predict a priori which of the eigenvalue problems are needed. Using a neural network model we can predict the geometric location where eigenvalue problems have to be solved, often reducing its number significantly. To obtain such an a priori classification, we use a mesh-independent sampling strategy which is comparable to an image recognition problem of the given coefficient function.

In a next step, we train a surrogate model which directly learns the necessary coarse basis functions themselves using again an image representation of the underlying coefficient function. Numerical results indicate the robustness of this approach.