Learning Interface Conditions in Domain Decomposition Solvers

Abstract

Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the optimal construction of these methods requires tedious analysis and is often available only in simplified, structured-grid settings, limiting their use for more complex problems. In this work, we generalize optimized Schwarz domain decomposition methods to unstructured-grid problems, using Graph Convolutional Neural Networks (GCNNs) and unsupervised learning to learn optimal modifications at subdomain interfaces. A key ingredient in our approach is an improved loss function, enabling effective training on relatively small problems, but robust performance on arbitrarily large problems, with computational cost linear in problem size. The performance of the learned linear solvers is compared with both classical and optimized domain decomposition algorithms, for both structured-and unstructured-grid problems.Preprint. Under review.

Domain Decomposition Learning Methods for Solving Elliptic Problems

“…Robin-Robin Algorithm via Solution-Oriented Learning Methods. Note that in addition to the deep learning analogue of overlapping Schwarz methods [29,35,30,42,44], the non-overlapping Robin-Robin algorithm [45,38] is also based on a direct exchange of solution value between neighbouring subdomains (see Figure 1 or algorithm 2.1). Moreover, as the decomposition leads to simpler and smoother functions to be learned on each subregion, it thus enables us to employ the standard solution-oriented learning methods [39,47,48,43] for the numerical solution of local problems.…”