Predicting the geometric location of critical edges in adaptive GDSW overlapping domain decomposition methods using deep learning

Overlapping GDSW domain decomposition methods are considered for diffusion problems in two dimensions discretized by finite elements. For a diffusion coefficient with high contrast, the condition number is usually dependent on it. A remedy is given by adaptive domain decomposition methods, where the coarse space is enhanced by additional coarse basis functions. These are chosen problem-dependently by solving small local eigenvalue problems. Here, the eigenvalue problems (EVPs) are associated with the edges of the domain decomposition interface; edges, where these EVPs have to be solved are denoted as critical edges. For many applications, not all edges are critical and the solution of the EVPs is not necessary. In an earlier work, a strategy to predict the location of critical edges, based on deep learning, has been proposed for adaptive FETI-DP, a class of nonoverlapping methods. In the present work, this strategy is successfully applied to adaptive GDSW; differences in the classification process for this overlapping method are described. Choosing the classification threshold has been a challenge in all these approaches. Here, for the first time, a heuristic based on the receiver operating characteristic (ROC) curve and the precision-recall graph is discussed. Results for a challenging realistic coefficient function are presented.