Parallel Scalability of Three-Level FROSch Preconditioners to 220000 Cores using the Theta Supercomputer

The parallel performance of the three-level fast and robust overlapping Schwarz (FROSch) preconditioners is investigated for linear elasticity. The FROSch framework is part of the Trilinos software library and contains a parallel implementation of different preconditioners with energy minimizing coarse spaces of generalized Dryja--Smith--Widlund type. The three-level extension is constructed by a recursive application of the FROSch preconditioner to the coarse problem. In this paper, the additional steps in the implementation in order to apply the FROSch preconditioner recursively are described in detail. Furthermore, it is shown that no explicit geometric information is needed in the recursive application of the preconditioner. In particular, the rigid body modes, including the rotations, can be interpolated on the coarse level without additional geometric information. Parallel results for a three-dimensional linear elasticity problem obtained on the Theta supercomputer (Argonne Leadership Computing Facility, Argonne, IL) using up to 220 000 cores are discussed and compared to results obtained on the SuperMUC-NG supercomputer (Leibniz Supercomputing Centre, Garching, Germany). Notably, it can be observed that a hierarchical communication operation in FROSch related to the coarse operator starts to dominate the computing time on Theta, which has a dragonfly interconnect, for 100 000 message passing interface (MPI) ranks or more. The same operation, however, scales well and stays within the order of a second in all experiments performed on SuperMUC-NG, which uses a fat tree network. Using hybrid MPI/OpenMP parallelization, the onset of the MPI communication problem on Theta can be delayed. Further analysis of the performance of FROSch on large supercomputers with dragonfly interconnects will be necessary.