On the Use of Block Low Rank Preconditioners for Primal Domain Decomposition Methods

Abstract

This article investigates the use of the block low rank (BLR) factorization, recently proposed in the MUMPS solver, to define efficient and cheap preconditioners for primal domain decomposition methods, such as the Balancing Domain Decomposition method (BDD) and its adaptive multipreconditioned variant. To be scalable, these methods are equipped with an augmentation projector built from the local preconditioners nullspaces. The determination of these nullspaces is a complex task in the case of ill conditioned system, the use of block low rank compression makes this task even more complex as MUMPS' automatic detection no longer works properly. Two alternatives based on incomplete factorization with a well-chosen Schur complement are proposed. Also, the first massively parallel implementation of the adaptive multipreconditioned BDD solver (AMPBDD) is introduced. The performance of the methods is assessed with two weak scalability studies on problems up to 24,576 cores and about 790 millions of unknowns, on the Sator and Topaze supercomputers. BLR preconditioning proves to be an interesting strategy both in terms of memory usage and time to solution for reasonably conditioned problems.