Scalable Lazy-update Multigrid Preconditioners

Abstract

Multigrid is one of the most effective methods for solving elliptic PDEs. It is algorithmically optimal and is robust when combined with Krylov methods. Algebraic multigrid is especially attractive due to its blackbox nature. This however comes at the cost of increased setup costs that can be significant in case of systems where the system matrix changes frequently making it difficult to amortize the setup cost. In this work, we investigate several strategies for performing lazy updates to the multigrid hierarchy corresponding to changes in the system matrix. These include delayed updates, value updates without changing structure, process local changes, and full updates. We demonstrate that in many cases, the overhead of building the AMG hierarchy can be mitigated for rapidly changing system matrices.