Asynchronous Multigrid Methods

Reducing synchronization in iterative methods for solving large sparse linear systems may become one of the most important goals for such solvers on exascale computers. Research in asynchronous iterative methods has primarily considered basic iterative methods. In this paper, we examine how multigrid methods can be executed asynchronously. We present models of asynchronous additive multigrid methods, and use these models to study the convergence properties of these methods. We also introduce two parallel algorithms for implementing asynchronous additive multigrid, the global-res and local-res algorithms. These two algorithms differ in how the fine grid residual is computed, where local-res requires less computation than global-res but converges more slowly. We compare two types of asynchronous additive multigrid methods: the asynchronous fast adaptive composite grid method with smoothing (AFACx) and additive variants of the classical multiplicative method (Multadd). We implement asynchronous versions of Multadd and AFACx in OpenMP and generate the prolongation and coarse grid matrices using the BoomerAMG package. Our experimental results show that asynchronous multigrid can exhibit grid-size independent convergence and can be faster than classical multigrid in terms of solve wall-clock time. We also show that asynchronous smoothing is the best choice of smoother for our test cases, even when only one smoothing sweep is used.