Reducing Communication in AMG for Reservoir Simulation

Abstract

Algebraic Multigrid (AMG) is an efficient multigrid method for solving large problems, using only the information provided by the underlying matrices. Unfortunately, on a parallel machine the performance of AMG can be negatively affected by dense communication patterns. The exchange of extensive data sets is generally caused by a high number of non-zero entries contained by the coarse grid operators. We discuss two solution strategies, that can be applied in order to improve the sparsity patterns of these operators: aggressive coarsening and the non-Galerkin method. Aggressive coarsening reduces the number of coarse grid variables by modifying the basic concepts of the standard coarsening scheme. The non-Galerkin approach is entirely different. While preserving the row sums, the non-Galerkin method removes less important entries of the coarse level operators, generated by AMG. In fact, it can be seen as an extension of the standard multigrid algorithm. We explore both techniques in the frame of reservoir simulation. We demonstrate that aggressive coarsening and non-Galerkin algorithm significantly reduce the total number of non-zero entries in the AMG hierarchy. Although aggressive coarsening shows high performance, in terms of execution time, on one processor core, it is less effective for parallel simulations. The non-Galerkin has not been tested in parallel, but its serial results are very promissing.