10. A Survey of Parallelization Techniques for Multigrid Solvers

Abstract

This paper surveys the techniques that are necessary for constructing computationally efficient parallel multigrid solvers. Both geometric and algebraic methods are considered. We first cover the sources of parallelism, including traditional spatial partitioning and more novel additive multilevel methods. We then cover the parallelism issues that must be addressed: parallel smoothing and coarsening, operator complexity, and parallelization of the coarsest grid solve.

Scaling Algebraic Multigrid Solvers: On the Road to Exascale

“…Our preparation of the AMG setup phase for exascale computing is a work in progress. In particular, the coarsening and interpolation algorithms may be quite complicated in parallel [5], and the long-distance variety [6] require a sizable amount of point-to-point communications. At this point, the interpolation routines in BoomerAMG are only partially threaded due to complexity, and none of the coarsening routines use any threading at all.…”

Fast and Efficient Methods for Circuit-based Automotive EMC Simulation

“…Even if the algorithm exhibits adequate parallelism an inadvantageous implementation can degrade the overall performance drastically [172].…”

“…For an efficient parallel implementation a good balance between setup times, convergence rates and cost per iteration is necessary. These features, in turn depend on operator's complexity, coarsening rates and smoother's effectiveness [172]. Applying algorithmic modifications, the communication exchange can be reduced at first glance.…”

An Object Oriented Parallel Finite Element Scheme for Computations of PDEs: Design and Implementation

“…Popular preconditioner are ILU, SOR, algebraic multigrid (AMG), geometric multigrid (GMG), etc. Among all, the multigrid method is very efficient, in particular for elliptic problems, and has O(n) complexity, where n is the number of equations in the algebraic system [2,3]. Even though the multigrid method can be used as an iterative solver, often it is used as a preconditioner for the GMRES or other iterative methods.…”