Parallel Algebraic Multigrid Research Articles

Parallel Algebraic Multigrid Methods on Distributed Memory Computers

Algebraic multigrid methods are well suited as preconditioners for iterative solvers. We consider linear systems of equations which are sparse and symmetric positive definite and which stem from a finite element discretization of a second order self-adjoint elliptic partial differential equation or a system of them. Since preconditioners based on algebraic multigrid are very efficient, additional speedup can only be achieved by parallelization. In this paper, we propose a general parallel algebraic multigrid algorithm for finite element discretizations based on domain decomposition ideas which is well suited for distributed memory computers. This paper pays special attention to the coarsening strategy which has to be adapted in the parallel case. Moreover, a general framework of data distribution gives rise to a construction scheme for the prolongation operators. Results of numerical studies on parallel computers with distributed memory are presented which show the high efficiency of the approach.

BoomerAMG: A parallel algebraic multigrid solver and preconditioner

Driven by the need to solve linear systems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigrid-like performance on unstructured grids. The sheer size of many modern physics and simulation problems has led to the development of massively parallel computers, and has sparked much research into developing algorithms for them. Parallelizing AMG is a difficult task, however. While much of the AMG method parallelizes readily, the process of coarse-grid selection, in particular, is fundamentally sequential in nature. We have previously introduced a parallel algorithm [A.J. Cleary, R.D. Falgout, V.E. Henson, J.E. Jones, in: Proceedings of the Fifth International Symposium on Solving Irregularly Structured Problems in Parallel, Springer, New York, 1998] for the selection of coarse-grid points, based on modifications of certain parallel independent set algorithms and the application of heuristics designed to insure the quality of the coarse grids, and shown results from a prototype serial version of the algorithm. In this paper we describe an implementation of a parallel AMG code, using the algorithm of A.J. Cleary, R.D. Falgout, V.E. Henson, J.E. Jones [in: Proceedings of the Fifth International Symposium on Solving Irregularly Structured Problems in Parallel, Springer, New York, 1998] as well as other approaches to parallelizing the coarse-grid selection. We consider three basic coarsening schemes and certain modifications to the basic schemes, designed to address specific performance issues. We present numerical results for a broad range of problem sizes and descriptions, and draw conclusions regarding the efficacy of the method. Finally, we indicate the current directions of the research.

Parallel algebraic multigrid based on subdomain blocking

The algebraic multigrid (AMG) approach provides a purely algebraic means to tackle the efficient solution of systems of equations posed on large unstructured grids, in 2D and 3D. While sequential AMG has been used for increasingly large problems (with several million unknowns), its application to even larger applications requires a parallel version. Since, in contrast to geometric multigrid, the hierarchy of coarser levels and the related operators develop dynamically during the setup phase of AMG, a direct parallelization is very complicated. Moreover, a “naive” parallelization would, in general, require unpredictable and highly complex communication patterns which seriously limit the achievable scalability, in particular of the costly setup phase. In this paper, we consider a classical AMG variant which has turned out be highly robust and efficient in solving large systems of equations corresponding to elliptic PDEs, discretized by finite differences or finite volumes. Based on a straightforward partitioning of variables (using one of the available algebraic partitioning tools such as Metis), a parallelization approach is proposed which minimizes the communication without sacrificing convergence in complex situations. Results will be presented for industrial CFD and oil-reservoir simulation applications on distributed memory machines, including PC-clusters.

A parallel algebraic multigrid solver for problems in solid mechanics discretisized by finite elements

This paper presents an algebraic multigrid solver which can also be applied as a preconditioner for the conjugate gradient method. The solver has been implemented in a parallel version of the finite element program FEAP, see Zienkiewicz O. C. and Taylor R. L. The Finite Element Method, volume 1. McGraw–Hill, London, 4th edition, 1989 [1]. The aim of the paper is to show the performance of these solvers on two different MIMD computers and to present a concept for porting a finite element code to a parallel machine of MIMD class. We discuss the parallel mesh generation and the parallel solution of problems in elasticity.