AMG based on compatible weighted matching for GPUs

Abstract

We describe main issues and design principles of an efficient implementation, tailored to recent generations of Nvidia Graphics Processing Units (GPUs), of an Algebraic MultiGrid (AMG) preconditioner previously proposed by one of the authors and already available in the open-source package BootCMatch: Bootstrap algebraic multigrid based on Compatible weighted Matching for standard CPU. The AMG method relies on a new approach for coarsening sparse symmetric positive definite (s.p.d.) matrices, named coarsening based on compatible weighted matching. It exploits maximum weight matching in the adjacency graph of the sparse matrix, driven by the principle of compatible relaxation, providing a suitable aggregation of unknowns which goes beyond the limits of the usual heuristics applied in the current methods. We adopt an approximate solution of the maximum weight matching problem, based on a recently proposed parallel algorithm, referred as the Suitor algorithm, and show that it allow us to obtain good quality coarse matrices for our AMG on GPUs. We exploit inherent parallelism of modern GPUs in all the kernels involving sparse matrix computations both for the setup of the preconditioner and for its application in a Krylov solver, outperforming preconditioners available in Nvidia AmgX library. We report results about a large set of linear systems arising from discretization of scalar and vector partial differential equations (PDEs).

AMG Preconditioners for Linear Solvers towards Extreme Scale

“…The method, named coarsening based on compatible weighted matching was first introduced in [22] and is already available in the sequential package described in [21]. A first parallel version of the method, exploiting fine-grained parallelism and specifically tailored for single GPU device is described in [7,8]. The method is independent of any heuristics or a priori information on the near kernel of A, i.e., the lower part of the range of eigenvalues of the system matrix A which is generally used to obtain good-quality aggregates, and it is a completely automatic procedure applicable to general s.p.d.…”

TEXTAROSSA: Towards EXtreme scale Technologies and Accelerators for euROhpc hw/Sw Supercomputing Applications for exascale

“…The new power-to-solution metrics requires a rethinking of many computational kernels of HPC applications looking for a trade-off between the reduction of the total energy and the minimization of the time-to-solution, promoting scalability. Within this context, extensions and improvements of high-performance algorithms and SW libraries for kernels in numerical linear algebra [44], [45] and graph computation, such as iterative [46], [47], [48], [49] and direct linear solvers, edge weighted graph matching, and fast multipole methods [50] will be deployed.…”

A General-Purpose AMG Linear Solver for High Performance Computing

“…Its parallelization does not present particular difficulties as mainly relies on standard sparse linear algebra operations such as sparse matrix by matrix and matrix by vector products or other basic tasks which are nowadays readily handled by the highly optimized Basic Linear Algebra Operations (BLAS) routines [5]. The only two tasks that present some difficulties from the parallelization viewpoint are the smoother set-up and application [2] and the coarsening stage [4,3], which have been deeply investigated by several authors in recent years.…”