AMG Preconditioners for Linear Solvers towards Extreme Scale

Abstract

Linear solvers for large and sparse systems are a key element of scientific applications, and their efficient implementation is necessary to harness the computational power of current computers. Algebraic MultiGrid (AMG) preconditioners are a popular ingredient of such linear solvers; this is the motivation for the present work where we examine some recent developments in a package of AMG preconditioners to improve efficiency, scalability and robustness on extreme scale problems. The main novelty is the design and implementation of a parallel coarsening algorithm based on aggregation of unknowns employing weighted graph matching techniques; this is a completely automated procedure, requiring no information from the user, and applicable to general symmetric positive definite (s.p.d.) matrices. The new coarsening algorithm improves in terms of numerical scalability at low operator complexity over decoupled aggregation algorithms available in previous releases of the package. The preconditioners package is built on the parallel software framework PSBLAS, which has also been updated to progress towards exascale. We present weak scalability results on one of the most powerful supercomputer in Europe for linear systems with sizes up to O(10 10 ) unknowns.

Preconditioning Richards Equations: spectral analysis and parallel solution at very large scale

“…In this work we use the recently proposed PSCToolkit 1 software framework for parallel sparse computations on current peta-scale supercomputers. PSCToolkit is composed of two main libraries, named PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms) [19,20], and AMG4PSBLAS (Algebraic MultiGrid Preconditioners for PSBLAS) [7]. PSBLAS implements all the main computational building blocks for iterative Krylov subspace linear solvers on parallel computers made of multiple nodes; a plugin for NVIDIA GPUs allows the exploitation of these devices in main sparse matrix and vector computations on hybrid architectures.…”

“…While this would guarantee the theoretical properties we look for, we also need to approximate the inverse of the symmetric and positive definite (spd for short) preconditioner with a matrix M . To this aim, we exploit some of the methods available in AMG4PSBLAS [7].…”

“…AMG4PSBLAS includes iterative methods for the approximation of the inverse of the preconditioner M , including domain decomposition techniques of Additive Schwarz (AS) type and AMG methods based on aggregation of unknowns; details on the methods and their parallel versions can be found in [21,22,7]. In the following we describe the main features of the preconditioners selected for the experiments discussed in section 6.…”

“…In AMG4PSBLAS two different parallel coarsening procedures are available; both employ disjoint aggregates of fine dofs to form the coarse dofs, and the prolongation matrices are piecewise-constant interpolation matrices (unsmoothed aggregation) or a smoothed variant thereof (smoothed aggregation). For details on the coarsening procedures implemented in AMG4PSBLAS we refer the readers to [22,7]. Here we note that in section 6 we refer to the two acronims: DSVMB: the smoothed aggregation scheme introduced in [23], and applied in a parallel setting by a decoupled approach, where each process applies the coarsening algorithm to its subset of dofs, ignoring interactions with dofs owned by other processes [22].…”

“…First, we discuss a scalable and efficient parallel solution of a root-finding algorithm; specifically, we employ a parallel version of the modified inexact Newton method with Krylov solvers and Armijo-Goldstein's line search, i.e., Newton-like algorithms where the Newton correction linear equations are solved by a Krylov subspace method. To this aim, we interface the KINSOL package available from the Sundials project [6] with the most recent version of libraries for solving sparse linear systems by parallel Krylov solvers coupled with Algebraic MultiGrid (AMG) preconditioners [7], currently being extended in the EoCoE-II EU-funded project.…”