AMG preconditioners for Linear Solvers towards Extreme Scale

TL;DR

This paper introduces a novel parallel coarsening algorithm for AMG preconditioners that enhances efficiency and scalability for solving extremely large sparse linear systems on supercomputers.

Contribution

The main contribution is a new automated parallel coarsening algorithm based on weighted graph matching, improving scalability and robustness for large-scale problems.

Findings

Achieved weak scalability on supercomputers with systems up to 10 billion unknowns.

Improved numerical scalability at low operator complexity.

Enhanced the AMG preconditioners package for exascale computing.

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 \texttt{PSBLAS}, which has also been updated to progress towards exascale. We present weak scalability results on one of the most powerful supercomputers in Europe, for linear systems with sizes up to $O(10^{10})$ unknowns.