Graph partitioning using matrix values for preconditioning symmetric positive definite systems

TL;DR

This paper introduces a spectral partitioning algorithm that considers matrix coefficients to improve preconditioning quality for symmetric positive definite systems, leading to faster convergence in iterative solutions.

Contribution

The paper proposes a novel spectral partitioning method that incorporates matrix coefficient information to enhance preconditioning effectiveness for symmetric positive definite systems.

Findings

Improved convergence rates with the new partitioning method.

Noticeable performance gains on test problems with large coefficient variations.

Enhanced preconditioning quality compared to standard algorithms.

Abstract

Prior to the parallel solution of a large linear system, it is required to perform a partitioning of its equations/unknowns. Standard partitioning algorithms are designed using the considerations of the efficiency of the parallel matrix-vector multiplication, and typically disregard the information on the coefficients of the matrix. This information, however, may have a significant impact on the quality of the preconditioning procedure used within the chosen iterative scheme. In the present paper, we suggest a spectral partitioning algorithm, which takes into account the information on the matrix coefficients and constructs partitions with respect to the objective of enhancing the quality of the nonoverlapping additive Schwarz (block Jacobi) preconditioning for symmetric positive definite linear systems. For a set of test problems with large variations in magnitudes of matrix coefficients, our numerical experiments demonstrate a noticeable improvement in the convergence of the resulting solution scheme when using the new partitioning approach.