A Novel Partitioning Method for Accelerating the Block Cimmino Algorithm

TL;DR

This paper introduces a new partitioning technique based on a graph model that improves the convergence rate of the block Cimmino algorithm by reducing inter-block inner products, leading to fewer iterations.

Contribution

The paper presents a novel row partitioning method that considers numerical orthogonality via a graph model, enhancing convergence of the block Cimmino algorithm.

Findings

Significant reduction in iteration count for convergence.

Improved eigenvalue spectrum of the iteration matrix.

Validated effectiveness on large sparse matrices.

Abstract

We propose a novel block-row partitioning method in order to improve the convergence rate of the block Cimmino algorithm for solving general sparse linear systems of equations. The convergence rate of the block Cimmino algorithm depends on the orthogonality among the block rows obtained by the partitioning method. The proposed method takes numerical orthogonality among block rows into account by proposing a row inner-product graph model of the coefficient matrix. In the graph partitioning formulation defined on this graph model, the partitioning objective of minimizing the cutsize directly corresponds to minimizing the sum of inter-block inner products between block rows thus leading to an improvement in the eigenvalue spectrum of the iteration matrix. This in turn leads to a significant reduction in the number of iterations required for convergence. Extensive experiments conducted on a large set of matrices confirm the validity of the proposed method against a state-of-the-art method.