A Transformation Approach that Makes SPAI, PSAI and RSAI Procedures Efficient for Large Double Irregular Nonsymmetric Sparse Linear Systems

TL;DR

This paper introduces a transformation method to efficiently precondition large double irregular sparse linear systems by converting them into multiple double regular sparse systems, enabling the effective use of SPAI, PSAI, and RSAI procedures.

Contribution

The paper proposes a novel transformation approach that makes SPAI, PSAI, and RSAI procedures practical for large double irregular sparse systems, which were previously computationally expensive.

Findings

Transformation approach significantly improves efficiency for large problems.

Numerical experiments show superior performance over standard methods.

Method effectively handles real-world double irregular sparse matrices.

Abstract

A sparse matrix is called double irregular sparse if it has at least one relatively dense column and row, and it is double regular sparse if all the columns and rows of it are sparse. The sparse approximate inverse preconditioning procedures SPAI, PSAI($tol$) and RSAI($tol$) are costly and even impractical to construct preconditioners for a large sparse nonsymmetric linear system with the coefficient matrix being double irregular sparse, but they are efficient for double regular sparse problems. Double irregular sparse linear systems have a wide range of applications, and 4.4\% of the nonsymmetric matrices in the Florida University collection are double irregular sparse. For this class of problems, we propose a transformation approach, which consists of four steps: (i) transform a given double irregular sparse problem into a small number of double regular sparse ones with the same coefficient matrix $\hat{A}$, (ii) use SPAI, PSAI($tol$) and RSAI($tol$) to construct sparse approximate inverses $M$ of $\hat{A}$, (iii) solve the preconditioned double regular sparse linear systems by Krylov solvers, and (iv) recover an approximate solution of the original problem with a prescribed accuracy from those of the double regular sparse ones. A number of theoretical and practical issues are considered on the transformation approach. Numerical experiments on a number of real-world problems confirm the very sharp superiority of the transformation approach to the standard approach that preconditions the original double irregular sparse problem by SPAI, PSAI($tol$) or RSAI($tol$) and solves the resulting preconditioned system by Krylov solvers.