A transformation approach that makes SPAI, PSAI and RSAI procedures efficient for large double irregular nonsymmetric sparse linear systems

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:348

Article

Jia, Zhongxiao¹  Kang, Wenjie¹ 

[1] Tsinghua Univ, Dept Math Sci, Beijing 100084, Peoples R China

关键词: Linear system;    Preconditioning;    Sparse approximate inverse;    Double irregular sparse;    Double regular sparse;    Transformation approach;   

DOI  :  10.1016/j.cam.2018.08.033

来源: Elsevier

【 摘 要 】

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 SPAT, 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 24.4% 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 (A) over cap, (ii) use SPAI, PSAI(tol) and RSAI(tol) to construct sparse approximate inverses M of (A) over cap, (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 SPAT, PSAI(tol) or RSAI(tol) and solves the resulting preconditioned system by Krylov solvers. (C) 2018 Elsevier B.V. All rights reserved.

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