A study on numerical solution methods with stable and high accuracy for large-scale linear systems

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K11342
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 60100:Computational science-related
• Research Institution: Osaka Electro-Communication University
• Principal Investigator: Itoh Shoji 大阪電気通信大学, 工学部, 特任准教授 (70333482)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: クリロフ部分空間法 / 双ランチョス / 前処理系 / 積型反復法

Outline of Final Research Achievements

In this study, we analyzed preconditioned bi-Lanczos iterative algorithms, which assume the existence of a dual system. By comparing the logical structures of these algorithms, we show that the direction of the preconditioned system can be switched by the construction and setting of the initial shadow residual vector. And we propose a changing over stopping criterion for the improved PCGS, that results in a higher accuracy than the conventional and the left-PCGS. Further, we proposed improved algorithms for preconditioned bi-Lanczos-type methods with residual norm minimization for the stable solution of systems of linear equations. In particular, preconditioned algorithms pertaining to the bi-conjugate gradient stabilized method (BiCGStab) and the generalized product-type method based on the BiCG (GPBiCG) have been improved. Numerical results showed the improvements with respect to the preconditioned BiCGStab, the preconditioned GPBiCG, and stopping criterion changeover.

Free Research Field

数値解析学

Academic Significance and Societal Importance of the Research Achievements

自然科学における様々な現象解明・予測や工学問題の解決や新技術開発において，多くの場合，大規模な線形方程式の求解に帰着される．そこでは，如何に安定かつ高精度に求解できるかが非常に重要である．近年，その様な線形方程式はクリロフ部分空間法に基づく反復解法を用いて求解されることが多く，求解性向上が期待される前処理併用による効果も大きい．ところが，前処理方法の設計が悪いと十分な精度で求解できない場合も少なくない．つまり，前処理付き解法の適切な設計が極めて重要である．本研究課題の学術的意義は，前処理付き解法による安定求解の本質的な原理の解明，および，高精度求解に向けた数理面からの追及である．