Innovations in the smoothing technique for linear iterative solvers and its application to optimization algorithms

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K18064
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 60100:Computational science-related
• Research Institution: Tokyo City University
• Principal Investigator: Aihara Kensuke 東京都市大学, 情報工学部, 准教授 (70735498)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 大規模連立一次方程式 / クリロフ部分空間法 / 平滑化技術 / 丸め誤差解析 / 積型BiCG法 / リーマニアン最適化 / ニュートン法 / レトラクション

Outline of Final Research Achievements

Krylov subspace methods are effective iterative solvers for large linear systems of equations. In this study, we have improved the smoothing technique for obtaining a smooth convergence behavior of the iterative methods. This new smoothing scheme enables the accuracy of the approximate solutions to be improved. We have also proposed a new framework named GPBiCGstab(L) which unifies several algorithms of the hybrid BiCG methods. Furthermore, we have studied on Riemannian optimization. We have developed an efficient Newton’s method, where Newton’s equation is solved efficiently using the Krylov subspace methods, and also designed a new retraction scheme based on the matrix decomposition.

Free Research Field

数値線形代数，数理最適化

Academic Significance and Societal Importance of the Research Achievements

大規模行列に関する数学的諸問題をコンピュータにより高速かつ高精度に解くための数値計算アルゴリズムの研究は，現代の科学技術計算において必要不可欠なものである．本研究は，その中でも最も基本的かつ重要な問題である連立一次方程式に着目し，既存手法の改良や新しい手法の開発を行ったものである．平滑化手法の革新や効果的な最適化アルゴリズムの確立といった本研究の成果は，関連する諸分野の発展に大きく貢献するものと考える．