Fast and accurate algorithms for solving eigenvalue problems

• Project/Area Number: 18K11343
• 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: Fukuoka Institute of Technology
• Principal Investigator: Miyata Takafumi 福岡工業大学, 情報工学部, 准教授 (90581645)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
• Keywords: 高速高精度アルゴリズム / 固有値問題 / 並列計算 / 計算科学

Outline of Final Research Achievements

We study numerical algorithms for solving eigenvalue problems.
(1) Large-scale eigenvalue problems arise in network analysis and are needed to be solved. We develop a fast algorithm for solving the problems. Numerical results on a parallel computer show that our algorithm can compute solutions of the problems faster compared to the existing algorithms. (2) We analyze the convergence property of the existing fast algorithm for solving eigenvalue problems. Our analysis clarifies the point that the algorithm is specialized for computing particular solutions of the problems. This result suggests that the algorithm and the other existing algorithms should be selectively used depending on target solutions. (3) We develop an iterative algorithms for solving large-scale generalized Hermitian eigenvalue problems. Our algorithm shows the faster convergence than the existing algorithms and thus requires lesser computational time.

Academic Significance and Societal Importance of the Research Achievements

固有値問題は応用分野に応じて多様な形式を有する複合的な工学問題であり、電子状態計算や構造解析、安定性解析など、様々な応用分野に現れる。特に近年の計算機の発達に伴って、扱う問題規模や複雑さが著しく増加しており、より速く、より正確に問題を解けるような高速高精度アルゴリズムの開発が求められる。
本研究は応用分野に現れる固有値問題を解くため、高速高精度アルゴリズムの研究に取り組んだ。本研究で開発を行ったアルゴリズムに関する研究は、応用諸分野に役立てるための基盤であり、今後益々発展に取り組む必要がある。

Research Products

• [Journal Article] A Riccati-type algorithm for solving generalized Hermitian eigenvalue problems (2021)
  • Author(s): Takafumi Miyata
  • Journal Title: The Journal of Supercomputing Volume: 77 Issue: 2 Pages: 2091-2102
  • DOI: 10.1007/s11227-020-03331-w
  • Peer Reviewed
• [Journal Article] On Correction-Based Iterative Methods for Eigenvalue Problems (2018)
  • Author(s): Takafumi Miyata
  • Journal Title: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences Volume: E101.A Issue: 10 Pages: 1668-1675
  • DOI: 10.1587/transfun.E101.A.1668
  • NAID: 130007494286
  • ISSN: 0916-8508, 1745-1337
  • Year and Date: 2018-10-01
  • Peer Reviewed
• [Journal Article] A heuristic search algorithm based on subspaces for PageRank computation (2018)
  • Author(s): Takafumi Miyata
  • Journal Title: The Journal of Supercomputing Volume: 74 Issue: 7 Pages: 3278-3294
  • DOI: 10.1007/s11227-018-2383-9
  • Peer Reviewed
• [Book] 数値線形代数の数理とHPC (2018)
  • Author(s): 宮田考史（分担執筆、pp.74-87）
  • Total Pages: 336
  • Publisher: 共立出版
  • ISBN: 9784320019553