Development of linear solvers on max-plus algebra and its applications

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03624
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Shibaura Institute of Technology
• Principal Investigator: Fukuda Akiko 芝浦工業大学, システム理工学部, 教授 (70609297)
• Co-Investigator(Kenkyū-buntansha): 渡邉 扇之介 福知山公立大学, 情報学部, 准教授 (80735316)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 超離散可積分系 / min-plus代数 / max-plus代数 / 固有値 / 箱玉系 / グラフ理論 / 交通流モデル / 超離散化

Outline of Final Research Achievements

(1) It is shown that eigenvalues for banded matrices over the min-plus algebra can be computed by the ultradiscrete hungry Toda equation of type I and the ultradiscrete hungry Lotka-Volterra system of type I. The ultradiscrete hungry Toda equation of type I can be interpreted as performing multiple bubble sort simultaneously. The evolution equation for convergence acceleration is obtained.(2) Conserved quantities for the box and ball system with numbered boxes and balls are derived, and the relationship with the hungry ε-BBS is also clarified.(3) The max-plus walk is introduced as a max-plus analogue of the quantum walk. An extended discrete Burgers equation was obtained based on the correlated random walk, and a new traffic flow model was derived by ultradiscretization.

Free Research Field

応用数学，可積分系，数値解析

Academic Significance and Societal Importance of the Research Achievements

これまで主に連続系や離散系で議論されてきた可積分アルゴリズムに関する理論を超離散系に拡充したことで，他の可積分アルゴリズム研究の今後の新たな展開が期待される。本研究で新たに得られたアルゴリズムはmin-plus代数上のアルゴリズムとして解釈されるが，min-plus/max-plus代数はスケジューリング問題や制御理論，離散事象システムなどへの応用が知られている。今後の実用化を目指したアルゴリズムの開発・改良により，これらの応用分野にとっても有益となり得る。