Versatile construction of highly accurate numerical methods based on potential theory

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K14241
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: The University of Tokyo
• Principal Investigator: Tanaka Ken'ichiro 東京大学, 大学院情報理工学系研究科, 准教授 (70610640)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 重み付きハーディ空間 / ポテンシャル論 / 関数近似公式 / 数値積分公式 / 凸エネルギー最小化問題 / 再生核ヒルベルト空間

Outline of Final Research Achievements

By setting some function spaces, we have established methods for constructing formulas for function approximation or numerical integration. For this purpose, we set some mathematical optimization problems providing sampling points for the formulas. In numerical analysis, various methods for constructing approximation formulas are known according to specific situations. In this study, we pursued a versatile framework for constructing formulas with good properties. We revealed a relationship between the construction problems and energy minimization problems in potential theory and used the relationships effectively.

Free Research Field

数値解析学

Academic Significance and Societal Importance of the Research Achievements

関数近似法や数値積分法は，様々な科学技術計算だけでなく，データ科学や統計的学習理論などとの関係が深い．数値解析学の分野では，様々な近似公式の構成法が知られているが，それらは個別の設定に応じて数学的工夫を凝らした方法となっており，最適な近似を得る汎用的原理の解明は難しい問題である．本研究は，道半ばではあるものの，この汎用的原理の解明に向けていくつかの成果を得たものと言える．