Research on constructions of numerical integration methods via determinantal point processes and its applications

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K17645
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Aichi Prefectural University
• Principal Investigator: Hirao Masatake 愛知県立大学, 情報科学部, 准教授 (90624073)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 行列式点過程 / 準モンテカルロデザイン系列 / 球面アンサンブル / 調和アンサンブル / デザイン理論 / 準モンテカルロ法 / Cubature公式

Outline of Final Research Achievements

Focusing on, e.g., spherical ensembles and harmonic ensembles, which are typical determinant point processes on the sphere, numerical integration performances and some applications are discussed.
For the former, we investigate the behavior of the worst error of numerical integration in some Sobolev spaces on the sphere, and show spherical ensembles asymptotically approximate QMC design sequences to Sobolev spaces on the two-dimensional sphere with smoothness 1 <s <2, which is confirmed that the determinant point process is useful in a specific function space. Furthermore, several availabilities of determinant point processes are suggested through the research on frame potentials and the approximation theory of the feature maps in the kernel method.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

Brauchart et al.(2014)で準モンテカルロデザイン系列が提案されて以降，行列式点過程を用いた確率的生成法の研究が行われているが，本研究はそれらの研究に対して先鞭をつけたものであり，多くの後続研究が現在行われている．また行列式点過程の応用として数値解析だけではなく，統計学，機械学習の問題も扱ったことで今後，様々な分野に広く発信できるものと予想している.