Theory and implementation of probabilistic numerical methods

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K22301
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: Kawai Reiichiro 東京大学, 大学院総合文化研究科, 教授 (20464258)
• Project Period (FY): 2020-09-11 – 2024-03-31
• Keywords: 確率数値解析 / 確率過程 / モンテカルロ法 / 統計力学 / 機械学習

Outline of Final Research Achievements

The focus centered on essential issues of stochastic models in numerical implementation, such as optimal execution of computational resources and optimal termination of iterative calculations. Specifically, the research aimed at establishing the foundational theory of probabilistic numerical analysis, convergence error analysis, and enhancing computational speed. The research content can be classified into two categories: probabilistic numerical methods independent of the model or problem setting, and those tailored to specific model based or problem setting specific approaches. Significant achievements were made in both realms throughout the research period based on theories such as Monte Carlo methods, variance reduction methods, stochastic gradient descent methods, infinitely divisible distributions, martingale theory, and Malliavin analysis. As a consequence, 20 articles were published in established peer-reviewed international journals.

Free Research Field

確率数値解析

Academic Significance and Societal Importance of the Research Achievements

実用レベルで求められる確率モデルは年々際限なく大規模かつ複雑になり、数値計算に頼らざるを得ない問題設定が大多数を占めるにいたり、数値手法の効果的な実装理論、収束保証、誤差評価の需要は恒久的に高まっていた。本研究から得られた多岐に渡る確率数値解析の基盤理論構築、収束誤差解析、そして計算速度向上は、自然科学や社会科学のあらゆる分野で用いられる確率モデルにおける計算資源の最適執行や反復計算の最適停止といった実装レベルにおいて必要不可欠な諸問題に対して、大きく、また普遍的に寄与できると考えている。