Numerical Analysis of Schroedinger's problem

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K03364
• 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: Tokyo Institute of Technology
• Principal Investigator: Nakano Yumiharu 東京工業大学, 情報理工学院, 准教授 (00452409)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: シュレディンガー問題 / 確率制御 / 拡散生成モデル / 確率微分方程式

Outline of Final Research Achievements

We developed a numerical method for the Schroedinger's problem based on a McKean-Vlasov type stochastic control problem and proved its rigorous convergence.
Furthermore, we studied the theoretical analysis of the diffusion generative model, which is now widely used as an image generation model. Specifically, we clarified the sufficient conditions for the convergence of the generated distribution of the Denoising Diffusion Probabilistic Models to the target distribution. In the convergence proofs known from existing studies, it has been unclear what conditions must be satisfied for the parameters of the forward time process to be successful, but in this study, we derived sufficient conditions for appropriate asymptotic behavior for these parameters.

Free Research Field

応用数学

Academic Significance and Societal Importance of the Research Achievements

シュレディンガー問題は近年，生成モデルの理論的基盤として注目されているものである．本研究では特に，シュレディンガー問題において初期分布が任意の場合に適当可能な新しい数値解法を提案し，理論的正当性も与えた．このことに応用数学としての学術的意義があるのはもちろん，新しい分布補間の手段を提示できたことで，例えば，画像から画像の生成など，新たな生成モデルの展開のための一助になることが期待できる．