A comprehensive analysis on the inverse problem of determining unknown coefficients of a differential equation which is a basis of a tomographic technology

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14344
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Kyoto University
• Principal Investigator: Kawagoe Daisuke 京都大学, 情報学研究科, 助教 (30848073)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 逆問題解析 / 偏微分方程式論 / 積分方程式 / 数値解析 / スペクトル解析

Outline of Final Research Achievements

This research project is a mathematical and numerical analysis aimed at the realization of Diffuse Optical Tomography (DOT), which is a next-generation non-invasive tomographic technique. The applicant has proposed an analytical method to solve an inverse problem to determine a coefficient of an integro-differential equation which is a mathematical model of DOT. He discussed the feasibility of this method through numerical experiments during the period. It was confirmed at the level of numerical experiments that the method works at least when the scattering effect is small (or the diameter of the domain is small) in two or three dimensional convex domains.

Free Research Field

数理解析学関連

Academic Significance and Societal Importance of the Research Achievements

DOT は, 近赤外光の生体に対する光学特性を利用した次世代の非侵襲的断層撮影技術であり, 医学的なメリットからその実現が期待されている. しかしながら, X 線や強磁場とは異なり, 生体内における近赤外光の伝播は散乱を伴うため, その実現が困難となっている. DOT は輸送方程式と呼ばれる微分積分方程式の係数決定逆問題と数理モデル化される. この係数決定逆問題に対して純粋数学的な観点からは多くの研究がなされてきたが, DOT の実現に繋がる解法は提案されてこなかった. 本研究課題は, 理論と応用とを結ぶ新たな学術の発露を担っている.