Studies on direct reconstruction methods and their numerical implementations for the solution of some inverse problems for partial differential equations

2014 Fiscal Year Final Research Report

• Project/Area Number: 23540173
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Okayama University of Science
• Principal Investigator: OHE TAKASHI 岡山理科大学, 理学部, 教授 (90258210)
• Co-Investigator(Kenkyū-buntansha): IKEHATA Masaru 広島大学大学院, 工学研究科, 教授 (90202910)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: 逆問題 / 偏微分方程式 / 数値解法 / 数値解析 / 応用数学

Outline of Final Research Achievements

In this research project, we mainly study direct numerical reconstruction method for inverse source problems for three-dimensional scalar wave equation, and inverse scattering problem for the Helmholtz equation.For inverse source problems for three-dimensional scalar wave equation, we develop numerical reconstruction methods for (a) fixed point wave sources, (b) moving point wave sources, and (c) slowly-moving dipole wave sources applying three types of reciprocity gap functionalｓ. We examine our method by some numerical experiments, and found that our method gives precise estimates for unknown sources.
For inverse scattering problem for the Helmholtz equation, we develop an enclosure method based on the logarithmic differential of the indicator function. We investigate the details of behaviors of the logarithmic differential of the indicator function, and show the effectiveness of our method for the estimataion the convex hull of unknown obstacles from both far and near field data.

Free Research Field

応用数学・数値解析