Stability of solution and reconstruction in inverse problems

反问题解和重构的稳定性

• 批准号: 11440025
• 负责人: YAMAMOTO Masahiro
• 金额: $ 7.55万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440025/
• 关键词: stability; inverse problem; impedance tomography; free boundary problem; finite element; multile precision; Kalman filter; 多倍長計算; 幾何形状決定; 係数決定; インピーダンス・トモグラフィ

This project aims at mathematical analysis and numerical analysis based on the mathematical analysis for inverse problems in applied sciences, Moreover we have studied related problems in applications, from the viewpoint of inverse problems, and prepared possible numerical methods.We conclude that by this project, we have established satisfactory theoretical results. Moreover, as for the development of numerical methods on the basis of the theoretical aspects, several methods have been proposed and tested for numerical examples. Still we have not compared the numerical results with real data directly from the real worlds such as works or plants. Therefore the continuation of this kind of projects are strongly demanded.For researches of related problems with inverse problems, the participants have achieved remarkable results by their own specialities.We have held several surveys concerning inverse problems by foreign researchers and taken part in international conferences on inverse problems for presenting the outputs of this project.With results obtained by this project, M. Yamamoto et al. have published a monograph on the mathematics and the numerical methods of inverse problems.Onishi, Tosaka, Iso, Kawarada, Miyoshi, Kimura, Ushijima and Kako have worked for inverse problems and related problems in applied sciences, from numerical points of view. Nakamura, Saitoh, Ikehata, Tanuma, Kubo have worked for theoretical aspects of inverse problems.

本课题以应用科学中逆问题的数学分析为基础进行数学分析和数值分析，并从逆问题的角度对应用中的相关问题进行了研究，准备了可能的数值方法，得出了满意的理论结果。此外，对于数值方法在理论方面的发展，提出了几种方法，并进行了数值算例验证。尽管如此，我们还没有将数值结果与直接来自真实世界的真实数据进行比较，例如作品或植物。对于与逆问题相关的问题的研究，参与者凭借各自的专长取得了显著的成果。我们多次举办关于反问题的外国研究人员的调查，并参加了国际反问题会议，以展示该项目的成果。出版了一部关于反问题的数学和数值方法的专著。Onishi、Tosaka、Iso、Kawa ada、Miyoshi、Kimura、Ushijima和Kako从数值的角度研究了应用科学中的反问题和相关问题。中村、齐藤、池田、田间、久保都致力于反问题的理论方面的工作。

项目成果

Bruckner,G.,Yamamoto M.: "On a noisy operator equation and the identification of point sources"ZAMM Z.Angew.Math.Mech. 80. 377-388 (2000)

布鲁克纳，G.，山本 M.：“关于噪声算子方程和点源的识别”ZAMM Z.Angew.Math.Mech。

TOSAKA, N., ONISHI, K. and YAMAMOTO, M.: "Mathematics and Solutions of Inverse Problems"Tokyo Daigaku Syuppan Kai.. (1999)

TOSAKA, N.、Onishi, K. 和 YAMAMOTO, M.：“数学和反问题的解决方案”东京大学 Syuppan Kai..（1999 年）

GORENFLO, R., YAMAMOTO, M.: "Operator theoretical approach of linear Abel integral equations of first kind"Japan J. Indutrial Appl. Math.. 16. 137-161 (1999)

GORENFLO, R., YAMAMOTO, M.：“第一类线性阿贝尔积分方程的算子理论方法”日本工业应用杂志。

G. Bruckner: "On an operator equation with noise in the operator and the right-hand side with application to an inverse vibration problem"Z. Angewandte Math. Mech.. (発表予定).

G. Bruckner：“关于算子和右侧带有噪声的算子方程及其在逆振动问题中的应用”Z. Angewandte Math..（即将介绍）。

Cheng,J.,Yamamoto,M.: "One New strategy for a priori choice of regularizing parameters in Tikhonov's regularization"Inverse Problems. 16. L31-L38 (2000)

Cheng，J.，Yamamoto，M.：“吉洪诺夫正则化中先验选择正则化参数的一种新策略”反问题。