Mathematical Sciences: Some Inverse Problems for Partial Differential Equations

数学科学：偏微分方程的一些反问题

• 批准号: 9002547
• 负责人: Victor Isakov
• 金额: $ 2.38万
• 依托单位: Wichita State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-15 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9002547&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Inverse; Problems

This project studies uniqueness and stability of solutions of well-known but ill-posed inverse problems. It considers the inverse problem of potential theory that is fundamental in geophysics, the inverse conductivity problem with applications in electrical tomography and in crack detection and a non-hyperbolic Cauchy problem for hyperbolic equations as well as related heat conduction parameter identification, earthquakes prediction and acoustic and seismic prospecting. The work will involve some classical (potential theory, Carleman type and energy estimates, orthogonality relations) and more recent methods ( the logarithmic convexity and the Calderon-Sylvester-Uhlmann's method). The aim is to solve fundamental questions, provide applications and derive new mathematical tools.

这个项目研究众所周知但不适定的反问题解的唯一性和稳定性。它考虑了地球物理中基本的位势理论的反问题，电导率反问题及其在电层析成像和裂纹检测中的应用，以及双曲型方程的非双曲型柯西问题，以及相关的热传导参数识别、地震预报和声学和地震勘探。这项工作将涉及一些经典的方法(位势理论、Carleman类型和能量估计、正交性关系)和较新的方法(对数凸性和Calderon-Sylvester-Uhlmann方法)。其目的是解决基本问题，提供应用程序，并推导出新的数学工具。