Inverse Boundary Value Problems in Partial Differential Equations

偏微分方程中的反边值问题

• 批准号: 1114944
• 负责人: Leo Tzou
• 金额: $ 5.11万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1114944&HistoricalAwards=false
• 关键词: Inverse; Boundary; Value; Problems; Partial

The PI proposes to study the mathematical theory behind non-invasive methods of detection via probing by electromagnetic waves. The goal of such methods is to determine the interior composition of an object by making only surface measurements and without drilling holes or taking samples. The theory has potential applications in the early detection of malignant breast tumors and can lead to more economical oil exploration techniques. The idea is to apply an electromagnetic wave on the surface of the object and analyze its interaction with the interior material via surface measurements. One of the objectives of this project is to show that when the surface measurements are taken on only a part of the surface one can still recover the interior composition of the material.The PI proposes to study non-invasive methods of detection from the point of view of inverse boundary value problems for partial differential equations (PDEs), where one solves a boundary value problem and measures the normal derivative of the solution at the boundary. The question is whether one can recover the coefficients of the PDE from these measurements. The PI proposes to address this problem for the conductivity equation and the Pauli-Dirac system. The PI proposes to use variational convergence methods and Carleman estimates for systems of PDEs.

PI 提议研究通过电磁波探测的非侵入性检测方法背后的数学理论。此类方法的目标是通过仅进行表面测量而不钻孔或取样来确定物体的内部成分。该理论在恶性乳腺肿瘤的早期检测方面具有潜在的应用，并且可以带来更经济的石油勘探技术。这个想法是在物体表面施加电磁波，并通过表面测量来分析其与内部材料的相互作用。该项目的目标之一是表明，当仅对表面的一部分进行表面测量时，仍然可以恢复材料的内部成分。PI建议从以下角度研究非侵入性检测方法：偏微分方程 (PDE) 的逆边值问题的视图，其中求解边值问题并测量边界处解的法态导数。问题是是否可以从这些测量中恢复偏微分方程的系数。 PI 建议通过电导率方程和泡利-狄拉克系统来解决这个问题。 PI 建议对偏微分方程系统使用变分收敛方法和卡尔曼估计。