Some Inverse Problems for Obstacles and Drift-Diffusion and Elasticity Systems

障碍物、漂移扩散和弹性系统的一些反问题

• 批准号: 1008902
• 负责人: Victor Isakov
• 金额: $ 28.5万
• 依托单位: Wichita State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-15 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1008902&HistoricalAwards=false
• 关键词: Some; Inverse; Problems; Obstacles; Drift

This project is concerned with analysis and design of numerical algorithms for reconstructing solutions and coefficients of partial differential equations from remote data. One of central topics is the study of increasing stability of the continuation of (acoustical and electromagnetic) waves. Another central topic is about uniqueness, stability, and methods of reconstruction of obstacles from boundary or scattering measurements, with emphasis on practical situations (in particular, on use of several frequencies or time data). The proposed research on drift-diffusion equations is concentrated around evaluation of doping profile in semiconductors and protein region in ion channels, as well finding volatility of option markets. The PI plans to obtain Carleman estimates for hyperbolic systems of anisotropic (including transversely isotropic) elasticity. All these research areas are challenging and central in applied mathematics. He will continue working on outstanding analytical problems, like uniqueness in the three-dimensional inverse conductivity problem with local data. Energy (including Carleman) estimates, harmonic and micro local analysis and potential theory will be main tools of this research. The research on increasing stability of continuation and of locating obstacles is crucial for enhancing resolution of remote sensing in a variety of civil and military applications. Finding the doped area of semiconductor devices is of value for quality control in computer industry, and monitoring of ion channels is of fundamental interest in biology and medicine. The study of nondestructive evaluation of elastic properties is quite important in aviation, car, and construction industries, geophysics, and medicine. The PI will train graduate students preparing them for challenges of science and industry. Goals of this project have important applications to engineering, geophysics, and medicine.

本计画系关于分析与设计从远距离资料重建偏微分方程式之解与系数之数值演算法。其中一个中心课题是研究（声波和电磁波）连续波的稳定性。另一个中心主题是关于唯一性，稳定性，以及从边界或散射测量重建障碍物的方法，重点是实际情况（特别是使用几个频率或时间数据）。 漂移-扩散方程的研究主要集中在半导体中的掺杂分布和离子通道中的蛋白质区域的评估以及期权市场的波动性。PI计划获得Carleman估计的各向异性（包括横观各向同性）弹性双曲系统。所有这些研究领域都是应用数学中具有挑战性和核心的。他将继续致力于突出的分析问题，如独特的三维逆电导率问题与当地的数据。 能量（包括Carleman）估计、调和和微局域分析以及势理论将是本研究的主要工具。 提高连续稳定性和障碍物定位的研究对于提高遥感在各种民用和军事应用中的分辨率至关重要。 确定半导体器件的掺杂区域对于计算机工业的质量控制具有重要价值，而离子通道的监测在生物学和医学中具有重要意义。弹性性能的无损评价研究在航空、汽车、建筑、物理、医学等领域具有重要意义。 PI将培养研究生，为他们应对科学和工业的挑战做好准备。 该项目的目标在工程、物理学和医学方面有重要的应用。