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

本课题是分析和设计从远程数据重建偏微分方程的解和系数的数值算法。中心课题之一是增加（声波和电磁波）延续的稳定性的研究。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中重建障碍物的方法，强调实际情况（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掺杂谱和离子通道中蛋白质区域的评估，以及期权市场的波动率。PI计划获得各向异性（包括横向各向同性）弹性双曲系统的Carleman估计。所有这些研究领域都是应用数学的核心和挑战。他将继续研究突出的分析问题，如局部数据三维逆电导率问题的唯一性。能量（包括Carleman）估计、谐波和微局部分析以及势理论将是本研究的主要工具。在各种民用和军事应用中，提高遥感图像的分辨率，提高障碍物的持续稳定性和定位稳定性的研究是至关重要的。寻找半导体器件的掺杂区域对计算机工业的质量控制有价值，离子通道的监测对生物学和医学有重要意义。弹性性能无损评价的研究在航空、汽车、建筑、地球物理和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、地球物理和医学方面具有重要的应用。

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.