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

该项目涉及从远程数据重建偏微分方程解和系数的数值算法的分析和设计。其中一个中心课题是研究提高(声波和电磁波)连续波的稳定性。另一个中心议题是关于根据边界或散射测量重建障碍物的唯一性、稳定性和方法，重点是实际情况(特别是使用几个频率或时间数据)。关于漂移扩散方程的研究主要集中在评估半导体和离子通道中蛋白质区域的掺杂分布，以及发现期权市场的波动性。PI计划获得各向异性(包括横向各向同性)弹性的双曲组的Carleman估计。所有这些研究领域在应用数学中都是具有挑战性的和核心的。他将继续致力于解决悬而未决的分析问题，例如利用本地数据解决三维电导率逆问题的唯一性问题。能量(包括Carleman)估计、谐和微观局域分析和势能理论将是本研究的主要工具。提高连续性稳定性和定位障碍物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。发现半导体器件的掺杂区域对于计算机行业的质量控制具有重要意义，而离子通道的监测是生物学和医学领域的重要研究内容。弹性性能无损评价的研究在航空、汽车、建筑、地球物理、医学等领域具有十分重要的意义。PI将培训研究生，为他们迎接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.