Mathematical Sciences: Some Inverse Problems in PDE

小行星9501050 本项目的基本目标是研究几个基本反问题的全局唯一性和稳定性。下一步的目标是发展一些有效的数值算法来解决这些问题。反问题一般是由附加边界或散射数据求偏微分方程中源项的系数。还感兴趣的是某些未知域进入边值问题。将研究椭圆型、抛物型和双曲型方程。因此，我们重点研究了逆电导率问题、逆散射问题和地震问题。 由单个或多个生成的附加数据 将考虑边界测量。将特别注意使用当地的边界数据，有限数量的散射频率，并识别非线性方程。本文采用了强有力的偏微分方程和复变方法，包括位势理论、Carleman型估计、复几何光学和Fourier积分算子。这项工作计划是一个深入的理论和随后的数值研究的挑战性应用问题，如声学，电气和地震prospecting在地球物理学和医学，以及裂缝检测和寻找物理本构关系的结果边界测量的物理场。 该项目致力于应用数学的挑战性和重要问题，将通过当代数学手段进行研究。预计这将回答几个问题的可能性评价的重要特征的物理和医学对象从相对便宜，非破坏性和可行的外部测量。对非经典和困难数学问题的深入理解将产生更好和可靠的数值算法。这必然导致从电、磁和超声波数据进行医疗诊断的有效方法，以及裂缝检测。寻找的方法（非线性） 在化学和工程中，控制若干复杂系统的定律有望提高数学模拟的可能性，并降低新技术的开发成本。 ***

9501050 Isakov The basic goal of this project is a study of global uniqueness and stabili ty for several fundamental inverse problems. The next aim is to develop some ef ficient numerical algorithms for their solution. Generally, an inverse problem consists in finding a coefficient of source term in a partial differential equa tion from an additional boundary or scattering data. Also of interest are cer- tain unknown domains entering a boundary value problem. Elliptic, parabolic, an d hyperbolic equations will be studied. Accordingly we focus on the inverse conductivity problem, inverse scattering and seismic problems. The additional data which are generated by single or many boundary measurements will be considered. A special attention will be paid to use of local boundary data, finite number of scattering frequencies, and to identification of non-linear equations. The powerful methods of PDE and Complex Variables are supposed to be used which include potential theory, Carleman type estimates, complex geometrical optics, and Fourier Integral Operators. The work is planned to be a deep theoretical and subsequent numerical investigation of challegning applied problems, like acoustic, electrical, and seismic prospec- ting in geophysics and medicine, as well as cracks detection and finding physi- cal constitutive laws from results of boundary measurements of physical fields. %%% This project is dedicated to challenging and important problems of applied mathematics which will be studied by contemporary mathematical means. This is expected to answer several questions about possibility of evaluation of important characteristics of physical and medical objects from relatively cheap ,non-destructive, and feasible exterior measurements. A deeper understanding of nonclassical and hard mathematical questions will generate better and reliable numerical algorithms. This must lead to efficient methods of medical diagnostic from electrical, magnetic, and u ltrasound data, of cracks detection. The method s of finding (non-linear) laws governing several complicated systems in che- mistry and engineering promises to enhance possibilities of mathematical simula tion and to reduce cost of developement of new technologies. ***