Mathematical Sciences: Some Inverse Problems in PDE

这个项目的基本目标是研究几个基本逆问题的全局唯一性和稳定性。下一个目标是开发一些有效的数值算法来解决这些问题。一般来说，反问题包括从附加的边界或散射数据中求出偏微分方程中源项的系数。同样令人感兴趣的是进入边值问题的某些未知域。椭圆型、抛物线型和双曲型方程将被研究。因此，我们重点研究了反电导率问题、反散射问题和地震问题。由单次或多次边界测量产生的额外数据将被考虑在内。将特别注意局部边界数据的使用，有限数目的散射频率，以及非线性方程的识别。应用了势理论、Carleman型估计、复几何光学和傅立叶积分算子等强大的偏微分方程和复变量方法。这项工作计划是对具有挑战性的应用问题进行深入的理论和后续数值研究，如地球物理学和医学中的声学，电气和地震勘探，以及裂缝检测和从物理场的边界测量结果中寻找物理本构定律。该项目致力于应用数学中具有挑战性和重要的问题，这些问题将通过当代数学手段进行研究。预计这将回答有关通过相对廉价、非破坏性和可行的外部测量来评估物理和医疗对象的重要特性的可能性的几个问题。对非经典和难数学问题的深入理解将产生更好、更可靠的数值算法。这必须导致有效的医学诊断方法，从电，磁和超声数据，裂纹检测。在化学和工程中寻找控制几个复杂系统的（非线性）规律的方法有望提高数学模拟的可能性，并降低开发新技术的成本。＊＊＊

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. ***