Mathematical Sciences: Some Inverse Problems in PDE

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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. ***