Some Inverse Problems in Elasticity, Financial Markets, and Scattering Theory

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Abstract: 0405976 Isakov, Wichita State University Some Inverse Problems in Elasticity, Financial Markets, and Scattering Theory We consider three important areas of inverse problems, where the main goal is to study uniqueness, stability, and numerical methods for reconstruction of coefficients of partial differential equations and systems from additional boundary data. The work on elasticity theory targets finding elastic parameters in isotropic and some anisotropic media from dynamical boundary data. We plan to attack fundamental open questions in case of zero (many boundary data) and special nonzero initial conditions. A close topic is uniqueness and increased stability in inverse problems for the Helmholtz equation, with emphasis on some challenging problems from scattering theory, like uniqueness of the potential and of an hard obstacle from the data at fixed frequency or at a fixed direction of an incident wave. The PI expects to use Carleman estimates, microlocal analysis, and potential theory, and to invent new approproate tools. Another area of research concerns determination of so-called volatility coefficient in the Black-Scholes model of option markets from market data. Here the emphasis will be on localization properties of the inverse problems, on volatility slowly changing with time, and on more complicated and realistic cases of American and index options.The results of research are expected to improve resolution of determination of interior properties of elastic materials and of interior of the Earth from noninvasive exterior measurements. The work on inverse option pricing should generate an effective and valuable tool of evaluation of current state of economy, especially of financial markets from current data, and hence to predict economical situation in near future. So the applications are to materials and manufacturing, environment, and civil infrastructure