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

本文研究了反问题的三个重要领域，其主要目标是研究从附加边界数据重构偏微分方程和系统系数的唯一性、稳定性和数值方法。弹性理论研究的目标是从各向同性和某些各向异性介质的动态边界数据中寻找弹性参数。我们计划在零（许多边界数据）和特殊的非零初始条件下攻击基本开放问题。一个密切的话题是唯一性和增加稳定性在反问题的亥姆霍兹方程，重点是一些具有挑战性的问题，从散射理论，如势能的唯一性和硬障碍的数据在固定频率或固定方向的入射波。PI期望使用Carleman估计、微局部分析和势能理论，并发明新的合适的工具。另一个研究领域是根据市场数据确定期权市场Black-Scholes模型中的所谓波动系数。这里的重点将放在逆问题的局部化特性、随时间缓慢变化的波动性，以及更复杂和现实的美国期权和指数期权案例上。研究结果有望提高通过非侵入性外部测量测定弹性材料内部性质和地球内部的分辨率。期权逆定价的研究应该能够从当前的数据中得出一个有效的、有价值的评估当前经济状况，特别是金融市场状况的工具，从而预测近期的经济形势。因此，应用范围包括材料和制造、环境和民用基础设施

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