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

摘要：0405976 Isakov，威奇托州立大学一些反问题的弹性，金融市场和散射理论我们考虑三个重要领域的反问题，其中的主要目标是研究唯一性，稳定性和数值方法重建系数的偏微分方程和系统从额外的边界数据。弹性理论的工作目标是从动态边界数据中找到各向同性和某些各向异性介质中的弹性参数。我们计划在零（许多边界数据）和特殊的非零初始条件的情况下攻击基本的开放问题。一个密切的主题是唯一性和增加的稳定性反问题的亥姆霍兹方程，强调一些具有挑战性的问题，从散射理论，如唯一性的潜力和硬障碍的数据在固定频率或在一个固定的方向入射波。PI希望使用Carleman估计，微局部分析和潜在理论，并发明新的适当工具。研究的另一个领域是从市场数据中确定期权市场的Black-Scholes模型中的所谓波动系数。这里的重点将是在本地化特性的反问题，波动率随时间缓慢变化，并在更复杂的和现实的情况下，美国和指数期权的研究结果，预计将提高分辨率的内部性质的弹性材料和地球内部的非侵入性外部测量的确定。逆期权定价的研究为从当前的数据中评估当前的经济状况，特别是金融市场的状况，从而预测未来的经济形势提供了一个有效的和有价值的工具。因此，这些应用是材料和制造业，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