Some inverse problems in elasticity, option pricing, semiconductors, and scattering theory.

该项目涉及偏微分方程反问题的四个重要领域。我们正在零和非零初始条件的情况下从动态边界数据中寻找各向同性和各向异性弹性理论系统的系数。我们关心系数确定的唯一性和稳定性。反问题普遍感兴趣的一个相关主题是当频率或漂移项很大时，提高亥姆霍兹型和抛物线方程的连续性和系数确定的稳定性。稳定性差是反问题应用的主要障碍之一，更好的稳定性非常有价值。主要的分析工具是卡尔曼估计、调和和微局域分析以及势理论。我们计划继续研究著名的 Black-Scholes 微分方程（抛物线型）的波动系数的识别，在重要的波动情况下（随时间呈线性）以及一篮子期权的情况。 一个新的成分将是积分几何的使用。最后，我们考虑使用反重力法和电导率方法来识别半导体模型中的电导率系数（掺杂分布），并且很少有部分边界测量和对偶问题。目标之一是设计快速可靠的算法来数值求解这些反问题。弹性方面的工作旨在提高地球、人体和工业材料（如钢铁）弹性特性测定的分辨率，这对于地球物理学、医学和工程至关重要。 反向期权定价的研究应该有助于监测和预测不久的将来的金融市场。寻找掺杂分布对于计算机芯片制造关键领域的质量控制非常重要。

The project is concerned with four important areas of inverse problems for partial differential equations. We are looking for coefficients of systems of isotropic and anisotropic elasticity theory from dynamical boundary data in case of zero and nonzero initial conditions. We are concerned with uniqueness and stability of determination of coefficients. A related topic of general interest for inverse problems is about increased stability of the continuation and of the determination of the coefficients for Helmholtz type and parabolic equations when frequency or drift terms are large. Poor stability is one of major obstacles for applications of inverse problems, and better stability is very valuable. The main analytical tools will be Carleman estimates, harmonic and microlocal analysis, and potential theory. We plan to continue research on identification of the volatility coefficient of the famous Black- Scholes differential equation (of parabolic type) in important cases of volatility which is linear in time and in case of basket options. A new ingredient will be use of integral geometry. Finally, we consider identification of a conductivity coefficient (doping profile) in semiconductor models by using methods of inverse gravimetry and conductivity with few partial boundary measurements and of dual problems. One of goals is designing fast and reliable algorithms for numerical solution of these inverse problems.The work on elasticity aims to improve resolution of determination of elastic properties of the Earth, of human body, and of material used in industry (like steel) which are crucial for geophysics, medicine, and engineering. Research on inverse option pricing should help with monitoring and predicting financial markets in near future. Looking for doping profile is important for quality control in crucial area of manufacturing of computer chips.