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

该项目涉及偏微分方程逆问题的四个重要领域。在零初始条件和非零初始条件的情况下，我们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。我们关心的是确定系数的唯一性和稳定性。反问题的一个普遍感兴趣的相关主题是当频率或漂移项较大时，Helmholtz型方程和抛物型方程的延拓和系数的确定的稳定性增加。稳定性差是反问题应用的主要障碍之一，而稳定性好是非常有价值的。主要的分析工具将是Carleman估计、调和和微局域分析以及位势理论。我们计划继续研究著名的Black-Scholes方程(抛物型)在时间线性的重要波动率情况下和篮子期权情况下的波动率系数的辨识问题。一个新的成分将是积分几何的使用。最后，我们考虑了半导体模型中的电导系数(掺杂分布)的识别问题，采用了反重量法和电导率的方法以及对偶问题。对逆向期权定价的研究将有助于在不久的将来对金融市场进行监测和预测。在计算机芯片制造的关键领域，寻找掺杂情况对于质量控制非常重要。

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.