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。我们正在寻找系数的各向同性和各向异性弹性理论系统的动态边界数据的情况下，零和非零的初始条件。我们关心的是系数确定的唯一性和稳定性。反问题的一个相关的主题是关于增加稳定性的延续和Helmholtz型和抛物型方程的系数的确定时，频率或漂移项是大的。稳定性差是反问题应用的主要障碍之一，而稳定性好的反问题是非常有价值的。主要的分析工具将是Carleman估计，谐波和微局部分析，和潜在的理论。我们计划继续研究著名的Black-Scholes微分方程（抛物型）的波动率系数的识别，在重要的情况下，波动率在时间上是线性的，在篮子期权的情况下。 一个新的组成部分将是使用积分几何。最后，我们考虑识别的电导率系数（掺杂配置文件）在半导体模型中，通过使用反重力和电导率与一些部分边界测量和对偶问题的方法。其中一个目标是设计快速可靠的算法来数值求解这些反问题。弹性力学的工作旨在提高地球、人体和工业材料（如钢）的弹性性质的测定分辨率，这些弹性性质对物理学、医学和工程学都是至关重要的。 反向期权定价的研究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，寻找掺杂分布对于质量控制是重要的。

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.