Some Theoretical and Applied Inverse Problems

本项目致力于研究一些重要反问题的唯一性和稳定性，特别是标量偏微分方程和（麦克斯韦和弹性）系统的扩散系数和传播速度的识别。工具将是新的Carleman型估计和控制理论与逆问题之间的相互作用。势能理论也可用于反重力和涡度问题的进一步研究。在多次边界测量的情况下，将利用greenleaf和Uhlmann最近构造的椭圆方程的局部解，努力解决反二次性问题中部分边界数据的基本唯一性问题。PI将尝试在反问题中寻找稳定性增加的情况，特别是从边界测量中获得非线性（空间无关的）椭圆和抛物方程恢复的Lipschitz条件稳定性估计，并量化高频平稳波勘探中稳定性增加的情况。理论结果将应用于期权市场波动率的识别。我们期望开发一种基于波动率建模的高效可靠的算法。这个问题对于期权市场的预测和根据当前市场数据评估经济稳定性具有重要的基础意义。另一个预期的应用是从实验数据中恢复基本的物理关系（在化学方程中，热传导）。由于新材料的发现和高温的使用，这个问题对当代工程越来越重要。反重力理论与现有的涡度模型相结合，有望在通过远距离压力测量来评估湍流（例如过去的飞机）方面取得进展。这一问题对提高大型机场的能力具有重要意义。该项目将涉及研究生，并激发他们对具有实际重要性的数学问题的兴趣。

This project is dedicated to the study of uniqueness and stability insome important inverse problems, in particular to identification of diffusion coefficientsand the speed of propagation for scalar partial differential equations and (Maxwell andelasticity)systems. Tools will be new Carleman type estimates andinteraction between control theory and inverse problems. Also potentialtheory is to be utilized in the further study of inverse gravimetry andvorticity problems. In case of many boundary measurements the effort will bemade to resolve the fundamental uniqueness question in the inverseconductivity problem with the data on a part of the boundary by usinglocalized solutions of elliptic equations recently constructed by Greenleafand Uhlmann. The PI will try to find cases of increased stability in inverseproblems, in particular obtaining Lipschitz conditional stability estimates forrecovery of nonlinear (space independent) elliptic and parabolic equationsfrom boundary measurements and quantifying increased stability inprospecting by stationary waves with higher frequency. Applications of theoretical results will be in identification of volatility in options markets. We expect to develop a very efficient and reliable algorithm based on the modeling of volatility. This problem is of fundamental importance for prediction of options markets and evaluation of economical stability from current market data. Another expected application is recovery of the fundamental physical relations (in chemical equations, heat conduction) from experimental data. This problem is of growing importance for contemporary engineering due to the discovery of new materials and the use of high temperatures.The theory of inverse gravimetry combined with available models of vorticity is expected to achieve progress in evaluation of turbulence (e.g. past aircrafts) from distant measurements of pressure. This problem has important implications for increasing capability of large airports. The project will involve graduate students and stimulate their interest in mathematical problems of practical importance.