Theory and Applications of Some Inverse Problems in PDE

PI打算继续他对某些重要逆问题的唯一性和稳定性的研究。偏微分方程理论的当代方法将被使用。这些包括Carleman型估计，傅里叶积分算子，势理论和一般泛函分析方法。重点是局部/部分边界测量的反问题，通过在简单线性变换中分离病态性来增加稳定性，非线性方程的恢复，以及反问题与最优控制理论之间的相互作用。基于该理论，PI计划为以下重要的应用反问题开发有效的数值算法。该项目的应用重点是：人脑活动部分的电磁探测问题；通过内部测量识别位于飞机机舱表面的噪声源；寻找复杂非线性物理系统的本构规律，包括多孔介质中的地下流动和化学反应；根据当前市场数据重建期权交易的波动系数。第二个问题对于设计噪音更小的汽车和中型飞机至关重要。第四部分是关于对影响当代社会生活的非常重要的金融领域的未来状态的预测。

The PI intends to continue his study of uniqueness and stability for certain important inverse problems. The contemporary methods of the theory of partial differential equations will be used. Those include Carleman type estimates, the Fourier integral operators, potential theory, and general functional analytic approach. An emphasis is on inverse problems with local/partial boundary measurements, on increasing stability by isolating ill-posedness in a simple linear transform, on recovery of non-linear equations, and on interaction between inverse problems and optimal control theory. Based on the theory the PI plans to develop efficient numerical algorithms for the following important applied inverse problems. The applied focus of the project will be on - the problem of electromagnetic detection of the active part of human brain- identifying a source of acoustical noise located on the surface of an aircraft cabin from interior measurements- finding constitutive laws for complicated non-linear physical systems, including underground flows in porous media and chemical reactions- reconstruction of the volatility coefficient of options trading from current market data.The second problem is of immediate importance in designing less noisy cars and midsize aircraft. The fourth one is about prediction of the future state of the very important financial area influencing the life of contemporary society.