Theory and Applications of Some Inverse Problems in PDE

PI打算继续研究某些重要反问题的唯一性和稳定性。将使用偏微分方程组理论的当代方法。其中包括Carleman估计、傅里叶积分算子、位势理论和一般泛函分析方法。重点是具有局部/部分边界测量的反问题，通过隔离简单线性变换中的不适定性来提高稳定性，恢复非线性方程，以及反问题与最优控制理论之间的相互作用。基于这一理论，PI计划为下列重要的应用反问题开发有效的数值算法。该项目的应用重点将是--人脑活动部分的电磁探测问题--根据内部测量识别飞机机舱表面的声噪声源--为复杂的非线性物理系统寻找本构定律，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。第二个问题对设计噪音较低的汽车和中型飞机具有直接重要性。第四，对影响当代社会生活的重要金融领域的未来状态进行预测。

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.