Mathematical Sciences: Parameter Estimation Problems with Discontinuous Coefficients

本课题的主题是具有不连续系数的流动方程的参数估计问题。在这些问题中，介质的参数需要通过间接测量来确定。具体来说，这个项目的目标是开发有效的算法来识别二维和三维问题中的不连续参数a(x)。建议的算法使用空间变量u（x,t）的有限元逼近，不需要参数变量a(x)的任何离散化。引入了特殊的平均方法来消除数值不稳定性。一般理论是基于抛物算子的g收敛性和同时最小化原理。二维问题的初步结果显示了成功识别的例子和矢量化的重要可能性。我们还研究了通过考虑额外的观测数据集进一步改进识别的问题。

The subject of this project is parameter estimation problems for flow equations with discontinuous coefficients. These are the problems where parameters of the medium are required to be determined by indirect measurements. Specifically the objective of this project is developing of efficient algorithms for the identification of discontinuous parameters a(x) in 2-D and 3-D problems. The suggested algorithm uses finite elements approximations of the space variable u(x,t) and does not require any discretization of the parameter variable a(x). Special averaging procedure is introduced to eliminate numerical instabilities. The general theory is based on G-convergence of parabolic operators and principle od simultaneous minimization. Preliminary results for 2- d problems show examples of successful identification and significant possibilities for vectorization. We also study the problem of further improvements in the identification by considering additional data sets of observations.