一类波动方程反问题的数值求解

项目名称： 一类波动方程反问题的数值求解

项目编号： No.10801098

项目类型： 青年科学基金项目

立项/批准年度： 2009

项目学科： 建筑科学

项目作者： 谢建利

作者单位： 上海交通大学

项目金额： 17万元

中文摘要： 本项目将研究一类波动方程的几个反问题的数值解法：在已知一类波动方程的解和部分系数的条件下，求方程或边界条件中未知的系数。由于反问题的不适定性，我们先提出合理的正则化方法，包括Tikhonov正则化和迭代正则化，接着对连续问题用有限元进行离散化，再构造梯度型的优化方法和多重网格技术对离散的问题进行数值求解，并给出数值方法的稳定性和收敛性分析，最后是数值试验和模拟。将求解正问题的现代数值方法应用于反问题和对反问题的求解作数值分析是本项目的创新之处。项目中将要考虑的一个反问题是重构方程的右端项，它同时是时间和空间的函数，这方面的研究以前几乎没有。项目的研究成果将有助于推动反问题数值解法的发展，进而有益于反问题的理论研究。

中文关键词： 反问题；正则化；梯度型方法；多重网格；数值分析

英文摘要： In this project , we will study the numerical solutions of some inverse problems for a class of wave equations. These inverse problems are about the identification of coefficients in the differential equation considered or in the boundary conditions, given the observation data of the solutions of the PDE and part of the coefficients in the PDE. Because of the ill-posedness of the inverse problems, we first propose some reasonable regularization formula including Tikhonov regularization and iterative regularization. We then discritize the continuous problems using finite element methods. Next we address some gradient-type optimization methods, combined with the multi-grid techniques, to solve the discrete optimization problems. And the stability and convergence analysis of the proposed numerical methods will be given. Finally numerical experiment or simulation will also be conducted. The main novelty in this research is the application of those modern methods developed for direct problems to the solution of inverse problems, and the numerical analysis for the inverse problems. One remarkable inverse problem, that will be studied in the project, is the reconstruction of the source term, which is a function of both time variable and spatial variable. Very few research works on this kind of inverse problems can be found. The results of this research will push forward the development on numerical solutions of inverse wave equations and thus benefit to the theoretical research of the inverse wave problems.

英文关键词： inverse problems; regularization; gradient-type methods; multi-grid methods; numerical analysis