椭圆偏微分方程参数识别问题的理论和数值研究

This project intends to study the parameter identification problems for elliptic partial differential equations (PDEs). The main contents are identifying the parameters in elliptic PDEs from additional boundary measurements. From the view of mathematics, the boundary of the domain is one dimension lower than the inner domain, so the inverse problem is generally ill-posed. By introducing the complex boundary value problem, we first match data in the problem domain rather than on the boundary. Then, the Tikhonov regularization method with various norms is used to reflect different properties of the coefficients. Besides, the finite element method is used to obtain the stable approximate solution and numerical algorithms will be constructed. The expected results of this project will not only enrich the theories of the inverse parameter problems for elliptic PDEs, but also promote its application in various fields, such as geophysical science. So the research has important theoretical significance and practical value.

本项目拟研究椭圆偏微分方程(PDEs)的参数识别问题, 开展以下工作: 通过额外的区域边界测量信息反演方程中的未知系数. 从数学角度出发, 由于区域边界比其内部低一维, 因此这类反问题一般是不适定的. 本项目通过引入复边值问题, 首先将边界信息的反演转化为区域信息的反演. 然后根据所需反演系数的特点考察在不同范数的Tikhonov正则化方法下的反演效果. 我们还将利用有限元方法找到其稳定的近似解并构造有效的数值算法进行模拟. 本项目的预期成果, 不仅能丰富偏微分方程系数反演问题的理论体系, 也能促进它在地球物理科学等领域的应用, 具有重要的理论意义和实用价值．

椭圆偏微分方程参数识别问题在数学物理和工程技术中有广泛的应用。本项目主要研究通过边界测量信息来反演椭圆偏微分方程左端项系数的问题。取得的主要研究成果如下：利用耦合复边界方法，我们成功反演了椭圆偏微分方程左端的零阶项系数。通过Tikhonov正则化方法和优化方法,给出了所需反演系数的近似解，然后利用有限元方法对问题进行离散并构造了数值算法。项目组发表一篇SCI论文，另一篇论文正在审稿中。研究成果丰富和发展了椭圆偏微分方程参数识别问题的理论和算法。