三维椭圆方程Cauchy问题的正则化方法

The Cauchy problem for elliptic equations is a common interesting subject of many engineering disciplines and industrial sectors. Many models can be transformed as it, such as inverse scattering, electrical impedance tomography, optical tomography, analytic continuation, and so on. The difficulty of solving this problem is that the solution of the problem does not depend continuously on the input data. The slight perturbation may cause solutions blasting. Currently, the two-dimensional case has been studied. But for three-dimensional case, due to the stronger degree of ill-posedness and the larger resulting matrix equation, it is more difficult to be solved. So far, there are only some conditional stability results and few effective regularization methods or efficient algorithms. Therefore, how to select the proper regularization parameter and deal with the discrete matrix equation, as well as how to create a fast and efficient algorithm are meaningful and quite difficult. In view of this, the project intends to use some regularization methods to study the Cauchy problem for 3D elliptic equations, It is expected to use deterministic methods to select the regularization parameters and obtain the convergence error estimates for a-priori and a-posteriori selection rules. Moreover, combining the regularization methods with preconditioned GMRES method will be tried. Meanwhile, some fast solvers with fast Fourier transform will be proposed to save the storage space and the computing time.

椭圆方程Cauchy问题在工程技术领域具有广泛应用，例如逆散射、电阻抗断层、光断层扫描和解析延拓等问题的模型都可转化为该问题。而解决此问题的难点在于其解并不连续依赖于输入数据，输入数据即使有微小的扰动都可能导致解的爆破。目前，二维情形已有一些结果，而对于三维情形，由于其不适定程度更高以及所得矩阵方程非常庞大等原因导致该问题的求解难度大增，目前只有一些条件稳定性的结果，很难见到有效的正则化方法和快速高效的算法。因此，如何选取合适的正则化参数、如何处理离散所得的矩阵方程、如何建立快速有效的算法都是很有意义和难度的问题。鉴于此，本项目拟用某些正则化方法来研究三维椭圆方程Cauchy问题，采用确定性方法来选取正则化参数，并分别对先验选取与后验选取规则给出相应的误差估计。同时尝试将正则化方法与预处理GMRES方法相结合，并借助于快速Fourier变换给出相应的快速算法来节省存储空间和计算时间。

椭圆方程Cauchy问题在工程技术领域具有广泛应用，例如逆散射、电阻抗断层、光断层扫描和解析延拓等问题的模型都可转化为该问题。而解决此问题的难点在于其解并不连续依赖于输入数据，输入数据即使有微小的扰动都可能导致解的爆破。关于二维情形已有一些研究结果，而对于三维情形，由于其不适定程度更高以及所得矩阵方程非常庞大等原因导致该问题的求解难度大增，之前只有一些条件稳定性的结果，很难见到有效的正则化方法和快速高效的算法。针对此，本项目采用拟边界值方法来研究三维椭圆方程Cauchy问题，采用确定性方法来选取正则化参数，并分别对先验选取与后验选取规则给出相应的误差估计。关于正方体区域，本项目将正则化方法与预处理GMRES方法相结合，并借助于快速Fourier变换给出相应的快速算法来节省存储空间和计算时间。对于一般的柱形区域，本项目给出了拟边界值方法，并得到相应的误差估计，后续工作将完善相应的数值结果。所得的理论结果将对其他不适定问题的处理提供一定的参考。