献给马富明教授60华诞 Helmholtz方程Cauchy问题的间接积分方程方法

INDIRECT BOUNDARY INTEGRAL EQUATION METHOD FOR THE CAUCHY PROBLEM OF THE HELMHOLTZ EQUATION

本文处理二维和三维Helmholtz方程的边界数据复原问题．通过利用位势理论近似问题的解，导出了解决Cauchy问题的一种非迭代积分方程方法．为了处理形成问题的不适定性，采用了Tikhonov正则化结合Morozov偏差原理的方法，并且给出了算法的收敛性和误差估计，最后给出了二维和三维的数值算例．通过数值算例我们检验了源点和边界之间距离的关系，算法关于噪声、源点数目的数值收敛性，稳定性．

In this paper, we examine the data completion problem of the Helmholtz equation in two-and three-dimensionals. Through the use of a layer approach to the solution, we devise a numerical method for approximating the solution of Cauchy problem. The proposed method is non-iterative and intrinsically handle the case of noisy and incompatible data. In order to cope with this ill-posed problem, our formulation is based on Tikhonov regularization in conjunction with the Morozov discrepancy principle associated with linear ill-posed inverse problems and leads to convergent scheme. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method. Two- and three-dimensional examples are given for checking the effectiveness of the proposed method. The numerical convergence, accuracy, and stability with respect to the number of source points, the distance between the pseudo and real boundary, and decreasing the amount of noise added into the input data, respectively, are also analyzed.