Laplace方程Cauchy问题的Tikhonov正则化方法

Tikhonov Regularization Method for the Cauchy Problem of the Laplace Equation

考虑了矩形区域上一个Laplace方程的Cauchy问题.对y=0时的Cauchy数据,以及x=0,x=π时的边界数据均已给出,要求0<y≤1时的解.对该不适定问题,文中用Tikhonov正则化方法构造正则化解,并证明了所得正则化解稳定地收敛于精确解.

In this paper, a Cauchy problem for the Laplace equation is considered in a rectangle domain. Cauchy data are given at y = 0 , and the boundary data are given at x = 0 and x = 7π . The solution for 0 〈 y ≤ 1 is sought. It is know that such problem is severely ill - posed. The authors use Tikhonov regularization. method to solve it. Convergence estimations are presented under an a - priori boundary assumption for the exact solution.