一种新的后验正则化方法求解Helmholtz方程的Cauchy问题

A New Posterior Regularization Method to Solve Cauchy Problem in Helmholtz Equation

由于Helmholtz方程的Cauchy问题的解不连续依赖于所给的Cauchy数据,Cauchy数据的一个小小扰动引起解有很大的变化,所以该问题是严重的不适定问题。为了解决该问题的不适定性,需要借助正则化方法进行求解,这种新的后验正则化方法的饱和效应使得随着解的光滑性假设的提高而提高其收敛率,令正则化近似解与精确解之间误差估计达到最优。根据正则化的最优理论,误差估计的阶数是最优的,这种新的正则化方法可以借助于傅里叶变换和逆变换实现。考虑在半带状区域上Helmholtz方程的Cauchy问题,提出一种新的后验正则化方法得到其正则化近似解,并通过偏差原理得到后验正则化参数选取法则及正则化近似解与精确解之间最优的Holder型收敛误差估计。

Due to the solution to Cauchy problem in Helmholtz equation is not continuously dependent on the given Cauchy data,a small disturbance in the Cauchy data can cause a significant change in the solution,so this is a serious ill-posed problem. In order to solve the ill posed nature of this problem,it is necessary to solve this problem by means of regularization method,and the saturation effect of this new posterior regularization method makes it more convergent with the increase of hypothesis of smoothness of the solution,so that the error estimation between the new regularization approximate solution and the precise value is optimal.According to the optimal theory of regularization,the order of error estimation is optimal,and this new regularization method can be realized by means of Fourier transformation and inverse transformation. In consideration of the Cauchy problem in Helmholtz equation in half-binding area,a kind of new posterior regularization method is proposed to obtain the approximate solution to regularization,and by means of the deviation principle,to gain the selection law of the posterior regularization parameters and the optimal Holder-type convergence error estimation between the approximate solution and the precise value of regularization.