In this paper, a Cauchy problem of Helmholtz equation in a multi-dimensional case is investigated. This problem is severely ill-posed and small perturbations to measurement data can result in large changes in the solution. A kind of operator regularization method is proposed. The stable error estimates are obtained in the $L^2-$norm and $H^r-$norm under the conditions that m is even, md>x, mk>1, and suitable choice of regular parameters. Error estimates show that the regularized solution depends continuously on the perturbation noisy data and wave number. Our method makes up the limitation of small waves. Three numerical experiments show that our proposed method is effective and stable, especially in the case of a large wave number.

本文研究了多维情况下亥姆霍兹方程的柯西问题。这个问题是严重不适定的，对测量数据的微小扰动会导致解的较大变化。提出了一种算子正则化方法。稳定的误差估计在${升}^2-$规范和 $H^r-$在m为偶数，md > x，mk >1的条件下范数，并且选择合适的正则参数。误差估计表明正则化解持续依赖于扰动噪声数据和波数。我们的方法弥补了小波的局限性。三个数值实验表明，我们提出的方法是有效且稳定的，尤其是在大波数的情况下。