We propose and study a data completion algorithm for recovering missing data from the knowledge of Cauchy data on parts of the same boundary. The algorithm is based on surface representation of the solution and is presented for the Helmholtz equation. This work is an extension of the data completion algorithm proposed by the two last authors where the case of data available of a closed boundary was studied. The proposed method is a direct inversion method robust with respect to noisy incompatible data. Classical regularization methods with discrepancy selection principles can be employed and automatically lead to a convergent schemes as the noise level goes to zero. We conduct 3D numerical investigations to validate our method on various synthetic examples.

中文翻译：

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基于部分柯西数据的表面势的亥姆霍兹方程的数据补全方法

我们提出并研究了一种数据补全算法，用于从相同边界部分的柯西数据知识中恢复缺失数据。该算法基于解的表面表示，并针对亥姆霍兹方程提出。这项工作是最后两位作者提出的数据完成算法的扩展，其中研究了封闭边界可用数据的情况。所提出的方法是对噪声不兼容数据鲁棒的直接反演方法。可以采用具有差异选择原则的经典正则化方法，并在噪声水平变为零时自动导致收敛方案。我们进行了 3D 数值研究，以在各种合成示例上验证我们的方法。