SIAM Journal on Imaging Sciences, Volume 13, Issue 2, Page 589-608, January 2020.
In this article we study the problem of recovering the initial data of the two-dimensional wave equation from Neumann measurements on a convex domain $\Omega\subset\mathbb{R}^2$ with smooth boundary. We derive an explicit inversion formula of a so-called back-projection type and deduce exact inversion formulas for circular and elliptical domains. In addition, for circular domains, we show that the initial data can also be recovered from any linear combination of its solution and its normal derivative on the boundary. Numerical results of our implementation of the derived inversion formulas are presented demonstrating their accuracy and stability.

中文翻译：

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来自Neumann迹线的二维波动方程的显式反演公式

SIAM影像科学杂志，第13卷，第2期，第589-608页，2020
年1月。在本文中，我们研究了从凸域$ \ Omega \上的Neumann测量值恢复二维波动方程的初始数据的问题。具有平滑边界的subset \ mathbb {R} ^ 2 $。我们推导了一个所谓的反投影类型的显式反演公式，并推导了圆形和椭圆域的精确反演公式。另外，对于圆形域，我们表明初始数据也可以从其解和边界上的正态导数的任何线性组合中恢复。给出了我们对导出的反演公式的实现的数值结果，证明了它们的准确性和稳定性。