We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the inverse problem to a system of boundary integral equations. We propose an iterative scheme that linearizes the equation using the Fr\'echet derivative of the forward operator. The application of special quadrature rules results to an ill-conditioned linear system which we solve using Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.

翻译：我们考虑的是重建封闭域内嵌入的洞穴的边界曲线的反面问题。 问题在波形方程式的两个维度中形成。 我们把拉盖尔变形与整体方程法结合起来, 并将反面问题降低到边界整体方程系统。 我们提出一个迭代方案, 利用远端操作器的Fr\'echet衍生物将方程线性线性线性线性线性系统进行线性线性规则的应用, 并使用Tikhonov的正规化来解决。 数字结果显示, 拟议的方法可以产生准确和稳定的重建。