We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables that is smooth outside the discontinuity lines and has a discontinuity of the first kind at each point of such lines. A uniform square grid with step τ is considered, and it is assumed that the mean values of a perturbed function over squares with side τ are known at each node of the grid. The perturbed function approximates the exact function in the space L₂(ℝ²). The perturbation level δ is known. To solve the problem under consideration, we design and study global discrete algorithms that are based on averaging procedures and approximate the discontinuity lines by a set of points of a uniform grid. The main result of the paper is the development of an approach to the global study of localization algorithms. We formulate conditions for the exact function, thus defining a class of correctness. Within this class, we perform a theoretical study of the proposed algorithms, introduce the characteristics to be estimated (the concept of approximating a set of discontinuity lines by a set of points of a uniform grid), and develop methods for deriving the estimates. To achieve this goal, we use a simplified statement: the discontinuity lines are straight line segments, and the proposed localization algorithm has the simplest thinning block. It is established that the localization error of the algorithm has order O(δ). Estimates of other important parameters characterizing the localization algorithm are given.

中文翻译：

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具有两个变量的函数的不连续线的全局定位问题

我们考虑将两个变量的函数的不连续线定位（找到位置）的不适定问题，这两个变量在不连续线的外部是平滑的，并且在此类线的每个点都具有第一种不连续性。考虑具有步长τ的均匀正方形网格，并假定在网格的每个节点处已知边为τ的正方形上的扰动函数的平均值。的扰动功能近似的空间中的确切功能大号₂（ℝ ²）。摄动水平δ是众所周知的。为了解决所考虑的问题，我们设计和研究了基于平均过程的全局离散算法，并通过一组均匀网格的点来近似不连续线。本文的主要结果是开发了一种全球研究定位算法的方法。我们为精确函数制定条件，从而定义一类正确性。在这一节课中，我们对提出的算法进行理论研究，介绍要估计的特征（通过一组均匀网格的点近似一组不连续线的概念），并开发得出估计的方法。为了实现这一目标，我们使用简化的语句：间断线是直线段，提出的定位算法具有最简单的细化块。建立算法的定位误差有序O（δ）。给出了表征定位算法的其他重要参数的估计。