ABSTRACT

We consider an ill-posed problem of localizing (finding the position of) discontinuity lines of a function of two variables, provided that the function of two variables is smooth outside the discontinuity lines and has a discontinuity of the first kind at each point on the line. There is a uniform grid with a step \(\tau\). It is assumed that we know the average values of the perturbed function on the square \(\tau\times\tau\) at each node of the grid. The perturbed function approximates the exact function in space \(L_2(\mathbb{R}^2)\). The perturbation level \(\delta\) is assumed to be known. Earlier, the authors investigated (obtained accuracy estimates of) the global discrete regularizing algorithms for approximating a set of discontinuity lines of a noisy function. However, stringent smoothness conditions were superimposed onto the discontinuity line. The main result of the present study is the improvement of localizing the accuracy estimation methods, which allows replacing the smoothness requirement with a weaker Lipschitz condition. Also, the conditions of separability are formulated in a more general form as compared to previous studies. In particular, it is found that the proposed algorithm makes it possible to obtain the localization accuracy of the order of \(O(\delta)\). Estimates of other important parameters characterizing the localization algorithm are also given.

中文翻译：

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噪声函数间断线定位方法的新精度估计

摘要

我们考虑一个不适当地的问题，即定位（确定两个变量的函数的不连续线的位置），条件是两个变量的函数在不连续线之外是平滑的，并且在变量的每个点上都具有第一种不连续性线。有一个带有步长\（\ tau \）的统一网格。假设我们知道网格每个节点上平方（\ tau \ times \ tau \）上摄动函数的平均值。摄动函数逼近空间\（L_2（\ mathbb {R} ^ 2）\）中的精确函数。摄动水平\（\ delta \）假定是已知的。早些时候，作者研究了全局离散正则化算法（获得了其精度估算），用于近似嘈杂函数的一组不连续线。但是，严格的平滑度条件叠加在不连续线上。本研究的主要结果是改进了精度估计方法的局域性，可以用较弱的Lipschitz条件代替平滑度要求。此外，与以前的研究相比，可分离性条件以更一般的形式制定。特别地，发现所提出的算法使得可以获得\（O（\ delta）\）量级的定位精度成为可能。。还给出了表征定位算法的其他重要参数的估计。