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Abstract <p>We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables, provided that outside the discontinuity lines the function satisfies a Lipschitz condition, and at each point on the lines there is a discontinuity of the first kind. For a uniform grid with step <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12258_2025_277_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau \)</EquationSource> <!--NumAnAp2503001Ageev-m1--> </InlineEquation>, it is assumed that at each node the mean values of a perturbed function on a square with side <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12258_2025_277_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau \)</EquationSource> <!--NumAnAp2503001Ageev-m2--> </InlineEquation> are known, and the perturbed function approximates the exact function in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12258_2025_277_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\({{L}_{2}}({{\mathbb{R}}^{2}}).\)</EquationSource> <!--NumAnAp2503001Ageev-m3--> </InlineEquation> The level of perturbation <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12258_2025_277_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta \)</EquationSource> <!--NumAnAp2503001Ageev-m4--> </InlineEquation> is assumed to be known. We propose a new approach based on a separation of the original noisy data to constructing regularizing algorithms for localizing the discontinuity lines. New algorithms are constructed for a class of functions with piecewise linear discontinuity lines and a convergence theorem with estimates of approximation accuracy is proved.</p>

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Regular Algorithms Based on a Separation of Perturbed Function Values for the Localization of Discontinuity Lines

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • A. L. Ageev,
  • T. V. Antonova

摘要

Abstract

We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables, provided that outside the discontinuity lines the function satisfies a Lipschitz condition, and at each point on the lines there is a discontinuity of the first kind. For a uniform grid with step \(\tau \) , it is assumed that at each node the mean values of a perturbed function on a square with side \(\tau \) are known, and the perturbed function approximates the exact function in \({{L}_{2}}({{\mathbb{R}}^{2}}).\) The level of perturbation \(\delta \) is assumed to be known. We propose a new approach based on a separation of the original noisy data to constructing regularizing algorithms for localizing the discontinuity lines. New algorithms are constructed for a class of functions with piecewise linear discontinuity lines and a convergence theorem with estimates of approximation accuracy is proved.