{"title":"On the Localization of Fractal Lines of Discontinuity from Noisy Data","authors":"","doi":"10.3103/s1066369x23090037","DOIUrl":null,"url":null,"abstract":"<span> <h3>Abstract</h3> <p>An ill-posed problem of localization (determining the position) of discontinuity lines of a function of two variables is considered: outside the discontinuity lines, the function is smooth, and, at each point on the line, it has a discontinuity of the first kind. Under the Lipschitz conditions on the discontinuity line, averaging procedures are constructed and global discrete regularizing algorithms of localization are studied. A parametric family of fractal lines is constructed for which all conditions can be checked analytically. A fractal having a large fractal dimension is indicated for which the efficiency of the constructed methods can be guaranteed.</p> </span>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"97 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s1066369x23090037","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
An ill-posed problem of localization (determining the position) of discontinuity lines of a function of two variables is considered: outside the discontinuity lines, the function is smooth, and, at each point on the line, it has a discontinuity of the first kind. Under the Lipschitz conditions on the discontinuity line, averaging procedures are constructed and global discrete regularizing algorithms of localization are studied. A parametric family of fractal lines is constructed for which all conditions can be checked analytically. A fractal having a large fractal dimension is indicated for which the efficiency of the constructed methods can be guaranteed.
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