Abstract:
We study the localization methods for the discontinuity lines of a noisy function of two variables. The function is assumed to have countably many discontinuity lines: finitely many discontinuity lines have “large” jump, and the jumps at the remaining discontinuity lines satisfy some smallness condition. It is required, from the noisy function and the error in L2, to determine the number and localize the position of the discontinuity lines that form the first set for the exact function. The problem under consideration belongs to the class of nonlinear ill-posed problems, and for solution we have to construct regularizing algorithms. We propose a simplified theoretical approach when conditions on the exact function are imposed in a narrow strip intersecting the discontinuity lines. We construct methods for the averaging and localization of discontinuity lines and obtain estimates for the accuracy of the localization.
Keywords:
ill-posed problem, regularization algorithm, localization of singularities, equation of the first kind, discontinuity line.
Citation:
A. L. Ageev, T. V. Antonova, “Methods for the approximating the discontinuity lines of a noisy function of two variables with countably many singularities”, Sib. Zh. Ind. Mat., 18:2 (2015), 3–11; J. Appl. Industr. Math., 9:3 (2015), 297–305
\Bibitem{AgeAnt15}
\by A.~L.~Ageev, T.~V.~Antonova
\paper Methods for the approximating the discontinuity lines of a~noisy function of two variables with countably many singularities
\jour Sib. Zh. Ind. Mat.
\yr 2015
\vol 18
\issue 2
\pages 3--11
\mathnet{http://mi.mathnet.ru/sjim877}
\crossref{https://doi.org/10.17377/sibjim.2015.18.201}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3549823}
\elib{https://elibrary.ru/item.asp?id=23598672}
\transl
\jour J. Appl. Industr. Math.
\yr 2015
\vol 9
\issue 3
\pages 297--305
\crossref{https://doi.org/10.1134/S1990478915030011}
Linking options:
https://www.mathnet.ru/eng/sjim877
https://www.mathnet.ru/eng/sjim/v18/i2/p3
This publication is cited in the following 3 articles:
T. I. Serezhnikova, Springer Proceedings in Mathematics & Statistics, 318, Mathematical Analysis With Applications, 2020, 417
V. Mezhuyev, O. M. Lytvyn, I. Pershyna, O. Nechuiviter, O. O. Lytvyn, “Algorithm for the reconstruction of the discontinuous structure of a body by its projections along mutually perpendicular lines”, Proceedings of 2018 7Th International Conference on Software and Computer Applications (ICSCA 2018), Assoc. Computing Machinery, 2018, 158–163
A. L. Ageev, T. V. Antonova, “High accuracy algorithms for approximation of discontinuity lines of a noisy function”, Proc. Steklov Inst. Math. (Suppl.), 303, suppl. 1 (2018), 1–11
Citation in format AMSBIB
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