Abstract:
For a problem of localizing the singularities (discontinuities of the first kind) of a noisy function in Lp (1⩽p<∞), new classes of regularizing methods are constructed. These methods determine the number of discontinuities and approximate their positions. Also, upper and lower bound of the localizing singularities and the separability threshold, is obtain. It is proved that the methods are order-optimal by accuracy as well as separability on some classes of functions with discontinuities.
Key words:
ill-posed problems, localization of singularities, discontinuities of the first kind, regularizing method, separability threshold.
This publication is cited in the following 12 articles:
Qusay Muzaffar, David Levin, Michael Werman, “Approximating a Function with a Jump Discontinuity—The High-Noise Case”, AppliedMath, 4:2 (2024), 561
V. I. Maksimov, Yu. S. Osipov, “On the identification of control failures by the dynamic regularization method”, Proc. Steklov Inst. Math. (Suppl.), 325, suppl. 1 (2024), S134–S146
A. L. Ageev, T. V. Antonova, “Estimates of characteristics of localization methods for discontinuities of the first kind of a noisy function”, J. Appl. Industr. Math., 13:1 (2019), 1–10
A. L. Ageev, T. V. Antonova, “Investigation of methods of localization of $q$-jumps and discontinities of firsth king of noisy function”, Russian Math. (Iz. VUZ), 63:7 (2019), 1–11
A. L. Ageev, T. V. Antonova, “Localization of boundaries for subsets of discontinuity points of noisy function”, Russian Math. (Iz. VUZ), 61:11 (2017), 10–15
Kurlikovskii D.V. Ageev A.L. Antonova T.V., “Research of a Threshold (Correlation) Method and Application For Localization of Singularities”, Sib. Electron. Math. Rep., 13 (2016), 829–848
A. L. Ageev, T. V. Antonova, “O diskretizatsii metodov lokalizatsii osobennostei zashumlennoi funktsii”, Tr. IMM UrO RAN, 21, no. 1, 2015, 3–13
Ageev A.L., Antonova T.V., “New Methods for the Localization of Discontinuities of the First Kind for Functions of Bounded Variation”, J. Inverse Ill-Posed Probl., 21:2 (2013), 177–191
A. L. Ageev, T. V. Antonova, “Approximation of discontinuity lines of a noisy function of two variables”, J. Appl. Industr. Math., 6:3 (2012), 269–279
A. L. Ageev, T. V. Antonova, “On the localization of singularities of the first kind for a function of bounded variation”, Proc. Steklov Inst. Math. (Suppl.), 280, suppl. 1 (2013), 13–25
T. V. Antonova, “Localization method for lines of discontinuity of approximately defined function of two variables”, Num. Anal. Appl., 5:4 (2012), 285–296
A. L. Ageev, T. V. Antonova, “O nekorrektno postavlennykh zadachakh lokalizatsii osobennostei”, Tr. IMM UrO RAN, 17, no. 3, 2011, 30–45
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