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 Trudy Inst. Mat. i Mekh. UrO RAN, 2011, Volume 17, Number 3, Pages 30–45 (Mi timm718)

On ill-posed problems of localization of singularities

A. L. Ageev, T. V. Antonova

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: Ill-posed problems of approximating (localizing) the positions of isolated singularities of a function of one variable are discussed. The function either is given with an error or is a solution to the convolution-type Fredholm integral equation of the first kind with an error in the right-hand side. The singularities can be $\delta$-functions, discontinuities of the first kind, or breakpoints. Earlier, the authors proposed an approach to deriving accuracy estimates for localization algorithms, which is similar to the classical approach of investigating methods on correctness classes. As a development of this theory, a general scheme of construction and investigation is proposed for regular method of localizing the singularities. The scheme can be used to uniformly derive many of the known results as well as new statements. Several classes of regularization methods generated by averaging kernels are considered. Estimates of localization accuracy and estimates of another important characteristic of the methods, namely, of the separability threshold, are obtained for the proposed methods. Lower estimates for the attainable accuracy and separability are obtained, which allows to establish the (order) optimality of the constructed methods on classes of functions with singularities for some problems.

Keywords: ill-posed problem, localization of singularities, regularizing method, separation threshold.

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Citation: A. L. Ageev, T. V. Antonova, “On ill-posed problems of localization of singularities”, Trudy Inst. Mat. i Mekh. UrO RAN, 17, no. 3, 2011, 30–45

Citation in format AMSBIB
\Bibitem{AgeAnt11} \by A.~L.~Ageev, T.~V.~Antonova \paper On ill-posed problems of localization of singularities \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2011 \vol 17 \issue 3 \pages 30--45 \mathnet{http://mi.mathnet.ru/timm718} \elib{http://elibrary.ru/item.asp?id=17870118} 

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This publication is cited in the following articles:
1. 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
2. D. V. Kurlikovskii, “Localization of discontinuities of the first kind methods for the solution of a convolution-type equation”, Russian Math. (Iz. VUZ), 58:3 (2014), 60–63
3. A. L. Ageev, T. V. Antonova, “O diskretizatsii metodov lokalizatsii osobennostei zashumlennoi funktsii”, Tr. IMM UrO RAN, 21, no. 1, 2015, 3–13
4. T. V. Antonova, “Methods of identifying a parameter in the kernel of the first kind equation of the convolution type on the class of functions with discontinuities”, Num. Anal. Appl., 8:2 (2015), 89–100
5. A. L. Ageev, T. V. Antonova, “Methods for the approximating the discontinuity lines of a noisy function of two variables with countably many singularities”, J. Appl. Industr. Math., 9:3 (2015), 297–305
6. A. L. Ageev, T. V. Antonova, “Discretization of a new method for localizing discontinuity lines of a noisy two-variable function”, Proc. Steklov Inst. Math. (Suppl.), 299, suppl. 1 (2017), 4–13
7. D. V. Kurlikovskii, A. L. Ageev, T. V. Antonova, “Issledovanie porogovogo (korrelyatsionnogo) metoda i ego prilozhenie k lokalizatsii osobennostei”, Sib. elektron. matem. izv., 13 (2016), 829–848
8. 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
9. 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
10. 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
11. A. L. Ageev, T. V. Antonova, “Issledovanie metodov lokalizatsii $q$-skachkov i razryvov pervogo roda zashumlennoi funktsii”, Izv. vuzov. Matem., 2019, no. 7, 3–14
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