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Статья
2017

Localization of subsets of discontinuity points of a noisy function


A. L. AgeevA. L. Ageev, T. V. AntonovaT. V. Antonova
Русская математика
https://doi.org/10.3103/S1066369X17110020
Abstract / Full Text

We consider an ill-posed problem of localization of discontinuities of the first kind of a one-dimensional function, when knowing only its approximation and the error level δ in the metric of L 2(−∞,+∞). We propose a new statement of the problem when all discontinuities are divisible into subsets, and the localization takes place for subsets of discontinuities. Assuming additionally that all discontinuities in each subset have jumps of one sign, we construct a new regular method that allows to determine the number of subsets of discontinuities, to approximate their boundaries, and to estimate the approximation accuracy.

Author information
  • Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, ul. S. Kovalevskoi 16, Ekaterinburg, 620990, RussiaA. L. Ageev & T. V. Antonova
References
  1. Winklerm, G., Wittich, O., Liebsher, V., Kempe, A. “Don’t Shed Tears Over Breaks”, Jahresber. Deutsch. Math.-Verein. 107, No. 2, 57–87 (2005).
  2. Sizikov, V.S. Mathematical Methods for Processing the Results of Measurements (Politekhnika, St.-Petersburg, 2001) [in Russian].
  3. Malla, S. Wavelets in Signal Processing (Mir, Moscow, 2005) [Russian translation].
  4. Introduction to Contour Analysis and its Applications to Processing Images and Signals, Ed. by Ya. A. Furman (Fizmatlit, Moscow, 2002) [in Russian].
  5. Ageev,A. L. and Antonova, T. V. “On a New Class of Ill-Posed Problems”, Izv. Ural. Gos. Univ., Mat. Mekh. Inform, 58(11), 24–42 (2008) [in Russian].
  6. Tikhonov, A. N. and Arsenin, V. Ya. Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1974) [in Russian].
  7. Ivanov, V. K., Vasin, V. V., and Tanana, V. P. The Theory of Linear Ill-Posed Problems and Their Applications (Nauka, Moscow, 1978) [in Russian].
  8. VasinV. V. and AgeevA. L. Ill-Posed Problems with a Priori Information (VSP, Utrecht, theNetherlands, 1995).
  9. Antonova, T. V. “Recovery of a FunctionWith a Finite Number of Discontinuities of the First Kind via Noisy Data”, RussianMathematics 45, No. 7, 63–66 (2001).
  10. Antonova T. V. “Solving Equations of the First Kind on Classes of Functions with Discontinuities”, Proc. Steklov Inst. Math., Mathematical Programming. Regularization and Approximation, Suppl. 1, 145–189 (2002).
  11. Ageev, A. L. and Antonova, T. V. “Problem on Separation of Singularities”, RussianMathematics 51, No. 11, 1–7 (2007).
  12. Ageev, A. L. and Antonova, T. V. “Regularizing Algorithms for Detecting Discontinuities in Ill-Posed Problems”, Zh. Vychisl. Mat. Mat. Fiz. 48, No. 8, 1362–1370 (2008).
  13. Antonova, T. V. “New Methods for Localizing Discontinuities of a Noisy Function”, Numer. Analysis Appl. 3, No. 4, 306–316 (2010).
  14. Ageev, A. L. and Antonova, T. V. “On Ill-Posed Problems of Localization of Discontinuities”, Tr. Inst.Mat. Mekh. UrO RAN 17, No. 3, 30–45 (2011) [in Russian].
  15. Ageev, A. L. and Antonova, T. V. “On the Localization of Discontinuities of the First Kind for a Function of Bounded Variation”, Tr. Inst. Mat. Mekh.UrORAN 18, No. 1, 56–68 (2012) [in Russian].
  16. Ageev, A. L. and 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, No. 2, 177–191 (2013)..
  17. Oudshoorn, C. G. M. “Asymptotically Minimax Estimation of a Function with Jumps”, Bernoulli 4 (1), 15–33 (1998).
  18. Korostelev, A. P. “On Minimax Estimation of a Discontinuous Signal”, Theory Probab. Appl. 32, No. 4, 727–730 (1987).