Abstract
The problem of detecting singularities (discontinuities of the first kind) of a noisy function in L 2 is considered. A wide class of regularizing algorithms that can detect discontinuities is constructed. New estimates of accuracy of determining the location of discontinuities are obtained and their optimality in terms of order with respect to the error level δ is proved for some classes of functions with isolated singularities. New upper bounds for the singularity separation threshold are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Tichonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1974; Winston, Washington, 1977).
V. K. Ivanov, V. V. Vasin, and V. P. Tanana, The Theory of Linear Ill-Posed Problems and Its Applications (Nauka, Moscow, 1978) [in Russian].
V. V. Vasin and A. L. Ageev, Ill-Posed Problems with a Priori Information (VSP, Utrecht, Netherlands, 1995).
A. L. Ageev and T. V. Antonova, “On Solution of Nonlinear with Respect to Parameter Equation of the First Kind on the Class of Discontinuous Functions,” J. Inverse Ill-Posed Probl. 7(1), 1–16 (1999).
A. L. Ageev, T. V. Antonova, T. E. Raikh, et al., “The Method of Separating Functionals in the Interpretation of Local Atomic Structure,” Mat. Modelir. 16(10), 81–92 (2004).
A. L. Ageev and M. E. Korshunov, “Image Processing and Detection of Object Location,” in Trudy Mezhdunarodnogoj seminara: Teoriya upravleniya i teoriya obobshchennykh reshenii uravnenii Gamil’tona-Yakobi, Yekaterinburg (Proc. of the Int. Workshop on the Theory of Generalized Solutions of Hamilton-Jacobi Equations) 2005 (Yekaterinburg, 2006), Vol. 2, pp. 61–65 [in Russian].
T. V. Antonova, “Recovery of Functions with a Finite Number of Simple Discontinuities from Noisy Data,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 7, 65–68 (2001).
T. V. Antonova, “Recovery of Function with Finite Number of Discontinuities by Noised Data,” J. Inverse Ill-Posed Probl. 10(2), 1–11 (2002).
T. V. Antonova, “Solution of Equations of the First Kind on Classes of Discontinuous Functions,” in Probl. teor. i prikl. matem. Trudy XXXI Regional’noi molodezhnoi konferentsii po problemam teoreticheskoi i prikladnoi matematiki, Yekaterinburg (Proc. of the XXXI Regional Youth Conf. on Theoretical and Applied Mathematics), 2000 (Ural Otd. Ross. Akad. Nauk, Yekaterinburg, 2000), pp. 30–31 [in Russian].
T. V. Antonova, “On the Solution of Integral Equations of the First Kind Nonlinear in Parameter in Classes of Distributions,” Zh. Vychisl. Mat. Mat. Fiz. 40, 819–831 (2000) [Comput. Math. Math. Phys. 40, 781–792 (2000)].
T. V. Antonova, Solution of Nonlinear Equations of the First Kind on on Classes of Discontinuous Functions, Available from VINITI, No. 2639, 2000.
T. V. Antonova, “Solving Equations of the First Kind on Classes of Functions with Singularities,” Proc. of the Steklov Inst. Math, Suppl. 1, 145–189 (2002).
S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, 1999; Mir, Moscow, 2005).
V. Yu. Terebizh, Introduction to the Statistical Theory of Inverse Problems (Fizmatlit, Moscow, 2005) [in Russian].
V. P. Kozlov, “On the Resolution of Spectral Devices: I. Statement of the Problem and Resolution Criteria,” Opt. Spektrosk. 16, 501–506 (1964).
V. P. Kozlov, “On the Resolution of Spectral Devices: II. Generalized Rsolving Power of a Spectral Device,” Opt. Spektrosk. 17, 278–283 (1964).
Author information
Authors and Affiliations
Additional information
Original Russian Text © A.L. Ageev, T.V. Antonova, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 8, pp. 1362–1370.
Rights and permissions
About this article
Cite this article
Ageev, A.L., Antonova, T.V. Regularizing algorithms for detecting discontinuities in ill-posed problems. Comput. Math. and Math. Phys. 48, 1284–1292 (2008). https://doi.org/10.1134/S0965542508080034
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542508080034