On ill-posed problems of localization of singularities

A. L. Ageev ; T. V. Antonova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 3, pp. 30-45

Résumé

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.

Détail

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

Bibliographie

[1] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1974, 223 pp. | MR | Zbl

[2] Ivanov V. K., Vasin V. V., Tanana V. P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978, 206 pp. | MR

[3] Vasin V. V., Ageev A. L., Ill-posed problems with a priori information, VSP, Utrecht, the Netherlands, 1995, 255 pp. | MR | Zbl

[4] Terebizh V. Yu., “Vosstanovlenie izobrazhenii pri minimalnoi apriornoi informatsii”, Uspekhi fiz. nauk, 165:2 (1995), 143–176

[5] Terebizh V. Yu., Vvedenie v statisticheskuyu teoriyu obratnykh zadach, Fizmatlit, M., 2005, 376 pp.

[6] Dzheims R., Opticheskie printsipy difraktsii rentgenovskikh luchei, IL, M., 1950, 572 pp.

[7] Stern E. A., “Theory of the extended x-ray absorption fine structure techniques”, Phys. Rev. B, 10:8 (1974), 3027–3037 | DOI

[8] Tychinskii V. P., “Mikroskopiya subvolnovykh struktur”, Uspekhi fiz. nauk, 66:11 (1996), 1219–1229

[9] Goncharskii A. V., Cherepaschuk A. M., Yagola A. G., Chislennye metody resheniya obratnykh zadach astrofiziki, Nauka, M., 1978, 336 pp. | MR

[10] Winkler G., Wittich O., Liebsher V., Kempe A., “Dont Shed Tears over Breaks”, Jahresber. Deutsch. Math.-Verein, 107:2 (2005), 57–87 | MR | Zbl

[11] Sizikov V. S., Matematicheskie metody obrabotki rezultatov izmerenii, Politekhnika, SPb., 2001, 240 pp. | MR

[12] Malla S., Veivlety v obrabotke signalov, Mir, M., 2005, 671 pp.

[13] Ya. A.Furman (red.), Vvedenie v konturnyi analiz i ego prilozheniya k obrabotke izobrazhenii i signalov, Fizmatlit, M., 2002, 596 pp.

[14] Kozlov V. P., “O razreshayuschei sposobnosti spektralnykh priborov. I. Postanovka zadachi i kriterii razresheniya”, Optika i spektroskopiya, 16:3 (1964), 501–506

[15] Kozlov V. P., “O razreshayuschei sposobnosti spektralnykh priborov. II. Obobschennaya razreshayuschaya sila spektralnogo pribora”, Optika i spektroskopiya, 17:2 (1964), 278–283

[16] Catarina G. M., “Oudshoorn Asymptotically minimax estimation of a function with jumps”, Bernoulli, 4:1 (1998), 15–33 | DOI | MR

[17] Korostelev A. P., “O minimaksnom otsenivanii razryvnogo signala”, Teoriya veroyatnostei i ee primeneniya, 32:4 (1987), 796–799 | MR | Zbl

[18] Ageev A. L., Antonova T. V., “O novom klasse nekorrektno postavlennykh zadach”, Izv. Ural. gos. un-ta. Matematika. Mekhanika. Informatika, 2008, no. 58(11), 27–45

[19] Antonova T. V., “Solving equations of the first kind on classes of functions with singularities”, Proc. Steklov Inst. Math., 2002, S145–S189 | MR | Zbl

[20] Ageev A. L., Antonova T. V., “O zadache razdeleniya osobennostei”, Izv. vuzov. Matematika, 2007, no. 11, 3–9 | MR

[21] Ageev A. L., Antonova T. V., “Otsenki snizu v zadachakh lokalizatsii osobennostei funktsii”, Problemy teoreticheskoi i prikladnoi matematiki, tr. 39-i Vseros. molodezh. konf., Ekaterinburg, 2008, 56–60

[22] Ageev A. L., Antonova T. V., “Localization algorithms for singularities of solution to convolution equation of the first kind”, J. Inverse Ill-Posed Probl., 16:7 (2008), 639–650 | DOI | MR | Zbl

[23] Ageev A. L., Antonova T. V., “Regulyariziruyuschie algoritmy vydeleniya razryvov v nekorrektnykh zadachakh”, Zhurn. vychisl. matematiki i mat. fiziki, 48:8 (2008), 1362–1370 | MR | Zbl

[24] Antonova T. V., “Regulyariziruyuschie algoritmy lokalizatsii izlomov zashumlennoi funktsii”, Trudy Instituta matematiki i mekhaniki UrO RAN, 15, no. 1, 2009, 44–58 | Zbl

[25] Antonova T. V., “Novye metody lokalizatsii razryvov zashumlennoi funktsii”, Sib. zhurn. vychisl. matematiki, 13:4 (2010), 375–386