Iterative processes of the Fej\'er type in ill-posed problems with a~prori information

V. V. Vasin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2009), pp. 3-24.

Résumé

In the latter thirty years, the solution of ill-posed problems with a priori information formed a separate field of research in the theory of ill-posed problems. We mean the class of problems, where along with the basic equation one has some additional data on the desired solution. Namely, one states some relations and constraints which contain important information on the object under consideration. As a rule, taking into account these data in a solution algorithm, one can essentially increase its accuracy for solving ill-posed (unstable) problems. It is especially important in the solution of applied problems in the case when a solution is not unique, because this approach allows one to choose a solution that meets the reality. In this paper we survey the methods for solving such problems. We briefly describe all relevant approaches (known to us), but we pay the main attention to the method proposed by us. This technique is based on the application of iterative processes of the Fejér type which admit a flexible and effective realization for a wide class of a priori constraints.

Détail

Keywords: ill-posed problem, Fejér mapping, iterative process, pseudo-contractive operator, regularizing algorithm.
Mots-clés : a priori information

Bibliographie

[1] Lavrentev M. M., “Ob integralnykh uravneniyakh pervogo roda”, DAN SSSR, 127:1 (1959), 31–33 | MR

[2] Lavrentev M. M., O nekotorykh nekorrektnykh zadachakh matematicheskoi fiziki, Izd-vo SO AN SSSR, Novosibirsk, 1962, 92 pp. | MR

[3] Ivanov V. K., “O lineinykh nekorrektnykh zadachakh”, DAN SSSR, 145:2 (1962), 270–272 | MR | Zbl

[4] Ivanov V. K., “O nekorrektno postavlennykh zadachakh”, Matem. sb., 61(103):2 (1963), 211–223 | MR | Zbl

[5] Tikhonov A. N., “O regulyarizatsii nekorrektno postavlennykh zadach”, DAN SSSR, 153:1 (1963), 49–52 | MR | Zbl

[6] Tikhonov A. N., “O reshenii nekorrektno postavlennykh zadach i metode regulyarizatsii”, DAN SSSR, 151:3 (1963), 501–504 | MR | Zbl

[7] Tikhonov A. N., “Ob ustoichivosti obratnykh zadach”, DAN SSSR, 39:5 (1944), 195–198

[8] Tikhonov A. N., Goncharskii A. V., Stepanov V. V., Yagola A. G., Regulyarizuyuschie algoritmy i apriornaya informatsiya, Nauka, M., 1983, 198 pp. | MR | Zbl

[9] Vasin V. V., Ageev A. L., Nekorrektnye zadachi s apriornoi informatsiei, UIF Nauka, Ekaterinburg, 1993, 263 pp. | MR

[10] Morozov V. A., Goldman N. P., Samarin M. K., “Metod deskriptivnoi regulyarizatsii i kachestvo priblizhennykh reshenii”, Inzh. fiz. zhurn., 38:6 (1977), 117–121

[11] Tikhonov A. N., Vasilev F. P., “Metody resheniya nekorrektnykh ekstremalnykh zadach”, Math. models and numerical methods, Banach center publ., 3, Warszava, 1978, 297–342 | MR | Zbl

[12] Acar R., Vogel C. R., “Analysis of bounded variation penalty method for ill-posed problems”, Inverse Problems, 10 (1994), 1217–1229 | DOI | MR | Zbl

[13] Chavent E., Kunish K., “Regularization of linear least squares problems by total bounded variation control”, Optimization and calculus of variations, 2 (1997), 359–376 | DOI | MR | Zbl

[14] Vogel C. R., Computational methods for inverse problems, SIAM, Philadelphia, 2002, 183 pp. | MR | Zbl

[15] Vasin V. V., “Iteratsionnye metody resheniya nekorrektnykh zadach s apriornoi informatsiei v gilbertovykh prostranstvakh”, Zhurn. vychisl. matem. i matem. fiz., 28:7 (1988), 971–980 | MR | Zbl

[16] Vasin V. V., Eremin I. I., Operatory i iteratsionnye protsessy feierovskogo tipa. Teoriya i prilozheniya, In-t kompyut. issled., RKhD, Moskva–Izhevsk, 2005, 199 pp.

[17] Vinokurov V. A., “Ob odnom neobkhodimom uslovii regulyarizuemosti po Tikhonovu”, DAN SSSR, 195:3 (1970), 530–531 | MR | Zbl

[18] Menikhes L. D., “O regulyarizuemosti otobrazhenii, obratnykh k integralnym operatoram”, DAN SSSR, 241:2 (1978), 625–629 | MR

[19] Bakushinskii A. B., Goncharskii A. V., Iterativnye metody resheniya nekorrektnykh zadach, Nauka, M., 1989, 128 pp. | MR

[20] Vasin V. V., Proksimalnyi algoritm s proektirovaniem v zadachakh vypuklogo programmirovaniya, Preprint, AN SSSR. Uralsk. nauchn. tsentr, In-t matem. i mekhan., Sverdlovsk, 1982, 47 pp.

[21] Eremin I. I., “Obobschenie relaksatsionnogo metoda Motskina–Agmona”, UMN, 20:2 (1965), 183–187 | MR | Zbl

[22] Eremin I. I., “O sistemakh neravenstv s vypuklymi funktsiyami v levykh chastyakh”, Izv. AN SSSR. Ser. matem., 30:2 (1966), 265–278 | MR | Zbl

[23] Eremin I. I., “Metody feierovskikh priblizhenii v vypuklom programmirovanii”, Matem. zametki, 3:2 (1968), 217–234 | MR

[24] Moreau J. J., “Proximite et dualite dans un espace Hilbertien”, Bull. Soc. Math. France, 93:2 (1965), 273–299 | MR | Zbl

[25] Martinet B., “Determination approachee d'un point fixe d'une application pseudo-contractante. Cas de l'application prox”, C. R. Acad. Sci. Paris Ser. A–B, 274 (1972), A163–A165 | MR

[26] Maruster S., “Quasi-nonexpansivity and two classical methods for solving nonlinear equations”, Proc. Amer. Math. Soc., 62:1 (1977), 119–123 | DOI | MR | Zbl

[27] Vasin V. V., Diskretizatsiya, iteratsionno-approksimatsionnye algoritmy resheniya neustoichivykh zadach i ikh prilozheniya, Dis. $\dots$ dokt. fiz.-matem. nauk, VTs SO RAN, Novosibirsk, 1985

[28] Fejér L., “Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen”, Math. Ann., 85:1 (1922), 41–48 | DOI | MR | Zbl

[29] Motzkin T. S., Schoenberg J. J., “The relaxation method for linear inequalities”, Canad. J. Math., 6:3 (1954), 393–404 | MR | Zbl

[30] Kaczmarz S., “Angenaherte Auflosung von Systemen linearer Gleichungen”, Bull. Inst. Acad. Pol. Sci. Leet., 35:4 (1937), 355–357

[31] Genel A., Lindenstrauss L., “An example concerning fixed point”, Israel J. Math., 22:1 (1975), 81–86 | DOI | MR | Zbl

[32] Rockafellar R. T., “Monotone operators and the proximal point algorithm”, SIAM J. Control and Optim., 14:5 (1976), 877–898 | DOI | MR | Zbl

[33] Halperin B., “Fixed points of nonexpansive maps”, Bull. Amer. Math. Soc., 73:6 (1967), 957–961 | DOI | MR

[34] Neubauer A., Scherzer O. A., “A convergence rate results for a steepest desceut method and minimal error method for the solution of nonlinear ill-posed problems”, J. Anal. Appl., 14:2 (1995), 369–378 | MR

[35] Scherzer O., “Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems”, J. Math. Anal. Appl., 194 (1995), 911–933 | DOI | MR | Zbl

[36] Engl H. W., Hanke M., Neubauer A., Regularization of inverse problems, Kluwer Acad. Publ., Dordrecht ets., 1996, 321 pp. | MR | Zbl

[37] Gilyazov S. F., Gol'dman N. L., Regularization of ill-posed problems by iteration methods, Kluwer Acad. Publ., Dordrecht ets., 2000, 340 pp. | MR | Zbl

[38] Kabanikhin S. I., Kowar R., Scherzer O., “On the Landweber iteration for the solution of a parameter identification problem in a hyperbolic partial differential equation of second order”, J. inv. ill-posed problems, 6:5 (1998), 403–430 | MR | Zbl

[39] Ageev A. L., Babanov Yu. A., Vasin V. V., Ershov N. V., “Postroenie regulyarizuyuschikh algoritmov po opredeleniyu struktury amorfnykh tel metodom rentgenospektralnogo strukturnogo analiza”, Chislennye i analitich. metody resheniya zadach mekhan. sploshnoi sredy, Sverdlovsk, 1981, 3–25 | MR | Zbl

[40] Ageev A. L., Vasin V. V., Bessonova E. N., Markushevich V. M., “Radiozondirovanie atmosfery na dvukh chastotakh. Algoritmicheskii analiz integralnogo uravneniya Fredgolma–Stiltesa”, Teor. problemy v geofizike, Vychisl. seismologiya, 29, Nauka, M., 1997, 100–118

[41] Giusti E., Minimal surfaces and functions of bounded variation, Birkhäuser, Basel, 1984, 240 pp. | MR | Zbl

[42] Vasin V. V., “Regularization and iterative approximation for linear ill-posed problems in the space of functions of bounded variation”, Proc. Stekl. Inst. Math., Suppl. 1 (2002), S225–S229 | MR

[43] Vasin V. V., Serezhnikova T. I., “Dvukhetapnyi metod approksimatsii negladkikh reshenii i vosstanovlenie zashumlennogo izobrazheniya”, Avtomatika i telemekhan., 2004, no. 3, 392–402

[44] Leonov A. S., “Kusochno-ravnomernaya regulyarizatsiya dvumernykh nekorrektnykh zadach s razryvnymi resheniyami: chislennyi analiz”, Zhurn. vychisl. matem. i matem. fiz., 39:12 (1999), 1939–1944 | MR

[45] Vasin V. V., Gribanov K. G., Zakharov V. I. et al., “Regular methods of solution of ill-posed problems for remote sensing of Earth atmosphere using high-resolution spectrometry”, Proc. SPIE, V. 6580 (Dec. 12, 2006), 2006, 65800T

[46] Gribanov K. G., Zakharov V. I., Vasin V. V., “Iterativnaya regulyarizatsiya v zadache opredeleniya $\mathrm{CO}_2$ v atmosfere po dannym sputnikovogo zondirovaniya”, Tez. dokl. Mezhdun. konf. “Algoritmicheskii analiz neustoichivykh zadach” (01–06 sent. 2008, Ekaterinburg), 119–120