A note on the modulus of continuity for ill-posed problems in Hilbert space

Bernd Hofmann   ; Peter Mathé

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 1, pp. 34-41

Résumé

The authors study linear ill-posed operator equations in Hilbert space. Such equations become conditionally well-posed by imposing certain smoothness assumptions, often given relative to the operator which governs the equation. Usually this is done in terms of general source conditions. Recently smoothness of an element was given in terms of properties of the distribution function of this element with respect to the self-adjoint associate of the underlying operator. In all cases the original ill-posed problem becomes well-posed, and properties of the corresponding modulus of continuity are of interest, specifically whether this is a concave function. The authors extend previous concavity results of a function related to the modulus of continuity, and obtained for compact operators in B. Hofmann, P. Mathé, and M. Schieck, Modulus of continuity for conditionally stable ill-posed problems in Hilbert space, J. Inverse Ill-Posed Probl. 16 (2008), no. 6, 567–585, to the general case of bounded operators in Hilbert space, and for recently introduced smoothness classes.

Détail

Keywords: ill-posed, individual smoothness, modulus of continuity.
Mots-clés : source conditions

Bibliographie

[1] Engl H. W., Hanke M., Neubauer A., Regularization of inverse problems, Math. and its Appl., 375, Kluwer Acad. Publ. Group, Dordrecht, 1996, 321 pp. | MR | Zbl

[2] Flemming J., Hofmann B., Math P., “Sharp converse results for the regularization error using distance functions”, Inverse Problems, 27:2 (2011), 025006, 18 pp. | DOI | MR | Zbl

[3] Hofmann B., Fleischer G., “Stability rates for linear ill-posed problems with compact and noncompact operators”, Z. Anal. Anwendungen., 18:2 (1999), 267–286 | MR | Zbl

[4] Hofmann B., Kindermann S., “On the degree of ill-posedness for linear problems with non-compact operators”, Methods Appl. Anal., 17:4 (2010), 445–462 | MR | Zbl

[5] Hofmann B., Mathé P., “Analysis of profile functions for general linear regularization methods”, SIAM J. Numer. Anal., 45:3 (2007), 1122–1141 | DOI | MR | Zbl

[6] Hofmann B., Mathé P., Schieck M., “Modulus of continuity for conditionally stable ill-posed problems in Hilbert space”, J. Inverse Ill-Posed Probl., 16:6 (2008), 567–585 | DOI | MR | Zbl

[7] Hofmann B., von Wolfersdorf L., “Some results and a conjecture on the degree of ill-posedness for integration operators with weights”, Inverse Problems, 21:2 (2005), 427–433 | DOI | MR | Zbl

[8] Ivanov V. K., Korolyuk T. I., “Error estimates for solutions of incorrectly posed linear problems”, USSR Comput. Math. Math. Phys., 9:1 (1969), 35–49 | DOI | MR | Zbl

[9] Ivanov V. K., Vasin V. V., Tanana V. P., Theory of linear ill-posed problems and its applications, Inverse and Ill-posed Probl. Ser., VSP, Utrecht, 2002, 218 pp. | MR | Zbl

[10] Lyusternik L. A., Sobolev V. I., Kratkii kurs funktsionalnogo analiza, Vyssh. Shkola, Moscow, 1982, 272 pp. | MR | Zbl

[11] Mathé P., Hofmann B., “Direct and inverse results in variable Hilbert scales”, J. Approx. Theory, 154:2 (2008), 77–89 | DOI | MR | Zbl

[12] Mathé P., Pereverzev S. V., “Geometry of linear ill-posed problems in variable Hilbert scales”, Inverse Problems, 19:3 (2003), 789–803 | DOI | MR | Zbl

[13] Mathé P., Pereverzev S. V., “Discretization strategy for linear ill-posed problems in variable Hilbert scales”, Inverse Problems, 19:6 (2003), 1263–1277 | DOI | MR | Zbl

[14] Nashed M. Z., “A new approach to classification and regularization of ill-posed operator equations”, Inverse and Ill-posed Problems, Notes Rep. Math. Sci. Engrg., 4, Acad. Press, Boston, 1987, 53–75 | MR

[15] Neubauer A., “On converse and saturation results for regularization methods”, Beiträge zur angewandten Analysis und {I}nformatik, Shaker, Aachen, 1994, 262–270 | MR | Zbl

[16] Neubauer A., “On converse and saturation results for Tikhonov regularization of linear ill-posed problems”, SIAM J. Numer. Anal., 34:2 (1997), 517–527 | DOI | MR | Zbl

[17] Reed M., Simon B., Methods of modern mathematical physics, v. I, Functional analysis, Acad. Press, New York, 1980, 400 pp. | MR | Zbl

[18] Rudin W., Functional analysis, McGraw-Hill Ser. in Higher Math., McGraw-Hill Book Co., New York, 1973, 397 pp. | MR | Zbl

[19] Tautenhahn U., Hamarik U., Hofmann B., Shao Y., Conditional stability estimates for illposed PDE problems by using interpolation, Preprint 2011-16, Preprint Ser. of Department of Math., Technische Universitat, Chemnitz, 2011, 40 pp. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-72654

[20] Vasin V. V., “The method of quasi-solutions by Ivanov is the effective method of solving ill-posed problems”, J. Inverse Ill-Posed Probl., 16:6 (2008), 537–552 | DOI | MR | Zbl

[21] Vasin V. V., Ageev A. L., Ill-posed problems with a priori information, Inverse and Ill-Posed Probl. Ser., VSP, Utrecht, 1995, 255 pp. | MR | Zbl

[22] Vasin V. V., Korotkii M. A., “Tikhonov regularization with nondifferentiable stabilizing functionals”, J. Inverse Ill-Posed Probl., 15:8 (2007), 853–865 | DOI | MR | Zbl

[23] Werner D., Funktionalanalysis, Springer-Verlag, Berlin, 2007, 531 pp. | MR