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Optimal stable solution of Cauchy problems for elliptic equations. (English) Zbl 0865.65076
The author considers ill-posed Cauchy problems for elliptic partial differential equations with linear densely defined selfadjoint positive definite operators. It is assumed that the initial data for the solution and its time derivative are noisy with a prescribed level of accuracy. Under some assumptions about the smoothness of the solution the conditions for best possible accuracy for identifying the solution from the noisy data are obtained. It is shown that the best possible accuracy depends either in a Hölder continuous way or in a logarithmic form on the noise level. Next, the author considers a special generalization of Tikhonov regularization methods for the above problems which permit one to attain optimal error bounds.

MSC:
65N15Error bounds (BVP of PDE)
35R25Improperly posed problems for PDE
35J25Second order elliptic equations, boundary value problems
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Full Text: DOI
References:
[1] Baumeister, J.: Stable Solution of Inverse Problems. Braunschweig: Vieweg 1987. · Zbl 0623.35008
[2] Engi, H. W.: Regularization methods for the stable solution of inverse problems. Surv. Math. Ind. 3 (1993), 71 - 143. · Zbl 0776.65043
[3] Groetsch, C. W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Boston: Pitman 1984. · Zbl 0545.65034
[4] Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Stuttgart: Teubner-Verlag 1986. · Zbl 0609.65065
[5] Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. New Haven: Yale University Press 1923. · Zbl 0049.34805
[6] Hanke, M. and P. C. Hansen: Regularization methods for large-scale problems. Surv. Math. Ind. 3 (1993), 253 - 315. · Zbl 0805.65058
[7] Hofmann, B.: Regularization for Applied Inverse and Ill-Posed Problems (Teubner-Texte zur Mathematik: Vol. 85). Leipzig: B. G. Teubner Verlagsges. 1986. · Zbl 0606.65038
[8] ivanov, V. K., Vasin, V. V. and V. P. Tanana: Theory of Linear Ill-Posed Problems (in Russian). Moscow: Nauka 1978.
[9] Krein, S. and Y. I. Petunin: Scales of Banach spaces. Russian Math. Surveys 21(1966), 85- 159.
[10] Lavrentiev, M. M.: Some Improperly Posed Problems in Mathematical Physics. Berlin: Springer-Verlag 1967. · Zbl 0149.41902
[11] Louis, A. K.: Inverse und schlecht gestellte Probleme. Stuttgart: Teubner-Verlag 1989. · Zbl 0667.65045
[12] Mair, B. A.: Tikhonov regularization for finitely and infinitely smoothing operators. SIAM J. Math. Anal. 25 (1994), 135 - 147. · Zbl 0819.65141 · doi:10.1137/S0036141092238060
[13] Schröter, T. and U. Tautenhahn: On the optimality of regularization methods for solving linear ill-posed problems. Z. Anal. Anw.. 13 (1994), 697 - 710. · Zbl 0826.65051
[14] Schröter, T. and U. Tautenhahn: On optimal regularization methods for the backward heat equation. Z. Anal. Anw. 15 (1996), 475 - 493. · Zbl 0848.65044 · doi:10.4171/ZAA/711
[15] Seidman, T. I.: ’Optimal Filtering’ for some ill-posed problems. In: Wave Propagation and Inversion (eds.: W. Fitzgibbon and M. Wheeler). Philadelphia: SIAM 1992, pp. 108 - 123. · Zbl 0781.35069
[16] Tautenhahn, U.: Optimality for linear ill-posed problems under general source conditions. Preprint IP 5/1996. Techn. Univ. Chem nitz-Zwickau (submitted). · Zbl 0907.65049 · doi:10.1080/01630569808816834
[17] Tikhonov, A. N. and V. Y. Arsenin: Solution of Ill-Posed Problems. New York: Wiley 1977.
[18] Vainikko, C. M.: On the optimality of methods for ill-posed problems. Z. Anal. Anw. 6 (1987), 351 - 362. · Zbl 0632.65061
[19] Vainikko, G. M. and A. Y. Veretennikov: Iteration Procedures in Ill-Posed Problems (in Russian). Moscow: Nauka 1986. Receiced 19.04.1996