Abstract
We propose a class ofa posteriori parameter choice strategies for Tikhonov regularization (including variants of Morozov's and Arcangeli's methods) that lead to optimal convergence rates toward the minimal-norm, least-squares solution of an ill-posed linear operator equation in the presence of noisy data.
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References
Nashed, M. Z., Editor,Generalized Inverses and Applications, Academic Press, New York, New York, 1976.
Engl, H. W.,Necessary and Sufficient Conditions for Convergence of Regularization Methods for Solving Linear Operator Equations of the First Kind, Numerical Functional Analysis and Optimization, Vol. 3, pp. 201–222, 1981.
Groetsch, C. W.,On the Asymptotic Order of Accuracy of Tikhonov Regularization, Journal of Optimization Theory and Applications, Vol. 41, pp. 293–298, 1983.
Engl, H. W.,On the Convergence of Regularization Methods for Ill-Posed Linear Operator Equations, Improperly-Posed Problems and Their Numerical Treatment, Edited by G. Hämmerlin and K. H. Hoffmann, Birkhäuser, Basel, Switzerland, pp. 81–95, 1983.
Groetsch, C. W.,The Parameter Choice Problem in Linear Regularization, Ill-Posed Problems: Theory and Practice, Edited by M. Z. Nashed (to appear).
Morozov, A. On the Solution of Functional Equations by the Method of Regularization, Soviet Mathematics Doklady, Vol. 7, pp. 414–417, 1966.
Arcangeli, R., Pseudo-Solution de l'Equation Ax=y, Comptes Rendus des Séances de l'Académie des Sciences, Paris, Série A, Vol. 263, pp. 282–285, 1966.
Groetsch, C. W.,The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, Massachusetts, 1984.
Groetsch, C. W.,Comments on Morozov's Discrepancy Principle, Improperly-Posed Problems and Their Numerical Treatment, Edited by G. Hämmerlin and K. H. Hoffmann, Birkhäuser, Basel, Switzerland, pp. 97–104, 1983.
Groetsch, C. W., andSchock, E.,Asymptotic Convergence Rate of Arcangeli's Method for Ill-Posed Problems, Applicable Analysis, Vol. 18, pp. 175–182, 1984.
Schock, E.,Parameter Choice by Discrepancy Principles for the Tikhonov Regularization of Ill-Posed Problems, Integral Equations and Operator Theory, Vol. 7, pp. 895–898, 1984.
Schock, E.,On the Asymptotic Order of Accuracy of Tikhonov Regularization, Journal of Optimization Theory and Applications, Vol. 44, pp. 95–104, 1984.
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Communicated by M. A. Golberg
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Engl, H.W. Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates. J Optim Theory Appl 52, 209–215 (1987). https://doi.org/10.1007/BF00941281
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DOI: https://doi.org/10.1007/BF00941281