An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates

Gfrerer, Helmut

doi:10.1090/S0025-5718-1987-0906185-4

An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates

Author: Helmut Gfrerer
Journal: Math. Comp. 49 (1987), 507-522
MSC: Primary 65J10; Secondary 47A50
DOI: https://doi.org/10.1090/S0025-5718-1987-0906185-4
MathSciNet review: 906185
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We propose an a posteriori parameter choice for ordinary and iterated Tikhonov regularization that leads to optimal rates of convergence towards the best approximate solution of an ill-posed linear operator equation in the presence of noisy data. Numerical examples are given.

Rémy Arcangeli, Pseudo-solution de l’équation $Ax=y$, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A282–A285 (French). MR 203457
R. Courant and D. Hilbert, Methoden der mathematischen Physik. I, Springer-Verlag, Berlin-New York, 1968 (German). Dritte Auflage; Heidelberger Taschenbücher, Band 30. MR 0344038
Heinz W. Engl, Necessary and sufficient conditions for convergence of regularization methods for solving linear operator equations of the first kind, Numer. Funct. Anal. Optim. 3 (1981), no. 2, 201–222. MR 627122, DOI https://doi.org/10.1080/01630568108816087
H. W. Engl, Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates, J. Optim. Theory Appl. 52 (1987), no. 2, 209–215. MR 879198, DOI https://doi.org/10.1007/BF00941281
Heinz W. Engl, On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems, J. Approx. Theory 49 (1987), no. 1, 55–63. MR 870549, DOI https://doi.org/10.1016/0021-9045%2887%2990113-4
Heinz W. Engl and Andreas Neubauer, An improved version of Marti’s method for solving ill-posed linear integral equations, Math. Comp. 45 (1985), no. 172, 405–416. MR 804932, DOI https://doi.org/10.1090/S0025-5718-1985-0804932-1
Gene H. Golub and Charles F. Van Loan, Matrix computations, Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1983. MR 733103
Charles W. Groetsch, Elements of applicable functional analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 55, Marcel Dekker, Inc., New York, 1980. MR 569746
C. W. Groetsch, J. T. King, and D. Murio, Asymptotic analysis of a finite element method for Fredholm equations of the first kind, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 1–11. MR 755337

C. W. Groetsch, "Comments on Morozov’s discrepancy principle," in Improperly Posed Problems and Their Numerical Treatment (G. Hämmerlin and K. H. Hoffmann, eds.), Birkhäuser, Basel, 1983.

C. W. Groetsch, On the asymptotic order of accuracy of Tikhonov regularization, J. Optim. Theory Appl. 41 (1983), no. 2, 293–298. MR 720775, DOI https://doi.org/10.1007/BF00935225
C. W. Groetsch, The theory of Tikhonov regularization for Fredholm equations of the first kind, Research Notes in Mathematics, vol. 105, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 742928
Charles W. Groetsch and Eberhard Schock, Asymptotic convergence rate of Arcangeli’s method for ill-posed problems, Applicable Anal. 18 (1984), no. 3, 175–182. MR 767499, DOI https://doi.org/10.1080/00036818408839519
J. Thomas King and David Chillingworth, Approximation of generalized inverses by iterated regularization, Numer. Funct. Anal. Optim. 1 (1979), no. 5, 499–513. MR 546129, DOI https://doi.org/10.1080/01630567908816031
J. T. Marti, An algorithm for computing minimum norm solutions of Fredholm integral equations of the first kind, SIAM J. Numer. Anal. 15 (1978), no. 6, 1071–1076. MR 512683, DOI https://doi.org/10.1137/0715071
V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl. 7 (1966), 414–417. MR 0208819

M. Z. Nashed (Editor), Generalized Inverses and Applications, Academic Press, New York, 1976.

E. Schock, "Approximate solution of ill-posed equations: arbitrarily slow convergence vs. superconvergence," in Improperly Posed Problems and Their Numerical Treatment (G. Hämmerlin and K. H. Hofmann, eds.), Birkhäuser, Basel, 1983.

E. Schock, On the asymptotic order of accuracy of Tikhonov regularization, J. Optim. Theory Appl. 44 (1984), no. 1, 95–104. MR 764866, DOI https://doi.org/10.1007/BF00934896
Eberhard Schock, Parameter choice by discrepancy principles for the approximate solution of ill-posed problems, Integral Equations Operator Theory 7 (1984), no. 6, 895–898. MR 774730, DOI https://doi.org/10.1007/BF01195873
T. I. Seidman, Nonconvergence results for the application of least-squares estimation to ill-posed problems, J. Optim. Theory Appl. 30 (1980), no. 4, 535–547. MR 572154, DOI https://doi.org/10.1007/BF01686719
J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR 557543