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
HTML articles powered by AMS MathViewer

by Helmut Gfrerer PDF

Math. Comp. 49 (1987), 507-522 Request permission

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.

References

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, Heidelberger Taschenbücher, Band 30, Springer-Verlag, Berlin-New York, 1968 (German). Dritte Auflage. 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 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 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 10.1016/0021-9045(87)90113-4
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 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

Improperly Posed Problems and Their Numerical Treatment

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 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 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 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 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

Generalized Inverses and Applications

Improperly Posed Problems and Their Numerical Treatment

E. Schock, On the asymptotic order of accuracy of Tikhonov regularization, J. Optim. Theory Appl. 44 (1984), no. 1, 95–104. MR 764866, DOI 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 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 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