The minimal error conjugate gradient method is a regularization method
HTML articles powered by AMS MathViewer
- by Martin Hanke
- Proc. Amer. Math. Soc. 123 (1995), 3487-3497
- DOI: https://doi.org/10.1090/S0002-9939-1995-1285994-5
- PDF | Request permission
Abstract:
The regularizing properties of the conjugate gradient iteration, applied to the normal equation of a linear ill-posed problem, were established by Nemirovskii in 1986. A seemingly more attractive variant of this algorithm is the minimal error method suggested by King. The present paper analyzes the regularizing properties of the minimal error method. It is shown that the discrepancy principle is no regularizing stopping rule for the minimal error method. Instead, a different stopping rule is suggested which leads to order-optimal convergence rates.References
- T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
- T. S. Chihara, Orthogonal polynomials and measures with end point masses, Rocky Mountain J. Math. 15 (1985), no. 3, 705–719. MR 813269, DOI 10.1216/RMJ-1985-15-3-705
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York-London, 1965. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin; Translated from the Russian by Scripta Technica, Inc; Translation edited by Alan Jeffrey. MR 0197789
- C. W. Groetsch, Generalized inverses of linear operators: representation and approximation, Monographs and Textbooks in Pure and Applied Mathematics, No. 37, Marcel Dekker, Inc., New York-Basel, 1977. MR 0458859 —, The theory of Tikhonov regularization for Fredholm equations of the first kind, Pitman, Boston, London, and Melbourne, 1984.
- Martin Hanke, Accelerated Landweber iterations for the solution of ill-posed equations, Numer. Math. 60 (1991), no. 3, 341–373. MR 1137198, DOI 10.1007/BF01385727
- Eberhard Schock (ed.), Beiträge zur angewandten Analysis und Informatik, Verlag Shaker, Aachen, 1994. Helmut Brakhage zu Ehren. [Dedicated to Helmut Brakhage on the occasion of his retirement]. MR 1287166
- J. T. King, A minimal error conjugate gradient method for ill-posed problems, J. Optim. Theory Appl. 60 (1989), no. 2, 297–304. MR 984986, DOI 10.1007/BF00940009
- Alfred Karl Louis, Inverse und schlecht gestellte Probleme, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], B. G. Teubner, Stuttgart, 1989 (German). MR 1002946, DOI 10.1007/978-3-322-84808-6 A. S. Nemirovskii, The regularization properties of the adjoint gradient method in ill-posed problems, USSR Comput. Math. and Math. Phys. 26 (1986), no. 2, 7-16.
- G. M. Vaĭnikko and A. Yu. Veretennikov, Iteratsionnye protsedury v nekorrektnykh zadachakh, “Nauka”, Moscow, 1986 (Russian). MR 859375
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 3487-3497
- MSC: Primary 65J10; Secondary 47A50, 65J20
- DOI: https://doi.org/10.1090/S0002-9939-1995-1285994-5
- MathSciNet review: 1285994