A discrepancy principle for Tikhonov regularization with approximately specified data
M. Thamban Nair ; Eberhard Schock
Annales Polonici Mathematici, Tome 69 (1998), p. 197-205 / Harvested from The Polish Digital Mathematics Library
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Many discrepancy principles are known for choosing the parameter α in the regularized operator equation (T*T+αI)xαδ=T*yδ, |y-yδ|≤δ, in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and T*yδ are approximated by Aₙ and zₙδ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results are applicable to the regularized projection method as well as to a degenerate kernel method considered by Groetsch (1990).

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:270770 
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M. Thamban Nair; Eberhard Schock. A discrepancy principle for Tikhonov regularization with approximately specified data. Annales Polonici Mathematici, Tome 69 (1998) pp. 197-205. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv69z3p197bwm/

[000] [1] H. W. Engl, Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates, J. Optim. Theory Appl. 52 (1987), 209-215. | Zbl 0586.65045

[001] [2] H. W. Engl and A. Neubauer, An improved version of Marti's method for solving ill-posed linear integral equations, Math. Comp. 45 (1985), 405-416. | Zbl 0578.65135

[002] [3] H. W. Engl and A. Neubauer, Optimal parameter choice for ordinary and iterated Tikhonov regularization, in: Inverse and Ill-Posed Problems, H. W. Engl and C. W. Groetsch (eds.), Academic Press, London, 1987, 97-125. | Zbl 0627.65060

[003] [4] S. George and M. T. Nair, Parameter choice by discrepancy principles for ill-posed problems leading to optimal convegence rates, J. Optim. Theory Appl. 13 (1994), 217-222. | Zbl 0820.47008

[004] [5] S. George and M. T. Nair, On a generalized Arcangeli's method for Tikhonov regularization with inexact data, Numer. Funct. Anal. Optim. 19 (1998), 773-787. | Zbl 0921.65045

[005] [6] H. Gfrerer, Parameter choice for Tikhonov regularization of ill-posed problems, in: Inverse and Ill-Posed Problems, H. W. Engl and C. W. Groetsch (eds.), Academic Press, London, 1987, 27-149. | Zbl 0631.65058

[006] [7] C. W. Groetsch, Comments on Morozov's discrepancy principle, in: Improperly Posed Problems and Their Numerical Treatment, G. Hammerline and K. H. Hoffmann (eds.), Birkhäuser, 1983, 97-104.

[007] [8] C. W. Groetsch, The Theory of Regularization for Fredholm Integral Equations of the First Kind, Pitman, London, 1984. | Zbl 0545.65034

[008] [9] C. W. Groetsch, Convergence analysis of a regularized degenerate kernel method for Fredholm integral equations of the first kind, Integral Equations Operator Theory 13 (1990), 67-75. | Zbl 0696.65094

[009] [10] C. W. Groetsch and J. Guacaneme, Regularized Ritz approximation for Fredholm equations of the first kind, Rocky Mountain J. Math. 15 (1985), 33-37. | Zbl 0598.65031

[010] [11] R. Kress, Linear Integral Equations, Springer, Heidelberg, 1989. | Zbl 0671.45001

[011] [12] B. V. Limaye, Spectral Perturbation and Approximation with Numerical Experiments, Proc. Centre for Math. Anal. Australian National Univ. 13, 1987. | Zbl 0647.47018

[012] [13] A. Neubauer, An a posteriori parameter choice for Tikhonov regularization in the presence of modelling error, Appl. Numer. Math. 14 (1988), 507-519. | Zbl 0698.65032

[013] [14] M. T. Nair, A generalization of Arcangeli's method for ill-posed problems leading to optimal convergence rates, Integral Equations Operator Theory 15 (1992), 1042-1046. | Zbl 0773.65038

[014] [15] M. T. Nair, A unified approach for regularized approximation method for Fredholm integral equations of the first kind, Numer. Funct. Anal. Optim. 15 (1994), 381-389. | Zbl 0802.65131

[015] [16] E. Schock, On the asymptotic order of accuracy of Tikhonov regularizations, J. Optim. Theory Appl. 44 (1984), 95-104. | Zbl 0531.65031

[016] [17] E. Schock, Parameter choice by discrepancy principle for the approximate solution of ill-posed problems, Integral Equations Operator Theory 7 (1984), 895-898. | Zbl 0558.47012