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1998 | 69 | 3 | 197-205
Tytuł artykułu

A discrepancy principle for Tikhonov regularization with approximately specified data

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EN
Abstrakty
EN
Many discrepancy principles are known for choosing the parameter α in the regularized operator equation $(T*T + αI)x_α^δ = T*y^δ$, $|y - y^δ| ≤ δ$, 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 $T*y^δ$ are approximated by Aₙ and $zₙ^δ$ 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).
Twórcy
  • Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India
  • Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Germany
Bibliografia
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [8] C. W. Groetsch, The Theory of Regularization for Fredholm Integral Equations of the First Kind, Pitman, London, 1984.
  • [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.
  • [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.
  • [11] R. Kress, Linear Integral Equations, Springer, Heidelberg, 1989.
  • [12] B. V. Limaye, Spectral Perturbation and Approximation with Numerical Experiments, Proc. Centre for Math. Anal. Australian National Univ. 13, 1987.
  • [13] A. Neubauer, An a posteriori parameter choice for Tikhonov regularization in the presence of modelling error, Appl. Numer. Math. 14 (1988), 507-519.
  • [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.
  • [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.
  • [16] E. Schock, On the asymptotic order of accuracy of Tikhonov regularizations, J. Optim. Theory Appl. 44 (1984), 95-104.
  • [17] E. Schock, Parameter choice by discrepancy principle for the approximate solution of ill-posed problems, Integral Equations Operator Theory 7 (1984), 895-898.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-apmv69z3p197bwm
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