A discrepancy principle for Tikhonov regularization with approximately specified data

M. Thamban Nair; Eberhard Schock

Annales Polonici Mathematici (1998)

  • Volume: 69, Issue: 3, page 197-205
  • ISSN: 0066-2216

Abstract

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Many discrepancy principles are known for choosing the parameter α in the regularized operator equation ( T * T + α I ) 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 z ₙ 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).

How to cite

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M. Thamban Nair, and Eberhard Schock. "A discrepancy principle for Tikhonov regularization with approximately specified data." Annales Polonici Mathematici 69.3 (1998): 197-205. <http://eudml.org/doc/270770>.

@article{M1998,
abstract = {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).},
author = {M. Thamban Nair, Eberhard Schock},
journal = {Annales Polonici Mathematici},
keywords = {ill-posed problems; minimal norm least-squares solution; Moore-Penrose inverse; Tikhonov regularization; discrepancy principle; optimal rate; linear ill-posed equations; Hilbert spaces; minimal norm solution},
language = {eng},
number = {3},
pages = {197-205},
title = {A discrepancy principle for Tikhonov regularization with approximately specified data},
url = {http://eudml.org/doc/270770},
volume = {69},
year = {1998},
}

TY - JOUR
AU - M. Thamban Nair
AU - Eberhard Schock
TI - A discrepancy principle for Tikhonov regularization with approximately specified data
JO - Annales Polonici Mathematici
PY - 1998
VL - 69
IS - 3
SP - 197
EP - 205
AB - 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).
LA - eng
KW - ill-posed problems; minimal norm least-squares solution; Moore-Penrose inverse; Tikhonov regularization; discrepancy principle; optimal rate; linear ill-posed equations; Hilbert spaces; minimal norm solution
UR - http://eudml.org/doc/270770
ER -

References

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  2. [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. Zbl0578.65135
  3. [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. Zbl0627.65060
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  13. [13] A. Neubauer, An a posteriori parameter choice for Tikhonov regularization in the presence of modelling error, Appl. Numer. Math. 14 (1988), 507-519. Zbl0698.65032
  14. [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. Zbl0773.65038
  15. [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. Zbl0802.65131
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  17. [17] E. Schock, Parameter choice by discrepancy principle for the approximate solution of ill-posed problems, Integral Equations Operator Theory 7 (1984), 895-898. Zbl0558.47012

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