# 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

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topM. 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] 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
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- [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.
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- [12] B. V. Limaye, Spectral Perturbation and Approximation with Numerical Experiments, Proc. Centre for Math. Anal. Australian National Univ. 13, 1987. Zbl0647.47018
- [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] 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] 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] 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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