Fractional Tikhonov regularization for linear discrete ill-posed problems

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A . This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residu...

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Bibliographic Details
Published in:BIT (Nordisk Tidskrift for Informationsbehandling) Vol. 51; no. 1; pp. 197 - 215
Main Authors: Hochstenbach, Michiel E., Reichel, Lothar
Format: Journal Article Conference Proceeding
Language:English
Published: Dordrecht Springer Netherlands 01.03.2011
Springer
Subjects:
Algebra
Algebraic geometry
Computational Mathematics and Numerical Analysis
Exact sciences and technology
Mathematics
Mathematics and Statistics
Nonlinear algebraic and transcendental equations
Numeric Computing
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Weighted residual norm
65R30
Discrepancy principle
65F10
65F22
Filter function
Solution norm constraint
Fractional Tikhonov
Ill-posed problem
Regularization
Error estimation
Iterative method
Tikhonov regularization
Penalty method
Non linear equation
Direct method
Linear equation
Ill posed problem
Transcendental equation
Numerical linear algebra
Least squares problem
Pseudoinverse
Equation system
Regularization method
Numerical analysis
Linear system
Matrix inversion
overdetermined system
Algebraic equation
ISSN:0006-3835, 1572-9125
Online Access:Get full text
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Summary:Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A . This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of the Moore-Penrose pseudoinverse of AA T as weighting matrix. Properties of this regularization method are discussed. Numerical examples illustrate that the proposed scheme for a suitable fractional power may give approximate solutions of higher quality than standard Tikhonov regularization.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-011-0313-9