×

Fractional regularization matrices for linear discrete ill-posed problems. (English) Zbl 1360.65113

Summary: The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices \(A^{\mathrm T}A\) (for Tikhonov regularization) and \(A\) (for Lavrentiev regularization) as regularization matrices, where \(A\) is the matrix that defines the linear discrete ill-posed problem. Both small- and large-scale problems are considered.

MSC:

65F22 Ill-posedness and regularization problems in numerical linear algebra
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems. Kluwer, Dordrecht · Zbl 0859.65054
[2] Groetsch CW (1984) The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, Boston · Zbl 0545.65034
[3] Hansen PC (2007) Regularization tools version 4.0 for Matlab 7.3. Numer Algorithms 46:189-194 · Zbl 1128.65029
[4] Klann E, Ramlau R (2008) Regularization by fractional filter methods and data smoothing. Inverse Probl 24:025018 · Zbl 1141.47009
[5] Girosi F, Jones M, Poggio T (1995) Regularization theory and neural network architecture. Neural Comput 7:219-269
[6] Liang ZP, Lauterbur PC (1994) An efficient method for dynamic magnetic resonance imaging. IEEE Trans Med Imaging 13:677-686
[7] Morigi S, Reichel L, Sgallari F (2006) An iterative Lavrentiev regularization method. BIT 46:589-606 · Zbl 1107.65038
[8] Hochstenbach ME, Reichel L (2011) Fractional Tikhonov regularization for linear discrete ill-posed problems. BIT 51:197-215 · Zbl 1215.65075
[9] Groetsch CW, Guacamene J (1987) Arcangeli’s method for Fredholm equations of the first kind. Proc Am Math Soc 99:256-260 · Zbl 0624.65132
[10] Calvetti D, Reichel L (2004) Tikhonov regularization with a solution constraint. SIAM J Sci Comput 26:224-239 · Zbl 1081.65033
[11] Hochstenbach ME, McNinch N, Reichel L (2012) Discrete ill-posed least-squares problems with a solution norm constraint. Linear Algebra Appl 436:3801-3818 · Zbl 1237.65039
[12] Lampe J, Rojas M, Sorensen DC, Voss H (2011) Accelerating the LSTRS algorithm. SIAM J Sci Comput 33:175-194 · Zbl 1368.65096
[13] Rojas M, Sorensen DC (2002) A trust-region approach to regularization of large-scale discrete forms of ill-posed problems. SIAM J Sci Comput 23:1842-1860 · Zbl 1006.86004
[14] Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, Baltimore · Zbl 0865.65009
[15] Hill RO Jr, Parlett BN (1992) Refined interlacing properties. SIAM J Matrix Anal Appl 13:239-247 · Zbl 0747.15004
[16] Baart ML (1982) The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned linear least-squares problems. IMA J Numer Anal 2:241-247 · Zbl 0484.65021
[17] Phillips DL (1962) A technique for the numerical solution of certain integral equations of the first kind. J ACM 9:84-97 · Zbl 0108.29902
[18] Wing GM (1991) A primer on integral equations of the first kind. SIAM, Philadelphia · Zbl 0753.45001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.