@article {15823,
	title = {Tikhonov Regularization and Total Least Squares},
	journal = {SIAM Journal on Matrix Analysis and Applications},
	volume = {21},
	year = {1999},
	month = {1999///},
	pages = {185 - 194},
	abstract = {Discretizations of inverse problems lead to systems of linear equations with a highly ill-conditioned coefficient matrix, and in order to computestable solutions to these systems it is necessary to apply regularization methods. We show how Tikhonov{\textquoteright}s regularization method, which in its original formulation involves a least squares problem, can be recast in a total least squares formulation suited for problems in which both the coefficient matrix and the right-hand side are known only approximately. We analyze the regularizing properties of this method and demonstrate by a numerical example that, in certain cases with large perturbations, the new method is superior to standard regularization methods.},
	keywords = {bidiagonalization, discrete ill-posed problems, regularization, total least squares},
	doi = {10.1137/S0895479897326432},
	url = {http://link.aip.org/link/?SML/21/185/1},
	author = {Golub, Gene H. and Hansen,Per Christian and O{\textquoteright}Leary, Dianne P.}
}
@article {15800,
	title = {Restoring Images Degraded by Spatially Variant Blur},
	journal = {SIAM Journal on Scientific Computing},
	volume = {19},
	year = {1998},
	month = {1998///},
	pages = {1063 - 1082},
	abstract = {Restoration of images that have been blurred by the effects of a Gaussian blurring function is an ill-posed but well-studied problem. Any blur that is spatially invariant can be expressed as a convolution kernel in an integral equation. Fast and effective algorithms then exist for determining the original image by preconditioned iterative methods. If the blurring function is spatially variant, however, then the problem is more difficult. In this work we develop fast algorithms for forming the convolution and for recovering the original image when the convolution functions are spatially variant but have a small domain of support. This assumption leads to a discrete problem involving a banded matrix. We devise an effective preconditioner and prove that the preconditioned matrix differs from the identity by a matrix of small rank plus a matrix of small norm. Numerical examples are given, related to the Hubble Space Telescope (HST) Wide-Field/Planetary Camera. The algorithms that we develop are applicable to other ill-posed integral equations as well.},
	keywords = {convolution, discrete ill-posed problems, first-kind integral equations, image restoration, regularization, spatially variant point spread function},
	doi = {10.1137/S106482759528507X},
	url = {http://link.aip.org/link/?SCE/19/1063/1},
	author = {Nagy,James G. and O{\textquoteright}Leary, Dianne P.}
}
@article {15796,
	title = {Regularization by Truncated Total Least Squares},
	journal = {SIAM Journal on Scientific Computing},
	volume = {18},
	year = {1997},
	month = {1997///},
	pages = {1223 - 1241},
	abstract = {The total least squares (TLS) method is a successful method for noise reduction in linear least squares problems in a number of applications. The TLS method is suited to problems in which both the coefficient matrix and the right-hand side are not precisely known. This paper focuses on the use of TLS for solving problems with very ill-conditioned coefficient matrices whose singular values decay gradually (so-called discrete ill-posed problems), where some regularization is necessary to stabilize the computed solution. We filter the solution by truncating the small singular values of the TLS matrix. We express our results in terms of the singular value decomposition (SVD) of the coefficient matrix rather than the augmented matrix. This leads to insight into the filtering properties of the truncated TLS method as compared to regularized least squares solutions. In addition, we propose and test an iterative algorithm based on Lanczos bidiagonalization for computing truncated TLS solutions.},
	keywords = {bidiagonalization, discrete ill-posed problems, regularization, total least squares},
	doi = {10.1137/S1064827594263837},
	url = {http://link.aip.org/link/?SCE/18/1223/1},
	author = {Fierro,R. D. and Golub, G. H and Hansen,P. C. and O{\textquoteright}Leary, Dianne P.}
}
