### Abstract

Solutions of numerically ill-posed least squares problems Ax ≈ b for A ∈ R^{mxn} by Tikhonov regularization are considered. For D ∈ R^{pxn}, the Tikhonov regularized least squares functional is given by J(σ) = \\Ax - b||^{2} _{w} + 1/σ^{2} ||D(x - x_{0}) \\^{2} _{2}where matrix W is a weighting matrix and X_{0} is given. Given a priori estimates on the covariance structure of errors in the measurement data b, the weighting matrix may be taken as W = W_{b} which is the inverse covariance matrix of the mean 0 normally distributed measurement errors e in b. If in addition X_{0} is an estimate of the mean value of x, and σ is a suitable statistically-chosen value, J evaluated at its minimizer x(σ) approximately follows a x ^{2} distribution with m̃ = m + p-n degrees of freedom. Using the generalized singular value decomposition of the matrix pair [W_{b} ^{1/2}AD], σ can then be found such that the resulting J follows this x^{2} distribution. But the use of an algorithm which explicitly relies on the direct solution of the problem obtained using the generalized singular value decomposition is not practical for large-scale problems. Instead an approach using the Golub-Kahan iterative bidiagonalization of the regularized problem is presented. The original algorithm is extended for cases in which X_{0} is not available, but instead a set of measurement data provides an estimate of the mean value of b. The sensitivity of the Newton algorithm to the number of steps used in the Golub-Kahan iterative bidiagonalization, and the relation between the size of the projected subproblem and a are discussed. Experiments presented contrast the efficiency and robustness with other standard methods for finding the regularization parameter for a set of test problems and for the restoration of a relatively large real seismic signal. An application for image deblurring also validates the approach for large-scale problems. It is concluded that the presented approach is robust for both small and large-scale discretely ill-posed least squares problems.

Original language | English (US) |
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Pages (from-to) | 3430-3445 |

Number of pages | 16 |

Journal | Computational Statistics and Data Analysis |

Volume | 54 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2010 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Computational Statistics and Data Analysis*,

*54*(12), 3430-3445. https://doi.org/10.1016/j.csda.2009.05.026