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Regularization parameter estimation for large-scale Tikhonov regularization using a priori information

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  • Renaut, Rosemary A.
  • Hnetynková, Iveta
  • Mead, Jodi

Abstract

Solutions of numerically ill-posed least squares problems for by Tikhonov regularization are considered. For , the Tikhonov regularized least squares functional is given by where matrix W is a weighting matrix and is given. Given a priori estimates on the covariance structure of errors in the measurement data , the weighting matrix may be taken as which is the inverse covariance matrix of the mean 0 normally distributed measurement errors in . If in addition is an estimate of the mean value of , and [sigma] is a suitable statistically-chosen value, J evaluated at its minimizer approximately follows a [chi]2 distribution with degrees of freedom. Using the generalized singular value decomposition of the matrix pair , [sigma] can then be found such that the resulting J follows this [chi]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 is not available, but instead a set of measurement data provides an estimate of the mean value of . 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 [sigma] 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.

Suggested Citation

  • Renaut, Rosemary A. & Hnetynková, Iveta & Mead, Jodi, 2010. "Regularization parameter estimation for large-scale Tikhonov regularization using a priori information," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3430-3445, December.
  • Handle: RePEc:eee:csdana:v:54:y:2010:i:12:p:3430-3445
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    References listed on IDEAS

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    1. Obereder, Andreas & Scherzer, Otmar & Kovac, Arne, 2007. "Bivariate density estimation using BV regularisation," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5622-5634, August.
    2. Qiu, Peihua, 2008. "A nonparametric procedure for blind image deblurring," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4828-4841, June.
    3. Benito, Monica & Pena, Daniel, 2007. "Detecting defects with image data," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6395-6403, August.
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    1. Lika, Konstadia & Augustine, Starrlight & Kooijman, Sebastiaan A.L.M., 2020. "The use of augmented loss functions for estimating dynamic energy budget parameters," Ecological Modelling, Elsevier, vol. 428(C).

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