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

0352046 - ÚI 2011 NL eng J - Journal Article
Renaut, R.A. - Hnětynková, Iveta - Mead, J.
Regularization parameter estimation for large-scale Tikhonov regularization using a priori information.
Computational Statistics and Data Analysis. Roč. 54, - (2010), s. 3430-3445. ISSN 0167-9473. E-ISSN 1872-7352
Keywords : ill-posed problems * Tikhonov regularization * chi2-distribution * Golub-Kahan iterative bidiagonalization * hybrid methods * Newton algorithm
Impact factor: 1.089, year: 2010

This paper is concerned with estimating the solutions of numerically ill-posed least squares problems through Tikhonov regularization. Given apriori estimates on the covariance structure of errors in the measurement data b, and a suitable statistically-chosen regularization parameter, the Tikhonov regularized least squares functional J approximately follows a chi2 distribution with M degrees of freedom. Using the generalized singular value decomposition a regularization parameter can then be found such that the resulting J follows this chi2 distribution, see Mead and Renaut (2008). Because the algorithm explicitly relies on the direct solution of the problem obtained using the generalized singular value decomposition it is not practical for large scale problems. Here the approach is extended for large scale problems through the use of the Newton iteration in combination with a Golub-Kahan iterative bidiagonalization of the regularized problem. We show that the presented approach is robust for both small and large scale discretely ill-posed least squares problems.
Permanent Link: http://hdl.handle.net/11104/0191648