REGULARIZATION PARAMETER ESTIMATION FOR LARGE SCALE TIKHONOV REGULARIZATION USING A PRIORI INFORMATION ROSEMARY A RENAUT * , IVETA HNETYNKOVA † , AND JODI MEAD ‡ Abstract. This paper is concerned with estimating the solutions of numerically ill-posed least squares problems through Tikhonov regularization. Given a priori estimates on the covariance structure of errors in the measurement data b, and a suitable statistically-chosen σ, the Tikhonov regularized least squares functional J (σ)= ‖Ax − b‖ 2 W b +1/σ 2 ‖D(x − x 0 )‖ 2 2 , evaluated at its minimizer x(σ), approximately follows a χ 2 distribution with ˜ m degrees of freedom. Here ˜ m = m + p − n for A ∈R m×n , D ∈R p×n , matrix W b is the inverse covariance matrix of the mean 0 normally distributed measurement errors e in b, and x 0 is an estimate of the mean value of x. Using the generalized singular value decomposition of the matrix pair [W 1/2 b AD], σ can then be found such that the resulting J follows this χ 2 distribution, 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. The algorithm is also 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 σ 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. We conclude that the presented approach is robust for both small and large scale discretely ill-posed least squares problems. Key words. ill-posed problems, Tikhonov regularization, χ 2 -distribution, Golub-Kahan iterative bidiagonaliza- tion, hybrid methods, Newton algorithm. AMS subject classifications. 15A09,15A29, 62F15, 62G08 1. Introduction. We are concerned with the solution of large-scale linear discrete ill- posed problems such as arise in many physical experiments associated, for example, with the discretization of integral equations [27, 9], and modeled by the ill-posed system of equations Ax = b. Matrix A ∈R m×n results from the underlying model discretization and a solution x ∈R n is desired for measurements b ∈R m , which are often noise-contaminated. An approximate solution ˆ x may be obtained by solving the weighted regularized least squares problem, with matrix D ∈R p×n , p ≤ n, chosen dependent on anticipated smoothness properties of the solution x, ˆ x = argmin J (x) = argmin{‖Ax − b‖ 2 W b + ‖D(x − x 0 )‖ 2 Wx }. (1.1) * Supported by NSF grants DMS 0513214 and DMS 0652833. Arizona State University, Department of Mathe- matics and Statistics, Tempe, AZ 85287-1804. Tel: 480-965-3795, Fax: 480-965-4160. Email: renaut@asu.edu † Supported by the research project MSM0021620839 financed by MSMT. Charles University in Prague, Faculty of Mathematics and Physics, and Institute of Computer Science, Academy of Sciences of the Czech Republic. Tel: +420-22191-3362, Fax: +420-22481-1036. Email: hnetynkova@scs.cas.cz ‡ Supported by NSF grant EPS 0447689, Boise State University, Department of Mathematicss, Boise, ID 83725- 1555. Tel: 208426-2432, Fax: 208-426-1354. Email: jmead@boisestate.edu