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

Abstract

Solutions of numerically ill-posed least squares problems A x ≈ b for A ∈ R m × n by Tikhonov regularization are considered. For D ∈ R p × n , the Tikhonov regularized least squares functional is given by J ( σ ) = ‖ A x − b ‖ W 2 + 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 χ 2 distribution with m ̃ = m + p − n degrees of freedom. Using the generalized singular value decomposition of the matrix pair [ W b 1 / 2 A D ] , σ can then be found such that the resulting J follows this χ 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 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. It is concluded that the presented approach is robust for both small and large-scale discretely ill-posed least squares problems.