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

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/σ² ||D(x - x₀) \\² ₂where matrix W is a weighting matrix and X₀ 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 X₀ 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 ^1/2AD], σ can then be found such that the resulting J follows this x² 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 X₀ 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.