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

Solutions of numerically ill-posed least squares problems Ax≈b for A∈Rm×nA∈Rm×n by Tikhonov regularization are considered. For D∈Rp×nD∈Rp×n, the Tikhonov regularized least squares functional is given by J(σ)=‖Ax−b‖W2+1/σ2‖D(x−x0)‖22 where matrix WW 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=Wb 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 σσ is a suitable statistically-chosen value, JJ evaluated at its minimizer x(σ) approximately follows a χ2χ2 distribution with m̃=m+p−n degrees of freedom. Using the generalized singular value decomposition of the matrix pair [Wb1/2AD], σσ can then be found such that the resulting JJ follows this χ2χ2 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 σσ 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.