Iterated Tikhonov regularization of large linear problems

Abstract

Many solution methods for linear discrete ill-posed problems with error-contaminated data (right-hand side) apply Tikhonov regularization to compute a meaningful approximate solution. This solution depends on a regularization parameter. It is well known that iterated Tikhonov regularization often determines an approximate solution of higher quality than (standard) Tikhonov regularization. We consider the situation when an estimate of the norm of the error in the data is known and would like to apply iterative Tikhonov regularization to determine an approximate solution that satisfies the discrepancy principle. This requires a suitable choice of a regularization parameter. The standard approach to determine this parameter is to compute solutions for several values of the regularization parameter and choose a computed approximate solution that satisfies the discrepancy principle. This paper discusses iterated Tikhonov regularization based on partial Golub-Kahan bidiagonalization and describes how the regularization parameter can be determined without computing several approximate solutions by using the connection between Golub-Kahan bidiagonalization and Gauss quadrature. This approach reduces the computational effort required to compute a desired solution.