The iterated Golub-Kahan-Tikhonov method

TL;DR

This paper analyzes an iterative regularization method combining Golub-Kahan bidiagonalization and Tikhonov regularization, providing error analysis and a new parameter choice approach, demonstrating improved accuracy over existing methods for large ill-posed problems.

Contribution

It introduces an iterated Golub-Kahan-Tikhonov method with comprehensive error analysis and a novel regularization parameter selection, outperforming standard and Arnoldi-based methods.

Findings

More accurate solutions than non-iterated Golub-Kahan-Tikhonov

Outperforms iterated Arnoldi-Tikhonov in accuracy

Provides a new approach for regularization parameter choice

Abstract

The Golub-Kahan-Tikhonov method is a popular solution technique for large linear discrete ill-posed problems. This method first applies partial Golub-Kahan bidiagonalization to reduce the size of the given problem and then uses Tikhonov regularization to compute a meaningful approximate solution of the reduced problem. It is well known that iterated variants of this method often yield approximate solutions of higher quality than the standard non-iterated method. Moreover, it produces more accurate computed solutions than the Arnoldi method when the matrix that defines the linear discrete ill-posed problem is far from symmetric. This paper starts with an ill-posed operator equation in infinite-dimensional Hilbert space, discretizes the equation, and then applies the iterated Golub-Kahan-Tikhonov method to the solution of the latter problem. An error analysis that addresses all discretization and approximation errors is provided. Additionally, a new approach for choosing the regularization parameter is described. This solution scheme produces more accurate approximate solutions than the standard (non-iterated) Golub-Kahan-Tikhonov method and the iterated Arnoldi-Tikhonov method.