Error estimates for Golub-Kahan bidiagonalization with Tikhonov regularization for ill-posed operator equations

Linear ill-posed operator equations arise in various areas of science and engineering. The presence of errors in the operator and the data often makes the computation of an accurate approximate solution difficult. In this paper, we compute an approximate solution of an ill-posed operator equation by first determining an approximation of the operators of generally fairly small dimension by carrying out a few steps of a continuous version of the Golub-Kahan bidiagonalization process to the noisy operator. Then Tikhonov regularization is applied to the low-dimensional problem so obtained and the regularization parameter is determined by solving a low-dimensional nonlinear equation. The effect of the errors incurred in each step of the solution process is analyzed. Computed examples illustrate the theory presented.

Automated summary

Linear ill-posed operator equations often appear in various fields of science and engineering. However, the presence of errors in the operator and data makes it challenging to compute an accurate approximate solution. In this paper, we propose a method that involves approximating the low-dimensional operators of the noisy operator using a continuous version of the Golub-Kahan bidiagonalization process. We then apply Tikhonov regularization to the obtained low-dimensional problem, with the regularization parameter determined by solving a low-dimensional nonlinear equation. Computed examples are provided to illustrate the theory presented in this paper.