Error estimates for Golub–Kahan bidiagonalization with Tikhonov regularization for ill–posed operator equations. (1st February 2023)

• Record Type: Journal Article
• Authors: Alqahtani, A; Ramlau, R; Reichel, L
• Is Part Of: Inverse problems. Volume 39:Number 2(2023)
• Publication Date: 2023-02-01
• Subjects: ill-posed problem -- inverse problem -- Golub–Kahan bidiagonalization -- Tikhonov regularization; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aca754
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: 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.