A Joint Bidiagonalization Based Algorithm for Large Scale Linear Discrete Ill-posed Problems in General-Form Regularization

TL;DR

This paper introduces a new iterative regularization algorithm based on joint bidiagonalization for large-scale linear ill-posed problems, which outperforms existing hybrid methods in accuracy and provides explicit filtered GSVD expansions.

Contribution

The paper develops a novel algorithm using joint bidiagonalization for general-form regularization, with explicit GSVD filters and proven semi-convergence, improving solution accuracy over previous hybrid methods.

Findings

The method exhibits semi-convergence with explicit GSVD filters.

Numerical experiments show improved accuracy over hybrid algorithms.

The algorithm effectively handles large-scale ill-posed problems.

Abstract

Based on the joint bidiagonalization process of a large matrix pair $\{A,L\}$, we propose and develop an iterative regularization algorithm for the large scale linear discrete ill-posed problems in general-form regularization: $\min\|Lx\| \ \mbox{{\rm subject to}} \ x\in\mathcal{S} = \{x|\ \|Ax-b\|\leq \tau\|e\|\}$ with a Gaussian white noise $e$ and $\tau>1$ slightly, where $L$ is a regularization matrix. Our algorithm is different from the hybrid one proposed by Kilmer {\em et al.}, which is based on the same process but solves the general-form Tikhonov regularization problem: $\min_x\left\{\|Ax-b\|^2+\lambda^2\|Lx\|^2\right\}$. We prove that the iterates take the form of attractive filtered generalized singular value decomposition (GSVD) expansions, where the filters are given explicitly. This result and the analysis on it show that the method must have the desired semi-convergence property and get insight into the regularizing effects of the method. We use the L-curve criterion or the discrepancy principle to determine $k^*$. The algorithm is simple and effective, and numerical experiments illustrate that it often computes more accurate regularized solutions than the hybrid one.