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

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.