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Based on the joint bidiagonalization process of a large matrix pair$\{A,L\}$, we propose and develop an iterative regularization algorithm for thelarge 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 oneproposed by Kilmer {\em et al.}, which is based on the same process but solvesthe general-form Tikhonov regularization problem:$\min_x\left\{\|Ax-b\|^2+\lambda^2\|Lx\|^2\right\}$. We prove that the iteratestake the form of attractive filtered generalized singular value decomposition(GSVD) expansions, where the filters are given explicitly. This result and theanalysis on it show that the method must have the desired semi-convergenceproperty and get insight into the regularizing effects of the method. We usethe L-curve criterion or the discrepancy principle to determine $k^*$. Thealgorithm is simple and effective, and numerical experiments illustrate that itoften computes more accurate regularized solutions than the hybrid one.

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