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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 the large scale linear discrete ill-posed problems in general-form regularization: $min|Lx| mbox{{rm subject to}} xinmathcal{S} = {x| |Ax-b|leq tau|e|}$ 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: $min_xleft{|Ax-b|^2+lambda^2|Lx|^2right}$. 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.
In this paper, we propose new randomization based algorithms for large scale linear discrete ill-posed problems with general-form regularization: ${min} |Lx|$ subject to ${min} |Ax - b|$, where $L$ is a regularization matrix. Our algorithms are inspi
Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space-dimensions. The matrix that defin
GMRES is one of the most popular iterative methods for the solution of large linear systems of equations that arise from the discretization of linear well-posed problems, such as Dirichlet boundary value problems for elliptic partial differential equ
Block coordinate descent (BCD) methods approach optimization problems by performing gradient steps along alternating subgroups of coordinates. This is in contrast to full gradient descent, where a gradient step updates all coordinates simultaneously.
Ill-posed linear inverse problems appear in many image processing applications, such as deblurring, super-resolution and compressed sensing. Many restoration strategies involve minimizing a cost function, which is composed of fidelity and prior terms