An iterative method LSMR is presented for solving linear systems Ax = b and leastsquares problems min ‖Ax−b‖2, with A being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation ATAx = ATb, so that the quantities ‖Ark‖ are monotonically decreasing (where rk = b−Axk is the re...

In this paper, We present the generalizations of the LSMR and NSCG methods for solving matrix equations. First, based on the LSMR algorithm, the Bl-LSMR and Gl-LSMR algorithms are derived by minimizing the Frobenius norm of residual matrix of normal equation. In addition, by extending the idea of LSMR algorithm, we also present the LSMR-M algorithm for solving the general coupled matrix equatio...

Large-Scale Multimedia Retrieval (LSMR) is the task to fast analyze a large amount of multimedia data like images or videos and accurately find the ones relevant to a certain semantic meaning. Although LSMR has been investigated for more than two decades in the fields of multimedia processing and computer vision, a more interdisciplinary approach is necessary to develop an LSMR system that is r...

We develop a hybrid iterative approach for computing solutions to large-scale illposed inverse problems via Tikhonov regularization. We consider a hybrid LSMR algorithm, where Tikhonov regularization is applied to the LSMR subproblem rather than the original problem. We show that, contrary to standard hybrid methods, hybrid LSMR iterates are not equivalent to LSMR iterates on the directly regul...

BACKGROUND In addition to an increased risk for chronic illnesses, obese individuals suffer from social stigmatization and discrimination, and severely obese people may experience greater risk of impaired psychosocial and physical functioning. Lower health-related quality of life (HRQOL) has been reported among obese persons seeking intensive treatment for their disease. To aid in the treatment...

lsmr (least squares minimal residual) is an iterative method for the solution of the linear system of equations and leastsquares problems. this paper presents a block version of the lsmr algorithm for solving linear systems with multiple right-hand sides. the new algorithm is based on the block bidiagonalization and derived by minimizing the frobenius norm of the resid ual matrix of normal equa...

it is well known that if the coefficient matrix in a linear system is large and sparse or sometimes not readily available, then iterative solvers may become the only choice. the block solvers are an attractive class of iterative solvers for solving linear systems with multiple right-hand sides. in general, the block solvers are more suitable for dense systems with preconditioner. in this paper,...

Unmatched pairs of forward and back projectors are common in X-ray CT computations for large-scale problems; they caused by the need fast algorithms that best utilize computer hardware, it is an interesting challenging task to develop easy-to-use these cases. Our approach use preconditioned GMRES, form AB- BA-GMRES algorithms, handle unmatched normal equations associated with pair. These simple...