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...

We study an interior-point gradient method for solving a class of so-called totally nonnegative least squares problems. At each iteration, the method decreases the residual norm along a diagonally scaled negative gradient direction with a special scaling. We establish the global convergence of the method, and present some numerical examples to compare the proposed method with a few similar meth...

The standard approaches to solving overdetermined linear systems Bx ≈ c construct minimal corrections to the data to make the corrected system compatible. In ordinary least squares (LS) the correction is restricted to the right hand side c, while in scaled total least squares (STLS) [14, 12] corrections to both c and B are allowed, and their relative sizes are determined by a real positive para...

In this paper, we analyze the limiting behavior of the weighted least squares problem minx∈<n Pp i=1 kDi(Aix − bi)k2, where each Di is a positive definite diagonal matrix. We consider the situation where the magnitude of the weights are drastically different block-wisely so that max(D1) ≥ min(D1) À max(D2) ≥ min(D2) À max(D3) ≥ . . . À max(Dp−1) ≥ min(Dp−1)À max(Dp). Here max(·) and min(·) repr...

We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated in a chosen finite-dimensional subspace. It has been shown that the stochastic Galerkin projection technique fails to minimize any measure of...

A weighted-norm first-order system least-squares (FOSLS) method for div/curl problems with edge singularities is presented. Traditional finite element methods, including least-squares methods, often suffer from a global loss of accuracy due to the influence of a nonsmooth solution near polyhedral edges. By minimizing a modified least-squares functional, optimal accuracy in weighted and non-weig...