for x, where A is a given n × n matrix, and b a given n-vector. We have many opportunities to solve linear equations with the same coefficient matrix and different right-hand sides (RHSs). Therefore, block Krylov subspace methods such as the block Bi-CG (Bl-BCG), block Bi-CGSTAB (Bl-BiCGSTAB) [2], block GMRES (Bl-GMRES) and block QMR (Bl-QMR) methods have been developed for solving block linear...

The total least squares (TLS) techniques, also called orthogonal regression and errors-in-variables modeling, see [15, 16], have been developed independently in several areas. For a given linear (orthogonally invariant) approximation problem AX ≈ B, where A ∈ Rm×n, B ∈ Rm×d, X ∈ Rn×d, the TLS formulation aims at a solution of a modified problem (A + E)X = B + G such that min ‖[G,E]‖F . The alge...

Most calculations in lattice Quantum Chromodynamics (QCD) involve the solution of a series of linear systems of equations with exceedingly large matrices and a large number of right hand sides. Iterative methods for these problems can be sped up significantly if we deflate approximations of appropriate invariant spaces from the initial guesses. Recently we have developed eigCG, a modification o...

Many parameter estimation problems involve with a parameter-dependent PDEs with multiple right hand sides. The computational cost and memory requirements of such problems increases linearly with the number of right hand sides. For many applications this is the main bottleneck of the computation. In this paper we show that problems with multiple right hand sides can be reformulated as stochastic...

We study systems of two elliptic equations, with right-hand sides with general power-like superlinear growth, and left-hand sides which are of Isaac’s or Hamilton-Jacobi-Bellman type (however our results are new even for linear lefthand sides). We show that under appropriate growth conditions such systems have positive solutions in bounded domains, and that all such solutions are bounded in the...

An efficient method for solution of systems of linear equations with many right hand sides is developed. The right hand sides are assumed to depend smoothly on a parameter. The equations are solved by an iterative method and a linear least squares approximation is used as initial guess. The work spent on the iterations is bounded independently of the number of right hand sides. The method is ap...

There exist two major problems in application of the conventional block BiCGSTAB method to the O(a)-improved Wilson-Dirac equation with multiple right-hand-sides: One is the deviation between the true and the recursive residuals. The other is the convergence failure observed at smaller quark masses for enlarged number of the right-hand-sides. The block BiCGGR algorithm which was recently propos...

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...