GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
CGS, a fast Lanczos-type solver for nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
On some parallel preconditioned CG schemes
Proceedings of a conference on Preconditioned conjugate gradient methods
Domain decomposition for parallel row projection algorithms
Applied Numerical Mathematics - II on Domain decomposition; Guest Editor: W. Proskurowski
SIAM Journal on Scientific and Statistical Computing
A block projection method for sparse matrices
SIAM Journal on Scientific and Statistical Computing - Special issue on iterative methods in numerical linear algebra
Row projection methods for large nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing - Special issue on iterative methods in numerical linear algebra
Block Lanczos techniques for accelerating the block Cimmino method
SIAM Journal on Scientific Computing
Relaxation methods for image reconstruction
Communications of the ACM
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Component-Averaged Row Projections: A Robust, Block-Parallel Scheme for Sparse Linear Systems
SIAM Journal on Scientific Computing
Efficient parallel implementation of iterative reconstruction algorithms for electron tomography
Journal of Parallel and Distributed Computing
Efficient controls for finitely convergent sequential algorithms
ACM Transactions on Mathematical Software (TOMS)
Solving PDEs in non-rectangular 3D regions using a collocation finite element method
Advances in Engineering Software
Row scaling as a preconditioner for some nonsymmetric linear systems with discontinuous coefficients
Journal of Computational and Applied Mathematics
Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers
Journal of Computational and Applied Mathematics
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Given a linear system Ax = b, one can construct a related “normal equations” system AATy = b, x = ATy. Björck and Elfving have shown that the SSOR algorithm, applied to the normal equations, can be accelerated by the conjugate gradient algorithm (CG). The resulting algorithm, called CGMN, is error-reducing and in theory it always converges even when the equation system is inconsistent and/or nonsquare. SSOR on the normal equations is equivalent to the Kaczmarz algorithm (KACZ), with a fixed relaxation parameter, run in a double (forward and backward) sweep on the original equations. CGMN was tested on nine well-known large and sparse linear systems obtained by central-difference discretization of elliptic convection-diffusion partial differential equations (PDEs). Eight of the PDEs were strongly convection-dominated, and these are known to produce very stiff systems with large off-diagonal elements. CGMN was compared with some of the foremost state-of-the art Krylov subspace methods: restarted GMRES, Bi-CGSTAB, and CGS. These methods were tested both with and without various preconditioners. CGMN converged in all the cases, while none of the preceding algorithm/preconditioner combinations achieved this level of robustness. Furthermore, on varying grid sizes, there was only a gradual increase in the number of iterations as the grid was refined. On the eight convection-dominated cases, the initial convergence rate of CGMN was better than all the other combinations of algorithms and preconditioners, and the residual decreased monotonically. The CGNR algorithm was also tested, and it was as robust as CGMN, but slower.