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
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
ACM Transactions on Mathematical Software (TOMS)
Row scaling as a preconditioner for some nonsymmetric linear systems with discontinuous coefficients
Journal of Computational and Applied Mathematics
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
Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number
Journal of Computational Physics
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CARP-CG is a conjugate gradient (CG) acceleration of CARP, which was introduced by Gordon and Gordon as a robust block-parallel scheme for sparse linear systems. CARP performs Kaczmarz (KACZ) row projections within the blocks, and the results from the separate blocks are merged by averaging, for each component, its updated values from the different blocks. The averaging operations are equivalent to a sequence of certain KACZ row projections in some superspace (the ''averaging projections''), and so CARP is equivalent to KACZ with cyclic relaxation parameters in that superspace. The CG-acceleration of CARP is based on a generalization of the (sequential) CGMN algorithm of Bjorck and Elfving, which accelerates KACZ by running it in a double sweep on the equations of a linear system, using a fixed relaxation parameter. CGMN is generalized to allow cyclic relaxation parameters, so the resulting method, called CGMNC, can be applied in the superspace. The averaging projections in the superspace can be done in any order, so CGMNC in the superspace can be implemented in the regular space by using CARP in a double sweep. The resulting algorithm, CARP-CG, is as robust as CARP but converges significantly faster. CARP-CG is compared with restarted GMRES, Bi-CGSTAB and CGS, with and without various preconditioners, on some stiff linear systems derived from convection dominated elliptic partial differential equations. The results indicate that CARP-CG is very robust and its runtime is very competitive with the other methods. A scaled version of CGNR was also tested, and it was as robust as CARP-CG, but slower.