Related papers: Iterated Gauss-Seidel GMRES
The parallel strong-scaling of Krylov iterative methods is largely determined by the number of global reductions required at each iteration. The GMRES and Krylov-Schur algorithms employ the Arnoldi algorithm for nonsymmetric matrices. The…
GMRES is one of the most popular iterative methods for the solution of large linear systems of equations that arise from the discretization of linear well-posed problems, such as Dirichlet boundary value problems for elliptic partial…
Communication, i.e., data movement, is a critical bottleneck for the performance of classical Krylov subspace method solvers on modern computer architectures. Variants of these methods which avoid communication have been introduced, which,…
We perform a backward stability analysis of preconditioned sketched GMRES [Nakatsukasa and Tropp, SIAM J. Matrix Anal. Appl, 2024] for solving linear systems $Ax=b$, and show that the backward stability at iteration $i$ depends on the…
In this paper, we develop a new Randomized Global Generalized Minimum Residual (RGlGMRES) algorithm for efficiently computing solutions to large scale linear systems with multiple right hand sides.The proposed method builds on a recently…
In this contribution, we study the numerical behavior of the Generalized Minimal Residual (GMRES) method for solving singular linear systems. It is known that GMRES determines a least squares solution without breakdown if the coefficient…
The Gauss-Seidel method has been used for more than 100 years as the standard method for the solution of linear systems of equations under certain restrictions. This method, as well as Cramer and Jacobi, is widely used in education and…
Communication-avoiding Krylov methods require solving small dense Gram systems at each outer iteration. We present a low-synchronization approach based on Forward Gauss--Seidel (FGS), which exploits the structure of Gram matrices arising…
Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and…
In [Hayami K, Sugihara M. Numer Linear Algebra Appl. 2011; 18:449--469], the authors analyzed the convergence behaviour of the Generalized Minimal Residual (GMRES) method for the least squares problem $ \min_{ {\bf x} \in {\bf R}^n} {\|…
Brown and Walker (1997) showed that GMRES determines a least squares solution of $ A x = b $ where $ A \in {\bf R}^{n \times n} $ without breakdown for arbitrary $ b, x_0 \in {\bf R}^n $ if and only if $A$ is range-symmetric, i.e. $ {\cal…
The generalized minimal residual (GMRES) algorithm is applied to image reconstruction using linear computed tomography (CT) models. The GMRES algorithm iteratively solves square, non-symmetric linear systems and it has practical application…
With a greedy strategy to construct control index set of coordinates firstly and then choosing the corresponding column submatrix in each iteration, we present a greedy block Gauss-Seidel (GBGS) method for solving large linear least squares…
A new variant of the GMRES method is presented for solving linear systems with the same matrix and subsequently obtained multiple right-hand sides. The new method keeps such properties of the classical GMRES algorithm as follows. Both bases…
With the emergence of mixed precision capabilities in hardware, iterative refinement schemes for solving linear systems $Ax=b$ have recently been revisited and reanalyzed in the context of three or more precisions. These new analyses show…
We present two minimum residual methods for solving sequences of shifted linear systems, the right-preconditioned shifted GMRES and shifted recycled GMRES algorithms which use a seed projection strategy often employed to solve multiple…
Consider solving large sparse range symmetric singular linear systems $ A {\bf x}= {\bf b} $ which arise, for instance, in the discretization of convection diffusion equations with periodic boundary conditions, and partial differential…
GMRES is a popular Krylov subspace method for solving linear systems of equations involving a general non-Hermitian coefficient matrix. The conventional bounds on GMRES convergence involve polynomial approximation problems in the complex…
A parameter-free method, namely the generalization of the Gauss-Seidel (GGS) method, is developed to solve generalized absolute value equations. Convergence of the proposed method is analyzed. Numerical results are given to demonstrate the…
The GMRES method is used to solve sparse, non-symmetric systems of linear equations arising from many scientific applications. The solver performance within a single node is memory bound, due to the low arithmetic intensity of its…