1,285 research outputs found
Deflated Iterative Methods for Linear Equations with Multiple Right-Hand Sides
A new approach is discussed for solving large nonsymmetric systems of linear
equations with multiple right-hand sides. The first system is solved with a
deflated GMRES method that generates eigenvector information at the same time
that the linear equations are solved. Subsequent systems are solved by
combining restarted GMRES with a projection over the previously determined
eigenvectors. This approach offers an alternative to block methods, and it can
also be combined with a block method. It is useful when there are a limited
number of small eigenvalues that slow the convergence. An example is given
showing significant improvement for a problem from quantum chromodynamics. The
second and subsequent right-hand sides are solved much quicker than without the
deflation. This new approach is relatively simple to implement and is very
efficient compared to other deflation methods.Comment: 13 pages, 5 figure
Hybrid algorithms for cyclically reduced convection-diffusion problems
We consider hybrid and adaptive iterative algorithms for cyclically-reduced discrete convection-diffusion problems. Hybrid algorithms combine via a two phase algorithm, iterative methods which require no a priori information about the coefficient matrix in the first phase with Chebyshev or Richardson iteration in the second phase. For two-dimensional convection-diffusion problems, central difference discretization is considered and the resulting linear system is reduced to approximately half its size by applying one step of cyclic reduction. We examine the numerical performance of the hybrid methods for solving the reduced systems. Our numerical experiments show that for the class of problems considered, an adaptive Chebyshev algorithm that uses modified moments to approximate the eigenvalues requires less work in most cases than the hybrid algorithms based on GMRES/Richardson methods
Some observations on weighted GMRES
We investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present new alternative implementations of the weighted Arnoldi algorithm which may be favorable in terms of computational complexity, and examine stability issues connected with these implementations. Two implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used
On the eigenvalues and eigenvectors of nonsymmetric saddle point matrices preconditioned by block triangular matrices
Block lower triangular and block upper triangular matrices are popular preconditioners for nonsymmetric saddle point matrices. In this note we show that a block lower triangular preconditioner gives the same spectrum as a block upper triangular preconditioner and that the eigenvectors of the two preconditioned systems are related
Error estimators and their analysis for CG, Bi-CG and GMRES
We present an analysis of the uncertainty in the convergence of iterative
linear solvers when using relative residue as a stopping criterion, and the
resulting over/under computation for a given tolerance in error. This shows
that error estimation is indispensable for efficient and accurate solution of
moderate to high conditioned linear systems (), where is
the condition number of the matrix. An error estimator for
iterations of the CG (Conjugate Gradient) algorithm was proposed more than two
decades ago. Recently, an error estimator was described for
the GMRES (Generalized Minimal Residual) algorithm which allows for
non-symmetric linear systems as well, where is the iteration number. We
suggest a minor modification in this GMRES error estimation for increased
stability. In this work, we also propose an error estimator
for A-norm and norm of the error vector in Bi-CG (Bi-Conjugate
Gradient) algorithm. The robust performance of these estimates as a stopping
criterion results in increased savings and accuracy in computation, as
condition number and size of problems increase
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