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Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 85 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
Any decreasing cycle–convergence curve is possible for restarted GMRES
, 907
"... Given a matrix order n, a restart parameter m (m < n), a decreasing positive sequence f(0)> f(1)>...> f(q) ≥ 0, where q < n/m, it is shown that there exits an nbyn matrix A and a vector r0 with ‖r0 ‖ = f(0) such that ‖rk ‖ = f(k), k = 1,..., q, where rk is the residual at cycle k ..."
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Given a matrix order n, a restart parameter m (m < n), a decreasing positive sequence f(0)> f(1)>...> f(q) ≥ 0, where q < n/m, it is shown that there exits an nbyn matrix A and a vector r0 with ‖r0 ‖ = f(0) such that ‖rk ‖ = f(k), k = 1,..., q, where rk is the residual at cycle k of restarted GMRES with restart parameter m applied to the linear system Ax = b, with initial residual r0 = b−Ax0. Moreover, the matrix A can be chosen to have any desired eigenvalues. We can also construct arbitrary cases of stagnation; namely, when f(0)> f(1)>...> f(i) = f(i + 1) ≥ 0 for any i < q. The restart parameter can be fixed or variable. 1
Noname manuscript No. (will be inserted by the editor) Some observations on weighted GMRES
"... Abstract 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 weight ..."
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Abstract 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 favourable in terms of computational complexity, and examine stability issues connected with these implementations. These 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.
Simpler Hybrid GMRES
, 2005
"... Abstract. Hybrid GMRES algorithms are effective for solving large nonsymmetric linear systems. GMRES is employed at the first phase to produce iterative polynomials, which will be used at the second phase to implement the Richardson iteration. In the process of GMRES, a least squares problem needs t ..."
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Abstract. Hybrid GMRES algorithms are effective for solving large nonsymmetric linear systems. GMRES is employed at the first phase to produce iterative polynomials, which will be used at the second phase to implement the Richardson iteration. In the process of GMRES, a least squares problem needs to be solved which involves an upper Hessenberg factorization. Instead of using GMRES, we may use simpler GMRES. Correspondingly, simpler hybrid GMRES algorithms are formulated. It is described how to construct the iterative polynomials from simpler GMRES. The new algorithms avoid the upper Hessenberg factorization so that they are easier to program and require a less amount of work. Numerical examples are conducted to illustrate the good performance of the new algorithms. Key words: linear systems, iterative methods, GMRES, hybrid algorithm. 1.