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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.
Recycling Krylov Subspaces for Sequences of Linear Systems
 SIAM J. Sci. Comput
, 2004
"... Many problems in engineering and physics require the solution of a large sequence of linear systems. We can reduce the cost of solving subsequent systems in the sequence by recycling information from previous systems. We consider two dierent approaches. For several model problems, we demonstrate tha ..."
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Cited by 74 (6 self)
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Many problems in engineering and physics require the solution of a large sequence of linear systems. We can reduce the cost of solving subsequent systems in the sequence by recycling information from previous systems. We consider two dierent approaches. For several model problems, we demonstrate that we can reduce the iteration count required to solve a linear system by a factor of two. We consider both Hermitian and nonHermitian problems, and present numerical experiments to illustrate the eects of subspace recycling.
The many proofs of an identity on the norm of oblique projections
 Numer. Algorithms
"... Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P 2 = P, which is neither null nor the identity, it holds that ‖P ‖ = ‖I − P ‖. This useful equality, while not widelyknown, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler o ..."
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Cited by 22 (1 self)
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Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P 2 = P, which is neither null nor the identity, it holds that ‖P ‖ = ‖I − P ‖. This useful equality, while not widelyknown, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler ones are presented.
Direct methods and ADIpreconditioned Krylov subspace methods for generalized Lyapunov equations
 Numer. Lin. Alg. Appl
"... Prepared using nlaauth.cls [Version: 2002/09/18 v1.02] ..."
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Cited by 14 (1 self)
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Prepared using nlaauth.cls [Version: 2002/09/18 v1.02]
SIMPLER BLOCK GMRES FOR NONSYMMETRIC SYSTEMS WITH MULTIPLE RIGHTHAND SIDES
"... Abstract. A Simpler Block GMRES algorithm is presented, which is a block version of Walker and Zhou’s Simpler GMRES. Similar to Block GMRES, the new algorithm also minimizes the residual norm in a block Krylov space at every step. Theoretical analysis shows that the matrixvalued polynomials constru ..."
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Cited by 8 (0 self)
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Abstract. A Simpler Block GMRES algorithm is presented, which is a block version of Walker and Zhou’s Simpler GMRES. Similar to Block GMRES, the new algorithm also minimizes the residual norm in a block Krylov space at every step. Theoretical analysis shows that the matrixvalued polynomials constructed by the new algorithm is the same as the original one. However, Simpler Block GMRES avoids the factorization of a block upper Hessenberg matrix. In consequence, it is much simpler to program and requires less work. Numerical experiments are conducted to illustrate the performance of the new block algorithm. Key words. linear systems, iterative methods, block methods, GMRES, Simpler GMRES AMS subject classifications. 65F10 1. Introduction. Block GMRES [13] and its variants [1, 6, 7] are effective for solving large nonsymmetric systems with multiple righthand sides of the form where is a nonsingular matrix of order, and
Efficient Deflation Methods applied on 3D Bubbly Flow Problems
, 2007
"... For various applications, it is wellknown that deflated ICCG is an efficient method to solve linear systems with an invertible coefficient matrix. Tang and Vuik [J. Comput. Appl. Math., 206 (2007), pp. 603– 614] proposed two equivalent variants of this deflated method, which can also solve linear ..."
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Cited by 8 (4 self)
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For various applications, it is wellknown that deflated ICCG is an efficient method to solve linear systems with an invertible coefficient matrix. Tang and Vuik [J. Comput. Appl. Math., 206 (2007), pp. 603– 614] proposed two equivalent variants of this deflated method, which can also solve linear systems with singular coefficient matrices that arise from the discretization of the Poisson equation with Neumann boundary conditions and discontinuous coefficients. In this paper, we also consider the original variant of DICCG in Vuik, Segal, and Meijerink [J. Comput. Phys., 152 (1999), pp. 385–403], that already proved its efficiency for invertible coefficient matrices. This variant appears to be theoretically equivalent to the first two variants, so that they all have the same convergence properties. Moreover, we show that the associated coarse linear systems within these variants can be solved both directly and iteratively. In applications with large grid sizes, the method with the iterative coarse solver can be substantially more efficient than the one with the standard direct coarse solver. Additionally, the results for stationary numerical experiments of Tang and Vuik [J. Comput. Appl. Math., 206 (2007), pp. 603–614] have only been given in terms of number of iterations. After discussing some implementation issues, we show in this paper that deflated ICCG is considerably faster than ICCG in the most test cases, by taking the computational time into account as well. Other 3D timedependent numerical experiments with falling droplets in air and rising air bubbles in water are performed, in order to show that deflated ICCG is also more efficient than ICCG in these cases, considering both the number of iterations and computational time.
KRYLOV SUBSPACE RECYCLING FOR SEQUENCES OF SHIFTED LINEAR SYSTEMS∗
, 2013
"... Abstract. Subspace recycling methods, a class of Krylov subspace deflation techniques, have been shown to have the potential to accelerate convergence of Krylov subspace methods. In particular, they can be useful when solving sequences of slowlychanging linear systems. We wish to extend such metho ..."
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Cited by 5 (1 self)
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Abstract. Subspace recycling methods, a class of Krylov subspace deflation techniques, have been shown to have the potential to accelerate convergence of Krylov subspace methods. In particular, they can be useful when solving sequences of slowlychanging linear systems. We wish to extend such methods to solve sequences of linear systems, where for each system, we also solve a family of shifted systems in which the coefficient matrices only differ by multiples of the identity from a base system matrix. In this work, we demonstrate the difficulty of extending recycling techniques to solve multiple shifted systems while maintaining the fixed storage property. As an alternative, we introduce a scheme which constructs approximate corrections to the solutions of the shifted systems at each cycle while only minimizing the base system residual. At convergence of the base system solution, we apply the method recursively to the remaining unconverged systems. The method is robust enough to be applied to sequences of systems where the base system changes slowly and the shifts differ for each base system. We present numerical examples involving systems arising in lattice quantum chromodynamics. Key words. Krylov subspace methods, subspace recycling, shifted linear systems, QCD 1. Introduction. We
Analytic models of the quantum harmonic oscillator
 Contemp. Math
, 1997
"... Abstract. There are many examples where nonorthogonality of a basis for Krylov subspace methods arises naturally. These methods usually require less storage or computational effort per iteration than methods using an orthonormal basis (optimal methods), but the convergence may be delayed. Truncated ..."
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Abstract. There are many examples where nonorthogonality of a basis for Krylov subspace methods arises naturally. These methods usually require less storage or computational effort per iteration than methods using an orthonormal basis (optimal methods), but the convergence may be delayed. Truncated Krylov subspace methods and other examples of nonoptimal methods have been shown to converge in many situations, often with small delay, but not in others. We explore the question of what is the effect of having a nonoptimal basis. We prove certain identities for the relative residual gap, i.e., the relative difference between the residuals of the optimal and nonoptimal methods. These identities and related bounds provide insight into when the delay is small and convergence is achieved. Further understanding is gained by using a general theory of superlinear convergence recently developed. Our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important, only the need to maintain linear independence is. Numerical examples illustrate our theoretical results.