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Deflated and augmented Krylov subspace methods: A framework for deflated . . .
, 2013
"... We present an extension of the framework of Gaul et al. (SIAM J. Matrix Anal. Appl. 34, 495–518 (2013)) for deflated and augmented Krylov subspace methods satisfying a Galerkin condition to more general Petrov–Galerkin conditions. The main goal is to apply the framework also to the biconjugate gra ..."
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Cited by 12 (2 self)
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We present an extension of the framework of Gaul et al. (SIAM J. Matrix Anal. Appl. 34, 495–518 (2013)) for deflated and augmented Krylov subspace methods satisfying a Galerkin condition to more general Petrov–Galerkin conditions. The main goal is to apply the framework also to the biconjugate gradient method (BiCG) and some of its generalizations, including BiCGStab approach does not depend on particular recurrences and thus simplifies the derivation of theoretical results. It easily leads to a variety of realizations by specific algorithms. We do not go into algorithmic details, but we show that for every method there are two different approaches for extending it by augmentation and deflation: one that explicitly takes care of the augmentation space in every step, and one that applies the unchanged basic algorithm to a projected problem but requires a correction step at the end. Both typically generate a Krylov space for a singular operator that is associated with the projected problem. The deflated biconjugate gradient requires two such Krylov spaces, but it also allows us to solve two dual linear systems at once. Deflated Lanczostype product methods fit in our new framework too. The question of how to extract the augmentation and deflation subspace is not addressed here.
Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple righthand sides
, 2009
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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
A flexible Krylov solver for shifted systems with application to oscillatory hydraulic tomography
 SIAM J. Sci. Comput
"... We discuss efficient solutions to systems of shifted linear systems arising in computations for oscillatory hydraulic tomography (OHT). The reconstruction of hydrogeological parameters such as hydraulic conductivity and specific storage using limited discrete measurements of pressure (head) obtai ..."
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Cited by 3 (1 self)
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We discuss efficient solutions to systems of shifted linear systems arising in computations for oscillatory hydraulic tomography (OHT). The reconstruction of hydrogeological parameters such as hydraulic conductivity and specific storage using limited discrete measurements of pressure (head) obtained from sequential oscillatory pumping tests, leads to a nonlinear inverse problem. We tackle this using the quasilinear geostatistical approach [15]. This method requires repeated solution of the forward (and adjoint) problem for multiple frequencies, for which we use flexible preconditioned Krylov subspace solvers specifically designed for shifted systems based on ideas in [13]. The solvers allow the preconditioner to change at each iteration. We analyze the convergence of the solver and perform an error analysis when an iterative solver is used for inverting the preconditioner matrices. Finally, we apply our algorithm to a challenging application taken from oscillatory hydraulic tomography to demonstrate the computational gains by using the resulting method. 1
University of Technology, The Netherlands. On the Convergence of GMRES with
, 2010
"... We consider the solution of large and sparse linear systems of equations by GMRES. Due to the appearance of unfavorable eigenvalues in the spectrum of the coefficient matrix, the convergence of GMRES may hamper. To overcome this, a deflated variant of GMRES can be used, which treats those unfavora ..."
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We consider the solution of large and sparse linear systems of equations by GMRES. Due to the appearance of unfavorable eigenvalues in the spectrum of the coefficient matrix, the convergence of GMRES may hamper. To overcome this, a deflated variant of GMRES can be used, which treats those unfavorable eigenvalues effectively. In the literature, several deflated GMRES variants are applied successfully to various problems, while a theoretical justification is often lacking. In contrast to deflated CG, the convergence of deflated GMRES seems to be harder to analyze and to understand. This paper presents some new theoretical insights into deflated GMRES based on Ainvariant deflation subspaces. Fundamental results regarding the convergence of deflated GMRES are proved in order to show the effectiveness and robustness of this method. Numerical experiments are provided to illustrate the theoretical results and to show some further properties of deflated GMRES. Consequently, practical variants of deflated GMRES from the literature can be better understood based on
Iranian Journal of Numerical Analysis and Optimization
"... The block LSMR algorithm for solving linear systems with multiple righthand sides F. Toutounian and M. Mojarrab LSMR (Least Squares Minimal Residual) is an iterative method for the solution of the linear system of equations and leastsquares problems. This paper presents a block version of the LSM ..."
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The block LSMR algorithm for solving linear systems with multiple righthand sides F. Toutounian and M. Mojarrab LSMR (Least Squares Minimal Residual) is an iterative method for the solution of the linear system of equations and leastsquares problems. This paper presents a block version of the LSMR algorithm for solving linear systems with multiple righthand sides. The new algorithm is based on the block bidiagonalization and derived by minimizing the Frobenius norm of the residual matrix of normal equations. In addition, the convergence of the proposed algorithm is discussed. In practice, it is also observed that the Frobenius norm of the residual matrix decreases monotonically. Finally, numerical experiments from real applications are employed to verify the eectiveness of the presented method.
On the Convergence of GMRES with Invariantsubspace Deflation
, 2010
"... We consider the solution of large and sparse linear systems of equations by GMRES. Due to the appearance of unfavorable eigenvalues in the spectrum of the coefficient matrix, the convergence of GMRES may hamper. To overcome this, a deflated variant of GMRES can be used, which treats those unfavorab ..."
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We consider the solution of large and sparse linear systems of equations by GMRES. Due to the appearance of unfavorable eigenvalues in the spectrum of the coefficient matrix, the convergence of GMRES may hamper. To overcome this, a deflated variant of GMRES can be used, which treats those unfavorable eigenvalues effectively. In the literature, several deflated GMRES variants are applied successfully to various problems, while a theoretical justification is often lacking. In contrast to deflated CG, the convergence of deflated GMRES seems to be harder to analyze and to understand. This paper presents some new theoretical insights into deflated GMRES based on Ainvariant deflation subspaces. Fundamental results regarding the convergence of deflated GMRES are proved in order to show the effectiveness and robustness of this method. Numerical experiments are provided to illustrate the theoretical results and to show some further properties of deflated GMRES. Consequently, practical variants of deflated GMRES from the literature can be better understood based on