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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.
Theory of inexact Krylov subspace methods and applications to scientific computing
, 2003
"... We provide a general frameworkfor the understanding of inexact Krylov subspace methods for the solution of symmetric and nonsymmetric linear systems of equations, as well as for certain eigenvalue calculations. This frameworkallows us to explain the empirical results reported in a series of CERFAC ..."
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Cited by 72 (7 self)
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We provide a general frameworkfor the understanding of inexact Krylov subspace methods for the solution of symmetric and nonsymmetric linear systems of equations, as well as for certain eigenvalue calculations. This frameworkallows us to explain the empirical results reported in a series of CERFACS technical reports by Bouras, Frayssé, and Giraud in 2000. Furthermore, assuming exact arithmetic, our analysis can be used to produce computable criteria to bound the inexactness of the matrixvector multiplication in such a way as to maintain the convergence of the Krylov subspace method. The theory developed is applied to several problems including the solution of Schur complement systems, linear systems which depend on a parameter, and eigenvalue problems. Numerical experiments for some of these scientific applications are reported.
A novel multigrid based preconditioner for heterogeneous Helmholtz problems
 SIAM J. Sci. Comput
, 2006
"... Abstract. An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtztype differential operator with a complex term. A multigrid iteration is used for approximately inverting the prec ..."
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Cited by 65 (6 self)
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Abstract. An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtztype differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner. Key words. Helmholtz equation, nonconstant high wavenumber, complex multigrid preconditioner, Fourier analysis
Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers
, 2010
"... The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Hel ..."
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Cited by 46 (6 self)
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The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The generalized minimal residual method (GMRES) solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the threedimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach. © 2011 Wiley Periodicals, Inc. 1
Flexible innerouter Krylov subspace methods
 SIAM J. NUMER. ANAL
, 2003
"... Flexible Krylov methodsrefersto a classof methodswhich accept preconditioning that can change from one step to the next. Given a Krylov subspace method, such as CG, GMRES, QMR, etc. for the solution of a linear system Ax = b, instead of having a fixed preconditioner M and the (right) preconditione ..."
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Cited by 34 (2 self)
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Flexible Krylov methodsrefersto a classof methodswhich accept preconditioning that can change from one step to the next. Given a Krylov subspace method, such as CG, GMRES, QMR, etc. for the solution of a linear system Ax = b, instead of having a fixed preconditioner M and the (right) preconditioned equation AM −1 y = b (Mx = y), one may have a different matrix, say Mk, at each step. In this paper, the case where the preconditioner itself is a Krylov subspace method is studied. There are several papers in the literature where such a situation is presented and numerical examples given. A general theory is provided encompassing many of these cases, including truncated methods. The overall space where the solution is approximated is no longer a Krylov subspace but a subspace of a larger Krylov space. We show how this subspace keeps growing as the outer iteration progresses, thus providing a convergence theory for these innerouter methods. Numerical tests illustrate some important implementation aspects that make the discussed innerouter methods very appealing in practical circumstances.
Preconditioned Iterative Solution of the 2D Helmholtz Equation
"... this paper is to precondition this linear system with a new preconditioner and then solve it iteratively using a Krylov subspace method. Numerical analysis shows the preconditioner to be eective on a simple 1D test problem, and results are presented showing considerable convergence acceleration ..."
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Cited by 28 (0 self)
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this paper is to precondition this linear system with a new preconditioner and then solve it iteratively using a Krylov subspace method. Numerical analysis shows the preconditioner to be eective on a simple 1D test problem, and results are presented showing considerable convergence acceleration for a number of dierent Krylov methods for more complex problems in 2D, as well as for the more general problem of harmonic disturbances to a nonstagnant steady ow
Preconditioning Helmholtz linear systems
, 2009
"... Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILUbased preconditioners ineffective. This paper present ..."
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Cited by 22 (1 self)
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Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILUbased preconditioners ineffective. This paper presents a study of ways to enhance standard preconditioners by altering the diagonal by imaginary shifts. Prior work indicates that modifying the diagonal entries during the incomplete factorization process, by adding to it purely imaginary values can improve the quality of the preconditioner in a substantial way. Here we propose simple algebraic heuristics to perform the shifting and test these techniques with the ARMS and ILUT preconditioners. Comparisons are made with applications stemming from the diffraction of an acoustic wave incident on a bounded obstacle (governed by the Helmholtz Wave Equation).
Axisymmetric Numerical Relativity
, 2005
"... Chapters 2, 3 and 6 contain work done in collaboration with my supervisor and published in a joint paper [119]. The dynamical shift conditions in chapter 6 are a later addition by myself. The remaining chapters are my own work. All computer programmes were written by myself unless otherwise stated. ..."
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Cited by 18 (8 self)
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Chapters 2, 3 and 6 contain work done in collaboration with my supervisor and published in a joint paper [119]. The dynamical shift conditions in chapter 6 are a later addition by myself. The remaining chapters are my own work. All computer programmes were written by myself unless otherwise stated. c○Oliver Rinne, 2005 This thesis is concerned with formulations of the Einstein equations in axisymmetric spacetimes which are suitable for numerical evolutions. The common basis for our formulations is provided by the (2+1)+1 formalism. General matter sources and rotational degrees of freedom are included. A first evolution system adopts elliptic gauge conditions arising from maximal slicing and conformal flatness. The numerical implementation is based on the finitedifference approach, using a Multigrid algorithm for the elliptic equations and the method of lines for the hyperbolic evolution equations.
Efficient Iterative Algorithms For The Stochastic Finite Element Method With Application To . . .
, 2002
"... In this study, we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions. We show that the solution can be represented in a com ..."
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Cited by 17 (2 self)
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In this study, we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions. We show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage, and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution. We also show how to generate a variety of statistical quantities from the computed solution. Finally, we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering.