A Flexible Inner-Outer Preconditioned GMRES Algorithm

Saad, Yousef

doi:10.1137/0914028

Cited by 1,199 publications

(830 citation statements)

References 4 publications

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“…The most efficient iterative solvers available today include Krylov subspace methods, such as GMRES and BICGSTAB [30], and multigrid methods. In our work, we employ a Flexible GMRES (FGMRES) solver [31] to solve the Poisson equation. Typically a GMRES solver is used together with a pre-conditioner, such as Jacobi, SOR, incomplete LU (ILU) decomposition [32], etc., in order to improve its robustness and efficiency.…”

Section: Solution Of the Pressure-correction Equationmentioning

confidence: 99%

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

Sotiropoulos

2007

Journal of Computational Physics

357

348

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A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

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Section: Solution Of the Pressure-correction Equationmentioning

confidence: 99%

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

Sotiropoulos

2007

Journal of Computational Physics

357

348

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“…[29] for a general introduction on Krylov subspace methods and to [29,Section 10] and [25,Section 9.4] for a review on flexible methods. The minimum residual norm GMRES method [26] has been extended by Saad [23] to allow variable preconditioning. The resulting algorithm known as FGMRES(m) relies on the Arnoldi relation…”

Section: General Settingmentioning

confidence: 99%

“…See also [1,Section 12.3] for additional references. Since then, numerous methods have been proposed to address the symmetric, non-symmetric or non-Hermitian cases; these include Flexible Conjugate Gradient [19], Flexible GMRES (FGMRES) [23], Flexible QMR [31] and GMRESR [34] among others. This class of methods is required when preconditioning with a different (possibly nonlinear) operator at each iteration of a subspace method is considered.…”

Section: Introductionmentioning

confidence: 99%

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Carvalho¹,

Gratton²,

Lago³

et al. 2011

SIAM J. Matrix Anal. & Appl.

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This work is concerned with the development and study of a minimum residual norm subspace method based on the Generalized Conjugate Residual method with inner Orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of non-symmetric or non-Hermitian linear systems. First we recall the main features of Flexible Generalized Minimum Residual with deflated restarting (FGMRES-DR), a recently proposed algorithm of the same family but based on the GMRES method. Next we introduce the new inner-outer subspace method named FGCRO-DR. A theoretical comparison of both algorithms is then made in the case of flexible preconditioning. It is proved that FGCRO-DR and FGMRES-DR are algebraically equivalent if a collinearity condition is satisfied. While being nearly as expensive as FGMRES-DR in terms of computational operations per cycle, FGCRO-DR offers the additional advantage to be suitable for the solution of sequences of slowly changing linear systems (where both the matrix and right-hand side can change) through subspace recycling. Numerical experiments on the solution of multidimensional elliptic partial differential equations show the efficiency of FGCRO-DR when solving sequences of linear systems.

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“…The approximate solution to Ac = r i then becomes the next direction for the outer approximate space. Note that flexible GMRES (FGMRES) method [18] can also be view as a method that approximates solutions to similar residual equations at each step. Different from GMRESR method, the inner and outer iterative methods in FGMRES are both GMRES.…”

Section: Acceleration Techniques For Gmresmentioning

confidence: 99%

Adaptively Accelerated Gmres Fast Fourier Transform Method for Electromagnetic Scattering

Xin¹,

Rui²

2008

PIER

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Abstract-The problem of electromagnetic scattering by 3D dielectric bodies is formulated in terms of a weak-form volume integral equation. Applying Galerkin's method with rooftop functions as basis and testing functions, the integral equation can be usually solved by Krylov-subspace fast Fourier transform (FFT) iterative methods. In this paper, the generalized minimum residual (GMRES)-FFT method is used to solve this integral equation, and several adaptive acceleration techniques are proposed to improve the convergence rate of the GMRES-FFT method. On several electromagnetic scattering problems, the performance of these adaptively accelerated GMRES-FFT methods are thoroughly analyzed and compared.

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A Flexible Inner-Outer Preconditioned GMRES Algorithm

Cited by 1,199 publications

References 4 publications

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Adaptively Accelerated Gmres Fast Fourier Transform Method for Electromagnetic Scattering

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