Hybrid $V$-cycle algebraic multilevel preconditioners

Vassilevski, Panayot S.

doi:10.1090/s0025-5718-1992-1122081-6

Cited by 15 publications

(6 citation statements)

References 15 publications

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“…Case (3) is of special importance for a further extension of the method to the multilevel case by two possible approaches; the smoothed aggregation algebraic multigrid (AMG) (cf., e.g. Reference [19]), and to (A)MG based on the so-called AMLI-cycle (in the spirit of References [20,21]). The latter multilevel extensions are subject of ongoing research [22].…”

Section: Aims and Main Resultsmentioning

confidence: 99%

Two‐level preconditioning of discontinuous Galerkin approximations of second‐order elliptic equations

Dobrev¹,

Lazarov²,

Vassilevski³

et al. 2006

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This paper reviews some known and proposes some new preconditioning methods for a number of discontinuous Galerkin (or DG) finite element approximations for elliptic problems of second order. Nested hierarchy of meshes is generally assumed. Our approach utilizes a general two-level scheme, where the finite element space for the DG method is decomposed into a subspace (viewed as an auxiliary or 'coarse' space), plus a correction which can be handled by a standard smoothing procedure. We consider three different auxiliary subspaces, namely, piecewise linear C 0 -conforming functions, piecewise linear functions that are continuous at the centroids of the edges/faces (Crouzeix-Raviart finite elements) and piecewise constant functions over the finite elements. To support the theoretical results, we also present numerical experiments for 3-D model problem showing uniform convergence of the constructed methods.(2) for any >0. Similarly, the symmetric bilinear forms A sym and A sew are coercive and bounded in V equipped with the norm (2) for >0 sufficiently large.The total number of degrees of freedom for the DG method is n h and n c,h is the number of degrees of freedom for the 'coarse' space of continuous FE.

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Section: Aims and Main Resultsmentioning

confidence: 99%

Two‐level preconditioning of discontinuous Galerkin approximations of second‐order elliptic equations

Dobrev¹,

Lazarov²,

Vassilevski³

et al. 2006

Numerical Linear Algebra App

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“…The AMLI-cycle was originally developed in [2,3,30] for multilevel preconditioners based on recursive 2 × 2 block incomplete factorizations, and was further adapted to multigrid methods in [31]. We do not bring additional contributions to that topic and therefore present only a short summary here, focusing on facts and practical aspects, whereas justifications are omitted and theoretical properties are stated without proof.…”

Section: The Amli-cyclementioning

confidence: 99%

“…A second advantage of the approach is that it fits well with the use of the socalled AMLI-cycle [2,3,30,31]. With this cycle, the iterative solution of the coarse system at each level is accelerated with a semi-iterative method based on Chebyshev polynomials.…”

mentioning

confidence: 99%

An Algebraic Multigrid Method with Guaranteed Convergence Rate

Napov¹,

Notay²

2012

SIAM J. Sci. Comput.

201

175

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We consider the iterative solution of large sparse symmetric positive definite linear systems. We present an algebraic multigrid method which has a guaranteed convergence rate for the class of nonsingular symmetric M-matrices with nonnegative row sum. The coarsening is based on the aggregation of the unknowns. A key ingredient is an algorithm that builds the aggregates while ensuring that the corresponding two-grid convergence rate is bounded by a user-defined parameter. For a sensible choice of this parameter, it is shown that the recursive use of the two-grid procedure yields a convergence independent of the number of levels, provided that one uses a proper AMLIcycle. On the other hand, the computational cost per iteration step is of optimal order if the mean aggregate size is large enough. This cannot be guaranteed in all cases but is analytically shown to hold for the model Poisson problem. For more general problems, a wide range of experiments suggests that there are no complexity issues and further demonstrates the robustness of the method. The experiments are performed on systems obtained from low order finite difference or finite element discretizations of second order elliptic partial differential equations (PDEs). The set includes twoand three-dimensional problems, with both structured and unstructured grids, some of them with local refinement and/or reentering corner, and possible jumps or anisotropies in the PDE coefficients.

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“…For symmetric positive definite (SPD) problems, more involved cycles have been proposed. Axelsson and Vassilevski introduced the algebraic multilevel iteration (AMLI)-cycle MG method [2,3,34], which uses Chebyshev polynomial to define the coarse-level solver. However, the AMLI-cycle MG method requires an accurate estimation of extreme eigenvalues on coarse levels to compute the coefficients of the Chebyshev polynomial, which may be difficult in practice.…”

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confidence: 99%

Momentum Accelerated Multigrid Methods

Niu,

2020

Preprint

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In this paper, we propose two momentum accelerated MG cycles. The main idea is to rewrite the linear systems as optimization problems and apply momentum accelerations, e.g., the heavy ball and Nesterov acceleration methods, to define the coarse-level solvers for multigrid (MG) methods. The resulting MG cycles, which we call them H-cycle (uses heavy ball method) and N-cycle (uses Nesterov acceleration), share the advantages of both algebraic multilevel iteration (AMLI)-and K-cycle (a nonlinear version of the AMLI-cycle). Namely, similar to the K-cycle, our H-and N-cycle do not require the estimation of extreme eigenvalues while the computational cost is the same as AMLI-cycle. Theoretical analysis shows that the momentum accelerated cycles are essentially special AMLI-cycle methods and, thus, they are uniformly convergent under standard assumptions. Finally, we present numerical experiments to verify the theoretical results and demonstrate the efficiency of H-and N-cycle.

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Hybrid $V$-cycle algebraic multilevel preconditioners

Cited by 15 publications

References 15 publications

Two‐level preconditioning of discontinuous Galerkin approximations of second‐order elliptic equations

Two‐level preconditioning of discontinuous Galerkin approximations of second‐order elliptic equations

An Algebraic Multigrid Method with Guaranteed Convergence Rate

Momentum Accelerated Multigrid Methods

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