An improved convergence analysis of smoothed aggregation algebraic multigrid

Brezina, Marian; Vaněk, Petr; Vassilevski, Panayot S.

doi:10.1002/nla.775

Cited by 34 publications

(53 citation statements)

References 11 publications

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“…Recent improvements of the convergence theory in [4], establishing the same convergence result as presented here, were also restricted to the smoothed aggregation method.…”

Section: Introductionmentioning

confidence: 74%

“…P r o o f. The proof given here is a generalization of the one given in [4]. Recall that both S k and I −λ…”

Section: Polynomial Smoothermentioning

confidence: 85%

“…(That is, for even γ > 0, we do not have to assume that S k is given by (3.11) and (3.8), we only need S k to be a polynomial in A k such that ̺(S k ) 1.) P r o o f. For the proof in the case of γ = 1 and S k given by (3.11) and (3.8), see [4], Lemma 6.2 and Proposition 7.3.…”

Section: Polynomial Smoothermentioning

confidence: 99%

“…The smoother we use was originally derived from a prolongator smoother in the context of the smoothed aggregation method [9], [7], [4], and the previously proved theory [9], [8] was also limited to that context. Recent improvements of the convergence theory in [4], establishing the same convergence result as presented here, were also restricted to the smoothed aggregation method.…”

Section: Introductionmentioning

confidence: 99%

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Nearly optimal convergence result for multigrid with aggressive coarsening and polynomial smoothing

Vaněk

Brezina

2013

Appl Math

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“…Recent improvements of the convergence theory in [4], establishing the same convergence result as presented here, were also restricted to the smoothed aggregation method.…”

Section: Introductionmentioning

confidence: 74%

“…P r o o f. The proof given here is a generalization of the one given in [4]. Recall that both S k and I −λ…”

Section: Polynomial Smoothermentioning

confidence: 85%

Section: Polynomial Smoothermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nearly optimal convergence result for multigrid with aggressive coarsening and polynomial smoothing

Vaněk

Brezina

2013

Appl Math

Self Cite

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“…(3.9) also shows that no inverses of A are involved when applying M −1 . The smoother defined above has the coercivity property (3.5) (see [9,10]). The following smoothing property also holds (see [9,10])…”

Section: Multigridmentioning

confidence: 99%

Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

Kalchev¹

2012

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This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-ρAMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigenproblems (corresponding to eigenvalues close to zero) form the coarse basis used to define the new two-level method of our interest. We illustrate the performance of this adaptive two-level procedure with a large set of numerical experiments that demonstrate its efficiency over building the solvers from scratch.

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An algebraic multilevel method for anisotropic elliptic equations based on subgraph matching

Brannick

Chen

Zikatanov

2012

Numerical Linear Algebra App

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SUMMARY This paper presents a strength of connection measure for algebraic multilevel algorithms for a class of linear systems corresponding to the graph Laplacian on a general graph. The coarsening in the multilevel algorithm is based on partitioning in subgraphs (using matching) of the underlying graph. Our idea is to define a local measure of the quality of the matching that follows from a commutative diagram we introduce, whose maximum gives an upper bound on the stability (energy seminorm) of the orthogonal projection on the coarse space. As an application, we focus on utilizing this measure as a tool for constructing coarse spaces for anisotropic diffusion problems. Specifically, we consider the diffusion equation with grid aligned as well as non‐grid‐aligned anisotropies in the diffusion coefficient and show that the strength of connection measure is able to appropriately capture the correct anisotropic behavior in both cases. We then study a coarsening algorithm that uses this measure in a greedy strategy to find the subgraph partitioning (set of aggregates). The process forms an initial set of subgraphs, each consisting of a single vertex, and then adds vertices to these subgraphs corresponding to the local direction of the anisotropy as determined by the proposed measure. Copyright © 2012 John Wiley & Sons, Ltd.

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An improved convergence analysis of smoothed aggregation algebraic multigrid

Cited by 34 publications

References 11 publications

Nearly optimal convergence result for multigrid with aggressive coarsening and polynomial smoothing

Nearly optimal convergence result for multigrid with aggressive coarsening and polynomial smoothing

Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

An algebraic multilevel method for anisotropic elliptic equations based on subgraph matching

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