Optimal Interpolation and Compatible Relaxation in Classical Algebraic Multigrid

Brannick, James; Cao, Fei; Kahl, Karsten; Falgout, Robert D.; Hu, Xiaozhe

doi:10.1137/17m1123456

Cited by 28 publications

(34 citation statements)

References 21 publications

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“…When applicable, AMG converges in linear complexity with the number of degrees-of-freedom (DOFs), and scales in parallel like O(log 2 (P )), up to hundreds of thousands of processors, P [6]. Originally, AMG was designed for essentially SPD linear systems, and convergence of AMG is relatively well understood in the SPD setting [10,11,25,59,62]. Nonsymmetric matrices pose unique difficulties for AMG in theory and in practice.…”

mentioning

confidence: 99%

Nonsymmetric Reduction-Based Algebraic Multigrid

Manteuffel¹,

Münzenmaier²,

Ruge³

et al. 2019

SIAM J. Sci. Comput.

104

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Algebraic multigrid (AMG) is often an effective solver for symmetric positive definite (SPD) linear systems resulting from the discretization of general elliptic PDEs, or the spatial discretization of parabolic PDEs. However, convergence theory and most variations of AMG rely on A being SPD. Hyperbolic PDEs, which arise often in large-scale scientific simulations, remain a challenge for AMG, as well as other fast linear solvers, in part because the resulting linear systems are often highly nonsymmetric. Here, a novel convergence framework is developed for nonsymmetric, reduction-based AMG, and sufficient conditions derived for 2 -convergence of error and residual. In particular, classical multigrid approximation properties are connected with reduction-based measures to develop a robust framework for nonsymmetric, reduction-based AMG.Matrices with block-triangular structure are then recognized as being amenable to reductiontype algorithms, and a reduction-based AMG method is developed for upwind discretizations of hyperbolic PDEs, based on the concept of a Neumann approximation to ideal restriction (nAIR). nAIR can be seen as a variation of local AIR ( AIR) introduced in previous work, specifically targeting matrices with triangular structure. Although less versatile than AIR, setup times for nAIR can be substantially faster for problems with high connectivity. nAIR is shown to be an effective and scalable solver of steady state transport for discontinuous, upwind discretizations, with unstructured meshes, and up to 6th-order finite elements, offering a significant improvement over existing AMG methods. nAIR is also shown to be effective on several classes of "nearly triangular" matrices, resulting from curvilinear finite elements and artificial diffusion.

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mentioning

confidence: 99%

Nonsymmetric Reduction-Based Algebraic Multigrid

Manteuffel¹,

Münzenmaier²,

Ruge³

et al. 2019

SIAM J. Sci. Comput.

104

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“…If RR T = I nc , then (4.2) reduces to (2.14). We mention that a similar expression of (4.2) has been given in [3].…”

Section: A New Expression For the Ideal Interpolation In Pmentioning

confidence: 57%

“…However, P opt itself is expensive to compute due to its columns consist of eigenvectors corresponding to small eigenvalues. Explicit form of P opt (or the optimal coarse-space) and the precise value E TG (P opt ) A can be found, e.g., in [24,3]. Recently, some interesting relationships between the optimal and ideal interpolations have been discussed by Brannick et al [3].…”

Section: Geometric Illustration Of the Measure µ Mmentioning

confidence: 99%

On the Ideal Interpolation Operator in Algebraic Multigrid Methods

Xu¹,

Zhang²

2018

SIAM J. Numer. Anal.

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Various algebraic multigrid algorithms have been developed for solving problems in scientific and engineering computation over the past decades. They have been shown to be well-suited for solving discretized partial differential equations on unstructured girds in practice. One key ingredient of algebraic multigrid algorithms is a strategy for constructing an effective prolongation operator. Among many questions on constructing a prolongation, an important question is how to evaluate its quality. In this paper, we establish new characterizations (including sufficient condition, necessary condition, and equivalent condition) of the so-called ideal interpolation operator. Our result suggests that, compared with common wisdom, one has more room to construct an ideal interpolation, which can provide new insights for designing algebraic multigrid algorithms. Moreover, we derive a new expression for a class of ideal interpolation operators.

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“…In line with [28] we plan to investigate the connection between the partial differential operator, its discretization and the smoothing scheme with the resulting covariance structure of the Gaussian fields. To some extent this development can be seen in line with the investigation of optimal interpolation in algebraic multigrid methods in [9], where an explicit influence of the smoother on the optimal construction of interpolation has been shown. Insight into this might allow us to translate the demonstrated potential for efficient adaptive algebraic multigrid constructions using a minimal amount of test vectors to more complex problems.…”

mentioning

confidence: 57%

“…In addition, as an implicit requirement, the sparsity of the coarse system of equations, given by P T AP has to be guaranteed in order to be able to apply the construction recursively and thus achieve optimal linear complexity. Based on the findings in [9], the complementarity of the smoothing iteration and the coarse-grid correction is equivalent to the requirement that range(P ) approximates the space spanned by eigenvectors of the error propagator of the smoother corresponding to eigenvalues close to 1, i.e., components that are slow to converge, also known as algebraically smooth error components [29].…”

mentioning

confidence: 99%

Coarsening in algebraic multigrid using Gaussian processes

Gottschalk¹,

Kahl²

2021

etna

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Multigrid methods have proven to be an invaluable tool to efficiently solve large sparse linear systems arising in the discretization of Partial Differential Equations (PDEs). Algebraic multigrid methods and in particular adaptive algebraic multigrid approaches have shown that multigrid efficiency can be obtained without having to resort to properties of the PDE. Yet the required setup of these methods poses a not negligible overhead cost. Methods from machine learning have attracted attention to streamline processes based on statistical models being trained on the available data. Interpreting algebraically smooth error as an instance of a Gaussian process, we develop a new, data driven approach to construct adaptive algebraic multigrid methods. Based on Gaussian a priori distributions, kriging interpolation minimizes the mean squared error of the a posteriori distribution, given the data on the coarse grid. Going one step further, we exploit the quantification of uncertainty in the Gaussian process model in order to construct efficient variable splittings. Using a semivariogram fit of a suitable covariance model we demonstrate that our approach yields efficient methods using a single algebraically smooth vector.

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Optimal Interpolation and Compatible Relaxation in Classical Algebraic Multigrid

Cited by 28 publications

References 21 publications

Nonsymmetric Reduction-Based Algebraic Multigrid

Nonsymmetric Reduction-Based Algebraic Multigrid

On the Ideal Interpolation Operator in Algebraic Multigrid Methods

Coarsening in algebraic multigrid using Gaussian processes

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