Accelerated multigrid for graph Laplacian operators

Dell'Acqua, Pietro; Frangioni, Antonio; Serra‐Capizzano, Stefano

doi:10.1016/j.amc.2015.08.033

Cited by 4 publications

(8 citation statements)

References 58 publications

(152 reference statements)

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“…Coarsening PQ and PV nodes. When applying the above smoother at a coarse level, we cycle through a sequence of coarse node groupings as in (10). If the group is of size 1, we simply apply the smoother as normal.…”

Section: Smoothingmentioning

confidence: 99%

“…Thus, Linear multigrid methods have also been applied to linear systems that do not arise from partial differential equations, and often do not even have an underlying geometry [7]. This has been most successful for graph Laplacian matrices, which are in many ways an algebraic analogue to the geometric Poisson problem [16,33,34,10]. Algebraic multigrid, with its ever expanding area of application and intent for extreme scale parallel scalability, is still an active area of research [14,6,46,17,4].…”

mentioning

confidence: 99%

“…If L is a graph Laplacian and one chooses R = P T , the matrix P T LP is also symmetric and positive-definite, and if P is piecewise-constant, then P T LP is a graph Laplacian as well. For general conditions for preserving the structure of graph Laplacian, see sections 3 and 4 of [10]. The choice of restriction and interpolation matrices is one of the many choices in developing a specific multigrid method, and a great deal of literature exists on this question (see, e.g., [44,7,13]).…”

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confidence: 99%

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A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations

Ponce¹,

Bindel²,

Vassilevski³

2018

SIAM J. Sci. Comput.

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Multigrid is a highly scalable class of methods most often used for solving large linear systems. In this paper we develop a nonlinear algebraic multigrid framework for the power flow equations, a complex quadratic system of the form diag(\bfitv)Y \bfitv = \bfits , where Y is approximately a complex scalar rotation of a real graph Laplacian. This is a standard problem that needs to be solved repeatedly during power grid simulations. A key difference between our multigrid framework and typical multigrid approaches is the use of a novel multiplicative coarse-grid correction to enable a dynamic multigrid hierarchy. We also develop a new type of smoother that allows one to coarsen together the different types of nodes that appear in power grid simulations. In developing a specific multigrid method, one must make a number of choices that can significantly affect the method's performance, such as how to construct the restriction and interpolation operators, what smoother to use, and how aggressively to coarsen. In this paper, we make simple but reasonable choices that result in a scalable and robust power flow solver. Experiments demonstrate this scalability and show that it is significantly more robust to poor initial guesses than current state-of-the-art solvers.

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Section: Smoothingmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations

Ponce¹,

Bindel²,

Vassilevski³

2018

SIAM J. Sci. Comput.

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“…More elaborated variants of interior point methods, i.e. primal-dual methods, nowadays are also applied routinely to MCF problems [8,9,10,11,16,31,33,34]. In the framework of IPMs, the main computational burden consists in the solution of a linear system of equations, involving the symmetric but possibly indefinite Jacobian arising within Newton's method applied to the KKT equations from the IPM.…”

Section: Introductionmentioning

confidence: 99%

Towards a multigrid method for the minimum-cost flow problem

Quaglino,

Krause

2016

Preprint

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We present a first step towards a multigrid method for solving the min-cost flow problem. Specifically, we present a strategy that takes advantage of existing black-box fast iterative linear solvers, i.e. algebraic multigrid methods. We show with standard benchmarks that, while less competitive than combinatorial techniques on small problems that run on a single core, our approach scales well with problem size, complexity, and number of processors, allowing for tackling large-scale problems on modern parallel architectures. Our approach is based on combining interior-point with multigrid methods for solving the nonlinear KKT equations via Newton's method. However, the Jacobian matrix arising in the Newton iteration is indefinite and its condition number cannot be expected to be bounded. In fact, the eigenvalues of the Jacobian can both vanish and blow up near the solution, leading to a significant slow-down of the convergence speed of iterative solvers -or to the loss of convergence at all. In order to allow for the application of multigrid methods, which have been originally designed for elliptic problems, we furthermore show that the occurring Jacobian can be interpreted as the stiffness matrix of a mixed formulation of the weighted graph Laplacian of the network, whose metric depends on the slack variables and the multipliers of the inequality constraints. Together with our regularization, this allows for the application of a black-box algebraic multigrid method on the Schur-complement of the system.

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“…It should also be noted that there are a few methods available for the problem with fast empirical running times; but with no equivalent guarantee on the theoretical worst-case running time: combinatorial multigrid (CMG) [25] and lean algebraic multigrid (LAMG) [26] are, as the names suggest, multigrid schemes; the work by Dell'Acqua et al [27,28] has evolved into a multi-iterative scheme combining multigrid and graph-based preconditioners (among others using spanning trees).…”

Section: Related Workmentioning

confidence: 99%

Engineering a Combinatorial Laplacian Solver: Lessons Learned

Hoske

Lukarski²,

Meyerhenke

et al. 2016

Algorithms

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Abstract:Linear system solving is a main workhorse in applied mathematics. Recently, theoretical computer scientists contributed sophisticated algorithms for solving linear systems with symmetric diagonally-dominant (SDD) matrices in provably nearly-linear time. These algorithms are very interesting from a theoretical perspective, but their practical performance was unclear. Here, we address this gap. We provide the first implementation of the combinatorial solver by Kelner et al. (STOC 2013), which is appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix (which is SDD) corresponds to a graph; solving symmetric Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning tree with the low-stretch property. The results of our experiments are ambivalent. While they confirm the predicted nearly-linear running time, the constant factors make the solver much slower for reasonable inputs than basic methods with higher asymptotic complexity. We were also not able to use the solver effectively as a smoother or preconditioner. Moreover, while spanning trees with lower stretch indeed reduce the solver's running time, we experience again a discrepancy in practice: in our experiments, simple spanning tree algorithms perform better than those with a guaranteed low stretch. We expect that our results provide insights for future improvements of combinatorial linear solvers.

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Accelerated multigrid for graph Laplacian operators

Cited by 4 publications

References 58 publications

A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations

A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations

Towards a multigrid method for the minimum-cost flow problem

Engineering a Combinatorial Laplacian Solver: Lessons Learned

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