Solving Graph Laplacian Systems Through Recursive Partitioning and Two-Grid Preconditioning

Ponce, Colin; Vassilevski, Panayot S.

doi:10.1137/15m1050872

SIAM J. Matrix Anal. & Appl.

2017

DOI: 10.1137/15m1050872

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Solving Graph Laplacian Systems Through Recursive Partitioning and Two-Grid Preconditioning

Colin Ponce

Panayot S. Vassilevski

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Cited by 2 publications

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“…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].…”

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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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A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations

Ponce¹,

Bindel²,

Vassilevski³

2018

SIAM J. Sci. Comput.

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Iterative Preconditioned Methods in Krylov Spaces: Trends of the 21st Century

Ильин

2021

Comput. Math. and Math. Phys.

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Solving Graph Laplacian Systems Through Recursive Partitioning and Two-Grid Preconditioning

Cited by 2 publications

References 8 publications

A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations

A Nonlinear Algebraic Multigrid Framework for the Power Flow Equations

Iterative Preconditioned Methods in Krylov Spaces: Trends of the 21st Century

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