Auxiliary Space Multigrid Method Based on Additive Schur Complement Approximation for Graph Laplacian

TL;DR

This paper introduces an algebraic auxiliary space multigrid method based on additive Schur complement approximation for solving graph Laplacian systems, offering a flexible, efficient, and structurally agnostic approach validated by numerical experiments.

Contribution

It presents a novel algebraic multigrid approach that uses additive Schur complement approximation, adaptable to general graphs without structural limitations.

Findings

Method is purely algebraic and easy to implement.

No restrictions on graph structure.

Computational complexity is straightforward to analyze.

Abstract

This research explores the application of the auxiliary space multigrid method (ASMG) that is based on additive Schur complement approximation (ASCA) to graph Laplacian matrices arising from general graphs. A major predicament when considering algebraic multigrid (AMG) methods on such graphs is the choice of a general coarsening strategy which has to be both cheap and effective. Such a strategy has been incorporated in the presented approach which in addition has several advantages. First, it is purely algebraic in its construction which makes the algorithm easy to implement. Furthermore, the approach requires no limitation on the graph's structure and itself can be adjusted to the particular problem. Last but not least, its computational complexity can be easily analysed. A demonstrative set of numerical experiments is presented.