Adaptive aggregation on graphs
Wenfang Xu, Ludmil T. Zikatanov
TL;DR
This paper extends a posteriori error estimation techniques from finite element methods to general graphs and Hilbert spaces, and uses these estimates to develop adaptive aggregation methods for graph Laplacians.
Contribution
It introduces a generalized a posteriori error estimator for graphs and Hilbert spaces, and applies it to create adaptive aggregation techniques for graph Laplacians.
Findings
The estimator effectively assesses aggregation quality.
The reshaping algorithm improves numerical results.
The approach is validated on multiple numerical examples.
Abstract
We generalize some of the functional (hyper-circle) a posteriori estimates from finite element settings to general graphs or Hilbert space settings. We provide several theoretical results in regard to the generalized a posteriori error estimators. We use these estimates to construct aggregation based coarse spaces for graph Laplacians. The estimator is used to assess the quality of an aggregation adaptively. Furthermore, a reshaping algorithm based is tested on several numerical examples.
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