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Mathematics of Computation

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Numerical integration on graphs: Where to sample and how to weigh


Authors: George C. Linderman and Stefan Steinerberger
Journal: Math. Comp. 89 (2020), 1933-1952
MSC (2010): Primary 05C50, 05C70, 35P05, 65D32
DOI: https://doi.org/10.1090/mcom/3515
Published electronically: January 29, 2020
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Abstract: Let $ G=(V,E,w)$ be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset $ W \subset V$ of vertices and weights $ a_w$ such that

$\displaystyle \frac {1}{\vert V\vert}\sum _{v \in V}^{}{f(v)} \sim \sum _{w \in W}{a_w f(w)}$    

for functions $ f:V \rightarrow \mathbb{R}$ that are ``smooth'' with respect to the geometry of the graph; here $ \sim $ indicates that we want the right-hand side to be as close to the left-hand side as possible. The main application comprises problems where $ f$ is known to vary smoothly over the underlying graph but is expensive to evaluate on even a single vertex. We prove an inequality showing that the integration problem can be rewritten as a geometric problem (``the optimal packing of heat balls''). We discuss how one would construct approximate solutions of the heat ball packing problem; numerical examples demonstrate the efficiency of the method.

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Additional Information

George C. Linderman
Affiliation: Program in Applied Mathematics, Yale University, New Haven, Connecticut 06511
Email: george.linderman@yale.edu

Stefan Steinerberger
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
Email: stefan.steinerberger@yale.edu

DOI: https://doi.org/10.1090/mcom/3515
Keywords: Graph, sampling, graph Laplacian, sampling, heat kernel, packing
Received by editor(s): March 20, 2018
Received by editor(s) in revised form: January 2, 2019, and September 1, 2019
Published electronically: January 29, 2020
Additional Notes: The first author was supported by NIH grant #1R01HG008383-01A1 (PI: Yuval Kluger) and U.S. NIH MSTP Training Grant T32GM007205.
The second author was supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.
Article copyright: © Copyright 2020 American Mathematical Society