Published on Mon Mar 19 2018

Numerical Integration on Graphs: where to sample and how to weigh

George C. Linderman, Stefan Steinerberger

Let be a finite, connected graph with weighted edges. We prove an inequality showing that the integration problem can be rewritten as a geometric problem. We discuss how one would construct approximate solutions of the heat ball packing problem.

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Abstract

Let be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset of vertices and weights such that $$ \frac{1}{|V|}\sum_{v \in V}^{}{f(v)} \sim \sum_{w \in W}{a_w f(w)}$$ for functions that are `smooth' with respect to the geometry of the graph. The main application are problems where is known to somehow depend on 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.