Numerical Integration on Graphs: where to sample and how to weigh
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Let G=(V,E,w) be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset W ⊂ V of vertices and weights a_w such that 1/|V|∑_v ∈ V^f(v)∼∑_w ∈ Wa_w f(w) for functions f:V →R that are `smooth' with respect to the geometry of the graph. The main application are problems where f 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.
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