2013/05/27 by Sergiy Borodachov, Douglas P. Hardin, Borodachov, S. V. +3
Mathematics · #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1305.6337
openalex publication_date 2013/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
Let A be a compact d-rectifiable set embedded in Euclidean space \RRp, d≤ p. For a given continuous distribution σ(x) with respect to d-dimensional Hausdorff measure on A, our earlier results provided a method for generating N-point configurations on A that have asymptotic distribution σ(x) as N→ ∞; moreover such configurations are "quasi-uniform" in the sense that the ratio of the covering radius to the separation distance is bounded independent of N. The method is based upon minimizing the energy of N particles constrained to A interacting via a weighted power law potential w(x,y)|x-y|-s, where s>d is a fixed parameter and w(x,y)=(σ(x)σ(y))^-(s/2d). Here we show that one can generate points on A with the above mentioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most rN=CN N-1/d from each other, with CN being a positive sequence tending to infinity arbitrarily slowly. To do this we minimize the energy with respect to a varying truncated weight vN(x,y)=Φ\(|x-y|/rN\)w(x,y), where Φ:(0,∞)→ [0,∞) is a bounded function with Φ(t)=0, t≥ 1, and limt→ 0+Φ(t)=1. This reduces, under appropriate assumptions, the complexity of generating N point `low energy' discretizations to order N CNd computations.