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• W2063360699 abstract "Let $$A$$ be a compact $$d$$ -rectifiable set embedded in Euclidean space $${mathbb R}^p, dle p$$ . For a given continuous distribution $$sigma (x)$$ with respect to a $$d$$ -dimensional Hausdorff measure on $$A$$ , our earlier results provided a method for generating $$N$$ -point configurations on $$A$$ that have an asymptotic distribution $$sigma (x)$$ as $$Nrightarrow infty $$ ; moreover, such configurations are “quasi-uniform” in the sense that the ratio of the covering radius to the separation distance is bounded independently 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 $$w(x,y)=left( sigma (x)sigma (y)right) ^{-({s}/{2d})}$$ . Here we show that one can generate points on $$A$$ with the aforementioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most $$r_N=C_N N^{-1/d}$$ from each other, with $$C_N$$ being a positive sequence tending to infinity arbitrarily slowly. To do this, we minimize the energy with respect to a varying truncated weight $$v_N(x,y)=Phi (|x-y|/r_N)cdot w(x,y)$$ , where $$Phi :(0,infty )rightarrow [0,infty )$$ is a bounded function with $$Phi (t)=0, tge 1$$ , and $$lim _{trightarrow 0^+}Phi (t)=1$$ . Under appropriate assumptions, this reduces the complexity of generating $$N$$ -point “low energy” discretizations to order $$N C_N^d$$ computations." @default.
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• W2063360699 date "2014-05-08" @default.
• W2063360699 modified "2023-09-29" @default.
• W2063360699 title "Low Complexity Methods For Discretizing Manifolds Via Riesz Energy Minimization" @default.
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• W2063360699 doi "https://doi.org/10.1007/s10208-014-9202-3" @default.
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