Low complexity methods for discretizing manifolds via Riesz energy minimization

Abstract

Let $A$ be a compact $d$-rectifiable set embedded in Euclidean space $\RR^p$, $d\le p$. For a given continuous distribution $\sigma (x)$ 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 $\sigma (x)$ as $N\to \infty$; 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 $w(x,y)={\left(\sigma (x)\sigma (y)\right)}^{-(s/2d)}$. 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 $r_N=C_N{N}^{-1/d}$ 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)=Φ\(\left|x-y\right|/r_N\)w(x,y)$, where $\Phi :(0,\infty )\to [0,\infty )$ is a bounded function with $\Phi (t)=0$, $t\ge 1$, and ${\lim }_{t\to 0^{+}}\Phi (t)=1$. This reduces, under appropriate assumptions, the complexity of generating $N$ point `low energy' discretizations to order $N{C}_N^d$ computations.

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