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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 methodfor generating $N$-point configurations on $A$ that have asymptoticdistribution $\sigma (x)$ as $N\to \infty$; moreover such configurations are"quasi-uniform" in the sense that the ratio of the covering radius to theseparation distance is bounded independent of $N$. The method is based uponminimizing the energy of $N$ particles constrained to $A$ interacting via aweighted power law potential $w(x,y)|x-y|^{-s}$, where $s>d$ is a fixedparameter 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 mentionedproperties keeping in the energy sums only those pairs of points that arelocated 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 weminimize the energy with respect to a varying truncated weight$v_N(x,y)=\Phi$\left|x-y\right|/r_N$w(x,y)$, where $\Phi:(0,\infty)\to[0,\infty)$ is a bounded function with $\Phi(t)=0$, $t\geq 1$, and $\lim_{t\to0^+}\Phi(t)=1$. This reduces, under appropriate assumptions, the complexity ofgenerating $N$ point `low energy' discretizations to order $N C_N^d$computations.

Low complexity methods for discretizing manifolds via Riesz energy minimization