Low Complexity Methods For Discretizing Manifolds Via Riesz Energy Minimization

Let A be a compact d-rectifiable set embedded in Euclidean space R, 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)). 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 = CNN −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 NC N computations.