Moment methods in energy minimization: New bounds for Riesz minimal energy problems

Moment Methods in Energy Minimization: New Bounds for Riesz Minimal Energy Problems

We use moment methods to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate the infinite dimensional optimization problems in this hierarchy by block diagonal semidefinite programs. For this we develop the necessary harmonic analysis for spaces consisting of subsets of another space, and we develop sym...

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 radi...

Energy Minimization Methods

MINIMAL RIESZ ENERGY POINT CONFIGURATIONS FOR RECTIFIABLE d-DIMENSIONAL MANIFOLDS

We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A ⊂ Rd′ that interact through the power law (Riesz) potential V = 1/r, where s > 0 and r is Euclidean distance in R ′ . With Es(A, N) denoting the minimal energy for such N -point configurations, we determine the asymptotic behavior (as N → ∞) of Es(A, N) for each fixed s ≥ d. Moreover, if A has positi...

Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets

Given a compact d-rectifiable set A embedded in Euclidean space and a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, we address the following question: how can one generate optimal configurations of N points on A that are “well-separated” and have asymptotic distribution ρ(x) as N → ∞? For this purpose we investigate minimal weighted Riesz energy points, that is, points...