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On the Global Optimality of Fibonacci Lattices in the Torus

Nicolas Nagel

TL;DR

The paper addresses the global optimality of point configurations on the torus with respect to tensor-product energies that unify discrepancy and quasi-Monte Carlo notions. It develops a linear programming bound using an arccos-transform and magic functions to certify lower bounds on energy and to identify equality cases. The main results prove the universal optimality of the 3-point lattice $\mathcal{T}^d$ in any dimension for a broad class of potentials, and establish that the 5-point Fibonacci lattice in 2D is globally optimal for a continuously parametrized family of potentials $c_p^2$, $0\le p\le 9$. These findings advance understanding of discrepancy-optimal point sets and have potential implications for quasi-Monte Carlo methods and related spaces, with several natural avenues for generalization to other lattices and product spaces.

Abstract

We use linear programming bounds to analyze point sets in the torus with respect to their optimality for problems in discrepancy theory and quasi-Monte Carlo methods. These concepts will be unified by introducing tensor product energies. We show that the canonical $3$-point lattice in any dimension is globally optimal among all $3$-point sets in the torus with respect to a large class of such energies. This is a new instance of universal optimality, a special phenomenon that is only known for a small class of highly structured point sets. In the case of $d=2$ dimensions it is conjectured that so-called Fibonacci lattices should also be optimal with respect to a large class of potentials. To this end we show that the $5$-point Fibonacci lattice is globally optimal for a continuously parametrized class of potentials relevant to the analysis fo the quasi-Monte Carlo method.

On the Global Optimality of Fibonacci Lattices in the Torus

TL;DR

The paper addresses the global optimality of point configurations on the torus with respect to tensor-product energies that unify discrepancy and quasi-Monte Carlo notions. It develops a linear programming bound using an arccos-transform and magic functions to certify lower bounds on energy and to identify equality cases. The main results prove the universal optimality of the 3-point lattice in any dimension for a broad class of potentials, and establish that the 5-point Fibonacci lattice in 2D is globally optimal for a continuously parametrized family of potentials , . These findings advance understanding of discrepancy-optimal point sets and have potential implications for quasi-Monte Carlo methods and related spaces, with several natural avenues for generalization to other lattices and product spaces.

Abstract

We use linear programming bounds to analyze point sets in the torus with respect to their optimality for problems in discrepancy theory and quasi-Monte Carlo methods. These concepts will be unified by introducing tensor product energies. We show that the canonical -point lattice in any dimension is globally optimal among all -point sets in the torus with respect to a large class of such energies. This is a new instance of universal optimality, a special phenomenon that is only known for a small class of highly structured point sets. In the case of dimensions it is conjectured that so-called Fibonacci lattices should also be optimal with respect to a large class of potentials. To this end we show that the -point Fibonacci lattice is globally optimal for a continuously parametrized class of potentials relevant to the analysis fo the quasi-Monte Carlo method.

Paper Structure

This paper contains 10 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preface on energy
  3. Quasi-Monte Carlo integration, discrepancy and diaphony
  4. Tensor product energies
  5. Main results
  6. Notation
  7. The linear programming bound
  8. Universal optimality of the $3$-point lattices
  9. The $5$-point Fibonacci lattice
  10. Summary and outlook

Key Result

Theorem 2

Let $d=2$ and let the potential $c = c_p^2$ be given by eq:potential_for_kpd. Then the $5$-point Fibonacci lattice minimizes $E_{c_p^2}$ among all $5$-point sets in $[0, 1)^2$ for the range $0 \leq p \leq 9$.

Theorems & Definitions (16)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Remark 5
  • Theorem 6
  • proof : Proof of Theorem \ref{['thm:continuous_bound']}
  • Corollary 7
  • proof
  • Definition 8
  • ...and 6 more