Discrete Energy on Rectifiable Sets | springerprofessional.de

Springer Professional

nach oben

2019 | Buch

Kapitel lesen Erstes Kapitel lesen

Discrete Energy on Rectifiable Sets

verfasst von: Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Verlag: Springer New York

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This book aims to provide an introduction to the broad and dynamic subject of discrete energy problems and point configurations. Written by leading authorities on the topic, this treatise is designed with the graduate student and further explorers in mind. The presentation includes a chapter of preliminaries and an extensive Appendix that augments a course in Real Analysis and makes the text self-contained. Along with numerous attractive full-color images, the exposition conveys the beauty of the subject and its connection to several branches of mathematics, computational methods, and physical/biological applications.

This work is destined to be a valuable research resource for such topics as packing and covering problems, generalizations of the famous Thomson Problem, and classical potential theory in Rd. It features three chapters dealing with point distributions on the sphere, including an extensive treatment of Delsarte–Yudin–Levenshtein linear programming methods for lower bounding energy, a thorough treatment of Cohn–Kumar universality, and a comparison of 'popular methods' for uniformly distributing points on the two-dimensional sphere. Some unique features of the work are its treatment of Gauss-type kernels for periodic energy problems, its asymptotic analysis of minimizing point configurations for non-integrable Riesz potentials (the so-called Poppy-seed bagel theorems), its applications to the generation of non-structured grids of prescribed densities, and its closing chapter on optimal discrete measures for Chebyshev (polarization) problems.

Anzeige

Inhaltsverzeichnis

Frontmatter

Chapter 0. An Overview: Discretizing Manifolds via Particle Interactions

Abstract

The problem of distributing points on a manifold (or discretizing a manifold) arises in many contexts that are of interest to the scientific community as well as in applied fields—statistical sampling, quadrature rules, information theory, coding techniques, computer-aided design, interpolation schemes, finite element tessellations, ground states of matter—to name but a few. Our goal is to address this problem from the perspective of particle interactions; namely, starting from a given formula for the pairwise interactions of N particles (points) that are confined to a given manifold A in the Euclidean space \(\mathbb R^{p},\) we wish to describe the structure of those configurations that arise when the N particles reach an equilibrium state (a state of minimal energy).

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 1. Preliminaries

Abstract

The purpose of this chapter is to recall some of the definitions, results, and principles of measure theory, function theory, and dimension theory, which we will need to present the material of this book. The theorems we cite are supported by proofs or by appropriate references to the literature.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 2. Basic Properties and Examples of Minimal Discrete Energy

Abstract

In this chapter, we begin our study of point configurations minimizing an energy defined in terms of pairwise interactions. We introduce the notions of minimal discrete energy along with some of their basic properties. We further provide several classical examples where optimal configurations can be determined implicitly or explicitly.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 3. Introduction to Best-Packing and Best-Covering

Abstract

In this chapter we discuss two fundamental problems of discrete geometry, the best-packing problem and the best-covering problem. In Section 3.1 the general best-packing problem is introduced. We show that it is the limiting case as \(s\rightarrow \infty \) of the Riesz s-energy problem, see Proposition 3.1.2. In that section we also estimate the minimal pairwise separation of an N-point s-energy minimizing configuration on a path connected compact set. Section 3.2 introduces the general best-covering problem and discusses the basic relationship between best-packing distance and mesh ratio on a given compact set.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 4. Continuous Energy and Its Relation to Discrete Energy

Abstract

This chapter deals with the continuous minimal energy problem on a compact set A in \(\mathbb R^p\) with respect to a symmetric and lower semicontinuous kernel K. The asymptotic behavior as N gets large of the discrete minimal energy problem on A (introduced in Section 2.1) is also studied when A has nonzero K-capacity; i.e., finite Wiener constant.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 5. Linear Programming Bounds and Universal Optimality on the Sphere

Abstract

This chapter is primarily devoted to linear programming methods for determining bounds and exact solutions for Riesz minimal energy, best-packing, and kissing number problems on the sphere \(S^d\subset \mathbb R^{d+1}\).

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 6. Asymptotics for Energy Minimizing Configurations on

Abstract

This chapter is devoted to large N asymptotic results for energy and point configurations on the multidimensional sphere \(S^d\) . We begin with a discussion of the property of uniform distribution on the sphere of a sequence of N-point configurations and provide necessary and sufficient conditions for such uniformity in terms of the notion of discrepancy.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 7. Some Popular Algorithms for Distributing Points on

Abstract

From star-charts to golf-ball dipples to testing radar in aircraft, well-placed points on the two-dimensional sphere \(S^2\subset \mathbb {R}^{3}\) have a vast number of practical applications. In this chapter, we describe the properties (such as equidistribution, covering, separation, quasi-uniformity, etc.) of such point configurations generated by commonly used methods, namely zonal equal-area points, generalized spiral points, Fibonacci points, HEALPix nodes, octahedral points, icosahedral points, cubed sphere nodes, Hammersley nodes, minimizing Coulomb and logarithmic energy points, radial icosahedral points, equal-area icosahedral nodes, maximal determinant nodes, and random points.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 8. Minimal Energy in the Hypersingular Case

Abstract

A fundamental result of the potential theory described in Theorem 4.2.2 asserts that for any infinite compact set A and lower semicontinuous kernel K on \(A\times A\), the N-point minimal K-energy satisfies.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 9. Minimal Energy Asymptotics in the “Harmonic Series” Case

Abstract

This chapter deals with the asymptotic behavior of the value of the minimal Riesz d-energy on compact subsets of d-dimensional \(C^1\) manifolds in \(\mathbb R^p\), \(d\le p\). The weak\(^*\)-limit distribution of sequences of energy minimizing configurations and lower estimates of their minimal pairwise separation can be also found here. Finally, a class of sequences of asymptotically d-energy minimizing configurations is constructed for sets of positive d-dimensional Lebesgue measure.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 10. Periodic Riesz and Gauss-Type Potentials

Abstract

In this chapter, we consider kernels of the form \(K(x, y):=F(x-y)\) for some F periodic with respect to a given lattice \(\varLambda \subset \mathbb {R}^d\); that is, \(F(x+v)=F(x)\) for all \(v\in \varLambda \). In Section 10.1, we consider periodic F defined by lattice sums of the form \(F(x)=\sum _{v\in \varLambda }f(x+v), \) for some lattice \(\varLambda \) and f with sufficient decay (see Definition 10.1.1) for the sum to converge absolutely (or to \(\infty \) unconditionally). Such sums represent the energy required to place a unit charge at x in the presence of unit charges located on \(\varLambda \) with pairwise interactions given by f. In this section, we also define and relate several notions of energy related to these periodic potentials.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 11. Configurations with Nonuniform Distribution

Abstract

In this chapter we discuss the following generalization of the minimal discrete energy problem. Let \((\subset \mathbb {R})\) be compact metric space and let \(w:A\times A\rightarrow [0,\infty ]\) be a given function which we will call the weight. For a given N-point configuration \(\omega _N=\{x_1,\ldots , x_N\}\) on A and a given number \(s>0\), define the (w, s)-energy of \(\omega _N\) by \( E^w_s(\omega _N):=\sum \limits _{i=1}^{N}\sum \limits _{j=1\atop j\ne i}^{N}\frac{w(x_i, x_j)}{|x_i - x_j|^s}. \)

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 12. Low-Complexity Energy Methods for Discretization

Abstract

One potential difficulty that arises in numerical generation of near-optimal configurations is the computational complexity of the energy sums and their derivatives. One way of dealing with this issue is to minimize weighted energy sums, where the weight varies with N and vanishes if the points x and y are further away from each other than a certain threshold distance \(r_N\). A natural question arises whether the limiting distribution of asymptotically optimal sequences of N-point configurations and the leading term of the minimal energy with the varying weight coincide with the ones of the energy with some fixed weight.

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 13. Best-Packing on Compact Sets

Abstract

In this chapter we study the behavior of the leading term as N gets large of the N-point best-packing distance

$$\begin{aligned}\delta _N(A)=\sup \limits _{\omega _N\subset A}\min \limits _{x, y\in \omega _N\atop x\ne y} \left| x-y\right| \end{aligned}$$

(defined earlier in Chapter 3) on a compact \((\mathcal H_d, d)\)-rectifiable set A in \(\mathbb R^p\) as well as the weak* limit distribution of point configurations \(\omega _N\) that attain the supremum in (13.0.1).

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Chapter 14. Optimal Discrete Measures for Potentials: Polarization (Chebyshev) Constants

Abstract

This chapter investigates optimal discrete measures from the perspective of a max-min problem for potentials on a given compact set A. More precisely, for a kernel \(K:A\times A \rightarrow \mathbb {R}\cup \{+\infty \}\), the so-called polarization (or Chebyshev) problem is the following: determine N-point configurations \(\{x_j\}_{j=1}^N\) on A so that the minimum of \(\sum _{j=1}^NK(x, x_j)\) for \(x\in A\) is as large as possible. Such optimization problems relate to the following practical question: if \(K(x, x_j)\) denotes the amount of a substance received at x due to an injector of the substance located at \(x_j\), what is the smallest number of like injectors and their optimal locations on A so that a prescribed minimal amount of the substance reaches every point of A?

Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Backmatter

Titel: Discrete Energy on Rectifiable Sets
verfasst von: Sergiy V. Borodachov
Douglas P. Hardin
Edward B. Saff
Copyright-Jahr: 2019
Verlag: Springer New York
Electronic ISBN: 978-0-387-84808-2
Print ISBN: 978-0-387-84807-5
DOI: https://doi.org/10.1007/978-0-387-84808-2

Premium Partner