A Survey on Spherical Designs: Existence, Numerical Constructions, and Applications

TL;DR

This survey comprehensively reviews the theory, numerical construction methods, and diverse applications of spherical designs, highlighting their importance across multiple scientific disciplines.

Contribution

It provides an extensive overview of existence results, construction algorithms, and practical applications of spherical designs, integrating theoretical and numerical perspectives.

Findings

Existence proofs for spherical designs across various dimensions.

Development of optimization-based and fast computational construction methods.

Applications in numerical integration, interpolation, signal processing, and solving PDEs.

Abstract

This paper provides a survey of spherical designs and their applications, with a particular emphasis on the perspective of ``numerical analysis''. A set \(X_N\) of \(N\) points on the unit sphere \(\mathbb{S}^d\) is called a \textit{spherical \(t\)-design} if the average value of any polynomial of degree at most \(t\) over \(X_N\) equals its average over the entire sphere. Spherical designs represent one of the most significant topics in the study of point distributions on spheres. They are deeply connected to algebraic combinatorics, discrete geometry, differential geometry, approximation theory, optimization, coding theory, quantum physics, and other fields, which have led to the development of profound and elegant mathematical theories. This article reviews fundamental theoretical results, numerical construction methods, and applied outcomes related to spherical designs. Key topics covered include existence proofs, optimization-based construction techniques, fast computational algorithms, and applications in interpolation, numerical integration, hyperinterpolation, signal and image processing, as well as numerical solutions to partial differential and integral equations.