The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere $\SS^q$ ($q\ge 2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well known formulas as Driscoll--Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178.
Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a usual compact space $\mathcal M$, Gröchenig introduced the weighted least squares polynomials and the least squares quadrature from pointwise samples of a function, and obtained approximation theorems and quadrature errors. In this paper we obtain approximation theorems and quadrature errors on the sphere which are optimal. We also give upper bounds of the operator norms of the weighted least squares operators.
This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree $n$ via hyperinterpolation. Hyperinterpolation of degree $n$ is a discrete approximation of the $L^2$-orthogonal projection of degree $n$ with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most $2n$. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz--Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be con…
Let $L_{q,μ},\, 1\le q<\infty, \ μ\ge0,$ denote the weighted $L_q$ space with the classical Jacobi weight $w_μ$ on the ball $\Bbb B^d$. We consider the weighted least $\ell_q$ approximation problem for a given $L_{q,μ}$-Marcinkiewicz-Zygmund family on $\Bbb B^d$. We obtain the weighted least $\ell_q$ approximation errors for the weighted Sobolev space $W_{q,μ}^r$, $r>(d+2μ)/q$, which are order optimal. We also discuss the least squares quadrature induced by an $L_{2,μ}$-Marcinkiewicz-Zygmund family, and get the quadrature errors for $W_{2,μ}^r$, $r>(d+2μ)/2$, which are also order optimal. Mean…
Numerical simulations with rigid particles, drops or vesicles constitute some examples that involve 3D objects with spherical topology. When the numerical method is based on boundary integral equations, the error in using a regular quadrature rule to approximate the layer potentials that appear in the formulation will increase rapidly as the evaluation point approaches the surface and the integrand becomes sharply peaked. To determine when the accuracy becomes insufficient, and a more costly special quadrature method should be used, error estimates are needed. In this paper we present quadratu…
The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the spheres can be used to create effective, stable finite difference methods based on radial basis functions (RBF-FD). For certain classes of PDEs this approach leads to precise convergence estimates for stencils which grow moderately with increasing discretization fineness.
The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz--Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the $q$-dimensional sphere $\mathbb{S}^q$, and investigate how well continuous $L_p$-norms of polynomials $f$ of maximum degree $n$ on the sphere $\mathbb{S}^q$ can be discretized by positively weighted $L_p$-sum of finitely many samples, and discuss the relationship between the offset between the continuou…
AstroParticule et Cosmologie, CNRS and Université Paris 7（粒子物理与宇宙学，法国国家科学研究中心与巴黎第七大学）；Laboratoire Probabilités et Modèles Aléatoires, CNRS and Universités Paris 6-7（概率与随机模型实验室，法国国家科学研究中心与巴黎第六至七大学）；Laboratoire du Traitement et de la Communication de l’Information, CNRS and Télécom Paris（信息与通信处理实验室，法国国家科学研究中心与巴黎电信学院）；Laboratoire Paul-Painlevé, Université Lille-1（保罗·潘勒韦实验室，里尔第一大学）
City University of Hong Kong（香港城市大学）；Xi’an Jiaotong University（西安交通大学）；University of Sydney（悉尼大学）
This paper proposes a distributed weighted regularized least squares algorithm (DWRLS) based on spherical radial basis functions and spherical quadrature rules to tackle spherical data that are stored across numerous local servers and cannot be shared with each other. Via developing a novel integral operator approach, we succeed in deriving optimal approximation rates for DWRLS and theoretically demonstrate that DWRLS performs similarly as running a weighted regularized least squares algorithm with the whole data on a large enough machine. This interesting finding implies that distributed lear…
