Optimal numerical integration on a sphere

Numerical Integration Over a Sphere*

Local numerical integration on the sphere

Many applications in geomathematics as well as bio–medical applications require the analysis of an unknown target function of a large amount of data, which can be modeled as data on a subset of the surface of a sphere. An important ingredient of this analysis is to develop numerical integration schemes (quadrature formulas) to integrate spherical polynomials of as high a degree as possible exac...

Extremal Systems of Points and Numerical Integration on the Sphere

This paper considers extremal systems of points on the unit sphere Sr ⊆ Rr+1, related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of dn = dim Pn points, where Pn is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S2 of degrees up to 191 (36, ...

Discrepancies of Point Sequences on the Sphere and Numerical Integration

where σ denotes the normalized surface measure on Sd and f is a continuous real valued function. As a general reference on Quasi-Monte Carlo methods we mention Niederreiter [22]. The problem of distributing points on the sphere is also related to constructive multivariate approximation, see Reimer [24]. For the recent literature on spherical problems concerned with approximation and numerical i...

Energy functionals, numerical integration and asymptotic equidistribution on the sphere

In this paper, we study the numerical integration of continuous functions on d-dimensional spheres S ⊆ R by equally weighted quadrature rules based at N ≥ 1 points on S which minimize a generalized energy functional. Examples of such points are configurations, which minimize energies for the Riesz kernel ‖x− y‖−s 0 < s ≤ d and logarithmic kernel − log ‖x− y‖. We deduce that extremal point confi...