QMC designs: Optimal order Quasi Monte Carlo integration schemes on the sphere

Abstract

We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space H s ( S d ) \mathbb {H}^s( \mathbb {S}^d) with smoothness parameter s > d / 2 s > d/2 defined over the unit sphere S d \mathbb {S}^d in R d + 1 \mathbb {R}^{d+1} . Focusing on N N -point configurations that achieve optimal order QMC error bounds (as is the case for efficient spherical designs), we are led to introduce the concept of QMC designs: these are sequences of N N -point configurations X N X_N on S d \mathbb {S}^d such that the worst-case error satisfies sup f ∈ H s ( S d ) , ‖ f ‖ H s ≤ 1 | 1 N ∑ x ∈ X N f ( x ) − ∫ S d f ( x ) d σ d ( x ) | = O ( N − s / d ) , N → ∞ , \begin{equation*} \sup _{\substack {f \in \mathbb {H}^s( \mathbb {S}^d ), \\ \| f \|_{\mathbb {H}^s} \leq 1}} \Bigg | \frac {1}{N} \sum _{\mathbf {x} \in X_N} f( \mathbf {x} ) - \int _{\mathbb {S}^d} f( \mathbf {x} ) \, \mathrm {d} \sigma _d( \mathbf {x} ) \Bigg | = \mathcal {O}\big ( N^{-s/d} \big ), \qquad N \to \infty , \end{equation*} with an implied constant that depends on the H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) -norm, but is independent of N N . Here σ d \sigma _d is the normalized surface measure on S d \mathbb {S}^d . We provide methods for generation and numerical testing of QMC designs. An essential tool is an expression for the worst-case error in terms of a reproducing kernel for the space H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) with s > d / 2 s > d/2 . As a consequence of this and a recent result of Bondarenko et al. on the existence of spherical designs with appropriate number of points, we show that minimizers of the N N -point energy for this kernel form a sequence of QMC designs for H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) . Furthermore, without appealing to the Bondarenko et al. result, we prove that point sets that maximize the sum of suitable powers of the Euclidean distance between pairs of points form a sequence of QMC designs for H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) with s s in the interval ( d / 2 , d / 2 + 1 ) {(d/2,d/2+1)} . For such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order Quasi Monte Carlo integration. Numerical experiments suggest that many familiar sequences of point sets on the sphere (equal area points, spiral points, minimal [Coulomb or logarithmic] energy points, and Fekete points) are QMC designs for appropriate values of s s . For comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute QMC designs for any s > d / 2 s>d/2 . If ( X N ) (X_N) is a sequence of QMC designs for H s ( S d ) \mathbb {H}^s( \mathbb {S}^d) , we prove that it is also a sequence of QMC designs for H s ′ ( S d ) \mathbb {H}^{s’}( \mathbb {S}^d) for all s ′ ∈ ( d / 2 , s ) s’\in (d/2,s) . This leads to the question of determining the supremum of such s s (here called the QMC strength of the sequence), for which we provide estimates based on computations for the aforementioned sequences.