Construction of designs on the 2-sphere
European Journal of Combinatorics
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Constructive polynomial approximation on the sphere
Journal of Approximation Theory
Hyperinterpolation on the sphere at the minimal projection order
Journal of Approximation Theory
Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces
SIAM Journal on Numerical Analysis
Quasi-Monte Carlo methods can be efficient for integration over products of spheres
Journal of Complexity
Optimal lower bounds for cubature error on the sphere S2
Journal of Complexity
Cubature over the sphere S2 in Sobolev spaces of arbitrary order
Journal of Approximation Theory
Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs
SIAM Journal on Numerical Analysis
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Building upon a recent existence result of Kuo and Sloan, this paper presents a component-by-component algorithm for constructing the m points of a quasi-Monte Carlo (QMC) rule for numerical integration over the d-fold product of unit spheres S^2@?R^3. Our construction is as follows: starting with a well-chosen generating point set of m points on S^2, the algorithm chooses a permutation of this generating point set for each sphere, one sphere at a time, so that the projection of the m QMC points onto each sphere is the same, and is just the generating point set but with the points occurring in a different order. Understandably, the quality of our QMC rule depends on the quality of both the generating point set and the successive permutations. This paper contains two key results. Firstly, assuming that the worst-case error for the generating point set in a certain Sobolev space satisfies a certain estimate, we prove inductively that the resulting QMC rule satisfies the existence result for the worst-case error bound in a d-dimensional weighted Sobolev space established non-constructively by Kuo and Sloan: specifically, the worst-case error of our QMC rule is bounded from above by cm^-^1^/^2, where c0 is independent of m and d, provided that the sum of the weights is bounded independently of d. Secondly, we show that the desired estimate for the generating point set on S^2 is automatically satisfied for m sufficiently large by a spherical n-design with m=O(n^2) points (if such spherical designs exist) and by a spherical n-design with m=O(n^3) points if slightly stronger assumptions are made on the smoothness of the weighted function space. The latter task involves techniques developed by Hesse and Sloan for numerical integration in Sobolev spaces on S^2. The construction cost for the component-by-component algorithm grows only linearly with d. However, a complete search over all m! permutations at each step of the construction is infeasible, thus a randomized version of the algorithm is recommended in practice.