Optimal order Quasi-Monte Carlo methods on the sphere

Abstract:

A Quasi-Monte Carlo rule approximates the integral of a, say continuous, function with respect to the uniform measure using the average of function values at well-chosen nodes. For example, such nodes may form spherical t-designs and thus integrate exactly all polynomials of degree t or less.

The quality of a sequence of node sets can be measured with respect to test functions from a smooth enough Sobolev space over the sphere - which intentionally is a reproducing kernel Hilbert space - by means of the worst-case error of the associated qMC rules. A perhaps amazing fact is that the worst-case error can be (almost) expressed in terms of the sum of certain powers of all mutual Euclidean distances of the integration nodes and has a geometrical interpretation as a discrepancy that generalizes the concept of the spherical cap L2-discrepancy.

It is known that spherical designs with essentially minimal(!) number of points achieve optimal order of convergence of these worst-case errors for any(!) such function space. The new concept of "approximate spherical designs" for specified smoothness s relaxes the requirement of "exact integration" but retains the optimal order of numerical integration up to at least the pre-scribed smoothness s. Interestingly, the usual low-discrepancy sequences on the sphere in 3-dimensional space are approximate spherical designs with smoothness almost 3/2 and it is an open question if the smoothness can be higher.

The talk is rounded off by discussing explicit constructions of point configurations on the sphere. This includes spherical digital nets and sequences, and spherical lattices derived from their flat counterparts by means of an area preserving map to the curved sphere.