Spherical Designs, Discrepancy and Numerical Integration

Spherical Designs, Discrepancy and Numerical Integration

A spherical design is a point configuration on the sphere, which yields exact equal-weight quadrature formulae for polynomials up to a given degree. Until now only very specific constructions for spherical designs are known. We establish connections to spherical cap discrepancy and show some general discrepancy bounds. Furthermore, we reformulate the problem of constructing designs as an optimi...

Discrepancy and numerical integration on metric measure spaces

We study here the error of numerical integration on metric measure spaces adapted to a decomposition of the space into disjoint subsets. We consider both the error for a single given function, and the worst case error for all functions in a given class of potentials. The main tools are the classical Marcinkiewicz– Zygmund inequality and ad hoc definitions of function spaces on metric measure sp...

Weighted discrepancy and numerical integration in function spaces

Discrepancy in Experimental Designs

Discrepancy, Integration and Tractability

The discrepancy function of a point distribution measures the deviation from the uniform distribution. Different versions of the discrepancy function capture this deviation with respect to different geometric objects. Via Koksma-Hlawka inequalities the norm of the discrepancy function in a function space is intimately connected to the worst case integration error of the quasi-Monte Carlo integr...