Activity: Talk or presentation › Talk at workshop, seminar or course › Science to science
A fundamental question in the study of numerical integration on the sphere and other manifolds concerns the characterization of 'good' node sets for numerical integration rules. Potential theoretic concepts and methods enable a reinterpretation of this question and provide a different angle for solution strategies.
A beautiful example is Stolarsky's Invariance Principle that combines concepts and ideas from either realm: the sum of all mutual distances among the nodes (discrete energy) plus the spherical cap $L_2$-discrepancy remains constant (continuous energy of the sphere) whatever the choice of nodes. In fact, the spherical cap $L_2$-discrepancy can be also understood as the worst-case numerical integration error for integrating functions from a Sobolev space seen as a certain reproducing kernel Hilbert space.
In this talk we explore further connections (spherical designs, QMC-designs, and uniform distribution of points in the square using external fields).