Adaptive RBF Cubature by Scattered Data on Spherical Polygons

In this work, we introduce an adaptive radial basis function (RBF) algorithm for numerical cubature over scattered data in a spherical polygon. In particular, we focus on the families of radial-power and thin-plate spline basis functions and we perform interpolation with an exponent parameter near-optimal for integration and data-driven. A further key ingredient of the algorithm is a novel adaptive routine for computing integrals over these domains. We make numerical experiments over various spherical polygons, even intricate ones as approximations of the African and Australian continents, highlighting the qualities of our approach. All the MATLAB routines used for this task are available as open-source codes.