This talk discusses kernel cubature rules. A kernel cubature rule is a numerical integration method that is worst-case optimal in the reproducing kernel Hilbert space induced by a user-specified positive-definite kernel. An equivalent Gaussian process formulation allows for interpreting kernel cubature rules as probabilistic numerical methods. We review i) non-approximate algorithms based on fully symmetric sets, sparse grids, and shift-invariant kernels that alleviate the characteristic cubic computational cost in the number of data points of kernel-based methods and ii) some connections between kernel cubature and classical polynomial numerical integration methods.
