Some parallels between classical and kernel quadrature

This talk draws three parallels between classical algebraic quadrature rules, that are exact for polynomials of low degree, and kernel (or Bayesian) quadrature rules: i) Computational efficiency. Construction of scalable multivariate algebraic quadrature rules is challenging whereas kernel quadrature necessitates solving a linear system of equations, quickly becoming computationally prohibitive. Fully symmetric sets and Smolyak sparse grids can be used to solve both problems. ii) Derivatives and optimal rules. Algebraic degree of a Gaussian quadrature rule cannot be improved by adding derivative evaluations of the integrand. This holds for optimal kernel quadrature rules in the sense that derivatives are of no help in minimising the worst-case error (or posterior integral variance). iii) Positivity of the weights. Essentially as a consequence of the preceding property, both the Gaussian and optimal kernel quadrature rules have positive weights (i.e., they are positive linear functionals).