Kernel quadrature by applying a point-wise gradient descent method to discrete energies

Abstract

We propose a method for generating nodes for kernel quadrature by a point-wise gradient descent method. For kernel quadrature, most methods for generating nodes are based on the worst case error of a quadrature formula in a reproducing kernel Hilbert space corresponding to the kernel. In typical ones among those methods, a new node is chosen among a candidate set of points in each step by an optimization problem with respect to a new node. Although such sequential methods are appropriate for adaptive quadrature, it is difficult to apply standard routines for mathematical optimization to the problem. In this paper, we propose a method that updates a set of points one by one with a simple gradient descent method. To this end, we provide an upper bound of the worst case error by using the fundamental solution of the Laplacian on $\mathbf{R}^{d}$. We observe the good performance of the proposed method by numerical experiments.