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

TL;DR

This paper introduces a novel point-wise gradient descent method for generating nodes in kernel quadrature, improving efficiency and performance over traditional sequential optimization approaches.

Contribution

It proposes a simple gradient descent approach for node generation in kernel quadrature, with theoretical error bounds and demonstrated numerical effectiveness.

Findings

The method achieves good numerical performance in experiments.

An upper bound of the worst case error is derived.

The approach simplifies node generation compared to traditional methods.

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.