Effective methods for obtaining good points for quadrature in reproducing kernel Hilbert spaces

Abstract

In this paper, we address the problem of numerical integration, which can be solved by kernel quadrature. Existing methods have limitations. In particular, the nodes are not well-balanced when their number is small. We propose two new methods for generating nodes for quadrature in reproducing kernel Hilbert spaces. By using the explicit formula for the error of the quadrature, we improve a set of a fixed number of sampling points with a tractable optimization algorithm. We provide a theoretical analysis of the convergence rate of the error of our first method. Numerical experiments show that our methods are effective.