2021/02/22 by Ken’ichiro Tanaka, Tanaka, Ken'ichiro
Computer Science · Mathematics · Physics and Astronomy · #41A55 #41A63 #65D30 #65D32 #65K05 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2102.10887
openalex publication_date 2021/02/22 · openalex created_date 2021/03/01 · openalex updated_date 2026/07/28
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 Rd. We observe the good performance of the proposed method by numerical experiments.