This paper presents new quadrature rules for functions in a reproducing kernel Hilbert space using nodes drawn by a sampling algorithm known as randomly pivoted Cholesky. The resulting computational procedure compares favorably to previous kernel quadrature methods, which either achieve low accuracy or require solving a computationally challenging sampling problem. Theoretical and numerical results show that randomly pivoted Cholesky is fast and achieves comparable quadrature error rates to more computationally expensive quadrature schemes based on continuous volume sampling, thinning, and recombination. Randomly pivoted Cholesky is easily adapted to complicated geometries with arbitrary kernels, unlocking new potential for kernel quadrature.

翻译：本文针对再生核希尔伯特空间中的函数，提出了一种利用随机枢轴乔列斯基抽样算法生成节点的全新求积规则。该计算流程相较于现有核求积方法具有显著优势——既有方法要么精度不足，要么需要求解计算代价高昂的抽样问题。理论与数值结果表明，随机枢轴乔列斯基算法不仅计算高效，还能达到与基于连续体积抽样、精简与重组等计算更复杂的求积方案相当的误差率。该算法可轻松适配具有任意核函数的复杂几何结构，为核求积方法的发展开辟了新前景。