Fast multilevel sparse Gaussian kernels for high-dimensional approximation and integration
Fast multilevel sparse Gaussian kernels for high-dimensional approximation and integration
复制标题
用于高维近似和积分的快速多级稀疏高斯核
DOI：
--
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
F. Usta
中科院分区：
文献类型：
--
作者：
Zhaonan Dong;E. Georgoulis;J. Levesley;F. Usta
针对高维函数插值和高维积分的数值积分问题，提出了一种在结构化稀疏网格上基于方向尺度张量积高斯核的快速多层算法.该算法基于最近的基于多级稀疏核的插值（MLSKI）方法（Georgoulis，Levesley & Subhan，J.（SIAM J. Sci. Comput.}，第35（2）页~ A815-A831，2013），特别关注用于高维函数$f：[0，1]^d的插值和积分问题的基于高斯的MLSKI的快速实现 omathbb{R}$，with $5le dle 10$. MLSKI插值过程被证明是插值的，并提出了一种快速实现。更具体地说，利用各向异性高斯核的张量积性质，在一系列分层等距节点上的一维基数基函数被预先计算到机器精度，从而将插值问题呈现为函数求值的线性组合的完全可并行的集合。提出了一种基于被积函数插值的数值积分算法。一系列的数值实验突出了所提出的算法的插值和积分高达10维的问题的适用性。
A fast multilevel algorithm based on directionally scaled tensor-product Gaussian kernels on structured sparse grids is proposed for interpolation of high-dimensional functions and for the numerical integration of high-dimensional integrals. The algorithm is based on the recent Multilevel Sparse Kernel-based Interpolation (MLSKI) method (Georgoulis, Levesley & Subhan, emph{SIAM J. Sci. Comput.}, 35(2), pp.~A815--A831, 2013), with particular focus on the fast implementation of Gaussian-based MLSKI for interpolation and integration problems of high-dimen-sional functions $f:[0,1]^d omathbb{R}$, with $5le dle 10$. The MLSKI interpolation procedure is shown to be interpolatory and a fast implementation is proposed. More specifically, exploiting the tensor-product nature of anisotropic Gaussian kernels, one-dimensional cardinal basis functions on a sequence of hierarchical equidistant nodes are precomputed to machine precision, rendering the interpolation problem into a fully parallelisable ensemble of linear combinations of function evaluations. A numerical integration algorithm is also proposed, based on interpolating the (high-dimensional) integrand. A series of numerical experiments highlights the applicability of the proposed algorithm for interpolation and integration for up to 10-dimensional problems.
影响因子：
--
作者：
Cliffe, K. A.;Giles, M. B.;Teckentrup, A. L.
通讯作者：
Teckentrup, A. L.