On weak tractability of the Clenshaw-Curtis Smolyak algorithm

被引：16

作者：

Hinrichs, Aicke ^{[1
]}

Novak, Erich ^{[2
]}

Ullrich, Mario ^{[2
]}

机构：

[1] Univ Rostock, Inst Math, D-18051 Rostock, Germany

[2] Univ Jena, Math Inst, D-07743 Jena, Germany

来源：

JOURNAL OF APPROXIMATION THEORY | 2014年 / 183卷

关键词：

DIFFERENTIABLE MULTIVARIATE FUNCTIONS; HIGH-DIMENSIONAL INTEGRATION; SPARSE GRIDS; SMOOTH FUNCTIONS; NUMERICAL-INTEGRATION; APPROXIMATION; INTERPOLATION;

D O I：

10.1016/j.jat.2014.03.012

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We consider the problem of integration of d-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw-Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm. (C) 2014 Elsevier Inc. All rights reserved.

引用

页码：31 / 44

页数：14