THE CURSE OF DIMENSIONALITY FOR NUMERICAL INTEGRATION OF SMOOTH FUNCTIONS

被引：27

作者：

Hinrichs, A. ^{[1
]}

Novak, E. ^{[2
]}

Ullrich, M. ^{[3
]}

Wozniakowski, H. ^{[4
,5
]}

机构：

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

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

[3] Univ Rome Tre, Dipartimento Matemat, I-00146 Rome, Italy

[4] Columbia Univ, Dept Comp Sci, New York, NY 10027 USA

[5] Univ Warsaw, Inst Appl Math, PL-02097 Warsaw, Poland

来源：

MATHEMATICS OF COMPUTATION | 2014年 / 83卷 / 290期

基金：

美国国家科学基金会;

关键词：

Curse of dimensionality; numerical integration; high dimensional numerical problems;

D O I：

10.1090/S0025-5718-2014-02855-X

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We prove the curse of dimensionality for multivariate integration of C-r functions: The number of needed function values to achieve an error epsilon is larger than c(r)(1 + gamma)(d) for epsilon <= epsilon(0), where cr, gamma > 0. The proofs are based on volume estimates for r = 1 together with smoothing by convolution. This allows us to obtain smooth fooling functions for r > 1.

引用

页码：2853 / 2863

页数：11

共 10 条

[1]

Abramowitz M., 1972, NBS APPL MATH SERIES, V55

[2]

Bakhvalov, 1959, VESTNIK MOSK MMAFH, V4, P3

[3] The curse of dimensionality for the class of monotone functions and for the class of convex functions [J].