ArticlePublisher preview available

Quasi-interpolation for high-dimensional function approximation

October 2024
Numerische Mathematik 156(5):1855-1885

DOI:10.1007/s00211-024-01435-6

Authors:

Wenwu Gao

Anhui University

Zhengjie Sun

Nanjing University of Science and Technology

The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rule at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and is capable of mitigating the curse of dimensionality for approximating high-dimensional functions.

Full grid and sparse grid at level 7 on [0,1]2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1]^{2}$$\end{document}, with 1292=16641\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$129^{2}=16641$$\end{document} and 1281 grid points, respectively

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Numerical results of approximating derivatives of fd∈W∞4([0,1]d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_d\in W^{4}_{\infty }([0,1]^{d})$$\end{document} on sparse grid

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Numerical results of approximating high-dimensional non-periodic function

…

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Numerische Mathematik (2024) 156:1855–1885

https://doi.org/10.1007/s00211-024-01435-6

Numerische

Mathemat ik

Quasi-interpolation for high-dimensional function

approximation

Wenwu Gao1·Jiecheng Wang2·Zhengjie Sun3·Gregory E. Fasshauer4

Received: 20 February 2022 / Revised: 9 May 2024 / Accepted: 19 September 2024 /

Published online: 5 October 2024

Abstract

The paper proposes a general quasi-interpolation scheme for high-dimensional func-

tion approximation. To facilitate error analysis, we view our quasi-interpolation as a

two-step procedure. In the ﬁrst step, we approximate a target function by a purpose-

built convolution operator (with an error term referred to as convolution error). In the

second step, we discretize the underlying convolution operator using certain quadra-

ture rule at the given sampling data sites (with an error term called discretization

error). The ﬁnal approximation error is obtained as an optimally balanced sum of

these two errors, which in turn views our quasi-interpolation as a regularization tech-

nique that balances convolution error and discretization error. As a concrete example,

we construct a sparse grid quasi-interpolation scheme for high-dimensional function

approximation. Both theoretical analysis and numerical implementations provide evi-

dence that our quasi-interpolation scheme is robust and is capable of mitigating the

curse of dimensionality for approximating high-dimensional functions.

This work was supported by NSFC (No.12271002, 12101310), Yong Outstanding Talents of Universities

of Anhui Province (No.2024AH020019), NSF of Jiangsu Province ( No.BK20210315), the Fundamental

Research Funds for the Central Universities (No.30923010912), and the Doctoral Research Startup

Project of University of South China (No.5524QD015).

BZhengjie Sun

sunzhengjie1218@163.com

Wen w u Ga o

wenwugao528@163.com

Jiecheng Wang

jxwjc05@163.com

Gregory E. Fasshauer

fasshauer@mines.edu

1School of Big Data and Statistics, Anhui University, Hefei, China

2School of Economic, Management and Law, University of South China, Hengyang, China

3School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing,

China

4Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80401,

USA

123

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Quasi-interpolation for high-dimensional function approximation

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