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On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification

Published online by Cambridge University Press:  03 June 2015

Zhen Gao  and
Tao Zhou
Zhen Gao*
Affiliation:
School of Mathematical Sciences, Ocean University of China, Qingdao, China
Tao Zhou*
Affiliation:
Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
Email:zhengao@ouc.edu.cn
Corresponding author.Email:tzhou@lsec.cc.ac.cn
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Abstract

In this work, we concern with the numerical comparison between different kinds of design points in least square (LS) approach on polynomial spaces. Such a topic is motivated by uncertainty quantification (UQ). Three kinds of design points are considered, which are the Sparse Grid (SG) points, the Monte Carlo (MC) points and the Quasi Monte Carlo (QMC) points. We focus on three aspects during the comparison: (i) the convergence properties; (ii) the stability, i.e. the properties of the resulting condition number of the design matrix; (iii) the robustness when numerical noises are present in function values. Several classical high dimensional functions together with a random ODE model are tested. It is shown numerically that (i) neither the MC sampling nor the QMC sampling introduce the low convergence rate, namely, the approach achieves high order convergence rate for all cases provided that the underlying functions admit certain regularity and enough design points are used; (ii)The use of SG points admits better convergence properties only for very low dimensional problems (say d ≤ 2); (iii)The QMC points, being deterministic, seem to be a good choice for higher dimensional problems not only for better convergence properties but also in the stability point of view.

Keywords

65C2065M70Least squarepolynomial approximationsuncertainty quantificationcondition number.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014 , pp. 365 - 381
Copyright
Copyright © Global Science Press Limited 2014

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