@Article{CiCP-16-365,
author = {Zhen Gao and Tao Zhou},
title = {On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification},
journal = {Communications in Computational Physics},
year = {2014},
volume = {16},
number = {2},
pages = {365--381},
abstract = {<p style="text-align: justify;">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 introduces 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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.130813.060214a},
url = {http://global-sci.org/intro/article_detail/cicp/7046.html}
}