Volume 16, Issue 2
On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification

Zhen Gao & Tao Zhou

Commun. Comput. Phys., 16 (2014), pp. 365-381.

Published online: 2014-08

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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 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.

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@Article{CiCP-16-365, author = {}, 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 = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.130813.060214a}, url = {http://global-sci.org/intro/article_detail/cicp/7046.html} }
TY - JOUR T1 - On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification JO - Communications in Computational Physics VL - 2 SP - 365 EP - 381 PY - 2014 DA - 2014/08 SN - 16 DO - http://doi.org/10.4208/cicp.130813.060214a UR - https://global-sci.org/intro/article_detail/cicp/7046.html KW - AB -

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.

Zhen Gao & Tao Zhou. (2020). On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification. Communications in Computational Physics. 16 (2). 365-381. doi:10.4208/cicp.130813.060214a
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