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

作　　者：Zhen Gao Tao Zhou 

机构地区：[1]School of Mathematical Sciences,Ocean University of China,Qingdao,China [2]Institute of Computational Mathematics,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China.

出　　处：《Communications in Computational Physics》2014年第7期365-381,共17页计算物理通讯（英文）

基　　金：supported by National Natural Science Foundation of China(11201441);the Natural Science Foundation of Shandong Province(ZR2012AQ003);and China Postdoctoral Science Foundation(2012M521374/2013T60684);The second author is supported by the National Natural Science Foundation of China(No.91130003 and No.11201461).

摘　　要：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.

关 键 词：Least square polynomial approximations uncertainty quantification condition number. 

分 类 号：O17[理学—数学]