On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification (Communications in Computational Physics, 2014)
In this work, we concern with the numerical comparison between differentkinds of design points in least square (LS) approach on polynomial spaces. Such atopic is motivated by uncertainty quantification (UQ). Three kinds of design points areconsidered, which are the Sparse Grid (SG) points, the Monte Carlo (MC) points andthe 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 resultingcondition number of the design matrix; (iii) the robustness when numerical noisesare present in function values. Several classical high dimensional functions togetherwith a random ODE model are tested. It is shown numerically that (i) neither the MCsampling nor the QMC sampling introduces the low convergence rate, namely, the approachachieves high order convergence rate for all cases provided that the underlyingfunctions admit certain regularity and enough design points are used; (ii)The use of SGpoints 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 forhigher dimensional problems not only for better convergence properties but also inthe stability point of view.