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This work is concerned with approximating multivariate functions in unboundeddomain by using discrete least-squares projection with random pointsevaluations. Particular attention are given to functions with random Gaussianor Gamma parameters. We first demonstrate that the traditional Hermite(Laguerre) polynomials chaos expansion suffers from the \textit{instability} inthe sense that an \textit{unfeasible} number of points, which is relevant tothe dimension of the approximation space, is needed to guarantee the stabilityin the least square framework. We then propose to use the Hermite/Laguerre {\emfunctions} (rather than polynomials) as bases in the expansion. Thecorresponding design points are obtained by mapping the uniformly distributedrandom points in bounded intervals to the unbounded domain, which involved amapping parameter $L$. By using the Hermite/Laguerre {\em functions} and aproper mapping parameter, the stability can be significantly improved even ifthe number of design points scales \textit{linearly} (up to a logarithmicfactor) with the dimension of the approximation space. Apart from thestability, another important issue is the rate of convergence. To speed up theconvergence, an effective scaling factor is introduced, and a principle forchoosing quasi-optimal scaling factor is discussed. Applications to parametricuncertainty quantification are illustrated by considering a random ODE modeltogether with an elliptic problem with lognormal random input.

On discrete least square projection in unbounded domain with random evaluations and its application to parametric uncertainty quantification