%0 Journal Article
%A Yan, Liang
%A Guo, Ling
%A Xiu, Dongbin
%D 2012
%I Begell House
%K stochastic collocation, Legendre polynomials, 𝓁1-minimization, multi-dimensional interpolation
%N 3
%P 279-293
%R 10.1615/Int.J.UncertaintyQuantification.2012003925
%T STOCHASTIC COLLOCATION ALGORITHMS USING 𝓁1-MINIMIZATION
%U https://www.dl.begellhouse.com/journals/52034eb04b657aea,3495dd1f6253a653,1447d224370655d6.html
%V 2
%X The idea of 𝓁1-minimization is the basis of the widely adopted compressive sensing method for function approximation.
In this paper, we extend its application to high-dimensional stochastic collocation methods. To facilitate practical
implementation, we employ orthogonal polynomials, particularly Legendre polynomials, as basis functions, and focus
on the cases where the dimensionality is high such that one can not afford to construct high-degree polynomial approximations.
We provide theoretical analysis on the validity of the approach. The analysis also suggests that using
the Chebyshev measure to precondition the 𝓁1-minimization, which has been shown to be numerically advantageous in
one dimension in the literature, may in fact become less efficient in high dimensions. Numerical tests are provided to
examine the performance of the methods and validate the theoretical findings.
%8 2012-03-29