Stochastic Collocation Algorithms Using `1-minimization

The idea of `1-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 `1-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.