Stochastic Collocation Methods via $\ell_1$ Minimization Using Randomized Quadratures

Abstract

We discuss the problem of approximating a multivariate function by a polynomial constructed via an $\ell_1$ minimization method using a randomly chosen subgrid of the corresponding tensor grid of Gaussian quadrature points. The input variables of the function are assumed to be independent random variables and thus the framework provides a nonintrusive way to construct the sparse polynomial chaos expansions, stemming from the motivating application of uncertainty quantification. We provide a theoretical analysis on the validity of the approach. The framework includes both the bounded measures, such as the uniform and the Chebyshev measures, and the unbounded measures which include the Gaussian measure. Several numerical examples are given to confirm the theoretical results.