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In this work, we discuss the problem of approximating a multivariate functionvia $\ell_1$ minimization method, using a random chosen sub-grid of thecorresponding tensor grid of Gaussian points. The independent variables of thefunction are assumed to be random variables, and thus, the framework provides anon-intrusive way to construct the generalized polynomial chaos expansions,stemming from the motivating application of Uncertainty Quantification (UQ). Weprovide theoretical analysis on the validity of the approach. The frameworkincludes both the bounded measures such as the uniform and the Chebyshevmeasure, and the unbounded measures which include the Gaussian measure. Severalnumerical examples are given to confirm the theoretical results.

Stochastic collocation methods via $L_1$ minimization using randomized quadratures