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We discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation. The problem stems from uncertainty quantification (UQ), where the independent variables of the functions are random variables with specified probability measure. We propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product Gaussian quadrature points. We analyze the stability properties of this method and prove that thedoi:10.1016/j.jcp.2015.06.042 fatcat:3m4afbdm3bb23byzzc3hw7lxoi