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We propose, theoretically investigate, and numerically validate an algorithmfor the Monte Carlo solution of least-squares polynomial approximation problemsin a collocation frame- work. Our method is motivated by generalized PolynomialChaos approximation in uncertainty quantification where a polynomialapproximation is formed from a combination of orthogonal polynomials. Astandard Monte Carlo approach would draw samples according to the density oforthogonality. Our proposed algorithm samples with respect to the equilibriummeasure of the parametric domain, and subsequently solves a weightedleast-squares problem, with weights given by evaluations of the Christoffelfunction. We present theoretical analysis to motivate the algorithm, andnumerical results that show our method is superior to standard Monte Carlomethods in many situations of interest.

A Christoffel function weighted least squares algorithm for collocation approximations