  We propose, theoretically investigate, and numerically validate an algorithm
for the Monte Carlo solution of least-squares polynomial approximation problems
in a collocation frame- work. Our method is motivated by generalized Polynomial
Chaos approximation in uncertainty quantification where a polynomial
approximation is formed from a combination of orthogonal polynomials. A
standard Monte Carlo approach would draw samples according to the density of
orthogonality. Our proposed algorithm samples with respect to the equilibrium
measure of the parametric domain, and subsequently solves a weighted
least-squares problem, with weights given by evaluations of the Christoffel
function. We present theoretical analysis to motivate the algorithm, and
numerical results that show our method is superior to standard Monte Carlo
methods in many situations of interest.
