  In this work, we combine the idea of data-driven polynomial chaos expansions
with the weighted least-square approach to solve uncertainty quantification
(UQ) problems. The idea of data-driven polynomial chaos is to use statistical
moments of the input random variables to develop an arbitrary polynomial chaos
expansion, and then use such data-driven bases to perform UQ computations. Here
we adopt the bases construction procedure by following
\cite{Ahlfeld_2016SAMBA}, where the bases are computed by using matrix
operations on the Hankel matrix of moments. Different from previous works, in
the postprocessing part, we propose a weighted least-squares approach to solve
UQ problems. This approach includes a sampling strategy and a least-squares
solver. The main features of our approach are two folds: On one hand, our
sampling strategy is independent of the random input. More precisely, we
propose to sampling with the equilibrium measure, and this measure is also
independent of the data-driven bases. Thus, this procedure can be done in prior
(or in a off-line manner). On the other hand, we propose to solve a Christoffel
function weighted least-square problem, and this strategy is quasi-linearly
stable -- the required number of PDE solvers depends linearly (up to a
logarithmic factor) on the number of (data-driven) bases. This new approach is
thus promising in dealing with a class of problems with epistemic
uncertainties. Several numerical tests are presented to show the effectiveness
of our approach.
