  In this work we propose and analyze a weighted proper orthogonal
decomposition method to solve elliptic partial differential equations depending
on random input data, for stochastic problems that can be transformed into
parametric systems. The algorithm is introduced alongside the weighted greedy
method. Our proposed method aims to minimize the error in a $L^2$ norm and, in
contrast to the weighted greedy approach, it does not require the availability
of an error bound. Moreover, we consider sparse discretization of the input
space in the construction of the reduced model; for high-dimensional problems,
provided the sampling is done accordingly to the parameters distribution, this
enables a sensible reduction of computational costs, while keeping a very good
accuracy with respect to high fidelity solutions. We provide many numerical
tests to asses the performance of the proposed method compared to an equivalent
reduced order model without weighting, as well as to the weighted greedy
approach, in both low and higher dimensional problems.
