  The estimation of probability density functions (PDF) using approximate maps
is a fundamental building block in computational probability. We consider
forward problems in uncertainty quantification: the inputs or the parameters of
a deterministic model are random with a known distribution. The scalar quantity
of interest is a fixed measurable function of the parameters, and is therefore
a random variable as a well. Often, the quantity of interest map is not
explicitly known and difficult to compute, and so the computational problem is
to design a good approximation (surrogate model) of the quantity of interest.
For the goal of approximating the {\em moments} of the quantity of interest,
there is a well developed body of research. One widely popular approach is
generalized Polynomial Chaos (gPC) and its many variants, which approximate
moments with spectral accuracy. However, it is not clear whether the quantity
of interest can be approximated with spectral accuracy as well. This result
does not follow directly from spectrally accurate moment estimation. In this
paper, we prove convergence rates for PDFs using collocation and Galerkin gPC
methods with Legendre polynomials in all dimensions. In particular, exponential
convergence of the densities is guaranteed for analytic quantities of interest.
In one dimension, we provide more refined results with stronger convergence
rates, as well as an alternative proof strategy based on optimal-transport
techniques.
