Spectral convergence of probability densities for forward problems in uncertainty quantification

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 an otherwise deterministic model are random with a known distribution. The scalar quantity of interest is a fixed function of the parameters, and can therefore be considered as a random variable as a well. Often, the quantity of interest map is not explicitly known, and so the computational problem is to find its ``right'' approximation (surrogate model). For the goal of approximating the {\em moments} of the quantity of interest, there is a developed body of research. One widely popular approach is generalized Polynomial Chaos (gPC) and its many variants, which approximate moments with spectral accuracy. But can the PDF of the quantity of interest be approximated with spectral accuracy? This is not directly implied by 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.