Spectral convergence of probability densities

The computation of probability density functions (PDF) using approximate maps (surrogate models) is a building block in such diverse fields as forward uncertainty quantification (UQ), sampling algorithms, learning, and inverse problems. In these settings, the probability measure of interest is induced by an unknown map from a known probability space to the real line, i.e., the measure of interest is a pushforward of another known measure. In computation, we do not know the true map, but only an approximate one. In the field of UQ, the generalized Polynomial Chaos (gPC) method is widely popular and yields excellent approximations of the map and its moment. But can the pushforward PDF be approximated with spectral accuracy as well? In this paper, we prove the first results of this kind. We provide convergence rates for PDFs using colocation and Galerkin gPC methods in all dimensions, guaranteeing exponential rates for analytic maps. In one dimension, we provide more refined results with stronger convergence rates, as well as an alternative proof strategy based on optimal-transport techniques.