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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 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.