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.

翻译：使用近似地图估计概率密度函数(PDF)是计算概率的一个基本基石。我们考虑在不确定性量化方面的前方问题:一个非决定性模型的输入或参数随已知分布而随机。利息的量量是参数的固定函数,因此可以视为随机变量。通常,利息的量并不明确知道,因此计算问题在于找到其“Right's might”(surogate 模型)的“Right's might”(surogate 模型) 。为了在利息数量中找到接近(emmoment),我们有一个发达的研究机构。一种广泛流行的方法是通用的多式混凝土(gPC)及其许多变量,这些变量与光谱准确度相近。但利息数量的PDFDF能否与光谱准确度的准确度相近? 光谱准确的时空估计并不直接暗示这一点。 在本文中,我们证明使用合位和加勒金(Galerkin gPC)方法与图伦卓多诺米亚各个层面相近。 特别是,以指数指数为指数的指数趋同的趋同性趋近的趋同度方法,以一个更精确的趋同度为保证,我们以最精确的精准的精准的精度提供了一种利率。