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{\em   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.
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.
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computation
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(surrogate models) is a
is a fundamental
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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
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
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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
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
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moments
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of the quantity of interest, there is a developed body of research. One widely popular approach is
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method is widely popular and yields excellent approximations of the map and its moment
and its many variants, which approximate moments with spectral accuracy
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pushforward PDF
PDF of the quantity of interest
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as well?
? This is not directly implied by spectrally accurate moment estimation.
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the first results of this kind. We provide
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colocation
collocation
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with Legendre polynomials
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, guaranteeing exponential rates for analytic maps
. In particular, exponential convergence of the densities is guaranteed for analytic quantities of interest