Density estimation in uncertainty propagation problems using a surrogate model

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The effect of uncertainties and noise on a quantity of interest (model output) is often better described by its probability density function (PDF) than by its moments. Although density estimation is a common task, the adequacy of approximation methods (surrogate models) for density estimation has not been analyzed before in the uncertainty-quantification literature. In this paper, we first show that standard surrogate models (such as generalized polynomial chaos), which are highly accurate for moment estimation, might completely fail to approximate the PDF, even for one-dimensional noise. This is because density estimation requires that the surrogate model accurately approximate the gradient of the quantity of interest and not just the quantity of interest itself. Hence, we develop a novel spline-based algorithm for density estimation whose convergence rate in Lq is polynomial in the sampling resolution. This convergence rate is better than that of standard statistical density estimation methods (such as histograms and kernel density estimators) at dimensions 1 ≤ d ≤ 5/2m, where m is the spline order. Furthermore, we obtain the convergence rate for density estimation with any surrogate model that approximates the quantity of interest and its gradient in L. Finally, we demonstrate our algorithm for problems in nonlinear optics and fluid dynamics.

Original languageEnglish
Pages (from-to)261-300
Number of pages40
JournalSIAM-ASA Journal on Uncertainty Quantification
Issue number1
StatePublished - 2020


FundersFunder number
United States-Israel Binational Science Foundation2016197
Israel Science Foundation177/13


    • Density estimation
    • Nonlinear dynamics
    • Probability density function
    • Spline
    • Surrogate model
    • Uncertainty quantification


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