Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

TL;DR

This paper introduces a novel spline-based density estimation method for uncertainty propagation, highlighting limitations of standard surrogate models and demonstrating improved convergence rates in practical nonlinear physics applications.

Contribution

The paper develops a new spline-based density estimation algorithm that outperforms traditional methods and analyzes the limitations of common surrogate models in density approximation.

Findings

Standard surrogate models may fail to approximate PDFs accurately.

The proposed spline-based method achieves polynomial convergence rates.

Application to nonlinear optics and fluid dynamics demonstrates effectiveness.

Abstract

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 (UQ) 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 approximates 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 $L^q$ 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 $1 \leq d\leq \frac{5}{2}m$, 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 $L^{\infty}$. Finally, we demonstrate our algorithm for problems in nonlinear optics and fluid dynamics.