Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model
Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model
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DOI：
10.1137/18m1205959
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发表时间：
2018-03
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影响因子：
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通讯作者：
A. Ditkowski;G. Fibich;A. Sagiv
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文献类型：
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作者：
A. Ditkowski;G. Fibich;A. Sagiv
不确定性和噪声对感兴趣量（模型输出）的影响通常可以通过概率密度函数 (PDF) 比矩来更好地描述。尽管密度估计是一项常见任务，但之前的不确定性量化（UQ）文献中尚未分析密度估计的近似方法（代理模型）的充分性。在本文中，我们首先表明，对于矩估计非常准确的标准代理模型（例如广义多项式混沌）可能完全无法近似 PDF，即使对于一维噪声也是如此。这是因为密度估计要求代理模型准确地近似感兴趣数量的梯度，而不仅仅是感兴趣数量本身。因此，我们开发了一种新的基于样条的密度估计算法，其收敛速度在$L^q$中是采样分辨率的多项式。此收敛速度优于维度 $1 \leq d\leq \frac{5}{2}m$ 的标准统计密度估计方法（例如直方图和核密度估计器），其中 $m$ 是样条阶数。此外，我们使用任何近似感兴趣数量及其梯度的代理模型获得密度估计的收敛率（$L^{\infty}$）。最后，我们展示了针对非线性光学和流体动力学问题的算法。
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.