Covariance estimation using h-statistics in Monte Carlo and multilevel Monte Carlo methods

IF 1.5 4区工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY

International Journal for Uncertainty Quantification Pub Date : 2024-07-01 DOI:10.1615/int.j.uncertaintyquantification.2024051528

Sharana Kumar Shivanand

引用次数: 0

Abstract

We present novel Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods to determine the unbiased covariance of random variables using h-statistics. The advantage of this procedure lies in the unbiased construction of the estimator's mean square error in a closed form. This is in contrast to conventional MC and MLMC covariance estimators, which are based on biased mean square errors defined solely by upper bounds, particularly within the MLMC. The numerical results of the algorithms are demonstrated by estimating the covariance of the stochastic response of a simple 1D stochastic elliptic PDE such as Poisson's model.

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在蒙特卡洛和多级蒙特卡洛方法中使用 h 统计量进行协方差估计

我们提出了新颖的蒙特卡罗（MC）和多级蒙特卡罗（MLMC）方法，利用 h 统计法确定随机变量的无偏协方差。这种方法的优势在于能以封闭形式无偏构建估计器的均方误差。这与传统的 MC 和 MLMC 协方差估计器形成了鲜明对比，后者基于仅由上界定义的有偏均方误差，尤其是在 MLMC 内。通过估计泊松模型等简单一维随机椭圆 PDE 的随机响应协方差，演示了算法的数值结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal for Uncertainty Quantification ENGINEERING, MULTIDISCIPLINARY-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

3.60

自引率

5.90%

发文量

期刊介绍： The International Journal for Uncertainty Quantification disseminates information of permanent interest in the areas of analysis, modeling, design and control of complex systems in the presence of uncertainty. The journal seeks to emphasize methods that cross stochastic analysis, statistical modeling and scientific computing. Systems of interest are governed by differential equations possibly with multiscale features. Topics of particular interest include representation of uncertainty, propagation of uncertainty across scales, resolving the curse of dimensionality, long-time integration for stochastic PDEs, data-driven approaches for constructing stochastic models, validation, verification and uncertainty quantification for predictive computational science, and visualization of uncertainty in high-dimensional spaces. Bayesian computation and machine learning techniques are also of interest for example in the context of stochastic multiscale systems, for model selection/classification, and decision making. Reports addressing the dynamic coupling of modern experiments and modeling approaches towards predictive science are particularly encouraged. Applications of uncertainty quantification in all areas of physical and biological sciences are appropriate.