High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Hongling Xie
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引用次数: 0

Abstract

There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.
多元高斯随机变量驱动随机微分方程密度估计的高阶谱方法
以前有一些工作是设计由多变量高斯随机变量驱动的随机偏微分方程(SPDE)密度估计的高效高阶数值方法。他们大多专注于提出具有独立随机变量的SPDE密度估计的数值方法,很少研究由多变量高斯随机变量驱动的SPDE的密度估计。在本文中,我们提出了一种基于gPC的密度估计的高阶算法,其中SPDE由多变量高斯随机变量驱动。我们的主要技术是:(1)采用高斯-施密特正交化建立了一个新的多元正交基;(2) 利用新构造的正交基,我们首先假设SPDE中的未知函数具有随机广义多项式混沌(gPC)展开,其次在多元高斯测度空间中实现SPDE的随机gPC展开,第三,我们获得并数值计算了展开系数的确定性微分方程;(3) 我们使用基于gPC的高阶算法进行密度估计和矩估计。我们将新提出的数值方法分别应用于已知的随机函数、随机一维波动方程和随机二维Schnakenberg模型。所有的随机方程都是由二元高斯随机变量驱动的。将其效率与基于已知随机函数的蒙特卡罗方法进行了比较。
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
Advances in Mathematical Physics
Advances in Mathematical Physics 数学-应用数学
CiteScore
2.40
自引率
8.30%
发文量
151
审稿时长
>12 weeks
期刊介绍: Advances in Mathematical Physics publishes papers that seek to understand mathematical basis of physical phenomena, and solve problems in physics via mathematical approaches. The journal welcomes submissions from mathematical physicists, theoretical physicists, and mathematicians alike. As well as original research, Advances in Mathematical Physics also publishes focused review articles that examine the state of the art, identify emerging trends, and suggest future directions for developing fields.
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