广义多项式混沌方法研究

项目名称： 广义多项式混沌方法研究

项目编号： No.11501427

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 史文杰

作者单位： 武汉纺织大学

项目金额： 18万元

中文摘要： gPC方法是随机计算中最流行的算法之一。gPC方法的鲁棒性、随机模型的gPC算法和新的随机算法的提出是当前计算数学研究的重要内容。本课题拟研究以下内容：1、从CV的角度考察gPC方法的鲁棒性。我们将研究CV对“维数诅咒”的贡献和讨论起源于有限维噪声假设的误差。我们将分析gPC方法的误差对CV的依赖性和研究gPC方法和Monte Carlo方法的对比性能。2、为广泛用于模拟现代纺织、生物生态、航空航天等领域中的不确定现象的几类随机时滞微分方程设计完整gPC算法，并且给出算法的误差估计。3、结合gPC方法和在Wick-Malliavin逼近中的若干最新成果，为PDF方法提出新的闭条件。总之，本课题的研究对不确定性量化和随机微分方程的数值解的研究具有重要的意义。

中文关键词： 广义多项式混沌方法；随机微分方程；随机时滞微分方程；不确定性量化；概率密度函数方法

英文摘要： The gPC method is one of the most popular approaches in stochastic computing. The investigation including the robustness of the gPC method and the gPC method for stochastic models as well as proposing new stochastic methods is now a key part of computational mathematics. So in this project, we will consider these problems: First, from the point of view of CV, we will consider the robustness of the gPC method. We will study the contribution of CV to “the curse of dimensionality”, and discuss the error arising from the finite-dimensional noise assumption. We will analyze the dependence of error of the gPC method on CV, and study the comparative performance of gPC and Monte Carlo methods. Second, we will extend the gPC method to solve several kinds of random delay differential equations which are used widely to model the undetermined phenomena from modern textiles, biology, ecology, aerospace and other fields. The error estimation of the gPC method will be derived. Third, with the gPC method and some recent advances in Wick-Malliavin approximation, we will propose a new closure for the probability density function method. All in all, this project is significant for uncertainty quantification and numerical solutions of random differential equations.

英文关键词： generalized polynomial chaos method;random differential equations;random delay differential equations;uncertainty quantification;probability density function method