几类随机发展方程的数值方法研究

发布时间：2018-06-17 20:01

本文选题：随机发展方程 + 随机偏微分方程　；参考：《华中科技大学》2016年博士论文

【摘要】：本论文考虑几类随机发展方程的数值方法,全文由两个部分组成。第一部分研究随机偏微分方程的数值方法,包括受乘性噪声驱动的随机弹性方程的随机指数积分子方法,非线性随机抛物方程的Parareal算法以及受无穷维分数阶Brown运动驱动的随机偏微分方程的Galerkin谱方法。第二部分研究随机常微分方程Milstein型方法的构造,并研究了方法的强收敛性和保指数均方稳定性。论文由以下六章组成：第一章介绍随机偏微分方程和随机常微分方程数值方法研究的发展历史与现状,并给出本文的主要内容和结构安排。第二章简要介绍无穷维维纳过程和可分Hilbert空间中无穷维随机积分的基础知识。第三章对于受乘性噪声驱动的随机弹性方程,我们在时间上采用随机指数积分子方法,空间上采用有限元方法。通过分别分析空间和时间的半离散误差获得全离散格式的误差估计,并证明了时间上的强收敛阶与方程解析解的时间正则性是一致的。第四章,我们将Parareal算法应用到一类非线性随机抛物方程。我们发现,当在粗网格上使用随机积分子、细网格上使用精确解算子时,对于受加性时空噪声驱动的随机抛物方程,Parareal算法的收敛速率是超线性的,而且收敛因子与噪声的正则性无关。进一步的数值试验还表明对于受乘性噪声驱动的随机抛物方程以及含有非全局Lipschitz漂移项的随机抛物方程,Parareal算法依然有效。第五章,对于受无穷维分数阶Brown运动驱动的随机偏微分方程,我们在空间上采用Galerkin谱方法,时间上采用线性隐式Euler方法,并利用方程解析解的正则性结果分析了数值方法的强收敛性。我们的结论表明数值方法的时空强收敛阶与解析解的时空正则性是一致的。第六章对于一类Ito型自治随机常微分方程,我们提出了两类两步Milstein方法,并分析了方法的护误差。我们证明了这两类格式都具有1阶强收敛阶,并且在步长满足一定限制的条件下,这两类格式都可以保持随机微分方程的指数均方稳定性,而且数值解的衰减速率会收敛到解析解的衰减速率。
[Abstract]:In this paper, we consider several numerical methods of stochastic evolution equations, which are composed of two parts. In the first part, we study the numerical methods of stochastic partial differential equations, including stochastic exponential product molecular methods for stochastic elastic equations driven by multiplicative noise. Parareal algorithm for nonlinear stochastic parabolic equations and Galerkin spectral method for stochastic partial differential equations driven by infinite dimensional fractional Brownian motion. In the second part, we study the construction of Milstein type methods for stochastic ordinary differential equations, and study the strong convergence and exponential mean-square stability of the methods. The thesis consists of six chapters: the first chapter introduces the history and present situation of the numerical methods of stochastic partial differential equations and stochastic ordinary differential equations, and gives the main contents and structure of this paper. In chapter 2, the basic knowledge of infinite dimensional random integral in separable Hilbert space is introduced briefly. In chapter 3, we use stochastic exponential product method in time and finite element method in space for stochastic elastic equations driven by multiplicative noise. By analyzing the semi-discrete error of space and time, the error estimates of the full discrete scheme are obtained, and it is proved that the order of strong convergence in time is consistent with the time regularity of the analytic solution of the equation. In chapter 4, we apply Parareal algorithm to a class of nonlinear stochastic parabolic equations. We find that the convergence rate of Parareal algorithm for stochastic parabolic equations driven by additive space-time noise is superlinear when random product molecules are used on rough meshes and exact solution operators are used on fine meshes. Moreover, the convergence factor is independent of the regularity of noise. Further numerical experiments show that the Parareal algorithm is still valid for stochastic parabolic equations driven by multiplicative noise and stochastic parabolic equations with non-global Lipschitz drift term. In chapter 5, for stochastic partial differential equations driven by infinite dimensional fractional Brownian motion, we use Galerkin spectral method in space and linear implicit Euler method in time. The strong convergence of the numerical method is analyzed by using the regularity of the analytical solution of the equation. Our results show that the order of strong convergence of the numerical method is consistent with the spatiotemporal regularity of the analytic solution. In chapter 6, for a class of Ito type autonomous stochastic ordinary differential equations, we propose two classes of two-step Milstein method and analyze the error of the method. It is proved that both schemes have a strong convergence order of order 1, and both schemes can maintain the exponential mean square stability of stochastic differential equations under the condition that the step size satisfies certain constraints. Moreover, the decay rate of the numerical solution will converge to the attenuation rate of the analytical solution.
【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

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