双曲型电报方程的随机求解方法研究

发布时间：2018-05-10 08:48

  本文选题：自适应重要性抽样蒙特卡罗方法 + 重要性抽样蒙特卡罗方法　； 参考：《内蒙古工业大学》2016年硕士论文

【摘要】：蒙特卡罗(MC)方法又称为随机模拟方法,是一种依赖于随机试验的模拟求解方法.蒙特卡罗方法在求解线性代数系统时,其收敛的速度不受该系统维数的影响.因而,蒙特卡罗方法可以很有效的处理高维问题.计算机的诞生,使随机试验过程变得更加快捷而有效,也使蒙特卡罗方法的优势更为突出.双曲型偏微分方程在工业技术、流体力学、经济金融等众多领域都有广泛的应用.电报方程是一种典型的双曲型偏微分方程,因研究均匀传输线上电压和电流的关系而被推导出来.该方程还可以刻画如人口动力系统、化学扩散问题和双曲热传导等物理现象.在实际应用中,相比普通的扩散方程,电报方程更适合刻画物理、化学以及生物等科学领域内的反应扩散问题.论文的核心思想是：将双曲型偏微分方程离散化,使之成为一个线性代数系统,利用蒙特卡罗方法随机模拟求解该线性代数系统.论文提出使用随机搜索方法求解一维二阶双曲型偏微分方程,并通过两个数值算例展示了随机搜索方法的有效性.使用重要性抽样蒙特卡罗方法、自适应重要性抽样蒙特卡罗方法与Gibbs抽样蒙特卡罗方法求解一维二阶双曲型电报方程,并将这三种方法与经典的马氏链蒙特卡罗方法进行比较.两个数值例子在运行时间与求解精度方面展示了重要性抽样蒙特卡罗方法、Gibbs抽样蒙特卡罗方法的有效性与自适应重要性抽样蒙特卡罗方法的高效性.
[Abstract]:Monte Carlo (MC) method, also known as stochastic simulation method, is a simulation method dependent on random test. The convergence rate of Monte Carlo method is not affected by the dimension of linear algebraic system. Therefore, Monte Carlo method can effectively deal with high dimensional problems. The birth of computer makes the process of random test more efficient and faster, and the advantage of Monte Carlo method more prominent. Hyperbolic partial differential equations are widely used in many fields, such as industrial technology, fluid mechanics, economy and finance. The Telegraph equation is a typical hyperbolic partial differential equation, which is derived from the study of the relationship between voltage and current on the uniform transmission line. The equation can also characterize physical phenomena such as population dynamic systems, chemical diffusion problems and hyperbolic heat conduction. In practical application, the Telegraph equation is more suitable to describe the reaction-diffusion problems in the fields of physics, chemistry and biology than the ordinary diffusion equation. The main idea of this paper is to discretize the hyperbolic partial differential equations into a linear algebraic system and to solve the linear algebraic system by Monte Carlo method. In this paper, a stochastic search method is proposed to solve one-dimensional second-order hyperbolic partial differential equations. Two numerical examples are given to demonstrate the effectiveness of the stochastic search method. The importance sampling Monte Carlo method, the adaptive importance sampling Monte Carlo method and the Gibbs sampling Monte Carlo method are used to solve one dimensional second order hyperbolic Telegraph equation. The three methods are compared with the classical Markov chain Monte Carlo method. Two numerical examples show the validity of the importance sampling Monte Carlo method and the high efficiency of the adaptive importance sampling Monte Carlo method.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82
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本文编号：1868640