【作者】 洪志敏；

【导师】 闫在在；

【作者基本信息】 内蒙古工业大学 ， 固体力学， 2013， 博士

【摘要】 蒙特卡罗方法又称随机抽样技巧或统计试验方法.半个多世纪以来,由于科学技术的发展和电子计算机的发明,这种方法作为一种独立的方法被提出来,并首先在核武器的试验与研制中得到了应用.蒙特卡罗方法是一种计算方法,但与一般数值计算方法有很大区别,它是以概率统计理论为基础的一种方法.由于蒙特卡罗方法可以较真实地描述事物的特点及实验过程,解决一些一般数值方法难以解决的问题,因而该方法的应用领域日趋广泛.蒙特卡罗方法并非只能用来解决具有随机过程的问题,它还能够求解诸如高维积分、矩阵求逆、具有初边值条件的偏微分方程、积分方程等确定性问题.蒙特卡罗方法作为一种数值计算方法,具有受几何条件限制小,收敛速度与问题的维数无关,误差容易确定,程序结构简单、灵活、易于实现的优点.但是它也同时具有收敛速度慢、误差具有概率性等缺点.针对蒙特卡罗方法的以上特点,在方程(积分方程、偏微分方程)求解问题中,结合运用蒙特卡罗方法与确定性数值解法,能够充分发挥蒙特卡罗方法的优点,这无论是在理论上还是在实际应用中都很有意义.本文为以后进一步深入细致地研究蒙特卡罗方法的应用打下了基础.本文研究了与土力学有关的Fredholm型和Volterra型线性积分方程,还研究了与热力学有关的非线性Fredholm型积分方程和抛物型偏微分方程.这些方程都有潜在的力学背景.本文的主要工作概括如下：1.对于积分方程求解问题,由于其本身不具有随机性,利用蒙特卡罗方法对其求解就需要构造一个合理的概率模型,使得这个概率模型的某项数字特征就是所求方程的解.基于这样的想法,在第二章和第五章,首先通过数值求积公式法以及方程未知函数的待定系数逼近法将积分方程化为线性代数方程组.其次,建立状态离散的马尔科夫随机过程概率模型,据此概率模型定义随机变量,使得随机变量的数学期望就是积分方程的解,文中利用蒙特卡罗方法对随机变量进行模拟.在第三章,首先利用逐次逼近法产生迭代形式.其次,据此迭代形式建立状态连续的马尔科夫随机过程概率模型,依此概率模型定义随机变量,且此随机变量的数学期望就是方程的迭代解.利用蒙特卡罗重要抽样方法随机模拟定义的随机变量,进而求得积分方程的数值近似解.在第四章,首先由逐次逼近法将积分方程化为求积分问题.其次,利用蒙特卡罗控制变量法求解积分,由于控制变量的合理选择,降低了因随机模拟所产生的概率误差.第二章至第五章数值算例的计算结果表明蒙特卡罗方法在求解上述问题时是非常灵活的,实现时程序结构设计简单且易于实现.而且,蒙特卡罗方法可以根据问题的需要单独计算出问题在给定区域内任一点的未知量或未知量在几点处的线性组合.2.在第六章,研究运用蒙特卡罗方法求解非线性积分方程问题.文中首先采用数值方法将非线性积分方程化为非线性代数方程组.其次,建立求解非线性方程组的优化问题,运用蒙特卡罗方法求解此优化问题.算例的随机模拟过程表明蒙特卡罗方法在求解此优化问题时是收敛、稳定、有效的.3.在第七章,抛物型偏微分方程初边值问题被研究,文中充分利用蒙特卡罗方法求解大型稀疏系统的优势,运用紧有限差分方法将偏微分方程离散化成具有稀疏系数矩阵的线性代数系统,建立状态离散的马尔科夫随机过程概率模型,理论证明了由此概率模型定义的随机变量的数学期望就是方程的近似数值解.算例实现表明,确定性计算方法与蒙特卡罗计算方法的有效结合可以提高计算结果的精度.本文最后给出了全文研究成果的总结,并对今后进一步的研究方向进行了展望.更多还原

【Abstract】 Monte Carlo (MC) method is also called random sampling technique or statisticaltest method. Over past half a century, with the development of science and technol-ogy and the invention of the computer, this method was proposed as an independentmethod, and then applied to the test and the development of nuclear weapons firstly.Monte Carlo method is a numerical method based on the theory of probability andstatistics. But it is very diferent with the general numerical methods. Monte Carlomethod can describe the characteristics of object and the process of experiment re-alistically, as well as solve some problems which the general numerical methods aredifcult to solve. Hence, the field of application of Monte Carlo method is extensiveincreasingly. Monte Carlo method can be used to solve not only the problems withrandom process, but also the deterministic problems, such as multidimensional inte-grals, inverse matrix, partial diferential equation and integral equation with the initialand bounded conditions, and so on.As a kind of numerical method, Monte Carlo method has a great many merits. Forexample, it has only a few moments that Monte Carlo method is limited to geometricrestrictions, the convergence speed is irrelevant to the dimensions of the problem,the error is easy to determine, the program structure is simple, flexible and easy toaccomplish. On the other hand, it has some drawbacks at the same time, such as aslow convergence speed and a probabilistic error etc.. So, in theory and practice it ismeaningful to combine Monte Carlo method with numerical method for solving thenumerical solutions of equations (integral equations and partial diferential equations)making the most use of its merits. This paper laid the foundation for further detailedresearch of Monte Carlo method application in the future.In this paper, we study the linear Fredholm and Volterra types Integral equationsrelated to soil mechanics and also discuss the nonlinear Fredholm type integral equationand parabolic partial diferential equation about thermodynamics. These equationshave the potential mechanical background. The main work of this dissertation can besummarized as follows:1. Because there is no randomness on integral equation problems, therefore, it is need to construct a reasonable probability model in order to the numerical character-istic just is the solution of integral equations. Based on this idea, in Chapters2and5,at first we use the Simpson quadrature formula and the Taylor series solution methodfor solving integral equations that yields linear system. In the second place Markovrandom process model with discrete state is established and then the mathematicalexpectation of random variable defined by this probability model is the solution ofintegral equations. Monte Carlo method is employed to simulate this established ran-dom process model. In Chapter3, first of all, we use the successive approximationsmethod for solving integral equations that yields iterative term. Secondly, Monte Car-lo method with importance sampling based on the simulation of a continuous randomprocess is employed to solve the solution of this iterative form. In Chapter4, firstly, wetransform the integral equations into integrating problem by the successive approxima-tions method. Secondly, Monte Carlo variance reduction techniques based on modifiedcontrol variate is used to deal with the generated multidimensional integral. The prob-ability error is reduced. In Chapters2,3,4,5, Numerical experiments indicate thatthe proposed Monte Carlo is more flexible and simple. Its program structure design issimple and easy to realize. And it can calculate independently an unknown functionalof the solution, in a given number of operations, at only one point of domain or thelinear combination of an unknown function at several points of domain.2. In Chapter6, Monte Carlo method for solving nonlinear integral equationsis researched. We intend to transform this kind equation into nonlinear algebraicsystem by numerical methods and the optimization problem is generated and thenMonte Carlo random search algorithm is employed to solve this optimization problem.Finally, some examples are given to demonstrate the validity and applicability of theproposed approach.3. In Chapter7, Monte Carlo method is provided to evaluate the temperaturedistribution of a one-dimensional linear parabolic partial diferential equation. Theadvantages of Monte Carlo method are took full advantage of solving the sparse system.Linear algebraic system with sparse coefcient matrix is given by using the compactfinite diference and Markov process model with the discrete state is established. Itis theoretically proved that mathematical expectation of defined random variable bythis probability model is the solution of this kind diferential equation. The computed results of numerical examples with the combined use of the CFD technique and MonteCarlo method are illustrated in order to show the efciency and the accuracy of theMonte Carlo method.In the end, all mentioned above are summarized and some conclusions are putforth.更多还原