本文選題：自適應(yīng)重要性抽樣蒙特卡羅方法 + 重要性抽樣蒙特卡羅方法��； 參考：《內(nèi)蒙古工業(yè)大學(xué)》2016年碩士論文

【摘要】：蒙特卡羅(MC)方法又稱(chēng)為隨機(jī)模擬方法,是一種依賴(lài)于隨機(jī)試驗(yàn)的模擬求解方法.蒙特卡羅方法在求解線性代數(shù)系統(tǒng)時(shí),其收斂的速度不受該系統(tǒng)維數(shù)的影響.因而,蒙特卡羅方法可以很有效的處理高維問(wèn)題.計(jì)算機(jī)的誕生,使隨機(jī)試驗(yàn)過(guò)程變得更加快捷而有效,也使蒙特卡羅方法的優(yōu)勢(shì)更為突出.雙曲型偏微分方程在工業(yè)技術(shù)、流體力學(xué)、經(jīng)濟(jì)金融等眾多領(lǐng)域都有廣泛的應(yīng)用.電報(bào)方程是一種典型的雙曲型偏微分方程,因研究均勻傳輸線上電壓和電流的關(guān)系而被推導(dǎo)出來(lái).該方程還可以刻畫(huà)如人口動(dòng)力系統(tǒng)、化學(xué)擴(kuò)散問(wèn)題和雙曲熱傳導(dǎo)等物理現(xiàn)象.在實(shí)際應(yīng)用中,相比普通的擴(kuò)散方程,電報(bào)方程更適合刻畫(huà)物理、化學(xué)以及生物等科學(xué)領(lǐng)域內(nèi)的反應(yīng)擴(kuò)散問(wèn)題.論文的核心思想是：將雙曲型偏微分方程離散化,使之成為一個(gè)線性代數(shù)系統(tǒng),利用蒙特卡羅方法隨機(jī)模擬求解該線性代數(shù)系統(tǒng).論文提出使用隨機(jī)搜索方法求解一維二階雙曲型偏微分方程,并通過(guò)兩個(gè)數(shù)值算例展示了隨機(jī)搜索方法的有效性.使用重要性抽樣蒙特卡羅方法、自適應(yīng)重要性抽樣蒙特卡羅方法與Gibbs抽樣蒙特卡羅方法求解一維二階雙曲型電報(bào)方程,并將這三種方法與經(jīng)典的馬氏鏈蒙特卡羅方法進(jìn)行比較.兩個(gè)數(shù)值例子在運(yùn)行時(shí)間與求解精度方面展示了重要性抽樣蒙特卡羅方法、Gibbs抽樣蒙特卡羅方法的有效性與自適應(yīng)重要性抽樣蒙特卡羅方法的高效性.
[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.
【學(xué)位授予單位】：內(nèi)蒙古工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O241.82
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本文編號(hào)：1868640