蒙特卡罗方法与拟蒙特卡罗方法解线性方程组

分别介绍蒙特卡罗方法和拟蒙特卡罗方法解线性方程组的基本原理,并对两种方法的误差和收敛速度进行讨论.提出误差由3方面造成:截断误差、方法本身、伪随机数序列和低差异序列分布不均匀.在收敛速度方面:蒙特卡罗法的收敛速度与问题的规模和模拟路径长度无关;拟蒙特卡罗方法的收敛与问题的规模无关,但与模拟路径长度有关.经过对两种方法适用的情况进行讨论及数据测试,认为在一般情况下应选择用拟蒙特卡罗方法解线性方程组.

The Monte Carlo Methods and Quasi Monte Carlo Methods for Systems of Linear Algebraic Equations

The Monte Carlo methods, Quasi-Monte Carlo methods for solving Systems of Linear Algebraic Equations (SLAE), and the error, the convergence rate of each method were analyzed. The error was caused by three aspects; the truncation error, the methods per se, the pseudo-random number and low-discrepancy sequences not uniform. In respect of convergence rate, Monte Carlo methods did not depend on the scale of problems and the number of walks. Quasi-Monte Carlo methods did not depend on the scale of problems either, but depended on the number of walks. In addition, the conditions of using the Monte Carlo methods, Quasi-Monte Carlo methods and the numerical experiments discussed, Quasi-Monte Carlo methods were chosen to solve Linear Algebraic Equations (LAE) in general condition.