蒙特卡罗方法在求解定积分中的应用研究

Research on the Application of Monte Carlo Method in Solving Definite Integrals

摘要: 随着计算机和机器学习的发展，蒙特卡罗方法(Monte–Carlo，简称MC)逐渐成为用于近似计算的重要统计抽样理论。本文主要研究了蒙特卡罗方法在求解定积分中的应用，基于Matlab软件实现了一些定积分的蒙特卡罗方法求解。将蒙特卡罗方法求解得到的近似解与传统方法求解得到的准确值作对比，验证了蒙特卡罗方法求解定积分的可行性和优越性。同时，分析了蒙特卡罗方法求解定积分误差产生的原因，并提出了增加样本量、选择抽样方法、调整随机数生成函数等改进措施，以提高算法的计算精度。

Abstract: With the development of computers and machine learning, Monte Carlo method (MC) has gradually become an important statistical sampling theory for approximate calculations. This paper mainly studies the application of Monte Carlo methods in solving definite integrals, and implements some Monte Carlo methods for solving definite integrals based on Matlab software. The comparison be-tween the approximate solution obtained by Monte Carlo method and the accurate value obtained by traditional methods verifies the feasibility and superiority of Monte Carlo method for solving definite integrals. Meanwhile, the reasons for the error of Monte Carlo method in solving the definite integral are analyzed, and improvement measures such as increasing the sample size, selecting the sampling method, and adjusting the random number generation function are proposed to improve the calculation accuracy of the algorithm.