第九章 Monte carlo积分
第九章 Monte Carlo积分
第九章 Monte carl积分 Monte carlo法的重要应用领之一:计算积分和多重积分 适用于求解: 1.被积函数、积分边界复杂,难以用解析方法或一般的数 值方法求解; 2.被积函数的具体形式未知,只知道由模拟返回的函数值 本章内容: 用 Monte carlo法求定积分的几种方法: 均匀投点法、期望值估计法、重要抽样法、半解析 法
第九章 Monte Carlo积分 Monte Carlo法的重要应用领域之一：计算积分和多重积分 适用于求解： 1. 被积函数、积分边界复杂，难以用解析方法或一般的数 值方法求解； 2. 被积函数的具体形式未知，只知道由模拟返回的函数值。 本章内容： 用Monte Carlo法求定积分的几种方法： 均匀投点法、期望值估计法、重要抽样法、半解析 法、…
第九章 Monte carlot积分 Goal: Evaluate an integral g(×)dx Why use random methods? Computation by deterministic quadrature can become expensive and inaccurate grid points add up quickly in high dimensions .o bad choices of grid may misrepresent g(x)
第九章 Monte Carlo积分 Goal: Evaluate an integral: = b a I g(x) dx Why use random methods? Computation by “deterministic quadrature” can become expensive and inaccurate. ❖ grid points add up quickly in high dimensions ❖ bad choices of grid may misrepresent g(x)
第九章 Monte carlo积分 a Monte Carlo method can be used to compute integral of any dimension d( d-fold integrals ERror comparison of d-fold integrals o Simpsons rule, approximating the integral of a function f using quadratic polynomials E∝N-1d∫(xhk=(x)=(x)+4/()+/( x1-x0 XI h o Monte Carlo method E∝N2 purely statistica, not rely on the dimension o Monte carlo method WINs. when d>>3
第九章 Monte Carlo积分 ❑ Monte Carlo method can be used to compute integral of any dimension d (d-fold integrals) ❑Error comparison of d-fold integrals Simpson’s rule,… d E N −1/ 2 1 − E N purely statistical, not rely on the dimension ! Monte Carlo method WINS, when d >> 3 Monte Carlo method approximating the integral of a function f using quadratic polynomials [ ( ) 4 ( ) ( )] 3 1 ( ) ( ) 0 1 2 0 0 0 f x dx f x dx h f x f x f x x h x x x = + + + x1 − x0 = x2 − x1 = h
第九章 Monte car积分 ☆Hit-or- Miss Method ☆ Sample Mean Method Variance Reduction Technique & Variance Reduction using Rejection Technique % Importance Sampling Method
第九章 Monte Carlo积分 ❖Hit-or-Miss Method ❖Sample Mean Method ❖Variance Reduction Technique ❖Variance Reduction using Rejection Technique ❖Importance Sampling Method