基于平均值法的二重积分数值模拟与方差缩减技术研究

Research on Double Integral Numerical Simulation and Variance Reduction Technique Based on Mean Value Method

摘要: 针对重积分计算困难的问题，将一重积分数值计算的平均值法推广到二重积分上。采用该方法对二重积分进行了数值算法得到了积分的估计值，在此基础上根据中心极限定理得到了二重积分的区间估计。利用分层抽样法对二重积分估计值的方差进行缩减，模拟结果表明，分层抽样法相比一般方法能使估计值的方差缩减率达到50%左右，由此可见，等比例分层抽样效果较好。

Abstract: In order to solve the problem of difficult calculation of multiple integrals, this paper extends the mean value method of numerical calculation of single integrals to double integrals. By using this method, the estimated value of the double integral is obtained by numerical algorithm. On this ba-sis, the interval estimation of the double integral is obtained according to the central limit theorem. The stratified sampling method is used to reduce the variance of the double integral estimate. The simulation results show that the stratified sampling method can reduce the variance of the estimate by about 50% compared with the general method. Therefore, the effect of equal proportional strati-fied sampling is better.