Monte Carlo Method Of Partial Differential Equation

In modern science,technology and engineering,there are a large number of partial differential equations.However,it is difficult to get an analytical solutions of partial differential equations,which makes a large number of scholars to turn to study their numerical solutions.While the stability and the convergence speed of Monte Carlo method are independent of dimensions,this method is a good tool to study the numerical solutions of partial differential equations.In this paper,Monte Carlo method and Multilevel Monte Carlo method of partial differential equation are studied as follows:The first chapter is the foreword.This chapter introduces the background and the research status at home and abroad of Monte Carlo method and multilevel Monte Carlo method for partial differential equations,and the work done in this paper.In the second chapter,the theoretical knowledge of Monte Carlo method is introduced.Monte Carlo method can be divided into four steps to solve the problem.Through error analysis,it is known that the maximum convergence speed of Monte Carlo method is O(n^-1/2).While the advantages and disadvantages of Monte Carlo method are simply summarized.In order to reduce the variance of Monte Carlo method,the dual random variable technique is introduced.In the third chapter,Monte Carlo method is used to solve the Poisson equation and the heat conduction equation.The random walk probability model is established,the theorem is proved,and the algorithm flow is given.The numerical solutions ??? of the Poisson equation and the heat conduction equation are obtained by four steps of constructing random walk probability model.The numerical experiments show that the numerical solutions ??? of the Poisson equation and the heat conduction equation converge with different spatial steps ?.On this basis,combined with the dual random variable technique to solve the numerical solutions ??? of the Poisson equation and the heat conduction equation,the numerical experiments show that the dual random variable technique can indeed reduce the variance of Monte Carlo method.In the fourth chapter,the theoretical knowledge of Multilevel Monte Carlo method is introduced.Multilevel Monte Carlo method is an improved method to reduce the calculation cost of Monte Carlo method.It is proved theoretically that the calculation cost of Multilevel Monte Carlo method is smaller than that of Monte Carlo method.The algorithm flow of Multilevel Monte Carlo is given.In the fifth chapter,Multilevel Monte Carlo method is used to solve the heat conduction equation.In Multilevel Monte Carlo method,in the level l=0,1,...,L,the random walk probability model of the heat conduction equation is constructed,the random variable_lh is determined,and the numerical solution ??? of Multilevel Monte Carlo method is calculated.The numerical experiments show that the calculation cost C_M of Multilevel Monte Carlo method is smaller than the calculation cost C_S of Monte Carlo method under any RMS error ?,which is to say that Multilevel Monte Carlo method reduces the calculation cost of Monte Carlo method,which is an improvement of Monte Carlo method.The sixth chapter summarizes the work of this paper and looks forward to the future research direction.