Efficient Monte Carlo Method For Parabolic Stochastic Partial Differential Equations

Parabolic equation is a kind of partial differential equation which is very important.Due to the uncertainty of materials and environment,it's diffusion coefficient and source terms are random.For the treatment of random terms,Monte Carlo method is a common method.In this paper,we studied the efficient algorithm for solving the stochastic parabolic equation,which combined the finite difference method and the finite element method with the Monte Carlo method.For the one-dimensional stochastic parabolic equation,we studied the multilevel Monte Carlo method combined with the central finite difference method to discretize,with analysis of it's computational complexity and error estimation.To solve the twodimensional stochastic parabolic equation,the ensemble quasi-Monte Carlo method is used.The Crank-Nicolson format is adopted for time dispersion and the finite element method is applied to space discretization.In this way,the stability and convergence of the discrete format is analyzed,ideal results have been achieved.To test the validity of the method and theory,we demonstrated two numerical experiments.Test 1 is a multilevel Monte Carlo method for solving one-dimensional random parabolic problems.Compared with the traditional Monte Carlo method,it has obvious advantages.Test 2 is the ensemble quasi-Monte Carlo method combined with the Crank-Nicolson finite element method for solving the two-dimensional random parabolic problem.Compared with the ensemble Monte Carlo method,a higher order of convergence and smaller errors have been obtained.