An Efficient Monte Carlo Method For Optimal Control Problems Of Stochastic Parabolic Equations

Partial differential equations and their optimal control theories have a wide range of applications in fields of finance,physics,seismology,etc.In recent years,scholars have made many achievements in deterministic partial differential equations related to optimal control theory.However,due to the real environment is in the process of constantly changing,which will be affected by random factors when solving the problem.It is obvious that the optimal control model of deterministic partial differential equations can not accurately describe the problem.Therefore,the stochastic optimal control problem and its theory analysis has become a current research hotspot.In this thesis,an ensemble Monte Carlo implicit Euler method is presented to solve the optimal control problem of the parabolic equation with random coefficients in the diffusion term.Under the condition of given sampling,the Lagrange multiplier method is used to obtain the optimality conditions.For the optimality conditions,the ensemble Monte Carlo implicit Euler method is used to discretize the state and co-state equations,and the control variables are updated with the variational discretization,and the error estimation of the optimal control problem is derived and the iterative solution algorithm is given.Numerical examples show that the ensemble Monte Carlo method does not affect the error convergence speed in the time direction and the space direction,and reduces the computational cost compared with the Monte Carlo method.The Monte Carlo method has a slow convergence rate,in order to make up for this shortcomings,we present a multi-level ensemble Monte Carlo implicit Euler method for solving the optimal control problem.A numerical example shows that the multi-level ensemble Monte Carlo implicit Euler method has more obvious advantages in saving computational cost.