一类随机抛物最优控制问题的集成蒙特卡洛隐式Euler法

An Ensemble Monte Carlo Implicit Euler Method for a Class of Stochastic Parabolic Optimal Control Problems

摘要: 针对扩散项带随机系数的演化方程随机最优控制问题，蒙特卡洛方法是一种非常重要的方法，针对随机量，每次抽样以后，按照以前确定问题的数值方法便可以求解，但是蒙特卡洛方法收敛速度慢，近似解误差要达到理想精度，需要大量的抽样，每次抽样需要计算一次逆矩阵，这样完成一个问题的求解，就需要计算大量的逆矩阵。我们针对一类随机抛物最优控制问题，提出一种数值求解集成蒙特卡洛隐式Euler法，即时间方向采用隐式Euler法，空间上采用线性有限元方法，随机量采用集成蒙特卡洛方法，控制量采用变分离散技巧，该方法针对所有抽样，只需要求一次逆矩阵，大大减少了计算量，通过误差分析和数值计算，该方法不影响时间方向和空间方向误差的收敛速度。

Abstract: Monte Carlo method is an important method for the stochastic optimal control problem of evolution equation with random coefficient in diffusion term. For random quantity, after each sampling, the problem can be solved using the numerical method for the corresponding deterministic problem. However, Monte Carlo method converges slowly, and the approximate solution error needs a large number of samples to achieve the desired accuracy. Each sample needs to calculate the inverse matrix. So a large number of inverse matrices need to be calculated to solve a problem. For a class of stochastic parabolic optimal control problems, we propose an ensemble Monte Carlo implicit Euler method. That is, the implicit Euler method is used for the time space, the linear finite element method is used for the space, an ensemble Monte Carlo method is used for the random coefficient, and the variational discretization technique is used for the control. This method only needs to calculate an inverse matrix once for all samples, which greatly reduces the cost. Through error analysis and numerical experiments, this method does not affect the convergence rate for time and physical space.