具有随机场系数偏微分方程最优控制问题自适应有限元方法研究

Efficient numerical methods for optimal control problem governed by PDEs with random field coefficients are becoming a new hot topic. However, compared with the deterministic optimal control, the works for computation of stochastic optimal control problems mainly focus on optimal condition and a priori error. To our best knowledge, there is still a lack of a posteriori error estimation and adaptive algorithm for stochastic optimal control problem at present. This project aims to study the adaptive finite element method for constrained optimal control problem governed by PDEs with random field coefficients. In particular, the following aspects are discussed. 1) Based on meshfree method, we would construct a fully discrete scheme for optimal control problem governed by elliptic equations with random field coefficients, in which the stochastic space is discretized by the unstructured meshfree stochastic Galerkin method and the physical space is discretized by the finite element method. The discrete optimal condition is deduced and a posteriori error estimations are derived. 2) We would study the local refinement and points arrangement method and build the adaptive algorithm for stochastic space and physical space to reduce the number of points and improve the computation efficiency. Finally, numerical tests are given to support the theory. 3) We would study the adaptive finite element method and deduce a posteriori error estimation for optimal control problem governed by parabolic equations with random field coefficients which applies the above theories have been developed. As it can be seen that the above work for the stochastic optimal control problems are of great importance.

具有随机场系数偏微分方程最优控制问题数值方法正成为新研究热点，但与确定情况相比，当前主要成果集中于最优性条件和先验估计，缺少对随机控制问题后验估计及自适应算法的研究。本项目拟研究一般控制集上具有随机场系数的偏微分方程最优控制问题自适应有限元方法。具体如下：1) 以无网格法为基础，对随机空间采用非结构化的无网格随机Galerkin离散，对物理空间采用经典的有限元离散，建立椭圆型随机偏微分方程最优控制问题的全离散格式，推导离散最优性条件，讨论格式收敛性获得相应后验误差估计；2) 研究随机空间的局部加密技术，探讨计算节点的构建方法，建立随机空间和物理空间的自适应算法，以期降低节点数量，提高计算效率，并通过数值模拟来验证理论分析的有效性；3)对抛物型随机偏微分方程最优控制问题，基于以上成果推导后验估计和设计自适应算法。本项目研究成果将进一步丰富随机偏微分方程最优控制的理论体系，具有一定的应用价值。

本项目主要采用无网格随机Galerkin和随机Collocation方法研究了具有随机场系数的随机椭圆型、随机对流扩散型微分方程最优控制问题的数值逼近和理论分析，发展其高效数值方法。主要研究内容有：(1)应用无网格随机Galerkin有限体积元研究控制受限的具有随机场系数的随机椭圆型微分方程最优控制问题的数值逼近和理论分析，发展其高效数值方法。首先推导出上述随机最优控制问题的最优性条件，运用无网格随机Galerkin有限体积元来离散上述最优性条件，得到相应随机最优控制问题的全离散格式，引入中间变量研究分析计算格式的收敛性并获得最优阶的先验误差估计。该工作率先系统地研究了上述最优控制问题的无网格随机Galerkin逼近和理论分析，提出了新的梯度算法并进行数值模拟；（2）采用随机Collocation方法研究控制受限的随机对流扩散型微分方程最优控制问题的数值逼近和理论分析，发展其高效数值方法。随机Collocation方法已经逐渐成为具有随机场系数的随机微分方程计算的重要工具。我们重点研究了采用随机Collocation结合间断有限元法处理随机对流占优方程最优控制问题的数值逼近方法和采用随机Collocation结合迎风有限体积元处理随机对流扩散方程最优控制问题的数值逼近方法，得到了逼近的先验误差估计。(3)研究了状态期望受限的随机椭圆型方程最优控制问题的随机Galerkin有限元方法。应用凸分析中次微分和Slater条件，首次得出状态受限随机最优控制问题的最优性条件，然后给出其随机Galerkin有限元逼近，推导出最优阶逼近误差先验估计，发展出新的梯度算法并进行了数值实验，该工作填补了状态受限随机最优控制问题的研究空白。