Numerical Method Of Optimal Control Problem Constrained By Elliptic SPDE

The deterministic optimal control theory is widely used to establish mathematical models,in the fields of engineering and natural science,e.g.hydrodynamics,economic model,image processing.By now,various and efficient numerical algorithms have been well studied.However,due to the uncertainty of parameters or the random error caused by measurement,the optimal control problem of complex system involving uncertainty quantification has attracting the attention of experts in related areas.In the past ten years,with the rapid development of high-performance computer processing capabilities,as well as the research needs of big data,deep learning and other fields,the theoretical and numerical algorithm research on such problems has become a research hotspot in the fields of computational mathematics and artificial intelligence.We main study the numerical solution of optimal control problems(OCPs)constrained by elliptic partial differential equation with random coefficients.Our research conclusions can be widely applied to optimization problems constrained by general SPDE.Firstly,the existence and uniqueness of the optimization problem is investigated,and Monte Carlo finite element method(MC-FEM)is used to discretize the optimal system.Then,following two numerical optimization methods were used and studied:(1).using classic Lagrange multiplier method to derive an equivalent optimization system,and then MC-FEM is applied to solve the system by determining the gradient of the objective functional;(2).using reduce method,that is,discrete the elliptic SPDE via MC-FEM,then plug the explicit form of solution into objective functional to eliminate the constrains to derive unconstrained OCP.Several numerical optimization methods including gradient descent method with fixed step,gradient descent method with line search method,Quasi-Newton method and trust region method were used.The resulted numerical optimization algorithms were presented.In the end,a two-dimensional elliptic SPDE control problem is taken as an example to compare the optimal results obtained by Lagrange method and reduce method.Our results show that the reduced method has faster convergent rate but lower computational complexity over Lagrange gradient method.