SGD-based Algorithm For A Distributed Optimal Control Problem

Many physical and engineering models involve uncertain data or uncertain parameters,i.e.,many realistic models are stochastic partial differential equations(SPDEs).There has been much progress in both the analysis and the numerical methods for these equations.This article is focusing on optimal control problems for stochatic elliptic partial differ-ential equations mathematically and computationally.More precisely,the plan of this thesis is as follows:Firstly,we prove the existence and uniqueness of the solution of minimization problems constrained by the SPDE.Next,we show the existence of a Lagrange multiplier and use Lagrange’s method to derive the stochastic optimality system of equations.Then,we use Monte Carlo finite element method to discrete the optimality system of equations,and adopt a stochastic gradient-based iterative algorithm to solve the discrete model.We also present the error estimates for the solution of the optimality system.After that,we simulate some real problems and the obtained numerical results to confirm our theoreti-cal analysis.Finally,we provide the generalization and application of stochastic gradient descent algorithm on optimal control problems.