A Study Of Numerical Methods For Some Classes Of PED-constrained Optimal Control Problems

In this paper we investigate some classes of PDE-constrained optimal control problems, including the numerical discretization methods for the partial differential equations (PDEs) involved in them. We give the theory analysis and numerical ex-amples to identify the efficiency of the methods. An efficient solver for the optimal solution of optimal control problems is critical.In the first part, we consider the optimal control problems constrained by deter-mined partial differential equations (PDEs). Firstly, we present a numerical method and analysis, based on the variational discretization concept, for optimal control prob-lems governed by elliptic PDEs with interfaces. The immersed finite element method (IFEM) is used to discretize the state equation required in the variational discretization approach. Optimal error estimates for the control, state and adjoint state are derived. Numerical examples are provided to confirm the theoretical results. Secondly, we present a high-order upwind finite volume element method (HUFVEM) to solve the optimal control problems governed by first-order hyperbolic equations. The method is efficient and easy for implementation. Both the semi-discrete error estimates and fully-discrete error estimates are derived. Optimal order error estimates in the sense of L2 norm are obtained. Numerical examples are provided to confirm the effectiveness of the method and the theoretical results.In the second part of this dissertation, we propose some numerical method-s for stochastic PDEs, which can be used in optimal control problems constrained by stochastic PDEs. Numerical solutions of elliptic PDEs with both random inputs and interfaces are considered. A sparse grid collocation algorithm based on the S-molyak construction is used. The numerical method consists of an immersed finite element discretization in the physical space and a Smolyak construction of the ex-treme of Chebyshev polynomials in the probability space. Convergence is verified and compared with the Monte Carlo simulations. We also present a framework for the construction of Monte Carlo finite volume element method (MCFVEM) for the convection-diffusion equation with a random diffusion coefficient. Statistic error is estimated analytically and experimentally. A Quasi Monte Carlo (QMC) technique with Sobol sequences is also used to accelerate convergence.