Numerical Methods For Optimal Control Problem Governed By PDEs With Random Field Coefficients

Efficient numerical methods for optimal control problem governed by PDEs with random field coefficients are becoming a new hot topic in the last ten years. However, compared with the deterministic optimal control, efficient computation of stochastic optimal control problems constrained by stochastic PDEs is still in its infancy, see the very recent work[20,49,81,48, 56,54]. These papers gave the optimality systems and numerical schemes based on the existence of Lagrange multiplier. However, to our best knowl-edge, we firstly gave numerical methods for the constrained optimal control problems governed by partial differential equations with random field coeffi-cients.We introduce the research background and the current research status of the stochastic optimal control problems in chapter 1.In chapter 2, a stochastic Galerkin approximation scheme is proposed for an optimal control problem governed by a parabolic PDE with random field coefficients. The objective functional is to minimize the expectation of a cost functional, and the deterministic control is of the obstacle constrained type. We firstly present the weak form for the stochastic optimal control problem and obtain the necessary and sufficient optimality conditions by utilizing Lions’lemma. Secondly, we use the Karhunen-Loeve (K-L) expansion as a main tool to convert stochastic optimality system to a finite dimension deter-ministic optimality system. Next, we establish a full discrete finite element approximation scheme for the optimality system through the discretization with respect to both the spatial space and the probability space by Galerkin method and with respect to time by the backward Euler scheme. Finally, we give the a priori error estimates for the state, the co-state and the control variables and present numerical examples to illustrate our theoretical results.In chapter 3, we deal with an optimal control problem governed by linear transient advection-diffusion equation with random field coefficients. The objective functional is to minimize the expectation of a cost functional, and the deterministic control is of the obstacle constrained type. As we all know, characteristic finite element method have been developed to perform stable computation[32], moreover, from the physical point of view, characteristic finite element finite method can better reflect the real movement. Then, a characteristic finite element approximation scheme combined with Galerkin method with respect to both the spatial space and the probability space and with respect to time by the backward Euler scheme is proposed in this chapter. We also obtain the necessary and sufficient optimality conditions by the well-known Lions’lemma and establish a scheme to approximate the optimality system. We derive some a priori error estimates for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.In chapter 4, we investigate the stochastic collocation approximation scheme for an optimal control problem governed by an elliptic PDE with random field coefficients. We obtain the necessary and sufficient optimal-ity conditions for the optimal control problem and establish a scheme to approximate the optimality system through the discretization by finite el-ements method for the spatial space and by stochastic collocation method for the probability space. We further investigate Smolyak approximation schemes, which are effective collocation strategies for smooth problems that depend on a moderately large number of random variables. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.In chapter 5, we discuss the stochastic collocation approximation scheme for an optimal control problem governed by parabolic PDE with random field coefficients and present the error estimations about numerical solutions for the state, the co-state and the control variables.