Stochastic Optimal Control Problems Based On Model Reduction Methods

In this doctoral dissertation,we mainly study elliptic eigenvalue problem and stochastic optimal control problem by using model reduction methods.With the model reduction methods,we aim to construct the optimization model corresponding to the eigenvalue problem and stochastic optimal control problem.Meanwhile,we attempt to reduce the computation cost as much as possible with a little loss of the accuracy and high computational efficiency.In recent years,the PDE eigenvalue problems have drawn more and more concentration in technology and engineering area.In the physical area,the eigenvalue problems are usually connected to the vibration,especially the resonance phenomenon.The eigenvalue problems are also widely applied in other engineering fields,such as astrophysics,oil reservoir simulations and electronic energy band structure calculations,etc.However,as a nonlinear problem,the numerical computation of eigenvalue problem needs much computation cost.For the multiscale eigenvalue problem,it is difficult to solve it on the finest mesh with the traditional numerical method.Furthermore,the computational cost is very high.Optimal control problems are often constrained with partial differential equations when modeling physical processes in sciences and engineering.To evaluate the physical model and the output of the model as accurately as possible,we will utilize high dimensional random variable to capture the uncertainty in the stochastic optimal control problems.Compared with deterministic optimal control problems,the numerical simulation for stochastic optimal control problems needs high computational cost and large memory space.For the multiscale optimal control problems,the numerical computation are more difficult.This may lead to the“curse-of-dimensionality”.To overcome the difficulty,for the multiscale eigenvalue problem,we will construct the optimization model of the elliptic PDE eigenvalue problem by using the generalized multiscale finite element method(GMs FEM).In the theoretic analysis,we give the error estimate of eigenvalue and eigenfunction.With two numerical examples,we show that the numerical reliability of generalized multiscale finite element method for the eigenvalue problem.To demonstrate the feasibility of model reduction methods applied to the stochastic optimal control problems,we will propose different model reduction methods for different optimal control problems.For the stochastic optimal control problems constrained by elliptic PDEs,we will propose the local-global model reduction method by combining the generalized multiscale finite element method and global reduced basis method in the fourth chapter.For the local-global model reduction method,we will analyze the existence and uniqueness of the optimal solutions of the reduced model.Moreover,a few numerical examples are provided to demonstrate the reliability and computation efficiency of the model reduction method.In the last chapter,to efficiently solve the stochastic optimal control problem controlled by stochastic parabolic equations,we will present the hybrid model reduction method based on the variable-separation method.With the hybrid model reduction,we will construct the stochastic basis functions and deterministic basis functions with low-fidelity models in the offline stage.For a given new random sample,we can quickly get the tensor-product optimal solutions in the online stage.For the hybrid model reduction,bi-model,multiscale coarse model and bi-fidelity model techniques are combined together to significantly reduce the computation complexity.In the last,a few of numerical examples are presented to confirm the performance the presented model reduction method.For the eigenvalue problem and stochastic optimal control problems,we utilize the model reduction methods to construct the corresponding optimal models.We also show the reliability and computation efficiency of the model reduction method by some numerical examples.