Optimal Control Of Stochastic Systems:Theory,Numerics,and Applications

With the growing popularity of stochastic modeling,the design of robust con-trollers has received significant attention in the past decade.This dissertation is con-cerned with the theoretical analysis and numerical investigations for the optimal control of stochastic systems,such as those that arise from uncertainty quantification and su-pervised deep learning tasks.To be specific,we firstly consider a distributed optimal control problem,in which the governing system is given by second-order elliptic equations with log-normal coef-ficients.To lessen the curse of dimensionality that originates from the representation of stochastic coefficients,the Monte Carlo finite element method is adopted for numerical discretization where a large number of sampled constraints are involved.For the solu-tion of such a large-scale optimization problem,stochastic gradient descent method is widely used but has slow convergence asymptotically due to its inherent variance.To remedy this problem,we adopt an averaged stochastic gradient descent method which performs stably even with the use of relatively large step sizes and small batch sizes.The application of multilevel Monte Carlo approach further improves the computing ef-ficiency,where sample size formulae at each level are derived by either minimizing the computational error subject to a given computational cost or minimizing the computa-tional cost subject to a given error tolerance.Numerical examples are given to illustrate capabilities of these methods.Note that in applications to subsurface contaminant transport and similar prob-lems,the permeability can vary over several orders of magnitude for the same type of porous media,we then present studies of stochastic homogenization of a distributed elliptic control problem with randomly and rapidly oscillating coefficients.Under the hypothesis that the coefficient functions are stationary and ergodic,we show the uniform convergence of solutions of such a robust control problem to the solution of a determin-istic effective control problem.Numerical experiments are carried out to validate and supplement our theoretical findings.In the meanwhile,much attention has been paid to the stochastic training of artifi-cial neural networks during the last few years,which is known as an effective regular-ization approach that helps improve the generalization capability of trained models.In the final part of this thesis,the method of modified equations is applied to show that the residual network and its variants with noise injection can be regarded as weak approxi-mations of stochastic differential equations.Such observations enable us to bridge the stochastic training processes with the optimal control of backward Kolmogorov's equa-tions.This not only offers a novel perspective on the effects of regularization from the loss landscape viewpoint but also sheds light on the design of more reliable and effi?cient stochastic training strategies.As an example,we propose a new way to utilize Bernoulli dropout within the plain residual network architecture and conduct experi-ments on a real-world image classification task to substantiate our theoretical findings.