具有随机场系数偏微分方程的最优控制问题数值方法

发布时间：2018-10-23 15:22

【摘要】：最近十年,具有随机场系数的最优控制问题数值解法正在变为一个新的研究热点,与确定性的情况相比,随机最优控制问题数值解正在起步阶段,目前关于这方面的工作并不多[20,49,81,48,56,54].而这些关于随机最优控制问题数值解的文章主要是利用Lagrange乘子法的有效性导出了最优性条件并给出了具体的数值格式.据我们所知,我们最先考虑了具有随机场系数的偏微分方程受限最优控制问题的数值解法并给出了具体的数值格式及误差估计.在本论文中,第一章我们主要介绍了问题的研究背景以及目前的研究现状.第二章我们给出了具有随机场系数的抛物方程受限最优控制问题的随机Galerkin逼近格式.首先,我们利用Lions'引理得到了随机控制问题的最优性条件,然后利用K-L展式将随机问题转化为一个有限维的确定性的最优控制问题,接下来我们给出了最优性条件的全离散格式并得到了状态变量、伴随状态变量和控制变量的先验误差估计,最后通过数值算例验证了我们的结论.第三章我们研究了具有随机场系数的对流扩散方程受限最优控制问题的随机Galerkin逼近格式.目标泛函是极小化一个消费泛函的数学期望值.众多周知,特征有限元方法[32]具有很好的计算稳定性,并且从实际的物理背景看,它更能真实的反映出运动本质.这一章我们结合特征线方法和随机Galerkin方法给出了最优控制问题的全离散格式,并得到了误差估计.第四章我们研究了具有随机场系数的椭圆方程受限最优控制问题的随机配置方法.我们给出了最优控制问题的最优性条件,得到了关于状态变量、伴随状态变量和控制变量的先验误差估计并利用算例验证了我们的结论.本章主要介绍了随机配置方法和随机Smolyak逼近格式,当解的光滑性比较好并且随机变量个数较大时,Smolyak逼近格式是一种非常高效的配置方法,能在保证较高精度的情况下大大减少配置点的数量.而对更一般的随机最优控制问题,当状态变量关于随机变量有某些不可导点或者奇点时,我们可以根据不可导点对概率空间作分解,在光滑区域上采用Smolyak逼近格式,在非光滑区域上通过增加配置点数量利用低次Lagrange插值函数作为基底函数进行逼近,也能在减少总的配置点数量情况下在整体上得到很好的逼近.第五章我们将随机配置方法应用到具有随机场系数的抛物方程受限最优控制问题上.给出了最优控制问题的全离散格式以及数值解的误差估计.
[Abstract]:In the last ten years, the numerical solution of the optimal control problem with random field coefficient is becoming a new research hotspot. Compared with the deterministic case, the numerical solution of the stochastic optimal control problem is in its infancy. At present, there is not much work in this area [20 / 49 / 81 / 48 / 56 / 54]. In these papers, the optimality conditions of stochastic optimal control problems are derived by using the validity of Lagrange multiplier method and the numerical schemes are given. As far as we know, we first consider the numerical solution of constrained optimal control problem of partial differential equations with random field coefficients, and give the numerical scheme and error estimate. In this paper, the first chapter mainly introduces the research background and current research situation. In chapter 2, we give a stochastic Galerkin approximation scheme for constrained optimal control of parabolic equations with random field coefficients. Firstly, we obtain the optimality conditions of stochastic control problems by using Lions' Lemma, and then transform the stochastic problem into a finite dimensional deterministic optimal control problem by K-L expansion. Then we give a fully discrete scheme of optimality conditions and obtain a priori error estimate for state variables, adjoint state variables and control variables. Finally, a numerical example is given to verify our conclusion. In chapter 3, we study the stochastic Galerkin approximation scheme for constrained optimal control of convection-diffusion equations with random field coefficients. A target functional is a mathematical expectation that minimizes a consumption functional. It is well known that the feature finite element method [32] has good computational stability, and from the actual physical background, it can truly reflect the essence of motion. In this chapter we combine the eigenline method with the stochastic Galerkin method to give the full discrete scheme of the optimal control problem and obtain the error estimates. In chapter 4, we study the stochastic collocation of constrained optimal control problems for elliptic equations with random field coefficients. We give the optimality conditions of the optimal control problem and obtain the priori error estimates for state variables, adjoint state variables and control variables, and verify our conclusion by an example. This chapter mainly introduces the stochastic collocation method and the stochastic Smolyak approximation scheme. When the solution is smooth and the number of random variables is large, the Smolyak approximation scheme is a very efficient collocation method. Can greatly reduce the number of configuration points while ensuring high accuracy. For a more general stochastic optimal control problem, when the state variables have some non-differentiable points or singularities of random variables, we can decompose the probabilistic space according to the non-differentiable points, and adopt the Smolyak approximation scheme on the smooth region. By increasing the number of collocation points in a non-smooth region, the low order Lagrange interpolation function can be used as a base function to approximate the total number of collocation points, and it can also be approximated well on the whole in the case of reducing the total number of collocation points. In chapter 5, we apply the stochastic collocation method to the constrained optimal control problem of parabolic equations with random field coefficients. The full discrete scheme of the optimal control problem and the error estimate of the numerical solution are given.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82;O232

【相似文献】