椭圆最优控制问题的优化施瓦兹方法

发布时间：2018-03-21 09:10

本文选题：优化施瓦兹方法　切入点：优化传输条件　出处：《东北师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：众所周知,无约束椭圆型偏微分方程最优控制问题等价于一个耦合二阶椭圆型偏微分方程组。在文章[J.D.BENAMOU,SIAM J.Numer.Anal.,33(1996),pp.2401 2416.]中,Benamou考虑了对这种耦合方程组使用施瓦兹区域分解算法进行求解,在求解过程中使用了Robin型以及耦合的传输条件,并且对不同传输条件所对应的算法做了严格的收敛性分析。本文以两子区域区域分解算法为例,推导出了施瓦兹迭代的最优传输条件,并依此提出使用Ventcell型以及双边Robin型传输条件加速施瓦兹迭代过程的收敛。同时,本文也分析了Benamou曾经考虑过的Robin型以及耦合型传输条件。为了获得更快的收敛速率,我们使用傅立叶分析获取子问题迭代的收敛因子,通过对收敛因子的优化,获得传输条件中松弛参数的优化选择,同时给出不同优化传输条件下算法的渐近收敛率。我们发现Tikhonov参数同时出现在优化参数以及相应的渐近收敛估计中,对施瓦兹区域分解算法的收敛性质具有显著影响。数学理论上,参数趋于零将导致问题趋于病态,但越小我们的算法收敛速度越快。最后,我们研究Tikhonov参数=?4的情况,这一情形是Benamou于1996年提出的一种计算策略。利用与前述相仿的研究手段,我们得到了=?4时Robin型以及耦合型传输条件的优化参数,并得到了接近于常数的收敛率估计。由于我们研究的耦合方程组包含了双调和方程作为特例,因而本文的研究工作对双调和方程施瓦兹方法的优化具有重要指导意义。在本文的最后,我们使用数值例子来检验理论结果。
[Abstract]:It is well known that the optimal control problem for unconstrained elliptic partial differential equations is equivalent to a coupled second-order elliptic partial differential equation system. In the paper [J.D.BENAMOUUUM J.Numer.Anal.Gun 33 / 1996 p. 2401], Benamou considered the use of Schwartz domain decomposition algorithm to solve the coupled equations. The Robin type and coupled transmission conditions are used in the solution, and the convergence of the algorithms corresponding to different transmission conditions is analyzed strictly. In this paper, two subdomain decomposition algorithms are taken as an example. In this paper, the optimal transmission conditions of Schwartz iteration are derived, and the convergence of the Schwartz iteration process is accelerated by using Ventcell type and bilateral Robin type transmission conditions. In this paper, we also analyze the Robin type and coupling transmission conditions considered by Benamou. In order to obtain faster convergence rate, we use Fourier analysis to obtain the iterative convergence factor of the subproblem, and optimize the convergence factor. The optimal selection of relaxation parameters in the transmission conditions is obtained, and the asymptotic convergence rates of the algorithms under different optimal transmission conditions are given. We find that the Tikhonov parameters appear simultaneously in the optimization parameters and the corresponding asymptotic convergence estimates. The convergence property of Schwartz domain decomposition algorithm is significantly affected. In mathematical theory, zero parameters will lead to the problem becoming ill-conditioned, but the smaller the algorithm is, the faster the convergence rate of our algorithm is. Finally, we study the Tikhonov parameters? In 1996, Benamou proposed a computational strategy. The optimal parameters of the 4:00 Robin type and coupling transmission conditions are obtained, and the convergence rate estimates close to the constant are obtained. Since the coupled equations we study contain the biharmonic equation as a special case, Therefore, the research work in this paper is of great significance to the optimization of the Schwartz method for biharmonic equations. At the end of this paper, we use numerical examples to test the theoretical results.
【学位授予单位】：东北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O232

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