两类离散的PDE约束优化问题的高效预处理方法

Partial differential equation (PDE) constrained optimization problems have been widely arisen in real-life applications and engineering designs, such as optimal design of the aircraft, furnace temperature control system, oil production hydraulic pressure control, etc. Due to the complexity of the problems, it is often difficult to find analytical solutions, so stable and efficient numerical solution is crucial. The main purpose of this project is to study fast and efficient precondition methods to solve two classes of discrete PDE constrained optimization problems. Based on the positive definiteness and spare structure of the subblocks of the coefficient matrices, we design sparse and efficient preconditioners by dimension reduction or dimension increasing technique for the coefficient matrix through parameterization techniques, and taking into consideration of the alternate direction iteration and dimension decomposition methods. We are concerned with the analysis about computational complexity of the preconditioners and spectral distributions of the preconditioned matrices. The selection of optimal or quasi-optimal parameters are also studied. Moreover, numerical experiments are presented to test the effectiveness and stabilities of the proposed algorithms, especially in the cases with fine grid meshes. This project will concern the efficient precondition methods for two classes of discrete PDE constrained optimization problems, and provide some new directions for the fast numerical solution of Cahn-Hilliard equation.

偏微分方程(PDE)约束优化问题广泛产生于飞机的最优型设计、炉温控制系统、石油采油水压控制等实际生产生活和工程设计中.考虑到实际问题的复杂性,通常很难求其解析解,因此稳定高效的数值求解就显得至关重要.本项目主要针对两类离散的PDE约束优化问题设计高效的预处理方法.利用优化问题离散后系数矩阵的子矩阵所具有的正定性质和稀疏结构,通过参数化技巧对系数矩阵进行降维或升维,并借鉴交替方向迭代和维数分解等技巧,设计稀疏的且易于实现的预处理子.从理论上分析预处理子的计算复杂性和预处理矩阵的特征值分布,研究(拟)最优松弛参数的选取方法,并结合数值实验验证预处理子的有效性和稳定性,特别是网格尺度较小的情形.研究成果将得到两类PDE约束优化问题的高效预处理子,并为Cahn-Hilliard方程的数值求解提供一定的思路.

本项目致力于研究泊松型及抛物型PDE约束优化问题经离散后得到的大型稀疏复对称线性系统的快速数值解法。通过分析复系数矩阵的特殊性质(如：矩阵W和T是否具有正定结构)，提出具有针对性的迭代法或自适应的预处理子。讨论迭代法的收敛条件以及相应预处理矩阵的谱性质，并分析迭代矩阵谱半径，研究(拟)最优松弛参数的选取或预处理的Krylov子空间方法迭代步数的上界。主要研究内容和结果如下：.(1) 针对泊松型问题得到的一类复对称正定线性系统，提出一个优化的结构预处理子，相应预处理矩阵的特征值是正实的且分布在(1/2，1]。当用于加速特定的Krylov子空间方法时，该预处理子可导出不依赖网格尺寸的稳定数值表现。.(2) 针对抛物型问题得到一类复对称不定线性系统，构造了一个满结构的块预处理子。该预处理子的求解仅涉及稀疏的对称正定子系统，且相应的预处理矩阵特征值全部聚簇在顶部半环。.(3) 针对抛物型问题得到一类复对称不定线性系统，利用矩阵的线性组合及PMHSS方法思想，构造了一类变形的PMHSS(VPMHSS)方法。理论分析表明该方法具有无条件收敛性质，且拟最优参数易于选择，可固定为1，同时给出易于计算的线性组合系数ε的表达式。数值实验表明了该方法的高效性和稳定性。