两类PDE控制离散矩阵的预处理及快速计算

Fast solving of large-scale system of linear equations is the core and basic questions for large-scale scientific computing, which has a very wide range of science and engineering application background, such as optimal control, computational fluid dynamics and so on. They are all need to solve a series of very high dimension structured linear system of equations after discretized. In this project, we will investigate exploringly efficient numerical algorithm and preconditioning technique for large-scale structured linear systems that arise in the time-dependent diffusion equation constrained problems and fractional diffusion equation control problems. At present, the research on this problem is very little, but it is urgent to study the high performance algorithms. Based on the previous work of our project team, to dig deeper into its structure of matrix, and combining the numerical scheme of PDE, characteristic properties of the structured matrix and a variety of means such as FFT and other. We will design and develop efficient preconditioner and fast algorithms with less CPU time and memory for solving fast two types of PDE-constrained optimization problems to provide some mathematcial theory and methods for practical applications. Research results have very high theoretical significance and can be to used in many practical application, such as aerospace, industrial, physical chemical and biological mechanics, electric, medical and financial areas and so on.

大规模线性方程组的快速求解是大规模科学计算的核心和基本问题，有着非常广泛的科学和工程应用背景，如最优控制、计算流体力学等问题，离散后都需要求解一系列超大型结构线性系统。本研究课题拟对两类PDE约束离散线性系统，即非定常扩散方程约束优化离散线性系统及非定常分数阶扩散方程控制离散线性系统进行深入研究。目前，类似的研究国内外刚刚起步。本项目拟在项目组的前期工作基础上，深入挖掘两类矩阵的特有结构，并结合微分方程数值求解方法、结构矩阵的特有性质、FFT等多种手段，针对两类结构矩阵开发高效预处理子及快速计算方法。为具有广泛应用背景的两类PDE约束优化问题的快速计算提供技术支撑。研究成果有很高的理论意义和实用价值，可为解决许多实际问题，服务于航空航天、工业、物理、电分析化学、生物力学、医学和金融等领域的实际需要。

偏微分方程约束问题在优化设计、航空航天、医学、水利科学、地球科学、工程技术等自然科学领域和经济金融等领域有广泛应用, 其快速和有效的数值算法研究是现代大规模科学计算的一个重要研究问题。本项目主要研究了两类PDE约束问题求解中的大规模离散线性或非线性方程组的快速算法及其理论，并利用问题来源模型进行了数值验证和分析。针对不同问题提出了对网格尺寸、模型参数和正则化参数均鲁棒的快速Krylov子空间求解器，取得了一系列重要的进展，课题组成员完成标注基金资助的高水平SCI论文33篇，论文发表在《Applied Numerical Mathematics》《Numerical Methods for Partial Differential Equations》《Journal of Computational and Applied Mathematics》《Calcolo》《Linear Algebra and its Applications》《Numerical Algorithms》等SCI期刊。项目得到的新技术和方法丰富并发展了大规模科学计算与工程应用的理论和方法。在人才培养方面，项目执行期间，4位博士生获得博士学位，8位硕士生获得硕士学位。项目组成员2人在项目执行期间也获得国家自然科学青年基金。项目研究达到了预期目标。