几类区域分解和凸优化算法及其在反问题中的应用

发布时间：2023-03-08 08:10

　　在过去的三十多年里,由于现实社会实际生产与实践应用的广泛迫切需要,在天气预报、大型飞机研制、油田勘探与开采等诸多领域,数学物理和工程计算问题的数值模拟规模日趋增大,其相应的计算工作陷入了前所未有的极大困境,同时也伴随着或导致了难以承受的工作量。如此状况,使得信息与计算科学工作者切身感受到解决超大规模计算问题的紧迫性,同时也凸显了对计算方法研究的重要性。此外,计算机科学技术的迅猛发展在从根本上改变了落后的计算工具的同时也逆向促进了相关数学基础理论的发展,使得计算数学成为数学学科的一个新兴而活跃的研究领域。本文主要研究以下两类优化问题:时间偏微分方程约束的最优控制问题和结构化的凸优化问题。从计算的角度来说,这两大类问题都具有很高的计算复杂度,并在实际工程计算中具有极其广泛的应用价值。作者主要基于区域分解方法和凸优化理论,构造相关问题的快速算法,证明每个算法的收敛性,并给出相应的数值算例。全文共分为六章。在第一章中,简要叙述与回顾了凸优化和反问题的相关文献,指出了当今科学计算困境的关键之处,并对超级计算机和区域分解方法进行了扼要介绍。特别地,建议读者阅读第二至第五各章的起始部分,这些部分同样...

【文章页数】：211 页

【学位级别】：博士

【文章目录】：
摘要
ABSTRACT
Chapter 1 Introduction
    1.1 Convex optimization
    1.2 Inverse problem
    1.3 Challenges in computational science
    1.4 Supercomputers and the domain decomposition method
    1.5 Contribution of this dissertation
Chapter 2 Proximal gradient algorithms for convex minimization
    2.1 Primal-dual fixed point algorithms
        2.1.1 Model problem and derivation of nested algorithms
        2.1.2 Main theorems
    2.2 Analysis of convergence
        2.2.1 Basic lemmas
        2.2.2 General convergence
        2.2.3 Linear convergence rates for special case
    2.3 Numerical experiments
Chapter 3 Explicit/implicit and Crank-Nicolson domain decomposition meth-ods for parabolic partial differential equation
    3.1 Model problem and DDM finite element schemes
        3.1.1 Model problem
        3.1.2 Domain decomposition schemes
        3.1.3 Main theorems
    3.2 Analysis of convergence
        3.2.1 Basic lemmas
        3.2.2 Proof of Theorem 3.1
        3.2.3 Proof of Theorem 3.2
    3.3 Numerical experiments
Chapter 4 Explicit/implicit domain decomposition method for optimal controlproblem
    4.1 Optimal control problem and optimality condition
        4.1.1 Model problem
        4.1.2 Optimality condition
    4.2 Finite element approximation based on domain decomposition
        4.2.1 Discretization
        4.2.2 Parallel iterative algorithm
        4.2.3 Main theorems
    4.3 Analysis of convergence
        4.3.1 Intial approximation
        4.3.2 Basic lemmas
        4.3.3 Existence of discretization and convergence of iterative algo-rithm
        4.3.4 Proof of a priori estimate
    4.4 Numerical experiments
Chapter 5 Non-iterative Domain decomposition methods for wave equation
    5.1 Model problem and DDM finite element procedures
        5.1.1 Model problem
        5.1.2 Standard finite element procedures
        5.1.3 Domain decomposition schemes
    5.2 Analysis of convergence
        5.2.1 Basic lemmas
        5.2.2 Proof of Theorem 5.1
        5.2.3 Proof of Theorem 5.2
    5.3 Numerical experiments
Chapter 6 Conclusion
Bibliography
作者简历及在学期间所取得得的科研成果
致谢



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