在過去的三十多年里,由于現(xiàn)實社會實際生產(chǎn)與實踐應用的廣泛迫切需要,在天氣預報、大型飛機研制、油田勘探與開采等諸多領(lǐng)域,數(shù)學物理和工程計算問題的數(shù)值模擬規(guī)模日趨增大,其相應的計算工作陷入了前所未有的極大困境,同時也伴隨著或?qū)е铝穗y以承受的工作量。如此狀況,使得信息與計算科學工作者切身感受到解決超大規(guī)模計算問題的緊迫性,同時也凸顯了對計算方法研究的重要性。此外,計算機科學技術(shù)的迅猛發(fā)展在從根本上改變了落后的計算工具的同時也逆向促進了相關(guān)數(shù)學基礎(chǔ)理論的發(fā)展,使得計算數(shù)學成為數(shù)學學科的一個新興而活躍的研究領(lǐng)域。本文主要研究以下兩類優(yōu)化問題:時間偏微分方程約束的最優(yōu)控制問題和結(jié)構(gòu)化的凸優(yōu)化問題。從計算的角度來說,這兩大類問題都具有很高的計算復雜度,并在實際工程計算中具有極其廣泛的應用價值。作者主要基于區(qū)域分解方法和凸優(yōu)化理論,構(gòu)造相關(guān)問題的快速算法,證明每個算法的收斂性,并給出相應的數(shù)值算例。全文共分為六章。在第一章中,簡要敘述與回顧了凸優(yōu)化和反問題的相關(guān)文獻,指出了當今科學計算困境的關(guān)鍵之處,并對超級計算機和區(qū)域分解方法進行了扼要介紹。特別地,建議讀者閱讀第二至第五各章的起始部分,這些部分同樣...

【文章頁數(shù)】：211 頁

【學位級別】：博士

【文章目錄】：
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
作者簡歷及在學期間所取得得的科研成果
致謝



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