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研究生： 曾郁潔
Yu-Chieh Tseng
論文名稱： Numerical Study of Algebraic Multigrid Methods
Numerical Study of Algebraic Multigrid Methodsfor Solving Linear/Nonlinear Elliptic Problems onSequential and Parallel Computers
指導教授： 黃楓南
Feng-Nan Hwang
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 100
語文別： 英文
論文頁數： 107
中文關鍵詞： 多重網格法 、橢圓問題 、平行計算
外文關鍵詞： Elliptic problems, Multigrid methods, Parallel computing
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    • 在現今做數值計算的趨勢中，多重網格法(Multigrid method)已是一個重要，不可或缺的數值方法，因為它的好處除了可降低迭代次數和計算時間之外，可平行化也是一個很大的優勢，這對專門研究平行計算的研究者們是一大福音。有關於多重網格法的發展已有一段時間，其效率及演算法的形式也是百家爭鳴。本文藉由對多重網格法的由來和其中發展出的演算法來解Poisson-Boltzmann Equations, Convection-Diffusion Equations等問題上的應用來探討多重網格法對於解其問題的效果及成本等等的結果，並觀察多重網格法的優缺點。透過了解多重網格法的特性，以期能用此特性來節省迭代次數和時間成本。用來解更多的大型線系統或大型的稀疏矩陣。

    In the nowadays, Multigrid method plays an important role in the trend of numerical computations.
    Besides of its advantages of decreasing the iterations and the computation time, parallelization is also a big advantage of the parallel computation, it brings the convience for those researchers who do the research about parallel computation. About the developement of the multigrid already exists for a period of time. Its efficiency and the form of algorithms also have many different versions. In this paper, we will discuss about the result of solving the Poisson-Boltzmann Equations, Convection-Diffusion Equations by using the numerical multigrid method, including the time cost and the effect of solving linear system after using multigird method. And recovering the disadvantages and advantages of multigrid method. Through understanding the concepts of multigrid method, we hope we can using this method to decrease the iterations and cost of time. And extending this method that can be used to solve more linear system problems or linear sparse matrix problems.

    Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Review of Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1 Algebraic Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 AMG algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Aggregation-based Algebraic Multigrid (AGMG) Method . . . . . . . . . 9 3.3.1 The solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . 13 3.3.2 Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Multilevel Method: A Parallel Smoothed Aggregation Multigrid Method . 17 3.4.1 Uncoupled Parallel Aggregation Scheme . . . . . . . . . . . . . 19 3.4.2 Maximally Independent Sets(MIS) Parallel Aggregation Scheme . 19 3.5 Newton-Krylov-Schwarz algorithm . . . . . . . . . . . . . . . . . . . . . 20 3.5.1 Inexact Newton Method with Backtracking (INB) . . . . . . . . . 20 3.5.2 Krylov Iterative Methods . . . . . . . . . . . . . . . . . . . . . . 20 3.5.3 Overlapping Schwarz Preconditioner . . . . . . . . . . . . . . . 21 4 Testing Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.1 Laplace Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.1.1 Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.1.2 Basic Solution Domain . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.4 Solution Graph of Laplace Equation . . . . . . . . . . . . . . . . 25 4.2 Anisotropic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2.1 Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2.2 Basic Solution Domain . . . . . . . . . . . . . . . . . . . . . . . 26 4.2.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2.4 Solution Graph of ANI Equation . . . . . . . . . . . . . . . . . . 28 4.3 Jump Coefficient Problem . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3.1 Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3.2 Basic Solution Domain . . . . . . . . . . . . . . . . . . . . . . . 29 4.3.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3.4 Solution Graph of Jump Coefficient Problem . . . . . . . . . . . 32 4.4 Convection-Diffusion Problems . . . . . . . . . . . . . . . . . . . . . . 33 4.4.1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4.2 Basic Solution Domains . . . . . . . . . . . . . . . . . . . . . . 34 4.4.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4.4 Solution Graph of Convection-Diffusion Equations . . . . . . . . 38 4.5 Poisson-Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5.1 The derivation of Poisson-Boltzmann Equations . . . . . . . . . . 44 4.5.2 Finite Element Method for Discretizing Poisson-Boltzmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.5.3 Basic Solution Domains . . . . . . . . . . . . . . . . . . . . . . 50 4.5.4 The solution graph . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5.5 Particles Interaction Force . . . . . . . . . . . . . . . . . . . . . 51 4.5.6 The computation of the non-dimensionalized electrostatic force acting on the charged particles . . . . . . . . . . . . . . . . . . . 54 4.6 Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.6.1 Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.6.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.6.3 Basic Solution Domain . . . . . . . . . . . . . . . . . . . . . . . 57 4.6.4 Solution Graphs of Poisson Equation . . . . . . . . . . . . . . . 57 5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.1 Sequential version: AGMG . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.1.1 Discussions and Results of Experiments . . . . . . . . . . . . . . 59 5.2 Parallel version: ML on PETSc . . . . . . . . . . . . . . . . . . . . . . . 63 5.2.1 Discussions and Results of Experiments . . . . . . . . . . . . . . 63 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Appendix about Indefinite Problems . . . . . . . . . . . . . . . . . . . . . . . . . 83 0.1 Helmholtz Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 0.1.1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 83 0.1.2 Basic Solution Domains . . . . . . . . . . . . . . . . . . . . . . 84 0.1.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 0.1.4 Solution Graphs of Helmholtz Equations . . . . . . . . . . . . . 86 0.2 Numerical Results of Helmholtz Problems . . . . . . . . . . . . . . . . . 91

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