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研究生： 程郁芬
Yu-Fen Cheng
論文名稱： A Parallel Two-level Polynomial Jacobi-Davidson Algorithm for Large Sparse Dissipative Acoustic Eigenvalue Problems
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： 71
中文關鍵詞： 阻尼 、平行計算 、有限元素法 、多項式特徵值問題 、聲波
外文關鍵詞： acoustic, polynomial eigenvalue problem, finite element method, Jacobi-Davidson method, additive Schwarz preconditioner, parallel computing, initial search space, precondition, damping
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    • 許多科學與工程上應用需要準確、快速、穩定和可拓展大型稀疏多項式特徵值問題(PEVPs)的數值解對於離散化的偏微分方程。根據數值結果顯示多項式Jacobi-Davidson演算法能夠有效率地對內部特徵值問題求解，因而被廣泛使用。多項式Jacobi-Davidson演算法是一個子空間法(subspace method)，從搜尋空間內提取合適的估計eigenpair並且透過解一個線性系統correction equation在JD的迭代去增加一個基底向量到search space。在本研究當中，我們提出一個新的two-level多項式JD演算法架構在additive Schwarz來解三次多項式特徵值對於噪音工程的應用問題。首先，我們建造搜尋空間利用粗網格之解為細網格的初始基底。另一方面，我們使用一個低成本並且有效率的preconditioner定義在粗網格的restricted additive Schwarz解線性系統correction equation，對於大型問題此方法在多重處理器的平行計算中扮演著重要角色。最後，經由數值結果得到論證，此演算法在平行叢集電腦具有穩健性和延展性。

    Many scientific and engineering applications require accurate, fast, robust, and scalable numerical solution of large sparse algebraic polynomial eigenvalue problems (PEVPs) arising from some appropriate discretization of partial differential equations. The polynomial Jacobi-Davidson (PJD) algorithm has been numerically shown as a promising approach for the PEVPs and has gained its popularity for finding their interior spectrum of the PEVPs. The PJD algorithm is a subspace method, which extracts the candidate approximate eigenpair from a search space and the space undated by embedding the solution of the correction equation at the JD iteration. In this research, we propose the two-level PJD algorithm for PEVPs with emphasis on the application of the dissipative acoustic cubic eigenvalue problem. The proposed two-level PJD algorithm is based on the Schwarz framework. The initial basis for the search space is constructed on the current level by using the solution of the same eigenvalue problem, but defined on the previous coarser grid. On the other hand, a low-cost and efficient preconditioner based on Schwarz framework, coarse restricted additive Schwarz (RAS_c) preconditioner for the correction equation, which plays a crucial role in parallel computing for large-scale problems by using a large number of processors. Some numerical examples obtained on a parallel cluster of computers are given to demonstrate the robustness and scalability of our PJD algorithm.

    List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Polynomial acoustic eigenvalue problems . . . . . . . . . . . . . . . . . . . 4 2.1 Dissipative acoustic eigenvalue problem . . . . . . . . . . . . . . . . . . . . 4 2.2 Galerkin nite element discretization . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Algebraic polynomial eigenvalue problems . . . . . . . . . . . . . . . . . . . 7 3 Two-level polynomial Jacobi-Davidson algorithm . . . . . . . . . . . . . . 10 3.1 One-level Jacobi-Davidson algorithm . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Two-level Jacobi-Davidson algorithm . . . . . . . . . . . . . . . . . . . . . . 12 3.2.1 Two-level approach for constructing initial search space algorithm . 13 3.2.2 One-level coarse restricted additive Schwarz precondition algorithm 14 4 Numerical results and discussions . . . . . . . . . . . . . . . . . . . . . . . . 16 4.1 Problems statement and setup for numerical experiments . . . . . . . . . . 16 4.2 Two-level approach for initial search space construction . . . . . . . . . . . 19 4.3 Two-level ASPJD algorithmic parameter tuning . . . . . . . . . . . . . . . . 24 4.3.1 Coarse grid solution quality . . . . . . . . . . . . . . . . . . . . . . . 25 4.3.2 Coarse grid size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3.3 Subdomain solution quality for RAS c coarse grid . . . . . . . . . . 27 4.3.4 Correction equation solver types and iterations . . . . . . . . . . . . 28 4.4 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.5 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A. Bilinear basis and the local sti ness and mass matrices . . . . . . . . . . . . 38 B. Analytical Solution for various boundary conditions . . . . . . . . . . . . . . 43 a. All Neumann boundary conditions . . . . . . . . . . . . . . . . . . . . . 43 b. All Dirichlet boundary conditions . . . . . . . . . . . . . . . . . . . . . . 45 c. Dirichlet (left), Neumann (others) boundary condition . . . . . . . . . . 50 d. Dirichlet (left), Robin (right), Neumann (others) boundary conditions . 52 C. NaN problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 D. Improve Ritz-value selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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