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研究生:沈廉智
研究生(外文):Lien-Chih Shen
論文名稱:Performance Comparison of Two PETSc-based Eigensolvers for Quadratic PDE Problems
論文名稱(外文):Performance Comparison of Two PETSc-based Eigensolvers for Quadratic PDE Problems
指導教授:黃楓南
指導教授(外文):Feng-Nan Hwang
學位類別:碩士
校院名稱:國立中央大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文出版年:2012
畢業學年度:100
語文別:英文
論文頁數:65
中文關鍵詞:二次特徵值問題SLEPcPETScPJDPack
外文關鍵詞:Quadratic Eigenvalue ProblemsQuadratic PDE ProblemsPJDPackSLEPcPerformance ComparisonPETSc
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在本論文中,我們將有系統的調查兩種用於解決二次特徵值問題(QEPs)的eigenpackage, 包含:the Scalable Library for Eigenvalue Problem Computations(SLEPc)與Parallel
Jacobi-Davidson Package(PJDPack)這兩個建構在Portable, Extensible Toolkit for Scientific Computation(PETSc)的Package. 對於這兩個eigenpackage最主要的差別在於SLEPc是使用linearization approach並且有多種不同的eigensolver去解決generalized後的特徵值問題. 而另一方面, PJDPack只有使用PJD演算法並且對於二次特徵值問題是使用直接解法. 為了能夠進行接下來的討論, 我們使用一個Matlab-based的工具, a collection of nonlinear eigenvalue problem (NLEVP)來製作大量具有差異性值的矩陣來做一些數值實驗並且用robustness, accuracy和efficiency來評估效率問題.
In this thesis, we systematically investigate the numerical performance of two eigenpackages for solving quadratic eigenvalue problems (QEPs), namely Scalable Library for Eigenvalue Problem Computations (SLEPc) and Parallel Jacobi-Davidson Package (PJDPack) are both in common built-on-top of Portable, Extensible, Toolkits for Scientific computation (PETSc) [3]. The major differeces between these two eigenpackages is that SLEPc adopts the linearization approach and provides several linear eigensolvers to solve the resulting companion GEPs. On the other hand, the PJD algorithm is the only kernel solver of PJDPack that targets directly the QEP. To draw the concrete conclusions, we generate a large number of test cases using a Matlab-based toolbox, a collection of nonlinear eigenvalue problem (NLEVP) with a diversity of matrix properties and conduct intense numerical experiments to evaluate the performance in terms of robustness, accuracy and efficiency.
Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Linearization approach in SLEPc . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Exterior spectrum case . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Interior spectrum case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Shift-and-invert-first-then-linearization . . . . . . . . . . . . . . 4
2.2.2 Linearization-first-then-shift-and-invert . . . . . . . . . . . . . . 6
2.3 Scaling technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Iterative solution of comparison GEP . . . . . . . . . . . . . . . . . . . . 8
2.5 The Eigenpackage of SLEPc . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5.1 Defining the Problem . . . . . . . . . . . . . . . . . . . . . . . . 11
3 JD algorithm in PJDPack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1 Test cases and environment . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2.1 The Shift-and-invert SLEPc QEP solver . . . . . . . . . . . . . . 23
4.2.2 Correction equation of PJDPack . . . . . . . . . . . . . . . . . . 24
4.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
The spectrum of all test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1 SLEPc QEP solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 PJDPack QEP solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3 NLEVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 The spectrum of all test cases . . . . . . . . . . . . . . . . . . . . . . . . 43
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