二次暨有理特徵值問題中高效能Arnoldi型態演算法

Abstract

本論文探討求解二次特徵值問題及有理特徵值之高效能Arnoldi型態演算法。其研究主題可分為兩部分：（一）流固系統中非線性特徵值問題之Arnoldi型態演算法之比較；（二）求解二次特徵值問題中的半正交廣義Arnoldi法。 我們探討並分析一個具有耗散聲能吸音牆密閉空間中聲場的阻尼振動模態。利用有限元素法，我們可由位移場的棱邊離散化將問題轉變為一個求解二次特徵值問題。另一方面，若考慮壓力節點的離散則會獲得一個有理特徵值問題。透過線性化的技巧，我們可將這兩個非線性特徵值問題分別改寫成型態為$Ax=\lambda x$的廣義特徵值問題。該問題可以用Arnoldi演算法處理兩種不同型態係數矩陣, $B^{-1}A$及$AB^{-1}$, 的標準特徵值問題。數值結果顯示利用Arnoldi法求解$AB^{-1}$具有較高的精準度。 對於求解二次特徵值問題中絕對值較靠近零之特徵值所對應的特徵對，我們發展了一個正交投影法－半正交廣義Arnoldi法。此外，我們更進一步提出可精化、可重啟動的半正交廣義Arnoldi法。相較於將二次特徵值問題線性化後再利用傳統隱式重啟動Arnoldi法求解，數值實驗顯示隱式重啟動半正交廣義Arnoldi法（不論是否有精化過程）具有極佳的收斂行為。

In this dissertation, we consider two themes related to Arnoldi-type algorithms for solving nonlinear eigenvalue problems.

We develop and analyze efficient methods for computing damped vibration modes of an acoustic fluid confined in a cavity, with absorbing walls capable of dissipating acoustic energy. The edge-based finite elements for the displacement field results in a quadratic eigenvalue problem. On the other hand, the discretization in terms of pressure nodal finite elements results in a rational eigenvalue problem. We use the linearization technique to transform these nonlinear eigenvalue problems, respectively, into generalized eigenvalue problems $Ax=\lambda x$ and apply Arnoldi algorithm to two different types of single matrices $B^{-1}A$ and $AB^{-1}$. Numerical accuracy shows that the application of Arnoldi on $AB^{-1}$ is better than that on $B^{-1}A$.

For computing a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems, we develop the semiorthogonal generalized Arnoldi method, an orthogonal projection technique. Furthermore, we propose refinable and restartable variations of this method to improve the accuracy and efficiency. Numerical examples demonstrate that the implicitly restarted semiorthogonal generalized Arnoldi method with or without refinement has superior convergence behaviors than the implicitly restarted Anoldi method applied to the linearized quadratic eigenvalue problem.