Jacobi-Davidson (JD) 演算法是一個解特徵值問題的迭代法，這些特徵值問題可以從一些工程或科學應用上的偏微分方程離散化所得來。在這篇論文中，我們針對量子點所產生的多項式特徵值問題求解。目前已有一些發展良好的平行套件，但是這些套件僅適合解決一般標準特徵值問題和廣義特徵值。所以我們利用 Portable, Extensible Toolkit for Scientific Computation (PETSc) 與 Scalable Library for Eigenvalue Problem Computations (SLEPc) 發展一個以 JD 為基礎的平行多用途科學計算軟體套件去解決多項式特徵值問題。JD 演算法是一種子空間法 (spbspace method)。在每次 JD 迭代，我們會利用 Rayleigh-Ritz procedure 從一個給定的 search space 去提取近似 Ritz-pair。如果近似的 Ritz-pair 不夠好，我們便需要透過解一個線性系統 correction equation 去增加一個基底向量到 search space。在 JD 演算法中，解 correction equation 是最 JD 演算法中昂貴的部份。所以對於迭代法去設計一個有效率的 preconditioner 是相當重要的。因此，我們提出並且研究 Schwarz 架構下的平行區域分割 preconditioner，這種 preconditioner 已被熟知並且廣泛利用在線性系統的求解，然而卻很少應用在特徵值問題上的研究。我們從五次與三次特徵值問題得到的數值結果顯示，在配合一些 Krylov 子空間法時，這樣的 preconditioner 是有效率並且能夠改進整體的 JD 收斂速率，進而在數百個處理器的平行電腦上得到相當卓越的效能。 Jacobi-Davidson (JD) algorithm is one of the most popular iterative method for solving large sparse eigenvalue problems (EVPs) obtained from the discretization of some partial differential equations in many engineering and scientific applications. In this thesis, we target polynomial EVPs solved by the JD algorithm arising in quantum dot simulations. Although several existing state-of-the-art parallel packages, they are only suitable for solving standard or generalized EVPs. Therefore, we are developing a parallel general-purpose scientific software package based on JD method for solving polynomial EVPs using two powerful scientific software libraries, namely Portable, Extensible Toolkit for Scientific Computation (PETSc) and Scalable Library for Eigenvalue Problem Computations (SLEPc). JD algorithm is a class of subspace methods. At each JD iteration, an approximate eigenpair is extracted from a given search space through the Rayleigh-Ritz procedure. If the approximate eigenpair is not good enough, one needs to enlarge the search space by adding a new basis vector, which is obtained by solving a large sparse linear system, known as the correction equations. In the JD algorithm, solving the correction equation is the most expensive part. Therefore, to design an efficient preconditioner for some iterative method becomes very crucial. Hence, we proposed and studied a parallel domain-decomposed preconditioner based on the Schwarz framework, which is wildly used and well-understood for solving linear systems, but less studied for solving EVPs. Our numerical results obtaining from quintic and cubic EVPs show that such efficient preconditioner in conjunction with some Krylov subspace method, such as GMRES, can improve the overall convergence rate of the JD algorithm so that exhibit superior performance on a parallel machine up to few hundred processors.
