在科學與工程領域中，時常會面對如何求解大型稀疏線性方程組的問題。隨著問題的尺度越大，礙於記憶體的限制，要在單一電腦下做計算處理變得很困難。為了使線性系統得以被平行得處理，目前不論直接法或是迭代法，一個常見的處理方式是將問題的變數與方程利用K向分割法做適當的排序，使矩陣具有塊狀箭頭形式，再以舒爾補方法將矩陣消為塊狀上三角矩陣。前者使得矩陣能被分為若干不相關之子線性方程組，故問題能夠在分散式記憶體系統下平行處理。

面對擁有良好性質的線性系統，我們可以透過迭代法搭配預處理去控制條件數，以及避免形成舒爾補矩陣。然而不幸的，面對一般的稀疏線性方程組，我們無法避免實際生成舒爾補矩陣。計算舒爾補矩陣時若是沒有注意計算與資料儲存的方式，在分散式系統中很容易導致過多的資料溝通(Communication) 。甚而在非分散式之平行系統架構中，也容易因舒爾補之計算開銷過大，而使負載平衡差異過大平行效果下降。然而我們發現，其實只要適當的安排矩陣的儲存方式與改變演算法中的計算順序，就能夠有效降低實作中需要的資料溝通，進而優化舒爾補方法的平行效率。

A common approach to solving a large sparse linear system in parallel is using the k-way
partition method to relabel the variables and equations so that the rearranged matrix has a
block arrow form, followed by Schur complement method to eliminate the resulting matrix
to have a block upper triangular form. In the first step, the matrix is constituted by several
independent submatrices so that they can be solved parallelly. However, if we do not
carefully deal with the processes in calculation and storage, large mount of data communication
between processes would occur in distributed memory system. In non-distributed
memory system, the efficiency of parallelization might be low since the unbalanced loading
of processes caused by the expensive calculation of forming Schur complement. In
this paper, we will propose a novel method which can reduce the communication in the
procedure of Schur complement simply by arranging the storagement and the method of
calculation, so we might have better efficiency of the parallelization.

論文審定書i
摘要ii
Abstract iii
List of Figures v
1 Introduction 1
1.1 Parallelization for sparse linear system . . . . . . . . . . . . . . . . . . . 1
1.2 The difficulty in balancing. . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Preliminaries 3
2.1 Schur Complement Method . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Graphs and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Rearrangement of variables and Equations . . . . . . . . . . . . . . . . . 7
3 General Sparse Linear System 9
3.1 Schur Complement Method in Distributed Memory Architecture . . . . . 9
3.2 Fill-in in the Schur Complement . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Algorithm with Communication Reduction . . . . . . . . . . . . . . . . 13
4 Numerical Method and Result 14
4.1 K-way Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Finding Separators and Permutation . . . . . . . . . . . . . . . . . . . . 15
4.3 SuperLU and LAPACK . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4 Algorithm with Communication Reduction . . . . . . . . . . . . . . . . 18
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Bibliography 20

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