利用等模調整器求解病態線性系統之研究

Abstract

文之想法乃藉由簡易求得之調整器來改善病態線性系統 中之給定矩陣 對於微小數值改變之敏感性，以達到數值解準確之結果，文章中將調整器分成兩類單側調整器以及雙側調整器，而任一種調整器當中又再因作用位置亦或是作用時機之差異又各有兩個調整器。 共軛梯度法於輕度病態之問題所求得之數值解，收斂速度快且準確性佳，因此本研究藉由不同調整器改善矩陣之病態程度，再使用共軛梯度法進行求解，由於目前並無任一調整器能夠適用於所有問題，本研究將藉由三個不同之正反算問題，來進行數值解比較，其中反算問題如反算柯西問題、反向熱傳導問題，正算問題則為線性希爾伯特問題。本研究藉由無網格法當中之基本解法將連續方程轉為線性方程以便於利用共軛梯度法求解。 不同調整器在面對正反算問題有不同之改善效果，於數值解誤差之改善或收斂速度之提升，可預期地，大部分之數值結果會較未經過任何調整器處理過就使用共軛梯度法進行求解之結果來得好。

In order to get the accurate numerical solutions of ill-posed linear systems we propose an equilibrated condtioning method to reduce the condition number of the given matrix by a simple idea. In the thesis we proposed two kinds of the equilibrated conditioners; one-side equilibrated conditioner and two-side equilibrated conditioner, and for each kind we consider the different acting position or acting timing to generate two conditioners. The conjugate gradient method(CGM) can get very well solution when the condition number is small; therefore, we try to use the different kind of conditioner to refine the ill-condition of the given matrix, and then use CGM to get the solution. We know that there is not any conditioner which is suitable for every problem, and we will use total three problems which include two inverse problems and one direct problem to verify our proposed methods, where Cauchy problem, Backward heat conduction problem and linear Hilbert problem to test them. We will discrete the problem into the linear system by using the method of fundamental solutions, which is one kind of meshless methods. Different conditioners when facing various problems which can obtain different effects, decreasing the numerical error or accelerating the convergence speed. Prospectively, most given matrix which is refined by conditioner which obtain the better numerical results than original one.