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博碩士論文 etd-0614106-030140 詳細資訊
Title page for etd-0614106-030140
論文名稱
Title
MQ徑向基底函數配置法對於偏微分方程之高精確度計算
High precision computations of multiquadric collocation method for partial differential equations
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
37
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2006-06-07
繳交日期
Date of Submission
2006-06-14
關鍵字
Keywords
無網格方法、誤差估計、指數收歛、最佳化形狀因子、條件數、任意精確度計算、MQ徑向基底函數配置法
meshless method, multiquadric collocation method, condition number, exponential convergence, arbitrary precision computation, error estimate, optimal shape factor
統計
Statistics
本論文已被瀏覽 5745 次,被下載 13
The thesis/dissertation has been browsed 5745 times, has been downloaded 13 times.
中文摘要
MQ徑向基底函數配置法由於它的誤差的指數收歛速率,因此對於解決偏微分方程式一個非常有效率的方法。它令人驚奇的具有兩種減低誤差的方式,除了傳統的加密網格點方式,還有令人想不到的只需要簡單的在基底函數上,增加形狀常數c,就可以達到相同的目的。重要的是,增加形狀常數c,並不會額外的造成計算上的負擔。數值結果被推測如果在不考慮捨入誤差、使用無限位精確度計算的情況下,c應該是趨近於 。因為捨入誤差會導致矩陣的不穩定造成很大的條件數,因此,沒辦法獲得無限位準確的解。因此,本篇論文主要是根據上述的推論,使用Mathematica這套軟體,因為它可以提供任意精確度的計算。
首先我們提出了較明確的誤差估計函數,從這個公式,給定一個有限的網格數,我們可以得到最佳化的形狀常數c,使得數值解的誤差達到最小。而且在真實的問題中,真實解常常是不知道的,因此,我們使用剩餘誤差去取代真實解,相同地我們的公式也是可以獲得的!因為並不是所有的問題都需要任意精確度的計算或著持續的密網格點,因此,我們也使用C++用16位精確度和加密網格點的方式與32位精確度和增加形狀常數德的方式,設法找到最適合的策略。最後,我們提供了一些數值實驗結果,來支持我們的結論!
Abstract
Multiquadric collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. More amazingly, there are two ways to reduce the error: the traditional way of refining the grid, and the unexpected way of simply increasing the value of shape constant $c$ contained in the multiquadric basis function, $sqrt{r^2 + c^2}$. The latter is accomplished without increasing computational cost. It has been speculated that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting $c
ightarrow infty$. The ability to obtain infinitely accurate solution is limited only by the roundoff error induced instability of matrix solution with large condition number. Using the arbitrary precision computation capability of {it Mathematica}, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented in this paper. A formula for a finite, optimal $c$ value that minimizes the solution error for a given grid size is obtained. Using residual errors, constants in error estimate and optimal $c$ formula can be obtained. These results are supported by numerical examples.
目次 Table of Contents
1 Introduction
2 The radial basis function collocation method
2.1 Radial basis function
2.2 Solving PDEs algorithm
3 Arbitrary precision computation
3.1 Maximum c
3.2 Error estimate
3.3 Optimal shape factor
3.4 Domain with different size
3.5 Residual errors plus "c-scheme"
4 Finite precision computation
4.1 Condition number
4.2 Quadruple precision +"c-scheme" or double precision +"h-scheme".
5 Conclusion
參考文獻 References
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