求解大規(guī)模線性離散不適定問(wèn)題的Arnoldi—Fractional Tikhonov正則化算法

張慧

摘要：隨著科技發(fā)展，不適定問(wèn)題出現(xiàn)在地球物理等多種領(lǐng)域。正則化方法是求解此類問(wèn)題近似解的有效算法。該文將Fractional Tikhonov正則化算法應(yīng)用于投影算法，提出求解大規(guī)模線性離散不適定問(wèn)題的Arnoldi-Fractional Tikhonov正則化算法。并進(jìn)一步提出廣義Arnoldi-Fractional Tikhonov正則化算法。最后，論文對(duì)所提出的算法編寫程序進(jìn)行數(shù)值試驗(yàn)比較。結(jié)果表明新算法是有效且具有優(yōu)勢(shì)的。

關(guān)鍵詞：不適定問(wèn)題；正則化方法；Fractional Tikhonov正則化方法；Arnoldi-Fractional Tikhonov正則化算法

中圖分類號(hào)：TP311 文獻(xiàn)標(biāo)識(shí)碼：A 文章編號(hào)：1009-3044（2016）02-0236-03

Abstract：With the development of the technology ，the ill-posed problems widely arise in many areas such as geophysics and so on. In this thesis， An Arnoldi-Fractional Tikhonov regularization method for large scale linear discrete ill-posed problems is presented via applying the Fractional Tikhonov regularization to the projection algorithm. Further more， the generalized Arnoldi-Fractional Tikhonov method are proposed in the follows. At last， this thesis conducts numerous classical numerical experiments on the improved methods proposed above. Numerical experiments and comparisons indicate that the new improved regularization methods are feasible and efficient.

Key words： ill-posed problems； regularization methods； the Fractional Tikhonov regularization method； the Arnoldi-Fractional Tikhonov regularization method

1 引言及主要結(jié)論

由表3的結(jié)果知，GAFT的計(jì)算結(jié)果精度比GAT的計(jì)算結(jié)果高，但是參數(shù)[α]的選取會(huì)直接影響到求解精度，并且不同矩陣所取的最合適的[α]不一樣。

根據(jù)以上三組數(shù)值試驗(yàn)的結(jié)果，可以看出：在Fractional Tikhonov正則化方法基礎(chǔ)上給出的Arnoldi-Fractional Tikhonov和Generalized-Arnoldi- Fractional Tikhonov正則化方法在求解大規(guī)模線性離散不適定問(wèn)題時(shí)是具有一定優(yōu)勢(shì)的，計(jì)算結(jié)果的精度更高一些。

5 結(jié)束語(yǔ)

對(duì)于大規(guī)模不適定問(wèn)題的求解難度在于其系數(shù)矩陣的奇異值分解計(jì)算量過(guò)大。首先將其投影到小規(guī)模子空間上，再對(duì)投影問(wèn)題的求解采用Fractional Tikhonov正則化方法，并推廣到廣義情形。數(shù)值試驗(yàn)表明新方法是具有一定優(yōu)勢(shì)的。但其中參數(shù)[α]的值影響解的精度，如何確定合適的[α]是下一步需要解決的問(wèn)題。

參考文獻(xiàn)：

[1] 黃光遠(yuǎn)，劉小軍.數(shù)學(xué)物理反問(wèn)題[M].濟(jì)南：山東科學(xué)技術(shù)出版社，1993.

[2] Kirsch A.An Introduction to the Mathematical Theory of Inverse Problems[M].New York：Springer- Verlag， 1996.

[3] 王彥飛.反演問(wèn)題的計(jì)算方法及其應(yīng)用[M].北京：高等教育出版社，2007.

[4] Tikhonov A N.Solution of incorrectly formulated problems and the regularization method[J]. Soviet. Math. Dokl.，1963（4）：1035-1038.

[5] Hochstenbach M E， Reichel. L. Fractional Tikhonov regularization for linear discrete ill-posed problems[J]. BIT， 2011（51）：197-215.

[6] Lewis B，Reichel L.Arnoldi-Tikhonov regularization methods[J].J.Comput.Appl.Math.， 2009，226： 92-102.

[7] Wilkinson J H. The Algebraic Eigenvalue problem[J].Oxford University Press.， 1965：104-108.

[8] Reichel L，Sgallari F，Ye Q.Tikhonov regularization based on generalized Krylov subspace methods[J]. Applied Numerical Mathematics.，2012（62）：1215-1228.

[9] 肖庭延，于滲根，王彥飛.反問(wèn)題數(shù)值解法[M].北京：科學(xué)出版社，2003.

[10] Morikuni K， Reichel L，Hayami K. FGMRES for linear discrete ill-posed problems[J]. Appl. Numer. Math.，2014（75）： 175-187.