【摘要】：隨著科學(xué)技術(shù)和生產(chǎn)水平的發(fā)展,不適定問題廣泛出現(xiàn)在地球物理、自動(dòng)控制等多種領(lǐng)域。正則化方法是求解此類問題近似解的有效算法。本文研究求解線性離散不適定問題的正則化算法。論文主要工作包括:首先,基于求解線性離散不適定問題的Tikhonov正則化方法和TSVD正則化方法,分析兩者的優(yōu)缺點(diǎn),提出了一種求解線性離散不適定問題的混合Tikhonov正則化算法。其次,將Fractional Tikhonov正則化算法應(yīng)用于投影算法,提出了求解大規(guī)模線性離散不適定問題的Arnoldi-Fractional Tikhonov正則化算法。再次,進(jìn)一步提出了廣義Arnoldi-Fractional Tikhonov正則化算法和限制值域的Arnoldi-Fractional Tikhonov正則化算法。而且,論文對(duì)所提出的算法都編寫程序?qū)崿F(xiàn)算法,針對(duì)經(jīng)典算例,進(jìn)行了數(shù)值試驗(yàn)和比較。數(shù)值試驗(yàn)結(jié)果表明了新算法是有效且具有優(yōu)勢(shì)的。
[Abstract]:With the development of science and technology and production level, ill-posed problems are widely used in many fields such as geophysics, automatic control and so on. Regularization method is an effective algorithm for solving approximate solutions of this kind of problems. In this paper, a regularization algorithm for solving linear discrete ill-posed problems is studied. The main work includes: firstly, based on the Tikhonov regularization method and TSVD regularization method for linear discrete ill-posed problems, a hybrid Tikhonov regularization algorithm is proposed to solve the linear discrete ill-posed problems by analyzing their advantages and disadvantages. Secondly, the Fractional Tikhonov regularization algorithm is applied to the projection algorithm, and a Arnoldi-Fractional Tikhonov regularization algorithm is proposed for solving large scale linear discrete ill-posed problems. Thirdly, the generalized Arnoldi-Fractional Tikhonov regularization algorithm and the Arnoldi-Fractional Tikhonov regularization algorithm are proposed. In addition, the algorithms are programmed to realize the algorithms, and numerical experiments and comparisons are carried out for classical examples. Numerical results show that the new algorithm is effective and has advantages.
【學(xué)位授予單位】：南京航空航天大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O241

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