本文關(guān)鍵詞：非線性不光滑問題的鄰近修正Landweber迭代法研究　出處：《大連海事大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

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【摘要】：本文主要研究了關(guān)于非線性不光滑問題的鄰近修正Landweber迭代法。在許多實(shí)際應(yīng)用中都會(huì)涉及到反問題,反問題最大的特點(diǎn)就是不適定性。由于反問題的不適定性,所以需要用正則化方法來得到真實(shí)解的穩(wěn)定近似解。在大規(guī)模非線性反問題中,通常采用迭代正則化方法,我們采用的迭代正則化方法是修正Landweber迭代法。然而,L~2范數(shù)罰函數(shù)可能會(huì)引起解過度光滑,解的過度光滑會(huì)導(dǎo)致與實(shí)際情況出現(xiàn)較大的偏差,為了克服這一困難,我們引入了稀疏正則化方法,增加一項(xiàng)L~1范數(shù)罰項(xiàng),但是,這樣會(huì)導(dǎo)致不光滑的情況。為了解決這一問題,我們引入了鄰近算子,將鄰近算子與修正Landweber迭代法相結(jié)合,用于解決非線性不光滑反問題。本文首先對(duì)鄰近算子和修正Landweber迭代法的定義及其相關(guān)性質(zhì)進(jìn)行了回顧。在此基礎(chǔ)上,將鄰近算子與修正Landweber迭代法相結(jié)合,得到了最終的鄰近修正Landweber迭代法。經(jīng)過分析,我們證明了該算法的收斂性,并得到了相應(yīng)的收斂率。最后,用該算法求解了參數(shù)識(shí)別問題,進(jìn)行了一維、二維數(shù)值實(shí)驗(yàn),驗(yàn)證了算法的有效性。理論分析和數(shù)值實(shí)驗(yàn)的結(jié)果表明:鄰近修正Landweber迭代法可以有效地解決非線性不光滑參數(shù)識(shí)別問題。
[Abstract]:In this paper, the neighborhood modified Landweber iterative method for nonlinear nonsmooth problems is studied. The inverse problem is involved in many practical applications. The biggest characteristic of inverse problem is ill-posed. Due to the ill-posed of inverse problem, we need to use regularization method to obtain the stable approximate solution of real solution. In large-scale nonlinear inverse problem. The iterative regularization method is usually used, and the modified Landweber iterative method is used. However, the penalty function of Ln 2 norm may cause the solution to be excessively smooth. In order to overcome this difficulty, we introduce a sparse regularization method to add a penalty term of L ~ (1) norm, but in order to overcome the problem, the excessive smoothness of the solution will lead to a large deviation from the actual situation. In order to solve this problem, we introduce the adjacent operator and combine the adjacent operator with the modified Landweber iterative method. In this paper, the definitions of adjacent operators and modified Landweber iterative methods and their related properties are reviewed. By combining the adjacent operator with the modified Landweber iterative method, the final neighborhood modified Landweber iterative method is obtained. After analysis, we prove the convergence of the algorithm. The corresponding convergence rate is obtained. Finally, the algorithm is used to solve the parameter identification problem, and the one-dimensional and two-dimensional numerical experiments are carried out. The results of theoretical analysis and numerical experiments show that the neighborhood modified Landweber iteration method can effectively solve the problem of nonlinear nonsmooth parameter identification.
【學(xué)位授予單位】：大連海事大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.6

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