两类不适定问题的正则化方法研究
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摘要
本论文主要讨论了两类不适定问题的正则化方法。一类是复分析中的数值解析延拓问题,这是一个线性的严重不适定问题。多年来研究进展不大,而且早期的结果基本上集中在条件稳定性上,近年来蓬勃发展的正则化理论及算法在这方面的应用则很少。本文讨论了一个带型区域上的数值解析延拓问题,我们用Fourier变换将其转化为一个线性算子方程,分析了该算子方程的不适定性,进而讨论了正则化方法的最优误差界。在此基础上,为了稳定地计算它,分别提出了一个修改的Tikhonov正则化方法和一个磨光正则化方法,并分别在先验与后验参数选取规则下给出误差估计,最后用数值例子验证了方法的有效性。另一类是对非线性不适定算子方程一般理论中的有关问题进行了研究,在对非线性条件的某种限制下,针对经典的Tikhonov正则化方法,在对数源条件下分别给出了先验参数选取和两个后验参数选取规则下的收敛速率,并对一个有重要应用背景的未知边界识别问题用所得结果进行了数值模拟,数值结果也体现了方法的有效性。
This thesis investigates the regularization methods for two classes of ill-posed problems. The first one is the problem of numerical analytic continuation in com-plex analysis, which is a linear and severely ill-posed problem. Over the years, advances in this research are not remarkable and the earlier results mostly focus on the conditional stability. However, it seems that there are few applications of modern theory of regularization methods which have been developed intensively in the last few decades. In this thesis, we consider the problem of numerical analytic continuation on a strip domain. Firstly, we transform it into a linear operator equation by using Fourier transform and analyze the ill-posedness. Moreover, the optimal error bound of regularization method is also given. Based on this, for stably computing it, we provide a modified Tikhonov regularization method and a mollification regularization method, and give error estimate under a-priori and a-posteriori parameter choice rules, respectively. Several numerical examples are provided, which show the two methods work effectively. The other kind is that some related problems in the general theory on a nonlinear ill-posed operator equa-tion are considered. Imposing some nonlinear conditions on the nonlinear operator, we provide a-priori and two a-posteriori parameter choice rules which are specific to Tikhonov regularization method under logarithmic-type source condition and give the convergence rates, respectively. In addition, an unknown boundary iden-tification problem, which has important application background, is also discussed. Numerical experiments support our theoretical results and show the effectiveness of the methods.
This thesis investigates the regularization methods for two classes of ill-posed problems. The first one is the problem of numerical analytic continuation in com-plex analysis, which is a linear and severely ill-posed problem. Over the years, advances in this research are not remarkable and the earlier results mostly focus on the conditional stability. However, it seems that there are few applications of modern theory of regularization methods which have been developed intensively in the last few decades. In this thesis, we consider the problem of numerical analytic continuation on a strip domain. Firstly, we transform it into a linear operator equation by using Fourier transform and analyze the ill-posedness. Moreover, the optimal error bound of regularization method is also given. Based on this, for stably computing it, we provide a modified Tikhonov regularization method and a mollification regularization method, and give error estimate under a-priori and a-posteriori parameter choice rules, respectively. Several numerical examples are provided, which show the two methods work effectively. The other kind is that some related problems in the general theory on a nonlinear ill-posed operator equa-tion are considered. Imposing some nonlinear conditions on the nonlinear operator, we provide a-priori and two a-posteriori parameter choice rules which are specific to Tikhonov regularization method under logarithmic-type source condition and give the convergence rates, respectively. In addition, an unknown boundary iden-tification problem, which has important application background, is also discussed. Numerical experiments support our theoretical results and show the effectiveness of the methods.
引文
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[37]M. Hanke, A regularization Levenberg-Marquardt scheme with applications to in-verse ground-water filtration problems, Inverse Problems 13,79-95,1997.
[38]M. Hanke, Regularizing properties of a truncated Newton-CG algorithm for non-linear inverse problems, Numer. Funct. Anal. Optim.18,971-993,1997.
[39]D.N. Hao and H. Shali, Stable analytic continuation by mollification and the fast Fourier transform, Method of Complex and Clifford Analysis (Proceedings of ICAM Hanoi,143-152,2004.
[40]D.N. Hao, A mollification method for ill-posed problems, Numer. Math.68,469-506, 1994.
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