具奇异或退化性质的二阶抛物型方程的系数反演问题

发布时间：2018-08-24 08:11

【摘要】：本文主要考虑具奇异或退化性质的二阶抛物型方程的系数反演问题,研究在适当的附加条件下解的唯一性和条件稳定性,正则化问题的解的存在性,唯一性,稳定性,收敛性,以及有效的数值重构方法。第一章,首先介绍了偏微分方程系数反问题的研究背景,其后引入了本文的数学模型,并详细阐述了研究动机和研究的主要困难。第二章,介绍了一些函数空间和相应的积分嵌入理论,以及二阶抛物型方程的适定性结果,这些结果在后面章节的证明中起到了重要作用。第三章,研究了一个利用终端观测值确定二阶抛物型方程的辐射系数的反问题。与通常的终端控制问题不同,这里的观测数据仅在某个固定方向上给出,而不是整个区域,这会导致抛物型方程的共轭理论在此并不适用。另外,由于方程的定解域是圆或扇形,在极坐标下定解域可转化为一个矩形,但同时也会造成方程的主项系数奇异。为了克服系数奇异的困难,我们引入了一些赋权的Sobolev空间。基于最优控制理论框架,原问题被转化为一个优化问题。我们首先证明了极小元的存在性,并导出了极小元所满足的必要条件。利用极小元所满足的必要条件,以及正问题解的一些先验估计结果,我们证明了极小元的唯一性和稳定性。最后,为了说明最优控制问题的解和原问题的解之间的差异,我们还证明了极小元的收敛性,并给出了收敛阶。第四章,研究了一个利用附加条件同时重构二阶退化抛物型方程的初值和源项系数的反问题。该问题的主要特征有两点:(i)方程的主项系数在定解区域的两端都退化为零;(ii)方程中包含两个独立的未知函数,因之这是一个多参数反演问题。系数的退化性一方面会造成方程在定解域的部分边界上缺失边界条件,另一方面还会导致方程的解没有足够的正则性。首先,我们利用Carleman估计和对数凸性方法证明了原问题解的唯一性和条件稳定性。由于原问题的不适定性,我们利用优化方法将原问题转化为一个最优控制问题,并建立了正则化解的存在性,必要条件和收敛性。由于控制泛函含有两个独立的未知函数,且二者的地位并不相同,我们无法应用抛物型方程的共轭理论,否则无法得到正则化解的全局唯一性。我们这里采用的是分项估计的方法,并通过对必要条件的细致分析,最终得到了正则化解的全局唯一性和稳定性。第五章,讨论了前一章中提出的反问题的数值重构。我们利用Landweber迭代算法来求反问题的数值解,其中的关键是求出正问题算子的共轭算子的具体形式。然而,由于两个未知函数的相互耦合,我们很难直接看出共轭算子的结构。为此,我们采用算子分解方法,通过将正问题算子分解为四个独立的算子,并分别求出对应的共轭算子,最后再组合在一起而得到了正问题算子的共轭算子。我们还进行了数值实验,并给出了典型的具体算例。数值实验表明我们的算法是稳定而有效的,两个未知函数都重构得很好。
[Abstract]:In this paper, we consider the inversion of coefficients for second-order parabolic equations with singular or degenerate properties. We study the uniqueness and conditional stability of solutions under appropriate additional conditions, the existence, uniqueness, stability, convergence and effective numerical reconstruction methods of regularization problems. In the first chapter, we introduce the coefficients of partial differential equations. In the second chapter, we introduce some function spaces and integral embedding theories, and the well-posed results of second-order parabolic equations. These results play an important role in the proof of the following chapters. In this paper, we study an inverse problem for determining the radiation coefficients of a second order parabolic equation by means of terminal observations. Unlike ordinary terminal control problems, the observed data are given only in a fixed direction, not in the whole region. This leads to the fact that the conjugate theory of the parabolic equation is not applicable here. In addition, the definite solution of the equation is given. In order to overcome the difficulty of coefficient singularity, we introduce some weighted Sobolev spaces. Based on the framework of optimal control theory, the original problem is transformed into an optimization problem. We prove the uniqueness and stability of the minimal element by using the necessary conditions satisfied by the minimal element and some prior estimates of the solution of the positive problem. Finally, we prove the difference between the solution of the optimal control problem and the solution of the original problem. In Chapter 4, we study an inverse problem of simultaneous reconstruction of the initial and source coefficients of Second Order Degenerate Parabolic Equations by using additional conditions. The main characteristics of this problem are as follows: (i) the principal coefficients of the equations degenerate to zero at both ends of the given solution region; (i i) the equations contain two independent unknown functions, because On the one hand, the degeneracy of coefficients leads to the absence of boundary conditions on some boundaries of the solution domain, and on the other hand, the solution of the equation does not have enough regularity. For the ill-posedness of the original problem, we use the optimization method to transform the original problem into an optimal control problem, and establish the existence, necessary conditions and convergence of the regularized solution. The global uniqueness and stability of the regularized solution can not be obtained. In the fifth chapter, the numerical reconstruction of the inverse problem proposed in the previous chapter is discussed. We use Landweber iterative algorithm to solve the inverse problem. The key to the numerical solution of the problem is to find the concrete form of the conjugate operator of the operator of the positive problem. However, because of the coupling of two unknown functions, it is difficult to see the structure of the conjugate operator directly. Finally, the conjugate operator of the positive problem operator is obtained by combining the conjugate operator. Numerical experiments are carried out and a typical example is given. Numerical experiments show that our algorithm is stable and effective, and both unknown functions are well reconstructed.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.26

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