关于两类方程系数反演问题的适定性研究
发布时间：2019-05-20 17:17
【摘要】：本文主要对椭圆、抛物型方程的系数反演问题进行了研究,这两类反问题不论在金融、物理、医学、地质探测和无线电传播领域还是在电磁金属成型技术中都有着极其广泛的应用。文章第一部分的椭圆方程是基于弗雷歇偏导,证明了所对应泛函的凸性,由泛函的这个性质可以得到解的唯一性;而文章中第二部分数学模型是一个抛物型方程,利用最优控制理论,探究了原问题对应的优化问题解的适定性。该问题的难点在于:首先,二阶抛物型方程中所需反演的是一个二阶项系数,这是一个强不适定且完全非线性的问题;同时,给出的附加条件并非通常意义下的终端观测值,而是积分平均意义的观测值,这种类型的附加条件会导致相应的控制泛函的极小元所满足的必要条件极为复杂。再者,由于所给的控制泛函没有凸性,一般情况下很难得到最优解的唯一性。通过仔细分析了极小元所满足的必要条件,并结合正问题的一些先验估计式,我们发现,当终端时刻T适当小时,可以证明极小元的局部唯一性和稳定性,这也是本文的主要工作。文章主要包含以下四个部分:首先引言部分讲述了反问题的背景,国内外的研究状况,以及反问题成长的一个历程。第一章从理论上重点分析了椭圆型方程的系数反演问题,首先介绍了椭圆方程的数学模型,由于反问题是不适定的,这时需要对椭圆方程的正问题进行能量估计,运用弗雷歇偏导理论对这个方程的能量估计式进行变形,得到了椭圆方程对应泛函的凸性,由泛函的这个性质可以得到椭圆方程解的唯一性。第二章是对抛物型方程的扩散系数反问题的研究,由于原问题的不适定性,将原问题转化为最优控制问题P,采用最优化方法对扩散系数反演问题进行了研究。再根据正问题的能量估计和相应的共轭方程的能量估计,然后运用能量估计式得到了最优解所满足的必要条件,最后在T比较小的情况下得到了最优解的唯一性和稳定性。第三章对抛物和椭圆型方程反问题的后续工作进行了总结与展望。
[Abstract]:In this paper, we mainly study the coefficient inversion of elliptical and parabola equations. these two kinds of inverse problems are in finance, physics and medicine. Geological exploration and radio propagation are also widely used in electromagnetic metal forming technology. In the first part of this paper, the elliptical equation is based on Frecher partial derivation, and the convexity of the corresponding functional is proved. The uniqueness of the solution can be obtained from this property of the functional. In the second part of the paper, the mathematical model is a parabola equation. By using the optimal control theory, the well-posedness of the solution of the optimization problem corresponding to the original problem is discussed. The difficulties of this problem are as follows: firstly, what needs to be inversed in the second-order parabola equation is a second-order term coefficient, which is a strongly ill-posed and completely nonlinear problem; At the same time, the additional conditions given are not terminal observations in the usual sense, but observations in the mean sense of integral. This type of additional conditions will lead to the very complex necessary conditions satisfied by the minimum elements of the corresponding control functional. Moreover, because the given control functional is not convex, it is difficult to obtain the uniqueness of the optimal solution in general. By carefully analyzing the necessary conditions satisfied by the minimum element and combining with some prior estimates of the positive problem, we find that when the terminal time T is properly small, the local uniqueness and stability of the minimum element can be proved. This is also the main work of this paper. The article mainly includes the following four parts: first of all, the introduction describes the background of the anti-problem, the research situation at home and abroad, and a course of the growth of the anti-problem. In the first chapter, the coefficient inversion problem of Elliptic equation is analyzed theoretically. Firstly, the mathematical model of Elliptic equation is introduced. Because the inverse problem is ill-posed, it is necessary to estimate the energy of the positive problem of Elliptic equation. The energy estimation formula of the equation is deformed by using Frecher partial derivation theory, and the convexity of the corresponding functional of the elliptical equation is obtained. from this property of the functional, the uniqueness of the solution of the elliptical equation can be obtained. In the second chapter, the inverse problem of diffusion coefficient of parabola equation is studied. because of the discomfort of the original problem, the original problem is transformed into the optimal control problem P, and the inverse problem of diffusion coefficient is studied by using the optimization method. Then according to the energy estimation of the positive problem and the energy estimation of the corresponding conjugated equation, then the necessary conditions for the optimal solution are obtained by using the energy estimation formula. Finally, the uniqueness and stability of the optimal solution are obtained when T is relatively small. In the third chapter, the follow-up work of inverse problems of parabola and Elliptic equations is summarized and prospected.
【学位授予单位】：兰州交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2481809
[Abstract]:In this paper, we mainly study the coefficient inversion of elliptical and parabola equations. these two kinds of inverse problems are in finance, physics and medicine. Geological exploration and radio propagation are also widely used in electromagnetic metal forming technology. In the first part of this paper, the elliptical equation is based on Frecher partial derivation, and the convexity of the corresponding functional is proved. The uniqueness of the solution can be obtained from this property of the functional. In the second part of the paper, the mathematical model is a parabola equation. By using the optimal control theory, the well-posedness of the solution of the optimization problem corresponding to the original problem is discussed. The difficulties of this problem are as follows: firstly, what needs to be inversed in the second-order parabola equation is a second-order term coefficient, which is a strongly ill-posed and completely nonlinear problem; At the same time, the additional conditions given are not terminal observations in the usual sense, but observations in the mean sense of integral. This type of additional conditions will lead to the very complex necessary conditions satisfied by the minimum elements of the corresponding control functional. Moreover, because the given control functional is not convex, it is difficult to obtain the uniqueness of the optimal solution in general. By carefully analyzing the necessary conditions satisfied by the minimum element and combining with some prior estimates of the positive problem, we find that when the terminal time T is properly small, the local uniqueness and stability of the minimum element can be proved. This is also the main work of this paper. The article mainly includes the following four parts: first of all, the introduction describes the background of the anti-problem, the research situation at home and abroad, and a course of the growth of the anti-problem. In the first chapter, the coefficient inversion problem of Elliptic equation is analyzed theoretically. Firstly, the mathematical model of Elliptic equation is introduced. Because the inverse problem is ill-posed, it is necessary to estimate the energy of the positive problem of Elliptic equation. The energy estimation formula of the equation is deformed by using Frecher partial derivation theory, and the convexity of the corresponding functional of the elliptical equation is obtained. from this property of the functional, the uniqueness of the solution of the elliptical equation can be obtained. In the second chapter, the inverse problem of diffusion coefficient of parabola equation is studied. because of the discomfort of the original problem, the original problem is transformed into the optimal control problem P, and the inverse problem of diffusion coefficient is studied by using the optimization method. Then according to the energy estimation of the positive problem and the energy estimation of the corresponding conjugated equation, then the necessary conditions for the optimal solution are obtained by using the energy estimation formula. Finally, the uniqueness and stability of the optimal solution are obtained when T is relatively small. In the third chapter, the follow-up work of inverse problems of parabola and Elliptic equations is summarized and prospected.
【学位授予单位】：兰州交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
【参考文献】
相关期刊论文 前1条
1 Zuicha DENG;Liu YANG;;An Inverse Problem of Identifying the Radiative Coefficient in a Degenerate Parabolic Equation[J];Chinese Annals of Mathematics(Series B);2014年03期
，本文编号：2481809
本文链接：https://www.wllwen.com/kejilunwen/yysx/2481809.html
