高维热传导方程热源识别问题的正则化方法和算法

发布时间：2018-07-04 16:12

本文选题：未知源识别 + 不适定问题　；参考：《兰州理工大学》2016年硕士论文

【摘要】：未知源项识别问题是反问题研究中一类重要问题,在实际问题中有重要的应用背景,如新能源的探测、污染源的寻找,尤其在我国北部雾霾如此严重的今天,对引起雾霾的源头寻找显得尤为重要.本文将利用三种不同的正则化方法研究三个不适定问题.第二章研究了含对流项的一维热传导方程只含有空间变量的热源识别反问题.在χ轴上以常速度运动着介质的扩散以及液体的热传导规律都可以用含对流项的热传导方程表示.本文利用高斯核磨光滑正则化方法给出正则解,在先验和后验两种正则化参数选取规则下,得到收敛的误差估计.第三章和第四章研究了高维热传导方程未知热源识别问题.我们主要研究了柱型区域和球型区域上的高维热传导方程的未知热源识别问题.本文在后验正则化参数选取规则下,分别利用简化的Tikhonov正则化方法和谱截断正则化方法,给出正则解与精确解之间H銉lder型误差估计,并利用各种不同类型的数值例子验证了我们方法的有效性.
[Abstract]:The problem of unknown source term identification is an important problem in the research of inverse problem. It has important application background in practical problems, such as the detection of new energy sources and the search of pollution sources, especially today when haze is so serious in the northern part of China. It is very important to find the source of haze. In this paper, three ill-posed problems are studied by using three different regularization methods. In chapter 2, the inverse problem of heat source identification for one-dimensional heat conduction equation with convection term is studied. The diffusion of the medium and the heat conduction of the liquid at constant velocity on the 蠂 axis can be expressed by the heat conduction equation with convection term. In this paper, the regular solution is obtained by using the Gao Si kernel grinding smooth regularization method, and the error estimates of convergence are obtained under the rules of selecting two regularization parameters a priori and a posteriori. In chapter 3 and chapter 4, the problem of identifying unknown heat source of high dimensional heat conduction equation is studied. In this paper, we mainly study the problem of identifying unknown heat sources for high dimensional heat conduction equations in cylindrical and spherical regions. In this paper, using the simplified Tikhonov regularization method and the spectral truncation regularization method, the H lder type error estimates between the regular solution and the exact solution are given under the rule of the selection of posterior regularization parameters. The effectiveness of our method is verified by various numerical examples.
【学位授予单位】：兰州理工大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82

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