项目名称: 具退化系数的发展型方程多参数反演问题的正则化理论和算法研究
项目编号: No.11261029
项目类型: 地区科学基金项目
立项/批准年度: 2013
项目学科: 数理科学和化学
项目作者: 杨柳
作者单位: 兰州交通大学
项目金额: 45万元
中文摘要: 具退化系数的发展型方程常常出现在人口预测与控制,多孔介质流体力学,以及金融数学等研究领域中。这些方程中的部分系数往往不能直接观测到,需要用间接的手段来标定,这就是我们所要研究的反问题。本项目研究利用某些附加条件同时重构退化抛物型方程的初值和源项系数的反问题。与普通的抛物型方程系数反问题不同,这里的方程在部分边界存在退化。方程的退化性一方面会导致在定解域的部分边界可能会缺少边界条件,另一方面会导致方程的解可能没有足够的正则性。本项目中的另一困难是模型的初值是未知的,这是一个严重不适定问题,方程的退化性并不能改变这一实质。我们的工作主要包括以下两个方面:1、理论方面:首先研究正问题的解的正则性,进而研究反问题的解的唯一性和条件稳定性,以及基于Tikhonov正则化理论框架下的最优控制解的适定性。2、数值模拟方面:在理论分析的基础上,设计稳定的迭代算法,进行数值试验,并作误差分析。
中文关键词: 初值;源项系数;多参数反演问题;正则化;稳定算法
英文摘要: Degenerate evolutionary equations often arise in many fields of research, such as population prediction and control, porous media fluid mechanics, financial mathematics, etc. Some coefficients in these equations can not be measured directly and need to be identified by indirect methods, which is the inverse problem we shall study. This project studies the inverse problem of simultaneously reconstructing the initial value and source coefficient in degenerate parabolic equations using some additional conditions. Being different from ordinary inverse coefficient problems in parabolic equations, there exists degeneracy on part of boundaries in the mathematical model. On one hand, the degeneracy may lead to the corresponding boundary conditions missing; on the other hand, it can also cause that the solution of the equation has no sufficient regularity. Another difficulty in the project is that the initial value of the model is unknown, which is known to be a severely ill-posed problem. The degeneracy of the equation can not change such the essence. Our work can be divided into two parts. In the theoretical part, we will study the regularity of the solution of forward problem and then investigate the uniqueness and conditional stability of the solution for the inverse problem. Correspondingly, on the basis of Tikhonov
英文关键词: initial value;source coefficient;inverse multi-parameters problem;regularization;stable algorithm