扩散方程的两类反问题的正则化方法和算法研究

In this project, we study two kinds inverse problems of diffusion equation. One is the identification of unknown source for diffusion equation. Now about some specific unknown source, the identification of unknown source has been formed a lot of results on the one dimensional diffusion equation, and plays a huge role for solving the practical problems. But for identifying the unknown source about one dimensional diffusion equation with nonlinear source term, high dimensional diffusion equation and the high dimensional fractional diffusion equation which has more practical application background, there are still little results. This topic about the identification of the unknown source on diffusion equation will be carried out as follows. Firstly, we will identify the nonlinear source term on the one dimensional diffusion equation. Secondly, under the posterior regularization parameter choice rule, we will identify the unknown source for high dimensional difffusion equation in special domain. Thirdly, we will identify the unknown source for high fractional diffusion equation in special domain. The other is that we determinate the unknown source and initial value simultaneously for high dimensional diffusion equation. Through studying two types of questions, we will give lots of new effective regularization methods, establish convergence error estimate, give the optimal error bounds analysis, and give stability and feasible algorithm. Implementation of this project will enrich the theory and regularization method of identifying the unknown source Moreover, we expect to give the guide for detecting the heat source, new energy and the pollution sources of factory and for controling the Fog weather of central and eastern regions of China.

本项目主要研究扩散方程的两类反问题：一类是扩散方程源项识别反问题。关于一维扩散方程的特殊源项识别已有很多结果，并在解决实际问题中发挥很大作用。但是关于一维扩散方程的非线性源项识别，高维扩散方程的源项识别以及更具有实际应用背景的高维分数阶扩散方程的源项识别，结果非常少。本课题对扩散方程的未知源识别问题展开如下研究： (1) 一维扩散方程的非线性源项识别；(2) 在后验正则化参数选取规则下，识别特殊区域上高维扩散方程的未知源; (3) 识别特殊区域上高维分数阶扩散方程的源项。另一类是同时反演特殊区域上的高维扩散方程的源项和初始状态。通过对这两类问题的研究，系统给出多种新的有效正则化方法，建立收敛误差估计式，进行最优误差界分析，构造稳定可行的算法。本项目的实施期望能给实际问题中的热源和新能源探测，以及工厂污染源的检测和我国中东部地区的雾霾天气控制起指导性作用。

随着全球工业的飞速发展以及人口的飞速增长，随之而来的是对人类生存环境的污染，尤其是最近几年我国中东部持续出现大范围的雾霾天气，给人类健康带来巨大的危害。另一方面随着全球人口的增大，生存环境的恶化，人类正面临着能源危机。因此如何有效的控制环境污染，寻找污染源，探索传染病毒的起源，以及如何寻求更有效的新能源是今后全球面临的两个重大问题。上面提到的这些实际问题归结到数学问题就是扩散方程未知项识别问题和初值识别反问题。本项目主要研究了反应扩散方程的两类反问题：一类是整数阶扩散方程的源项识别和初值识别反问题，一类是分数阶反应扩散方程的源项识别和初值识别反问题。除此之外，我们还研究了非齐次Helmholtz方程柯西问题，Poisson 方程源项识别以及修正的Helmholtz源项识别和柯西问题。综上，通过对本项目研究，系统给出多种新的有效正则化方法，建立收敛误差估计式，进行最优误差界分析，构造稳定可行的算法。本项目的实施期望能给实际问题中的热源和新能源探测，传染病毒源头的探索，以及工厂污染源的检测和我国中东部地区的雾霾天气控制起指导性作用。