用Tikhonov正则化方法同时反演对流扩散方程的对流速度和源函数

Simultaneously inverting the convection velocity and source function of covection-diffusion equations by using the Tikhonov regularization method

摘要:

在给定两个附加观测数据的条件下, 本文基于Tikhonov正则化方法研究了对流扩散方程的对流速度和源函数的同时反演问题. 鉴于原问题是一个初始值非零的对流扩散方程, 本文通过将初始值转化为源项得到了一个组合源项, 首先将原问题转化为一个具有齐次条件的对流扩散问题. 由于所得问题是不适定的, 本文进而利用Tikhonov正则化方法构建了相应的极小化目标泛函, 得到了问题最优解的存在性和所满足的必要条件. 最后, 对终端时刻较小的特殊情形, 本文证明了最优解的唯一性和稳定性.

Abstract:

In this paper, given additionally two observation data, the inverse problem of simultaneously inverting the convection velocity and source function of the convection diffusion equations is studied. The original problem belongs to a class of convection-diffusion equations with non-zero initial value. First, by transforming the information of the initial value into a source fcuntion and then combining it with the original source function, we transform the original problem into a convection-diffusion problem with homogeneous conditions. Further, to handle the ill-posedness of the new problem, we construct the corresponding minimization objective function by using the Tikhonov regularization method, and the existence and necessary conditions for the optimal solution of the new problem are discussed. Finally, for the special case of small terminal time, the uniqueness and stability of the optimal solution are obtained.