二维时间反向热传导问题的Fourier正则化

Fourier Regularization of the Two-Dimensional Time-Reverse Heat Conduction Problem

摘要: 本文用Fourier正则化的方法求解了二维时间反向热传导问题，即在无界区域中通过终值时刻的数据来确定温度的分布。该问题在图像处理方面有着广泛的应用。本文先用Fourier变换求出了问题的精确解，发现该反问题是严重不适定的。为了解决这一反问题，用Fourier正则化的方法构造出了问题的Fourier正则化解，并在先验条件的假设下给出了其精确解与正则化近似解之间的对数型误差估计。最后由偏差原理给出了近似解的后验误差估计。

Abstract: In this paper, the two-dimensional time reverse heat transfer problem is solved by the method of Fourier regularization, that is, the temperature distribution is determined by the data at the final value in the unbounded region. This problem has a wide range of applications in image processing. In this paper, the exact solution of the problem is first solved by using the Fourier transform, and it is found that the inverse problem is seriously ill-posed. In order to solve this inverse problem, the Fourier regular solution of the problem is constructed by the Fourier regularization method, and the logarithmic error estimation between the exact solution and the regularization approximate solution is given under the assumption of the prior conditions. Finally, the posterior error estimation of the approximate solution is given by the bias principle.