二维时间反向热传导问题的两种正则化方法及后验误差估计

Two Regularization Methods for the Two-Dimensional Time-Inverse Heat Conduction Problem and Its Posterior Error Estimation

本文讨论了一类二维时间反向热传导问题,它从终值时刻 的温度分布来反演初始时刻的温度分布。该问题在图像处理方面有重要应用。这是一个严重不适定问题,它的解在一定条件下不连续依赖于数据。针对传统正则化方法的缺陷,本文采用拟逆正则化方法和分数次Tikhonov正则化方法,来恢复解对数据的依赖性。同时我们还给出了两种方法相应的后验参数选取规则及其正则解与精确解的误差估计。

The time-inverse heat conduction problem was concerned in two-dimensional space, which retrieves the temperature distribution from the temperature distribution at the final moment. This problem has important application in image processing. This is a serious ill-posed problem, i.e. its solution is not continuously dependent on the data under certain conditions. For the defects of traditional regularization methods, the quasi-reversibility regularization method and the fractional Tikhonov regularization method are proposed to restore the dependence of the solution on the data. Meanwhile, the errors between the approximate solutions and the exact solution for the ill-posed problem are estimated, and the posteriori regularization parameter selection rules are given.