时间分数阶倒向扩散问题的全变分正则化

Total Variation Regularization for a Backward Time-fractional Diffusion Problem

考虑在二维情况下一个时间分数阶倒向扩散问题,将这个模型看成是图像模糊问题且模糊过程为缓慢扩散的.为了避免物体图像在边界处过度光滑造成的影响,我们采用全变分正则化方法将这个不适定问题适定化,并且对这个最优化问题进行研究,然后讨论了最优化问题极小元的存在唯一性以及稳定性.最后运用Bregman迭代以及分离的Bregman迭代方法快速地实现其数值解,减弱了大量的计算.

Consider a two-dimensional backward problem for a time-fractional diffusion process , which can be seen as image de-blurring where the blurring process is assumed to be slow diffusion .In order to avoid the influence of the boundary of the object image caused by the over-smoothing,we use total variation regularization method to turn the ill-posed problems to the well posed problem ,and study the optimization problem ,discuss the existence and uniqueness and stability of the minimizer of the optimization problem .Finally,we use Bregman iteration and split Bregman iteration achieve the numerical scheme quickly ,the amount of computation is weakened .