POD Method for the Inverse Initial Value Problem of Two‐Dimensional Time‐Fractional Diffusion Equation and Application to Image Deblurring

ABSTRACT In this paper, we mainly investigate the backward problem for the two‐dimensional time‐fractional diffusion equation by the proper orthogonal decomposition (POD) method, and also apply this method to solve the image deblurring problem. The approach constructs a low‐dimensional basis by solution snapshots to accelerate the reconstruction of the initial field from noisy terminal time measurements. Besides, we give the Tikhonov regularization method and obtain the convergence analysis about the regularization solutions. Finally, we propose several numerical experiments to show the efficiency and accuracy of our method.