Exponential Tikhonov Regularization Method for an Inverse Source Problem in a Sub-diffusion Equation

Abstract

In this paper, it is considered an inverse space-dependent source problem of time-space fractional diffusion equation from the noisy final data in a bounded domain. Such a problem is mildly ill-posed. A new regularization method called the exponential Tikhonov method with a parameter $\gamma$ is utilized to solve the problem, and its convergence rates are analyzed under an a-priori and an a-posteriori regularization parameter choice rule. A novel result indicates that the optimal convergence rate can be obtained and it is independent of the regularity information of the unknown source term when $\gamma$ is less than or equal to zero. However, when $\gamma$ is greater than zero, the optimal convergence rate depends on the value of $\gamma$ related to the regularity of the unknown source but it does not have convergence saturation limit and can theoretically approach 1, which is superior to Tikhonov's regularization framework using the usual Sobolev space norm as a penalty term in a minimized functional. Numerical examples show that the proposed regularization method is effective and stable, and both parameter choice rules work well.