An optimal control approach for determining the source term in fractional diffusion equation by different cost functionals

Abstract

This work is devoted to the mathematical analysis of an inverse source problem governed by a time-fractional diffusion equation. The aims of this paper are to identify the source function from additional data based on a regularized optimal control approach, and to determine the regularization parameters using bi-level optimization. To do this, firstly we formulate our inverse problem to an optimal control one with two types of fidelity terms, which are the Least-Squares fitting and L1 norm. Secondly, we establish the existence of the minimizer for the optimal control problem corresponding to the L1 cost function, since it is the general case. Thirdly, we present some numerical results for the different cases of the cost function. These results are based on solving the optimal control model by gradient method for the L2 case and the primal-dual method for non-smooth L1 norm. Finally, we propose an adaptive selection strategy for the regularization parameter based on the bi-level optimization method, and we give some numerical results to show the effectiveness of the proposed approach.