Implementation of the ADMM approach to constrained optimal control problem with a nonlinear time-fractional diffusion equation

Abstract

In this paper, we study the inverse problem of identifying the parameters in a nonlinear subdiffusion model from an observation defined in the given $ \Omega_1 $ subset of $ \Omega $. The nonlinear subdiffusion model involves a Caputo fractional derivative of order $ \alpha\in (0,1) $ in time. To address our model, we first examine the regularity of the solution for the direct problem using the Mittag-Leffler function. To investigate our inverse parameter problem, we reformulate first it in to Least-Squares optimization problem. Then, we establish the existence of the optimal solution and prove the convexity of the considered cost function by using its first derivative. To solve this problem numerically, we adapt a recent method in the literature known as the alternating direction method of multiplier (ADMM) which we establish its convergence. In order to show the effectiveness of the proposed method we present some numerical experiments.