Nonsmooth Optimization Method for Determining Nonsmooth Potential Parameter in Nonlinear Subdiffusion Equation

A determination of the nonsmooth potential parameter in a nonlinear subdiffusion model using terminal observation through a nonsmooth optimal control approach is proposed. This is achieved by converting the inverse problem to an optimal control problem with the fidelity and regularization terms in the cost functional represented by the L1 and TV norms, respectively, in order to account for the nonsmoothness of the potential. The existence of a minimizer for the optimal control problem is established and numerical solutions are obtained using the primal- dual method. The effectiveness of the proposed method is demonstrated via numerical results in comparison with the gradient descent method.