A well-posed parameter identification for nonlocal diffusion problems

This paper concerns the problem of identifying the diffusion parameter in the nonlocal diffusion model. The parameter identification problem is formulated as an optimal control problem, and the objective of the control is a cost functional formulated by the energy functional. By using the nonlocal vector calculus, in a manner analogous to the local partial differential equations counterpart, we proved that the cost functional is a strictly convex functional with a unique global minimizer. Moreover, the existence and uniqueness of parameter identification are further demonstrated. Finally, one-dimensional numerical experiments are given to illustrate our theoretical results and show that continuous and discontinuous parameters can be estimated.