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The paper presents a numerical method for identifying discontinuous conductivities in elliptic equations from boundary observations. The solutions to this inverse problem are obtained through a constrained optimization problem, where the cost functional is a combination of the Kohn-Vogelius and Total Variation functionals. Instead of regularizing the Total Variation stabilization functional, which is commonly used in the literature, we introduce an Alternating Direction Method of Multipliers to preserve the favorable properties of non-smoothness and convexity. The discretization is carried out using a mixed finite element/volume method, while the numerical solutions are iteratively computed using a variant of the Uzawa algorithm. We show the surjectivity of the derivatives of the constraints related to the discrete optimization problem and derive a source condition for the discrete inverse problem. We then investigate the convergence analysis and establish the convergence rate. Finally, we conclude with some numerical experiments to illustrate the efficiency of the proposed method.

This paper presents the SIR space-time model, which is a coupled reaction–diffusion system with nonlinear Robin boundary conditions. These boundary conditions are supposed to lock the border (no outflow neither immigration nor migration) when the number of infected individuals explode, and this may be considered as an automatic containment or lock-down. In practice, we can precise some threshold for the number of infected individuals and when it is reached the model locks the region automatically. This work provides a thorough study of the presented model, including the existence and uniqueness of the solution, its boundedness and its asymptotic behaviour. We end with some numerical experiments performed on the basis of the finite difference approach and Newton’s method to highlight and validate the theoretical results.