Reconstruction of a Space-Dependent Coefficient in a Linear Benjamin–Bona–Mahony Type Equation

In this paper, we consider the problem of reconstructing a space-dependent coefficient in a linear Benjamin–Bona–Mahony (BBM)-type equation from a single measurement of its solution at a given time. We analyze the well-posedness of the forward initial-boundary value problem and characterize the inverse problem as a constrained optimization one. Our objective consists on reconstructing the variable coefficient in the BBM equation by minimizing an appropriate regularized Tikhonov-type functional constrained by the BBM equation. The well-posedness of the forward problem is studied and approximated numerically by combining a finite-element strategy for spatial discretization using the Python-FEniCS package, together with a second-order implicit scheme for time stepping. The minimization process of the Tikhonov-regularization adopted is performed by using an iterative L-BFGS-B quasi-Newton algorithm as described for instance by Byrd et al. (1995) and Zhu et al. (1997). Numerical simulations are presented to demonstrate the robustness of the proposed method with noisy data. The local stability and uniqueness of the solution to the constrained optimization problem for a fixed value of the regularization parameter are also proved and illustrated numerically.