Regularized inverse determination of the tension coefficient in a wave equation

This study addresses the inverse determination of an unknown tension coefficient in a one-dimensional wave equation from final-time observations. Since coefficient reconstruction is inherently ill-posed, a quadratic Tikhonov regularization term incorporating a prior (initial guess) coefficient is employed to stabilize the minimization. Existence and uniqueness of the regularized minimizer are established. The minimizer is computed by a gradient-type method, where the Fréchet derivative of the objective is obtained via an adjoint problem. For numerical realization, a hybrid scheme that combines a Galerkin approximation with a Laplace-transform technique is used. The methodology is illustrated on two benchmark examples together with a systematic sensitivity study with respect to the regularization parameter and the initial guess; a noise-perturbation protocol is also outlined to assess robustness. The presented framework provides a practical and stable approach for inverse coefficient problems governed by wave equations.