Parameter identification for the linear wave equation with Robin boundary condition

Abstract

We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

Parameter Identification of a Linear Wave Equation From Experimental Boundary Data

“…More recently, in [1], an inverse problem for the linear wave equation subject to Dirichlet's and Robin's boundary conditions is considered. Using the method of characteristics, the authors identify both the domain length and the Robin's parameters.…”

“…Algorithm 2: Adjoint method for finding a minimum with enforced stability Input : A set of measured data Y and the associate input U Output: An optimized value of the parametersp[b] 1 We set an initial guessp[0], a fixed number of iteration n > 1 and a fixed number M > 1 of loops to enforce the stability;…”