In this talk, we consider the problem of exact boundary control of the wave equation and its numerical approximation. A systematic method to construct the control is given by the so-called the Hilbert Uniqueness Method (HUM) introduced by J. L. Lions in 1988. A crucial step in HUM is the solution of an operator equation in order to obtain the initial values of the associated adjoint wave equation. A conjugate gradient algorithm, due to R. Glowinski (1992), for the numerical solution of the operator equation when coupled with appropriate discretisation procedures for the adjoint wave equation yields a numerical approximation of HUM.
It has been shown in the literature that the controllability of the wave equation is equivalent to the observability of the adjoint wave equation, e.g. E. Zuazua, (2005). Hence, it is essential that the numerical solution of the adjoint wave equation involved in the conjugate gradient based method for HUM be observable. Unfortunately, the standard finite difference, finite element and spectral discretisation procedures yield pathological results due to the presence of spurious high frequency Fourier modes (Infante and Zuazua, 1999). Therefore, various techniques, such as the use of bi-grids, applying filtering procedures etc. are proposed to get a uniformly observable numerical solution of the adjoint wave equation with respect to the discretisation parameters.
In this talk, we consider the numerical approximation of the adjoint via the Legendre Spectral Galerkin scheme and review the spectrum of the discrete differentiation operator. Following T. Boulmezaoud and J. Urquiza (2006), we study the non uniform observability of the numerical solution obtained by the spectral Galerkin approximation and explore some cures to recover the uniform observability.
Date and Time : 31-08-2017 14:00
Venue : PSB 2214
Speaker : Bijal Amalani, IISER TVM