We consider a control problem for the wave equation: Given the initial state, find a specific boundary condition, called a control, that steers the system to a desired final state. The Hilbert uniqueness method (HUM) is a mathematical method for the solution of such control problems. It builds on the duality between the control system and its adjoint system, and these systems are connected via a so-called controllability operator. In this project, we are concerned with the numerical approximation of HUM control for the one-dimensional wave equation. We study two semi-discretizations of the wave equation: a linear finite element method (L-FEM) and a discontinuous Galerkin-FEM (DG-FEM). The controllability operator is discretized with both L-FEM and DG-FEM to obtain a HUM matrix. We show that formulating HUM in a sine basis is beneficial for several reasons: (i) separation of low and high frequency waves, (ii) close connection to the dispersive relation, (iii) simple and effective filtering. The dispersive behavior of a discretization is very important for its ability to solve control problems. We demonstrate that the group velocity is determining for a scheme’s success in relation to HUM. The vanishing group velocity for high wavenumbers results in a dramatic decay of the corresponding eigenvalues of the HUM matrix and thereby also in a huge condition number. We show that, provided sufficient filtering, the phase velocity decides the accuracy of the computed controls. DG-FEM shows very suitable for the treatment of control problems. The good dispersive behavior is an important virtue and a decisive factor in the success over L-FEM. Increasing the order of DG-FEM even give results of spectral accuracy. The field of control is closely related to other fields of mathematics among these are inverse problems. As an example, we employ a HUM solution to an inverse source problem for the wave equation: Given boundary measurements for a wave problem with a separable source, find the spatial part of the source term. The reconstruction formula depends on a set of HUM eigenfunction controls; we suggest a discretization and show its convergence. We compare results obtained by L-FEM controls and DG-FEM controls. The reconstruction formula is seen to be quite sensitive to control inaccuracies which indeed favors DG-FEM over L-FEM.
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Knudsen, Kim, Hansen, Per Christian, Pedersen, Michael