1 Applied functional analysis, Department of Mathematics, Technical University of Denmark2 Department of Mathematics, Technical University of Denmark3 Scientific Computing, Department of Informatics and Mathematical Modeling, Technical University of Denmark4 Department of Informatics and Mathematical Modeling, Technical University of Denmark5 Department of Applied Mathematics and Computer Science, Technical University of Denmark
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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Hansen, Per Christian, Pedersen, Michael, Knudsen, Kim