Inverse acoustic problems arise, e.g., when one wants to compute the source which is responsible for a measured acoustic field. Here, the acoustic field is measured near the sound source, and the problem is to reconstruct the acoustic field on the source surface. This reconstruction process is an inverse problem which can only be solved by means of the use of appropriate regularization algorithms.
The current project uses a formulation based on an integral equation formulation that involves a boundary element description of the source surface. When this integral equation is discretized, it leads to a linear system of equations with a large, dense, and ill-conditioned coefficient matrix. This system, in turn, is solved by means of appropriate discrete regularization algorithms.
These images show a test measurement setup, in which a small loudspeaker is placed on a regular tyre. The microphone array used for measuring the acoustic field is seen to the left of the tyre, and the location of the loudspeaker is shown by the red arrow. The goal is now to reconstruct the vibrations on the tyre, generated by the loudspeaker, from the measured acoustic field.
The solutions, i.e., the vibrations on the tyre, consist of linear combinations of certain fundamental vibrations or modes, which are computed by means of the singular value decomposition (SVD) of the coefficient matrix. The images above show a low-frequency mode and a high-frequency mode computed by means of the SVD.
When the reconstruction is computed, regularization is necessary in order ensure that measurement noise does not cause the solution to be dominated by the high-frequency modes. In this project, we use Tikhonov regularization, and as the regularizing functional we use the acoustic energy of the source. This ensures the computation of a physically meaningful regularized solution.
These two images - as well as the image on top of the homepage - show a particular reconstructed solution, computed by means of the boundary element method combined with Tikhonov regularization. The three images show the same solution viewed from three different positions. The computed vibration is correctly located at the position where the loudspeaker is placed on the tyre.
The quality of the reconstructions depends on the proper choice of a regularization parameter (that balances solution smoothness and fit to measured data). Click here to see a bad reconstruction with too many high-frequency modes. Algorithms for automatic choice of the regularization parameter is a topic of current research at IMM.