Adaptive Wavelet and Frame Techniques for Acoustic BEM

The boundary element method (BEM) is a commonly used tool to numerically solve the Helmholtz equation. With BEM, problems in unbounded domains can be treated and only the surfaces of objects have to be discretized. However, the matrices generated by BEM are usually dense and their dimension grows with the frequency, because in order to guarantee a sufficiently accurate solution, the grid size of the discretization has to be dependent on the wave number. Therefore it is not feasible to solve acoustic problems for high frequencies using BEM without matrix compression techniques like wavelet or fast multipole methods. Wavelet methods provide at least two basic advantages during the numerical treatment of integral equations. Firstly, wavelets allow for suitable preconditioning strategies that result in uniformly bounded condition numbers of the system matrices and secondly, wavelets can be designed as localized functions with vanishing moments, which can be used to design efficient compression strategies. Based on wavelet expansions, reliable and efficient error estimators can be constructed, which will be used in BIOTOP to develop adaptive strategies. The resulting algorithms are asymptotically optimal in the sense that the convergence rate of the best N-term approximation is realized. To further increase matrix compression, BEM is combined with frames. Frames are a generalization of bases and offer more flexible construction procedures, thus they can be adapted more flexible to the problem at hand. The newly developed alpha-modulation frames for example are well suited to sparsely represent signals that contain both, oscillatory components as well as isolated singularities. Therefore, alpha-modulation frames have high potential to sparsely represent solutions of the Helmholtz equation. To test the algorithms developed in BIOTOP they are used to solve a problem relevant for practical applications, namely the calculation of head related transfer functions (HRTFs). HRTFs describe the filtering effect of head, torso and especially the outer ear for incoming sounds in humans. HRTFs can be applied to generate virtual 3D- sound fields. Measurements of these filter functions require special equipment, thus a fast and stable way to numerically calculate HRTFs is of great interest. The mesh that is used in the BEM calculations will be created by a 3D-scan of a human head. It will be necessary to calculate HRTFs up to frequencies of 16 kHz, thus the grid has to be very fine and the dimension of the system matrix can go up to tens of thousands. The newly calculated results will be compared with already calculated HRTFs (using a fast multipole BEM code) as well as with measured HRTFs. BIOTOP combines the experience of three established research groups in Germany, Austria and Switzerland (D-A- CH) to create an innovative approach to efficiently solve acoustic boundary value problems.

The main goal of BIOTOP was the development of efficient methods to simulate the propagation of acoustic waves, e.g. for calculating the noise-reduction effect of noise barriers or for deriving individualized filter functions for 3D audio. Because the acoustic field contains oscillating components, the computing time and the memory consumption grow with the frequency. Besides problems caused by high frequencies, a higher accuracy of the methods is necessary close to corners and edges of the scatterer which, in turn, also increases the number of unknowns. In BIOTOP the theoretical foundations as well as issues around practical implementation of computer code based on special building blocks (= ansatz functions) for representing the unknown solution were investigated. To deal with the problems around edges a special type of building block called wavelets was used. Wavelets have the property that they can `zoom` in on regions (e.g. corners) where the resolution needs to be finer and thus allow the computer code to adaptively change the accuracy of the ansatz functions in certain regions. To deal with problems at high frequencies so called frames are investigated, in particular, $\alpha$-modulation frames. Frames can be constructed to include oscillating components and have been widely used in signal processing, but they are only rarely used in other areas of mathematics and engineering. Compared to regular ansatz function, frames can be overcomplete, which means that there are several ways to combine the elements of the given frame to achieve the same solution, which is rather unusual in current engineering applications. In the project it was shown that: a) computer programs based on wavelets can be very efficient to calculate acoustic scattering at objects with corners and that the newly developed algorithms can automatically adapt to the given geometry and b) that frames have great potential to efficiently deal with oscillations which provides the bases for computer code that adapts itself to different frequency. In BIOTOP a new mathematical theory dealing with the stability and construction methods for different type of frames has been developed as well as first numerical experiments with frames based on simple generating functions have been preformed. In that respect BIOTOP also functioned as a bridge between pure mathematical theory and engineering applications.