| Abstract |
In BIOTOP, we will develop new methods to model 3D acoustic wave problems, especially the calculation of head related transfer functions (HRTFs), which play an important role in the localization of sound sources in 3D. Head related transfer functions describe the filtering effect of the head, the torso, and especially the outer ear (pinna) on incoming sounds and can be used to describe and simulate spatial hearing in virtual acoustics.
For the simulation of acoustic problems, the boundary element method (BEM) is an important tool. Unfortunately, the matrices generated by BEM are usually dense and their dimension grows with the wavenumber/frequency. To overcome this obstruction, we will design efficient variants of the boundary element method based on adaptive wavelet and frame methods. They will be particularly tuned to the requirements in acoustics, thus allowing an efficient computation of sound fields also for high frequencies. We will combine wavelet compression strategies with adaptive techniques and design new frames adapted to the problem at hand, and investigate the mathematical and numerical properties. Our approach is motivated by the fact, that besides the basic compression property, wavelets provide the following principle advantage: smoothness norms such as Besov or Sobolev norms are equivalent to weighted sequence norms of wavelet expansion coefficients. This fact gives rise to preconditioning strategies that produce uniformly bounded condition numbers. Moreover, these norm equivalences can be used to design adaptive numerical algorithms that are guaranteed to converge optimally in the sense that they realize the order of the best-N-term approximation. This property is known to hold for a wide range of problems including operators of negative order.
BIOTOP is a multi-disciplinary project by necessity, involving mathematics, numerics and acoustical modeling. As a D-A-CH project, it combines the expertise of the research groups from Germany (adaptive schemes, regularity theory), Austria (frames, calculation of HRTFs, acoustic modeling) and Switzerland (wavelet BEM, adaptivity).
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 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 use 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 for 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) toacoustic boundary |
