Adaptive Wavelet and Frame Techniques for Acoustic BEM

在BIOTOP中，我们将开发新的方法来模拟三维声波问题，特别是头部相关传递函数（hrtf）的计算，它在三维声源定位中起着重要作用。头部相关传递函数描述了头部，躯干，特别是外耳（耳廓）对传入声音的过滤效果，可用于描述和模拟虚拟声学中的空间听力。对于声学问题的模拟，边界元法（BEM）是一个重要的工具。然而，边界元法生成的矩阵密度较大，且其维数随波数/频率的增加而增大。为了克服这一障碍，我们将基于自适应小波方法和帧的概念设计有效的边界元方法变体。它们将特别适应声学方面的要求，从而允许对高频声场进行有效的计算。我们将把小波压缩策略与自适应技术结合起来，设计适合手头问题的新框架，并研究其数学和数值性质。BIOTOP是一个必要的多学科项目，涉及数学、数值和声学建模。作为一个D-A-CH项目，它结合了来自德国（自适应方案，规则理论），奥地利（框架，HRTFs计算，声学建模）和瑞士（小波BEM，适应性）的研究小组的专业知识。

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 the boundary element method are 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 methods and the concept of frames. 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. 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).