The discretisation of boundary integral equations for the scalar Helmholtz equation leads to large dense linear systems. Efficient boundary element methods (BEM), such as the fast multipole method (FMM) and H-matrix based methods, focus on structured low-rank approximations of subblocks in these systems. It is known that the ranks of these subblocks increase with the wavenumber. We explore a data-sparse representation of BEM-matrices valid for a range of frequencies, based on extracting the known phase of the Green's function. Algebraically, this leads to a Hadamard product of a frequency matrix with an H-matrix. We show that the frequency dependency of this H-matrix can be determined using a small number of frequency samples, even for geometrically complex three-dimensional scattering obstacles. We describe an efficient construction of the representation by combining adaptive cross approximation with adaptive rational approximation in the continuous frequency dimension. We show that our data-sparse representation allows to efficiently sample the full BEM-matrix at any given frequency, and as such it may be useful as part of an efficient sweeping routine.

中文翻译：

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由3D标量Helmholtz方程产生的BEM矩阵的频率提取

标量亥姆霍兹方程的边界积分方程的离散化导致大型密集线性系统。高效的边界元方法（BEM），例如快速多极方法（FMM）和基于H矩阵的方法，专注于这些系统中子块的结构化低秩逼近。已知这些子块的等级随波数增加。基于提取格林函数的已知相位，我们探索了适用于一定频率范围的BEM矩阵的数据稀疏表示。代数上，这导致频率矩阵与H矩阵的Hadamard乘积。我们表明，即使对于几何复杂的三维散射障碍物，也可以使用少量频率样本来确定此H矩阵的频率依赖性。我们通过在连续频率维度上将自适应交叉逼近与自适应有理逼近相结合来描述表示的有效构造。我们表明，我们的数据稀疏表示可以在任何给定的频率上有效地采样整个BEM矩阵，因此，它可以用作有效扫描例程的一部分。