Compression method for boundary integral matrices with translation-invariant kernel functions

提出的项目旨在开发边界积分方程的数值方法，可以处理出现的非常精细的网格，例如，在处理复杂的几何形状或高频散射时。该项目主要关注两种方法：首先，我们打算利用核函数的平移不变性来减少存储需求。虽然这在粒子方法的背景下是常见的做法，例如对于快速多极方法，但在边界元方法领域似乎很少，因为基函数的支持的更复杂的形状使得难以使用简单的对分技术。其次，我们的目标是调整代数再压缩算法，使它们也能利用平移不变性。特别是对于更复杂的核函数，例如高频散射，这些再压缩算法非常重要，因为它们将存储需求减少了两个数量级以上，并允许我们在负担得起的硬件上处理精细网格。与现有方法相比，新技术不需要特殊的结构网格，而是能够处理相当普遍的非结构化网格，例如，在处理静电、静磁或散射应用时。

The proposed project aims to develop numerical methods for boundary integral equations that can handle the very fine meshes that appear, e.g., when dealing with complicated geometries or high-frequency scattering.The project focuses on two approaches: first, we intend to take advantage of the translation-invariance of the kernel function to reduce the storage requirements. While this is common practice in the context of particle methods, e.g., for fast multipole methods, is appears to be rare in the field of boundary element methods, since the more complicated shapes of the supports of the basis functions make it difficult to use simple bisection techniques. Second, we aim to adapt algebraic recompression algorithms to allow them to also take advantage of translation-invariance. Particularly for more complicated kernel functions arising, e.g., for high-frequency scattering, these recompression algorithms are very important, since they reduce the storage requirements by more than two orders of magnitude and allow us to treat fine meshes on affordable hardware.Compared to existing methods, the new technique does not require specially structure meshes, but will be able to handle fairly general unstructured meshes that appear, e.g., when dealing with applications in electrostatics, magnetostatics, or scattering.