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

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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.