Compressing H^2 Matrices for Translationally Invariant Kernels

摘要

$H^{2}$ matrices provide compressed representations of the matrices obtained when discretizing surface and volume integral equations. The memory costs associated with storing $H^{2}$ matrices for static and low-frequency applications are $O(N)$. However, when the $H^{2}$ representation is constructed using sparse samples of the underlying matrix, the translation matrices in the $H^{2}$ representation do not preserve any translational invariance present in the underlying kernel. In some cases, this can result in an $H^{2}$ representation with relatively large memory requirements. This paper outlines a method to compress an existing $H^{2}$ matrix by constructing a translationally invariant $H^{2}$ matrix from it. Numerical examples demonstrate that the resulting representation can provide significant memory savings.