High-frequency asymptotic compression of dense BEM matrices

High-frequency scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically large and dense, with the elements representing interactions between all parts of a scattering obstacle. However, the matrix represents the action of an oscillatory integral operator on an oscillatory density function, and at high frequencies this action is highly localized in nature. Asymptotic methods for high-frequency problems are based on extracting the phase of the density function via ray tracing or similar techniques. This results in a much smaller matrix, but this approach is very expensive for scattering obstacles with complex geometries. In this paper, we explore methods to exploit the asymptotic localization of the operator directly. We introduce sparsity in the original discretization matrix by modifying the Green's function with well-chosen smooth window functions. A relatively simple technique automatically decides where to place these windows for general geometries, thus providing valuable asymptotic information. We illustrate that such asymptotic compression can be achieved even for near-trapping domains and multiple scattering obstacles. Furthermore, we show robustness of the technique even for low-order BEM discretizations using piecewise linears or cubics in 2D, and preliminary results for 3D problems are encouraging. The cost of a matrix-vector product with respect to a full matrix is decreased after asymptotic compression, and in addition it appears that the condition number of the sparser matrix is much improved as well. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional discretization.nrpages: 25status: publishe