Approximation of the high-frequency Helmholtz kernel by nested directional interpolation
This work provides a theoretical foundation for efficient numerical methods in wave scattering and related domains, though it is an incremental improvement over existing interpolation-based approaches.
The paper presents an approximation scheme for highly oscillatory kernel functions like the 2D and 3D Helmholtz kernels, achieving exponential convergence in polynomial degree and enabling multilevel techniques. The analysis supports fast methods for Helmholtz integral operators with polylogarithmic-linear complexity.
We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and postmultiplication by plane waves. It is shown to converge exponentially in the polynomial degree and supports multilevel approximation techniques. Our convergence analysis may be employed to establish exponential convergence of certain classes of fast methods for discretizations of the Helmholtz integral operator that feature polylogarithmic-linear complexity.