Fourier Based Fast Multipole Method for the Helmholtz Equation
This work improves the efficiency and error analysis of the fast multipole method for solving Helmholtz boundary integral equations, which is important for computational acoustics and electromagnetics.
The paper presents a Fourier-based fast multipole method for the Helmholtz equation that replaces spherical harmonics with Fourier basis functions, accelerating time-critical stages via fast Fourier transforms and providing simplified error analysis with sharp error bounds. Numerical verification confirms the error bounds and optimizations reduce quadrature points and transfer function cost.
The fast multipole method (FMM) has had great success in reducing the computational complexity of solving the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM that uses Fourier basis functions rather than spherical harmonics. By modifying the transfer function in the precomputation stage of the FMM, time-critical stages of the algorithm are accelerated by causing the interpolation operators to become straightforward applications of fast Fourier transforms, retaining the diagonality of the transfer function, and providing a simplified error analysis. Using Fourier analysis, constructive algorithms are derived to a priori determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and verified numerically. Various optimizations are considered to reduce the number of quadrature points and reduce the cost of computing the transfer function.