Fourier-Based Fast Multipole Method for the Helmholtz Equation

Abstract

The fast multipole method (FMM) has had great success in reducing the computa-4 tional complexity of solving the boundary integral form of the Helmholtz equation. We present a 5 formulation of the Helmholtz FMM that uses Fourier basis functions rather than spherical harmonics. 6 By modifying the transfer function in the precomputation stage of the FMM, time-critical stages of 7 the algorithm are accelerated by causing the interpolation operators to become straightforward ap-8 plications of fast Fourier transforms, retaining the diagonality of the transfer function, and providing 9 a simplified error analysis. Using Fourier analysis, constructive algorithms are derived to a priori 10 determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and 11 verified numerically. Various optimizations are considered to reduce the number of quadrature points 12 and reduce the cost of computing the transfer function.13

Multilevel Fast Multipole Algorithm With Global and Local Interpolators

“…• For out-to-in translation the radiation pattern must be interpolated to translation size, which contains around four times the samples of storage size, i.e., about half of the samples of the algorithm that utilizes the local interpolator. • Error control and stability are excellent, but unfortunately asymptotic CPU-time cost of matrix-vector product is [12], which is higher than the that can be obtained with the local interpolator. However, on lower levels (or smaller problems) the global interpolator based on trigonometric polynomial expansion remains very competitive.…”

“…This method was first proposed by Sarvas in 2003 [11]. The approach allows the storage of the sampled radiation pattern components to take place in significantly reduced size, and the method has been developed further [12], [13]. The version presented in [13] reduced the number of sample points to roughly half compared to the original version by Sarvas, by simply reducing the sample rate in the polar regions in azimuthal direction.…”

“…The version presented in [13] reduced the number of sample points to roughly half compared to the original version by Sarvas, by simply reducing the sample rate in the polar regions in azimuthal direction. In [12] the authors employed a similar approach.…”

SURFACE AND VOLUME INTEGRAL EQUATION METHODS FOR TIME-HARMONIC SOLUTIONS OF MAXWELL'S EQUATIONS (Invited Paper)

“…One alternative for local interpolators is the global interpolator based on trigonometric polynomial expansions [120,128,129]. Such presentations have several useful properties: Conversion between sample values and coefficients of the expansion can be performed effectively and accurately with fast Fourier transform (FFT).…”

“…Figure 7 illustrates solutions (acoustic surface pressure) of broadband MLFMA (scalar Helmholtz SIE solver) utilizing global interpolators at two frequencies 2.5 Hz and 2.5 kHz corresponding to 22 MHz and 2.2 GHz in EM with the same wavelength as in acoustics. The major challenges of the discussed global interpolator, along with the higher asymptotic cost O(N log 2 N ) [129], are that the construction of the translators is a heavy process and implementation in distributed computing system is challenging.…”