Truncated multigrid versus pre-corrected FFT/AIM for bioelectromagnetics: When is O(N) better than O(NlogN)?

The effectiveness of multigrid and fast Fourier transform (FFT) based methods are investigated for accelerating the solution of volume integral equations encountered in bioelectromagnetics (BIOEM) analysis. The typical BIOEM simulation is in the mixed-frequency regime of analysis because the field variations in the simulation domain are dictated by a combination of the free space wavelength, geometrical features, and the wavelengths/skin depths in tissues. In this case, multigrid-based methods (when appropriately truncated at high-frequency levels) can achieve O(N) complexity that is asymptotically superior to the O(NlogN) complexity of FFT-based ones. Nevertheless, the constant in front of their asymptotic complexity estimate is larger and their accuracy-efficiency tradeoffs are different. Numerical experiments are performed to compare these methods and the results show that multigrid-based methods begin to outperform FFT-based ones for N~10³.