The Fast Multipole Method and Fourier Convolution for the Solution of Acoustic Scattering on Regular Volumetric Grids

Area of Science:

• Computational physics
• Acoustics
• Numerical methods

Background:

• Solving large-scale, three-dimensional acoustic scattering problems is computationally intensive.
• Existing methods struggle with complex or unknown scattering geometries.
• Iterative inverse scattering and imaging techniques often yield data on regular grids.

Purpose of the Study:

• To apply the fast multipole method (FMM) to large-scale 3D acoustic scattering problems.
• To leverage regular grid arrangements for improved computational efficiency.
• To demonstrate the method's suitability for geometries reconstructed from imaging data.

Main Methods:

• Utilizing the fast multipole method (FMM) for acoustic scattering.
• Defining inhomogeneous objects on a regular grid.
• Employing fast Fourier transforms (FFTs) for Green's function convolutions in neighboring interactions.
• Analyzing storage and computation scaling with the number of scattering elements.

Main Results:

• The FMM, when applied to regular grids, achieves linear scaling of storage and computation with the number of scattering elements.
• FFTs significantly reduce the computational cost of finest-level FMM interactions.
• The method's efficiency increases with a higher number of scattering elements per finest-level box.
• The approach effectively mitigates the dependence of FMM cost on finest-level box size.

Conclusions:

• The composite method, combining FMM with regular grids and FFTs, offers an efficient solution for large-scale acoustic scattering.
• This approach is particularly advantageous for scattering problems with unknown or reconstructed geometries.
• The findings demonstrate a practical and scalable computational strategy for acoustic imaging and inverse problems.