Reduced-rank approximations to the far-field transform in the gridded fast multipole method

Abstract

The fast multipole method (FMM) has been shown to have a reduced computational dependence on the size of finest-level groups of elements when the elements are positioned on a regular grid and FFT convolution is used to represent neighboring interactions. However, transformations between plane-wave expansions used for FMM interactions and pressure distributions used for neighboring interactions remain significant contributors to the cost of FMM computations when finest-level groups are large. The transformation operators, which are forward and inverse Fourier transforms with the wave space confined to the unit sphere, are smooth and well approximated using reduced-rank decompositions that further reduce the computational dependence of the FMM on finest-level group size. The adaptive cross approximation (ACA) is selected to represent the forward and adjoint far-field transformation operators required by the FMM. However, the actual error of the ACA is found to be greater than that predicted using traditional estimates, and the ACA generally performs worse than the approximation resulting from a truncated singular-value decomposition (SVD). To overcome these issues while avoiding the cost of a full-scale SVD, the ACA is employed with more stringent accuracy demands and recompressed using a reduced, truncated SVD. The results show a greatly reduced approximation error that performs comparably to the full-scale truncated SVD without degrading the asymptotic computational efficiency associated with ACA matrix assembly.

Comparison of temporal and spectral scattering methods using acoustically large breast models derived from magnetic resonance images

“…(3) are obtained efficiently using the fast multipole method. 22,23 There are advantages and disadvantages to both methods. The k-space method produces a complete temporal solution that can be Fourier transformed to obtain monochromatic solutions at any spatial frequency.…”

“…Both of these aspects of the computed models present unique challenges that require novel formulations and implementations of acoustic scattering solvers. The FMM 22,23 and the k-space method 21,24,25 have each been substantially enhanced to improve efficiency and also to extend the scope of solvable problems characterized by segmented magnetic resonance images. Combinations of these prior improvements made possible the case studies presented here of previously infeasible FMM and k-space solutions to large-scale, realistic scattering problems.…”

Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation

“…Therefore, singular value decomposition (SVD) can be used to compress this matrix [42] and BLAS can be used to maximize the cache utilization [39]. Some methods use a combination of these techniques like Chebychev with SVD [38] and planewave with adaptive cross approximation (ACA) and SVD [59]. The use of SVD is a systematic and optimal way of achieving what the variable expansion order techniques in the previous paragraph were trying to do manually.…”

Design of a low profile conformal array for transcranial ultrasound imaging

“…Scattered field data is resulted from the subtraction of incident field data from the total field data. These scattered field data is then used for UT problem which can later be mathematically formulated as an inverse scattering problem which is a nonlinear problem [143], [148], [149], [157], [158].…”