The fast multipole method and Fourier convolution for the solution of acoustic scattering on regular volumetric grids

Hesford, Andrew J.; Waag, Robert C.

doi:10.1016/j.jcp.2010.07.025

Cited by 12 publications

(16 citation statements)

References 31 publications

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“…One general category of the application is the Fourier-based physics equation solver and analysis [2], [3], [26], [27], [7]. For example, when solving the volume integral equations in electromagnetics through the well-known conjugate-gradient fast Fourier transform method, the computation of the Fourier transform for the electric current density are needed.…”

Section: Numerical Resultsmentioning

confidence: 99%

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An Accurate Conformal Fourier Transform Method for 3D Discontinuous Functions

Zhu

Liu

2015

IEEE Trans. Antennas Propagat.

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Fourier transform of discontinuous functions are often encountered in computational electromagnetics and other areas. In this work, a highly accurate, fast conformal Fourier transform (CFT) algorithm is proposed to evaluate the finite Fourier transform of 3D discontinuous functions. A curved tetrahedron mesh combined with curvilinear coordinate transform, instead of the Cartesian grid, is adopted to flexibly model an arbitrary shape of the discontinuity boundary. This enables us to take full advantages of high order interpolation and Gaussian quadrature methods to achieve highly accurate Fourier integration results with a low sampling density. The 3D nonuniform fast Fourier transform (NUFFT) helps to keep the complexity of the proposed algorithm to that similar to the traditional 3D FFT algorithm. Therefore, the proposed CFT algorithm can achieve order of magnitude higher accuracy than 3D FFT with lower sampling density and similar computation time. The convergence is proved and verified.

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Section: Numerical Resultsmentioning

confidence: 99%

“…Fourier transform (FT), as a most important tool for spectral analyses, is often encountered in computational physics, including areas such as electromagnetics [1], [2], [3], [4], image processing [5], [6] and acoustics [7], [8].…”

Section: Introductionmentioning

confidence: 99%

An Accurate Conformal Fourier Transform Method for 3D Discontinuous Functions

Zhu

Liu

2015

IEEE Trans. Antennas Propagat.

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“…(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.…”

Section: Methodsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Hesford

Tillett

Astheimer

et al. 2014

The Journal of the Acoustical Society of America

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Accurate and efficient modeling of ultrasound propagation through realistic tissue models is important to many aspects of clinical ultrasound imaging. Simplified problems with known solutions are often used to study and validate numerical methods. Greater confidence in a timedomain k-space method and a frequency-domain fast multipole method is established in this paper by analyzing results for realistic models of the human breast. Models of breast tissue were produced by segmenting magnetic resonance images of ex vivo specimens into seven distinct tissue types. After confirming with histologic analysis by pathologists that the model structures mimicked in vivo breast, the tissue types were mapped to variations in sound speed and acoustic absorption. Calculations of acoustic scattering by the resulting model were performed on massively parallel supercomputer clusters using parallel implementations of the k-space method and the fast multipole method. The efficient use of these resources was confirmed by parallel efficiency and scalability studies using large-scale, realistic tissue models. Comparisons between the temporal and spectral results were performed in representative planes by Fourier transforming the temporal results. An RMS field error less than 3% throughout the model volume confirms the accuracy of the methods for modeling ultrasound propagation through human breast.

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“…The acoustic wave equation can be written as 4,[29][30][31] qðrÞr Á ½q À1 ðrÞrpðrÞ þk 2 ðrÞpðrÞ ¼ ÀSðrÞ;…”

Section: Problem Formulationmentioning

confidence: 99%

Ultrasound tomography for simultaneous reconstruction of acoustic density, attenuation, and compressibility profiles

Mojabi

LoVetri

2015

The Journal of the Acoustical Society of America

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A fast and efficient forward scattering solver is developed for use in ultrasound tomography. The solver is formulated so as to enable the calculation of scattering from large and relatively high-contrast objects with inhomogeneous physical properties that vary simultaneously in acoustic attenuation, compressibility, and density. It is based on the method of moments in conjunction with a novel implementation of the conjugate gradient algorithm which requires the use of the adjoints of the scattering operators. The solver takes advantage of the symmetric block Toeplitz matrix with symmetric Toeplitz blocks property of the Green's function matrix to increase efficiency and only stores the first row of this matrix to reduce memory requirements. This row is then used for the matrix-vector multiplication using the fast Fourier transform technique, thus, resulting in the computational complexity of O(n log n). The marching-on-source technique is also used to provide a good initial guess which allows the conjugate gradient technique to converge faster than initializing with an arbitrary guess. This feature is important in tomographic inversion algorithms which require that the object to be imaged be interrogated via several incident fields. Forward scattering and inversion examples, based on the Conjugate Gradient Least Squares regularized Born Iterative Method, are shown, in two-dimensions, for objects varying in all three physical properties.

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The fast multipole method and Fourier convolution for the solution of acoustic scattering on regular volumetric grids

Cited by 12 publications

References 31 publications

An Accurate Conformal Fourier Transform Method for 3D Discontinuous Functions

An Accurate Conformal Fourier Transform Method for 3D Discontinuous Functions

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

Ultrasound tomography for simultaneous reconstruction of acoustic density, attenuation, and compressibility profiles

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