New trends in computational electromagnetics

Chew, Weng Cho; Dai, Qi I.; Liu, Qin S.; Xia, Tian; Roth, Thomas E.; Gan, Hong-Seng; Liu, Aiyin; Chen, Shu C.; Hidayetoğlu, Mert; Jiang, Li Jun; Sun, Sheng; Hwu, Wen mei

doi:10.1049/sbew533e_ch2

Cited by 4 publications

(5 citation statements)

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“…This method may be interpreted as a variant of well-established "point-and-shoot" strategies used to accelerate the translation of multipole expansions for wideband Fast Multipole Methods for the Helmholtz equation in three dimensions [6,7]. We end this section with a brief discussion of the scalar case as well as its extension to the Stokes equations.…”

Section: Fast Near-singular Evaluationmentioning

confidence: 99%

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Boundary integral equation analysis for suspension of spheres in Stokes flow

Corona¹,

Veerapaneni²

2018

Journal of Computational Physics

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We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical expressions for evaluating the operators away from the boundary. When two particle are located close to each other, we use a truncated series expansion to compute the hydrodynamic interaction. On the other hand, we use the standard spectrally accurate quadrature scheme to evaluate smooth integrals on the far-field, and accelerate the resulting discrete sums using the fast multipole method (FMM). We employ this discretization scheme to analyze several boundary integral formulations of interest including those arising in porous media flow, active matter and magneto-hydrodynamics of rigid particles. We provide numerical results verifying the accuracy and scaling of their evaluation.

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Section: Fast Near-singular Evaluationmentioning

confidence: 99%

“…2), in order to translate wave expansions from children to parent (M2M) and viceversa (L2L). For a detailed derivation we refer the reader to Chapter 5 of [7] and [19]. The key result (Thm.…”

Section: Translation Operators For Solid Spherical Harmonicsmentioning

confidence: 99%

Boundary integral equation analysis for suspension of spheres in Stokes flow

Corona¹,

Veerapaneni²

2018

Journal of Computational Physics

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“…The approximation is constructed by truncating the expansion after p terms. Bases that deliver exponential convergence have been constructed for the Laplace [39], Helmholtz [17,18], Maxwell [16], and Gaussian [40,77,51,63,41] kernels. Efficient approximations have also been carried out using the SVD of the kernel function [45,35] and in a basis of Chebyshev polynomials [32,33].…”

Section: Fig 21mentioning

confidence: 99%

“…The Fast Gauss Transform [40,77,51,41] is a variant of the FMM for the Gaussian kernel. Similar approaches have been applied to solving the kernel summation problem for the Helmholtz [17,18] and Maxwell equations [16].…”

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confidence: 99%

Far-field compression for fast kernel summation methods in high dimensions

March

Biros

2017

Applied and Computational Harmonic Analysis

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We consider fast kernel summations in high dimensions: given a large set of points in d dimensions (with d 3) and a pair-potential function (the kernel function), we compute a weighted sum of all pairwise kernel interactions for each point in the set. Direct summation is equivalent to a (dense) matrix-vector multiplication and scales quadratically with the number of points. Fast kernel summation algorithms reduce this cost to log-linear or linear complexity.Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by constructing approximate representations of interactions of points that are far from each other. In algebraic terms, these representations correspond to low-rank approximations of blocks of the overall interaction matrix. Existing approaches require an excessive number of kernel evaluations with increasing d and number of points in the dataset.To address this issue, we use a randomized algebraic approach in which we first sample the rows of a block and then construct its approximate, low-rank interpolative decomposition. We examine the feasibility of this approach theoretically and experimentally. We provide a new theoretical result showing a tighter bound on the reconstruction error from uniformly sampling rows than the existing state-of-the-art. We demonstrate that our sampling approach is competitive with existing (but prohibitively expensive) methods from the literature. We also construct kernel matrices for the Laplacian, Gaussian, and polynomial kernels -all commonly used in physics and data analysis. We explore the numerical properties of blocks of these matrices, and show that they are amenable to our approach. Depending on the data set, our randomized algorithm can successfully compute low rank approximations in high dimensions. We report results for data sets with ambient dimensions from four to 1,000.

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“…The fast multipole method (FMM) [12] is one of the most important fast algorithms for kernel summation, in which target and source points are hierarchically divided as well-separated sets, and on each set, the kernel function is low-rank approximated by using multipole expansions. In the original FMM, kernel function is approximated by analytical tools (either with addition theorems of special functions or Taylor expansions) [12,4,6,10,5]. To overcome the difficulties when analytic formulation of kernel functions is not available, various semi-analytic [1,11,18] and algebraic FMMs [21,22,23] were developed in recent decades.…”

Section: Introductionmentioning

confidence: 99%