The Chebyshev fast Gauss and nonuniform fast Fourier transforms and their application to the evaluation of distributed heat potentials

Veerapaneni, Shravan; Biros, George

doi:10.1016/j.jcp.2008.05.003

Cited by 9 publications

(9 citation statements)

References 26 publications

(52 reference statements)

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“…In this paper, we restrict our attention to layer potentials and assume that the volume forcing term F (x, t) = 0. (Volume integrals are discussed, for example, in [14,25].) For the Dirichlet problem (α = 1 and β = 0 in (1.3)), taking the limit as a point x ∈ Ω(t) approaches a point x o ∈ Γ(t), we obtain the integral equation…”

Section: γ(τ ) (13)mentioning

confidence: 99%

High Order Accurate Methods for the Evaluation of Layer Heat Potentials

Li¹,

Greengard²

2009

SIAM J. Sci. Comput.

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Abstract. We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly decaying in space. We show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which we refer to as "geometrically induced stiffness," but that rules based on product integration of the full heat kernel in time are robust. When combined with previously developed fast algorithms for the evolution of the "history part" of layer potentials, diffusion processes in complex, moving geometries can be computed accurately and in nearly optimal time.

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Section: γ(τ ) (13)mentioning

confidence: 99%

High Order Accurate Methods for the Evaluation of Layer Heat Potentials

Li¹,

Greengard²

2009

SIAM J. Sci. Comput.

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“…Once σ or µ is known, (1.7), (1.9) can be used to evaluate the solution at time t = ∆t. This yields a one-step marching method for the heat equation that is both stable and robust (see, for example, [1,5,7,8,13,18,21,30,31]). …”

Section: Introductionmentioning

confidence: 99%

“…Some notable prior work on continuous (volume) fast transforms includes [27], which describes a triangulation-based adaptive refinement method, and [31], which makes use of a high-order, adaptive, quad-tree-based discretization. As in [31], our approach relies on an adaptive quadtree with high-order Chebyshev grids on leaf nodes, but we carry out a modified version of the FGT on the quad-tree itself. This requires a somewhat more complicated implementation, following that of the hierarchical fast multipole method (FMM) [14].…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Fast Gauss Transform in Two Dimensions

Wang

Greengard²

2018

SIAM J. Sci. Comput.

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Abstract.A variety of problems in computational physics and engineering require the convolution of the heat kernel (a Gaussian) with either discrete sources, densities supported on boundaries, or continuous volume distributions. We present a unified fast Gauss transform for this purpose in two dimensions, making use of an adaptive quad-tree discretization on a unit square which is assumed to contain all sources. Our implementation permits either free-space or periodic boundary conditions to be imposed, and is efficient for any choice of variance in the Gaussian.

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“…Discrete sums of the form (1) are encountered in a variety of disciplines including computational physics, machine learning, computational finance and computer graphics [1], [2], [3], [4], [5]. The motivation for the present work comes from potential theory [6] applied to solving linear constant-coefficient parabolic partial differential equations (PDEs).…”

Section: Introductionmentioning

confidence: 99%

Parallel Fast Gauss Transform

Sampath

Sundar

Veerapaneni

2010

2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis

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We present fast adaptive parallel algorithms to compute the sum of N Gaussians at N points. Direct sequential computation of this sum would take O(N 2 ) time. The parallel time complexity estimates for our algorithms are O N np for uniform point distributions and O N np log N np + np log np for nonuniform distributions using np CPUs. We incorporate a planewave representation of the Gaussian kernel which permits "diagonal translation". We use parallel octrees and a new scheme for translating the plane-waves to efficiently handle nonuniform distributions. Computing the transform to six-digit accuracy at 120 billion points took approximately 140 seconds using 4096 cores on the Jaguar supercomputer at the Oak Ridge National Laboratory.Our implementation is kernel-independent and can handle other "Gaussian-type" kernels even when an explicit analytic expression for the kernel is not known. These algorithms form a new class of core computational machinery for solving parabolic PDEs on massively parallel architectures.

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The Chebyshev fast Gauss and nonuniform fast Fourier transforms and their application to the evaluation of distributed heat potentials

Cited by 9 publications

References 26 publications

High Order Accurate Methods for the Evaluation of Layer Heat Potentials

High Order Accurate Methods for the Evaluation of Layer Heat Potentials

An Adaptive Fast Gauss Transform in Two Dimensions

Parallel Fast Gauss Transform

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