A free-space adaptive FMM-Based PDE solver in three dimensions

Langston, M. Harper; Greengard, Leslie; Zorin, Denis

doi:10.2140/camcos.2011.6.79

Cited by 51 publications

(51 citation statements)

References 54 publications

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“…As such, we are working to incorporate R-Stream [22], a High Level Compiler for embedded computing and parallel processing of algorithms, to generate higher quality implementations of the GPU portion. Further optimizations involve exploiting inherent symmetries in the FMM structure as in [19] as well as providing a pure single-precision option for the entire algorithm for additional efficiency when lower numerical accuracy is desired.…”

Section: Discussionmentioning

confidence: 99%

Re-Introduction of communication-avoiding FMM-accelerated FFTs with GPU acceleration

Langston

Baskaran

Meister

et al. 2013

2013 IEEE High Performance Extreme Computing Conference (HPEC)

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As distributed memory systems grow larger, communication demands have increased. Unfortunately, while the costs of arithmetic operations continue to decrease rapidly, communication costs have not. As a result, there has been a growing interest in communication-avoiding algorithms for some of the classic problems in numerical computing, including communication-avoiding Fast Fourier Transforms (FFTs). A previously-developed low-communication FFT, however, has remained largely out of the picture, partially due to its reliance on the Fast Multipole Method (FMM), an algorithm that typically aids in accelerating dense computations. We have begun an algorithmic investigation and re-implementation design for the FMM-accelerated FFT, which exploits the ability to tune precision of the result (due to the mathematical nature of the FMM) to reduce power-burning communication and computation, the potential benefit of which is to reduce the energy required for the fundamental transform of digital signal processing. We reintroduce this algorithm as well as discuss new innovations for separating the distinct portions of the FMM into a CPUdedicated process, relying on inter-processor communication for approximate interactions, and a GPU-dedicated process for dense interactions with no communication. 1

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Section: Discussionmentioning

confidence: 99%

Re-Introduction of communication-avoiding FMM-accelerated FFTs with GPU acceleration

Langston

Baskaran

Meister

et al. 2013

2013 IEEE High Performance Extreme Computing Conference (HPEC)

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“…A key component of our solver is the efficient evaluation of the convolution operators G and D in (2). In 2D, a fast scheme was proposed in [14] and a 3D extension is discussed in [20]. We use a scheme that uses a novel algorithm for traversing near and far-field integrals.…”

Section: Related Workmentioning

confidence: 99%

A Volume Integral Equation Stokes Solver for Problems with Variable Coefficients

Malhotra

Gholami

Biros

2014

SC14: International Conference for High Performance Computing, Networking, Storage and Analysis

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We present a novel numerical scheme for solving the Stokes equation with variable coefficients in the unit box. Our scheme is based on a volume integral equation formulation. Compared to finite element methods, our formulation decouples the velocity and pressure, generates velocity fields that are by construction divergence free to high accuracy and its performance does not depend on the order of the basis used for discretization. In addition, we employ a novel adaptive fast multipole method for volume integrals to obtain a scheme that is algorithmically optimal. Our scheme supports non-uniform discretizations and is spectrally accurate. To increase per node performance, we have integrated our code with both NVIDIA and Intel accelerators. In our largest scalability test, we solved a problem with 20 billion unknowns, using a 14-order approximation for the velocity, on 2048 nodes of the Stampede system at the Texas Advanced Computing Center. We achieved 0.656 petaFLOPS for the overall code (23% efficiency) and one petaFLOPS for the volume integrals (33% efficiency). As an application example, we simulate Stokes flow in a porous medium with highly complex pore structure using a penalty formulation to enforce the no slip condition.

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“…Instead, we will assume that the right-hand side (the function f in (1.12)) is specified on a level-restricted quad-tree. These data structures have been shown to be extremely effective for elliptic volume integrals [2,6,10,19,12,22]. For the sake of simplicity, we assume that the source distribution f in (1.12) is supported in the unit box D, centered at the origin.…”

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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A free-space adaptive FMM-Based PDE solver in three dimensions

Cited by 51 publications

References 54 publications

Re-Introduction of communication-avoiding FMM-accelerated FFTs with GPU acceleration

Re-Introduction of communication-avoiding FMM-accelerated FFTs with GPU acceleration

A Volume Integral Equation Stokes Solver for Problems with Variable Coefficients

An Adaptive Fast Gauss Transform in Two Dimensions

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