A Parallel Directional Fast Multipole Method

Benson, Austin R.; Poulson, Jack; Tran, Kenneth; Engquist, Björn; Ying, Lexing

doi:10.1137/130945569

Cited by 14 publications

(11 citation statements)

References 44 publications

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“…Noting that, for any surface Γ N , there are only sixty-four boxes overall on level d = 3 of the algorithm (and, in general, even fewer relevant boxes), we see that a definite limit exists on the parallelism achievable by this approach. The method presented in [14] uses this strategy in an MPI context, and it is therefore subject to such a hard limitation on achievable parallelism (although in a somewhat mitigated form, owing to the characteristics of that algorithm, as discussed in Section 1). To avoid such limitations we consider an alternate OpenMP parallelization strategy specifically enabled by the characteristics of the IFGF algorithm, as described in what follows.…”

Section: Openmp Parallelizationmentioning

confidence: 99%

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Massively Parallelized Interpolated Factored Green Function Method

Bauinger¹,

Bruno²

2021

Preprint

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This paper presents a parallel implementation of the "Interpolated Factored Green Function" (IFGF) method introduced recently for the accelerated evaluation of discrete integral operators arising in wave scattering and other areas (Bauinger and Bruno, Jour. Computat. Phys., 2021). On the basis of the hierarchical IFGF interpolation strategy, the proposed (hybrid MPI-OpenMP) parallel implementation results in highly efficient data communication, and it exhibits in practice excellent parallel scaling up to large numbers of cores-without any hard limitations on the number of cores concurrently employed in an efficient manner. Moreover, on any given number of cores, the proposed parallel approach preserves the O(N log N ) computing cost inherent in the sequential version of the IFGF algorithm. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and it is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. A variety of numerical results presented in this paper illustrate the character of the proposed parallel algorithm, including excellent weak and strong parallel scaling properties in all cases considered-for problems of up to 4, 096 wavelengths in electrical size, and scaling tests spanning from 1 compute core to all 1, 680 cores available in the HPC cluster used.

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Section: Openmp Parallelizationmentioning

confidence: 99%

“…FMM-type algorithms. Indeed, in contrast to the incremental propagation and surface evaluation approach inherent in the IFGF method, previous acceleration methods rely on the FFT algorithmwhich, as discussed in Section 1, leads to inefficiencies in the upper portions of the corresponding octree structures [2,14].…”

Section: Openmp Parallelizationmentioning

confidence: 99%

Massively Parallelized Interpolated Factored Green Function Method

Bauinger¹,

Bruno²

2021

Preprint

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“…The memory savings that the fast algorithms provide stem from the fact that the far-field part of the interaction matrix is replaced with the operators in (6). These are the same for groups that have the same position relative to each other.…”

Section: Elsementioning

confidence: 99%

“…The resulting performance is typically a bit better for volume formulations then for boundary formulations, since the computational density is higher in the former case. A particular issue for the MLFMA formulation of electromagnetic scattering problems is that the work per element (group) in the tree data structure increases with the level, and additional partitioning strategies are needed for the coarser part of the structure [6,30,56].…”

Section: State Of the Artmentioning

confidence: 99%

Parallelization of Hierarchical Matrix Algorithms for Electromagnetic Scattering Problems

Larsson¹,

Zafari²,

Righero³

et al. 2019

Lecture Notes in Computer Science

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Numerical solution methods for electromagnetic scattering problems lead to large systems of equations with millions or even billions of unknown variables. The coefficient matrices are dense, leading to large computational costs and storage requirements if direct methods are used. A commonly used technique is to instead form a hierarchical representation for the parts of the matrix that corresponds to far-field interactions. The overall computational cost and storage requirements can then be reduced to O(N log N). This still corresponds to a large-scale simulation that requires parallel implementation. The hierarchical algorithms are rather complex, both regarding data dependencies and communication patterns, making parallelization non-trivial. In this chapter, we describe two classes of algorithms in some detail, we provide a survey of existing solutions, we show results for a proof-of-concept implementation, and we provide various perspectives on different aspects of the problem. The list of authors is organized into three subgroups, Larsson and Zafari (coordination and proof-of-concept implementation), Righero, Francavilla, Giordanengo, Vipiana, and Vecchi (definition of and expertise relating to the application), Kessler, Ancourt, and Grelck (perspectives and parallel expertise).

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“…We emphases that these algebraic representations are not only valid for linear elliptic operators, but also valid for operators associated with low-to-medium frequency Helmholtz equations and radial basis function kernel matrices. When operators admit high-frequency property, the low-rank structure appears in a very different way comparing to that in all aforementioned fast algorithms, and are also well-studied by the community [8,9,13,14,[31][32][33][34]38].…”

Section: Introductionmentioning

confidence: 99%

Distributed-Memory $\mathscr{H}$-Matrix Algebra I: Data Distribution and Matrix-Vector Multiplication

Li¹,

Poulson²,

Ying³

2021

CSIAM-AM

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We introduce a data distribution scheme for H-matrices and a distributedmemory algorithm for H-matrix-vector multiplication. Our data distribution scheme avoids an expensive Ω(P 2 ) scheduling procedure used in previous work, where P is the number of processes, while data balancing is well-preserved. Based on the data distribution, our distributed-memory algorithm evenly distributes all computations among P processes and adopts a novel tree-communication algorithm to reduce the latency cost. The overall complexity of our algorithm is O N log N P +αlogP+ βlog 2 P for H-matrices under weak admissibility condition, where N is the matrix size, α denotes the latency, and β denotes the inverse bandwidth. Numerically, our algorithm is applied to address both two-and three-dimensional problems of various sizes among various numbers of processes. On thousands of processes, good parallel efficiency is still observed.

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A Parallel Directional Fast Multipole Method

Cited by 14 publications

References 44 publications

Massively Parallelized Interpolated Factored Green Function Method

Massively Parallelized Interpolated Factored Green Function Method

Parallelization of Hierarchical Matrix Algorithms for Electromagnetic Scattering Problems

Distributed-Memory $\mathscr{H}$-Matrix Algebra I: Data Distribution and Matrix-Vector Multiplication

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