Communication Complexity of the Fast Multipole Method and its Algebraic Variants

Yokota, Rio; Turkiyyah, George; Keyes, David E.

doi:10.14529/jsfi140104

Cited by 5 publications

(2 citation statements)

References 42 publications

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“…However, the reduction of communication from exploiting hierarchy is significant in distributed-memory implementations. As shown in [6], the same efficiencies can be achieved for H-matrices because the latter are naturally implemented with the same dual tree traversal structure as FMM, the principal difference being the need to store basis vectors for the compressed weak interactions.…”

Section: A Renaissance In Computational Linear Algebramentioning

confidence: 99%

Hierarchical algorithms on hierarchical architectures

Keyes

Ltaief

Turkiyyah

2020

Phil. Trans. R. Soc. A.

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A traditional goal of algorithmic optimality, squeezing out flops, has been superseded by evolution in architecture. Flops no longer serve as a reasonable proxy for all aspects of complexity. Instead, algorithms must now squeeze memory, data transfers, and synchronizations, while extra flops on locally cached data represent only small costs in time and energy. Hierarchically low-rank matrices realize a rarely achieved combination of optimal storage complexity and high-computational intensity for a wide class of formally dense linear operators that arise in applications for which exascale computers are being constructed. They may be regarded as algebraic generalizations of the fast multipole method. Methods based on these hierarchical data structures and their simpler cousins, tile low-rank matrices, are well proportioned for early exascale computer architectures, which are provisioned for high processing power relative to memory capacity and memory bandwidth. They are ushering in a renaissance of computational linear algebra. A challenge is that emerging hardware architecture possesses hierarchies of its own that do not generally align with those of the algorithm. We describe modules of a software toolkit, hierarchical computations on manycore architectures, that illustrate these features and are intended as building blocks of applications, such as matrix-free higher-order methods in optimization and large-scale spatial statistics. Some modules of this open-source project have been adopted in the software libraries of major vendors. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

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Section: A Renaissance In Computational Linear Algebramentioning

confidence: 99%

Hierarchical algorithms on hierarchical architectures

Keyes

Ltaief

Turkiyyah

2020

Phil. Trans. R. Soc. A.

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“…Finally, a downsweep phase expands the nodes of y, multiplying them by the bases from the corresponding levels of U to produce the final output vector y. It is also worth noting that these phases of the hgemv computation are very closely related to the phases of the fast multipole method, of which H 2 -matrices can be regarded as an algebraic generalization [65].…”

Section: Hierarchical Matrix Vector Multiplicationmentioning

confidence: 99%

Hierarchical Matrix Operations on GPUs

Boukaram¹,

Turkiyyah

Keyes³

2019

ACM Trans. Math. Softw.

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Hierarchical matrices are space- and time-efficient representations of dense matrices that exploit the low-rank structure of matrix blocks at different levels of granularity. The hierarchically low-rank block partitioning produces representations that can be stored and operated on in near-linear complexity instead of the usual polynomial complexity of dense matrices. In this article, we present high-performance implementations of matrix vector multiplication and compression operations for the H 2 variant of hierarchical matrices on GPUs. The H 2 variant exploits, in addition to the hierarchical block partitioning, hierarchical bases for the block representations and results in a scheme that requires only O ( n ) storage and O ( n ) complexity for the mat-vec and compression kernels. These two operations are at the core of algebraic operations for hierarchical matrices, the mat-vec being a ubiquitous operation in numerical algorithms while compression/recompression represents a key building block for other algebraic operations, which require periodic recompression during execution. The difficulties in developing efficient GPU algorithms come primarily from the irregular tree data structures that underlie the hierarchical representations, and the key to performance is to recast the computations on flattened trees in ways that allow batched linear algebra operations to be performed. This requires marshaling the irregularly laid out data in a way that allows them to be used by the batched routines. Marshaling operations only involve pointer arithmetic with no data movement and as a result have minimal overhead. Our numerical results on covariance matrices from 2D and 3D problems from spatial statistics show the high efficiency our routines achieve over 550GB/s for the bandwidth-limited matrix-vector operation and over 850GFLOPS/s in sustained performance for the compression operation on the P100 Pascal GPU.

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Finite Element Discretizations for Variable-Order Fractional Diffusion Problems

2023

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We present a finite element discretization scheme for multidimensional fractional diffusion problems with spatially varying diffusivity and fractional order. We consider the symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded domains. A number of challenges are encountered when discretizing these equations. The first comes from the heterogeneous kernel singularity in the fractional integral operator. The second comes from the formally dense discrete operator with its quadratic growth in memory footprint and arithmetic operations. An additional challenge comes from the need to handle volume conditions—the generalization of classical local boundary conditions to the nonlocal setting. Satisfying these conditions requires that the effect of the whole domain, including both the interior and exterior regions, be computed on every interior point in the discretization. Performed directly, this would result in quadratic complexity. In order to address these challenges, we propose a strategy that decomposes the stiffness matrix into three components. The first is a sparse matrix that handles the singular near-field separately, and is computed by adapting singular quadrature techniques available for the homogeneous case to the case of spatially variable order. The second component handles the remaining smooth part of the near-field as well as the far-field, and is approximated by a hierarchical $${\mathcal {H}}^2$$ H 2 matrix that maintains linear complexity in storage and operations. The third component handles the effect of the global mesh at every node, and is written as a weighted mass matrix whose density is computed by a fast-multipole type method. The resulting algorithm has therefore overall linear space and time complexity. Analysis of the consistency of the stiffness matrix is provided and numerical experiments are conducted to illustrate the convergence and performance of the proposed algorithm.

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Communication Complexity of the Fast Multipole Method and its Algebraic Variants

Cited by 5 publications

References 42 publications

Hierarchical algorithms on hierarchical architectures

Hierarchical algorithms on hierarchical architectures

Hierarchical Matrix Operations on GPUs

Finite Element Discretizations for Variable-Order Fractional Diffusion Problems

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