Scaling Algebraic Multigrid Solvers: On the Road to Exascale

Baker, Allison H.; Falgout, Robert D.; Gamblin, Todd; Kolev, Tzanio V.; Schulz, Martin; Yang, Ulrike Meier

doi:10.1007/978-3-642-24025-6_18

“…For the Laplace and Laplace-like operators that commonly arise in elliptic, time-implicit parabolic, and frequency-domain hyperbolic PDEs, algebraic multigrid (AMG) scales efficiently in proportion to available memory on hundreds of thousands of rigidly schedulable, tightly coupled cores for systems with billions of unknowns [2]. However, for constant coefficient problems the fast multipole method (FMM) is asymptotically superior in complexity and tolerates less synchronization.…”

Section: Exascale Features Of the Fast Multipole Methodsmentioning

confidence: 99%

Communication Complexity of the Fast Multipole Method and its Algebraic Variants

Yokota

¹

,

Turkiyyah

²

,

Keyes

³

2014

JSFI

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A combination of hierarchical tree-like data structures and data access patterns from fast multipole methods and hierarchical low-rank approximation of linear operators from H-matrix methods appears to form an algorithmic path forward for efficient implementation of many linear algebraic operations of scientific computing at the exascale. The combination provides asymptotically optimal computational and communication complexity and applicability to large classes of operators that commonly arise in scientific computing applications. A convergence of the mathematical theories of the fast multipole and H-matrix methods has been underway for over a decade. We recap this mathematical unification and describe implementation aspects of a hybrid of these two compelling hierarchical algorithms on hierarchical distributed-shared memory architectures, which are likely to be the first to reach the exascale. We present a new communication complexity estimate for fast multipole methods on such architectures. We also show how the data structures and access patterns of H-matrices for low-rank operators map onto those of fast multipole, leading to an algebraically generalized form of fast multipole that compromises none of its architecturally ideal properties.

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“…After the local iterations, the updated values are communicated. This approach is inspired by the well known hybrid relaxation schemes [9,8]. The obtained algorithm, visualized in Figure 1, can be written as a component-wise update of the solution approximation to form…”

Section: Mathematical Backgroundmentioning

confidence: 99%

GPU-Accelerated Asynchronous Error Correction for Mixed Precision Iterative Refinement

Anzt

¹

,

Łuszczek

²

,

Dongarra

³

et al. 2012

Euro-Par 2012 Parallel Processing

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Abstract. In hardware-aware high performance computing, block-asynchronous iteration and mixed precision iterative refinement are two techniques that may be used to leverage the computing power of SIMD accelerators like GPUs in the iterative solution of linear equation systems. Although they use a very different approach for this purpose, they share the basic idea of compensating the convergence properties of an inferior numerical algorithm by a more efficient usage of the provided computing power. In this paper, we analyze the potential of combining both techniques. Therefore, we derive a mixed precision iterative refinement algorithm using a block-asynchronous iteration as an error correction solver, and compare its performance with a pure implementation of a blockasynchronous iteration and an iterative refinement method using double precision for the error correction solver. For matrices from the University of Florida Matrix collection, we report the convergence behaviour and provide the total solver runtime using different GPU architectures.

show abstract

“…After the local iterations, the updated values are communicated. This approach is inspired by the well known hybrid relaxation schemes [9,8]. The obtained algorithm, visualized in Figure 1, can be written as a component-wise update of the solution approximation to form x…”

Section: Mathematical Backgroundmentioning

confidence: 99%

GPU-Accelerated Asynchronous Error Correction for Mixed Precision Iterative Refinement

Antz

¹

,

Łuszczek

²

,

Dongarra

³

et al. 2011

View full text Add to dashboard Cite

Abstract. In hardware-aware high performance computing, block-asynchronous iteration and mixed precision iterative refinement are two techniques that may be used to leverage the computing power of SIMD accelerators like GPUs in the iterative solution of linear equation systems. Although they use a very different approach for this purpose, they share the basic idea of compensating the convergence properties of an inferior numerical algorithm by a more efficient usage of the provided computing power. In this paper, we analyze the potential of combining both techniques. Therefore, we derive a mixed precision iterative refinement algorithm using a block-asynchronous iteration as an error correction solver, and compare its performance with a pure implementation of a blockasynchronous iteration and an iterative refinement method using double precision for the error correction solver. For matrices from the University of Florida Matrix collection, we report the convergence behaviour and provide the total solver runtime using different GPU architectures.

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Scaling Algebraic Multigrid Solvers: On the Road to Exascale

Cited by 37 publications

References 13 publications

Communication Complexity of the Fast Multipole Method and its Algebraic Variants

Communication Complexity of the Fast Multipole Method and its Algebraic Variants

GPU-Accelerated Asynchronous Error Correction for Mixed Precision Iterative Refinement

GPU-Accelerated Asynchronous Error Correction for Mixed Precision Iterative Refinement

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