A Block-Asynchronous Relaxation Method for Graphics Processing Units

  • Hartwig Anzt (Author)
  • Jack Dongarra (Author)
  • Vincent Heuveline (Author)
    Engineering Mathematics and Computing Lab (EMCL), Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University
  • Stanimire Tomov (Author)

Abstract

In this paper, we analyze the potential of asynchronous relaxation methods on Graphics Processing Units (GPUs). For this purpose, we developed a set of asynchronous iteration algorithms in CUDA and compared them with a parallel implementation of synchronous relaxation methods on CPU-based systems. For a set of test matrices taken from the University of Florida Matrix Collection we monitor the convergence behavior, the average iteration time and the total time-to-solution time. Analyzing the results, we observe that even for our most basic asynchronous relaxation scheme, despite its lower convergence rate compared to the Gauss-Seidel relaxation (that we expected), the asynchronous iteration running on GPUs is still able to provide solution approximations of certain accuracy in considerably shorter time than Gauss-Seidel running on CPUs. Hence, it overcompensates for the slower convergence by exploiting the scalability and the good fit of the asynchronous schemes for the highly parallel GPU architectures. Further, enhancing the most basic asynchronous approach with hybrid schemes -- using multiple iterations within the "subdomain" handled by a GPU thread block and Jacobi-like asynchronous updates across the "boundaries", subject to tuning various parameters -- we manage to not only recover the loss of global convergence but often accelerate convergence of up to two times (compared to the standard but difficult to parallelize Gauss-Seidel type of schemes), while keeping the execution time of a global iteration practically the same. This shows the high potential of the asynchronous methods not only as a stand alone numerical solver for linear systems of equations fulfilling certain convergence conditions but more importantly as a smoother in multigrid methods. Due to the explosion of parallelism in todays architecture designs, the significance and the need for asynchronous methods, as the ones described in this work, is expected to grow.

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The Engineering Mathematics and Computing Lab (EMCL), directed by Prof. Dr. Vincent Heuveline, is a research group at the Interdisciplinary Center for Scientific Computing (IWR).

The EMCL Preprint Series contains publications that were accepted for the Preprint Series of the EMCL and are planned to be published in journals, books, etc. soon.

The EMCL Preprint Series was published under the roof of the Karlsruhe Institute of Technology (KIT) until April 30, 2013. As from May 01, 2013 it is published under the roof of Heidelberg University.

Published
2011-07-01
Language
en