DocumentCode :
2996773
Title :
A Block-Asynchronous Relaxation Method for Graphics Processing Units
Author :
Anzt, Hartwig ; Tomov, Stanimire ; Dongarra, Jack ; Heuveline, Vincent
Author_Institution :
Karlsruhe Inst. of Technol., Karlsruhe, Germany
fYear :
2012
fDate :
21-25 May 2012
Firstpage :
113
Lastpage :
124
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 "sub domain" 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 today\´s architecture designs, the significance and the need for asynchronous methods, as the ones described in this work, is expected to grow.
Keywords :
Jacobian matrices; graphics processing units; iterative methods; parallel architectures; CPU-based system; CUDA; GPU thread block; Gauss-Seidel relaxation; Jacobi-like asynchronous updates; asynchronous iteration algorithm; average iteration time; block-asynchronous relaxation method; convergence behavior; global convergence; graphics processing unit; highly parallel GPU architecture; total time-to-solution time; Convergence; Educational institutions; Equations; Graphics processing unit; Jacobian matrices; Linear systems; Synchronization; Asynchronous Relaxation; Chaotic Iteration; Graphics Processing Units (GPUs); Jacobi Method;
fLanguage :
English
Publisher :
ieee
Conference_Titel :
Parallel and Distributed Processing Symposium Workshops & PhD Forum (IPDPSW), 2012 IEEE 26th International
Conference_Location :
Shanghai
Print_ISBN :
978-1-4673-0974-5
Type :
conf
DOI :
10.1109/IPDPSW.2012.11
Filename :
6270632
Link To Document :
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