WEIGHTED BLOCK-ASYNCHRONOUS RELAXATION FOR GPU-ACCELERATED SYSTEMS

In this paper, we analyze the potential of using weights for block-asynchronous re- laxation methods on GPUs. For this purpose, we introduce dierent weighting techniques similar to those applied in block-smoothers for multigrid methods. Having proven a sucient convergence condition for the weighted block-asynchronous iteration, we analyze the performance of the algo- rithms implemented using CUDA and compare them with weighted synchronous relaxation schemes like SOR. For test matrices taken from the University of Florida Matrix Collection we report the convergence behavior and the total runtime for the dierent weighting techniques. Analyzing the results, we observe that using weights may accelerate the convergence rate of block-asynchronous iteration considerably. This shows the high potential of using weights in block-asynchronous iter- ation for numerically solving linear systems of equations fullling certain convergence conditions. While component-wise relaxation methods are seldom directly applied to linear equation systems, using them as smoother in a multigrid framework they often provide an important contribution to nite element solvers. Since the parallelization potential of the classical smoothers like SOR and Gauss-Seidel is usually very limited, replacing them with block-asynchronous smoothers may have a considerable impact on the overall multigrid performance. Due to the explosion of parallelism in to- day's architecture designs, the signicance and the need for highly parallel asynchronous smoothers, as the ones described in this work, is expected to grow.

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