Weighted Block - Asynchronous Relaxation for Gpu - Accelerated Systems ∗

In this paper, we analyze the potential of using weights for block-asynchronous relaxation methods on GPUs. For this purpose, we introduce different weighting techniques similar to those applied in block-smoothers for multigrid methods. Having proven a sufficient convergence condition for the weighted block-asynchronous iteration, we analyze the performance of the algorithms 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 different 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 iteration for numerically solving linear systems of equations fulfilling 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 finite 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 today’s architecture designs, the significance and the need for highly parallel asynchronous smoothers, as the ones described in this work, is expected to grow.