Weighted block-asynchronous iteration on 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. For test matrices taken from the University of Florida Matrix Collection we report the convergence behavior and the total runtime for the different techniques. Analyzing the results, we observe that using weights may accelerate the convergence rate of block-asynchronous iteration considerably. While component-wise relaxation methods are seldom directly applied to systems of linear equations, 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 by weighted block-asynchronous smoothers may be beneficial to the overall multigrid performance. Due to the increase of heterogeneity in today's architecture designs, the significance and the need for highly parallel asynchronous smoothers is expected to grow.