With the advent of high performance computers with massively parallel processors, it has become very important to develop scalable methods such as AMG. One of AMG's most important components is the smoother. The most effective smoothers, such as Gauss-Seidel or even multiplicative Schwarz smoothers are often sequential in nature, whereas Jacobi or block Jacobi-like smoothers often fail, unless an appropriate smoothing parameter is used and even then their convergence is often slow. Many efforts to parallelize Gauss-Seidel have been made. Possible variants include the use of multi-coloring techniques or hybrid smoothers. Multi coloring techniques are difficult to implement and can be inefficient, if too many colors are involved, which is often the case on the coarser levels on AMG. Hybrid schemes use a sequential smoother on each processor, but update in a Jacobi like fashion across processor boundaries, and therefore often have similar disadvantages as Jacobi like smoothers.
In this talk, we investigate the use of relaxation parameters in hybrid smoothers. We analyze their influence on the smoothing property, describe a procedure that automatically determines good parameters on each level of AMG and present numerical results that show significant improvement over AMG with unrelaxed hybrid smoothers.
*This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract number W-7405-Eng-48.