Bus coupled systems for multigrid algorithms

Abstract

Speedup and efficiency of some simple parallel multigrid algorithms for a class of bus coupled systems are investigated. We consider some basic multigrid methods (V-cycle, W-cycle) with regular grid generation and without local refinements. Our bus coupled systems consist of many independent processors each with its own local memory. A typical example for our abstract bus concept is a ring bus. The investigation of such systems is restricted to hierarchical orthogonal systems. Simple orthogonal bus systems, tree structures and mixed types are included in our general model. It can be shown that all systems are of identical suitability if the tasks are sufficiently large. The smaller however the degree of parallelism of an algorithm is, the clearer are the differences in the performance of the various systems. We can classify the most powerful systems and systems with lower performance but better technical properties. Complexity investigations enabled us to evaluate the different systems. These investigations are complemented by simulations based on the different parallel algorithms.

In general, the order of the speedup depends only on a few parameters, such as the dimension of the problem, the cycle type and the dimension of the system. The constant factors in our asymptotical expressions for the speedup depend on many parameters, especially on those of the processors and buses. We investigate these relations by simulation of some typical examples. The simulation also clarifies under which circumstances the asymptotical rules are useful for the description of system behavior.