Efficient solutions of partial differential equations require a match between the algorithm and the underlying architecture. The new chip-multiprocessors, CMPs (a.k.a. multicore), feature low intra-chip communication cost and smaller per-thread caches compared to previous systems. From an algorithmic point of view this means that data locality issues become more important than communication overheads. This may require re-evaluation of many existing algorithms.
We have investigated parallel implementations of multigrid methods using a temporally blocked, naturally ordered, smoother implementation. Compared with the standard multigrid solution based on the two-color red-black algorithm, we improve the data locality often as much as ten times, while our use of a fine-grained locking scheme keeps the parallel efficiency high.
While our algorithm initially was inspired by CMPs, it was surprising to see our OpenMP multigrid implementation run up to 40 percent faster than the standard red-black algorithm on an 8-way SMP system. Studying the smoother part of the algorithm in isolation often shows it performing two iterations at the same time as a single iteration with an ordinary red-black smoother. Running our smoother on a 32-thread UltraSPARC T1 (Niagara) SMT/CMP and a simulated 32-way CMP demonstrates the communication cost of our algorithm to be low on such architectures.
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