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Numerical Properties and Scalability of s-Step Preconditioned Conjugate Gradient Methods
Descriptions-step Preconditioned Conjugate Gradient (PCG) variants for iteratively solving large sparse linear systems reduce the number of global synchronization points of standard PCG by a factor of O(s). Despite improving scalability on large-scale parallel computers, they have worse numerical properties than standard PCG. Choosing a suitable basis type for the s-step basis matrices is known to potentially improve numerical stability strongly. The s-step method proposed first in the literature was designed to only use the monomial basis. We generalize this method to support arbitrary basis types, denoting our new method as sPCG.
Moreover, we theoretically and experimentally compare all s-step PCG methods. To the best of our knowledge, this is the first comprehensive comparison in the literature. Our theoretical analysis, strong scaling experiments with a synthetic test problem, and runtime experiments with real-world problems confirm that our novel sPCG algorithm achieves higher speedup over standard PCG than existing s-step algorithms.
Moreover, we theoretically and experimentally compare all s-step PCG methods. To the best of our knowledge, this is the first comprehensive comparison in the literature. Our theoretical analysis, strong scaling experiments with a synthetic test problem, and runtime experiments with real-world problems confirm that our novel sPCG algorithm achieves higher speedup over standard PCG than existing s-step algorithms.