Krylov solvers are key kernels in many large-scale science and engineering applications for solving sparse linear systems. Applications running at scale can experience significant slowdown due to factors such as network congestion, off-node congestion, network distance, and performance variation across processes. Performance models can help us better understand factors limiting performance, however simple models fail to capture slowdowns often occurring at scale and performance variation across multiple runs of the same code. This work develops performance models that capture behavior found at scale and uses these models to guide optimizations for Krylov solvers and related kernels using both blocking and non-blocking communication for structured grid problems at scale. We use detailed performance analysis with network performance counters to show how network behavior relates to observed performance and guide the development of performance models that capture the runtime impact of network congestion, network distance, communication and computation overlap, and process mappings. These models guide us to optimize kernels using MPI protocol changes, node-aware communication, and topology-aware communication. The resulting tools and analysis provide us with a better understanding of how to improve performance at scale that can benefit a wider range of applications.

The preconditioned conjugate gradient method (PCG) is a popular method for solving linear systems at scale. PCG requires frequent blocking allreduce collective operations that can limit performance at scale. We investigate PCG variations designed to reduce communication costs by decreasing the number of allreduces and by overlapping communication with computation using a non-blocking allreduce. These variations include two methods we have developed, non-blocking PCG and 2-step pipelined PCG, and pipelined PCG from Ghysels and Vanroose. Performance modeling for communication and computation costs shows the expected performance of these methods. Weak and strong scaling experiments on up to 128k cores show that scalable PCG methods can outperform standard PCG at scale. We observe that the fastest method varies depending on the work per core, suggesting we need a suite of scalable solvers to obtain the best performance. Experiments with multiple preconditioners and linear systems show the robustness of these methods.