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Mixed-precision computing has the potential to significantly reduce the cost of exascale computations, but determining when and how to implement it in programs can be challenging. In this article, we propose a methodology for enabling mixed-precision with the help of computer arithmetic tools, roofline model, and computer arithmetic techniques. As case studies, we consider Nekbone (Nek5000 developers), a mini-application for the Computational Fluid Dynamics (CFD) solver Nek5000 (Fischer et al.), and a modern Neko (Jansson et al., 2024) CFD application. With the help of the Verificarlo (Denis et al., 2016) tool and computer arithmetic techniques, we introduce a strategy to address stagnation issues in the preconditioned Conjugate Gradient method in Nekbone and apply these insights to implement a mixed-precision version of Neko. We evaluate the derived mixed-precision versions of these codes by combining metrics in three dimensions: accuracy, time-to-solution, and energy-to-solution. Notably, mixed-precision in Nekbone reduces time-to-solution by roughly 1.62x and energy-to-solution by 2.43x on MareNostrum 5, while in the real-world Neko application, the gain is up to 1.3x in both time and energy, with the accuracy that matches double-precision results.

Mixed-precision computing has the potential to significantly reduce the cost of exascale computations, but determining when and how to implement it in programs can be challenging. In this article, we consider Nekbone, a mini-application for the Computational Fluid Dynamics (CFD) solver Nek5000, as a case study, and propose a methodology for enabling mixed-precision with the help of computer arithmetic tools and roofline model. We evaluate the derived mixed-precision program by combining metrics in three dimensions: accuracy, time-to-solution, and energy-to-solution. Notably, the introduction of mixed-precision in Nekbone, reducing time-to-solution by 40.7% and energy-to-solution by 47% on 128 MPI ranks without sacrificing the accuracy.

In this document, the EuroHPC JU Center of Excellence in Exascale CFD (CEEC) aims to provide users/ application developers with a brief overview of possibilities, limitations, and best practices for measuring energy consumption on European HPC systems. CEEC is working  to reduce the energy footprint of its consortium codes on such systems by applying novel algorithmic solutions. However, in initially exploring options for collecting energy measurements on both local and European HPC systems, we found no single approach for energy measurements and the process of taking these measurements comparatively more difficult than measuring time-to-solution with e.g. basic start-end time calls. This difficulty often stems from a requirement for privileged access to specific hardware counters. Mitigation strategies for this restriction exist and enable users to collect the energy metric, but they are not widely known. We describe these strategies followed by concrete examples from CEEC on how to harvest the energy measurements. We believe this will help to increase awareness and thus utilization of energy consumption measurements in the application development process.

Furthermore, we describe several other important issues: 1) granularity and overhead of measurements since energy=power x time and 2) what is included (there multiple factors) in the number delivered by a tool/ framework/ workload manager. We strive to be concise and precise aiming to provide a glimpse of energy measurement methods as well as many references for further exploration. Our takeaway messages are

• The community/ data centers need to facilitate energy measurements on the European HPC systems and teach the community how to conduct such measurements.
• The community/ data centers need to provide transparent and easy-to-use guides on each (at least large) European HPC system, outlining the ways to collect energy measurements.

In CEEC, we are taking the first steps towards spreading these messages, aiming to create a larger consortium including experts and data centers, who can contribute to and update this document. Explore and stay tuned!

Parallel implementations of Krylov subspace methods often help to accelerate the procedure of finding an approximate solution of a linear system. However, such parallelization coupled with asynchronous and out-of-order execution often makes more visible the non-associativity impact in floating-point operations. These problems are even amplified when communication-hiding pipelined algorithms are used to improve the parallelization of Krylov subspace methods. Introducing reproducibility in the implementations avoids these problems by getting more robust and correct solutions. This paper proposes a general framework for deriving reproducible and accurate variants of Krylov subspace methods. The proposed algorithmic strategies are reinforced by programmability suggestions to assure deterministic and accurate executions. The framework is illustrated on the preconditioned BiCGStab method and its pipelined modification, which in fact is a distinctive method from the Krylov subspace family, for the solution of non-symmetric linear systems with message-passing. Finally, we verify the numerical behavior of the two reproducible variants of BiCGStab on a set of matrices from the SuiteSparse Matrix Collection and a 3D Poisson’s equation.

In this article, we propose an accuracy-assuring technique for finding a solution for unsymmetric linear systems. Such problems are related to different areas such as image processing, computer vision, and computational fluid dynamics. Parallel implementation of Krylov subspace methods speeds up finding approximate solutions for linear systems. In this context, the refined approach in pipelined BiCGStab enhances scalability on distributed memory machines, yielding to substantial speed improvements compared to the standard BiCGStab method. However, it’s worth noting that the pipelined BiCGStab algorithm sacrifices some accuracy, which is stabilized with the residual replacement technique. This paper aims to address this issue by employing the ExBLAS-based reproducible approach. We validate the idea on a set of matrices from the SuiteSparse Matrix Collection.

Parallel implementations of Krylov subspace algorithms often help to accelerate the procedure to find the solution of a linear system. However, from the other side, such parallelization coupled with asynchronous and out-of-order execution often enlarge the non-associativity of floating-point operations. This results in non-reproducibility on the same or different settings. This paper proposes a general framework for deriving reproducible and accurate variants of a Krylov subspace algorithm. The proposed algorithmic strategies are reinforced by programmability suggestions to assure deterministic and accurate executions. The framework is illustrated on the preconditioned BiCGStab method for the solution of non-symmetric linear systems with message-passing. Finally, we verify the two reproducible variants of PBiCGStab on a set matrices from the SuiteSparse Matrix Collection and a 3D Poisson’s equation.