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1. Efficient exascale discretizations: High-order finite element methods

   Kolev, T.; Fischer, P.; Min, M.; Dongarra, J.; Brown, J.; Dobrev, V.; Warburton, T.; Tomov, S.; Shephard, M.; Abdelfattah, A.; et al (January 2021, The international journal of high performance computing applications)

   Efficient exploitation of exascale architectures requires rethinking of the numerical algorithms used in many large-scale applications. These architectures favor algorithms that expose ultra fine-grain parallelism and maximize the ratio of floating point operations to energy intensive data movement. One of the few viable approaches to achieve high efficiency in the area of PDE discretizations on unstructured grids is to use matrix-free/partially assembled high-order finite element methods, since these methods can increase the accuracy and/or lower the computational time due to reduced data motion. In this paper we provide an overview of the research and development activities in the Center for Efficient Exascale Discretizations (CEED), a co-design center in the Exascale Computing Project that is focused on the development of next-generation discretization software and algorithms to enable a wide range of finite element applications to run efficiently on future hardware. CEED is a research partnership involving more than 30 computational scientists from two US national labs and five universities, including members of the Nek5000, MFEM, MAGMA and PETSc projects. We discuss the CEED co-design activities based on targeted benchmarks, miniapps and discretization libraries and our work on performance optimizations for large-scale GPU architectures. We also provide a broad overview of research and development activities in areas such as unstructured adaptive mesh refinement algorithms, matrix-free linear solvers, high-order data visualization, and list examples of collaborations with several ECP and external applications. 

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2. GPU algorithms for efficient exascale discretizations

   Abdelfattah, A.; Barra, V.; Beams, N.; Bleile, R.; Brown J.; Camier, J.; Carson R.; Chalmers, N.; Dobrev, V.; Dudouit, Y.; et al (September 2021, Parallel computing)

   In this paper, we describe the research and development activities in the Center for Efficient Exascale Discretization within the US Exascale Computing Project, targeting state-of-the-art high-order finite-element algorithms for high-order applications on GPU-accelerated platforms. We discuss the GPU developments in several components of the CEED software stack, including the libCEED, MAGMA, MFEM, libParanumal, and Nek projects. We report performance and capability improvements in several CEED-enabled applications on both NVIDIA and AMD GPU systems. 

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3. Multi-Level Analysis of Compiler-Induced Variability and Performance Tradeoffs

   https://doi.org/10.1145/3307681.3325960  

   Bentley, Michael; Briggs, Ian; Gopalakrishnan, Ganesh; Ahn, Dong H.; Laguna, Ignacio; Lee, Gregory L.; Jones, Holger E. (January 2019, HPDC '19 Proceedings of the 28th International Symposium on High-Performance Parallel and Distributed Computing)

   Successful HPC software applications are long-lived. When ported across machines and their compilers, these applications often produce different numerical results, many of which are unacceptable. Such variability is also a concern while optimizing the code more aggressively to gain performance. Efficient tools that help locate the program units (files and functions) within which most of the variability occurs are badly needed, both to plan for code ports and to root-cause errors due to variability when they happen in the field. In this work, we offer an enhanced version of the open-source testing framework FLiT to serve these roles. Key new features of FLiT include a suite of bisection algorithms that help locate the root causes of variability. Another added feature allows an analysis of the tradeoffs between performance and the degree of variability. Our new contributions also include a collection of case studies. Results on the MFEM finite-element library include variability/performance tradeoffs, and the identification of a (hitherto unknown) abnormal level of result-variability even under mild compiler optimizations. Results from studying the Laghos proxy application include identifying a significantly divergent floating-point result-variability and successful root-causing down to the problematic function over as little as 14 program executions. Finally, in an evaluation of 4,376 controlled injections of floating-point perturbations on the LULESH proxy application, we showed that the FLiT framework has 100% precision and recall in discovering the file and function locations of the injections all within an average of only 15 program executions. 

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4. Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise

   https://doi.org/10.3934/dcdss.2021071  

   Feng, Xiaobing; Ma, Shu (January 2022, Discrete & Continuous Dynamical Systems - S)

   This paper is concerned with fully discrete finite element approximations of a stochastic nonlinear Schrödinger (sNLS) equation with linear multiplicative noise of the Stratonovich type. The goal of studying the sNLS equation is to understand the role played by the noises for a possible delay or prevention of the collapsing and/or blow-up of the solution to the sNLS equation. In the paper we first carry out a detailed analysis of the properties of the solution which lays down a theoretical foundation and guidance for numerical analysis, we then present a family of three-parameters fully discrete finite element methods which differ mainly in their time discretizations and contains many well-known schemes (such as the explicit and implicit Euler schemes and the Crank-Nicolson scheme) with different combinations of time discetization strategies. The prototypical \begin{document}$$ \theta $$\end{document}-schemes are analyzed in detail and various stability properties are established for its numerical solution. An extensive numerical study and performance comparison are also presented for the proposed fully discrete finite element schemes. 

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5. Evolution of the SLATE linear algebra library

   https://doi.org/10.1177/10943420241286531  

   Gates, Mark; Abdelfattah, Ahmad; Akbudak, Kadir; Al_Farhan, Mohammed; Alomairy, Rabab; Bielich, Daniel; Burgess, Treece; Cayrols, Sébastien; Lindquist, Neil; Sukkari, Dalal; et al (September 2024, The International Journal of High Performance Computing Applications)

   SLATE (Software for Linear Algebra Targeting Exascale) is a distributed, dense linear algebra library targeting both CPU-only and GPU-accelerated systems, developed over the course of the Exascale Computing Project (ECP). While it began with several documents setting out its initial design, significant design changes occurred throughout its development. In some cases, these were anticipated: an early version used a simple consistency flag that was later replaced with a full-featured consistency protocol. In other cases, performance limitations and software and hardware changes prompted a redesign. Sequential communication tasks were parallelized; host-to-host MPI calls were replaced with GPU device-to-device MPI calls; more advanced algorithms such as Communication Avoiding LU and the Random Butterfly Transform (RBT) were introduced. Early choices that turned out to be cumbersome, error prone, or inflexible have been replaced with simpler, more intuitive, or more flexible designs. Applications have been a driving force, prompting a lighter weight queue class, nonuniform tile sizes, and more flexible MPI process grids. Of paramount importance has been building a portable library that works across several different GPU architectures – AMD, Intel, and NVIDIA – while keeping a clean and maintainable codebase. Here we explore the evolving design choices and their effects, both in terms of performance and software sustainability. 

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