The efficient utilization of mixed-precision numerical linear algebra algorithms can offer attractive acceleration to scientific computing applications. Especially with the hardware integration of low-precision special-function units designed for machine learning applications, the traditional numerical algorithms community urgently needs to reconsider the floating point formats used in the distinct operations to efficiently leverage the available compute power. In this work, we provide a comprehensive survey of mixed-precision numerical linear algebra routines, including the underlying concepts, theoretical background, and experimental results for both dense and sparse linear algebra problems.

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As parallel computers approach the exascale, power efficiency in Highperformance computing (HPC) systems is of increasing concern. Exploiting both, the hardware features, and algorithms is an effective solution to achieve power efficiency, and address the energy constraints in modern and future HPC systems. In this work, we present a novel design and implementation of an energy efficient solution for dense linear system of equations, which are at the heart of largescale HPC applications. The proposed energy efficient linear system solvers are based on two main components: (1) iterative refinement techniques, and (2) reduced precision computing features in the modern accelerators and co-processors. While most of the energy efficiency approaches aim to reduce the consumption with a minimal performance penalty, our method improves both, the performance and the energy-efficiency. Compared to highly optimised linear system solvers, our kernels are up to 2× faster to deliver the same accuracy solution, and reduce the energy consumption up to half on Intel KNL architectures. By using efficiently the tensor cores available in the NVIDIA V100 PCIe GPUs, the speedups can be up to 4× with more than 80% reduction on the energy consumption.

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.