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.
The development of NekRS, a GPU-oriented thermal-fluids simulation code based on the spectral element method (SEM) is described. For performance portability, the code is based on the open concurrent compute abstraction and leverages scalable developments in the SEM code Nek5000 and in libParanumal, which is a library of high-performance kernels for high-order discretizations and PDE-based miniapps. Critical performance sections of the Navier-Stokes time advancement are addressed. Performance results on several platforms are presented, including scaling to 27,648 V100s on OLCF Summit, for calculations of up to 60B gridpoints.
Performance tests and analyses are critical to effective HPC software development and are central components in the design and implementation of computational algorithms for achieving faster simulations on existing and future computing architectures for large-scale application problems. In this paper, we explore performance and space-time trade-offs for important compute-intensive kernels of large-scale numerical solvers for PDEs that govern a wide range of physical applications. We consider a sequence of PDE- motivated bake-off problems designed to establish best practices for efficient high-order simulations across a variety of codes and platforms. We measure peak performance (degrees of freedom per second) on a fixed number of nodes and identify effective code optimization strategies for each architecture. In addition to peak performance, we identify the minimum time to solution at 80% parallel efficiency. The performance analysis is based on spectral and p-type finite elements but is equally applicable to a broad spectrum of numerical PDE discretizations, including finite difference, finite volume, and h-type finite elements.
Electrostatic gating and optical pumping schemes enable efficient time modulation of graphenes free carrier density, or Drude weight. We develop a theory for plasmon propagation in graphene under temporal modulation. When the modulation is on the timescale of the plasmonic period, we show that it is possible to create a backwards-propagating or standing plasmon wave and to amplify plasmons. The theoretical models show very good agreement with direct Maxwell simulations.
