Search

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., 22 pages, 18 figures

MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing efficiency. MFEM's goal is to provide application scientists with access to cutting-edge algorithms for high-order finite element meshing, discretizations and linear solvers, while enabling researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel and GPU-accelerated settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library., Comment: 36 pages, 21 figures

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., 25 pages, 54 figures

The C++ programming language is not only a keystone of the high-performance-computing ecosystem but has proven to be a successful base for portable parallel-programming frameworks. As is well known, C++ programmers use templates to specialize algorithms, thus allowing the compiler to generate highly-efficient code for specific parameters, data structures, and so on. This capability has been limited to those specializations that can be identified when the application is compiled, and in many critical cases, compiling all potentially-relevant specializations is not practical. ClangJIT provides a well-integrated C++ language extension allowing template-based specialization to occur during program execution. This capability has been implemented for use in large-scale applications, and we demonstrate that just-in-time- compilation-based dynamic specialization can be integrated into applications, often requiring minimal changes (or no changes) to the applications themselves, providing significant performance improvements, programmer-productivity improvements, and de- creased compilation time.

Summary The basic principles of the discrete duality and nonlinear monotone finite volume methods are combined in order to obtain a new monotone nonlinear finite volume method for the approximation of diffusion operators on general meshes. Numerical results highlight both the second-order accuracy of this method on general meshes and its capability to deal with challenging anisotropic diffusion problems on various computational domains. Copyright © 2016 John Wiley & Sons, Ltd.

We present a matrix-free flow solver for high-order finite element discretizations of the incompressible Navier-Stokes and Stokes equations with GPU acceleration. For high polynomial degrees, assembling the matrix for the linear systems resulting from the finite element discretization can be prohibitively expensive, both in terms of computational complexity and memory. For this reason, it is necessary to develop matrix-free operators and preconditioners, which can be used to efficiently solve these linear systems without access to the matrix entries themselves. The matrix-free operator evaluations utilize GPU-accelerated sum-factorization techniques to minimize memory movement and maximize throughput. The preconditioners developed in this work are based on a low-order refined methodology with parallel subspace corrections. The saddle-point Stokes system is solved using block-preconditioning techniques, which are robust in mesh size, polynomial degree, time step, and viscosity. For the incompressible Navier-Stokes equations, we make use of projection (fractional step) methods, which require Helmholtz and Poisson solves at each time step. The performance of our flow solvers is assessed on several benchmark problems in two and three spatial dimensions., 20 pages, 11 figures