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The growing discrepancy between CPU computing power and memory bandwidth drives more and more numerical algorithms into a bandwidth-
bound regime. One example is the overlapping Schwarz smoother, a highly effective building block for iterative multigrid solution of elliptic equations with higher order finite elements. Two options of reducing the required
memory bandwidth are sparsity exploiting storage layouts and representing matrix entries with reduced precision in floating point or fixed point
format. We investigate the impact of several options on storage demand and contraction rate, both analytically in the context of subspace correction methods and numerically at an example of solid mechanics. Both perspectives agree on the favourite scheme: fixed point representation of Cholesky factors in nested dissection storage.
The effectiveness of ahead-of-time compiler optimization heavily depends on the amount of available information at compile time. Input-specific information that is only available at runtime cannot be used, although it often determines loop counts, branching predicates and paths, as well as memory-access patterns. It can also be crucial for generating efficient SIMD-vectorized code. This is especially relevant for the many-core architectures paving the way to exascale computing, which are more sensitive to code-optimization. We explore the design-space for using input-specific information at compile-time and present KART, a C++ library solution that allows developers tocompile, link, and execute code (e.g., C, C++ , Fortran) at application runtime. Besides mere runtime compilation of performance-critical code, KART can be used to instantiate the same code multiple times using different inputs, compilers, and options. Other techniques like auto-tuning and code-generation can be integrated into a KART-enabled application instead of being scripted around it. We evaluate runtimes and compilation costs for different synthetic kernels, and show the effectiveness for two real-world applications, HEOM and a WSM6 proxy.
Solving partial differential equations on unstructured grids is a cornerstone of engineering and scientific computing. Nowadays, heterogeneous parallel platforms with CPUs, GPUs, and FPGAs enable energy-efficient and computationally demanding simulations. We developed the HighPerMeshes C++-embedded Domain-Specific Language (DSL) for bridging the abstraction gap between the mathematical and algorithmic formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different parallel programming and runtime models on the other hand. Thus, the HighPerMeshes DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HighPer-Meshes DSL, and demonstrate its usage with three examples, a Poisson and monodomain problem, respectively, solved by the continuous finite element method, and the discontinuous Galerkin method for Maxwell’s equation. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters is presented. Finally, the achievable performance and scalability are demonstrated for a typical example problem on a multi-core CPU cluster.
Effective vectorization is becoming increasingly important for high performance and energy efficiency on processors with wide SIMD units. Compilers often require programmers to identify opportunities for vectorization, using directives to disprove data dependences. The OpenMP 4.x SIMD directives strive to provide portability. We investigate the ability of current compilers (GNU, Clang, and Intel) to generate SIMD code for microbenchmarks that cover common patterns in scientific codes and for two kernels from the VASP and the MOM5/ERGOM application. We explore coding strategies for improving SIMD performance across different compilers and platforms (Intel® Xeon® processor and Intel® Xeon Phi™ (co)processor). We compare OpenMP* 4.x SIMD vectorization with and without vector data types against SIMD intrinsics and C++ SIMD types. Our experiments show that in many cases portable performance can be achieved. All microbenchmarks are available as open source as a reference for programmers and compiler experts to enhance SIMD code generation.
A classical technique to vectorize code that contains control flow is a control-flow to data-flow conversion. In that approach statements are augmented with masks that denote whether a given vector lane participates in the statement’s execution or idles. If the scheduling of work to vector lanes is performed statically, then some of the vector lanes will run idle in case of control flow divergences or varying work intensities across the loop iterations. With an increasing number of vector lanes, the likelihood of divergences or heavily unbalanced work assignments increases and static scheduling leads to a poor resource utilization. In this paper, we investigate different approaches to dynamic SIMD vector lane scheduling using the Mandelbrot set algorithm as a test case. To overcome the limitations of static scheduling, idle vector lanes are assigned work items dynamically, thereby minimizing per-lane idle cycles. Our evaluation on the Knights Corner and Knights Landing platform shows, that our approaches can lead to considerable performance gains over a static work assignment. By using the AVX-512 vector compress and expand instruction, we are able to further improve the scheduling.
The Vienna Ab initio Simulation Package (VASP) is a widely used electronic structure code that originally exploits process-level parallelism through the Message Passing Interface (MPI) for work distribution within and across nodes.
Architectural changes of modern parallel processors urge programmers to address thread- and data-level parallelism as well to benefit most from the available compute resources within a node.
We describe for VASP how to approach for an MPI + OpenMP parallelization including data-level parallelism through OpenMP SIMD constructs together with a generic high-level vector coding scheme.
We can demonstrate an improved scalability of VASP and more than 20% gain over the MPI-only version, as well as a 2x increased performance of collective operations using the multiple-endpoint MPI feature.
The high-level vector coding scheme applied to VASP's general gradient approximation routine gives up 9x performance gain on AVX512 platforms with the Intel compiler.
Programs that process linearly indexed fields with
structured element types in a data-parallel way usually suffer from the fact that compilers fail to generate efficient code if the selected data layout appears inappropriate for the chosen
target architecture. If their internal heuristics cannot proof a performance gain from a data-parallel execution, compilers may fall back to scalar code generation. Data access through proxy types together with a customized container is one means to assist the compiler in generating efficient machine code in these cases without changing the user code. We present an automated proxy-type generator (using Clang’s LibTooling) and a configurable C++ container that supports different data
layouts in a transparent way.
Application Performance on a Cray XC30 Evaluation System with Xeon Phi Coprocessors at HLRN-III
(2015)
OpenCL: There and Back Again
(2015)