Accelerating incompressible flow computations with a Pthreads-CUDA implementation on small-footprint multi-GPU platforms

Graphics processor units (GPU) that are originally designed for graphics rendering have emerged as massively-parallel “co-processors” to the central processing unit (CPU). Small-footprint multi-GPU workstations with hundreds of processing elements can accelerate compute-intensive simulation science applications substantially. In this study, we describe the implementation of an incompressible flow Navier–Stokes solver for multi-GPU workstation platforms. A shared-memory parallel code with identical numerical methods is also developed for multi-core CPUs to provide a fair comparison between CPUs and GPUs. Specifically, we adopt NVIDIA’s Compute Unified Device Architecture (CUDA) programming model to implement the discretized form of the governing equations on a single GPU. Pthreads are then used to enable communication across multiple GPUs on a workstation. We use separate CUDA kernels to implement the projection algorithm to solve the incompressible fluid flow equations. Kernels are implemented on different memory spaces on the GPU depending on their arithmetic intensity. The memory hierarchy specific implementation produces significantly faster performance. We present a systematic analysis of speedup and scaling using two generations of NVIDIA GPU architectures and provide a comparison of single and double precision computational performance on the GPU. Using a quad-GPU platform for single precision computations, we observe two orders of magnitude speedup relative to a serial CPU implementation. Our results demonstrate that multi-GPU workstations can serve as a cost-effective small-footprint parallel computing platform to accelerate computational fluid dynamics (CFD) simulations substantially.     

Parallel strategies for 2D Discrete Wavelet Transform in shared memory systems and GPUs

In this work, we analyze the behavior of several parallel algorithms developed to compute the two-dimensional discrete wavelet transform using both OpenMP over a multicore platform and CUDA over a GPU. The proposed parallel algorithms are based on both regular filter-bank convolution and lifting transform with small implementations changes focused on both the memory requirements reduction and the complexity reduction. We compare our implementations against sequential CPU algorithms and other recently proposed algorithms like the SMDWT algorithm over different CPUs and the Wippig&Klauer algorithm over a GTX280 GPU. Finally, we analyze their behavior when algorithms are adapted to each architecture. Significant execution times improvements are achieved on both multicore platforms and GPUs. Depending on the multicore platform used, we achieve speed-ups of 1.9 and 3.4 using two and four processes, respectively, when compared to the sequential CPU algorithm, or we obtain speed-ups of 7.1 and 8.9 using eight and ten processes. Regarding GPUs, the GPU convolution algorithm using the GPU shared memory obtains speed-ups up to 20 when compared to the CPU sequential algorithm.     

Exploiting the capabilities of modern GPUs for dense matrix computations

We present several algorithms to compute the solution of a linear system of equations on a graphics processor (GPU), as well as general techniques to improve their performance, such as padding and hybrid GPU‐CPU computation. We compare single and double precision performance of a modern GPU with unified architecture, and show how iterative refinement with mixed precision can be used to regain full accuracy in the solution of linear systems, exploiting the potential of the processor for single precision arithmetic. Experimental results on a GTX280 using CUBLAS 2.0, the implementation of BLAS for NVIDIA^® GPUs with unified architecture, illustrate the performance of the different algorithms and techniques proposed. Copyright © 2009 John Wiley & Sons, Ltd.     

CUDA-enabled Sparse Matrix–Vector Multiplication on GPUs using atomic operations

Existing formats for Sparse Matrix–Vector Multiplication (SpMV) on the GPU are outperforming their corresponding implementations on multi-core CPUs. In this paper, we present a new format called Sliced COO (SCOO) and an efficient CUDA implementation to perform SpMV on the GPU using atomic operations. We compare SCOO performance to existing formats of the NVIDIA Cusp library using large sparse matrices. Our results for single-precision floating-point matrices show that SCOO outperforms the COO and CSR format for all tested matrices and the HYB format for all tested unstructured matrices on a single GPU. Furthermore, our dual-GPU implementation achieves an efficiency of 94% on average. Due to the lower performance of existing CUDA-enabled GPUs for atomic operations on double-precision floating-point numbers the SCOO implementation for double-precision does not consistently outperform the other formats for every unstructured matrix. Overall, the average speedup of SCOO for the tested benchmark dataset is 3.33 (1.56) compared to CSR, 5.25 (2.42) compared to COO, 2.39 (1.37) compared to HYB for single (double) precision on a Tesla C2075. Furthermore, comparison to a Sandy-Bridge CPU shows that SCOO on a Fermi GPU outperforms the multi-threaded CSR implementation of the Intel MKL Library on an i7-2700 K by a factor between 5.5 (2.3) and 18 (12.7) for single (double) precision.     

Direct volume rendering of unstructured tetrahedral meshes using CUDA and OpenMP

Direct volume visualization is an important method in many areas, including computational fluid dynamics and medicine. Achieving interactive rates for direct volume rendering of large unstructured volumetric grids is a challenging problem, but parallelizing direct volume rendering algorithms can help achieve this goal. Using Compute Unified Device Architecture (CUDA), we propose a GPU-based volume rendering algorithm that itself is based on a cell projection-based ray-casting algorithm designed for CPU implementations. We also propose a multicore parallelized version of the cell-projection algorithm using OpenMP. In both algorithms, we favor image quality over rendering speed. Our algorithm has a low memory footprint, allowing us to render large datasets. Our algorithm supports progressive rendering. We compared the GPU implementation with the serial and multicore implementations. We observed significant speed-ups that, together with progressive rendering, enables reaching interactive rates for large datasets.     

A discontinuous Galerkin method with block cyclic reduction solver for simulating compressible flows on GPUs

An optimized implementation of a block tridiagonal solver based on the block cyclic reduction (BCR) algorithm is introduced and its portability to graphics processing units (GPUs) is explored. The computations are performed on the NVIDIA GTX480 GPU. The results are compared with those obtained on a single core of Intel Core i7-920 (2.67 GHz) in terms of calculation runtime. The BCR linear solver achieves the maximum speedup of 5.84x with block size of 32 over the CPU Thomas algorithm in double precision. The proposed BCR solver is applied to discontinuous Galerkin (DG) simulations on structured grids via alternating direction implicit (ADI) scheme. The GPU performance of the entire computational fluid dynamics (CFD) code is studied for different compressible inviscid flow test cases. For a general mesh with quadrilateral elements, the ADI-DG solver achieves the maximum total speedup of 7.45x for the piecewise quadratic solution over the CPU platform in double precision.     

基于非结构网格隐式算法的GPU加速研究

针对非结构网格隐式算法在GPU上的加速效果不佳的问题,通过分析GPU的架构及并行模式,研究并实现了基于非结构网格格点格式的隐式LU-SGS算法的GPU并行加速.通过采用RCM和Metis网格重排序（重组）方法,优化非结构网格的数据局部性,改善非结构网格的隐式算法在GPU上的并行加速效果.通过三维机翼算例验证了本文实现的正确性及效率.结果表明两种网格重排序（重组）方法分别得到了63%和69%的加速效果提高.优化后的LU-SGS隐式GPU并行算法获得了相较于CPU串行算法27倍的加速比,充分说明了本文方法的高效性.     

Optimized FFT computations on heterogeneous platforms with application to the Poisson equation

We develop optimized multi-dimensional FFT implementations on CPU–GPU heterogeneous platforms for the case when the input is too large to fit on the GPU global memory, and use the resulting techniques to develop a fast Poisson solver. The solver involves memory bound computations for which the large 3D data may have to be transferred over the PCIe bus several times during the computation. We develop a new strategy to decompose and allocate the computation between the GPU and the CPU such that the 3D data is transferred only once to the device memory, and the executions of the GPU kernels are almost completely overlapped with the PCI data transfer. We were able to achieve significantly better performance than what has been reported in previous related work, including over 145 GFLOPS for the three periodic boundary conditions (single precision version), and over 105 GFLOPS for the two periodic, one Neumann boundary conditions (single precision version). The effective bidirectional PCIe bus bandwidth achieved is 9–10 GB/s, which is close to the best possible on our platform. For all the cases tested, the single 3D data PCIe transfer time, which constitutes a lower bound on what is possible on our platform, takes almost 70% of the total execution time of the Poisson solver.     

GPU computing with Kaczmarz’s and other iterative algorithms for linear systems

The graphics processing unit (GPU) is used to solve large linear systems derived from partial differential equations. The differential equations studied are strongly convection-dominated, of various sizes, and common to many fields, including computational fluid dynamics, heat transfer, and structural mechanics. The paper presents comparisons between GPU and CPU implementations of several well-known iterative methods, including Kaczmarz’s, Cimmino’s, component averaging, conjugate gradient normal residual (CGNR), symmetric successive overrelaxation-preconditioned conjugate gradient, and conjugate-gradient-accelerated component-averaged row projections (CARP-CG). Computations are preformed with dense as well as general banded systems. The results demonstrate that our GPU implementation outperforms CPU implementations of these algorithms, as well as previously studied parallel implementations on Linux clusters and shared memory systems. While the CGNR method had begun to fall out of favor for solving such problems, for the problems studied in this paper, the CGNR method implemented on the GPU performed better than the other methods, including a cluster implementation of the CARP-CG method.     

Simulation of one-layer shallow water systems on multicore and CUDA architectures

The numerical solution of shallow water systems is useful for several applications related to geophysical flows, but the big dimensions of the domains suggests the use of powerful accelerators to obtain numerical results in reasonable times. This paper addresses how to speed up the numerical solution of a first order well-balanced finite volume scheme for 2D one-layer shallow water systems by using modern Graphics Processing Units (GPUs) supporting the NVIDIA CUDA programming model. An algorithm which exploits the potential data parallelism of this method is presented and implemented using the CUDA model in single and double floating point precision. Numerical experiments show the high efficiency of this CUDA solver in comparison with a CPU parallel implementation of the solver and with respect to a previously existing GPU solver based on a shading language.     

CUDA optimization strategies for compute- and memory-bound neuroimaging algorithms

As neuroimaging algorithms and technology continue to grow faster than CPU performance in complexity and image resolution, data-parallel computing methods will be increasingly important. The high performance, data-parallel architecture of modern graphical processing units (GPUs) can reduce computational times by orders of magnitude. However, its massively threaded architecture introduces challenges when GPU resources are exceeded. This paper presents optimization strategies for compute- and memory-bound algorithms for the CUDA architecture. For compute-bound algorithms, the registers are reduced through variable reuse via shared memory and the data throughput is increased through heavier thread workloads and maximizing the thread configuration for a single thread block per multiprocessor. For memory-bound algorithms, fitting the data into the fast but limited GPU resources is achieved through reorganizing the data into self-contained structures and employing a multi-pass approach. Memory latencies are reduced by selecting memory resources whose cache performance are optimized for the algorithm's access patterns. We demonstrate the strategies on two computationally expensive algorithms and achieve optimized GPU implementations that perform up to 6× faster than unoptimized ones. Compared to CPU implementations, we achieve peak GPU speedups of 129× for the 3D unbiased nonlinear image registration technique and 93× for the non-local means surface denoising algorithm.     

Parallel hyperbolic PDE simulation on clusters: Cell versus GPU

Increasingly, high-performance computing is looking towards data-parallel computational devices to enhance computational performance. Two technologies that have received significant attention are IBM's Cell Processor and NVIDIA's CUDA programming model for graphics processing unit (GPU) computing. In this paper we investigate the acceleration of parallel hyperbolic partial differential equation simulation on structured grids with explicit time integration on clusters with Cell and GPU backends. The message passing interface (MPI) is used for communication between nodes at the coarsest level of parallelism. Optimizations of the simulation code at the several finer levels of parallelism that the data-parallel devices provide are described in terms of data layout, data flow and data-parallel instructions. Optimized Cell and GPU performance are compared with reference code performance on a single x86 central processing unit (CPU) core in single and double precision. We further compare the CPU, Cell and GPU platforms on a chip-to-chip basis, and compare performance on single cluster nodes with two CPUs, two Cell processors or two GPUs in a shared memory configuration (without MPI). We finally compare performance on clusters with 32 CPUs, 32 Cell processors, and 32 GPUs using MPI. Our GPU cluster results use NVIDIA Tesla GPUs with GT200 architecture, but some preliminary results on recently introduced NVIDIA GPUs with the next-generation Fermi architecture are also included. This paper provides computational scientists and engineers who are considering porting their codes to accelerator environments with insight into how structured grid based explicit algorithms can be optimized for clusters with Cell and GPU accelerators. It also provides insight into the speed-up that may be gained on current and future accelerator architectures for this class of applications.

Program summary

Program title: SWsolverCatalogue identifier: AEGY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GPL v3No. of lines in distributed program, including test data, etc.: 59 168No. of bytes in distributed program, including test data, etc.: 453 409Distribution format: tar.gzProgramming language: C, CUDAComputer: Parallel Computing Clusters. Individual compute nodes may consist of x86 CPU, Cell processor, or x86 CPU with attached NVIDIA GPU accelerator.Operating system: LinuxHas the code been vectorised or parallelized?: Yes. Tested on 1-128 x86 CPU cores, 1-32 Cell Processors, and 1-32 NVIDIA GPUs.RAM: Tested on Problems requiring up to 4 GB per compute node.Classification: 12External routines: MPI, CUDA, IBM Cell SDKNature of problem: MPI-parallel simulation of Shallow Water equations using high-resolution 2D hyperbolic equation solver on regular Cartesian grids for x86 CPU, Cell Processor, and NVIDIA GPU using CUDA.Solution method: SWsolver provides 3 implementations of a high-resolution 2D Shallow Water equation solver on regular Cartesian grids, for CPU, Cell Processor, and NVIDIA GPU. Each implementation uses MPI to divide work across a parallel computing cluster.Additional comments: Sub-program numdiff is used for the test run.     

交替方向隐式CFD解法器的GPU并行计算及其优化

交替方向隐格式(ADI)是常见的偏微分方程离散格式之一,目前对ADI格式在计算流体力学（CFD）实际应用中的GPU并行工作开展较少。从一个有限体积CFD应用出发,通过分析ADI解法器的特点和计算流程,基于统一计算架构(CUDA)编程模型设计了基于网格点与网格线的两类细粒度GPU并行算法,讨论了若干性能优化方法。在天河-1A系统上,采用128×128×128网格规模的单区结构网格算例,无粘项、粘性项及ADI迭代计算的GPU并行性能相对于单CPU核,分别取得了100.1、40.1和10.3倍的加速比,整体ADI CFD解法器的GPU并行加速比为17.3     

Multi-physics bi-directional evolutionary topology optimization on GPU-architecture

Topology optimization has proven to be viable for use in the preliminary phases of real world design problems. Ultimately, the restricting factor is the computational expense since a multitude of designs need to be considered. This is especially imperative in such fields as aerospace, automotive and biomedical, where the problems involve multiple physical models, typically fluids and structures, requiring excessive computational calculations. One possible solution to this is to implement codes on massively parallel computer architectures, such as graphics processing units (GPUs). The present work investigates the feasibility of a GPU-implemented lattice Boltzmann method for multi-physics topology optimization for the first time. Noticeable differences between the GPU implementation and a central processing unit (CPU) version of the code are observed and the challenges associated with finding feasible solutions in a computational efficient manner are discussed and solved here, for the first time on a multi-physics topology optimization problem. The main goal of this paper is to speed up the topology optimization process for multi-physics problems without restricting the design domain, or sacrificing considerable performance in the objectives. Examples are compared with both standard CPU and various levels of numerical precision GPU codes to better illustrate the advantages and disadvantages of this implementation. A structural and fluid objective topology optimization problem is solved to vary the dependence of the algorithm on the GPU, extending on the previous literature that has only considered structural objectives of non-design dependent load problems. The results of this work indicate some discrepancies between GPU and CPU implementations that have not been seen before in the literature and are imperative to the speed-up of multi-physics topology optimization algorithms using GPUs.

     

Recent progress and challenges in exploiting graphics processors in computational fluid dynamics

The progress made in accelerating simulations of fluid flow using GPUs, and the challenges that remain, are surveyed. The review first provides an introduction to GPU computing and programming, and discusses various considerations for improved performance. Case studies comparing the performance of CPU- and GPU-based solvers for the Laplace and incompressible Navier–Stokes equations are performed in order to demonstrate the potential improvement even with simple codes. Recent efforts to accelerate CFD simulations using GPUs are reviewed for laminar, turbulent, and reactive flow solvers. Also, GPU implementations of the lattice Boltzmann method are reviewed. Finally, recommendations for implementing CFD codes on GPUs are given and remaining challenges are discussed, such as the need to develop new strategies and redesign algorithms to enable GPU acceleration.     

Large-scale solutions of three-dimensional compressible flows using the parallel N3S-MUSCL solver

We are concerned here with the parallelisation of the N3S-MUSCL industrial code which aims to solve the three-dimensional compressible Navier–Stokes equations by implementing a mixed finite element/finite volume formulation on unstructured tetrahedral meshes. Defining a good strategy for the parallelisation of an unstructured mesh based solver is a challenge, particularly when one aims at reaching a high level of performance while maintaining portability of the source code between scalar, vector and parallel machines. The parallelisation strategy adopted in this study combines mesh partitioning techniques and a message-passing programming model. The targeted parallel architectures are of MIMD type.     

Combinatorial Bidirectional Path‐Tracing for Efficient Hybrid CPU/GPU Rendering

This paper presents a reformulation of bidirectional path‐tracing that adequately divides the algorithm into processes efficiently executed in parallel on both the CPU and the GPU. We thus benefit from high‐level optimization techniques such as double buffering, batch processing, and asyncronous execution, as well as from the exploitation of most of the CPU, GPU, and memory bus capabilities. Our approach, while avoiding pure GPU implementation limitations (such as limited complexity of shaders, light or camera models, and processed scene data sets), is more than ten times faster than standard bidirectional path‐tracing implementations, leading to performance suitable for production‐oriented rendering engines.     

Practical CFD Simulations on Programmable Graphics Hardware using SMAC†

The explosive growth in integration technology and the parallel nature of rasterization‐based graphics APIs (Application Programming Interface) changed the panorama of consumer‐level graphics: today, GPUs (Graphics Processing Units) are cheap, fast and ubiquitous. We show how to harness the computational power of GPUs and solve the incompressible Navier‐Stokes fluid equations significantly faster (more than one order of magnitude in average) than on CPU solvers of comparable cost. While past approaches typically used Stam's implicit solver, we use a variation of SMAC (Simplified Marker and Cell). SMAC is widely used in engineering applications, where experimental reproducibility is essential. Thus, we show that the GPU is a viable and affordable processor for scientific applications. Our solver works with general rectangular domains (possibly with obstacles), implements a variety of boundary conditions and incorporates energy transport through the traditional Boussinesq approximation. Finally, we discuss the implications of our solver in light of future GPU features, and possible extensions such as three‐dimensional domains and free‐boundary problems.     

A review and comparative study of upwind biased schemes for compressible flow computation. Part III: Multidimensional extension on unstructured grids

Summary The edge based Galerkin finite element formulation is used as the basic building block for the construction of multidimensional generalizations, on unstructured grids, of several higher order upwind biased procedures originally designed for the solution of the 1D compressible Euler system of equations. The use of a central type discretization for the viscous flux terms enables the simulation of multidimensional flows governed by the laminar compressible Navier Stokes equations. Numerical issues related to the development and implementation of multidimensional solution algorithms are considered. A number of inviscid and viscous flow simulations, in different flow regimes, are analyzed to enable the reader to assess the performance of the surveyed formulations.     

异构集群上的宏基因组聚类优化

宏基因组基因聚类是筛选致病基因的新型方法,其依赖于海量的测序数据、有效的聚类算法以及高效的计算机来实现。相关系数矩阵的计算是进行聚类前必须完成的操作,占总计算量的比重较大。以某基因库为例,包含1300个样本、每样本百万基因的数据,单线程运行需要27年。充分发挥多核CPU的潜力,利用GPU加速卡强大的计算能力,将程序扩展到多节点集群上运行,是重要而迫切的工作。在仔细分析算法的基础上,首先针对单CPU节点和单GPU卡做了高效实现,获得了接近理想的加速比;然后利用缓存优化进一步提升性能;最后使用负载均衡方法在MPI线程间分发计算任务,实现了良好的扩展。相比未优化的单线程程序,16节点CPU获得了238.8倍的加速,6 块GPU卡获得了263.8倍的加速。