Graphics Processing Unit Card Research Articles

GPU-enabled back-propagation artificial neural network for digit recognition in parallel

In this paper, we show that the GPU (graphics processing unit) can be used not only for processing graphics, but also for high speed computing. We provide a comparison between the times taken on the CPU and GPU to perform the training and testing of a back-propagation artificial neural network. We implemented two neural networks for recognizing handwritten digits; one consists of serial code executed on the CPU, while the other is a GPU-based version of the same system which executes in parallel. As an experiment for performance evaluation, a system for neural network training on the GPU is developed to reduce training time. The programming environment that the system is based on is CUDA which stands for compute unified device architecture, which allows a programmer to write code that will run on an NVIDIA GPU card. Our results over an experiment of digital image recognition using neural network confirm the speed-up advantages by tapping on the resources of GPU. Our proposed model has an advantage of simplicity, while it shows on par performance with the state-of-the-arts algorithms.

GPU Compute Unified Device Architecture (CUDA)-based Parallelization of the RRTMG Shortwave Rapid Radiative Transfer Model

Radiative transfer of electromagnetic radiation through a planetary atmosphere is computed using an atmospheric radiative transfer model (RTM). One RTM is the rapid RTM (RRTM), which calculates both longwave and shortwave atmospheric radiative fluxes and heating rates. Broadband radiative transfer code for general circulation model (GCM) applications, rapid RTM for global (RRTMG), is based on the single-column reference code, RRTM. The focus of this paper is on the RRTMG shortwave (RRTMG_SW) model. Due to its accuracy, RRTMG_SW has been implemented operationally in many weather forecast and climate models. In this paper, we examine the feasibility of using graphics processing units (GPUs) to accelerate the RRTMG_SW for a massive amount of atmospheric profiles. In recent years, GPUs have emerged as a low-cost, low-power, and a very high-performance alternative to conventional central processing units (CPUs). GPUs can provide a substantial improvement in RRTMG speed by supporting the parallel computation of large numbers of independent radiative calculations in separate atmospheric profiles. A GPU-compatible version of RRTMG was implemented and thorough testing was performed to ensure that the original level of accuracy is retained. Our results show that GPUs can provide significant speedup over conventional CPUs. In particular, Nvidia’s Tesla K40 GPU card can provide a speedup of ${{202}} \times $ compared to its single-threaded Fortran counterpart running on Intel Xeon E5-2603 CPU, whereas the speedup for four CPU cores, on one CPU socket, with respect to 1 CPU core is ${{5}}.{{6}} \times $ .

Performance analysis of parallel gravitational N-body codes on large GPU clusters

We compare the performance of two very different parallel gravitational N-body codes for astrophysical simulations on large Graphics Processing Unit (GPU) clusters, both of which are pioneers in their own fields as well as on certain mutual scales - NBODY6++ and Bonsai. We carry out benchmarks of the two codes by analyzing their performance, accuracy and efficiency through the modeling of structure decomposition and timing measurements. We find that both codes are heavily optimized to leverage the computational potential of GPUs as their performance has approached half of the maximum single precision performance of the underlying GPU cards. With such performance we predict that a speed-up of 200 – 300 can be achieved when up to 1k processors and GPUs are employed simultaneously. We discuss the quantitative information about comparisons of the two codes, finding that in the same cases Bonsai adopts larger time steps as well as larger relative energy errors than NBODY6++, typically ranging from 10 – 50 times larger, depending on the chosen parameters of the codes. Although the two codes are built for different astrophysical applications, in specified conditions they may overlap in performance at certain physical scales, thus allowing the user to choose either one by fine-tuning parameters accordingly.

A GPU OpenCL based cross-platform Monte Carlo dose calculation engine (goMC)

Monte Carlo (MC) simulation has been recognized as the most accurate dose calculation method for radiotherapy. However, the extremely long computation time impedes its clinical application. Recently, a lot of effort has been made to realize fast MC dose calculation on graphic processing units (GPUs). However, most of the GPU-based MC dose engines have been developed under NVidia’s CUDA environment. This limits the code portability to other platforms, hindering the introduction of GPU-based MC simulations to clinical practice. The objective of this paper is to develop a GPU OpenCL based cross-platform MC dose engine named goMC with coupled photon–electron simulation for external photon and electron radiotherapy in the MeV energy range. Compared to our previously developed GPU-based MC code named gDPM (Jia et al Phys. Med. Biol. 57 7783–97), goMC has two major differences. First, it was developed under the OpenCL environment for high code portability and hence could be run not only on different GPU cards but also on CPU platforms. Second, we adopted the electron transport model used in EGSnrc MC package and PENELOPE’s random hinge method in our new dose engine, instead of the dose planning method employed in gDPM. Dose distributions were calculated for a 15 MeV electron beam and a 6 MV photon beam in a homogenous water phantom, a water-bone-lung-water slab phantom and a half-slab phantom. Satisfactory agreement between the two MC dose engines goMC and gDPM was observed in all cases. The average dose differences in the regions that received a dose higher than 10% of the maximum dose were 0.48–0.53% for the electron beam cases and 0.15–0.17% for the photon beam cases. In terms of efficiency, goMC was ~4–16% slower than gDPM when running on the same NVidia TITAN card for all the cases we tested, due to both the different electron transport models and the different development environments. The code portability of our new dose engine goMC was validated by successfully running it on a variety of different computing devices including an NVidia GPU card, two AMD GPU cards and an Intel CPU processor. Computational efficiency among these platforms was compared.

A GPU Implementation of an Explicit Compact FDTD Algorithm with a Digital Impedance Filter for Room Acoustics Applications

In recent years, computational engineering has undergone great changes due to the development of the graphics processing unit (GPU) technology. For example, in room acoustics, the wave-based methods, that formerly were considered too expensive for 3-D impulse response simulations, are now chosen to exploit the parallel nature of GPU devices considerably reducing the execution time of the simulations. There exist contributions related to this topic that have explored the performance of different GPU algorithms; however, the computational analysis of a general explicit model that incorporates algorithms with different neighboring orders and a general frequency dependent impedance boundary model has not been properly developed. In this paper, we present a GPU implementation of a complete room acoustic model based on a family of explicit finite-difference time-domain (FDTD) algorithms. We first develop a strategy for implementing a frequency independent (FI) impedance model which is free from thread divergences and then, we extend the model adding a digital impedance filter (DIF) boundary subroutine able to compute the acoustic pressure of different nodes such as corners or edges without an additional performance penalty. Both implementations are validated and deeply analyzed by performing different 3-D numerical experiments. Finally, we define a performance metric which is able to objectively measure the computing throughput of a FDTD implementation using a simple number. The robustness of this metric allows us to compare algorithms even if these have been run in different GPU cards or have been formulated with other explicit models.

Multi-phase SPH modelling of violent hydrodynamics on GPUs

This paper presents the acceleration of multi-phase smoothed particle hydrodynamics (SPH) using a graphics processing unit (GPU) enabling large numbers of particles (10–20 million) to be simulated on just a single GPU card. With novel hardware architectures such as a GPU, the optimum approach to implement a multi-phase scheme presents some new challenges. Many more particles must be included in the calculation and there are very different speeds of sound in each phase with the largest speed of sound determining the time step. This requires efficient computation. To take full advantage of the hardware acceleration provided by a single GPU for a multi-phase simulation, four different algorithms are investigated: conditional statements, binary operators, separate particle lists and an intermediate global function. Runtime results show that the optimum approach needs to employ separate cell and neighbour lists for each phase. The profiler shows that this approach leads to a reduction in both memory transactions and arithmetic operations giving significant runtime gains. The four different algorithms are compared to the efficiency of the optimised single-phase GPU code, DualSPHysics, for 2-D and 3-D simulations which indicate that the multi-phase functionality has a significant computational overhead. A comparison with an optimised CPU code shows a speed up of an order of magnitude over an OpenMP simulation with 8 threads and two orders of magnitude over a single thread simulation. A demonstration of the multi-phase SPH GPU code is provided by a 3-D dam break case impacting an obstacle. This shows better agreement with experimental results than an equivalent single-phase code. The multi-phase GPU code enables a convergence study to be undertaken on a single GPU with a large number of particles that otherwise would have required large high performance computing resources.

134 MR Augmented Cardiopulmonary Exercise Testing – a Novel Way of Assessing Cardiovascular Function

Introduction Reduced exercise capacity is a common feature of many cardiovascular diseases. Quantitative assessment of exercise capacity is usually achieved by measuring peak oxygen consumption (VO2). However, measuring peak VO2 alone neglects the different components of reduced exercise capacity: namely reduced cardiac output (CO) and oxygen extraction (ΔcO2). A better approach would be to simultaneously measure VO2 and CO and then calculate ΔcO2. This could be achieved using MR augmented cardiopulmonary exercise testing (MR-CPET). The aims of this study were to demonstrate: 1) MR-CPET is feasible and well tolerated, 2) peak VO2 in the MR scanner correlates with conventional peak VO2 and 3) variation in peak VO2 is related to both peak CO and peak oxygen extraction (ΔcO2) as calculated by the Fick equation. Method 17 healthy volunteers (21–55 years) underwent MR-CPET. Exercise was performed on an MR-compatible ergometer (Lode, Groningen, The Netherlands) and VO2 was assessed using a commercial respiratory gas analyser (Ultima, MedGraphics, St. Paul, USA) with a modified sampling tube that was MR compatible. Set-up for MR-CPET is shown in [Figure 1][1]. Aortic flow was continuously measured using a previously validated UNFOLD-SENSE spiral PCMR sequence. Images were reconstructed using a graphical processing unit card and analysed using an in-house plug-in for OsiriX software. Conventional CPET was also performed within 2 weeks of MR-CPET. For both tests, participants were asked to rate i) concern ii) comfort and iii) perceived helplessness. ![Abstract 134 Figure 1][2] Abstract 134 Figure 1 Set-up for MR-CPET: a) subject in exercise position on MR compatible ergometer b) subject with facemask attached to MR compatible umbilicus passing through the wave-guide Results 15 out of 17 volunteers completed exercise; exclusions were due to claustrophobia (n = 1) and inability to master exercise technique (n = 1). Reported concern and discomfort was higher with MR-CPET, although still within acceptable limits. Peak VO2, peak VCO2 and VE showed strong correlation between conventional CPET and MR-CPET: VO2 peak (r = 0.94, p < 0.001); VCO2 (r = 0.87, p < 0.001); VE (r = 0.88, p < 0.001). Resting and peak values VO2, CO, HR, SV and ΔcO2 are shown in [Table 1][3]. Multiple linear regression analysis demonstrated that both peak CO and ΔcO2 were independent predictors of peak VO2 measured during MR-CPET (beta = 0.73 and 0.38 respectively, p < 0.001) and conventional CPET (beta = 0.78 and 0.28 respectively, p < 0.001). ![Abstract 134 Table 1][2] Abstract 134 Table 1 Values at rest and Peak VO2 obtained at MR-CPET Conclusion MR-CPET is feasible, well tolerated and demonstrates physiology not apparent with conventional CPET. In this study, we have shown that MR-CPET allows assessment of the differing contributions of CO and ΔcO2 to variation in peak VO2. We believe that will be useful in understanding the origin of reduced exercise capacity in cardiac disease. [1]: #F1 [2]: pending:yes [3]: #F2

A GPU‐based preconditioned Newton‐Krylov solver for flexible multibody dynamics

SummaryThis paper describes an approach to numerically approximate the time evolution of multibody systems with flexible (compliant) components. Its salient attribute is that at each time step, both the formulation of the system equations of motion and their numerical solution are carried out using parallel computing on graphics processing unit cards. The equations of motion are obtained using the absolute nodal coordinate formulation, yet any other multibody dynamics formalism would fit equally well the overall solution strategy outlined herein. The implicit numerical integration method adopted relies on a Newton–Krylov methodology and a parallel direct sparse solver to precondition the underlying linear system. The proposed approach, implemented in a software infrastructure available under an open‐source BSD‐3 license, leads to improvements in overall simulation times of up to one order of magnitude when compared with matrix‐free parallel solution approaches that do not use preconditioning. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Directive-based GPU programming for computational fluid dynamics

Directive-based programming of graphics processing units (GPUs) has recently appeared as a viable alternative to using specialized low-level languages such as CUDA C and OpenCL for general-purpose GPU programming. This technique, which uses “directive” or “pragma” statements to annotate source codes written in traditional high-level languages, is designed to permit a unified code base to serve multiple computational platforms. In this work we analyze the popular OpenACC programming standard, as implemented by the PGI compiler suite, in order to evaluate its utility and performance potential in computational fluid dynamics (CFD) applications. We examine the process of applying the OpenACC Fortran API to a test CFD code that serves as a proxy for a full-scale research code developed at Virginia Tech; this test code is used to asses the performance improvements attainable for our CFD algorithm on common GPU platforms, as well as to determine the modifications that must be made to the original source code in order to run efficiently on the GPU. Performance is measured on several recent GPU architectures from NVIDIA and AMD (using both double and single precision arithmetic) and the accelerator code is benchmarked against a multithreaded CPU version constructed from the same Fortran source code using OpenMP directives. A single NVIDIA Kepler GPU card is found to perform approximately 20× faster than a single CPU core and more than 2× faster than a 16-core Xeon server. An analysis of optimization techniques for OpenACC reveals cases in which manual intervention by the programmer can improve accelerator performance by up to 30% over the default compiler heuristics, although these optimizations are relevant only for specific platforms. Additionally, the use of multiple accelerators with OpenACC is investigated, including an experimental high-level interface for multi-GPU programming that automates scheduling tasks across multiple devices. While the overall performance of the OpenACC code is found to be satisfactory, we also observe some significant limitations and restrictions imposed by the OpenACC API regarding certain useful features of modern Fortran (2003/8); these are sufficient for us to conclude that it would not be practical to apply OpenACC to our full research code at this time due to the amount of refactoring required.

OpenACC acceleration of an unstructured CFD solver based on a reconstructed discontinuous Galerkin method for compressible flows

SummaryAn OpenACC directive‐based graphics processing unit (GPU) parallel scheme is presented for solving the compressible Navier–Stokes equations on 3D hybrid unstructured grids with a third‐order reconstructed discontinuous Galerkin method. The developed scheme requires the minimum code intrusion and algorithm alteration for upgrading a legacy solver with the GPU computing capability at very little extra effort in programming, which leads to a unified and portable code development strategy. A face coloring algorithm is adopted to eliminate the memory contention because of the threading of internal and boundary face integrals. A number of flow problems are presented to verify the implementation of the developed scheme. Timing measurements were obtained by running the resulting GPU code on one Nvidia Tesla K20c GPU card (Nvidia Corporation, Santa Clara, CA, USA) and compared with those obtained by running the equivalent Message Passing Interface (MPI) parallel CPU code on a compute node (consisting of two AMD Opteron 6128 eight‐core CPUs (Advanced Micro Devices, Inc., Sunnyvale, CA, USA)). Speedup factors of up to 24× and 1.6× for the GPU code were achieved with respect to one and 16 CPU cores, respectively. The numerical results indicate that this OpenACC‐based parallel scheme is an effective and extensible approach to port unstructured high‐order CFD solvers to GPU computing. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Implementation of Multi-GPU Based Lattice Boltzmann Method for Flow Through Porous Media

AbstractThe lattice Boltzmann method (LBM) can gain a great amount of performance benefit by taking advantage of graphics processing unit (GPU) computing, and thus, the GPU, or multi-GPU based LBM can be considered as a promising and competent candidate in the study of large-scale fluid flows. However, the multi-GPU based lattice Boltzmann algorithm has not been studied extensively, especially for simulations of flow in complex geometries. In this paper, through coupling with the message passing interface (MPI) technique, we present an implementation of multi-GPU based LBM for fluid flow through porous media as well as some optimization strategies based on the data structure and layout, which can apparently reduce memory access and completely hide the communication time consumption. Then the performance of the algorithm is tested on a one-node cluster equipped with four Tesla C1060 GPU cards where up to 1732 MFLUPS is achieved for the Poiseuille flow and a nearly linear speedup with the number of GPUs is also observed.

Towards the clinical implementation of iterative low-dose cone-beam CT reconstruction in image-guided radiation therapy: cone/ring artifact correction and multiple GPU implementation.

Compressed sensing (CS)-based iterative reconstruction (IR) techniques are able to reconstruct cone-beam CT (CBCT) images from undersampled noisy data, allowing for imaging dose reduction. However, there are a few practical concerns preventing the clinical implementation of these techniques. On the image quality side, data truncation along the superior-inferior direction under the cone-beam geometry produces severe cone artifacts in the reconstructed images. Ring artifacts are also seen in the half-fan scan mode. On the reconstruction efficiency side, the long computation time hinders clinical use in image-guided radiation therapy (IGRT). Image quality improvement methods are proposed to mitigate the cone and ring image artifacts in IR. The basic idea is to use weighting factors in the IR data fidelity term to improve projection data consistency with the reconstructed volume. In order to improve the computational efficiency, a multiple graphics processing units (GPUs)-based CS-IR system was developed. The parallelization scheme, detailed analyses of computation time at each step, their relationship with image resolution, and the acceleration factors were studied. The whole system was evaluated in various phantom and patient cases. Ring artifacts can be mitigated by properly designing a weighting factor as a function of the spatial location on the detector. As for the cone artifact, without applying a correction method, it contaminated 13 out of 80 slices in a head-neck case (full-fan). Contamination was even more severe in a pelvis case under half-fan mode, where 36 out of 80 slices were affected, leading to poorer soft tissue delineation and reduced superior-inferior coverage. The proposed method effectively corrects those contaminated slices with mean intensity differences compared to FDK results decreasing from ∼497 and ∼293 HU to ∼39 and ∼27 HU for the full-fan and half-fan cases, respectively. In terms of efficiency boost, an overall 3.1 × speedup factor has been achieved with four GPU cards compared to a single GPU-based reconstruction. The total computation time is ∼30 s for typical clinical cases. The authors have developed a low-dose CBCT IR system for IGRT. By incorporating data consistency-based weighting factors in the IR model, cone/ring artifacts can be mitigated. A boost in computational efficiency is achieved by multi-GPU implementation.

Parallel computing on graphics processing units and heterogeneous platforms

Parallel computing on graphics processing units and heterogeneous platforms

An optimized GPU implementation of a 2D free surface simulation model on unstructured meshes

This work is related with the implementation of a finite volume method to solve the 2D Shallow Water Equations on Graphic Processing Units (GPU). The strategy is fully oriented to work efficiently with unstructured meshes which are widely used in many fields of Engineering. Due to the design of the GPU cards, structured meshes are better suited to work with than unstructured meshes. In order to overcome this situation, some strategies are proposed and analyzed in terms of computational gain, by means of introducing certain ordering on the unstructured meshes. The necessity of performing the simulations using unstructured instead of structured meshes is also justified by means of some test cases with analytical solution.

High performance MRI simulations of motion on multi-GPU systems.

BackgroundMRI physics simulators have been developed in the past for optimizing imaging protocols and for training purposes. However, these simulators have only addressed motion within a limited scope. The purpose of this study was the incorporation of realistic motion, such as cardiac motion, respiratory motion and flow, within MRI simulations in a high performance multi-GPU environment.MethodsThree different motion models were introduced in the Magnetic Resonance Imaging SIMULator (MRISIMUL) of this study: cardiac motion, respiratory motion and flow. Simulation of a simple Gradient Echo pulse sequence and a CINE pulse sequence on the corresponding anatomical model was performed. Myocardial tagging was also investigated. In pulse sequence design, software crushers were introduced to accommodate the long execution times in order to avoid spurious echoes formation.The displacement of the anatomical model isochromats was calculated within the Graphics Processing Unit (GPU) kernel for every timestep of the pulse sequence. Experiments that would allow simulation of custom anatomical and motion models were also performed. Last, simulations of motion with MRISIMUL on single-node and multi-node multi-GPU systems were examined.ResultsGradient Echo and CINE images of the three motion models were produced and motion-related artifacts were demonstrated. The temporal evolution of the contractility of the heart was presented through the application of myocardial tagging. Better simulation performance and image quality were presented through the introduction of software crushers without the need to further increase the computational load and GPU resources. Last, MRISIMUL demonstrated an almost linear scalable performance with the increasing number of available GPU cards, in both single-node and multi-node multi-GPU computer systems.ConclusionsMRISIMUL is the first MR physics simulator to have implemented motion with a 3D large computational load on a single computer multi-GPU configuration. The incorporation of realistic motion models, such as cardiac motion, respiratory motion and flow may benefit the design and optimization of existing or new MR pulse sequences, protocols and algorithms, which examine motion related MR applications.

Ray-Tracing of GNSS Signal Through the Atmosphere Powered by CUDA, HMPP and GPUs Technologies

The ray-tracing of signals emitted by the Global Navigation Satellite Systems (GNSS) is implemented on Graphics Processing Units (GPU) by two parallel programming techniques: the “Compute Unified Device Architecture” (CUDA) C-language and the directives for “Hybrid Multicore Parallel Programming” (HMPP) developed by CAPS Entreprise. The signal propagation is obtained by the numerical integration of the differential system derived from the eikonal equation by the Runge-Kutta method. The computation of atmospheric delays on GPU has to preserve the millimeter accuracy using the double precision arithmetic. Four versions describe how the ray-tracing of 8,100 rays was optimized for the Fermi architecture. As referring to a single-core single-threaded CPU version, accelerations ranging from 20 to 50 times are progressively obtained when the software enhancements gradually harness the hardware capabilities. The versions HMPP and CUDA provide exactly the same accelerations. HMPP further provides an easy implementation for multiple kind of GPU cards. A speed-up of 75 times versus the CPU version is finally reached when the ray-tracing algorithm is applied to 130,000 rays.

GWEGA: GPU-accelerated WEGA for molecular superposition and shape comparison.

Virtual screening of a large chemical library for drug lead identification requires searching/superimposing a large number of three-dimensional (3D) chemical structures. This article reports a graphic processing unit (GPU)-accelerated weighted Gaussian algorithm (gWEGA) that expedites shape or shape-feature similarity score-based virtual screening. With 86 GPU nodes (each node has one GPU card), gWEGA can screen 110 million conformations derived from an entire ZINC drug-like database with diverse antidiabetic agents as query structures within 2 s (i.e., screening more than 55 million conformations per second). The rapid screening speed was accomplished through the massive parallelization on multiple GPU nodes and rapid prescreening of 3D structures (based on their shape descriptors and pharmacophore feature compositions).

Rapid hologram generation utilizing layer-based approach and graphic rendering for realistic three-dimensional image reconstruction by angular tiling

An approach of rapid hologram generation for the realistic three-dimensional (3-D) image reconstruction based on the angular tiling concept is proposed, using a new graphic rendering approach integrated with a previously developed layer-based method for hologram calculation. A 3-D object is simplified as layered cross-sectional images perpendicular to a chosen viewing direction, and our graphics rendering approach allows the incorporation of clear depth cues, occlusion, and shading in the generated holograms for angular tiling. The combination of these techniques together with parallel computing reduces the computation time of a single-view hologram for a 3-D image of extended graphics array resolution to 176 ms using a single consumer graphics processing unit card.

Multimodality imaging and state-of-art GPU technology in discriminating benign from malignant breast lesions on real time decision support system

The aim of this study was to design a pattern recognition system for assisting the diagnosis of breast lesions, using image information from Ultrasound (US) and Digital Mammography (DM) imaging modalities. State-of-art computer technology was employed based on commercial Graphics Processing Unit (GPU) cards and parallel programming. An experienced radiologist outlined breast lesions on both US and DM images from 59 patients employing a custom designed computer software application. Textural features were extracted from each lesion and were used to design the pattern recognition system. Several classifiers were tested for highest performance in discriminating benign from malignant lesions. Classifiers were also combined into ensemble schemes for further improvement of the system's classification accuracy. Following the pattern recognition system optimization, the final system was designed employing the Probabilistic Neural Network classifier (PNN) on the GPU card (GeForce 580GTX) using CUDA programming framework and C++ programming language. The use of such state-of-art technology renders the system capable of redesigning itself on site once additional verified US and DM data are collected. Mixture of US and DM features optimized performance with over 90% accuracy in correctly classifying the lesions.

GPU implementation of a parallel two‐list algorithm for the subset‐sum problem

SUMMARYThe subset‐sum problem is a well‐known non‐deterministic polynomial‐time complete (NP‐complete) decision problem. This paper proposes a novel and efficient implementation of a parallel two‐list algorithm for solving the problem on a graphics processing unit (GPU) using Compute Unified Device Architecture (CUDA). The algorithm is composed of a generation stage, a pruning stage, and a search stage. It is not easy to effectively implement the three stages of the algorithm on a GPU. Ways to achieve better performance, reasonable task distribution between CPU and GPU, effective GPU memory management, and CPU–GPU communication cost minimization are discussed. The generation stage of the algorithm adopts a typical recursive divide‐and‐conquer strategy. Because recursion cannot be well supported by current GPUs with compute capability less than 3.5, a new vector‐based iterative implementation mechanism is designed to replace the explicit recursion. Furthermore, to optimize the performance of the GPU implementation, this paper improves the three stages of the algorithm. The experimental results show that the GPU implementation has much better performance than the CPU implementation and can achieve high speedup on different GPU cards. The experimental results also illustrate that the improved algorithm can bring significant performance benefits for the GPU implementation. Copyright © 2014 John Wiley &amp; Sons, Ltd.