Fukang Yin

Performance evaluation of hybrid programming patterns for large CPU/GPU heterogeneous clusters

Abstract The CPU/GPU heterogeneous clusters are important platforms for high performance computing applications. However, there are many challenges for efficiently performing the scientific and engineering legacy code on these heterogeneous systems. In this paper, we endeavor to address the programming-model issue by combining the existing models (i.e., MPI, OpenMP and CUDA). First, two hybrid programming patterns are presented, namely the MPI + CUDA and MPI + OpenMP/CUDA . Second, three kernels (i.e., EP, CG and MG) of the NAS parallel benchmarks (NPBs), which are abstracted from many legacy computational fluid dynamics applications, are implemented with the above two patterns. Third, these hybrid implementations are executed on the TianHe-1A supercomputer, and the corresponding experimental results show that significant performance improvement can be achieved with the above patterns. Finally, a detailed performance analysis about the two hybrid patterns is performed and some guidelines for porting the legacy code onto large-scale heterogeneous CPU/GPU clusters are also given.

A Coupled Method of Laplace Transform and Legendre Wavelets for Lane-Emden-Type Differential Equations

A coupled method of Laplace transform and Legendre wavelets is presented to obtain exact solutions of Lane-Emden-type equations. By employing properties of Laplace transform, a new operator is first introduced and then its Legendre wavelets operational matrix is derived to convert the Lane-Emden equations into a system of algebraic equations. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The results show that the proposed method is very effective and easy to implement.

Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions

A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

Fractional Variational Iteration Method versus Adomian’s Decomposition Method in Some Fractional Partial Differential Equations

A comparative study is presented about the Adomian’s decomposition method (ADM), variational iteration method (VIM), and fractional variational iteration method (FVIM) in dealing with fractional partial differential equations (FPDEs). The study outlines the significant features of the ADM and FVIM methods. It is found that FVIM is identical to ADM in certain scenarios. Numerical results from three examples demonstrate that FVIM has similar efficiency, convenience, and accuracy like ADM. Moreover, the approximate series are also part of the exact solution while not requiring the evaluation of the Adomian’s polynomials.

Couple of the Variational Iteration Method and Fractional-Order Legendre Functions Method for Fractional Differential Equations

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid “noise terms” is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique.

Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations

This paper develops a modified variational iteration method coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs). The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.

A General Iteration Formula of VIM for Fractional Heat- and Wave-Like Equations

A general iteration formula of variational iteration method (VIM) for fractional heat- and wave-like equations with variable coefficients is derived. Compared with previous work, the Lagrange multiplier of the method is identified in a more accurate way by employing Laplace’s transform of fractional order. The fractional derivative is considered in Jumarie’s sense. The results are more accurate than those obtained by classical VIM and the same as ADM. It is shown that the proposed iteration formula is efficient and simple.

Communication and Memory Access Latency Characteristics of CPU/GPU Heterogeneous Cluster

CPU/GPU heterogeneous computing embraces a rapid development in recent years. Considering that there are huge differences between CPU and GPU, CPU/GPU heterogeneous computing still faces many challenges. Therefore, collaborative features of fine-grained and coarse-grained parallelism are necessary to be explored in software designing. This paper takes a comprehensive study both on the CPU/GPU heterogeneous clusters hardware and program execution characteristics. After performing OSU Micro-Benchmark (OMB) test on the TH-1A system, we got the communication bandwidth of inter nodes, intra nodes and memory access latency results between CPU and GPU. Finally, we designed experiments to complete IS and FT benchmarks of NPB suite on TH-1A. The results showed that we can get desired results on CPU/GPU heterogeneous cluster when the problem was computation intensive and with relatively large problem scale. The results also provide practical principles for designing parallel computing model of CPU/GPU heterogeneous cluster in our future work.

GPU Computing Using Concurrent Kernels: A Case Study

With the rapid evolution of processor architectures, more attention has been paid to the hardware-oriented numeric applications. Based on the newly released Fermi architecture, we investigate the approach to accelerate high performance computing (HPC) applications with concurrent kernels. We concentrated on two performance factors, namely the launching order of concurrent kernels and the kernel granularity. Extensive experiments show that the launching order of concurrent kernels can hardly affect application performance. Particularly, we identify the heuristics of kernel granularity that may result in the best performance, i.e. the occupancy of each kernel should be in the interval [30%, 50%].

Notes and correspondence on ensemble-based three-dimensional variational filters

Several ensemble-based three-dimensional variational (3D-Var) filters are compared. These schemes replace the static background error covariance of the traditional 3D-Var with the ensemble forecast error covariance, but generate analysis ensemble anomalies (perturbations) in different ways. However, it is demonstrated in this paper that they are all theoretically equivalent to the ensemble transformation Kalman filter (ETKF). Furthermore, a new method named EnPSAS is presented. The analysis shows that EnPSAS has a small condition number and can apply covariance localization more easily than other ensemble-based 3D-Var methods.