Iterative Solver for Linear System Obtained by Edge Element: Variable Preconditioned Method With Mixed Precision on GPU

The variable preconditioned (VP) Krylov subspace method on multi Graphics Processing Unit (GPU) is numerically investigated. Besides, the linear system obtained by finite element method with an edge element is adopted for the problem. The results of computations show that VP conjugate gradient method on multi GPU demonstrated significant achievement than that of CPU. Especially, VP conjugate gradient method on multi GPU is 4.35 times faster than that of CPU. However, transmission rate between the PC using Gigabit Ethernet is the bottleneck of the performance.

The variable preconditioned (VP) Krylov subspace method with communication avoiding (CA) technique is adopted for the solver of a linear system obtained from electromagnetic analysis, and the numerical features are investigated. A massive communication time between processing units (PUs) or graphics PU/many integrated core is a big issue for parallelization efficiency that needs to be solved. Although k-skip Krylov subspace method is one solution for the issue, convergence property of the method is becoming worse. For this reason, we adopt the k-skip Krylov subspace method as inner-loop solver of the VP Krylov subspace method. The result of computation shows that the VP Krylov subspace method with CA technique is very effective for the linear system obtained from electromagnetic analysis.

Most Computational Fluid Dynamics (CFD) applications solve the Naiver-Stokes equations using various discretization methods like Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), etc. The most compute-intensive process in CFD algorithms is the pressure Poisson solver. Solving the pressure Poisson equation requires the solution of a set of linear simultaneous equations. As the problem size/complexity increases, so does the effort and time invested in solving the equations. This study presents fast and robust iterative solvers and the acceleration of the same using the Graphics Processing Unit (GPU) architecture. Serial versions of the codes are implemented in the C language and the parallelization on GPU is achieved using Compute Unified Device Architecture (CUDA). For this study, a two-dimensional heat conduction problem is considered. FDM is used for discretization. Iterative solvers Conjugate Gradient and Multi Grid methods have been implemented in serial and parallel. The performance enhancement with single GPU and CPU system over a single CPU in terms of computing time is reported. It is seen that multigrid algorithms are superior in convergence and nearly 13 times speed-up has been obtained through GPU acceleration using CUDA, for a grid size of 2048 X 2048. The tests have been done on a cluster with Intel(R) Xeon(R) CPU E5-2670 CPU and NVIDIA K10 GPU.

During the last decades, explicit approximate inverse preconditioning methods have been used for efficiently solving sparse linear systems on multiprocessor systems. The effectiveness of explicit approximate inverse preconditioning schemes relies on the use of efficient preconditioners that are close approximants to the coefficient matrix and are fast to compute in parallel. A new parallel computational technique is proposed for the parallelization of the explicit preconditioned conjugate gradient type method on a Graphics Processing Unit (GPU). The proposed parallel methods have been implemented using Compute Unified Device Architecture (CUDA) developed by NVIDIA. The inherently parallel operations between vectors and matrices involved in the explicit preconditioned biconjugate conjugate gradient type schemes exhibit significant amounts of loop-level parallelism because of the matrix–vector and the vector–vector products that can lead to high performance gain on the GPU systems, specifically designed for such computations. Finally, numerical results for the performance of the explicit preconditioned biconjugate conjugate gradient type method for solving characteristic two dimensional boundary value problems, using the finite difference method, on a massive multiprocessor interface on a GPU are presented. The CUDA implementation issues of the proposed method are also discussed.

Summary form only given. Numerical analysis of electromagnetic scattering has been widely studied in the past. The method of moments (MoM) is suitable to simulate scattering from an object in an open region. In order to solve problems with complicated geometries and materials, the finite element method (FEM) is preferred. However, when solving unbounded problems with FEM, the use of absorbing boundary conditions or perfectly matched layers will result in truncation errors. To combine the advantages of both MoM and FEM, the finite element-boundary integral (FE-BI) method was developed and widely used (J. M. Jin, The Finite Element Method in Electromagnetics, Second Edition. New York: Wiley, 2002). This algorithm is not only able to deal with highly complicated geometry and material, but also able to eliminate the truncation error. However, since the MoM has O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2)</sup> computational and storage complexities, the computational cost of the BI-related algorithm is still prohibitively high when solving large-scale scattering problems. To alleviate this problem, the multilevel fast multipole algorithm (MLFMA) has been incorporated into the FE-BI method (J. Liu and J. M. Jin, IEEE Trans. Antennas Propag., vol. 51, pp.1157-1167, Jun. 2003). Another important technique to accelerate the computation is to resort to parallel computers. Recently, the graphics processing unit (GPU), as one of many-core computing systems, has received intensive attention from the scientific computing community due to its low price, a massive number of processors, and the resulting high computational efficiency. In this work, we discuss the GPU implementation of the FE-BI algorithm for solving electromagnetic scattering problems. First, the assembly of the BI-related matrix is parallelized on GPU by NVIDIA's compute unified device architecture (CUDA). A massive number of threads are assigned to carry out the assembly, and each thread computes one element in the BI-related matrix. Second, both the direct and iterative solvers are implemented on GPU. For the direct solver, the Intel's math kernel library (MKL) is used for LU decomposition, and the CUSPARSE (a set of basic linear algebra subroutines used for handling sparse matrices) is employed to speedup forward and backward substitutions. For the iterative solver, the high performance CUDA Basic Linear Algebra Subprograms (BLAS) is utilized for the matrix-vector multiplication and the vector-vector multiplication. Extensive numerical experiments are conducted to validate the implementation and demonstrate the achievable computational speedup.

A wide class of numerical methods needs to solve a linear system, where the matrix pattern of non-zero coefficients can be arbitrary. These problems can greatly benefit from highly multithreaded computational power and large memory bandwidth available on graphics processor units (GPUs), especially since dedicated general purpose APIs such as close-to-metal (CTM) (AMD–ATI) and compute unified device architecture (CUDA) (NVIDIA) have appeared. CUDA even provides a BLAS implementation, but only for dense matrices (CuBLAS). Other existing linear solvers for the GPU are also limited by their internal matrix representation. This paper describes how to combine recent GPU programming techniques and new GPU dedicated APIs with high performance computing strategies (namely block compressed row storage (BCRS), register blocking and vectorization), to implement a sparse general-purpose linear solver. Our implementation of the Jacobi-preconditioned conjugate gradient algorithm outperforms by up to a factor of 6.0 × leading-edge CPU counterparts, making it attractive for applications which are content with single precision.

Parallelization of the finite-difference time-domain method for room acoustics modelling based on CUDA

For semiempirical electronic structure methods, solving the Roothaan-Hall equations to determine the one-electron density matrix is generally the computational bottleneck. Therefore, alternatives have been proposed to directly solve for the one-electron density matrix without the need to solve for the orbitals first. In this work, we implement an efficient dense linear algebra implementation of Niklasson's density matrix purification schemes using graphics processing units (GPUs). The computational bottleneck in these methods is the matrix-matrix multiplication needed to construct the purification polynomials, which can be accelerated by using GPUs. Of particular interest in this work is the use of consumer-grade GPUs that thrive on algorithms that maximize the amount of single-precision (FP32) operations carried out. Therefore, we present a tailored mixed-precision (MP) scheme to leverage much of the FP32 performance of these GPUs without sacrificing numerical accuracy in the self-consistent field (SCF) calculations. We demonstrate that our MP implementation is faster than LAPACK (Intel oneMKL DSYGVD) and cuSOLVER DSYGVD diagonalization-based density matrix builds for molecules with more than 1000 basis functions in combination with the semiempirical GFN2-xTB method. At the same time, the numerical precision of the energies and gradients is not significantly impacted by the MP scheme compared to a full double-precision (FP64) treatment. This gives access to significant accelerations of semiempirical calculations on commodity computing hardware. Going further, we show that our asynchronous GPU implementation enables running multiple SCFs in parallel on a single GPU, which enables us to leverage our implementation for accelerating state-of-the-art conformational sampling procedures that are based on molecular dynamics and metadynamic simulations.

<strong class="journal-contentHeaderColor">Abstract.</strong> With semiconductor technology gradually approaching its physical and thermal limits, Graphics processing unit (GPU) is becoming an attractive solution in many scientific applications due to their high performance. This paper presents an application of GPU accelerators in air quality model. We endeavor to demonstrate an approach that runs a PPM solver of horizontal advection (HADVPPM) for air quality model CAMx on GPU clusters. Specifically, we first convert the HADVPPM to a new Compute Unified Device Architecture C (CUDA C) code to make it computable on the GPU (GPU-HADVPPM). Then, a series of optimization measures are taken, including reducing the CPU-GPU communication frequency, increasing the size of data computation on GPU, optimizing the GPU memory access, and using thread and block indices in order to improve the overall computing performance of CAMx model coupled with GPU-HADVPPM (named as CAMx-CUDA model). Finally, a heterogeneous, hybrid programming paradigm is presented and utilized with the GPU-HADVPPM on GPU clusters with Massage Passing Interface (MPI) and CUDA. Offline experiment results show that running GPU-HADVPPM on one NVIDIA Tesla K40m and NVIDIA Tesla V100 GPU can achieve up to 845.4x and 1113.6x acceleration. By implementing a series of optimization schemes, the CAMx-CUDA model resulted in a 29.0x and 128.4x improvement in computational efficiency using a GPU accelerator card on a K40m and V100 cluster, respectively. In terms of the single-module computational efficiency of GPU-HADVPPM, it can achieve 1.3x and 19.4x speedup on NVIDIA Tesla K40m GPU and NVIDA Tesla V100 GPU respectively. The multi-GPU acceleration algorithm enables 4.5x speedup with 8 CPU cores and 8 GPU accelerators on V100 cluster.

<strong class="journal-contentHeaderColor">Abstract.</strong> With semiconductor technology gradually approaching its physical and thermal limits, Graphics processing unit (GPU) is becoming an attractive solution in many scientific applications due to their high performance. This paper presents an application of GPU accelerators in air quality model. We endeavor to demonstrate an approach that runs a PPM solver of horizontal advection (HADVPPM) for air quality model CAMx on GPU clusters. Specifically, we first convert the HADVPPM to a new Compute Unified Device Architecture C (CUDA C) code to make it computable on the GPU (GPU-HADVPPM). Then, a series of optimization measures are taken, including reducing the CPU-GPU communication frequency, increasing the size of data computation on GPU, optimizing the GPU memory access, and using thread and block indices in order to improve the overall computing performance of CAMx model coupled with GPU-HADVPPM (named as CAMx-CUDA model). Finally, a heterogeneous, hybrid programming paradigm is presented and utilized with the GPU-HADVPPM on GPU clusters with Massage Passing Interface (MPI) and CUDA. Offline experiment results show that running GPU-HADVPPM on one NVIDIA Tesla K40m and NVIDIA Tesla V100 GPU can achieve up to 845.4x and 1113.6x acceleration. By implementing a series of optimization schemes, the CAMx-CUDA model resulted in a 29.0x and 128.4x improvement in computational efficiency using a GPU accelerator card on a K40m and V100 cluster, respectively. In terms of the single-module computational efficiency of GPU-HADVPPM, it can achieve 1.3x and 19.4x speedup on NVIDIA Tesla K40m GPU and NVIDA Tesla V100 GPU respectively. The multi-GPU acceleration algorithm enables 4.5x speedup with 8 CPU cores and 8 GPU accelerators on V100 cluster.

Turbo Similarity Searching (TSS) is a two phases searching procedure that has been proven by previous works as one of the best searching method on chemical databases. TSS however consumes lots of computation time due to the number of searches and fusion carried out in the second phase of its procedure. Hence, TSS would not be able to cater the increase in the chemical database size due to this limitation. With the emergence of the parallel technology, this research looks into accelerating TSS on the widely-used many-cores i.e. the Graphics Processing Unit (GPU) and multi-cores platform. This would not only solve the computational time issue but also the cost as GPUs can be obtained at a lower cost. Hence, the implementation of TSS will help the medicinal chemist to execute the virtual screening in an accurate and fast manner. This study investigates the best possible method to parallelize TSS via experimentation of three parallel designs; two designs were implemented on GPU platform using the Compute Unified Device Architecture (CUDA) API namely CUDA 1 and CUDA 2 whilst one design was implemented on multi-core platform using OpenMP API. The CUDA 1 design had shown tremendous speedup and low GPU-memory utilization as compared to CUDA 2 design. In general observation, the parallel CUDA 1 was 131 times faster than sequential and 51 times faster than parallel OpenMP. This leads to the conclusion that CUDA 1 design as the best parallel design for TSS.

Based on the principle of QR loop iterations, this paper implements a parallel algorithm based on the hardware of GPU (Graphic Process Unit) by using routines from CUDA (Computer Unified Device Architecture) to find the eigenvalues of general matrices. CPU and GPU computing card form a client-server computing framework. Here, CPU can be regarded as a client and GPU card is considered as a computing server. For the experiment environment, this paper chooses a GPU card with the model of NVIDIA GeForce GTX460 as a server side and a CPU with the model of Intel Core i5-760 quad-core as a client side. Win7 64-bit is selected as the operating system. The parallel implementation consists of two parts:PA H and PA QR. PA H is a procedure that transforms a matrix A into the Hessenberg matrix B. PA QR is the actual parallel algorithm of the QR iterations that is imposed on the Hessenberg matrix B for finding eigenvalues of the general matrix A. The speedup ratio of the proposed algorithm is jarless when the number of iterations becomes greater. The experimental results show that the parallel implementation with CUDA on GPU only makes use of less running time than traditional sequential algorithms. The speedup ratio of PA H is between 1.79 and 7.81. The speedup ratio of PA QR is between 3.24 and 118.9. Especially, when the order of general matrix equals 8192, the amount of iterations becomes 10000. The speedup ratio of the PA-H and the PA QR can run up to 7.81 and 118.9 respectively.

The small slop approximation (SSA) is an accurate method to calculate the electromagnetic (EM) scattering properties of rough surfaces. However, its computational complexity restricts its application to smaller domains and there is always the need for speedup in very large cases using pure central processing units (CPUs) hardware. With the development of graphics processing units (GPUs), more processors are dedicated to perform independent calculations. In addition, NVIDIA introduced a parallel computing platform, compute unified device architecture (CUDA), which provides researchers an easy way to use processors on GPU. To calculate EM scattering properties on GPU, we reformulate the SSA method with CUDA to take advantage of GPU threads. Because each thread executes synchronously and deals with a corresponding point data of rough surface, the CUDA-based SSA method calculates faster than the pure-CPU equivalent. To overcome memory limitations, the data of large rough surface are stored on hard disk. Moreover, a subsidiary thread is used to deal with the process of data transmission between the memory and the hard disk and reduce transmitting time further. The factors, block size, data transfers, and register, are also discussed in the optimization of the CUDA application. Test cases running on a NVIDIA GTX 460 GPU indicate that two orders of magnitude speedup, including file input and output, is obtained with our new formulation.

Watershed analysis, as a fundamental component of digital terrain analysis, is based on the Digital Elevation Model (DEM), which is a grid (raster) model of the Earth surface and topography. Watershed analysis consists of computationally and data intensive computing algorithms that need to be implemented by leveraging parallel and high-performance computing methods and techniques. In this paper, the Multiple Flow Direction (MFD) algorithm for watershed analysis is implemented and evaluated on multi-core Central Processing Units (CPU) and many-core Graphics Processing Units (GPU), which provides significant improvements in performance and energy usage. The implementation is based on NVIDIA CUDA (Compute Unified Device Architecture) implementation for GPU, as well as on OpenACC (Open ACCelerators), a parallel programming model, and a standard for parallel computing. Both phases of the MFD algorithm (i) iterative DEM preprocessing and (ii) iterative MFD algorithm, are parallelized and run over multi-core CPU and GPU. The evaluation of the proposed solutions is performed with respect to the execution time, energy consumption, and programming effort for algorithm parallelization for different sizes of input data. An experimental evaluation has shown not only the advantage of using OpenACC programming over CUDA programming in implementing the watershed analysis on a GPU in terms of performance, energy consumption, and programming effort, but also significant benefits in implementing it on the multi-core CPU.

Many-core framework, represented by Graphic Processing Unit (GPU), has been becoming more popular in scientific computing and data analysis, and Compute Unified Device Architecture (CUDA) allows developers to carry out computation on GPUs more conveniently. Matrix operation is one of the most important application scenarios of GPU computation and CUDA, widely appearing in domains like graph applications and artificial intelligence. This paper focuses on accelerating matrix power operation (MPO) process, which appears in many engineering problems and may be time-consuming when the scale of matrixes is large, with GPUs and CUDA. In this paper, three techniques are used to speed up the MPO process: matrix parallel reduction, matrix parallel multiplication and CUDA dynamic parallelism. The experiments were carried out on NVIDIA RTX 2080 Super and Intel(R) Core(TM) i7-10750H. The code is compiled and executed in Visual Studio 2019 community, the GPU driver version is 461.40, the CUDA version is 11.1, and the NVCC version is 11.1.74. With all three techniques, the MPO process can get hundreds of times speedup compared with the sequential multiplication process executed by Eigen library, even though the code of algorithm is hardly optimized. Moreover, with the number of power increasing, the efficiency of algorithm can be even higher because of the relatively fixed time consumed in data transmission and GPU memory allocation.