In this paper, we present a scalable, numerically stable, high-performance tridiagonal solver. The solver is based on the SPIKE algorithm for partitioning a large matrix into small independent matrices, which can be solved in parallel. For each small matrix, our solver applies a general 1-by-1 or 2-by-2 diagonal pivoting algorithm, which is also known to be numerically stable. Our paper makes two major contributions. First, our solver is the first numerically stable tridiagonal solver for GPUs. Our solver provides comparable quality of stable solutions to Intel MKL and Matlab, at speed comparable to the GPU tridiagonal solvers in existing packages like CUSPARSE. It is also scalable to multiple GPUs and CPUs. Second, we present and analyze two key optimization strategies for our solver: a high-throughput data layout transformation for memory efficiency, and a dynamic tiling approach for reducing the memory access footprint caused by branch divergence.
(Chang, Li-Wen and Stratton, John A. and Kim, Hee-Seok and Hwu, Wen-mei W.: “A scalable, numerically stable, high-performance tridiagonal solver using GPUs”, Supercomputing 2012. [WWW])
This paper presents an efficient technique for fast generation of sparse systems of linear equations arising in computational electromagnetics in a finite element method using higher order elements. The proposed approach employs a graphics processing unit (GPU) for both numerical integration and matrix assembly. The performance results obtained on a test platform consisting of a Fermi GPU (1x Tesla C2075) and a CPU (2x twelve-core Opterons), indicate that the GPU implementation of the matrix generation allows one to achieve speedups by a factor of 81 and 19 over the optimized single-and multi-threaded CPU-only implementations, respectively.
(Adam Dziekonski et al., “Finite Element Matrix Generation on a GPU”, Progress In Electromagnetics Research 128:249-265, 2012. [PDF])
amgcl is a simple and generic algebraic multigrid (AMG) hierarchy builder. Supported coarsening methods are classical Ruge-Stuben coarsening, and either plain or smoothed aggregation. The constructed hierarchy is stored and used with help of one of the supported backends including VexCL, ViennaCL, and CUSPARSE/Thrust.
With help of amgcl, solution of a large sparse system of linear equations may be easily accelerated through OpenCL, CUDA, or OpenMP technologies. Source code of the library is publicly available under MIT license at https://github.com/ddemidov/amgcl.
The latest release 1.4.0 of the free open-source linear algebra library ViennaCL features the following highlights:
- Two computing backends in addition to OpenCL: CUDA and OpenMP
- Improved performance for (Block-) ILU0/ILUT preconditioners
- Optional level scheduling for ILU substitutions on GPUs
- Mixed-precision CG solver
- Initializer types from Boost.uBLAS (unit_vector, zero_vector, etc.)
Any contributions of fast CUDA or OpenCL computing kernels for future releases of ViennaCL are welcome! More information is available at http://viennacl.sourceforge.net.
Accelerating numerical algorithms for solving sparse linear systems on parallel architectures has attracted the attention of many researchers due to their applicability to many engineering and scientific problems. The solution of sparse systems often dominates the overall execution time of such problems and is mainly solved by iterative methods. Preconditioners are used to accelerate the convergence rate of these solvers and reduce the total execution time. Sparse Approximate Inverse (SAI) preconditioners are a popular class of preconditioners designed to improve the condition number of large sparse matrices and accelerate the convergence rate of iterative solvers for sparse linear systems. We propose a GPU accelerated SAI preconditioning technique called GSAI, which parallelizes the computation of this preconditioner on NVIDIA graphic cards. The preconditioner is then used to enhance the convergence rate of the BiConjugate Gradient Stabilized (BiCGStab) iterative solver on the GPU. The SAI preconditioner is generated on average 28 and 23 times faster on the NVIDIA GTX480 and TESLA M2070 graphic cards respectively compared to ParaSails (a popular implementation of SAI preconditioners on CPU) single processor/core results. The proposed GSAI technique computes the SAI preconditioner in approximately the same time as ParaSails generates the same preconditioner on 16 AMD Opteron 252 processors.
(Maryam Mehri Dehnavi, David Fernandez, Jean-Luc Gaudiot and Dennis Giannacopoulos: “Parallel Sparse Approximate Inverse Preconditioning on Graphic Processing Units”, IEEE Transactions on Parallel and Distributed Systems (to appear). [DOI])
In this work, we evaluate OpenCL as aprogramming tool for developing performance-portable applications for GPGPU. While the Khronos group developed OpenCL with programming portability in mind, performance is not necessarily portable. OpenCL has required performance-impacting initializations that do not exist in other languages such as CUDA. Understanding these implications allows us to provide a single library with decent performance on a variety of platforms. We choose triangular solver (TRSM) and matrix multiplication (GEMM) as representative level 3 BLAS routines to implement in OpenCL. We profile TRSM to get the time distribution of the OpenCL runtime system. We then provide tuned GEMM kernels for both the NVIDIA Tesla C2050 and ATI Radeon 5870, the latest GPUs offered by both companies. We explore the benefits of using the texture cache, the performance ramifications of copying data into images, discrepancies in the OpenCL and CUDA compilers’ optimizations, and other issues that affect the performance. Experimental results show that nearly 50% of peak performance can be obtained in GEMM on both GPUs in OpenCL. We also show that the performance of these kernels is not highly portable. Finally, we propose the use of auto-tuning to better explore these kernels’ parameter space using search harness.
(Peng Du, Rick Weber, Piotr Luszczek, Stanimire Tomov, Gregory Peterson, Jack Dongarra, “From CUDA to OpenCL: Towards a performance-portable solution for multi-platform GPU programming”, Parallel Computing 38(8):391–407, Aug. 2012. [DOI] [early techreport])
A wide range of applications in engineering and scientific computing are involved in the acceleration of the sparse matrix vector product (SpMV). Graphics Processing Units (GPUs) have recently emerged as platforms that yield outstanding acceleration factors. SpMV implementations for GPUs have already appeared on the scene. This work is focused on the ELLR-T algorithm to compute SpMV on GPU architecture, its performance is strongly dependent on the optimum selection of two parameters. Therefore, taking account that the memory operations dominate the performance of ELLR-T, an analytical model is proposed in order to obtain the auto-tuning of ELLR-T for particular combinations of sparse matrix and GPU architecture. The evaluation results with a representative set of test matrices show that the average performance achieved by auto-tuned ELLR-T by means of the proposed model is near to the optimum. A comparative analysis of ELLR-T against a variety of previous proposals shows that ELLR-T with the estimated configuration reaches the best performance on GPU architecture for the representative set of test matrices.
(Francisco Vázquez and José Jesús Fernández and Ester M. Garzón: “Automatic tuning of the sparse matrix vector product on GPUs based on the ELLR-T approach”, Parallel Computing 38(8), 408-420, Aug. 2012. [DOI])
A novel algorithm for computing the incomplete-LU and Cholesky factorization with 0 fill-in on a graphics processing unit (GPU) is proposed. It implements the incomplete factorization of the given matrix in two phases. First, the symbolic analysis phase builds a dependency graph based on the matrix sparsity pattern and groups the independent rows into levels. Second, the numerical factorization phase obtains the resulting lower and upper sparse triangular factors by iterating sequentially across the constructed levels. The Gaussian elimination of the elements below the main diagonal in the rows corresponding to each single level is performed in parallel. The numerical experiments are also presented and it is shown that the numerical factorization phase can achieve on average more than 2.8x speedup over MKL, while the incomplete-LU and Cholesky preconditioned iterative methods can achieve an average of 2x speedup on GPU over their CPU implementation.
(Maxim Naumov. Parallel Incomplete-LU and Cholesky Factorization in the Preconditioned Iterative Methods on the GPU, NVIDIA Technical Report NVR-2012-003. May 2012.)
We study the use of a GPU for the numerical approximation of the curvature dependent flows of graphs – the mean-curvature flow and the Willmore flow. Both problems are often applied in image processing where fast solvers are required. We approximate these problems using the complementary finite volume method combined with the method of lines. We obtain a system of ordinary differential equations which we solve by the Runge–Kutta–Merson solver. It is a robust solver with an automatic choice of the integration time step. We implement this solver on CPU but also on GPU using the CUDA toolkit. We demonstrate that the mean-curvature flow can be successfully approximated in single precision arithmetic with the speed-up almost 17 on the Nvidia GeForce GTX 280 card compared to Intel Core 2 Quad CPU. On the same card, we obtain the speed-up 7 in double precision arithmetic which is necessary for the fourth order problem – the Willmore flow of graphs. Both speed-ups were achieved without affecting the accuracy of the approximation. The article is structured in such way that the reader interested only in the implementation of the Runge–Kutta–Merson solver on the GPU can skip the sections containing the mathematical formulation of the problems.
(Oberhuber T., Suzuki A., Žabka V.: “The CUDA implementation of the method of lines for the curvature dependent flows”, Kybernetika 47(2):251–272, 2011. [PDF])
Partial differential equations are typically solved by means of finite difference, finite volume or finite element methods resulting in large, highly coupled, ill-conditioned and sparse (non-)linear systems. In order to minimize the computing time we want to exploit the capabilities of modern parallel architectures. The rapid hardware shifts from single core to multi-core and many-core processors lead to a gap in the progression of algorithms and programming environments for these platforms — the parallel models for large clusters do not fully utilize the performance capability of the multi-core CPUs and especially of the GPUs. Software stack needs to run adequately on the next generation of computing devices in order to exploit the potential of these new systems. Moving numerical software from one platform to another becomes an important task since every parallel device has its own programming model and language. The greatest challenge is to provide new techniques for solving (non-)linear systems that combine scalability, portability, fine-grained parallelism and flexibility across the assortment of parallel platforms and programming models. The goal of this thesis is to provide new fine-grained parallel algorithms embedded in advanced sparse linear algebra solvers and preconditioners on the emerging multi-core and many-core technologies.
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