Abstract:

Communicating data within the graphic processing unit (GPU) memory system and between the CPU and GPU are major bottlenecks in accelerating Krylov solvers on GPUs. Communication-avoiding techniques reduce the communication cost of Krylov subspace methods by computing several vectors of a Krylov subspace “at once,” using a kernel called “matrix powers.” The matrix powers kernel is implemented on a recent generation of NVIDIA GPUs and speedups of up to 5.7 times are reported for the communication-avoiding matrix powers kernel compared to the standards prase matrix vector multiplication (SpMV) implementation.

(M. Mehri Dehnavi, Y. El-Kurdi, J. Demmel and D. Giannacopoulos: “Communication-Avoiding Krylov Techniques on Graphic Processing Units”, IEEE Transactions on Magnetics 49(5):1749-1752, May 2013. [DOI])

PARALUTION is a library for sparse iterative methods with special focus on multi-core and accelerator technology such as GPUs. In particular, it incorporates fine-grained parallel preconditioners designed to expolit modern multi-/many-core devices. Based on C++, it provides a generic and flexible design and interface which allow seamless integration with other scientific software packages. The library is open source and released under GPL. Key features are:

• OpenMP, CUDA and OpenCL support
• No special hardware/library requirement
• Portable code and results across all hardware
• Many sparse matrix formats
• Various iterative solvers/preconditioners
• Generic and robust design
• Plug-in for the finite element package Deal.II
• Documentation: user manual (pdf), reports, doxygen

More information, including documentation and case studies, is available at http://www.paralution.com.

Abstract:

The paper presents techniques for generating very large finite-element matrices on a multicore workstation equipped with several graphics processing units (GPUs). To overcome the low memory size limitation of the GPUs, and at the same time to accelerate the generation process, we propose to generate the large sparse linear systems arising in finite-element analysis in an iterative manner on several GPUs and to use the graphics accelerators concurrently with CPUs performing collection and addition of the matrix fragments using a fast multithreaded procedure. The scheduling of the threads is organized in such a way that the CPU operations do not affect the performance of the process, and the GPUs are idle only when data are being transferred from GPU to CPU. This approach is verified on two workstations: the first consists of two 6-core Intel Xeon X5690 processors with two Fermi GPUs: each GPU is a GeForce GTX 590 with two graphics processors and 1.5 GB of fast RAM; the second workstation is equipped with two Tesla C2075 boards carrying 6 GB of RAM each and two 12-core Opteron 6174s. For the latter setup, we demonstrate the fast generation of sparse finite-element matrices as large as 10 million unknowns, with over 1 billion nonzero entries. Comparing with the single-threaded and multithreaded CPU implementations, the GPU-based version of the algorithm based on the ideas presented in this paper reduces the finite-element matrix-generation time in double precision by factors of 100 and 30, respectively.

(Dziekonski, A., Sypek, P., Lamecki, A. and Mrozowski, M.: “Generation of large finite-element matrices on multiple graphics processors”. International Journal on Numerical Methoths in Engineering, 2012, in press. [DOI])

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.

Abstract:

We present new format for storing sparse matrices on GPU. We compare it with several other formats including CUSPARSE which is today probably the best choice for processing of sparse matrices on GPU in CUDA. Contrary to CUSPARSE which works with common CSR format, our new format requires conversion. However, multiplication of sparse-matrix and vector is significantly faster for many matrices. We demonstrate it on set of 1 600 matrices and we show for what types of matrices our format is profitable.

(Heller M., Oberhuber T.: “Improved Row-grouped CSR Format for Storing of Sparse Matrices on GPU”, Proceedings of Algoritmy 2012, 2012, Handlovičová A., Minarechová Z. and Ševčovič D. (ed.), pages 282-290, ISBN 978-80-227-3742-5) [ARXIV preprint]

Abstract:

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])

Abstract:

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])

Abstract:

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.)

SpeedIT provides a set of accelerated solvers for sparse linear systems of equations. The library supports C/C++ and Fortran, and it can be used with OpenFOAM to accelerate CFD simulations. SpeedIT 2.1 contains two new preconditioners:

• Algebraic Multigrid with Smoothed Aggregation (AMG)
• Approximate Inverse (AINV)

OpenFOAM simulations on the GPU can be up to 3.5x faster compared to CG and DIC/DILU preconditioners on the CPU and up to 1.6x faster if you run GAMG.

See the SpeedIT website and blog for more details.