1,728 research outputs found
Iteration-fusing conjugate gradient
This paper presents the Iteration-Fusing Conjugate Gradient (IFCG) approach which is an evolution of the Conjugate Gradient method that consists in i) letting computations from different iterations to overlap between them and ii) splitting linear algebra kernels into subkernels to increase concurrency and relax data-dependencies. The paper presents two ways of applying the IFCG approach: The IFCG1 algorithm, which aims at hiding the cost of parallel reductions, and the IFCG2 algorithm, which aims at reducing idle time by starting computations as soon as possible. Both IFCG1 and IFCG2 algorithms are two complementary approaches aiming at increasing parallel performance. Extensive numerical experiments are conducted to compare the IFCG1 and IFCG2 numerical stability and performance against four state-of-the-art techniques. By considering a set of representative input matrices, the paper demonstrates that IFCG1 and IFCG2 provide parallel performance improvements up to 42.9% and 41.5% respectively and average improvements of 11.8% and 7.1% with respect to the best state-of-the-art techniques while keeping similar numerical stability properties. Also, this paper provides an evaluation of the IFCG algorithms' sensitivity to system noise and it demonstrates that they run 18.0% faster on average than the best state-of-the-art technique under realistic degrees of system noise.This work has been supported by the Spanish Government (Severo Ochoa grants SEV2015-0493), by the Spanish Ministry of Science and Innovation (contracts TIN2015-65316) , by Generalitat de Catalunya
(contracts 2014-SGR-1051 and 2014-SGR-1272) and by the IBM/BSC Deep Learning Center Initiative.Peer ReviewedPostprint (author's final draft
Iteration-fusing conjugate gradient for sparse linear systems with MPI + OmpSs
In this paper, we target the parallel solution of sparse linear systems via iterative Krylov subspace-based method enhanced with a block-Jacobi preconditioner on a cluster of multicore processors. In order to tackle large-scale problems, we develop task-parallel implementations of the preconditioned conjugate gradient method that improve the interoperability between the message-passing interface and OmpSs programming models. Specifically, we progressively integrate several communication-reduction and iteration-fusing strategies into the initial code, obtaining more efficient versions of the method. For all these implementations, we analyze the communication patterns and perform a comparative analysis of their performance and scalability on a cluster consisting of 32 nodes with 24 cores each. The experimental analysis shows that the techniques described in the paper outperform the classical method by a margin that varies between 6 and 48%, depending on the evaluation
Iteration-fusing conjugate gradient for sparse linear systems with MPI + OmpSs
In this paper, we target the parallel solution of sparse linear systems via iterative Krylov subspace-based method enhanced with a block-Jacobi preconditioner on a cluster of multicore processors. In order to tackle large-scale problems, we develop task-parallel implementations of the preconditioned conjugate gradient method that improve the interoperability between the message-passing interface and OmpSs programming models. Specifically, we progressively integrate several communication-reduction and iteration-fusing strategies into the initial code, obtaining more efficient versions of the method. For all these implementations, we analyze the communication patterns and perform a comparative analysis of their performance and scalability on a cluster consisting of 32 nodes with 24 cores each. The experimental analysis shows that the techniques described in the paper outperform the classical method by a margin that varies between 6 and 48%, depending on the evaluation.This research was partially supported by the H2020 EU FETHPC Project 671602 “INTERTWinE.” The researchers from Universidad Jaume I were sponsored by Project TIN2017-82972-R of the Spanish Ministerio de EconomĂa y Competitividad. Maria Barreda was supported by the POSDOC-A/2017/11 project from the Universitat Jaume I.Peer ReviewedPostprint (author's final draft
Matrix-free GPU implementation of a preconditioned conjugate gradient solver for anisotropic elliptic PDEs
Many problems in geophysical and atmospheric modelling require the fast
solution of elliptic partial differential equations (PDEs) in "flat" three
dimensional geometries. In particular, an anisotropic elliptic PDE for the
pressure correction has to be solved at every time step in the dynamical core
of many numerical weather prediction models, and equations of a very similar
structure arise in global ocean models, subsurface flow simulations and gas and
oil reservoir modelling. The elliptic solve is often the bottleneck of the
forecast, and an algorithmically optimal method has to be used and implemented
efficiently. Graphics Processing Units have been shown to be highly efficient
for a wide range of applications in scientific computing, and recently
iterative solvers have been parallelised on these architectures. We describe
the GPU implementation and optimisation of a Preconditioned Conjugate Gradient
(PCG) algorithm for the solution of a three dimensional anisotropic elliptic
PDE for the pressure correction in NWP. Our implementation exploits the strong
vertical anisotropy of the elliptic operator in the construction of a suitable
preconditioner. As the algorithm is memory bound, performance can be improved
significantly by reducing the amount of global memory access. We achieve this
by using a matrix-free implementation which does not require explicit storage
of the matrix and instead recalculates the local stencil. Global memory access
can also be reduced by rewriting the algorithm using loop fusion and we show
that this further reduces the runtime on the GPU. We demonstrate the
performance of our matrix-free GPU code by comparing it to a sequential CPU
implementation and to a matrix-explicit GPU code which uses existing libraries.
The absolute performance of the algorithm for different problem sizes is
quantified in terms of floating point throughput and global memory bandwidth.Comment: 18 pages, 7 figure
- …