163 research outputs found
Alternating-Direction Line-Relaxation Methods on Multicomputers
We study the multicom.puter performance of a three-dimensional Navier–Stokes solver based on alternating-direction line-relaxation methods. We compare several multicomputer implementations, each of which combines a particular line-relaxation method and a particular distributed block-tridiagonal solver. In our experiments, the problem size was determined by resolution requirements of the application. As a result, the granularity of the computations of our study is finer than is customary in the performance analysis of concurrent block-tridiagonal solvers. Our best results were obtained with a modified half-Gauss–Seidel line-relaxation method implemented by means of a new iterative block-tridiagonal solver that is developed here. Most computations were performed on the Intel Touchstone Delta, but we also used the Intel Paragon XP/S, the Parsytec SC-256, and the Fujitsu S-600 for comparison
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
Simulating the Behaviour of the Human Brain on NVIDIA GPU: cuHinesBatch & cuThomasBatch implementations
Understand the human brain is one of the century challenges. On this work we are going to achieve a small step towards this objective presenting a novel data layout in order to compute more efficiently the Hines algorithm on GPU. A more general tridiagonal solver is going to be presented too
Some fast elliptic solvers on parallel architectures and their complexities
The discretization of separable elliptic partial differential equations leads to linear systems with special block triangular matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconsistant coefficients. A method was recently proposed to parallelize and vectorize BCR. Here, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches, including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational complexity lower than that of parallel BCR
A Three-Level Parallelisation Scheme and Application to the Nelder-Mead Algorithm
We consider a three-level parallelisation scheme. The second and third levels
define a classical two-level parallelisation scheme and some load balancing
algorithm is used to distribute tasks among processes. It is well-known that
for many applications the efficiency of parallel algorithms of the second and
third level starts to drop down after some critical parallelisation degree is
reached. This weakness of the two-level template is addressed by introduction
of one additional parallelisation level. As an alternative to the basic solver
some new or modified algorithms are considered on this level. The idea of the
proposed methodology is to increase the parallelisation degree by using less
efficient algorithms in comparison with the basic solver. As an example we
investigate two modified Nelder-Mead methods. For the selected application, a
few partial differential equations are solved numerically on the second level,
and on the third level the parallel Wang's algorithm is used to solve systems
of linear equations with tridiagonal matrices. A greedy workload balancing
heuristic is proposed, which is oriented to the case of a large number of
available processors. The complexity estimates of the computational tasks are
model-based, i.e. they use empirical computational data
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