1,267 research outputs found
Exponential Integrators on Graphic Processing Units
In this paper we revisit stencil methods on GPUs in the context of
exponential integrators. We further discuss boundary conditions, in the same
context, and show that simple boundary conditions (for example, homogeneous
Dirichlet or homogeneous Neumann boundary conditions) do not affect the
performance if implemented directly into the CUDA kernel. In addition, we show
that stencil methods with position-dependent coefficients can be implemented
efficiently as well.
As an application, we discuss the implementation of exponential integrators
for different classes of problems in a single and multi GPU setup (up to 4
GPUs). We further show that for stencil based methods such parallelization can
be done very efficiently, while for some unstructured matrices the
parallelization to multiple GPUs is severely limited by the throughput of the
PCIe bus.Comment: To appear in: Proceedings of the 2013 International Conference on
High Performance Computing Simulation (HPCS 2013), IEEE (2013
Mapping the Arnold web with a GPU-supercomputer
The Arnold diffusion constitutes a dynamical phenomenon which may occur in
the phase space of a non-integrable Hamiltonian system whenever the number of
the system degrees of freedom is . The diffusion is mediated by a
web-like structure of resonance channels, which penetrates the phase space and
allows the system to explore the whole energy shell. The Arnold diffusion is a
slow process; consequently the mapping of the web presents a very
time-consuming task. We demonstrate that the exploration of the Arnold web by
use of a graphic processing unit (GPU)-supercomputer can result in distinct
speedups of two orders of magnitude as compared to standard CPU-based
simulations.Comment: 7 pages, 4 figures, a video supplementary provided at
http://www.physik.uni-augsburg.de/~seiberar/arnold/Energy15_HD_frontNback.av
A -mode integrator for solving evolution equations in Kronecker form
In this paper, we propose a -mode integrator for computing the solution
of stiff evolution equations. The integrator is based on a d-dimensional
splitting approach and uses exact (usually precomputed) one-dimensional matrix
exponentials. We show that the action of the exponentials, i.e. the
corresponding batched matrix-vector products, can be implemented efficiently on
modern computer systems. We further explain how -mode products can be used
to compute spectral transformations efficiently even if no fast transform is
available. We illustrate the performance of the new integrator by solving
three-dimensional linear and nonlinear Schr\"odinger equations, and we show
that the -mode integrator can significantly outperform numerical methods
well established in the field. We also discuss how to efficiently implement
this integrator on both multi-core CPUs and GPUs. Finally, the numerical
experiments show that using GPUs results in performance improvements between a
factor of 10 and 20, depending on the problem
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