538 research outputs found
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
Status and Future Perspectives for Lattice Gauge Theory Calculations to the Exascale and Beyond
In this and a set of companion whitepapers, the USQCD Collaboration lays out
a program of science and computing for lattice gauge theory. These whitepapers
describe how calculation using lattice QCD (and other gauge theories) can aid
the interpretation of ongoing and upcoming experiments in particle and nuclear
physics, as well as inspire new ones.Comment: 44 pages. 1 of USQCD whitepapers
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Matrix probing: a randomized preconditioner for the wave-equation Hessian
This paper considers the problem of approximating the inverse of the
wave-equation Hessian, also called normal operator, in seismology and other
types of wave-based imaging. An expansion scheme for the pseudodifferential
symbol of the inverse Hessian is set up. The coefficients in this expansion are
found via least-squares fitting from a certain number of applications of the
normal operator on adequate randomized trial functions built in curvelet space.
It is found that the number of parameters that can be fitted increases with the
amount of information present in the trial functions, with high probability.
Once an approximate inverse Hessian is available, application to an image of
the model can be done in very low complexity. Numerical experiments show that
randomized operator fitting offers a compelling preconditioner for the
linearized seismic inversion problem.Comment: 21 pages, 6 figure
Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units
In geophysical applications, the interest in leastsquares
migration (LSM) as an imaging algorithm is
increasing due to the demand for more accurate solutions
and the development of high-performance computing. The
computational engine of LSM in this work is the numerical
solution of the 3D Helmholtz equation in the frequency
domain. The Helmholtz solver is Bi-CGSTAB preconditioned
with the shifted Laplace matrix-dependent multigrid
method. In this paper, an efficient LSM algorithm is presented
using several enhancements. First of all, a frequency
decimation approach is introduced that makes use of redundant
information present in the data. It leads to a speedup of
LSM, whereas the impact on accuracy is kept minimal. Secondly,
a new matrix storage format Very Compressed Row
Storage (VCRS) is presented. It not only reduces the size of
the stored matrix by a certain factor but also increases the
efficiency of the matrix-vector computations. The effects of
lossless and lossy compression with a proper choice of the
compression parameters are positive. Thirdly, we accelerate
the LSM engine by graphics cards (GPUs). A GPU is used
as an accelerator, where the data is partially transferred to
a GPU to execute a set of operations or as a replacement,
where the complete data is stored in the GPU memory. We
demonstrate that using the GPU as a replacement leads to
higher speedups and allows us to solve larger problem sizes.
Summarizing the effects of each improvement, the resulting
speedup can be at least an order of magnitude compared to
the original LSM method
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