58,811 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
Logarithmic link smearing for full QCD
A Lie-algebra based recipe for smoothing gauge links in lattice field theory
is presented, building on the matrix logarithm. With or without hypercubic
nesting, this LOG/HYL smearing yields fat links which are differentiable w.r.t.
the original ones. This is essential for defining UV-filtered ("fat link")
fermion actions which may be simulated with a HMC-type algorithm. The effect of
this smearing on the distribution of plaquettes and on the residual mass of
tree-level O(a)-improved clover fermions in quenched QCD is studied.Comment: 29 pages, 7 figures; v2: improved text, includes comparison of
APE/EXP/LOG with optimized parameters, 3 references adde
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