Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph