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<P><A HREF = "Section_accelerate.html">Return to Section accelerate overview</A>
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<H4>5.3.4 KOKKOS package
</H4>
<P>The KOKKOS package was developed primaritly by Christian Trott
(Sandia) with contributions of various styles by others, including
Sikandar Mashayak (UIUC), Stan Moore (Sandia), and Ray Shan (Sandia).
The underlying Kokkos library was written
primarily by Carter Edwards, Christian Trott, and Dan Sunderland (all
Sandia).
</P>
<P>The KOKKOS package contains versions of pair, fix, and atom styles
that use data structures and macros provided by the Kokkos library,
which is included with LAMMPS in lib/kokkos.
</P>
<P>The Kokkos library is part of
<A HREF = "http://trilinos.sandia.gov/packages/kokkos">Trilinos</A> and can also
be downloaded from <A HREF = "https://github.com/kokkos/kokkos">Github</A>. Kokkos is a
templated C++ library that provides two key abstractions for an
application like LAMMPS. First, it allows a single implementation of
an application kernel (e.g. a pair style) to run efficiently on
different kinds of hardware, such as a GPU, Intel Phi, or many-core
chip.
</P>
<P>The Kokkos library also provides data abstractions to adjust (at
compile time) the memory layout of basic data structures like 2d and
3d arrays and allow the transparent utilization of special hardware
load and store operations. Such data structures are used in LAMMPS to
store atom coordinates or forces or neighbor lists. The layout is
chosen to optimize performance on different platforms. Again this
functionality is hidden from the developer, and does not affect how
the kernel is coded.
</P>
<P>These abstractions are set at build time, when LAMMPS is compiled with
the KOKKOS package installed. This is done by selecting a "host" and
"device" to build for, compatible with the compute nodes in your
machine (one on a desktop machine or 1000s on a supercomputer).
</P>
<P>All Kokkos operations occur within the context of an individual MPI
task running on a single node of the machine. The total number of MPI
tasks used by LAMMPS (one or multiple per compute node) is set in the
usual manner via the mpirun or mpiexec commands, and is independent of
Kokkos.
</P>
<P>Kokkos provides support for two different modes of execution per MPI
task. This means that computational tasks (pairwise interactions,
neighbor list builds, time integration, etc) can be parallelized for
one or the other of the two modes. The first mode is called the
"host" and is one or more threads running on one or more physical CPUs
(within the node). Currently, both multi-core CPUs and an Intel Phi
processor (running in native mode, not offload mode like the
USER-INTEL package) are supported. The second mode is called the
"device" and is an accelerator chip of some kind. Currently only an
NVIDIA GPU is supported via Cuda. If your compute node does not have
a GPU, then there is only one mode of execution, i.e. the host and
device are the same.
</P>
<P>When using the KOKKOS package, you must choose at build time whether
you are building for OpenMP, GPU, or for using the Xeon Phi in native
mode.
</P>
<P>Here is a quick overview of how to use the KOKKOS package:
</P>
<UL><LI>specify variables and settings in your Makefile.machine that enable OpenMP, GPU, or Phi support
<LI>include the KOKKOS package and build LAMMPS
<LI>enable the KOKKOS package and its hardware options via the "-k on" command-line switch use KOKKOS styles in your input script
</UL>
<P>The latter two steps can be done using the "-k on", "-pk kokkos" and
"-sf kk" <A HREF = "Section_start.html#start_7">command-line switches</A>
respectively. Or the effect of the "-pk" or "-sf" switches can be
duplicated by adding the <A HREF = "package.html">package kokkos</A> or <A HREF = "suffix.html">suffix
kk</A> commands respectively to your input script.
</P>
<P><B>Required hardware/software:</B>
</P>
<P>The KOKKOS package can be used to build and run LAMMPS on the
following kinds of hardware:
</P>
<UL><LI>CPU-only: one MPI task per CPU core (MPI-only, but using KOKKOS styles)
<LI>CPU-only: one or a few MPI tasks per node with additional threading via OpenMP
<LI>Phi: on one or more Intel Phi coprocessors (per node)
<LI>GPU: on the GPUs of a node with additional OpenMP threading on the CPUs
</UL>
<P>Kokkos support within LAMMPS must be built with a C++11 compatible
compiler. For example, gcc 4.7.2 or later.
</P>
<P>Note that Intel Xeon Phi coprocessors are supported in "native" mode,
not "offload" mode like the USER-INTEL package supports.
</P>
<P>Only NVIDIA GPUs are currently supported.
</P>
<P>IMPORTANT NOTE: For good performance of the KOKKOS package on GPUs,
you must have Kepler generation GPUs (or later). The Kokkos library
exploits texture cache options not supported by Telsa generation GPUs
(or older).
</P>
<P>To build the KOKKOS package for GPUs, NVIDIA Cuda software must be
installed on your system. See the discussion above for the USER-CUDA
and GPU packages for details of how to check and do this.
</P>
<P><B>Building LAMMPS with the KOKKOS package:</B>
</P>
<P>You must choose at build time whether to build for OpenMP, Cuda, or
Phi.
</P>
<P>You can do any of these in one line, using the src/Make.py script,
described in <A HREF = "Section_start.html#start_4">Section 2.4</A> of the manual.
Type "Make.py -h" for help. If run from the src directory, these
commands will create src/lmp_kokkos_omp, lmp_kokkos_cuda, and
lmp_kokkos_phi. Note that the OMP and PHI options use
src/MAKE/Makefile.mpi as the starting Makefile.machine. The CUDA
option uses src/MAKE/OPTIONS/Makefile.kokkos_cuda.
</P>
<PRE>Make.py -p kokkos -kokkos omp -o kokkos_omp -a file mpi
Make.py -p kokkos -kokkos cuda arch=31 -o kokkos_cuda -a file kokkos_cuda
Make.py -p kokkos -kokkos phi -o kokkos_phi -a file mpi
</PRE>
<P>Or you can follow these steps:
</P>
<P>CPU-only (run all-MPI or with OpenMP threading):
</P>
<PRE>cd lammps/src
make yes-kokkos
make g++ KOKKOS_DEVICES=OpenMP
</PRE>
<P>Intel Xeon Phi:
</P>
<PRE>cd lammps/src
make yes-kokkos
make g++ KOKKOS_DEVICES=OpenMP KOKKOS_ARCH=KNC
</PRE>
<P>CPUs and GPUs:
</P>
<PRE>cd lammps/src
make yes-kokkos
make cuda KOKKOS_DEVICES=Cuda
</PRE>
<P>These examples set the KOKKOS-specific OMP, MIC, CUDA variables on the
make command line which requires a GNU-compatible make command. Try
"gmake" if your system's standard make complains.
</P>
<P>IMPORTANT NOTE: If you build using make line variables and re-build
LAMMPS twice with different KOKKOS options and the *same* target,
e.g. g++ in the first two examples above, then you *must* perform a
"make clean-all" or "make clean-machine" before each build. This is
to force all the KOKKOS-dependent files to be re-compiled with the new
options.
</P>
<P>You can also hardwire these make variables in the specified machine
makefile, e.g. src/MAKE/Makefile.g++ in the first two examples above,
with a line like:
</P>
<PRE>KOKKOS_ARCH = KNC
</PRE>
<P>Note that if you build LAMMPS multiple times in this manner, using
different KOKKOS options (defined in different machine makefiles), you
do not have to worry about doing a "clean" in between. This is
because the targets will be different.
</P>
<P>IMPORTANT NOTE: The 3rd example above for a GPU, uses a different
machine makefile, in this case src/MAKE/Makefile.cuda, which is
included in the LAMMPS distribution. To build the KOKKOS package for
a GPU, this makefile must use the NVIDA "nvcc" compiler. And it must
have a KOKKOS_ARCH setting that is appropriate for your NVIDIA
hardware and installed software. Typical values for KOKKOS_ARCH are given
below, as well
as other settings that must be included in the machine makefile, if
you create your own.
</P>
<P>IMPORTANT NOTE: Currently, there are no precision options with the
KOKKOS package. All compilation and computation is performed in
double precision.
</P>
<P>There are other allowed options when building with the KOKKOS package.
As above, they can be set either as variables on the make command line
or in Makefile.machine. This is the full list of options, including
those discussed above, Each takes a value shown below. The
default value is listed, which is set in the
lib/kokkos/Makefile.kokkos file.
</P>
<P>#Default settings specific options
#Options: force_uvm,use_ldg,rdc
</P>
<UL><LI>KOKKOS_DEVICES, values = <I>OpenMP</I>, <I>Serial</I>, <I>Pthreads</I>, <I>Cuda</I>, default = <I>OpenMP</I>
<LI>KOKKOS_ARCH, values = <I>KNC</I>, <I>SNB</I>, <I>HSW</I>, <I>Kepler</I>, <I>Kepler30</I>, <I>Kepler32</I>, <I>Kepler35</I>, <I>Kepler37</I>, <I>Maxwell</I>, <I>Maxwell50</I>, <I>Maxwell52</I>, <I>Maxwell53</I>, <I>ARMv8</I>, <I>BGQ</I>, <I>Power7</I>, <I>Power8</I>, default = <I>none</I>
<LI>KOKKOS_DEBUG, values = <I>yes</I>, <I>no</I>, default = <I>no</I>
<LI>KOKKOS_USE_TPLS, values = <I>hwloc</I>, <I>librt</I>, default = <I>none</I>
<LI>KOKKOS_CUDA_OPTIONS, values = <I>force_uvm</I>, <I>use_ldg</I>, <I>rdc</I>
</UL>
<P>KOKKOS_DEVICE sets the parallelization method used for Kokkos code
(within LAMMPS). KOKKOS_DEVICES=OpenMP means that OpenMP will be
used. KOKKOS_DEVICES=Pthreads means that pthreads will be used.
KOKKOS_DEVICES=Cuda means an NVIDIA GPU running CUDA will be used.
</P>
<P>If KOKKOS_DEVICES=Cuda, then the lo-level Makefile in the src/MAKE
directory must use "nvcc" as its compiler, via its CC setting. For
best performance its CCFLAGS setting should use -O3 and have a
KOKKOS_ARCH setting that matches the compute capability of your NVIDIA
hardware and software installation, e.g. KOKKOS_ARCH=Kepler30. Note
the minimal required compute capability is 2.0, but this will give
signicantly reduced performance compared to Kepler generation GPUs
with compute capability 3.x. For the LINK setting, "nvcc" should not
be used; instead use g++ or another compiler suitable for linking C++
applications. Often you will want to use your MPI compiler wrapper
for this setting (i.e. mpicxx). Finally, the lo-level Makefile must
also have a "Compilation rule" for creating *.o files from *.cu files.
See src/Makefile.cuda for an example of a lo-level Makefile with all
of these settings.
</P>
<P>KOKKOS_USE_TPLS=hwloc binds threads to hardware cores, so they do not
migrate during a simulation. KOKKOS_USE_TPLS=hwloc should always be
used if running with KOKKOS_DEVICES=Pthreads for pthreads. It is not
necessary for KOKKOS_DEVICES=OpenMP for OpenMP, because OpenMP
provides alternative methods via environment variables for binding
threads to hardware cores. More info on binding threads to cores is
given in <A HREF = "Section_accelerate.html#acc_8">this section</A>.
</P>
<P>KOKKOS_ARCH=KNC enables compiler switches needed when compling for an
Intel Phi processor.
</P>
<P>KOKKOS_USE_TPLS=librt enables use of a more accurate timer mechanism
on most Unix platforms. This library is not available on all
platforms.
</P>
<P>KOKKOS_DEBUG is only useful when developing a Kokkos-enabled style
within LAMMPS. KOKKOS_DEBUG=yes enables printing of run-time
debugging information that can be useful. It also enables runtime
bounds checking on Kokkos data structures.
</P>
<P>KOKKOS_CUDA_OPTIONS are additional options for CUDA.
</P>
<P>For more information on Kokkos see the Kokkos programmers' guide here:
/lib/kokkos/doc/Kokkos_PG.pdf.
</P>
<P><B>Run with the KOKKOS package from the command line:</B>
</P>
<P>The mpirun or mpiexec command sets the total number of MPI tasks used
by LAMMPS (one or multiple per compute node) and the number of MPI
tasks used per node. E.g. the mpirun command in MPICH does this via
its -np and -ppn switches. Ditto for OpenMPI via -np and -npernode.
</P>
<P>When using KOKKOS built with host=OMP, you need to choose how many
OpenMP threads per MPI task will be used (via the "-k" command-line
switch discussed below). Note that the product of MPI tasks * OpenMP
threads/task should not exceed the physical number of cores (on a
node), otherwise performance will suffer.
</P>
<P>When using the KOKKOS package built with device=CUDA, you must use
exactly one MPI task per physical GPU.
</P>
<P>When using the KOKKOS package built with host=MIC for Intel Xeon Phi
coprocessor support you need to insure there are one or more MPI tasks
per coprocessor, and choose the number of coprocessor threads to use
per MPI task (via the "-k" command-line switch discussed below). The
product of MPI tasks * coprocessor threads/task should not exceed the
maximum number of threads the coproprocessor is designed to run,
otherwise performance will suffer. This value is 240 for current
generation Xeon Phi(TM) chips, which is 60 physical cores * 4
threads/core. Note that with the KOKKOS package you do not need to
specify how many Phi coprocessors there are per node; each
coprocessors is simply treated as running some number of MPI tasks.
</P>
<P>You must use the "-k on" <A HREF = "Section_start.html#start_7">command-line
switch</A> to enable the KOKKOS package. It
takes additional arguments for hardware settings appropriate to your
system. Those arguments are <A HREF = "Section_start.html#start_7">documented
here</A>. The two most commonly used
options are:
</P>
<PRE>-k on t Nt g Ng
</PRE>
<P>The "t Nt" option applies to host=OMP (even if device=CUDA) and
host=MIC. For host=OMP, it specifies how many OpenMP threads per MPI
task to use with a node. For host=MIC, it specifies how many Xeon Phi
threads per MPI task to use within a node. The default is Nt = 1.
Note that for host=OMP this is effectively MPI-only mode which may be
fine. But for host=MIC you will typically end up using far less than
all the 240 available threads, which could give very poor performance.
</P>
<P>The "g Ng" option applies to device=CUDA. It specifies how many GPUs
per compute node to use. The default is 1, so this only needs to be
specified is you have 2 or more GPUs per compute node.
</P>
<P>The "-k on" switch also issues a "package kokkos" command (with no
additional arguments) which sets various KOKKOS options to default
values, as discussed on the <A HREF = "package.html">package</A> command doc page.
</P>
<P>Use the "-sf kk" <A HREF = "Section_start.html#start_7">command-line switch</A>,
which will automatically append "kk" to styles that support it. Use
the "-pk kokkos" <A HREF = "Section_start.html#start_7">command-line switch</A> if
you wish to change any of the default <A HREF = "package.html">package kokkos</A>
optionns set by the "-k on" <A HREF = "Section_start.html#start_7">command-line
switch</A>.
</P>
<PRE>host=OMP, dual hex-core nodes (12 threads/node):
mpirun -np 12 lmp_g++ -in in.lj # MPI-only mode with no Kokkos
mpirun -np 12 lmp_g++ -k on -sf kk -in in.lj # MPI-only mode with Kokkos
mpirun -np 1 lmp_g++ -k on t 12 -sf kk -in in.lj # one MPI task, 12 threads
mpirun -np 2 lmp_g++ -k on t 6 -sf kk -in in.lj # two MPI tasks, 6 threads/task
mpirun -np 32 -ppn 2 lmp_g++ -k on t 6 -sf kk -in in.lj # ditto on 16 nodes
</PRE>
<P>host=MIC, Intel Phi with 61 cores (240 threads/phi via 4x hardware threading):
mpirun -np 1 lmp_g++ -k on t 240 -sf kk -in in.lj # 1 MPI task on 1 Phi, 1*240 = 240
mpirun -np 30 lmp_g++ -k on t 8 -sf kk -in in.lj # 30 MPI tasks on 1 Phi, 30*8 = 240
mpirun -np 12 lmp_g++ -k on t 20 -sf kk -in in.lj # 12 MPI tasks on 1 Phi, 12*20 = 240
mpirun -np 96 -ppn 12 lmp_g++ -k on t 20 -sf kk -in in.lj # ditto on 8 Phis
</P>
<PRE>host=OMP, device=CUDA, node = dual hex-core CPUs and a single GPU:
mpirun -np 1 lmp_cuda -k on t 6 -sf kk -in in.lj # one MPI task, 6 threads on CPU
mpirun -np 4 -ppn 1 lmp_cuda -k on t 6 -sf kk -in in.lj # ditto on 4 nodes
</PRE>
<PRE>host=OMP, device=CUDA, node = dual 8-core CPUs and 2 GPUs:
mpirun -np 2 lmp_cuda -k on t 8 g 2 -sf kk -in in.lj # two MPI tasks, 8 threads per CPU
mpirun -np 32 -ppn 2 lmp_cuda -k on t 8 g 2 -sf kk -in in.lj # ditto on 16 nodes
</PRE>
<P>Note that the default for the <A HREF = "package.html">package kokkos</A> command is
to use "full" neighbor lists and set the Newton flag to "off" for both
pairwise and bonded interactions. This typically gives fastest
performance. If the <A HREF = "newton.html">newton</A> command is used in the input
script, it can override the Newton flag defaults.
</P>
<P>However, when running in MPI-only mode with 1 thread per MPI task, it
will typically be faster to use "half" neighbor lists and set the
Newton flag to "on", just as is the case for non-accelerated pair
styles. You can do this with the "-pk" <A HREF = "Section_start.html#start_7">command-line
switch</A>.
</P>
<P><B>Or run with the KOKKOS package by editing an input script:</B>
</P>
<P>The discussion above for the mpirun/mpiexec command and setting
appropriate thread and GPU values for host=OMP or host=MIC or
device=CUDA are the same.
</P>
<P>You must still use the "-k on" <A HREF = "Section_start.html#start_7">command-line
switch</A> to enable the KOKKOS package, and
specify its additional arguments for hardware options appopriate to
your system, as documented above.
</P>
<P>Use the <A HREF = "suffix.html">suffix kk</A> command, or you can explicitly add a
"kk" suffix to individual styles in your input script, e.g.
</P>
<PRE>pair_style lj/cut/kk 2.5
</PRE>
<P>You only need to use the <A HREF = "package.html">package kokkos</A> command if you
wish to change any of its option defaults, as set by the "-k on"
<A HREF = "Section_start.html#start_7">command-line switch</A>.
</P>
<P><B>Speed-ups to expect:</B>
</P>
<P>The performance of KOKKOS running in different modes is a function of
your hardware, which KOKKOS-enable styles are used, and the problem
size.
</P>
<P>Generally speaking, the following rules of thumb apply:
</P>
<UL><LI>When running on CPUs only, with a single thread per MPI task,
performance of a KOKKOS style is somewhere between the standard
(un-accelerated) styles (MPI-only mode), and those provided by the
USER-OMP package. However the difference between all 3 is small (less
than 20%).
<LI>When running on CPUs only, with multiple threads per MPI task,
performance of a KOKKOS style is a bit slower than the USER-OMP
package.
<LI>When running on GPUs, KOKKOS is typically faster than the USER-CUDA
and GPU packages.
<LI>When running on Intel Xeon Phi, KOKKOS is not as fast as
the USER-INTEL package, which is optimized for that hardware.
</UL>
<P>See the <A HREF = "http://lammps.sandia.gov/bench.html">Benchmark page</A> of the
LAMMPS web site for performance of the KOKKOS package on different
hardware.
</P>
<P><B>Guidelines for best performance:</B>
</P>
<P>Here are guidline for using the KOKKOS package on the different
hardware configurations listed above.
</P>
<P>Many of the guidelines use the <A HREF = "package.html">package kokkos</A> command
See its doc page for details and default settings. Experimenting with
its options can provide a speed-up for specific calculations.
</P>
<P><B>Running on a multi-core CPU:</B>
</P>
<P>If N is the number of physical cores/node, then the number of MPI
tasks/node * number of threads/task should not exceed N, and should
typically equal N. Note that the default threads/task is 1, as set by
the "t" keyword of the "-k" <A HREF = "Section_start.html#start_7">command-line
switch</A>. If you do not change this, no
additional parallelism (beyond MPI) will be invoked on the host
CPU(s).
</P>
<P>You can compare the performance running in different modes:
</P>
<UL><LI>run with 1 MPI task/node and N threads/task
<LI>run with N MPI tasks/node and 1 thread/task
<LI>run with settings in between these extremes
</UL>
<P>Examples of mpirun commands in these modes are shown above.
</P>
<P>When using KOKKOS to perform multi-threading, it is important for
performance to bind both MPI tasks to physical cores, and threads to
physical cores, so they do not migrate during a simulation.
</P>
<P>If you are not certain MPI tasks are being bound (check the defaults
for your MPI installation), binding can be forced with these flags:
</P>
<PRE>OpenMPI 1.8: mpirun -np 2 -bind-to socket -map-by socket ./lmp_openmpi ...
Mvapich2 2.0: mpiexec -np 2 -bind-to socket -map-by socket ./lmp_mvapich ...
</PRE>
<P>For binding threads with the KOKKOS OMP option, use thread affinity
environment variables to force binding. With OpenMP 3.1 (gcc 4.7 or
later, intel 12 or later) setting the environment variable
OMP_PROC_BIND=true should be sufficient. For binding threads with the
KOKKOS pthreads option, compile LAMMPS the KOKKOS HWLOC=yes option, as
discussed in <A HREF = "Sections_start.html#start_3_4">Section 2.3.4</A> of the
manual.
</P>
<P><B>Running on GPUs:</B>
</P>
<P>Insure the -arch setting in the machine makefile you are using,
e.g. src/MAKE/Makefile.cuda, is correct for your GPU hardware/software
(see <A HREF = "Section_start.html#start_3_4">this section</A> of the manual for
details).
</P>
<P>The -np setting of the mpirun command should set the number of MPI
tasks/node to be equal to the # of physical GPUs on the node.
</P>
<P>Use the "-k" <A HREF = "Section_commands.html#start_7">command-line switch</A> to
specify the number of GPUs per node, and the number of threads per MPI
task. As above for multi-core CPUs (and no GPU), if N is the number
of physical cores/node, then the number of MPI tasks/node * number of
threads/task should not exceed N. With one GPU (and one MPI task) it
may be faster to use less than all the available cores, by setting
threads/task to a smaller value. This is because using all the cores
on a dual-socket node will incur extra cost to copy memory from the
2nd socket to the GPU.
</P>
<P>Examples of mpirun commands that follow these rules are shown above.
</P>
<P>IMPORTANT NOTE: When using a GPU, you will achieve the best
performance if your input script does not use any fix or compute
styles which are not yet Kokkos-enabled. This allows data to stay on
the GPU for multiple timesteps, without being copied back to the host
CPU. Invoking a non-Kokkos fix or compute, or performing I/O for
<A HREF = "thermo_style.html">thermo</A> or <A HREF = "dump.html">dump</A> output will cause data
to be copied back to the CPU.
</P>
<P>You cannot yet assign multiple MPI tasks to the same GPU with the
KOKKOS package. We plan to support this in the future, similar to the
GPU package in LAMMPS.
</P>
<P>You cannot yet use both the host (multi-threaded) and device (GPU)
together to compute pairwise interactions with the KOKKOS package. We
hope to support this in the future, similar to the GPU package in
LAMMPS.
</P>
<P><B>Running on an Intel Phi:</B>
</P>
<P>Kokkos only uses Intel Phi processors in their "native" mode, i.e.
not hosted by a CPU.
</P>
<P>As illustrated above, build LAMMPS with OMP=yes (the default) and
MIC=yes. The latter insures code is correctly compiled for the Intel
Phi. The OMP setting means OpenMP will be used for parallelization on
the Phi, which is currently the best option within Kokkos. In the
future, other options may be added.
</P>
<P>Current-generation Intel Phi chips have either 61 or 57 cores. One
core should be excluded for running the OS, leaving 60 or 56 cores.
Each core is hyperthreaded, so there are effectively N = 240 (4*60) or
N = 224 (4*56) cores to run on.
</P>
<P>The -np setting of the mpirun command sets the number of MPI
tasks/node. The "-k on t Nt" command-line switch sets the number of
threads/task as Nt. The product of these 2 values should be N, i.e.
240 or 224. Also, the number of threads/task should be a multiple of
4 so that logical threads from more than one MPI task do not run on
the same physical core.
</P>
<P>Examples of mpirun commands that follow these rules are shown above.
</P>
<P><B>Restrictions:</B>
</P>
<P>As noted above, if using GPUs, the number of MPI tasks per compute
node should equal to the number of GPUs per compute node. In the
future Kokkos will support assigning multiple MPI tasks to a single
GPU.
</P>
<P>Currently Kokkos does not support AMD GPUs due to limits in the
available backend programming models. Specifically, Kokkos requires
extensive C++ support from the Kernel language. This is expected to
change in the future.
</P>
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