On Sat, Jul 29, 2017 at 7:01 PM, Oleksandr Koshkarov
<[email protected] <mailto:[email protected]>> wrote:
Thank you for the response,
Now that you said about additional communication for fft, I want
to ask another question:
What if I disregard FFT and will compute convolution directly, I
will rewrite it as vector-matrix-vector product
I would caution you here. The algorithm is the most important thing
to get right up front. In this case, the effect of
dimension will be dominant, and anything that does not scale well with
dimension will be thrown away, perhaps
quite quickly, as you want to solve bigger problems.
Small calculation: Take a 6-dim box which is 'n' on a side, so that N
= n^6. FFT is N log N, whereas direct evaluation of
a convolution is N^2. So if we could do n = 100 points on a side with
FFT, you can do, assuming the constants are about
equal,
100^6 (log 100^6) = x^12
1e12 (6 2) = x^12
x ~ 10
Past versions of PETSc had a higher dimensional DA construct (adda in
prior versions), but no one used it so we removed it.
I assume you are proposing something like
https://arxiv.org/abs/1306.4625
or maybe
https://arxiv.org/abs/1610.00397
The later is interesting since I think you could use a DMDA for the
velocity and something like DMPlex for the space.
Thanks,
Matt
x^T A x (where T - is transpose, x - is my state vector, and A is
a matrix which represents convolution and probably it will be not
very sparse).
Yes - I will do more work (alghorithm will be slower), but will I
win if we take the parallelization into account - less
communication + probably more scalable algorithm?
Best regards,
Alex.
On 07/29/2017 05:23 PM, Barry Smith wrote:
DMDA provides a simple way to do domain decomposition of
structured grid meshes in the x, y, and z direction, that is
it makes it easy to efficiently chop up vectors in all 3
dimensions, allowing very large problems easily. Unfortunately
it only goes up to 3 dimensions and attempts to produce
versions for higher dimensions have failed.
If you would like to do domain decomposition across all six
of your dimensions (which I think you really need to do to
solve large problems) we don't have a tool that helps with
doing the domain decomposition. To make matters more
complicated, likely the 3d mpi-fftw that need to be applied
require rejiggering the data across the nodes to get it into a
form where the ffts can be applied efficiently.
I don't think there is necessarily anything theoretically
difficult about managing the six dimensional domain
decomposition I just don't know of any open source software
out there that helps to do this. Maybe some PETSc users are
aware of such software and can chime in.
Aside from these two issues :-) PETSc would be useful for
you since you could use our time integrators and nonlinear
solvers. Pulling out subvectors to process on can be done with
with either VecScatter or VecStrideScatter, VecStrideGather
etc depending on the layout of the unknowns, i.e. how the
domain decomposition is done.
I wish I had better answers for managing the 6d domain
decomposition
Barry
On Jul 29, 2017, at 5:06 PM, Oleksandr Koshkarov
<[email protected] <mailto:[email protected]>> wrote:
Dear All,
I am a new PETSc user and I am still in the middle of the
manual (I have finally settled to choose PETSc as my main
numerical library) so I am sorry beforehand if my
questions would be naive.
I am trying to solve 6+1 dimensional Vlasov equation with
spectral methods. More precisely, I will try to solve
half-discretized equations of the form (approximate form)
with pseudospectral Fourier method:
(Equations are in latex format, the nice website to see
them is https://www.codecogs.com/latex/eqneditor.php
<https://www.codecogs.com/latex/eqneditor.php>)
\frac{dC_{m,m,p}}{dt} =\\
\partial_x \left ( a_n C_{n+1,m,p} +b_n C_{n,m,p} +c_n
C_{n-1,m,p} \right ) \\
+ \partial_y \left ( a_m C_{n,m+1,p} +b_m C_{n,m,p} +c_m
C_{n,m-1,p} \right ) \\
+ \partial_z \left ( a_p C_{n,m,p+1} +b_p C_{n,m,p} +c_p
C_{n,m,p-1} \right ) \\
+ d_n E_x C_{n-1,m,p} + d_m E_x C_{n,m-1,p} + d_p E_x
C_{n,m,p-1} \\
+ B_x (e_{m,p} C_{n,m-1,p-1} + f_{m,p}C_{n,m-1,p+1} +
\dots) + B_y (\dots) + B_z (\dots)
where a,b,c,d,e,f are some constants which can depend on
n,m,p,
n,m,p = 0...N,
C_{n,m,p} = C_{n,m,p}(x,y,z),
E_x = E_x(x,y,z), (same for E_y,B_x,...)
and fields are described with equation of the sort (linear
pdes with source terms dependent on C):
\frac{dE_x}{dt} = \partial_y B_z - \partial_z B_x +
(AC_{1,0,0} + BC_{0,0,0}) \\
\frac{dB_x}{dt} = -\partial_y E_z + \partial_z E_x
with A,B being some constants.
I will need fully implicit CrankâNicolson method, so my
plan is to use SNES or TS where
I will use one MPI PETSc vector which describes the state
of the system (all, C, E, B), then I can evolve linear
part with just matrix multiplication.
The first question is, should I use something like DMDA?
if so, it is only 3dimensional but I need 6 dimensional
vectots? Will it be faster then matrix multiplication?
Then to do nonlinear part I will need 3d mpi-fftw to
compute convolutions. The problem is, how do I extract
subvectors from full big vector state? (what is the best
approach?)
If you have some other suggestions, please feel free to share
Thanks and best regards,
Oleksandr Koshkarov.
--
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which
their experiments lead.
-- Norbert Wiener
http://www.caam.rice.edu/~mk51/ <http://www.caam.rice.edu/%7Emk51/>
