The PETSc DMDA is not lightweight

Note: this post was originally written by David Ketcheson.

In this PETSc-users thread, my good friend Matt Knepley claims that the DMDA object is lightweight, and suggests that it is okay to make them willy- nilly. Unfortunately, my experience indicates otherwise. But words are cheap; let’s investigate.

%load_ext memory_profiler

Memory profiling with memory_profiler

I’ve written a little script that just creates a DA and 3 vecs. In order to make use of memory_profiler, I’ll call it with system commands. But I’ve pasted the contents here for completeness. If you’re running the notebook, you don’t need to run the next cell.

# contents of da.py
from petsc4py import PETSc
import sys

@profile
def foo(size=128,ndim=3,dof=1):
    da = PETSc.DA().create(sizes=[size]*ndim,dof=dof)
    q1 = da.createGlobalVec()
    q2 = da.createGlobalVec()
    q3 = da.createGlobalVec()

if __name__ == '__main__':
    foo(int(sys.argv[1]),int(sys.argv[2]),int(sys.argv[3]))

Notice that the command line arguments are

1. The size of the grid
2. The number of dimensions
3. The number of DOFs

Now let’s run some tests.

The cost of a DA

!python -m memory_profiler da.py 128 3 1

Filename: da.py

Line #    Mem usage    Increment   Line Contents
================================================
     4   23.473 MiB    0.000 MiB   @profile
     5                             def foo(size=128,ndim=3,dof=1):
     6   72.395 MiB   48.922 MiB       da = PETSc.DA().create([size]*ndim,dof)
     7   72.480 MiB    0.086 MiB       q1 = da.createGlobalVec()
     8   88.484 MiB   16.004 MiB       q2 = da.createGlobalVec()
     9  104.488 MiB   16.004 MiB       q3 = da.createGlobalVec()

The third column (“increment”) is the key here – it shows the increase in total memory usage after executing each line.

The first thing you notice is that the DA is NOT lightweight. It takes as much memory as 3 Vecs! But that will look cheap in a moment.

Notice also that the first vec seems to use no memory. I think this is because memory_profiler just tracks peak usage, and the DA creation creates and destroys a vec.

In what follows, I’ve used sed to extract just the essential lines of the output.

Let’s try that again, with 2 DOFs.

!python -m memory_profiler da.py 128 3 2 / | sed -n -e 7p -e 10p

     6  296.215 MiB  272.945 MiB       da = PETSc.DA().create([size]*ndim,dof)
     9  328.309 MiB   32.004 MiB       q3 = da.createGlobalVec()

Whoa! Now the DA is 8.5 times as large as a Vec!

As expected, the Vec is just twice as large. What if we keep increasing the number of DOFs?

!python -m memory_profiler da.py 128 3 3 / | sed -n -e 7p -e 10p
!python -m memory_profiler da.py 128 3 4 / | sed -n -e 7p -e 10p

     6  408.219 MiB  384.949 MiB       da = PETSc.DA().create([size]*ndim,dof)
     9  504.316 MiB   48.004 MiB       q3 = da.createGlobalVec()
     6  520.219 MiB  496.949 MiB       da = PETSc.DA().create([size]*ndim,dof)
     9  648.316 MiB   64.004 MiB       q3 = da.createGlobalVec()

It’s clear that each additional DOF increases the size of the DA by 112 MB. That’s more than double the size of the whole DA we had with dof=1. And the size of the Vec only increases (appropriately) by 16 MB. So for large numbers of DOFs, the DA will be as large as 7 Vecs.

2D and 1D results

The same phenomenon is evident in two dimensions:

!python -m memory_profiler da.py 1024 2 1 / | sed -n -e 7p -e 10p
!python -m memory_profiler da.py 1024 2 2 / | sed -n -e 7p -e 10p
!python -m memory_profiler da.py 1024 2 3 / | sed -n -e 7p -e 10p

     6   48.371 MiB   24.895 MiB       da = PETSc.DA().create([size]*ndim,dof)
     9   64.469 MiB    8.004 MiB       q3 = da.createGlobalVec()
     6  160.398 MiB  136.922 MiB       da = PETSc.DA().create([size]*ndim,dof)
     9  176.492 MiB   16.004 MiB       q3 = da.createGlobalVec()
     6  216.402 MiB  192.926 MiB       da = PETSc.DA().create([size]*ndim,dof)
     9  264.500 MiB   24.004 MiB       q3 = da.createGlobalVec()

Again we see that:

• The 1-DOF DA costs 3 Vecs
• The 2-DOF DA costs about 8.5 Vecs
• Asymptotically, a DA with many DOFs costs 7 Vecs

What about 1D?

!python -m memory_profiler da.py 1000000 1 1 / | sed -n -e 7p -e 10p
!python -m memory_profiler da.py 1000000 1 2 / | sed -n -e 7p -e 10p
!python -m memory_profiler da.py 1000000 1 3 / | sed -n -e 7p -e 10p

     6   47.164 MiB   23.688 MiB       da = PETSc.DA().create([size]*ndim,dof)
     9   70.168 MiB    7.648 MiB       q3 = da.createGlobalVec()
     6   77.684 MiB   54.207 MiB       da = PETSc.DA().create([size]*ndim,dof)
     9  123.582 MiB   15.285 MiB       q3 = da.createGlobalVec()
     6  100.570 MiB   77.094 MiB       da = PETSc.DA().create([size]*ndim,dof)
     9  169.355 MiB   22.914 MiB       q3 = da.createGlobalVec()

These results are relatively sane: regardless of the number of DOFs, the ratio is 3-3.5.

Conclusion

So DAs are not lightweight. They cost as much as 3-8.5 Vecs, and their cost increases with the number of DOFs. It is not clear to me a priori why their cost should even depend on the number of DOFs.

In PyClaw, this means that the DAs always occupy much more memory than the Vecs themselves! This is becoming a serious bottleneck for our current 3D simulations on Shaheen.