---
authors: XG
plotly: true
---

# Parallelism in the DFPT formalism

## Atomic displacements, homogeneous electric fields, strain.

This tutorial aims at showing how to use the parallelism for all the properties
that are computed on the basis of the density-functional perturbation theory
(DFPT) part of ABINIT, i.e.:

  * responses to atomic displacements (to compute phonon frequencies, band structures, ... )
  * responses to homogeneous electric fields (dielectric constant, Born effective charges, Infra-red characteristics )
  * responses to strain (elastic constants, piezoelectric constants ...)

Such computations are realized when one of the input variables [[rfphon]],
[[rfelfd]] or [[rfstrs]] are non-zero, which activates [[optdriver]]=1.
You are supposed to be well-familiarized with such calculations before starting
the present tutorial. See the input variables described in [[varset:dfpt]] and
the tutorial [Response-Function 1](/tutorial/rf1) and subsequent tutorials.

You will learn about the basic implementation of parallelism for DFPT
calculations, then will execute a very simple, quick calculation for one
dynamical matrix for FCC aluminum, whose scaling is limited to a few dozen
computing cores, then you will execute a calculation whose scaling is much
better, but that would take a few hours in sequential, using a provided input file.

For the last section of that part, you would be better off having access
to more than 100 computing cores, although you might also change the input
parameters to adjust to the machine you have at hand. For the other parts of
the tutorial, a 16-computing-core machine is recommended, in order to perform
the scalability measurements.

You are supposed to know already some basics of parallelism in ABINIT,
explained in the tutorial [A first introduction to ABINIT in parallel](/tutorial/basepar).

This tutorial should take less than two hours to be done if a powerful parallel
computer is available.

[TUTORIAL_README]

## 1 The structure of the parallelism for the DFPT part of ABINIT

**1.1.** Let us recall first the basic structure of typical DFPT calculations summarized in box 1.

![Schema 1](paral_dfpt_assets/DFPT_Structure.png)

The step 1 is done in a preliminary ground-state calculation (either by an
independent run, or by computing them using an earlier dataset before the DFPT calculation).
The parallelisation of this step is examined in a separate tutorial.

The step 2 and step 3 are the time-consuming DFPT steps, to which the present
tutorial is dedicated, and for which the implementation of the parallelism
will be explained. They generate different files, and in particular, one (or several) DDB file(s).
As explained in related tutorials (see e.g.
[Response-Function 1](/tutorial/rf1)), several perturbations are usually treated in
one dataset (hence the "Do for each perturbation"-loop in step 2 of this
Schema). For example, in one dataset, although one considers only one phonon
wavevector, all the primitive atomic displacements for this wavevector (as
determined by the symmetries) can be treated in turn.

The step 4 refers to MRGDDB and ANADDB post-processing of the DDB generated by
steps 2 and 3. These are not time-consuming sections. No parallelism is needed.

**1.2.** The equations to be considered for the computation of the first-order
wavefunctions and potentials (step 2 of box 1), which is the really time-consuming part
of any DFPT calculation, are presented in box 2.

![Schema 2](paral_dfpt_assets/DFPT_Equations.png)

The parallelism currently implemented for the DFPT part of ABINIT is based on
the parallel distribution of the most important arrays: the set of first-order
and ground-state wavefunctions.
While each such wavefunction is stored completely in the memory linked to one
computing core (there is no splitting of different parts of one wavefunction
between different computing cores - neither in real space nor in reciprocal
space), different such wavefunctions might be distributed over different
computing cores. This easily achieves a combined k point, spin and band index
parallelism, as explained in box 3.

![Schema 3](paral_dfpt_assets/1WFs.png)

**1.3.** The parallelism over k points, band index and spin for the set of
wavefunctions is implemented in all relevant steps, except for the reading
(initialisation) of the ground-state wavefunctions (from an external file).
Thus, the most CPU time-consuming parts of the DFPT computations are parallelized.
The following are not parallelized:

  * input of the set of ground-state wavefunctions
  * computing the first-order change of potential from the first-order density,
    and similar operations that do not depend on the bands or k points.

In the case of small systems, the maximum achievable speed-up will be limited
by the input of the set of ground-state wavefunctions. For medium to large
systems, the maximum achievable speed-up will be determined by the operations
that do not depend on the k point and band indices.
Finally, the distribution of the set of ground-state wavefunctions to the
computing cores that need them is partly parallelized, as no parallelism over
bands can be exploited.

*Before continuing you might work in a different subdirectory as for the other
tutorials. Why not work_paral_dfpt?*

All the input files can be found in the *\$ABI_TESTS/tutoparal/Input* directory.
You might have to adapt them to the path of the directory in which you have decided to perform your runs.
You can compare your results with reference output files located in *\$ABI_TESTS/tutoparal/Refs*.

In the following, when "(mpirun ...) abinit" appears, you have to use a
specific command line according to the operating system and architecture of
the computer you are using. This can be for instance: mpirun -np 16 abinit

## 2 Computation of one dynamical matrix (q =0.25 -0.125 0.125) for FCC aluminum

We start by treating the case of a small systems, namely FCC aluminum, for
which there is only one atom per unit cell. Of course, many k points are needed, since this is a metal.

**2.1.** The first step is the pre-computation of the ground state
wavefunctions. This is driven by the files *tdfpt_01.abi*.
You should edit it and examine it.

{% dialog tests/tutoparal/Input/tdfpt_01.abi %}

One relies on a k-point grid of 8x8x8 x 4 shifts (=2048 k points), and 5 bands.
The k-point grid sampling is well converged, actually.
For this ground-state calculation, symmetries can be used to reduce
drastically the number of k points: there are 60 k points in the irreducible
Brillouin zone (this cannot be deduced from the examination of the input file, though).
In order to treat properly the phonon calculation, the number of bands is larger than the
default value, that would have given [[nband]]=3. Indeed, several of the unoccupied bands 
plays a role in the response calculations in the case of etallic occupations.
For example, the acoustic sum rule might be largely violated when too few unoccopied 
bands are treated. 

This calculation is very fast, actually.
You can launch it:

    mpirun -n 4  abinit tdfpt_01.abi > log &

A reference output file is available in *\$ABI_TESTS/tutoparal/Refs*, under
the name *tdfpt_01.abo*. It was obtained using 4 computing cores, and took a few seconds.

**2.2.** The second step is the DFPT calculation, see the file *tdfpt_02.abi*

{% dialog tests/tutoparal/Input/tdfpt_02.abi %}

There are three perturbations (three atomic displacements). For the two first
perturbations, no symmetry can be used, while for the third, two symmetries
can be used to reduce the number of k points to 1024. Hence, for the perfectly
scalable sections of the code, the maximum speed up is 5120 (=1024 k points *
5 bands), if you have access to 5120 computing cores. However, the sequential
parts of the code dominate at a much, much lower value. Indeed, the sequential
parts is actually a few percents of the code on one processor, depending on
the machine you run. The speed-up might saturate beyond 8...16 (depending on
the machine). Note that the number of processors that you use for this second step
is independent of the number of processors that you used for the first step.
The only relevant information from the first step is the *_WFK file.

First copy the output of the ground-state calculation so that it can be used
as the input of the DFPT calculation:

    cp tdfpt_01.o_WFK tdfpt_02.i_WFK

(A _WFQ file is not needed, as all GS wavefunctions at k+q are present in the GW wavefuction at k).
Then, you can launch the calculation:

    mpirun -n 4 abinit tdfpt_02.abi > tdfpt_02.log &

A reference output file is given in *\$ABI_TESTS/tutoparal/Refs*, under the name
*tdfpt_02.abo*. Edit it, and examine some information.
The calculation has been made with four computing cores:

```
-   mpi_nproc: 4, omp_nthreads: -1 (-1 if OMP is not activated)
```

The wall clock time is less than 50 seconds :

```
-
- Proc.   0 individual time (sec): cpu=         28.8  wall=         28.9

================================================================================

 Calculation completed.
.Delivered   0 WARNINGs and   0 COMMENTs to log file.
+Overall time at end (sec) : cpu=        115.4  wall=        115.8
```

The major result is the phonon frequencies:

      Phonon wavevector (reduced coordinates) :  0.25000 -0.12500  0.12500
     Phonon energies in Hartree :
       6.521506E-04  7.483301E-04  1.099648E-03
     Phonon frequencies in cm-1    :
    -  1.431305E+02  1.642395E+02  2.413447E+02

**2.3.** Because this test case is quite fast, you should play a bit with it.
In particular, run it several times, with an increasing number of computing
cores (let us say, up to 40 computing cores, at which stage you should have
obtained a saturation of the speed-up).
You should be able to obtain a decent speedup up to 8 processors, then the gain becomes more and more marginal.
Note however that the result is independent (to an exquisite accuracy) of the number of computing cores that is used

Let us explain the timing section. It is present a bit before the end of the output file:

    -
    - For major independent code sections, cpu and wall times (sec),
    -  as well as % of the time and number of calls for node 0-
    -<BEGIN_TIMER mpi_nprocs = 4, omp_nthreads = 1, mpi_rank = 0>
    - cpu_time =           28.8, wall_time =           28.9
    -
    - routine                        cpu     %       wall     %      number of calls  Gflops    Speedup Efficacity
    -                                                                  (-1=no count)
    - fourwf%(pot)                  14.392  12.5     14.443  12.5         170048      -1.00        1.00       1.00
    - nonlop(apply)                  2.787   2.4      2.802   2.4         125248      -1.00        0.99       0.99
    - nonlop(forces)                 2.609   2.3      2.620   2.3          64000      -1.00        1.00       1.00
    - fourwf%(G->r)                  2.044   1.8      2.052   1.8          47124      -1.00        1.00       1.00
    - dfpt_vtorho-kpt loop           1.174   1.0      1.177   1.0             21      -1.00        1.00       1.00
    - getgh1c_setup                  1.111   1.0      1.114   1.0           8960      -1.00        1.00       1.00
    - mkffnl                         1.087   0.9      1.092   0.9          20992      -1.00        1.00       1.00
    - projbd                         1.023   0.9      1.036   0.9         286336      -1.00        0.99       0.99
    - dfpt_vtowfk(contrib)           0.806   0.7      0.805   0.7             -1      -1.00        1.00       1.00
    - others (120)                  -2.787  -2.4     -2.825  -2.4             -1      -1.00        0.99       0.99
    -<END_TIMER>
    -
    - subtotal                      24.246  21.0     24.317  21.0                                  1.00       1.00

    - For major independent code sections, cpu and wall times (sec),
    - as well as % of the total time and number of calls
    
    -<BEGIN_TIMER mpi_nprocs = 4, omp_nthreads = 1, mpi_rank = world>
    - cpu_time =         115.3, wall_time =         115.6
    -
    - routine                         cpu     %       wall     %      number of calls Gflops    Speedup Efficacity
    -                                                                  (-1=no count)
    - fourwf%(pot)                  51.717  44.8     51.909  44.9         679543      -1.00        1.00       1.00
    - dfpt_vtorho:MPI               12.267  10.6     12.292  10.6             84      -1.00        1.00       1.00
    - nonlop(apply)                 10.092   8.8     10.149   8.8         500343      -1.00        0.99       0.99
    - nonlop(forces)                 9.520   8.3      9.562   8.3         256000      -1.00        1.00       1.00
    - fourwf%(G->r)                  7.367   6.4      7.398   6.4         187320      -1.00        1.00       1.00
    - dfpt_vtorho-kpt loop           4.182   3.6      4.193   3.6             84      -1.00        1.00       1.00
    - getgh1c_setup                  3.955   3.4      3.967   3.4          35840      -1.00        1.00       1.00
    - mkffnl                         3.911   3.4      3.929   3.4          83968      -1.00        1.00       1.00
    - projbd                         3.735   3.2      3.780   3.3        1144046      -1.00        0.99       0.99
    - dfpt_vtowfk(contrib)           2.754   2.4      2.748   2.4             -4      -1.00        1.00       1.00
    - getghc-other                   1.861   1.6      1.790   1.5             -4      -1.00        1.04       1.04
    - pspini                         0.861   0.7      0.865   0.7              4      -1.00        0.99       0.99
    - newkpt(excl. rwwf   )          0.754   0.7      0.757   0.7             -4      -1.00        1.00       1.00
    - others (116)                 -14.132 -12.3    -14.220 -12.3             -1      -1.00        0.99       0.99
    -<END_TIMER>

    - subtotal                      98.845  85.7     99.120  85.7                                  1.00       1.00

It is made of two groups of data. The first one corresponds to the analysis of
the timing for the computing core (node) number 0. The second one is the sum
over all computing cores of the data of the first group. Note that there is a
factor of four between these two groups, reflecting that the load balance is good.

Let us examine the second group of data in more detail. It corresponds to a
decomposition of the most time-consuming parts of the code. Note that the
subtotal is 85.7 percent, thus the statistics is not very accurate, as it should be close to 100%.
Actually, as of ABINIT v9, there is must be a bug in the timing decomposition, since, 
e.g. there is a negative time announced for the "others" subroutines.

Anyhow, several of the most time-consuming parts are directly related to application of the Hamiltonian to wavefunctions,
namely, fourwf%(pot) (application of the local potential, which implies two Fourier transforms), nonlop(apply) 
(application of the non-local part of the Hamiltonian), nonlop(forces) (computation of the non-local part of the
interatomic forces), fourwf%(G->r) (fourier transform needed to build the first-order density). Also, quite noticeable
is dfpt_vtorho:MPI , synchronisation of the MPI parallelism. 

A study of the speed-up brought by the k-point parallelism for this simple test case
gives the following behaviour, between 1 and 40 cores:

<div id="plotly_plot" style="width:90%;height:450px;"></div>
<script>
$(function() {
    Plotly.newPlot(document.getElementById('plotly_plot'), 
        [{ x: [1, 2, 4, 6, 8, 12, 16, 24, 32, 40], y: [1, 2, 4, 6, 8, 12, 16, 24, 32, 40], name: 'Ideal'},
         { x: [1, 2, 4, 6, 8, 12, 16, 24, 32, 40], y: [1, 1.92, 3.75, 5.50, 7.17, 9.34, 11.1, 14.4, 17.8, 19.7], name: 'Observed' }], 
        { title: "Parallel speed-up", 
          xaxis: {title:'Number of cores'} }
    );
});
</script>

At 8 processors, one gets a speed-up of 7.17, which is quite decent (about 90% efficiency),
but 16 processors, the speed-up is only 11.1 (about 70% efficiency), and the efficiency is below 50% at 40 processors,
for a speed-up of 19.7 ..

## 3 Computation of one perturbation for a slab of 29 atoms of barium titanate

**3.1.** This test, with 29 atoms, is slower, but highlights other aspects 
of the DFPT parallelism than the Al FCC case. 
It consists in the computation of one perturbation at [[qpt]] 0.0 0.25
0.0 for a 29 atom slab of barium titanate, artificially terminated by a double
TiO<sub>2</sub> layer on each face, with a reasonable k-point sampling of the Brillouin zone.

The symmetry of the system and perturbation will allow to decrease this
sampling to one quarter of the Brillouin zone. E.g. with the k-point sampling
[[ngkpt]] 4 4 1, there will be actually 4 k-points in the irreducible Brillouin
zone for the Ground state calculations. For the DFPT case, only one (binary) symmetry
will survive, so that, after the calculation of the frozen-wavefunction part
(for which no symmetry is used), the self-consistent part will be done with 8
k points in the corresponding irreducible Brillouin zone. Beyond 8 cores, the parallelisation
will be done also on the bands. This will prove to be much more dependent
on the computer architecture than the (low-communication) parallelism over k-points.

With the sampling 8 8 1, there will be 32 k points in the irreducible Brillouin zone for the DFPT
case. This will allow potentially to use efficiently a larger number of processors, provided the
computer architecture and network is good enough.

There are 116 occupied bands. For the ground state calculation, 4 additional conduction
bands will be explicitly treated, which will allow better SCF stability thanks to [[iprcel]] 45.
Note that the value of [[ecut]] that is used in the
present tutorial is too low to obtain physical results (it should be around 40 Hartree).
Also, only one atomic displacement is considered, so that the phonon frequencies
delivered at the end of the run are meaningless.

As in the previous case, a preparatory ground-state calculation is needed.
We use the input variable [[autoparal]]=1 . It does not delivers the best repartition of
processors among [[np_spkpt]], [[npband]] and [[npfft]], but achieves a decent repartition, usually within a factor of two.
With 24 processors, it selects [[np_spkpt]]=4 (optimal), [[npband]]=3 and [[npfft]]=2, while [[npband]]=6 and [[npband]]=1 would do better.
For information, the speeup going from 24 cores to 64 cores is 1.76, not quite the increase of number of processor (2.76).
Anyhow, the topics of the tutorial is not the GS calculation.

The input files are provided, in the directory *\$ABI_TESTS/tutoparal/Input*.
The preparatory step is driven by *tdfpt_03.abi*. The real
(=DFPT) test case is driven by *tdfpt_04.abi*. The
reference output files are present in *\$ABI_TESTS/tutoparal/Refs*:
*tdfpt_03_MPI24.abo* and *tdfpt_04_MPI24.abo*. The naming convention is such that the
number of cores used to run them is added after the name of the test: the
*tdfpt_03.abi* files are run with 24 cores.
The preparatory step takes about 3 minutes, and the DFPT step takes about
3 minutes as well.

{% dialog tests/tutoparal/Input/tdfpt_03.abi tests/tutoparal/Input/tdfpt_04.abi %}

You can run now these test cases. For tdfpt_03, with [[autoparal]]=1, 
you will be able to run on different numbers of processors compatible with [[nkpt]]=4,
[[nband]]=120 and [[ngfft]]=[30 30 192], detected by ABINIT. Alternatively, you might decide to explicitly 
define [[np_spkpt]], [[npband]] and [[npfft]].
At variance, for tdfpt_04, no adaptation of the input file is
needed to be able to run on an arbitrary number of processors.
To launch the ground-state computation, type:

    mpirun -n 24 abinit tdfpt_03.abi > log &

then copy the output of the ground-state calculation so that it can be used as
the input of the DFPT calculation:

    mv tdfpt_03o_WFK.nc tdfpt_04i_WFK.nc

and launch the calculation:

    mpirun -n 24 abinit tdfpt_04.abi > log &

Now, examine the obtained output file for test 04, especially the timing.

In the reference file *\$ABI_TESTS/tutoparal/Refs/tdfpt_04_MPI24.abo*,
with 24 computing cores, the timing section delivers:

    - For major independent code sections, cpu and wall times (sec),
    -  as well as % of the time and number of calls for node 0-
    -<BEGIN_TIMER mpi_nprocs = 24, omp_nthreads = 1, mpi_rank = 0>
    - cpu_time =          159.9, wall_time =          160.0
    -
    - routine                        cpu     %       wall     %      number of calls  Gflops    Speedup Efficacity
    -                                                                  (-1=no count)
    - projbd                        46.305   1.2     46.345   1.2          11520      -1.00        1.00       1.00
    - nonlop(apply)                 42.180   1.1     42.183   1.1           5760      -1.00        1.00       1.00
    - dfpt_vtorho:MPI               25.087   0.7     25.085   0.7             30      -1.00        1.00       1.00
    - fourwf%(pot)                  22.435   0.6     22.436   0.6           6930      -1.00        1.00       1.00
    - nonlop(forces)                 5.485   0.1      5.486   0.1           4563      -1.00        1.00       1.00
    - fourwf%(G->r)                  4.445   0.1      4.446   0.1           2340      -1.00        1.00       1.00
    - pspini                         2.311   0.1      2.311   0.1              1      -1.00        1.00       1.00

    <...>

    - For major independent code sections, cpu and wall times (sec),
    - as well as % of the total time and number of calls

    -<BEGIN_TIMER mpi_nprocs = 24, omp_nthreads = 1, mpi_rank = world>
    - cpu_time =        3826.4, wall_time =        3828.0
    -
    - routine                         cpu     %       wall     %      number of calls Gflops    Speedup Efficacity
    -                                                                  (-1=no count)
    - projbd                      1238.539  32.4   1239.573  32.4         278400      -1.00        1.00       1.00
    - nonlop(apply)               1012.783  26.5   1012.982  26.5         139200      -1.00        1.00       1.00
    - fourwf%(pot)                 544.047  14.2    544.201  14.2         167040      -1.00        1.00       1.00
    - dfpt_vtorho:MPI              450.196  11.8    450.170  11.8            720      -1.00        1.00       1.00
    - nonlop(forces)               131.081   3.4    131.129   3.4         108576      -1.00        1.00       1.00
    - fourwf%(G->r)                107.885   2.8    107.930   2.8          55680      -1.00        1.00       1.00
    - pspini                        54.945   1.4     54.943   1.4             24      -1.00        1.00       1.00

    <...>

    - others (99)                  -98.669  -2.6    -99.848  -2.6             -1      -1.00        0.99       0.99
    -<END_TIMER>

    - subtotal                    3569.873  93.3   3570.325  93.3                                  1.00       1.00

You will notice that the run took about 160 seconds (wall clock time)..
The sum of the major independent code sections is reasonably
close to 100%. You might now explore the behaviour of the CPU time for
different numbers of compute cores (consider values below and above 24
processors). Some time-consuming routines will benefit from the parallelism, some other will not.

The kpoint + band parallelism will efficiently work for many important sections
of the code: projbd, fourwf%(pot), nonlop(apply), fourwf%(G->r).
In this test, the product nkpt (the effective number of k points for the current
perturbation) times nband is 8*120=960. Of course, the total speed-up will
saturate well below this value, as there are some non-parallelized sections of the code.

In the above-mentioned list, the kpoint+band parallelism cannot be exploited
(or is badly exploited) in several sections of the code : "dfpt_vtorho:MPI",
about 12 percents of the total time of the run on 24 processors, "pspini", about 1.4 percent. 
This amounts to about 1/8 of the total.
However, the scalability of the band parallelisation is rather poor, and effective saturation
in this case already happens at 16 processor.

A study of the speed-up brought by the combined k-point and band parallelism for this test case
on a 2 AMD EPYC 7502 machine (2 CPUS, each with 32 cores)
gives the following behaviour, between 1 and 40 cores:

<div id="plotly_plot2" style="width:90%;height:450px;"></div>
<script>
$(function() {
    Plotly.newPlot(document.getElementById('plotly_plot2'), 
        [{ x: [1, 2, 4, 6, 8, 12, 16, 24, 32, 40], y: [1, 2, 4, 6, 8, 12, 16, 24, 32, 40], name: 'Ideal'},
         { x: [1, 2, 4, 6, 8, 12, 16, 24, 32, 40], y: [1, 1.97, 3.93, 3.86, 7.67, 7.53, 14.77, 16.44, 15.47, 14.37], name: 'Observed' }],
        { title: "Parallel speed-up", 
          xaxis: {title:'Number of cores'} }
    );
}); 
</script>

The additional band parallelism is very efficient when running with 16 cores, bringing 14.77 speed-up,
while using only the k point parallelism, with 6 cores, gives 7.67 speed-up. However, the behaviour
is disappointing beyond 16 cores, or even for a number of processors which is not a multiple of 8.

Such a behaviour might be different on your machine.

**3.2.** A better parallelism can be seen if the number of k-points is brought
back to a converged value (8x8x1). Again, 
Try this if you have more than 100 processors at hand.

Set in your input files *tdfpt_03.abi* and *tdfpt_04.abo* :

       ngkpt 8 8 1    ! This should replace ngkpt 4 4 1

and launch again the preliminary step, then the DFPT step. Then, you can
practice the DFPT calculation by varying the number of computing cores. For
the latter, you could even consider varying the number of self-consistent
iterations to see the initialisation effects (small value of nstep), or target
a value giving converged results ([[nstep]] 50 instead of [[nstep]] 18). The energy
cut-off might also be increased (e.g. [[ecut]] 40 Hartree gives a much better
value). Indeed, with a large value of k points, and large value of nstep, you
might be able to obtain a speed-up of more than one hundred for the DFPT
calculation, when compared to a sequential run (see below). Keep track of the
time for each computing core number, to observe the scaling.

On a machine with a good communication network, the following results were observed in 2011.
The Wall clock timing decreases from

    - Proc.   0 individual time (sec): cpu=       2977.3  wall=       2977.3

with 16 processors to

    - Proc.   0 individual time (sec): cpu=        513.2  wall=        513.2

with 128 processors.

The next figure presents the speed-up of a typical calculation with increasing
number of computing cores (also, the efficiency of the calculation).

![Schema 1](paral_dfpt_assets/Speedup.jpeg)

Beyond 300 computing cores, the sequential parts of the code start to dominate.
With more realistic computing parameters ([[ecut]] 40), they dominate only beyond 600 processors.

However, on the same (recent, but with slow connection beyond 32 cores) computer than for the [[ngkpt]] 4 4 1 case,
the saturation sets in already beyond 16 cores, with the following behaviour (the reference is taken with respect to the timing at 4 processors):

<div id="plotly_plot3" style="width:90%;height:450px;"></div>
<script>
$(function() {
    Plotly.newPlot(document.getElementById('plotly_plot3'),
        [{ x: [1,4, 8, 12, 16, 24, 32, 40], y: [1, 4, 8, 12, 16, 24, 32, 40], name: 'Ideal'},
         { x: [4, 8, 12, 16, 24, 32, 40], y: [4, 7.86, 10.46, 14.92, 13.82, 16.15, 15.27], name: 'Observed' }],
        { title: "Parallel speed-up",
          xaxis: {title:'Number of cores'} }
    );
});
</script>

Thus, it is very important that you gain some understanding of the scaling of your typical runs
for your particular computer, and that you know the parameters (especially [[nkpt]]) of your calculations. 
Up to 4 or 8 cores, the ABINIT scaling will usually be very good, if
k-point parallelism is possible. In the range between 10 and 100 cores, the speed-up might still be good,
but this will depend on details.

This last example is the end of the present tutorial. 
The basics of the current implementation of the parallelism for the DFPT part of ABINIT have been explained,
then you have explored two test cases: one for a small cell
materials, with lots of k points, and another one, medium-size, in which the k
point and band parallelism must be used. It might reveal efficient, but this will depend on the detail of your calculation
and your computer architecture.