Parallelism for the ground state¶

Explore the various features of the KGB parallelization scheme¶

This tutorial will indicate how to perform GS calculations on hundreds processors using ABINIT.

You will learn how to use some keywords related to the “KGB” parallelization scheme where “K” stands for “k-point”, “G” refers to the wavevector of a planewave, and “B” stands for a “Band”. On the one hand, you will see how to improve the speedup of your calculations and, on the other hand, how to increase the convergence speed of a self consistent field calculation.

This lesson should take about 1.5 hour and requires access to at least a 200 CPU core parallel computer.

You are supposed to know already some basics of parallelism in ABINIT, explained in the tutorial A first introduction to ABINIT in parallel. On the contrary, the present tutorial is not a prerequisite for other tutorials about parallelism in ABINIT.

1 Introduction¶

Before continuing you might work in a different subdirectory, as for the other lessons. Why not “work_paral_gspw”?

All the input files can be found in the ~abinit/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 ~abinit/tests/tutoparal/Refs.

In the following, when “run ABINIT over nn CPU cores” 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 -n nn abinit < abinit.files


or the use of a specific submission file. Some scripts are given as examples in the directory ~abinit/doc/tutorial/lesson_paral_gspw/. You can adapt them to your own calculations.

When the size of the system increases up to 100 or 1000 atoms, it is usually impossible to perform ab initio calculations with a single computing core. This is because the basis sets used to solve the problem (PWs, bands, …) increase proportionally (linearly, as the square, or even exponentially…). The trouble has two origins:

• the memory i.e. the amount of data stored in RAM
• the computation time, with specific bottlenecks which are for example the eigensolver, the 3dim-FFTs…

Therefore, it is generally mandatory to adopt a parallelization strategy. That is to say: (i) to distribute the data in a MPI-sense on a large number of processors, and (ii) to parallelize the routines responsible for the increase of the computation time.

In this tutorial, we will go beyond the simple k-point and spin parallelism explained in the tutorial A first introduction to ABINIT in parallel and show:

• how to improve performance by using a large number of processors, even when the number of k-points is not large,

• how to decrease the computation time for a given number of processors. The aim is twofold:

1. reduce the time needed to perform one electronic iteration (to improve the efficiency)
2. reduce the number of electronic iterations (to improve the convergence)
• how to use other keywords or features related to the KGB parallelization scheme.

The tests are performed on a system with 108 atoms of gold; this is a benchmark used for a long time during the development and the implementation of the KGB parallelization. With respect to the input file used for the benchmark, the cutoff energy is strongly reduced in this tutorial, for practical reasons. For real tests, you can see the results (in particular the scaling) in:

• the publication concerning the KGB parallelisation: F. Bottin, S. Leroux, A. Knyazev, G. Zerah, Comput. Mat. Science 42, 329, (2008) “Large scale ab initio calculations based on three levels of parallelization “(available on Arxiv.org).

• the Abinit paper: X. Gonze, B. Amadon, P.-M. Anglade, J.-M. Beuken, F. Bottin, P. Boulanger, F. Bruneval,’ D. Caliste, R. Caracas, M. Cote, T. Deutsch, L. Genovese, Ph. Ghosez, M. Giantomassi, S. Goedecker, D.R. Hamann, P. Hermet, F. Jollet, G. Jomard, S. Leroux, M. Mancini, S. Mazevet, M.J.T. Oliveira, G. Onida, Y. Pouillon, T. Rangel, G.-M. Rignanese, D. Sangalli, R. Shaltaf, M. Torrent, M.J. Verstraete, G. Zerah, J.W. Zwanziger , Computer Phys. Commun. 180, 2582-2615 (2009). “ABINIT : First-principles approach of materials and nanosystem properties.”

• the presentation of F. Bottin at the ABINIT workshop 2007 (Monday 29, session 2).

You are strongly suggested to read these documents before beginning this tutorial. You might learn a lot of useful things.

However, even if you scan them with attention, you won’t learn the answer to the most frequently asked question. Why this parallelization is named KGB? We don’t know. Some people say that the reason comes from the K-point, the plane waves G and the Bands, but you can imagine everything you want.

2 A simple way to begin…¶

One of the most simple way to launch the KGB parallelization in ABINIT is to add just one input variable to the sequential input file. This is paral_kgb and controls everything concerning the KGB parallelization: the use of the LOBPCG eigensolver (wfoptalg=4 or 14) of A. Knyazev, the parallel 3dim-FFT (fftalg=401) written by S. Goedecker, and some other tricks… At the time of writing, you still have to define the number or processors needed on each level of the KGB parallelization: npband, npfft and npkpt.

If ABINIT is not yet able to handle directly the number of cores and to launch from scratch an efficient distribution of processors, you can already use this parallelization as a black box. It is possible to get a good estimation of the most efficient processor distribution by performing a simple sequential run. This is done by simply adding to the input file (in addition to paral_kgb=1) autoparal=1 and max_ncpus=”the maximum number of processors you want”.

Note

max_ncpus is a new variable introduced in Abinit version 7.6.2. Prior to this version, one had to use paral_kgb=-“the maximum number of processors you want” to specify the maximum number of processors.

In order to do that, copy the file tests/tutoparal/Input/tgspw_01.in and the related tgspw_01.files file in your working directory.

Then run ABINIT on 1 CPU core.
At the end of the log file tgspw_01.log, you will see:

   npimage|       npkpt|    npspinor|       npfft|      npband|     bandpp |       nproc|      weight|
1 - >    1|   1 ->    1|   1 ->    1|   1 ->   22|   1 ->  108|   1 ->   65|   8 ->  108|   1 ->  108|
1|           1|           1|          12|           9|           1|         108|      55.13 |
1|           1|           1|          10|           9|           1|          90|      54.98 |
1|           1|           1|           9|           9|           1|          81|      54.85 |
1|           1|           1|           9|          12|           1|         108|      54.37 |
1|           1|           1|          12|           9|           2|         108|      52.86 |
......................


A weight is assigned to each distribution of processors. As indicated in the documentation, you are advised to select a processor distribution with a higher weight. If we just focus on npband and npfft, we see that for 108 processors the recommended distribution is (12x9).

In a second step you can launch ABINIT in parallel on 108 processors by changing your input file as follows:

- paral_kgb 1 autoparal 1 max_ncpus 108
+ paral_kgb 1 npband 12 npfft 9


You can now perform your calculations using the KGB parallelization in ABINIT. But somehow, you did it without understanding how you got the result…

3 … which is however coherent with a more sophisticated method¶

In this part we will try to recover the previous result, but with a more comprehensive understanding of what is happening. As shown above, the couple (npbandxnpfft) of input variables can have various values: (108x1), (54x2), (36x3), (27x4), (18x6) and (12x9). But also (9x12) … which is not indicated. In order to perform these seven calculations you can use the input file tgspw_02.in (and tgspw_02.files) and change the line corresponding to the processor distribution. A first calculation with:

+ npband 108 npfft 1


A second one with another distribution:

- npband 108 npfft 1
+ npband  54 npfft 2


And so on… Alternatively, this can be performed using a shell script including:

>> cp tgspw_02.in tmp.file
>> echo "npband 108 npfft 1" >> tgspw_02.in
>> mpirun -n 108 abinit < tgspw_02.files
>> cp tgspw_02.out tgspw_02.108-01.out
>> cp tmp.file tgspw_02.in
>> echo "npband 54 npfft 2" >> tgspw_02.in
>> ...


By reference to the couple (npbandxnpfft), all these results are named: tgspw_02.108-01.out, tgspw_02.054-02.out, tgspw_02.036-03.out, tgspw_02.027-04.out, tgspw_02.018-06.out, tgspw_02.012-09.out and tgspw_02.009-12.out. The timing of each calculation can be retrieved using the shell command:

>> grep Proc *out
>> tgspw_02.009-12.out:- Proc.   0 individual time (sec): cpu=         88.3  wall=         88.3
>> tgspw_02.012-09.out:- Proc.   0 individual time (sec): cpu=         75.2  wall=         75.2
>> tgspw_02.018-06.out:- Proc.   0 individual time (sec): cpu=         63.7  wall=         63.7
>> tgspw_02.027-04.out:- Proc.   0 individual time (sec): cpu=         69.9  wall=         69.9
>> tgspw_02.036-03.out:- Proc.   0 individual time (sec): cpu=        116.0  wall=        116.0
>> tgspw_02.054-02.out:- Proc.   0 individual time (sec): cpu=        104.7  wall=        104.7
>> tgspw_02.108-01.out:- Proc.   0 individual time (sec): cpu=        141.5  wall=        141.5


As far as the timing is concerned, the best distributions are then the ones proposed above in section 2.: that is to say the couples (18x6) and (27x4). So the prediction was pretty good.

Up to now, we have not learned more than in section 1.. We have so far only considered the timing (the efficiency) of one electronic step, or 10 electronic steps as this is limited in the input file. However, when the npband value is modified, the size of the block in LOBPCG changes, and finally the solutions of this blocked eigensolver are also affected. In other words, we never had in mind that the convergence of these calculations is also strongly important. One calculation can be the quickest on one step but the slowest at the end of the convergence because it takes many more steps. In order to see this without performing any additional calculations, we can have a look at the degree of convergence at the end of the calculations we already have. The last iterations of the SCF loop give:

>> grep "ETOT 10" *.out
>> tgspw_02.009-12.out: ETOT 10  -3754.4454784191    -1.549E-03 7.222E-05 1.394E+00
>> tgspw_02.012-09.out: ETOT 10  -3754.4458434046    -7.875E-04 6.680E-05 2.596E-01
>> tgspw_02.018-06.out: ETOT 10  -3754.4457793663    -1.319E-03 1.230E-04 6.962E-01
>> tgspw_02.027-04.out: ETOT 10  -3754.4459048995    -1.127E-03 1.191E-04 5.701E-01
>> tgspw_02.036-03.out: ETOT 10  -3754.4460493339    -1.529E-03 7.121E-05 3.144E-01
>> tgspw_02.054-02.out: ETOT 10  -3754.4460393029    -1.646E-03 7.096E-05 7.284E-01
>> tgspw_02.108-01.out: ETOT 10  -3754.4464631635    -6.162E-05 2.151E-05 7.457E-02


The last column indicates the convergence of the density (or potential) residual. You can see that this quantity is the smallest when npband is the highest. This result is expected: the convergence is better when the size of the block is as large as possible, so need for re-orthogonalization among band processor pools is minimized. But the best convergence is obtained for the (108x1) distribution… when the worst timing is measured.

So, you face a dilemma. The calculation with the smallest number of iterations (the best convergence) is not the best concerning the timing of one iteration (the best efficiency), and vice versa… So you have to check both of these features for all the processor distributions. On one hand, the timing of one iteration and, on the other hand, the number of iterations needed to converge. The best choice is a compromise between them, not necessarily the independent optima.

In the following we will choose the (27x4) couple, because this one definitively offers more guarantees concerning the convergence and the timing even if the (18x6) one is slightly quicker per electronic step.

Note: you can verify that the convergence is not changed when the npfft value is modified. The same results will be obtained, step by step.

4 Meaning of bandpp: part 1 (convergence)¶

We have seen in the previous section that the best convergence is obtained when the size of the block is the largest. This size is related to the npband input variable. But not only. It is possible to increase the size of the block without increasing drastically the number of band processors. This means that it’s possible to decrease the number of electronic steps without increasing strongly the timing of one electronic step. For systems with peculiar convergence, when some trouble leads the calculation to diverge, this is not just useful but indispensable to converge at all.

The input variable enabling an increasing of the size block without increasing the number of band processors is named bandpp. The size block is then defined as: bandppxnpband. In the following, we keep the same input file as previously and add:

- nstep 10
+ nstep 20
+ npband 27 npfft 4


You can copy the input file tgspw_03.in then run ABINIT over 108 cores with tgspw_02.027-04.out, bandpp=1 and bandpp=2. The output files will be named tgspw_03.bandpp1.out and tgspw_03.bandpp2.out, respectively. A comparison between these two files shows that the convergence is better in the second case. The convergence is even achieved before the input file limit of 20 electronic iterations. Quod Erat Demonstrandum:

For a given number of processors, it is possible to improve the convergence by increasing bandpp.

We can also compare the result obtained for the (27x4) distribution and bandpp=2 with the (54x2) one and bandpp=1. Use the same input file and add:

- npband 27 npfft 4
+ npband 54 npfft 2 bandpp 1


Then run ABINIT over 108 cores and copy the output file to tgspw_03.054-02.out. Perform a diff (with vimdiff for example) between the two output files tgspw_03.bandpp1.out and tgspw_03.054-02.out. You can convince yourself that the two calculations (54x2) with bandpp=1 and (27x4) with bandpp=2, give exactly the same convergence. This result is expected, since the sizes of the block are equal (to 54) and the number of FFT processors npfft does not affect the convergence.

It is possible to modify the distribution of processors, without changing the convergence, by reducing npband and increasing bandpp proportionally.

5 Meaning of bandpp: part 2 (efficiency)¶

In the previous section, we showed that the convergence doesn’t change if bandpp and npband change in inverse proportions. What about the influence of bandpp if you fix the distribution? Two cases have to be treated separately.

You can see, in the previous calculations of section 4, that the timing increases when bandpp increases:

>> grep Proc tgspw_03.bandpp1.out tgspw_03.bandpp2.out
>> tgspw_03.bandpp1.out:- Proc.   0 individual time (sec): cpu=        121.4  wall=        121.4
>> tgspw_03.bandpp2.out:- Proc.   0 individual time (sec): cpu=        150.7  wall=        150.7


while there are fewer electronic iterations for bandpp=2 (19) than for bandpp=1 (20). If you perform a diff between these two files, you will see that the increase in time is essentially due to the section “zheegv-dsyegv”.

>> grep "zheegv-dsyegv" tgspw_03.bandpp1.out tgspw_03.bandpp2.out
>> tgspw_03.bandpp1.out:- zheegv-dsyegv               1321.797  10.1   1323.215  10.1         513216
>> tgspw_03.bandpp2.out:- zheegv-dsyegv               5166.002  31.8   5164.574  31.8         244944


The “zheegv-dsyegv” is a part of the LOBPCG algorithm which is performed in sequential, so the same calculation is done on each processor. In the second calculation, the size of the block being larger (27x2=54) than in the first (27), the computational time of this diagonalization is more expensive. To sum up, the timing of a single electronic step increases by increasing bandpp, but the convergence improves.

Do not increase too much the bandpp value, unless you decrease proportionally npband or if you want to improve the convergence whatever the cost in total timing.

The only exception is when istwfk=2, i.e. when real wavefunctions are employed. This occurs when the Gamma point alone is used to sample the Brillouin Zone. You can use the input file tgspw_04.in in order to check that. The input is modified with respect to the previous input files in order to be more realistic and use only the Gamma point. Add bandpp=1,2,4 or 6 in the input file tgspw_04.in and run ABINIT in each case over 108 cores. You will obtain four output files named tgspw_04.bandpp1.out, tgspw_04.bandpp2.out, tgspw_04.bandpp4.out and tgspw_04.bandpp6.out in reference to bandpp. If you compare the outputs of these calculations:

>> grep Proc *out
>> tgspw_04.bandpp1.out:- Proc.   0 individual time (sec): cpu= 61.4  wall= 61.4
>> tgspw_04.bandpp2.out:- Proc.   0 individual time (sec): cpu= 49.0  wall= 49.0
>> tgspw_04.bandpp4.out:- Proc.   0 individual time (sec): cpu= 62.5  wall= 62.5
>> tgspw_04.bandpp6.out:- Proc.   0 individual time (sec): cpu= 75.3  wall= 75.3


you can see that the timing decreases for bandpp=2 and increases thereafter. This behaviour comes from the FFTs. For bandpp=2, the real wavefunctions are associated in pairs in the complex FFTs, leading to a reduction by a factor of 2 of the timing in this part (you can see this reduction by a diff of the output files). Above bandpp=2, there is longer any gain in the FFTs, whereas some significant losses in computational time appear in the “zheegv-dsyegv” section.

When calculations are performed at the Gamma point, you are strongly encouraged to use bandpp=2… or more if you need to improve the convergence whatever the timing.

6 The KGB parallelization¶

Up to now, we only performed a GB parallelization. This implies parallelization over 2 levels of PWs or over 2 levels of bands and FFTs, for different sections of the code (see the paper or presentation). If the system has more than 1 k-point, one can add a third level of parallelization and perform a real KBG parallelization. There is no additional difficulty in adding processors on this level. In order to explain the procedure, we restart with the same input file that was used in section 3 and add a denser M-P grid (see the input file tgspw_05.in). In this case, the system has 4 k-points in the IBZ so the calculation can be parallelized over (at most) 4 k-point processors. This is done using the npkpt input variable:

+ npkpt 4


This implies we use four times more processors than before, so run ABINIT over 432 CPU cores. The timing obtained in the output file tgspw_05.out:

>> grep Proc tgspw_05.out
>> - Proc.   0 individual time (sec): cpu=         87.3  wall=         87.3


is quasi-identical to the one obtained for 1 k-point (69.9 sec, see the output file tgspw_02.027-04.out. This means that a calculation 4 times larger (due to an increase of the number of k-points) gives approximatively the same human time if you parallelize over all the k-points. You have just re-derived a well established result: the scaling (the speedup) is quasi-linear on the k-point level.

When you want to parallelize a calculation, begin by the k-point level, then follow with the band level (up to npband=50 typically) then finish by the FFT level.

Here, the timing obtained for the output tgspw_05.out leads to a hypothetical speedup of 346, which is good, but not 432 as expected if the scaling was linear as a function of the number of the k-point processors. Indeed, in order to be comprehensive, we have to mention that the timing obtained in this output is slightly longer (17 sec. more) than the one obtained in tgspw_02.027-04.out. Compare the time spent in all the routines. A first clue comes from the timing done below the “vtowfk level”, which contains 99.xyz% of sequential processor time:

>> grep "vtowfk   " tgspw_05.out tgspw_02.027-04.out
>> tgspw_05.out:- vtowfk          26409.565  70.7  26409.553  70.7  4320
>> tgspw_05.out:- vtowfk          26409.565  70.7  26409.553  70.7  4320
>> tgspw_02.027-04.out:- vtowfk    6372.940  84.8   6372.958  84.8  1080
>> tgspw_02.027-04.out:- vtowfk    6372.940  84.8   6372.958  84.8  1080


We see that the KGB parallelization performs really well, since the wall time spent within vtowfk is approximatively equal: 26409/432 ~ 6372/108. So, the speedup is quasi-linear below vtowfk. The problem comes from parts above vtowfk which are not parallelized and are responsible for the negligible (1-99.xyz)% of time spent in sequential. These parts are no longer negligible when you parallelize over hundreds of processors.

You can also prove this using the percentages rather than the values of the overall times (shown above): the time spent in vtowfk corresponds to 84.8% of the overall time when you don’t parallelize over k-points, and only 70.7% when you parallelize. This means you lose time above vtowfk in this case.

This behaviour is related to the Amdhal’s law: “The speedup of a program using multiple processors in parallel computing is limited by the time needed for the sequential fraction of the program. For example, if a program needs 20 hours using a single processor core, and a particular portion of 1 hour cannot be parallelized, while the remaining promising portion of 19 hours (95%) can be parallelized, then regardless of how many processors we devote to a parallelized execution of this program, the minimum execution time cannot be less than that critical 1 hour. Hence the speedup is limited up to 20.”

In our case, the part above the loop over k-point in not parallelized by the KGB parallelization. Even if this part is very small, less than 1%, when the biggest part below is strongly (perfectly) parallelized, this small part determines an upper bound for the speedup.

To do in the future: Discuss convergence and wfoptalg, nline