Life after the serial code

Life after the serial code

Let’s imagine you have a working code and now you are producing some results. What do you feel? You are very happy, of course, but you feel the urge to improve your code. It’s always a good idea to parallelize the code, and in the meantime you can think about the visualization, how to impress your colleagues and friends.

Parallelization

In my previous blog post I wrote about the already existing MPI parallelization of the code. Briefly, we have to solve the eigenvalue-equation of the Fock-matrix that has huge dimensions, but the matrix is -fortunately- block diagonal. The blocks are labelled with a k index, and each block is diagonalized by a different process. According to the original project plan, my task would have been the further development of the parallelization to make the code even faster. What have I done instead? I extended a code with a subroutine that performs a complicated computation for a lot of grid points. Yes, electron density computation made the program running longer. Well, it’s time for parallelization.

We get several electronic orbitals for each k value, and the electron density is computed for each orbital. I decided to implement the MPI parallelization of the electron density computation the same way as it was done for the Fock matrix diagonalization. We use the master-slave system: If we have N processes, the main program is carried out by the Master process (rank=0) that calls the Slave processes (rank=1,2,…,N-1) at the parallel regions and collects the data from the Slaves at the end.

Each process does the computation for a different k value. We distribute the work using a counter variable that is 0 at the beginning. A process does the upcoming k value if its rank is equal to the actual counter value. If a process accepts a k value, it increases the counter value with 1, or updates it to 0 if it was N-1. This way, process 0 does k=1, proc. 1 does k=2, proc. N-1 does k=N, then proc 0 does k=N+1, proc. 1 does k=N+2, and so on. The whole algorithm can be described something like this:

Master: Hey Slaves, wake up! We are computing electron density!
Slaves: Slaves are ready, Master, and waiting for instructions.
Master: Turn on your radios, I am broadcasting the data and variables.
Slaves: Broadcasted information received, entering k-loop.
From now, the Master and the Slaves do the same:
Everybody: Is my rank equal to the counter? If yes, I do the the actual k and update the counter; if not, then I will just wait.
At the end of the k-loop:
Slaves: Master, we finished the work. Slaves say goodbye and exit.
Master: Thank you Slaves, Master also leaves electron density computation.

The first implementation of parallelization

There is a problem with the current parallelization: you cannot utilize more processors than the number of k values. It is very effective for large nanotubes with lots of k values, but you cannot use the potentials of the supercomputer for small systems.

Next idea: Let’s put the parallelization in the inner loop, the loop over the orbitals with the same k. In this case we distribute the individual orbitals among the processes. If we have M orbitals in each block, the first M processes start working on the k=1 orbitals, the next process gets the first orbital for k=2. This way we can distribute the work more evenly and utilize more processors.

The second implementation

I tested the parallelization on a nanotube model that has 32 different k values. The speedup as a function of the number of the processors is shown in the figure below. The blue points stop at 32, because we cannot use more processors than the number of k values. The speedup is a linear function of the processor until about 64, but after that the speedup does not grow that much with the number of processors.

Test of the parallelization

Visualization

Motto: 1 figure = 1 000 words = 1 000 000 data points

Before I started this project I thought that the visualization part will be the easiest one, but I was really wrong. It can be very difficult to produce an image that captures the essence of the scientific result and pretty as well. (I have even seen a Facebook page dedicated to particularly ugly plots. I cannot find it now, but it’s better that way.) I do not have a lot of experience with visualization, as it was only lab reports for me. For lab report all we had to do was to make a scatter plot, fit a function to the points, explain the outliers and -most importantly- have proper axes labels with physical quantities in italic and with proper units. The result? A correct, but boring plot and the best mark for the lab report.

Now I am trying to make figures that can show the results and capture the attention, too. If you have a good idea, any advice is greatly welcomed.

The first test system of the was the benzene molecule, and I computed the electron density only in the plane of the molecule. I plotted the results with Wolfram Mathematica using ListPlot3D and ListDensityPlot.

Electron density of benzene orbital using ListPlot3D in Mathematica
The same benzene orbital but plotted with ListDensityPlot

But electron density is a function is space! How can I plot the four-dimensional data? The answer is simple, let’s make cuts along the x-z plane, and make a plot for each cut. Here you can find 1231 cuts for the R=(6,0) nanotube. I apologize if you really tried to click on the “link”, it was a bad joke. I do not expect you to reconstruct the 3D data from plane cuts, because there are better ways of visualization.

What we can plot is the electron density isosurface, a surface where the electron density is equal to a given value. Quantum chemical program packages can make such plots if we give them the input in the proper format. I have x-y-z coordinate values and the data, and I convert this to Gaussian cube file format, then I use the Visual Molecular Dynamics program to make the isosurface plots. Here you can see some examples for the R=(6,0) nanotube.

This is all for now, I hope I could tell you something interesting. I will come back with the final blog post next week.

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Hello Everyone, my name is Irén and I am from Budapest. I have just completed my MSc in Chemistry at the Eötvös Loránd University, and now I am applying for the PhD School of Chemistry there. I have always been interested in mathematics and chemistry but I discovered programming only at the university (it was a love at first sight). This is why theoretical computational chemistry became my main topic of interest. This summer I will be staying in Bratislava where I will perform electronic structure computations of nanotubes and optimize the code. I have applied for the Summer of HPC program to study more about parallel programming, to peek into new fields of science and to meet new people.

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