# Parallelization of ATK calculations¶

**Version:** 2016.1

ATK uses a range of parallelization techniques to distribute computational workload over a number of computing cores (CPUs). This may reduce the total wall-clock time, either directly or by reducing the peak memory requirement. We will here explore the basics of how to parallelize ATK calculations, both for bulk systems and for NEGF device calculations.

ATK offers **multi-level parallelization** and will by **default** try to distribute
the computing workload in the best way possible. However, manual tuning of parallelization
settings may sometimes be beneficial. In this case, the ATK user needs to be aware
of an important distinction between two very different ways to “measure” the effect
of parallelization:

Efficiency: | If efficiency is the goal of a particular parallelization scheme (which it often is),
the speed-up per extra CPU should be as high as possible. That is, a calculation
that takes 1 hour in serial (on a single CPU) should ideally take 15 minutes
on 4 CPUs. Parallelization efficiency is therefore important if efficient use of
CPU resources is more important than the wall-clock time spent on the calculation. |
---|---|

Time-to-result: | If the goal of parallelizing over several CPUs is to get the calculation to
finish as fast as possible, one will usually choose to compromise on efficiency.
For example, parallelizing a bulk calculation with 4 k-points over 40 CPUs
will rarely give a speed-up of 40 compared to running the calculation on 1 CPU.
However, it will in most cases still lead to a significant reduction of the
wall-clock time, by a factor of maybe 10-20 depending on the system.
This may therefore result in a less efficient parallelization scheme,
when comparing the total CPU-time to the wall-clock time, but may be preferred
if getting the result quickly is more important than very efficient use of
computing resources. |

## Unit-of-work¶

When distributing an entire ATK calculation over a number of CPUs, the smallest “package” of computational workload that is conveniently assigned to a single CPU is one “unit-of-work”. Importantly, this entity is not the same for bulk and device (NEGF) calculations.

Bulks | Devices (NEGF) |
---|---|

1 k-point |
1 contour point |

For a bulk calculation, the natural unit-of-work is one single *k*-point: A bulk
calculation with 4 irreducible *k*-points is efficiently parallelized over 4 CPUs,
and ATK will **automatically** assign one *k*-point to each CPU, provided 4 CPUs are available.

For device calculations, one NEGF contour integration point is the fundamental unit-of-work.
Contributions to the density matrix from all transverse *k*-points (\(N_k\),
orthogonal to the transport direction) must be evaluated at each contour energy point
(\(N_e\), usually equal to 48), so the total workload consists of
\(N_k \times N_e\) units of work. ATK will **automatically** distribute these
over the available CPUs as efficiently as possible. Obviously, the contour integration
is fully distributed if \(N_k \times N_e\) CPUs are available (one contour point
on each CPU).

## Parallelization levels in ATK¶

The ATK calculator engines contain several layers of algorithms that can benefit from parallelization in different ways. In general, ATK can parallelize over both MPI processes and threads, but it is often most efficient to use only one of these methods for a given calculation. In general MPI is most efficient for ATK-DFT, while threading can be used for ATK-ForceField.

Note

We plan on improving this behavior for ATK 2017, so threading becomes just as effective as MPI for ATK-DFT.

### ATK-DFT¶

For regular ATK-DFT calculations, it is very efficient to parallelize over k-points, and this will always be prioritized. In all cases, ATK will automatically determine the optimal parallelization strategy based on the available CPU cores. Additionally, it will print a warning if the specifics of the calculation makes it impossible to use an optimal parallelization with the current number of CPU cores. In this case, it will be more efficient to change the number of CPU cores to better match the calculation, such as making sure the number of CPU cores is a divisor of the total number of k-points.

Warning

The automatic parallelization scheme in ATK is built on the assumption that you have enough (infinite) memory available. You should therefore check the settings for memory-intensive calculations to ensure that the calculations run with optimal efficiency.

In more advanced calculations, such as NEB and AKMC, further levels of parallelization can be used to increase efficiency. We will get back to this at the end of this document.

## Bulk calculations¶

As mentioned above, the fundamental unit-of-work for a bulk calculation is one single
*k*-point. Default parallelization settings will automatically assign 1 *k*-point
per MPI process, if possible. However, 2 *k*-points cannot be optimally distributed
on 3 cores (1 core would be idle), but they can actually be distributed on 4 cores
by assigning 2 cores to work on each *k*-point. This is illustrated in the figure below.

The matrix shows how ATK will **automatically** distribute the workload in the
case of a bulk calculation with 2 irreducible *k*-points. Each filled circle
represents a *k*-point, i.e., one unit-of-work. Blue, red, and
green colors
indicate that 1, 2, and 3 processors are used per *k*-point, respectively.
The black dashes indicate that one CPU will be idle if the calculation is
parallelized over 3 or 5 processors.

### Processes per *k*-point¶

The matrix in Fig. 18 shows that choosing the number of computing
cores, \(N_\mathrm{MPI}\), to be either 1 or a multiple of the number of *k*-points,
\(N_k\), leads to a parallelization scheme that utilizes all the allocated
cores. With 4, 6, 8, etc. cores, the workload for each *k*-point is **automatically**
shared by several cores. However, this kind of parallelization can also be selected
**manually** by the user.

The `DiagonalizationSolver`

calculates the density matrix by direct diagonalization
of the Hamiltonian matrix, and the solver accepts the parameter `processes_per_kpoint`

,
which sets the number of MPI processes assigned to work on each *k*-point.
In the matrix in Fig. 18, `processes_per_kpoint`

is 1, 2,
and 3 for blue, red, and green circles, respectively.

While `processes_per_kpoint = Automatic`

is the default, it is easy to set it
using the **Script Generator**. Open the
**Calculator** settings window, and select *Algorithm Parameters*.
Make sure `DiagonalizationSolver`

is chosen as eigenvalue solver, and click
the **Parameters** button to open a small window that allows you to manually set
the number of processes per *k*-point.

### Exercise I: Non-default number of processes per *k*-point¶

- Open the VNL
**Builder**, click , and add the`Silver`

bulk configuration to the Stash. - Use the plugin to double the size of the configuration along all directions (2x2x2 repetition).
- Send the silver bulk to the
**Script Generator**, and double-click the**New Calculator**icon to add the default ATK-DFT calculator to the script. - Double-click the added calculator to open the calculator settings window.
Select
*Algorithm Parameters*and click the**Parameters**button. - Then change the number of processes per
*k*-point from`Automatic`

to`2`

, and click*OK*. Also close the calculator settings window.

Tip

The `DiagonalizationSolver`

also accepts the parameter
`bands_above_fermi_level`

, which is by default set to `All`

bands.
Explicitly setting this parameter to a positive integer can often be used to speed
up the ATK-DFT calculation. However, be aware that too small values of
`bands_above_fermi_level`

will lead to inaccuracies in the diagonalization routine.

Finally, send the script to the

**Editor**to see how the parameter`processes_per_kpoint`

is set in the script. The script will look like shown below.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

# -*- coding: utf-8 -*- # ------------------------------------------------------------- # Bulk Configuration # ------------------------------------------------------------- # Set up lattice lattice = FaceCenteredCubic(8.1714*Angstrom) # Define elements elements = [Silver, Silver, Silver, Silver, Silver, Silver, Silver, Silver] # Define coordinates fractional_coordinates = [[ 0. , 0. , 0. ], [ 0. , -0. , 0.5], [ 0. , 0.5, -0. ], [ 0. , 0.5, 0.5], [ 0.5, 0. , -0. ], [ 0.5, 0. , 0.5], [ 0.5, 0.5, -0. ], [ 0.5, 0.5, 0.5]] # Set up configuration bulk_configuration = BulkConfiguration( bravais_lattice=lattice, elements=elements, fractional_coordinates=fractional_coordinates ) # ------------------------------------------------------------- # Calculator # ------------------------------------------------------------- k_point_sampling = MonkhorstPackGrid( na=3, nb=3, nc=3, ) numerical_accuracy_parameters = NumericalAccuracyParameters( k_point_sampling=k_point_sampling, ) density_matrix_method = DiagonalizationSolver( processes_per_kpoint=2, ) algorithm_parameters = AlgorithmParameters( density_matrix_method=density_matrix_method, ) calculator = LCAOCalculator( numerical_accuracy_parameters=numerical_accuracy_parameters, algorithm_parameters=algorithm_parameters, ) bulk_configuration.setCalculator(calculator) nlprint(bulk_configuration) bulk_configuration.update() nlsave('Silver.nc', bulk_configuration)

Notice lines 41–43, where

`processes_per_kpoint=2`

is set for the`DiagonalizationSolver`

. This defines the density matrix method that is passed on to the`AlgorithmParameters`

object, which is eventually given as an option for the`LCAOCalculator`

.If you have MPI installed on your machine, you should try to run the calculation with parallelization over 2 cores. Use either the

**Job Manager**, or run the script directly from**Command-line**.

```
$ mpiexec -n 2 atkpython Silver.py > Silver.log
```

In any case, you will the following

**DiagonalizationSolver parallelization report**in the ATK job logfile, here named`Silver.log`

:+------------------------------------------------------------------------------+ | DiagonalizationSolver parallelization report. | +------------------------------------------------------------------------------+ | Total number of processes: 2 | | Total number of k-points: 14 | | Processes per k-point: 2 | | Number of process groups: 1 | +------------------------------------------------------------------------------+

### Speed-up and peak memory reduction¶

The figure below illustrates how the calculation wallclock time and peak memory
requirement of a simple ATK-DFT calculation may both be reduced by using MPI
parallelization. The bulk calculation has 4 irreducible *k*-points.

In Fig. 19, blue triangles indicates wallclock time and red circles indicate the peak memory requirement on each core. Horizontal blue lines indicate 100%, 50%, 25%, and 12.5% of the wallclock time without parallelization (1 MPI process). They indicate perfect efficiency for 1, 2, 4, and 8 MPI processes.

It is clear that a significant speed-up is gained with 2–4 MPI processes, and that
using 4 cores (such that each *k*-point fits onto a single core) reduces the wallclock
time by almost a factor of 4. However, 5–7 MPI processes bring no
further performance gains, because the extra cores are largely idle (there are only
4 *k*-points to distribute). A significant speed-up occurs with 8 MPI processes
(on 8 cores), where the value of `processes_per_kpoint`

automatically switches
from `1`

to `2`

, such that all 8 cores can do useful work. With this in mind,
the green circles indicate the most efficient choices for the number of MPI processes.

Regarding peak memory, the most significant memory reduction happens already with
2 MPI processes. The smallest peak memory is obtained with `processes_per_kpoint`

larger than 1, i.e., 8 MPI processes. Bear in mind that the amount of peak memory
reduction obtainable by workload distribution is highly system dependent. Large systems
will usually offer better opportunities for reducing the memory footprint this way.

## NEGF calculations¶

In a SurfaceConfiguration or in a DeviceConfiguration, the system is effectively treated as infinite along the C cell vector, and as periodic along the A and B cell vectors. The 3D Brillouin zone typical of a BulkConfiguration is thus projected onto a 2D Brillouin zone associated with the periodically repeated system in A and B. The electronic structure of the system is thus solved in a way analogous to a BulkConfiguration in A and B, but in a different way along C.

The `GreensFunction`

method uses a block tridiagonal inversion method used to calculate the Green’s function
and lesser Green’s function at each at each \(k\)-point and at each energy \(\varepsilon\).
The lesser Green’s function is then used to obtain the full density matrix of the system \(\hat{D}\):

where \(G^< (\varepsilon,k)\) is the lesser Green’s function. In ATK, the meaning of \(\varepsilon\) depends on whether the system is at equilibrium or out-of-equilibrium:

**Equilibrium**: \(\varepsilon\) is a complex energy and the Green’s function is obtained using**complex contour integration****Non-equilibrium**: \(\varepsilon\) is a real energy and the Green’s function is obtained using**real contour integration**

In practice, the integration is carried out considering a double weighted sum over two discrete sets of \(k\)-points \(N_k = [k_1,\ldots,k_\mathrm{n}]\) and energies \(M_\varepsilon = [\varepsilon_1,\ldots,\varepsilon_\mathrm{m}]\)

Note

Since the evaluation of the Green’s function at non-equilibrium is done using real contour integration, many more \(\varepsilon\)-points are needed! This also means the calculation can potentially scale to many more processes.

The evaluation of each element \(G^<_{i,j}\) is the computationally intensive part of the whole procedure
and takes approximatively the same time for both *i* and *j*, so the double sum can be conveniently rewritten as a single sum over a
generalized set of **contour points** \(N_k + M_\varepsilon = [x_1,\ldots,x_\mathrm{n+m}]\), which refer
both to energies and \(k\)-points:

Each contour point can be regarded as the **unit of work** in a NEGF calculation.

### Automatic distribution of contour points¶

By default, ATK tries to distribute the contour points in a NEGF calculation on as many processes as possible. The evaluation of each contour point is done independently, which implies the following distribution rules for \(N_k \times M_\varepsilon\) contour points:

- \(N_{MPI} = N_k \times M_\varepsilon\) : An optimal distribution of 1 contour point per process is achieved
- \(N_{MPI} <= N_k \times M_\varepsilon\): ATK will calculate groups of \(N_{MPI}\) processes stepwise. If \(N_{MPI}\) is not a dividend of \(N_k \times M_\varepsilon\), some processes will be partially idle.
- \(N_{MPI} > N_k \times M_\varepsilon\) : There will not be any improvement compared to the ideal case, but more memory will be available for each contour point.

### Processes per contour point¶

The option `processes_per_contour_point`

allows to control the **number of processes used to solve
each individual unit of work**. In a device or surface calculation, it can be set in the
**Script Generator** by opening the **Calculator** settings window, and selecting
*Device Algorithm Parameters*. Make sure `GreensFunction`

is chosen as the method, and click
the **Parameters** button to open a small window that allows you to manually set the number of processes
per contour point.

The default value `processes_per_contour_point = 1`

provides the maximum efficiency in the calculation.
For very large calculations, setting `processes_per_contour_point`

to a number larger than 1 allows you
to increase the available memory per individual unit of work beyond that available for a single process.
As an example, suppose you have \(N_k \times M_\varepsilon\) contour points and \(N_k \times M_\varepsilon\)
processes, each one having a maximum memory available which is only half of that required to solve a contour point.
In this case, by setting `processes_per_contour_point = 2`

, two groups of \((N_k \times M_\varepsilon)/2\) will be
solved in a stepwise fashion by using two cores per unit of work, effectively doubling the memory
available per contour point.

### Speedup in a device calculation¶

Fig. 21 shows the parallel scaling of a NEGF calculation of a Ag/Si(100)
interface, similar to the one considered in the paper *“General atomistic approach for modeling metal–semiconductor interfaces
using density functional theory and non-equilibrium Green’s function”* [cSMB+16]. The system is
calculated at equilibrium using \(N_k\) = 10 \(k\)-points in the irreducible 2D Brillouin zone, and
\(M_\varepsilon\) = 48 \(\varepsilon\)-points on the complex contour for the evaluation of the equilibrium Green’s
function. This results in a total of \(N_k \times M_\varepsilon\) = 480 contour points.

The number of processes \(N_{MPI}\) has been chosen to be always a divisors of 480, to avoid the presence of
idle processes during the calculation.
It can be seen that the speedup is close ideal for all the different number of processes considered. As
expected, the total walltime for `processes_per_contour_point = 2`

(`processes_per_contour_point = 4`

)
is around two (four) times larger that for `processes_per_contour_point = 1`

using the same number of
processes, because 2 (4) blocks of 240 (120) units of work are solved stepwise. However, the memory available
per unit of work is two (four) times larger for `processes_per_contour_point = 2`

(`processes_per_contour_point = 4`

) compared to `processes_per_contour_point = 1`

.

## Examples of multi-level parallelisms in ATK¶

In the preceding sections, we have discussed the basics of how MPI parallelism
can be used to distribute the workload of individual ATK calculations for bulks and devices.
However, ATK also offers advanced multi-level parallelism, where the workload of
**composite calculations** may be distributed from the highest level of a hierarchic
workflow all the way down to the lowest level, where only one unit-of-work is left.

Such composite calculations include nudged elastic band (NEB) calculations, IV-curve calculations, adaptive kinetic Monte Carlo (AKMC) simulations, and Crystal Structure Prediction using a genetic algorithm, amongst others.

All such methods require a workflow involving calculations for several different
structures at the same time, and often quite many of them. Multi-level parallelization
makes it possible to run several different calculations for several different structures
**at the same time**, thereby reducing the time-to-result.

Multi-level parallelism is of course **automatically enabled** if the number of assigned
MPI processes is sufficient. However, the **Parallel Parameters** calculator settings
may be used to manually set the desired scheme for multi-level distribution of the
workload.

### Nudged elastic band calculations¶

Consider first an example of ATK multi-level parallelization of NEB calculations,
here using 64 MPI processes in total. The calculation for each of the 4 NEB images has 2 irreducible
*k*-points. The figure below shows how the full NEB calculation is parallelized over
first images and then over *k*-points for each image.

**Parallelization level I**: 64 CPUs are distributed over 4 groups with 16 CPUs for each group, where each group is assigned to a single NEB image. Setting the number of CPUs per group is automatically done by ATK.If needed, you may explicitly set the number of MPI processes per NEB image in the VNL Scripter (Processes per NEB image), or set it in the script as follows:

`parallel_parameters = ParallelParameters(processes_per_neb_image=16)`

.**Parallelization level II**: If the ATK calculator for the NEB calculation uses \(N_{k}\)*k*-points for the Brillouin zone integration, e.g., \(N_{k} = 2\), each image group can be divided into 2 groups with 8 CPUs per*k*-point. ATK automatically sets this parallelization.If needed, you may also set the number of MPI processes per

*k*-point manually in the VNL Scripter ( ).**Parallelization level III**: ATK automatically sets this parallelization.

### I-V curve calculations¶

Computing an I-V curve requires a number of NEGF device calculations at different biases. Multi-level parallelism is therefore used to distribute the workload over bias points and NEGF contour points for each bias.

**Parallelization level I**: 192 CPUs are distributed over 2 groups with 96 CPUs per group, where each group is assigned to a single bias calculation. Setting the number of CPUs per group is automatically done by ATK.If needed, you may explicitly set the number of MPI processes per bias point in the VNL Scripter (Processes per bias point), or set it in the script as follows:

`parallel_parameters = ParallelParameters(processes_per_bias_point=96)`

.**Parallelization level II**: If the ATK calculator for the I-V Curve calculation uses \(N_{e}=48\) energy points for the NEGF contour integration, and \(N_{k}=2\)*k*-points for the 2D Brillouin zone integration at a given energy, each bias group can be divided into 96 groups with 1 CPU per contour (*k*, energy) point. ATK automatically sets this parallelization.If needed, you may also set the number of MPI processes per contour point manually in the script as follows:

`parallel_parameters = ParallelParameters(processes_per_bias_point=96, processes_per_contour_point=1)`

.

### Adaptive Kinetic Monte Carlo simulations¶

The AKMC method is an obvious candidate for multi-level parallelization. A number of saddle searches must be carried out, and ATK will by default try to fully distribute the workload. We will here consider AKMC calculations using ATK-DFT, as efficient parallelization is most important in this case, though AKMC with ATK-ForceField also benefits from parallelization.

**Parallelization level I**: 192 CPUs are distributed over 48 groups with 4 CPU per group, where each group is assigned to a single saddle point search. Setting the number of CPUs per group is automatically done by ATK.If needed, you may explicitly set the number of MPI processes per saddle search in the VNL Scripter (Processes per saddle search), or set it in the script as follows:

`parallel_parameters = ParallelParameters(processes_per_saddle_search=4)`

.**Parallelization level II**: If the ATK calculator for the AKMC calculation uses \(N_{k}\)*k*-points for the Brillouin zone integration, e.g., \(N_{k} = 2\), each image group can be divided into 2 groups with 2 CPUs per*k*-point. ATK automatically sets this parallelization.If needed, you may also set the number of MPI processes per

*k*-point manually in the VNL Scripter ( ).**Parallelization level III**: ATK automatically sets this parallelization.

### Crystal Structure Prediction¶

A genetic algorithm relies on continuously evolving and improving improving generations of candidate minimum-energy configurations. Each generation consists of several (sometimes hundreds or thousands) of different configurations that all need to be geometry optimized. Multi-level parallelism will therefore give a significant speed-up for such calculations in most cases.

**Parallelization level I**: 192 CPUs are distributed over 48 groups with 4 CPUs per group, where each group is assigned to an individual configuration calculation. Setting the number of CPUs per group is automatically done by ATK.If needed, you may explicitly set the number of MPI processes per individual configuration in the VNL Scripter (Processes per individual), or set it in the script as follows:

`parallel_parameters = ParallelParameters(processes_per_individual=4)`

.**Parallelization level II**: If the ATK calculator for the Crystal Structure Prediction calculation uses \(N_{k}\)*k*-points for the Brillouin zone integration, e.g., \(N_{k} = 2\), each image group can be divided into 2 groups with 2 CPUs per*k*-point. ATK automatically sets this parallelization.If needed, you may also set the number of MPI processes per

*k*-point manually in the VNL Scripter ( ).**Parallelization level III**: ATK automatically sets this parallelization.

## References¶

[cSMB+16] | D. Stradi, U. Martinez, A. Blom, M. Brandbyge, and K. Stokbro. General atomistic approach for modeling metal-semiconductor interfaces using density functional theory and nonequilibrium green’s function. Phys. Rev. B, 93:155302, Apr 2016. doi:10.1103/PhysRevB.93.155302. |