=Paper=
{{Paper
|id=Vol-2507/392-396-paper-72
|storemode=property
|title=Parallel Algorithms for Studying the System of Long Josephson Junctions
|pdfUrl=https://ceur-ws.org/Vol-2507/392-396-paper-72.pdf
|volume=Vol-2507
|authors=Maxim Bashashin,Andrey Nechaevskiy,Dmitry Podgainy,Ilhom Rahmonov,Yury Shukrinov,Oksana Streltsova,Elena Zemlyanaya,Maxim Zuev
}}
==Parallel Algorithms for Studying the System of Long Josephson Junctions==
https://ceur-ws.org/Vol-2507/392-396-paper-72.pdf
Proceedings of the 27th International Symposium Nuclear Electronics and Computing (NEC’2019)
Budva, Becici, Montenegro, September 30 – October 4, 2019
PARALLEL ALGORITHMS FOR STUDYING THE SYSTEM
OF LONG JOSEPHSON JUNCTIONS
M. Bashashin1,3, A. Nechaevskiy1, D. Podgainy1, I. Rahmonov2,
Yu. Shukrinov2,3, O. Streltsova1,3, E. Zemlyanaya1,3, M. Zuev1
1
Laboratory of Information Technologies, JINR, 6 Joliot-Curie St., Dubna, 141980, Russia
2
Bogoliubov Laboratory of Theoretical Physics, JINR, 6 Joliot-Curie St., Dubna, 141980, Russia
3
Dubna State University, 19 Universitetskaya St., Dubna, 141980, Russia
E-mail: zuevmax@jinr.ru
The results on studying the efficiency of parallel implementations of the computing scheme for
calculating the current-voltage characteristics of the system of long Josephson junctions are presented
in the paper [1-4]. The following parallel implementations were developed: the OpenMP
implementation for computing on systems with shared memory, the CUDA implementation for
computing on Nvidia graphics processors. The development, debugging and profiling of parallel
applications were performed on the education and testing polygon of the HybriLIT heterogeneous
computing platform, while computations were carried out on the “Govorun” supercomputer [2].
Keywords: long Josephson junctions, parallel computations, HPC
Maxim Bashashin, Andrey Nechaevskiy, Dmitry Podgainy, Ilhom Rahmonov, Yury Shukrinov,
Oksana Streltsova, Elena Zemlyanaya, Maxim Zuev
Copyright © 2019 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
392
Proceedings of the 27th International Symposium Nuclear Electronics and Computing (NEC’2019)
Budva, Becici, Montenegro, September 30 – October 4, 2019
1. Formulation of the problem
A generalized model that takes into account the inductive and capacitive coupling between
long Josephson junctions (LJJs) is considered [1]. The system of N coupled LJJs is supposed to consist
of superconducting (S) and intermediate dielectric (I) layers with a length L (Fig. 1).
Figure 1. System of coupled long Josephson Figure 2. Schematic representation of the
junctions dependence of the current on time
Taking into account the capacitive and inductive coupling between the contacts, the phase
dynamics of the system N LJJs is described by the initial-boundary problem for the system of
differential equations relative to the difference of phases l x, t and voltage Vl x, t at each l contact
(l 1, 2, N ) . In a dimensionless form, the system of equations has the form [1]:
1 V1
C V ,
t 2 V
, V 2 , 0 x L, t 0,
V 1 V I t ;
2
V
t x 2 N N
where is the inductive coupling matrix, С is the capacitive coupling matrix:
1 S 0 0 S Dc sc 0 0 sc
0 S 1 S 0 , C 0 sc Dc sc 0 ,
S 0 0 S 1 s
c 0 0 sc D c
β is the dissipation parameter, the inductive coupling parameter S takes a value in the interval
0 S 0.5 ; Dc is the effective electric JJ thickness normalized by the thickness of the dielectric
layer; sc is the capacitive coupling parameter, I t is the external current.
The system of equations is supplemented with zero initial and boundary conditions:
0, t l L, t
Vl 0, t Vl L, t l 0, l 1, 2 N .
x x
The problem when boundary conditions in the direction x were defined by the external magnetic field
was also considered [3].
When calculating the current-voltage characteristics (CVC), the dependence of the current on
time was selected in the form of steps (a schematic representation is given in Figure 2), i.e. the
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Proceedings of the 27th International Symposium Nuclear Electronics and Computing (NEC’2019)
Budva, Becici, Montenegro, September 30 – October 4, 2019
problem is solved at the constant current I I j , while the found functions l , Vl l 1, 2 N are
taken as initial conditions to solve the problem for the current I j 1 .
2. Computing scheme
A uniform grid by the spatial variable (the number of grid nodes NX) was built for the
numerical solution of the initial-boundary problem. In the system of equations (1), the second-order
derivative by the coordinate x is approximated using three-point finite-difference formulas on the
discrete grid with the uniform step x . The obtained system of differential equations relative to the
values l , Vl l 1, 2 N in nodes of the discrete grid by x is solved by the fourth-order Runge-Kutta
method.
To calculate CVC, averaging Vl x, t over the coordinate and time is performed. To do this, at
each time step, the integration of voltage over the coordinate using the Simpson method and the
averaging are carried out
L
1
Vl Vl x, t dx,
L0
then the voltage is averaged over time using the formula
T
max
1
Vl Vl t dt
Tmax Tmin Tmin
and summed by all JJs. To integrate over time, the rectangle method is used.
3. Parallel scheme
When numerically solving the initial-boundary problem by the fourth-order Runge-Kutta
method over the time variable, at each time layer the Runge-Kutta coefficients K i can be found
independently (in parallel) for all NX nodes of the spatial grid and for all N Josephson junctions.
Meanwhile, the coefficients Ki i 1, 2,3, 4 are defined one by one (sequentially). Thus, the
parallelization is efficiently performed on the NX N points. When carrying out averaging in CVC
computing, the calculation of integrals can be performed in parallel as well.
4. Parallel implementations
To speed up CVC computing, parallel implementations of the computing scheme described
above were developed. The results on studying the efficiency of parallel implementations performed at
the values of the following parameters: L 10 , I min 0 , I max 1.1 , 0.2 , N 1 , Tmax 200 ,
t 0.04 – are presented below; the number of nodes by the spatial variable is NX 20048 .
4.1 OpenMP implementation
The computations were performed:
on computing nodes with processors Intel Xeon Phi 7290 (KNL: 16GB, 1.50 GHz, 72 cores, 4
threads per core supported – total 288 logical cores) and the Intel compiler (Intel Cluster Studio
18.0.1 20171018);
on dual-processor computing nodes with processors Intel Xeon E5-2695 (Broadwell; 45 MB
Cache, 2.1 GHz, 18 cores, 2 threads per core supported – total 72 logical cores per node);
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Proceedings of the 27th International Symposium Nuclear Electronics and Computing (NEC’2019)
Budva, Becici, Montenegro, September 30 – October 4, 2019
on dual-processor computing nodes with processors Intel Xeon Gold 6154 (Skylake; 24.75 MB
Cache, 3.00 GHz, 18 cores, 2 threads per core supported – total 72 logical cores per node);
on dual-processor computing nodes with processors Intel Xeon Platinum 8268 (Cascade Lake;
35.75 MB Cache, 2.9 GHz, 24 cores, 2 threads per core supported – total 96 logical cores per
node).
The graphs of the dependence of the calculation speedup obtained using the parallel algorithm:
T
S 1,
Tn
(where T1 is the computation time using one core, Tn is the time of computations on n-logical cores)
on the number of threads, the number of which is equal to the number of logical cores, and the graph
of the dependence of efficiency of using computing cores by the parallel algorithm:
T
E 1 100%,
nTn
characterizing the scalability of the parallel algorithm, are presented below.
Figure 3 shows the dependence of speedup on the number of threads when performing
computations on nodes with KNL without instructions AVX-512 and using instructions AVX-512,
while Figure 4 illustrates the dependence of efficiency of these computations. It is noteworthy that the
use of this instruction allowed us to reduce the computation time in 1.8 times.
Figure 3. Graph of the dependence of speedup of Figure 4. Graph of the dependence of efficiency
parallel computing on the number of threads of parallel computing on the number of threads
The computation time on CPU Intel Xeon E5-2695, Intel Xeon Gold 6154 and Intel Xeon
Platinum 8268 is presented in Figure 5.
Figure 5. Computation time on CPU
4.2 CUDA implementation
A CUDA implementation of the parallel algorithm was developed for computing on Nvidia
graphics accelerators. The parallel reduction algorithm using shared memory was applied for
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Proceedings of the 27th International Symposium Nuclear Electronics and Computing (NEC’2019)
Budva, Becici, Montenegro, September 30 – October 4, 2019
calculating integrals. The computation time on the graphics accelerators Nvidia Tesla K40 and Nvidia
Tesla K80 is presented in Figure 6.
Figure 6. Computation time on GPU
5. Comparative analysis of parallel implementations
For the above parameter values to calculate CVC of LJJs, the best computation time on nodes
with the KNL processor is 2108.911 minutes on 150 OpenMP threads with instructions AVX-512;
using Nvidia K80 instead of Nvidia K40 reduced the computation time by 1.08 times or the above
parameter values for calculating CVC of LJJs.
When comparing the processors Intel Xeon E5-2695, Intel Xeon Gold 6154 and Intel Xeon
Platinum 8268, the minimal computation time is 83.23 seconds on Intel Xeon Platinum 8268, and the
speedup of computing reached 2.15 times in comparison with Intel Xeon E5-2695.
Acknowledgements
The study was supported by the Russian Science Foundation (the project № 18-71-10095).
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