March  2021, 14(3): 1111-1122. doi: 10.3934/dcdss.2020383

Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs

1. 

Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Praha 2,120 00, Czech Republic

2. 

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA, Nuclear Physics Institute, Czech Academy of Sciences, Řež 25068, Czech Republic

3. 

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA

4. 

Faculty of Information Technology, Czech Technical University, Prague 16000, Czech Republic, Aerospace Research and Test Establishment, Prague 19905, Czech Republic

5. 

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA

* Corresponding author

Received  January 2019 Revised  February 2020 Published  March 2021 Early access  June 2020

Starting from the matrix elements of a nucleon-nucleon potential operator provided in a basis of spherical harmonic oscillator functions, we present an algorithm for expressing a given potential operator in terms of irreducible tensors of the SU(3) and SU(2) groups. Further, we introduce a GPU-based implementation of the latter and investigate its performance compared with a CPU-based version of the same. We find that the CUDA implementation delivers speedups of 2.27x – 5.93x.

Citation: Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383
References:
[1]

Y. Akiyama and J. P. Draayer, A user's guide to Fortran programs for Wigner and Racah coefficients of SU$_3$, Comp. Phys. Comm, 5 (1973), 405-406. 

[2]

T. Dytrych, K. D. Launey, J. P. Draayer, P. Maris, J. P. Vary, E. Saule, U. Catalyurek, M. Sosonkina, D. Langr and M. A. Caprio, Collective modes in light nuclei from first principles, Phys. Rev. Lett., 111 (2013), 252501. doi: 10.1103/PhysRevLett.111.252501.

[3]

T. DytrychP. MarisK. D. LauneyJ. P. DraayerJ. VaryD. LangrE. SauleM. A. CaprioU. Catalyurek and M. Sosonkina, Efficacy of the SU(3) scheme for ab initio large-scale calculations beyond the lightest nuclei, Comp. Phys. Comm., 207 (2016), 202-210.  doi: 10.2172/1326837.

[4]

H. T. Johansson and C. Forssén, Fast and accurate evaluation of Wigner $3j$, $6j$, and $9j$ symbols using prime factorization and multiword integer arithmetic, SIAM J. Sci. Comput., 38 (2016), A376–A384. doi: 10.1137/15M1021908.

[5]

K. D. LauneyT. Dytrych and J. P. Draayer, Symmetry-guided large-scale shell-model theory, Prog. Part. Nucl. Phys., 89 (2016), 101-136.  doi: 10.1016/j.ppnp.2016.02.001.

[6]

M. F. O'Reilly, A closed formula for the product of irreducible representations of SU(3), J. Math. Phys., 23 (1982), 2022-2028.  doi: 10.1063/1.525258.

show all references

References:
[1]

Y. Akiyama and J. P. Draayer, A user's guide to Fortran programs for Wigner and Racah coefficients of SU$_3$, Comp. Phys. Comm, 5 (1973), 405-406. 

[2]

T. Dytrych, K. D. Launey, J. P. Draayer, P. Maris, J. P. Vary, E. Saule, U. Catalyurek, M. Sosonkina, D. Langr and M. A. Caprio, Collective modes in light nuclei from first principles, Phys. Rev. Lett., 111 (2013), 252501. doi: 10.1103/PhysRevLett.111.252501.

[3]

T. DytrychP. MarisK. D. LauneyJ. P. DraayerJ. VaryD. LangrE. SauleM. A. CaprioU. Catalyurek and M. Sosonkina, Efficacy of the SU(3) scheme for ab initio large-scale calculations beyond the lightest nuclei, Comp. Phys. Comm., 207 (2016), 202-210.  doi: 10.2172/1326837.

[4]

H. T. Johansson and C. Forssén, Fast and accurate evaluation of Wigner $3j$, $6j$, and $9j$ symbols using prime factorization and multiword integer arithmetic, SIAM J. Sci. Comput., 38 (2016), A376–A384. doi: 10.1137/15M1021908.

[5]

K. D. LauneyT. Dytrych and J. P. Draayer, Symmetry-guided large-scale shell-model theory, Prog. Part. Nucl. Phys., 89 (2016), 101-136.  doi: 10.1016/j.ppnp.2016.02.001.

[6]

M. F. O'Reilly, A closed formula for the product of irreducible representations of SU(3), J. Math. Phys., 23 (1982), 2022-2028.  doi: 10.1063/1.525258.

Figure 1.  Data layout of buffer used for transfer of the Wigner coupling coefficients and quantum numbers of irreducible tensors computed on CPU to GPU
Figure 2.  Data layout of buffer holding records with triplets of references to records in buffer $ B_W $. Each such triplet thus represents one irreducible tensor
Table3 
Table 1.  Performance results obtained on the Blue Waters system. Notation $ +1 $ in the column with MPI processes denotes one master process which just assigns HO shells to other MPI processes. The master process is not taken into account for efficiency evaluation
$ N_{\max} $ MPI procs. CPU only CPU+GPU
Time [s] Efficiency Time [s] Speed-up
$ 8 $ 7+1 295.3 75.3 3.92
15+1 137.6 1 36.8 3.73
31+1 66.5 1 17.5 3.79
63+1 39.4 0.83 9.23 4.26
127+1 32.8 0.56 6.35 5.17
255+1 31.0 0.52 5.22 5.94
$ 10 $ 7+1 2219 648 3.42
15+1 1034 1 318 3.24
31+1 499 1 151 3.28
63+1 248 0.99 74 3.32
127+1 165 0.74 43 3.75
255+1 138 0.59 32 4.25
$ 12 $ 7+1 13083 4493 2.91
15+1 6097 1 2116 2.88
31+1 2943 1 1054 2.79
63+1 1447 1 515 2.80
127+1 776 0.92 269 2.88
255+1 565 0.68 169 3.33
$ 14 $ 5+1 64865 26104 2.48
15+1 30227 1 12204 2.47
31+1 14596 1 5944 2.45
63+1 7179 1 2932 2.44
127+1 3581 0.99 1461 2.45
255+1 2142 0.83 838 2.55
$ N_{\max} $ MPI procs. CPU only CPU+GPU
Time [s] Efficiency Time [s] Speed-up
$ 8 $ 7+1 295.3 75.3 3.92
15+1 137.6 1 36.8 3.73
31+1 66.5 1 17.5 3.79
63+1 39.4 0.83 9.23 4.26
127+1 32.8 0.56 6.35 5.17
255+1 31.0 0.52 5.22 5.94
$ 10 $ 7+1 2219 648 3.42
15+1 1034 1 318 3.24
31+1 499 1 151 3.28
63+1 248 0.99 74 3.32
127+1 165 0.74 43 3.75
255+1 138 0.59 32 4.25
$ 12 $ 7+1 13083 4493 2.91
15+1 6097 1 2116 2.88
31+1 2943 1 1054 2.79
63+1 1447 1 515 2.80
127+1 776 0.92 269 2.88
255+1 565 0.68 169 3.33
$ 14 $ 5+1 64865 26104 2.48
15+1 30227 1 12204 2.47
31+1 14596 1 5944 2.45
63+1 7179 1 2932 2.44
127+1 3581 0.99 1461 2.45
255+1 2142 0.83 838 2.55
Table 2.  Performance results obtained for 32 MPI processes running on two 16-core IBM Power9 CPUs and a single NVIDIA V100 GPU
$ N_{\max} $ CPU only CPU+GPU
Time [s] Time [s] Speed-up
8 41.65 15.78 2.63
10 274.14 97.99 2.79
12 1649.7 611.1 2.69
14 7761.9 3407.6 2.27
$ N_{\max} $ CPU only CPU+GPU
Time [s] Time [s] Speed-up
8 41.65 15.78 2.63
10 274.14 97.99 2.79
12 1649.7 611.1 2.69
14 7761.9 3407.6 2.27
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