Logo PTI Logo FedCSIS

Proceedings of the 16th Conference on Computer Science and Intelligence Systems

Annals of Computer Science and Information Systems, Volume 25

An Optimized Technique for Wigner Kernel Estimation

, , ,

DOI: http://dx.doi.org/10.15439/2021F84

Citation: Proceedings of the 16th Conference on Computer Science and Intelligence Systems, M. Ganzha, L. Maciaszek, M. Paprzycki, D. Ślęzak (eds). ACSIS, Vol. 25, pages 235–238 ()

Full text

Abstract. We study an optimized Adaptive Monte Carlo algorithm for the Wigner kernel - an important problem in quantum mechanics. We will compare the results with the basic adaptive approach and other stochastic approaches for computing the Wigner kernel represented by difficult multidimensional integrals in dimension $d$ up to 12. The higher cases $d>12$ will  be considered for the first time. A comprehensive study and an analysis of the computational complexity of the optimized Adaptive MC algorithm under consideration has also been presented.

References

  1. Berntsen J., Espelid T.O., Genz A. (1991) An adaptive algorithm for the approximate calculation of multiple integrals, ACM Trans. Math. Softw. 17: 437–451.
  2. Dimov I. (2008) Monte Carlo Methods for Applied Scientists, New Jersey, London, Singapore, World Scientific, 291 p., ISBN-10 981-02-2329-3.
  3. Dimov I., Karaivanova A., Georgieva R., Ivanovska S. (2003) Parallel Importance Separation and Adaptive Monte Carlo Algorithms for Multiple Integrals, Springer Lecture Notes in Computer Science, 2542, 99–107.
  4. Dimov I., Georgieva R. (2010) Monte Carlo Algorithms for Evaluating Sobol’ Sensitivity Indices. Math. Comput. Simul. 81(3): 506–514.
  5. Ermakov S.M. (1985) Monte Carlo Methods and Mixed Problems, Nauka, Moscow.
  6. Feynman R.P. (1948) Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. 20.
  7. Sellier J.M. (2015) A signed particle formulation of non-relativistic quantum mechanics, Journal of Computational Physics 297: 254—265.
  8. Sellier J.M., Dimov I. (2016) On a full Monte Carlo approach to quantum mechanics, Physica A: Statistical Mechanics and its Applications Volume 463: 45–62.
  9. Sellier J.M., Dimov I. (2014) The many-body Wigner Monte Carlo method for time-dependent ab-initio quantum simulations, J. Comput. Phys. 273: 589–597.
  10. Sellier J.M., Nedjalkov M., Dimov I. (2015) An introduction to applied quantum mechanics in the Wigner Monte Carlo formalism, Physics Reports Volume 577: 1–34.
  11. Shao S., Lu T., Cai W. (2011) Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport. Commun. Comput. Phys., 9: 711–739.
  12. Shao S. and Sellier J.M. (2015) Comparison of deterministic and stochastic methods for time-dependent Wigner simulations. J. Comput. Phys., 300: 167–185.
  13. Todorov, V., Dimov, I., Georgieva, R., & Dimitrov, S. (2019). Adaptive Monte Carlo algorithm for Wigner kernel evaluation. Neural Computing and Applications, 1-12.
  14. Xiong Y., Chen Z., Shao S (2016) An advective-spectral-mixed method for time-dependent many-body Wigner simulations. SIAM J. Sci. Comput., to appear, [https://arxiv.org/abs/1602.08853].
  15. Wigner E. (1932) On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40: 749.