Adaptive Monte Carlo algorithm for Wigner kernel evaluation

  • Venelin TodorovEmail author
  • Ivan Dimov
  • Rayna Georgieva
  • Stoyan Dimitrov
Original Article


In this paper, we study numerically various approaches, namely an adaptive Monte Carlo algorithm, a particular rank-1 lattice algorithm based on generalized Fibonacci numbers and a Monte Carlo algorithm based on Latin hypercube sampling for computing multidimensional integrals. We compare the performance of the algorithms over three case studies—multidimensional integrals from Bayesian statistics, the so-called Genz test functions and the Wigner kernel—an important issue in quantum mechanics represented by multidimensional integrals. A comprehensive study and an analysis of the computational complexity of the algorithms under consideration has been presented. Adaptive strategy is well-established as an efficient and reliable tool for multidimensional integration of integrands functions with computational peculiarities like peaks. The presented adaptive Monte Carlo algorithm gives reliable results in computing the Wigner kernel by a stochastic approach that has significantly lower computational complexity than the existing deterministic approaches.


Multidimensional integration Adaptive Monte Carlo algorithm Fibonacci lattice sets Latin hypercube sampling Wigner kernel 

Mathematics Subject Classification

65C05 65U05 65F10 65Y20 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.


  1. 1.
    Antonov I, Saleev V (1979) An economic method of computing \(LP_{\tau }\)-sequences. USSR Comput Math Phys 19:252–256CrossRefGoogle Scholar
  2. 2.
    Baraniuk RG, Jones DL (1993) A signal-dependent time-frequency representation: optimal kernel design. IEEE Trans Signal Process 41(4):1589–1602CrossRefGoogle Scholar
  3. 3.
    Berntsen J, Espelid TO, Genz A (1991) An adaptive algorithm for the approximate calculation of multiple integrals, ACM Trans. Math Softw 17:437–451CrossRefGoogle Scholar
  4. 4.
    Bratley P, Fox B (1988) Algorithm 659: implementing Sobol’s Quasirandom sequence generator. ACM Trans Math Softw 14(1):88–100MathSciNetCrossRefGoogle Scholar
  5. 5.
    Briol F-X, Oates CJ, Girolami M, Osborne MA, Sejdinovic D (2019) Probabilistic integration: a role in statistical computation? Stat Sci 34(1):1–22MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cull P, Holloway JL (1989) Computing Fibonacci numbers quickly. Inf Process Lett 32(3):143–149MathSciNetCrossRefGoogle Scholar
  7. 7.
    Davis PJ, Rabinowitz P (1984) Methods of numerical integration, 2nd edn. Academic Press, LondonzbMATHGoogle Scholar
  8. 8.
    Dimov I (2008) Monte Carlo methods for applied scientists. World Scientific, New Jersey, p 291zbMATHGoogle Scholar
  9. 9.
    Dimov I, Karaivanova A, Georgieva R, Ivanovska S, (2003) Parallel Importance Separation and Adaptive Monte Carlo Algorithms for Multiple Integrals, 5th Int. conf. on NMA, (August 2002) Borovets, Bulgaria, Springer Lecture Notes in Computer Science, 2542. Springer-Verlag, Berlin, Heidelberg, New York, pp 99–107CrossRefGoogle Scholar
  10. 10.
    Dimov I, Georgieva R (2010) Monte Carlo algorithms for evaluating Sobol’ sensitivity indices. Math Comput Simul 81(3):506–514MathSciNetCrossRefGoogle Scholar
  11. 11.
    Eglajs V, Audze P (1977) New approach to the design of multifactor experiments: problems of dynamics and strengths. 35 (in Russian). Riga: Zinatne Publishing House pp 104–107Google Scholar
  12. 12.
    Ermakov SM (1985) Monte Carlo methods and mixed problems. Nauka, MoscowGoogle Scholar
  13. 13.
    Feynman RP (1948) Space-time approach to non-relativistic quantum mechanics. Rev Mod Phys: 20Google Scholar
  14. 14.
    Fox B (1986) Algorithm 647: implementation and relative efficiency of Quasirandom sequence generators. ACM Trans Math Softw 12(4):362–376CrossRefGoogle Scholar
  15. 15.
    Genz A (1984) Testing multidimensional integration routines. Tools, Methods and Languages for Scientific and Engineering Computation, pp 81–94Google Scholar
  16. 16.
    Hua LK, Wang Y (1981) Applications of number theory to numerical analysis. Springer, BerlinzbMATHGoogle Scholar
  17. 17.
    Jarosz W (2008) Efficient Monte Carlo methods for light transport in scattering media, PhD dissertation, UCSDGoogle Scholar
  18. 18.
    Joe S, Kuo F (2003) Remark on algorithm 659: implementing Sobol’s Quasirandom sequence generator. ACM Trans Math Softw 29(1):49–57MathSciNetCrossRefGoogle Scholar
  19. 19.
    Karaivanova A, Dimov I, Ivanovska S (2001) A quasi-monte carlo method for integration with improved convergence. LNCS 2179:158–165zbMATHGoogle Scholar
  20. 20.
    Larkin FM (1972) Gaussian measure in Hilbert space and applications in numerical analysis. Rocky Mt J Math 2(3):379–421MathSciNetCrossRefGoogle Scholar
  21. 21.
    Larkin FM (1974) Probabilistic error estimates in spline interpolation and quadrature. In: IFIP Congress, pp 605–609Google Scholar
  22. 22.
    Li Wei, Lingyun Lu, Xie Xiaotian, Yang Ming (2017) A novel extension algorithm for optimized Latin hypercube sampling. J Stat Comput Simul 87(13):2549–2559. MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lin S (2011) Algebraic methods for evaluating integrals in Bayesian statistics, Ph.D. dissertation, UC BerkeleyGoogle Scholar
  24. 24.
    Lin S, Sturmfels B, Xu Z (2009) Marginal likelihood integrals for mixtures of independence models. J Mach Learn Res 10:1611–1631MathSciNetzbMATHGoogle Scholar
  25. 25.
    McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245. MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Metropolis N, Ulam S (1949) The Monte Carlo method. J Am Stat Assoc 44(247):335–341CrossRefGoogle Scholar
  27. 27.
    Minasny BB, McBratney AB (2006) A conditioned Latin hypercube method for sampling in the presence of ancillary information. J Comput Geosci Arch 32(9):1378–1388CrossRefGoogle Scholar
  28. 28.
    Minasny B, McBratney AB (2010) Conditioned Latin hypercube sampling for calibrating soil sensor data to soil properties, Chapter: Proximal Soil Sensing, Progress in Soil Science pp 111–119CrossRefGoogle Scholar
  29. 29.
    O’Hagan A (1991) Bayes–Hermite quadrature. J Stat Plan Inference 29(3):245–260MathSciNetCrossRefGoogle Scholar
  30. 30.
    Schurer R (2001) Parallel high-dimensional integration: Quasi-monte carlo versus adaptive cubature rules, Computational Science, ICCS 2001. Springer, BerlinzbMATHGoogle Scholar
  31. 31.
    Sellier JM (2015) A signed particle formulation of non-relativistic quantum mechanics. J Comput Phys 297:254–265MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sellier JM, Dimov I (2016) On a full Monte Carlo approach to quantum mechanics. Phys A Stat Mech Appl 463:45–62MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sellier JM, Dimov I (2014) The many-body Wigner Monte Carlo method for time-dependent ab-initio quantum simulations. J Comput Phys 273:589–597MathSciNetCrossRefGoogle Scholar
  34. 34.
    Sellier JM, Nedjalkov M, Dimov I (2015) An introduction to applied quantum mechanics in the Wigner Monte Carlo formalism. Phys Rep 577:1–34MathSciNetCrossRefGoogle Scholar
  35. 35.
    Shao S, Lu T, Cai W (2011) Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport. Commun Comput Phys 9:711–739MathSciNetCrossRefGoogle Scholar
  36. 36.
    Shao S, Sellier JM (2015) Comparison of deterministic and stochastic methods for time-dependent Wigner simulations. J Comput Phys 300:167–185MathSciNetCrossRefGoogle Scholar
  37. 37.
    Sloan IH, Kachoyan PJ (1987) Lattice methods for multiple integration: theory, error analysis and examples, SIAM. J Numer Anal 24:116–128MathSciNetCrossRefGoogle Scholar
  38. 38.
    Sloan IH, Joe S (1994) Lattice methods for multiple integration. Oxford University Press, OxfordzbMATHGoogle Scholar
  39. 39.
    Sobol I (1973) Numerical methods Monte Carlo. Nauka, MoscowzbMATHGoogle Scholar
  40. 40.
    Sobol IM (1967) “Distribution of points in a cube and approximate evaluation of integrals”. Zh. Vych. Mat. Mat. Fiz. 7: 784–802 (in Russian); U.S.S.R Comput. Maths. Math. Phys. 7:86–112Google Scholar
  41. 41.
    Sobol IM (1989) Quasi-Monte Carlo methods. In: Dimov IT, Sendov BI (eds) International youth workshop on Monte Carlo methods and parallel algorithms. World Scientific, Singapore, pp 75–81Google Scholar
  42. 42.
    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, [arXiv:1602.08853]
  43. 43.
    Wang Y, FJ Hickernell (2002) An historical overview of lattice point setsGoogle Scholar
  44. 44.
    Wigner E (1932) On the quantum correction for thermodynamic equilibrium. Phys Rev 40:749CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Venelin Todorov
    • 1
    • 2
    Email author
  • Ivan Dimov
    • 2
  • Rayna Georgieva
    • 2
  • Stoyan Dimitrov
    • 3
  1. 1.Information Modeling DepartmentInstitute of Mathematics and Informatics, Bulgarian Academy of SciencesSofiaBulgaria
  2. 2.Department of Parallel AlgorithmsInstitute of Information and Communication Technologies, Bulgarian Academy of SciencesSofiaBulgaria
  3. 3.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

Personalised recommendations