Skip to main content
Springer Nature Link
Log in

Advanced stochastic approaches for Sobol’ sensitivity indices evaluation

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

Sensitivity analysis is a modern promising technique for studying large systems such as ecological systems. The main idea of sensitivity analysis is to evaluate and predict (through computer simulations on large mathematical models) the measure of the sensitivity of the model’s output to the perturbations of some input parameters, and it is a technique for refining the mathematical model. The main problem in the sensitivity analysis is the evaluation of total sensitivity indices. The mathematical formulation of this problem is represented by a set of multidimensional integrals. In this work, some new stochastic approaches for evaluating Sobol’ sensitivity indices of the unified Danish Eulerian model have been presented. For the first time, a special type of digital nets and lattice rules are applied for multidimensional sensitivity analysis and their advantages are discussed. A comparison of accuracy of eight stochastic approaches for evaluating Sobol’ sensitivity indices is performed. The obtained results will be important and useful for the surveyed scientists (physicists, chemicals, meteorologists) to make a comparative classification of the input parameters with respect to their influence on the concentration of the pollutants of interest.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Antonov I, Saleev V (1979) An economic method of computing \(LP_{\tau }\)-sequences. USSR Comput Math Phys 19:252–256

    MATH  Google Scholar 

  2. Archer G, Saltelli A, Sobol’ I (1997) Sensitivity measures, ANOVA-like techniques and the use of bootstrap. J Stat Comput Simul 58:99–120

    MATH  Google Scholar 

  3. Atanassov E, Durchova M (2003) Generating and testing the modified Halton sequences. LNCS 2542:91–98

    MathSciNet  MATH  Google Scholar 

  4. Bahvalov N (1959) On the approximate computation of multiple integrals. In: Vestnik Moscow State University, Series on Mathematics, Mechanics, vol 4, pp 3–18

  5. Bakhvalov N (2015) On the approximate calculation of multiple integrals. J Complex 31(4):502–516

    MATH  Google Scholar 

  6. Berntsen J, Espelid TO, Genz A (1991) An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans Math Softw 17:437–451

    MathSciNet  MATH  Google Scholar 

  7. Bratley P, Fox B (1988) Algorithm 659: implementing Sobol’s quasirandom sequence generator. ACM Trans Math Softw 14(1):88–100

    MathSciNet  MATH  Google Scholar 

  8. Caflisch RE (1998) Monte Carlo and quasi-Monte Carlo methods. Acta Numer 7:1–49

    MathSciNet  MATH  Google Scholar 

  9. van der Corput J (1935) Verteilungsfunktionen I & II, Nederl. Akad. Wetensch. Proceedings, vol 38, pp 813–820, 1058–1066

  10. Csomós P, Faragó I, Havasi A (2005) Weighted sequential splittings and their analysis. Comput Math Appl 50:1017–1031

    MathSciNet  MATH  Google Scholar 

  11. Cukier R, Fortuin C, Shuler K, Petschek A, Schaibly J (1973) Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I. Theory. J Chem Phys 59:3873–3878

    Google Scholar 

  12. Dick J, Pillichshammer F (2010) Digital nets and sequences. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  13. Dimitriu G (2019) Global sensitivity analysis for a chronic myelogenous leukemia model: proceedings of 9th international conference NMA’2018, Borovets, Bulgaria, August 20–24, 2018, LNCS 11189, Springer, Jan

  14. Dimov IT (2007) Monte Carlo methods for applied scientists. World Scientific, London

    MATH  Google Scholar 

  15. Dimov IT, Atanassov E (2007) Exact error estimates and optimal randomized algorithms for integration. LNCS 4310:131–139

    MATH  Google Scholar 

  16. Dimov I, Georgieva R (2010) Monte Carlo algorithms for evaluating Sobol’ sensitivity indices. Math Comput Simul 81(3):506–514. https://doi.org/10.1016/j.matcom.2009.09.005 ISSN 0378-4754

    Article  MathSciNet  MATH  Google Scholar 

  17. Dimov IT, Georgieva R, Ostromsky TZ, Zlatev Z (2013) Variance-based sensitivity analysis of the unified Danish Eulerian model according to variations of chemical rates. In: Dimov I, Faragó I, Vulkov L (eds) Proceedings of NAA 2012, LNCS 8236. Springer, New York, pp 247–254

  18. Dimov IT, Georgieva R, Ostromsky Tz, Zlatev Z (2013) Sensitivity studies of pollutant concentrations calculated by UNI-DEM with respect to the input emissions. Central Eur J Math Numer Methods Large Scale Sci Comput 11(8):1531–1545

    MathSciNet  MATH  Google Scholar 

  19. Dimov I, Ostromsky Tz, Zlatev Z (2005) Challenges in using splitting techniques for large-scale environmental modeling, In: Faragó I, Georgiev K, Havasi Á (eds) Advances in air pollution modeling for environmental security, NATO Science Series, vol 54, Springer, New York, pp 115–132

  20. Dimov IT, Georgieva R, Ostromsky TZ, Zlatev Z (2013) Advanced algorithms for multidimensional sensitivity studies of large-scale air pollution models based on Sobol sequences. Comput Math Appl 65(3):338–351

    MathSciNet  MATH  Google Scholar 

  21. Dimov I, Zlatev Z (1997) Testing the sensitivity of air pollution levels to variations of some chemical rate constants. Notes Numer Fluid Mech 62:167–175

    MATH  Google Scholar 

  22. Faure H (1982) Discrépances de suites associèes à un systéme de numération (en dimension s). Acta Arith 41:337–351

    MathSciNet  MATH  Google Scholar 

  23. Ferretti F, Saltelli A, Tarantola S (2016) Trends in sensitivity analysis practice in the last decade. J Sci Total Environ 568:666–670

    Google Scholar 

  24. Fidanova S (2004) Convergence proof for a Monte Carlo method for combinatorial optimization problems. In: international conference on computational science. Springer, Berlin, pp 523–530

  25. Fox B (1986) Algorithm 647: implementation and relative efficiency of quasirandom sequence generators. ACM Trans Math Softw 12(4):362–376

    MATH  Google Scholar 

  26. Freeman T, Bull J (1997) A comparison of parallel adaptive algorithms for multi-dimensional integration, In: Proceedings of 8th SIAM conference on parallel processing for scientific computing

  27. Georgiev I, Kandilarov J (2009) An immersed interface FEM for elliptic problems with local own sources. AIP Conf Proc 1186:335–342

    MathSciNet  Google Scholar 

  28. Gery M, Whitten G, Killus J, Dodge M (1989) A photochemical kinetics mechanism for urban and regional scale computer modelling. J Geophys Res 94(D10):12925–12956

    Google Scholar 

  29. Goda T, Suzuki K, Yoshiki T (2016) Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration. J Complex 33:30–54

    MathSciNet  MATH  Google Scholar 

  30. Graham IG, Kuo FY, Nichols J, Scheichl R, Schwab Ch, Sloan IH (2015) QMC FE methods for PDEs with log-normal random coefficients. Numer Math 131:329–368

    MathSciNet  MATH  Google Scholar 

  31. Guldberg CM, Waage P (1899) Experiments concerning chemical affinity; German translation by Abegg in Ostwald’s Klassiker der Exacten Wissenschaften, no. 104, Wilhelm Engelmann, Leipzig, pp 10–125

  32. Gurov TV, Whitlock PA (2002) An efficient backward Monte Carlo estimator for solving of a quantum kinetic equation with memory kernel. Math Comput Simul 60:85

    MathSciNet  MATH  Google Scholar 

  33. Haber S (1983) Parameters for integrating periodic functions of several variables. Math Comput 41(163):115–129

    MathSciNet  MATH  Google Scholar 

  34. Halton J (1960) On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer Math 2:84–90

    MathSciNet  MATH  Google Scholar 

  35. Halton J, Smith GB (1964) Algorithm 247: radical-inverse quasi-random point sequence. Commun ACM 7:701–702

    Google Scholar 

  36. Hamdad H, Pézerat Ch, Gauvreau B, Locqueteau Ch, Denoual Y (2019) Sensitivity analysis and propagation of uncertainty for the simulation of vehicle pass-by noise. In: Applied acoustics, vol 149. Elsevier, pp 85–98

  37. Havasi Á, Bartholy J, Faragó I (2001) Splitting method and its application in air pollution modeling. Időjárás 105(1):39–58

    Google Scholar 

  38. Hesterberg T (1995) Weighted average importance sampling and defensive mixture distributions. Technometrics 37(2):185–194

    MATH  Google Scholar 

  39. Homma T, Saltelli A (1996) Importance measures in global sensitivity analysis of nonlinear models. Reliab Eng Syst Saf 52:1–17

    Google Scholar 

  40. Hua LK, Wang Y (1981) Applications of number theory to numerical analysis. Springer, New York

    MATH  Google Scholar 

  41. Joe S, Kuo F (2003) Remark on algorithm 659: implementing Sobol’s quasirandom sequence generator. ACM Trans Math Softw 29(1):49–57

    MathSciNet  MATH  Google Scholar 

  42. Kalos MA, Whitlock PA (1986) Monte Carlo methods, volume 1: basics. Wiley, New York

    MATH  Google Scholar 

  43. Kandilarov J, Koleva M, Vulkov L (2008) A second-order cartesian grid finite volume technique for elliptic interface, Lecture Notes in Computer Science, no 4818. Springer, Berlin, pp 679–687

  44. Karaivanova A (1997) Adaptive Monte Carlo methods for numerical integration. Math Balk 11:391–406

    MathSciNet  MATH  Google Scholar 

  45. Karaivanova A, Dimov I (1998) Error analysis of an adaptive Monte Carlo method for numerical integration. Math Comput Simul 47:201–213

    MathSciNet  Google Scholar 

  46. Korobov NM (1959) Approximate evaluation of repeated integrals. Dokl Akad Nauk SSSR 124:1207–1210

    MathSciNet  MATH  Google Scholar 

  47. Korobov NM (1960) Properties and calculation of optimal coefficients (Russian). Sov Math Dokl 1:696–700

    MATH  Google Scholar 

  48. Korobov NM (1963) Number-theoretical methods in approximate analysis. Fizmatgiz, Moscow

    Google Scholar 

  49. Kuo FY, Nuyens D (2016) Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients: a survey of analysis and implementation. Found Comput Math 16(6):1631–1696

    MathSciNet  MATH  Google Scholar 

  50. McKay M (1995) Evaluating prediction uncertainty, Technical report NUREG/CR-6311, US Nuclear Regulatory Commission and Los Alamos National Laboratory

  51. Nedjalkov M, Dimov I, Rossi F, Jacoboni C (1996) Convergency of the Monte Carlo algorithms for the solution of the wigner quantum-transport equation. Math Comput Model 23(8/9):159

    MathSciNet  MATH  Google Scholar 

  52. Niederreiter H (1992) Random number generation and quasi-Monte Carlo methods, CBMS-NSF regional conference series in applied mathematics, vol 63. SIAM, Philadelphia

    Google Scholar 

  53. Niederreiter H (1978) Existence of good lattice points in the sense of Hlawka. Monatsh. Math 86:203–219

    MathSciNet  MATH  Google Scholar 

  54. Nuyens D (2007) Fast construction of good lattice rules, PhD Thesis, Leuven,

  55. Nuyens D (2017) The magic point shop of QMC point generators and generating vectors. https://people.cs.kuleuven.be/~dirk.nuyens/qmc-generators/

  56. Ostromsky Tz, Dimov IT, Georgieva R, Zlatev Z (2012) Parallel computation of sensitivity analysis data for the Danish Eulerian model. In: Proceedings of 8th international conference LSSC’11, LNCS-7116. Springer, pp 301–309

  57. Ostromsky Tz, Dimov IT, Georgieva R, Zlatev Z (2011) Air pollution modelling, sensitivity analysis and parallel implementation. Int J Environ Pollut 46(1/2):83–96

    MATH  Google Scholar 

  58. Ostromsky Tz, Dimov IT, Marinov P, Georgieva R, Zlatev Z (2011) Advanced sensitivity analysis of the Danish Eulerian Model in parallel and grid environment, In: Proceedings of 3rd international conference AMiTaNS’11, AIP conference proceedings, vol 1404, pp 225–232

  59. Ostromsky TZ, Zlatev Z (2002) flexible two-level parallel implementations of a large air pollution model. In: Dimov I, Lirkov I, Margenov S, Zlatev Z (eds) Numerical methods and applications, LNCS-2542. Springer, New York, pp 545–554

    Google Scholar 

  60. Owen A (1995) Monte Carlo and quasi-Monte Carlo methods in scientific computing, volume 106, Lecture Notes in Statistics, pp 299–317

  61. Paskov SH (1994) Computing high dimensional integrals with applications to finance, Technical report CUCS-023-94, Columbia University

  62. Poryazov S, Saranova E, Ganchev I (2018) Conceptual and analytical models for predicting the quality of service of overall telecommunication systems. Autonomous control for a reliable internet of services. Springer, Cham, pp 151–181

    Google Scholar 

  63. Sabelfeld K (1991) Monte Carlo methods in boundary value problems. Springer, Berlin

    Google Scholar 

  64. Saltelli A (2002) Making best use of model valuations to compute sensitivity indices. Comput Phys Commun 145:280–297

    MATH  Google Scholar 

  65. Saltelli A, Chan K, Scott M (2000) Sensitivity analysis. Probability and statistics series. Wiley, New York

    Google Scholar 

  66. Saltelli A, Tarantola S, Chan K (1999) A quantitative model-independent method for global sensitivity analysis of model output. Source Technometr Arch 41(1):39–56

    Google Scholar 

  67. Saltelli A, Tarantola S, Campolongo F, Ratto M (2004) Sensitivity analysis in practice: a guide to assessing scientific models. Halsted Press, New York

    MATH  Google Scholar 

  68. Sharygin IF (1963) A lower estimate for the error of quadrature formulas for certain classes of functions. Zh Vychisl Mat i Mat Fiz 3:370–376

    Google Scholar 

  69. Sloan IH, Kachoyan PJ (1987) Lattice methods for multiple integration: theory, error analysis and examples. SIAM J Numer Anal 24:116–128

    MathSciNet  MATH  Google Scholar 

  70. Sloan IH, Joe S (1994) Lattice methods for multiple Integration. Oxford University Press, Oxford

    MATH  Google Scholar 

  71. Sloan IH, Reztsov AV (2002) Component-by-component construction of good lattice rules. Math Comput 71:263–273

    MathSciNet  MATH  Google Scholar 

  72. Sobol IM (1973) Monte Carlo numerical methods. Nauka, Moscow (in Russian)

    MATH  Google Scholar 

  73. Sobol’ IM (1993) Sensitivity estimates for nonlinear mathematical models. Math Model Comput Exp 1:407–414

    MathSciNet  MATH  Google Scholar 

  74. Sobol’ IM (1990) Sensitivity estimates for nonlinear mathematical models. Mat Model 2(1):112–118

    MathSciNet  MATH  Google Scholar 

  75. Sobol’ IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55(1–3):271–280

    MathSciNet  MATH  Google Scholar 

  76. Sobol’ IM (2003) Theorem and examples on high dimensional model representation. Reliab Eng Syst Saf 79:187–193

    Google Scholar 

  77. Sobol IM, Tarantola S, Gatelli D, Kucherenko S, Mauntz W (2007) Estimating the approximation error when fixing unessential factors in global sensitivity analysis. Reliab Eng Syst Saf 92:957–960

    Google Scholar 

  78. Valkov I, Atanassov K, Doukovska L (2017) Generalized nets as a tool for modelling of the urban bus transport. In: Christiansen H, Jaudoin H, Chountas P, Andreasen T, Legind Larsen H (eds) Flexible query answering systems, FQAS 2017. Lecture Notes in Computer Science, vol 10333. Springer, Cham, pp 276–285

  79. Veach E, Guibas L (1995) Optimally combining sampling techniques for Monte Carlo rendering, In: Computer graphics proceedings, pp 419–428

  80. Wang Y, Hickernell FJ (2000) An historical overview of lattice point sets. In: Monte Carlo and quasi-Monte Carlo methods 2000, proceedings of a conference held at Hong Kong Baptist University, China

  81. Zlatev Z (1995) Computer treatment of large air pollution models. KLUWER Academic Publishers, Dordrecht

    Google Scholar 

  82. Zlatev Z, Dimov I (2006) Computational and numerical challenges in environmental modelling. Elsevier, Amsterdam

    MATH  Google Scholar 

  83. Zlatev Z, Dimov I, Georgiev K (1994) Modeling the long-range transport of air pollutants. IEEE Comput Sci Eng 1(3):45–52

    Google Scholar 

  84. Zlatev Z, Dimov I, Georgiev K (1996) Three-dimensional version of the Danish Eulerian model. Z Angew Math Mech 76(S4):473–476

    MATH  Google Scholar 

Download references

Acknowledgements

Venelin Todorov is supported by the National Program—2020 “Young scientists and Postdoctoral candidates” of the Bulgarian Ministry of Education and Science. Stoyan Apostolov and Yuri Dimitrov are supported by the Bulgarian National Science Fund under Young Scientists Project KP-06-M32/2-17.12.2019 “Advanced Stochastic and Deterministic Approaches for Large-Scale Problems of Computational Mathematics.” The work is also partially supported by the National Scientific Program “Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICT in SES),” Contract No DO1–205/23.11.2018, financed by the Ministry of Education and Science in Bulgaria and by the Bulgarian National Science Fund under Project DN 12/5-2017 “Efficient Stochastic Methods and Algorithms for Large-Scale Problems.”

Author information

Authors and Affiliations

  1. Information Modeling Department, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Block 8, 1113, Sofia, Bulgaria

    Venelin Todorov

  2. Department of Parallel Algorithms, Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 25 A, 1113, Sofia, Bulgaria

    Venelin Todorov, Ivan Dimov, Tzvetan Ostromsky & Rayna Georgieva

  3. Faculty of Mathematics and Informatics, Sofia University, 1126, Sofia, Bulgaria

    Stoyan Apostolov

  4. University of Forestry, Sofia, Bulgaria

    Yuri Dimitrov

  5. Department of Environmental Science, Aarhus University, Roskilde, Denmark

    Zahari Zlatev

Authors
  1. Venelin Todorov
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Ivan Dimov
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Tzvetan Ostromsky
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Stoyan Apostolov
    View author publications

    You can also search for this author inPubMed Google Scholar

  5. Rayna Georgieva
    View author publications

    You can also search for this author inPubMed Google Scholar

  6. Yuri Dimitrov
    View author publications

    You can also search for this author inPubMed Google Scholar

  7. Zahari Zlatev
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Venelin Todorov.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Todorov, V., Dimov, I., Ostromsky, T. et al. Advanced stochastic approaches for Sobol’ sensitivity indices evaluation. Neural Comput & Applic 33, 1999–2014 (2021). https://doi.org/10.1007/s00521-020-05074-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-020-05074-4

Keywords

Mathematics Subject Classification