Validated Enclosure of Uncertain Nonlinear Equations Using SIVIA Monte Carlo

  • Nisha Rani MahatoEmail author
  • Luc Jaulin
  • S. Chakraverty
  • Jean Dezert
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


The dynamical systems in various science and engineering problems are often governed by nonlinear equations (differential equations). Due to insufficiency and incompleteness of system information, the parameters in such equations may have uncertainty. Interval analysis serves as an efficient tool for handling uncertainties in terms of closed intervals. One of the major problems with interval analysis is handling “dependency problems” for computation of the tightest range of solution enclosure or exact enclosure. Such dependency problems are often observed while dealing with complex nonlinear equations. In this regard, initially, two test problems comprising interval nonlinear equations are considered. The Set Inversion via Interval Analysis (SIVIA) along with the Monte Carlo approach is used to compute the exact enclosure of the test problems. Further, the efficiency of the proposed approach has also been verified for solving nonlinear differential equation (Van der Pol oscillator) subject to interval initial conditions.


Uncertain nonlinear equations Nonlinear oscillator Dependency problem SIVIA Monte Carlo Contractor 


  1. 1.
    Akbari M, Ganji D, Majidian A, Ahmadi A (2014) Solving nonlinear differential equations of vanderpol, rayleigh and duffing by agm. Front Mech Eng 9(2):177–190CrossRefGoogle Scholar
  2. 2.
    Alefeld G, Herzberger J (2012) Introduction to interval computation. Academic Press, LondonzbMATHGoogle Scholar
  3. 3.
    Chabert G, Jaulin L (2009) Contractor programming. Artif Intell 173:1079–1100MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chakraverty S, Mahato NR, Karunakar P, Rao TD (2019) Advanced numerical and semi-analytical methods for differential equations. Wiley, HobokenGoogle Scholar
  5. 5.
    Computer assisted proofs in dynamics group (capd).
  6. 6.
  7. 7.
    Dezert J, Han D, Tacnet JM (2017) Multi-criteria decision-making with imprecise scores and BF-TOPSIS. In: Information Fusion (Fusion), 2017 20th International Conference on. pp 1–8. International Society of Information Fusion (ISIF)Google Scholar
  8. 8.
    Gerald CF (2004) Applied numerical analysis. Pearson Education IndiaGoogle Scholar
  9. 9.
    Hansen ER (1975) A generalized interval arithmetic. In: Interval mathematics. Springer, pp 7–18Google Scholar
  10. 10.
    Jaulin L, Kieffer M, Didrit O, Walter E (2001) Applied interval analysis: with examples in parameter and state estimation, robust control and robotics, vol 1. Springer, LondonCrossRefGoogle Scholar
  11. 11.
    Krämer W (2006) Generalized intervals and the dependency problem. In: PAMM: proceedings in applied mathematics and mechanics, vol 6. Wiley Online Library, pp 683–684Google Scholar
  12. 12.
    Moore RE (1962) Interval arithmetic and automatic error analysis in digital computing. Ph.D. Dissertation, Department of Mathematics, Stanford UniversityGoogle Scholar
  13. 13.
    Moore RE (1979) Methods and applications of interval analysis, vol 2. SiamGoogle Scholar
  14. 14.
    Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. SIAM Publications, Philadelphia, PACrossRefGoogle Scholar
  15. 15.
    Sandretto JAd, Chapoutot A, Mullier O, Dynibex.
  16. 16.
    Stolfi J, De Figueiredo L (2003) An introduction to affine arithmetic. Trends Appl Comput Math 4(3):297–312MathSciNetzbMATHGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Mathematics, National Institute of Technology RourkelaRourkelaIndia
  2. 2.ENSTA-Bretagne, Lab-STICC, CNRS 6285BrestFrance
  3. 3.The French Aerospace LabPalaiseauFrance

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