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On the interaction domain reconstruction in the weighted thermostatted kinetic framework

  • Carlo BiancaEmail author
  • Marco Menale
Regular Article
  • 8 Downloads

Abstract.

This paper is devoted to the modeling of out-of-equilibrium complex living systems by means of the weighted thermostatted kinetic theory framework. The weighted mathematical framework is based on the definition and interaction of different functional subsystems each of them able to express a specific strategy. The time evolution of the functional subsystems is described by nonlinear partial integro-differential equations with quadratic type nonlinearity coupled with a thermostat in order to ensure the reaching of nonequilibrium stationary states. In particular the weighted framework is based on the definition of the weighted interactions which are modeled by introducing an interaction domain. This paper focuses on the interaction domain reconstruction by employing the methods of the inverse theory and the information theory. Specifically the solution of different inverse problems based on the knowledge of global weighted measurements related to the system is investigated. An optimization problem based on the maximum entropy principle of Shannon is analyzed and the existence of the interaction domain is proven by employing fixed-point arguments. Applications to living systems, such as social systems and crowd dynamics, and further research directions are outlined in the last section of the paper.

References

  1. 1.
    A.V. Kryazhimskii, Y.S. Osipov, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions (Gordon and Breach, London, 1995)Google Scholar
  2. 2.
    A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer, New York, 1996)Google Scholar
  3. 3.
    A. Asanov, Regularization, Uniqueness and Existence of Solutions of Volterra Equations of the First Kind (VSP, Utrecht, 1998)Google Scholar
  4. 4.
    A.L. Bughgeim, Volterra Equations and Inverse Problems (VSP, Utrecht, 1999)Google Scholar
  5. 5.
    F. Yaman, V.G. Yakhno, R. Potthast, Math. Probl. Eng. 2013, 303154 (2013)Google Scholar
  6. 6.
    C.R. Smith, W.T. Grandy (Editors), Maximum-Entropy and Bayesian Methods in Inverse Problems, in Fundamental Theories of Physics (Reidel, Dordrecht, 1985)Google Scholar
  7. 7.
    M. Dashti, A.M. Stuart, SIAM J. Numer. Anal. 48, 322 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, Philadelphia, 2005)Google Scholar
  9. 9.
    J. Llibre, R. Ramyrez, Inverse Problems in Ordinary Differential Equations and Applications (Birkhauser, 2016)Google Scholar
  10. 10.
    V. Isakov, Inverse Problems for Partial Differential Equations (Springer, 1998)Google Scholar
  11. 11.
    Yu.E. Anikonov, Inverse Problems for Kinetic and other Evolution Equations, in Inverse and Ill-Posed Problems Series (Utrecht, 2001)Google Scholar
  12. 12.
    M.V. Neshchadim, Selcuk J. Appl. Math. 4, 87 (2003)Google Scholar
  13. 13.
    G. Bal, J. Phys.: Conf. Ser. 124, 012001 (2008)Google Scholar
  14. 14.
    C. Bianca, A. Kombargi, Appl. Math. Inf. Sci. 11, 1463 (2017)CrossRefGoogle Scholar
  15. 15.
    C. Bianca, M. Ferrara, L. Guerrini, J. Glob. Optim. 58, 389 (2014)CrossRefGoogle Scholar
  16. 16.
    J.H. Holland, J. Syst. Sci. Complex. 19, 1 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Y. Bar-Yam, Dynamics of Complex Systems: Studies in Nonlinearity (Westview Press, 2003)Google Scholar
  18. 18.
    K.F. Gauss, J. Reine Angew. Math. 4, 232 (1829)MathSciNetCrossRefGoogle Scholar
  19. 19.
    E.T. Jayne, Phys. Rev. 106, 620 (1957)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    C.E. Shannon, Bell Syst. Tech. J. 27, 379 (1948)CrossRefGoogle Scholar
  21. 21.
    C.E. Shannon, Bell Syst. Tech. J. 27, 623 (1948)CrossRefGoogle Scholar
  22. 22.
    R. Kleeman, Entropy 13, 612 (2011)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Y. Liu, C. Liu, D. Wang, Entropy 13, 211 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    M. Batty, Geograph. Anal. 42, 395 (2010)CrossRefGoogle Scholar
  25. 25.
    A. Mohammad-Djafari, Entropy 17, 3989 (2015)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    A. Granas, J. Dugundji, Fixed point theory (Springer Science & Business Media, New York, 2013)Google Scholar
  27. 27.
    M.B. Short, P.J. Bratingham, A.L. Bertozzi, Proc. Natl. Acad. Sci. 107, 3961 (2010)ADSCrossRefGoogle Scholar
  28. 28.
    P.A. Jones, P.J. Brantigam, L.R. Chayes, Math. Models Methods Appl. Sci. 20, 1397 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    R.G. Fryer jr., Ration. Soc. 19, 335 (2017)CrossRefGoogle Scholar
  30. 30.
    C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys. 81, 591 (2009)ADSCrossRefGoogle Scholar
  31. 31.
    C. Bianca, V. Coscia, Appl. Math. Lett. 24, 149 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Y.-Q. Jiang, P. Zhang, S.C. Wong, R.-X. Liu, Physica A 389, 4623 (2010)ADSCrossRefGoogle Scholar
  33. 33.
    C. Dogbe, Appl. Math. Inf. Sci. 7, 29 (2013)MathSciNetCrossRefGoogle Scholar
  34. 34.
    U. Weidmann, U. Kirsch, M. Schreckenberg (Editors), Pedestrian and Evacuation Dynamics (Springer International Publishing, Switzerland, 2014)Google Scholar
  35. 35.
    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, V. Zdravkovic, Proc. Natl. Acad. Sci. 105, 1232 (2008)ADSCrossRefGoogle Scholar
  36. 36.
    R. Burger, A. Garcia, M. Kunik, Math. Models Methods Appl. Sci. 18, 1741 (2008)MathSciNetCrossRefGoogle Scholar
  37. 37.
    S.K. Nemirovskii, Phys. Rev. B 77, 214509 (2008)ADSCrossRefGoogle Scholar
  38. 38.
    O. Esmer, Information Theory, Entropy and Urban Spatial Structure (LAP Lambert Academic Publishing, Saarbrucken, 2011)Google Scholar
  39. 39.
    C. Bianca, Math. Comput. Model. 51, 72 (2010)CrossRefGoogle Scholar
  40. 40.
    H.W. Engl, K. Kunisch, A. Neubauer, Inverse Prob. 5, 523 (1989)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire Quartz EA 7393, École Supérieure d’Ingénieurs en Génie ÉlectriqueProductique et Management IndustrielCergy Pontoise CedexFrance
  2. 2.Laboratoire de Recherche en Eco-innovation Industrielle et Energétique, École Supérieure d’Ingénieurs en Génie ÉlectriqueProductique et Management IndustrielCergy Pontoise CedexFrance
  3. 3.Dipartimento di Matematica e FisicaUniversità degli Studi della Campania “L. Vanvitelli”CasertaItaly

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