Uncertainty Quantification for Kinetic Models in Socio–Economic and Life Sciences

Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 14)


Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic. The construction of kinetic models describing the above processes, however, has to face the difficulty of the lack of fundamental principles since physical forces are replaced by empirical social forces. These empirical forces are typically constructed with the aim to reproduce qualitatively the observed system behaviors, like the emergence of social structures, and are at best known in terms of statistical information of the modeling parameters. For this reason the presence of random inputs characterizing the parameters uncertainty should be considered as an essential feature in the modeling process. In this survey we introduce several examples of such kinetic models, that are mathematically described by nonlinear Vlasov and Fokker–Planck equations, and present different numerical approaches for uncertainty quantification which preserve the main features of the kinetic solution.



The research that led to the present survey was partially supported by the research grant Numerical methods for uncertainty quantification in hyperbolic and kinetic equations of the group GNCS of INdAM. MZ acknowledges support from GNCS and “Compagnia di San Paolo” (Torino, Italy).


  1. 1.
    S.M. Ahn, S.Y. Ha, Stochastic flocking dynamics of the Cucker–Smale model with multiplicative white noises. J. Math. Phys. 51(10), 103–301 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    G. Ajmone Marsan, N. Bellomo, M. Egidi, Towards a mathematical theory of complex socio–economical systems by functional subsystems representation. Kinet. Relat. Model. 1(2), 249–278 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    G. Albi, L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics. Multiscale Model. Simul. 11(1), 1–29 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. Albi, M. Herty, L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus. Commun. Math. Sci. 13(6), 1407–1429 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    G. Albi, L. Pareschi, M. Zanella, Uncertainty quantification in control problems for flocking models. Math. Probl. Eng. 2015, 1–14 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Albi, L. Pareschi, G. Toscani, M. Zanella, Recent advances in opinion modeling: control and social influence, in Active Particles Vol.1: Theory, Methods, and Applications, ed. by N. Bellomo, P. Degond, E. Tadmor (Birkhäuser–Springer, Berlin, 2017), pp. 49–98Google Scholar
  7. 7.
    G. Albi, L. Pareschi, M. Zanella, Opinion dynamics over complex networks: kinetic modelling and numerical methods. Kinet. Relat. Model. 10(1), 1–32 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, V. Zdravkovic, Empirical investigation of starling flocks: a benchmark study in collective animal behavior. Anim. Behav. 76(1), 201–215 (2008)CrossRefGoogle Scholar
  9. 9.
    A.B.T. Barbaro, P. Degond, Phase transition and diffusion among socially interacting self-propelled agents. Discrete Continuous Dyn. Syst. Ser. B 19, 1249–1278 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A.B.T. Barbaro, J.A. Cañizo, J.A. Carrillo, P. Degond, Phase transitions in a kinetic model of Cucker–Smale type. Multiscale Model. Simul. 14(3), 1063–1088 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    N. Bellomo, J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems. Math. Models Methods Appl. Sci. 22(1), 1140006 (2012)Google Scholar
  12. 12.
    N. Bellomo, B. Piccoli, A. Tosin, Modeling crowd dynamics from a complex system viewpoint. Math. Models Methods Appl. Sci. 22(suppl 2), 1230004 (2012)Google Scholar
  13. 13.
    M. Bennoune, M. Lemou, L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics. J. Comput. Phys. 227, 3781–3803 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Bessemoulin-Chatard, F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34, 559–583 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Bongini, M. Fornasier, M. Hansen, M. Maggioni, Inferring interaction rules from observations of evolutive systems I: the variational approach. Math. Models Methods Appl. Sci. 27, 909 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    S. Boscarino, F. Filbet, G. Russo, High order semi-implicit schemes for time dependent partial differential equations. J. Sci. Comput. 68, 975–1001 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    C. Buet, S. Dellacherie, On the Chang and Cooper numerical scheme applied to a linear Fokker-Planck equation. Commun. Math. Sci. 8(4), 1079–1090 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    C. Buet, S. Cordier, V. Dos Santos, A conservative and entropy scheme for a simplified model of granular media. Transp. Theory Stat. Phys. 33(2), 125–155 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    M. Burger, J.A. Carrillo, M.-T. Wolfram, A mixed finite element method for nonlinear diffusion equations. Kinet. Relat. Models 3, 59–83 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    R.E. Caflisch, Monte Carlo and Quasi Monte Carlo methods. Acta Numer. 7, 1–49 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J.A. Carrillo, G. Toscani, Exponential convergence toward equilibrium for homogeneous Fokker–Planck–type equations. Math. Methods Appl. Sci. 21, 1269–1286 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    J.A. Carrillo, R.J. McCann, C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Revista Matemática Iberoamericana 19, 971–1018 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    J.A. Carrillo, M. Fornasier, J. Rosado, G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker–Smale model. SIAM J. Math. Anal. 42(1), 218–236 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    J.A. Carrillo, M. Fornasier, G. Toscani, F. Vecil, Particle, kinetic and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio–Economic and Life Sciences (Birkhauser, Boston, 2010), pp. 297–336zbMATHGoogle Scholar
  25. 25.
    J.A. Carrillo, Y.-P. Choi, M. Hauray, The derivation of swarming models: mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, vol. 553, CISM International Centre for Mechanical Sciences (Springer, Heidelberg, 2014), pp. 1–46Google Scholar
  26. 26.
    J.A. Carrillo, A. Chertock, Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. Commun. Comput. Phys. 17, 233–258 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    C. Cercignani, The Boltzmann Equation and its Applications (Springer, New York, 1988)CrossRefzbMATHGoogle Scholar
  28. 28.
    C. Chainais-Hillairet, A. Jüngel, S. Schuchnigg, Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities. ESAIM Math. Model. Numer. Anal. 50(1), 135–162 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    J.S. Chang, G. Cooper, A practical difference scheme for Fokker–Planck equations. J. Comput. Phys. 6(1), 1–16 (1970)CrossRefzbMATHGoogle Scholar
  30. 30.
    A. Chertock, S. Jin, A. Kurganov, An operator splitting based stochastic Galerkin method for the one–dimensional compressible Euler equations with uncertainty (Preprint, 2016)Google Scholar
  31. 31.
    H. Cho, D. Venturi, G.E. Karniadakis, Numerical methods for high–dimensional probability density function equations. J. Comput. Phys. 305(15), 817–837 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Y.-P. Choi, S.-Y. Ha, Z. Li, Emergent dynamics of the Cucker–Smale flocking model and its variants, in Active Particles, Volume 1, eds. by N. Bellomo, P. Degond, E. Tadmor. Modeling and Simulation in Science, Engineering and Technology (Birkhäuser, Cham, 2017), pp. 299–331Google Scholar
  33. 33.
    S. Cordier, L. Pareschi, G. Toscani, On a kinetic model for a simple market economy. J. Stat. Phys. 120(1), 253–277 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    E. Cristiani, B. Piccoli, A. Tosin, Modeling self–organization in pedestrian and animal groups from macroscopic and microscopic viewpoints, in Mathematical Modeling of Collective Behavior in Socio–Economic and Life Sciences, ed. by G. Naldi, L. Pareschi, G. Toscani. Modeling and Simulation in Science, Engineering and Technology (Birkhäuser, Boston, 2010), pp. 337–364Google Scholar
  35. 35.
    E. Cristiani, B. Piccoli, A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul. 9(1), 155–182 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    N. Crouseilles, M. Lemou, An asymptotic preserving scheme based on a micro–macro decomposition for collisional Vlasov equation: diffusion and high–field scaling limits. Kinet. Relat. Model. 4(2), 441–477 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    N. Crouseilles, G. Dimarco, M. Lemou, Asymptotic preserving and time diminishing schemes for rarefied gas dynamic. Kinet. Relat. Model. 10, 643–668 (2017)MathSciNetzbMATHGoogle Scholar
  38. 38.
    F. Cucker, S. Smale, Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    P. Degond, L. Pareschi, G. Russo, (eds.), Modeling and Computational Methods for Kinetic Equations, Modeling and Simulation in Science, Engineering and Technology (Birkhäuser Boston Inc., Boston, 2004)Google Scholar
  40. 40.
    P. Degond, J.-G. Liu, S. Motsch, V. Panferov, Hydrodynamic models of self-organized dynamics: derivation and existence theory. Methods Appl. Anal. 20(2), 89–114 (2013)MathSciNetzbMATHGoogle Scholar
  41. 41.
    P. Degond, J.-G. Liu, C. Ringhofer, Evolution in a non–conservative economy driven by local Nash equilibria. Philos. Trans. A Math. Phys. Eng. Sci. 372(2028), 20130394 (2014)Google Scholar
  42. 42.
    B. Després, G. Poëtte, D. Lucor, Robust uncertainty propagation in systems of conservation laws with the entropy closure method, in Uncertainty Quantification in Computational Fluid Dynamics. Lecture Notes in Computational Science and Engineering, vol. 92 (Springer, Berlin, 2010), pp. 105–149Google Scholar
  43. 43.
    G. Dimarco, L. Pareschi, Numerical methods for kinetic equations. Acta Numer. 23, 369–520 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    G. Dimarco, L. Pareschi, Variance reduction Monte Carlo methods for uncertainty quantification in the Boltzmann equation and related problems (Preprint, 2018)Google Scholar
  45. 45.
    G. Dimarco, Q. Li, B. Yan, L. Pareschi, Numerical methods for plasma physics in collisional regimes. J. Plasma Phys. 81(1), 305810106 (2015)Google Scholar
  46. 46.
    G. Dimarco, L. Pareschi, M. Zanella, Micro-Macro generalized polynomial chaos techniques for kinetic equations. (Preprint, 2018)Google Scholar
  47. 47.
    A. Dimits, W. Lee, Partially linearized algorithms in gyrokinetic particle simulation. J. Comput. Phys. 107(2), 309–323 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    M.R. D’Orsogna, Y.L. Chuang, A.L. Bertozzi, L. Chayes, Self-propelled particles with soft-core interactions: patterns, stability and collapse. Phys. Rev. Lett. 96, 104302 (2006)CrossRefGoogle Scholar
  49. 49.
    R. Duan, M. Fornasier, G. Toscani, A kinetic flocking model with diffusion. Commun. Math. Phys. 300, 95–145 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    D.A. Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int. J. Numer. Methods Eng. 21, 1129–1148 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    B. Düring, M.-T. Wolfram, Opinion dynamics: inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 471, 20150345 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    B. Düring, P. Markowich, J.-F. Pietschmann, M.-T. Wolfram, Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A Math. Phys. Eng. Sci. 465(2112), 3687–3708 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    F. Filbet, L. Pareschi, T. Rey, On steady–state preserving spectral methods for the homogeneous Boltzmann equation. Comptes Rendus Mathematique 353(4), 309–314 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    G. Furioli, A. Pulvirenti, E. Terraneo, G. Toscani, Fokker-Planck equations in the modeling of socio–economic phenomena. Math. Models Methods Appl. Sci. 27(1), 115–158 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    M.B. Giles, Multilevel Monte Carlo methods. Acta Numer. 24, 259–328 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws. Exponential-Fit, Well-Balanced and Asymptotic-Preserving. SEMA SIMAI Springer Series (Springer, Berlin, 2013)Google Scholar
  57. 57.
    S. Gottlieb, C.W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    S.-Y. Ha, E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Relat. Models 3(1), 415–435 (2008)MathSciNetzbMATHGoogle Scholar
  59. 59.
    S.-Y. Ha, K. Lee, D. Levy, Emergence of time–asymptotic flocking in a stochastic Cucker–Smale system. Commun. Math. Sci. 7(2), 453–469 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Problems. Springer Series in Comput. Mathematics, Vol. 8, Springer-Verlag 1987, Second revised edition 1993.Google Scholar
  61. 61.
    J. Hu, S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty. J. Comput. Phys. 315, 150–168 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    J. Hu, S. Jin, D. Xiu, A stochastic Galerkin method for Hamilton–Jacobi equations with uncertainty. SIAM J. Sci. Comput. 37(5), A2246–A2269 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, in Lecture Notes for Summer School on Methods and Models of Kinetic Theory, (M&MKT), Porto Ercole (Grosseto, Italy) Riv. Mat. Univ. Parma. 3(2), 177–216 (2012)Google Scholar
  64. 64.
    S. Jin, D. Xiu, X. Zhu, A well-balanced stochastic Galerkin method for scalar hyperbolic balance laws with random inputs. J. Sci. Comput. 67, 1198–1218 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Y. Katz, K. Tunstrøm, C.C. Ioannou, C. Huepe, I.D. Couzin, Inferring the structure and dynamics of interactions in schooling fish. Proc. Natl. Acad. Sci. U. S. A. 108(46), 18720–18725 (2011)CrossRefGoogle Scholar
  66. 66.
    E.W. Larsen, C.D. Levermore, G.C. Pomraning, J.G. Sanderson, Discretization methods for one-dimensional Fokker–Planck operators. J. Comput. Phys. 61(3), 359–390 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    M. Lemou, L. Mieussens, A new asymptotic preserving scheme based on micro–macro formulation for linear kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 31(1), 334–368 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    O. Le Maitre, O.M. Knio, Spectral Methods for Uncertainty Quantification: with Applications to Computational Fluid Dynamics. Scientific Computation (Springer, Dordrechat, 2010)CrossRefzbMATHGoogle Scholar
  69. 69.
    T.-P. Liu, S.-H. Yu, Boltzmann equation: micro–macro decomposition and positivity of shock profiles. Commun. Math. Phys. 246(1), 133–179 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    D. Matthes, A. Jüngel, G. Toscani, Convex Sobolev inequalities derived from entropy dissipation. Arch. Ration. Mech. Anal. 199(2), 563–596 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    M. Mohammadi, A. Borzì, Analysis of the Chang–Cooper discretization scheme for a class of Fokker-Planck equations. J. Numer. Math. 23(3), 271–288 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    G. Naldi, L. Pareschi, G. Toscani, (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (Birkhäuser, Boston, 2010)zbMATHGoogle Scholar
  73. 73.
    L. Pareschi, T. Rey, Residual equilibrium schemes for time dependent partial differential equations. Computers & Fluids 156, 329–342 (2017)MathSciNetCrossRefGoogle Scholar
  74. 74.
    L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods (Oxford University Press, Oxford, 2013)zbMATHGoogle Scholar
  75. 75.
    L. Pareschi, M. Zanella, Structure–preserving schemes for nonlinear Fokker–Planck equations and applications. J. Sci. Comput. 1–26 (2017)Google Scholar
  76. 76.
    L. Pareschi, M. Zanella, Structure–preserving schemes for mean–field equations of collective behavior. Proceedings of the 16th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Aachen 2016, to appearGoogle Scholar
  77. 77.
    P. Pettersson, G. Iaccarino, J. Nordström, A stochastic Galerkin method for the Euler equations with Roe variable transformation. J. Comput. Phys. 257, 481–500 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    P. Pettersson, G. Iaccarino, J. Nordström, Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties. Mathematical Engineering (Springer, Berlin, 2015)CrossRefzbMATHGoogle Scholar
  79. 79.
    G. Poëtte, B. Després, D. Lucor, Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228(7), 2443–2467 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    H. Risken, The Fokker–Planck Equation. Methods of Solution and Applications, 2nd edn. (Springer, Berlin, 1989)Google Scholar
  81. 81.
    H.L. Scharfetter, H.K. Gummel, Large signal analysis of a silicon Read diode oscillator. IEEE Trans. Electron Devices 16, 64–77 (1969)CrossRefGoogle Scholar
  82. 82.
    E. Sonnendrucker, Numerical methods for Vlasov equations. Technical report, MPI TU Munich, 2013Google Scholar
  83. 83.
    G. Toscani, Entropy production and the rate of convergence to equilibrium for the Fokker–Planck equation. Q. Appl. Math. LVII(3), 521–541 (1999)Google Scholar
  84. 84.
    G. Toscani, Kinetic models of opinion formation. Commun. Math. Sci. 4(3), 481–496 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    G. Toscani, C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203(3), 667–706 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  86. 86.
    C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Mechanics, ed. by S. Friedlander, D. Serre, vol. I (North–Holland, Amsterdam, 2002), pp. 71–305Google Scholar
  87. 87.
    A.A. Vlasov, Many–Particle Theory and its Application to Plasma. Russian Monographs and Text on Advanced Mathematics and Physics, vol. VII (Gordon and Breach, Science Publishers, Inc., New York, 1961)Google Scholar
  88. 88.
    D. Xiu, Numerical Methods for Stochastic Computations (Princeton University Press, Princeton, 2010)zbMATHGoogle Scholar
  89. 89.
    D. Xiu, J.S. Hesthaven, High–order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    D. Xiu, G.E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 614–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  91. 91.
    B. Yan, A hybrid method with deviational particles for spatial inhomogeneous plasma. J. Comput. Phys. 309, 18–36 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    Y. Zhu, S. Jin, The Vlasov-Poisson-Fokker-Planck system with uncertainty and a one-dimensional asymptotic-preserving method. SIAM Multiscale Model. Simul. 15(4), pp. 1502–1529.Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.University of FerraraFerraraItaly
  2. 2.Politecnico di TorinoTorinoItaly

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