August  2018, 11(4): 859-889. doi: 10.3934/krm.2018034

Local sensitivity analysis for the Cucker-Smale model with random inputs

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

3. 

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA

Received  December 2017 Revised  January 2018 Published  April 2018

Fund Project: The work of S.-Y. Ha was supported by National Research Foundation of Korea(NRF-2017R1A2B2001864), and the work of S. Jin was supported by NSF grants DMS-1522184 and DMS-1107291, and the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin.

We present pathwise flocking dynamics and local sensitivity analysis for the Cucker-Smale(C-S) model with random communications and initial data. For the deterministic communications, it is well known that the C-S model can model emergent local and global flocking dynamics depending on initial data and integrability of communication function. However, the communication mechanism between agents is not a priori clear and needs to be figured out from observed phenomena and data. Thus, uncertainty in communication is an intrinsic component in the flocking modeling of the C-S model. In this paper, we provide a class of admissible random uncertainties which allows us to perform the local sensitivity analysis for flocking and establish stability to the random C-S model with uncertain communication.

Citation: Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic and Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
References:
[1]

S. M. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.

[2]

S. M. Ahn and S. -Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.

[3]

G. Albi, L. Pareschi and M. Zanella, Uncertain quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14pp. doi: 10.1155/2015/850124.

[4]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.

[5]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences, in Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, (2010), 297-336.

[6]

J. A. Carrillo, L. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Preprint.

[7]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Methods in Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.

[8]

Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Theory, Models, Applications, Series: Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhauser-Springer, 1 (2017), 299-331.

[9]

F. Cucker and J.-G Dong, On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.  doi: 10.1142/S0218202516500639.

[10]

F. Cucker and J.-G Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.

[11]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.

[12]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[13]

P. Degond and S. Motsch, Large-scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1022.  doi: 10.1007/s10955-008-9529-8.

[14]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.

[15]

S. -Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear in Kinetic Relat. Models.

[16]

S.-Y. HaB. Kwon and M.-J. Kang, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Mod. Meth. Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.

[17]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.

[18]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[19]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[20]

J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168.  doi: 10.1016/j.jcp.2016.03.047.

[21]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, in Uncertainty Quantification for Kinetic and Hyperbolic Equations, (eds S. Jin and L. Pareschi), 14 (2018), 193-229. doi: 10.1007/978-3-319-67110-9_6.

[22]

J. HuS. Jin and D. Xiu, A stochastic Galerkin method for Hamiltonian-Jacobi equations with uncertainty, SIAM. J. Sci. Comput., 37 (2015), 2246-2269.  doi: 10.1137/140990930.

[23]

H. -N. Huang, S. A. M. Marcantognini and N. J. Young, Chain rules for higher derivatives, The Mathematical Intelligencer, 28 (2006), 61-69, http://ambio1.leeds.ac.uk/~nicholas/abstracts/FaadiBruno3.pdf. doi: 10.1007/BF02987158.

[24]

S. Jin, J. -G. Liu and Z. Ma, Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro-macro decomposition based asymptotic preserving method, Research in Math. Sci., 4 (2017), Paper No. 15, 25 pp. doi: 10.1186/s40687-017-0105-1.

[25]

S. Jin and L. Liu, An asymptotic-preserving stochastic Galerkin method for the semicondutor Boltzmann equation with random inputs and diffusive scalings, Multiscale Model. Simu., 15 (2017), 157-183.  doi: 10.1137/15M1053463.

[26]

S. Jin, M. -B. Tran and E. Zuazua, A local sensivity analysis for a damped wave equation with random initial input, Preprint.

[27]

S. JinD. Xiu and X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, J. Comput. Phys., 289 (2015), 35-52.  doi: 10.1016/j.jcp.2015.02.023.

[28]

S. JinD. Xiu and X. Zhu, A well-balanced stochastic Galerkin method for scalar hyperbolic balance laws with random inputs, J. Sci. Comput., 67 (2016), 1198-1218.  doi: 10.1007/s10915-015-0124-2.

[29]

S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple Scales, To appear in SIAM J. Math. Anal.

[30]

M.-J. Kang and A. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542.

[31]

T. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.

[32]

T. KarperA. Mellet and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243.  doi: 10.1137/120866828.

[33]

T. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227-242. doi: 10.1007/978-3-642-39007-4_11.

[34]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.

[35]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertainty Quantification, 5 (2017), 1193-1219.  doi: 10.1137/16M1106675.

[36]

L. Liu and S. Jin, Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs, Preprint.

[37]

S. Motsch and E. Tadmor, Heterophilious dynamics: Enhanced Consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[38]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[39]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105. 

[40]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. of Guidance, Control and Dynamics, 32 (2009), 527-537. 

[41]

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensivity analysis, Global sensivity analysis, The Primer, (2008), 1-51.

[42]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719.  doi: 10.1137/060673254.

[43]

R. Shu and S. Jin, Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime, Preprint.

[44]

E. Tadmor, Mathematical aspects of self-organized dynamics: consensus, emergence of leaders, and social hydrodynamics, SIAM News, 48 (2015).

[45]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

[46]

T. VicsekE. Ben-Jacob CzirókI. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[47]

D. Xiu, Numerical Methods fo Stochastic Computations, Princeton University Presss, 2010.

[48]

D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM. J. Scientific Computiong, 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.

show all references

References:
[1]

S. M. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.

[2]

S. M. Ahn and S. -Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.

[3]

G. Albi, L. Pareschi and M. Zanella, Uncertain quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14pp. doi: 10.1155/2015/850124.

[4]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.

[5]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences, in Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, (2010), 297-336.

[6]

J. A. Carrillo, L. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Preprint.

[7]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Methods in Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.

[8]

Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Theory, Models, Applications, Series: Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhauser-Springer, 1 (2017), 299-331.

[9]

F. Cucker and J.-G Dong, On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.  doi: 10.1142/S0218202516500639.

[10]

F. Cucker and J.-G Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.

[11]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.

[12]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[13]

P. Degond and S. Motsch, Large-scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1022.  doi: 10.1007/s10955-008-9529-8.

[14]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.

[15]

S. -Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear in Kinetic Relat. Models.

[16]

S.-Y. HaB. Kwon and M.-J. Kang, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Mod. Meth. Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.

[17]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.

[18]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[19]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[20]

J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168.  doi: 10.1016/j.jcp.2016.03.047.

[21]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, in Uncertainty Quantification for Kinetic and Hyperbolic Equations, (eds S. Jin and L. Pareschi), 14 (2018), 193-229. doi: 10.1007/978-3-319-67110-9_6.

[22]

J. HuS. Jin and D. Xiu, A stochastic Galerkin method for Hamiltonian-Jacobi equations with uncertainty, SIAM. J. Sci. Comput., 37 (2015), 2246-2269.  doi: 10.1137/140990930.

[23]

H. -N. Huang, S. A. M. Marcantognini and N. J. Young, Chain rules for higher derivatives, The Mathematical Intelligencer, 28 (2006), 61-69, http://ambio1.leeds.ac.uk/~nicholas/abstracts/FaadiBruno3.pdf. doi: 10.1007/BF02987158.

[24]

S. Jin, J. -G. Liu and Z. Ma, Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro-macro decomposition based asymptotic preserving method, Research in Math. Sci., 4 (2017), Paper No. 15, 25 pp. doi: 10.1186/s40687-017-0105-1.

[25]

S. Jin and L. Liu, An asymptotic-preserving stochastic Galerkin method for the semicondutor Boltzmann equation with random inputs and diffusive scalings, Multiscale Model. Simu., 15 (2017), 157-183.  doi: 10.1137/15M1053463.

[26]

S. Jin, M. -B. Tran and E. Zuazua, A local sensivity analysis for a damped wave equation with random initial input, Preprint.

[27]

S. JinD. Xiu and X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, J. Comput. Phys., 289 (2015), 35-52.  doi: 10.1016/j.jcp.2015.02.023.

[28]

S. JinD. Xiu and X. Zhu, A well-balanced stochastic Galerkin method for scalar hyperbolic balance laws with random inputs, J. Sci. Comput., 67 (2016), 1198-1218.  doi: 10.1007/s10915-015-0124-2.

[29]

S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple Scales, To appear in SIAM J. Math. Anal.

[30]

M.-J. Kang and A. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542.

[31]

T. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.

[32]

T. KarperA. Mellet and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243.  doi: 10.1137/120866828.

[33]

T. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227-242. doi: 10.1007/978-3-642-39007-4_11.

[34]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.

[35]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertainty Quantification, 5 (2017), 1193-1219.  doi: 10.1137/16M1106675.

[36]

L. Liu and S. Jin, Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs, Preprint.

[37]

S. Motsch and E. Tadmor, Heterophilious dynamics: Enhanced Consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[38]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[39]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105. 

[40]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. of Guidance, Control and Dynamics, 32 (2009), 527-537. 

[41]

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensivity analysis, Global sensivity analysis, The Primer, (2008), 1-51.

[42]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719.  doi: 10.1137/060673254.

[43]

R. Shu and S. Jin, Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime, Preprint.

[44]

E. Tadmor, Mathematical aspects of self-organized dynamics: consensus, emergence of leaders, and social hydrodynamics, SIAM News, 48 (2015).

[45]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

[46]

T. VicsekE. Ben-Jacob CzirókI. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[47]

D. Xiu, Numerical Methods fo Stochastic Computations, Princeton University Presss, 2010.

[48]

D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM. J. Scientific Computiong, 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.

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