Papers
Topics
Authors
Recent
Assistant
AI Research Assistant
Well-researched responses based on relevant abstracts and paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses.
Gemini 2.5 Flash
Gemini 2.5 Flash 131 tok/s
Gemini 2.5 Pro 46 tok/s Pro
GPT-5 Medium 26 tok/s Pro
GPT-5 High 32 tok/s Pro
GPT-4o 71 tok/s Pro
Kimi K2 192 tok/s Pro
GPT OSS 120B 385 tok/s Pro
Claude Sonnet 4.5 36 tok/s Pro
2000 character limit reached

On the mean-field limit of the Cucker-Smale model with Random Batch Method (2407.21297v1)

Published 31 Jul 2024 in math.NA, cs.NA, math.AP, math.CA, and math.PR

Abstract: In this work, we focus on the mean-field limit of the Random Batch Method (RBM) for the Cucker-Smale model. Different from the classical mean-field limit analysis, the chaos in this model is imposed at discrete time and is propagated to discrete time flux. We approach separately the limits of the number of particles $N\to\infty$ and the discrete time interval $\tau\to 0$ with respect to the RBM, by using the flocking property of the Cucker-Smale model and the observation in combinatorics. The Wasserstein distance is used to quantify the difference between the approximation limit and the original mean-field limit. Also, we combine the RBM with generalized Polynomial Chaos (gPC) expansion and proposed the RBM-gPC method to approximate stochastic mean-field equations, which conserves positivity and momentum of the mean-field limit with random inputs.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (37)
  1. Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives. Mathematical Models and Methods in Applied Sciences, 29(10):1901–2005, 2019.
  2. A quest toward a mathematical theory of the dynamics of swarms. Mathematical Models and Methods in Applied Sciences, 27(04):745–770, 2017.
  3. Nicola Bellomo and J Soler. On the mathematical theory of the dynamics of swarms viewed as complex systems. Mathematical Models and Methods in Applied Sciences, 22(supp01):1140006, 2012.
  4. A kinetic equation for granular media. ESAIM: Mathematical Modelling and Numerical Analysis, 31(5):615–641, 1997.
  5. Biology of synchronous flashing of fireflies, 1966.
  6. Asymptotic flocking dynamics for the kinetic Cucker–Smale model. SIAM Journal on Mathematical Analysis, 42(1):218–236, 2010.
  7. Particle based gpc methods for mean-field models of swarming with uncertainty. Comuunications in Computational Physics, 25(2):508–531, 2019.
  8. Propagation of chaos: A review of models, methods and applications. i. models and methods. Kinetic and Related Models, 15(6):895–1015, 2022.
  9. Propagation of chaos: A review of models, methods and applications. ii. applications. Kinetic and Related Models, 15(6):1017–1173, 2022.
  10. Emergent behavior in flocks. IEEE Transactions on automatic control, 52(5):852–862, 2007.
  11. John M Danskin. The theory of max-min, with applications. SIAM Journal on Applied Mathematics, 14(4):641–664, 1966.
  12. Continuum limit of self-driven particles with orientation interaction. Mathematical Models and Methods in Applied Sciences, 18(supp01):1193–1215, 2008.
  13. Synchronization in complex networks of phase oscillators: A survey. Automatica, 50(6):1539–1564, 2014.
  14. On the rate of convergence in wasserstein distance of the empirical measure. Probability theory and related fields, 162(3-4):707–738, 2015.
  15. Uniform-in-time error estimate of the random batch method for the Cucker–Smale model. Mathematical Models and Methods in Applied Sciences, 31(06):1099–1135, 2021.
  16. Uniform stability of the Cucker-Smale model and its application to the mean-field limit. Kinetic and Related Models, 11(5):1157–1181, 2018.
  17. Collective synchronization of classical and quantum oscillators. EMS Surveys in Mathematical Sciences, 3(2):209–267, 2016.
  18. A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Communications in Mathematical Sciences, 7(2):297–325, 2009.
  19. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic and Related Models, 1(3):415–435, 2008.
  20. Pierre-Emmanuel Jabin. A review of the mean field limits for vlasov equations. Kinetic and Related models, 7(4):661–711, 2014.
  21. On the mean field limit of the Random Batch Method for interacting particle systems. Science China Mathematics, pages 1–34, 2022.
  22. Random batch methods (rbm) for interacting particle systems. Journal of Computational Physics, 400:108877, 2020.
  23. On the Random Batch Method for second order interacting particle systems. Multiscale Modeling & Simulation, 20(2):741–768, 2022.
  24. Uncertainty quantification for hyperbolic and kinetic equations. Springer, 2017.
  25. A simple control law for uav formation flying. Technical report, Technical Report 2002-38, Institute for Systems Research, 2002.
  26. Yoshiki Kuramoto. Self-entrainment of a population of coupled non-linear oscillators. In International Symposium on Mathematical Problems in Theoretical Physics: January 23–29, 1975, Kyoto University, Kyoto/Japan, pages 420–422. Springer, 1975.
  27. Regularized dirac delta functions for phase field models. International journal for numerical methods in engineering, 91(3):269–288, 2012.
  28. A new model for self-organized dynamics and its flocking behavior. Journal of Statistical Physics, 144:923–947, 2011.
  29. On the mean field limit for Cucker-Smale models. arXiv preprint arXiv:2011.12584, 2020.
  30. Charles S Peskin. Mathematical aspects of heart physiology. Courant Inst. Math, 1975.
  31. Control to flocking of the kinetic Cucker–Smale model. SIAM Journal on Mathematical Analysis, 47(6):4685–4719, 2015.
  32. Flocks, herds, and schools: A quantitative theory of flocking. Physical review E, 58(4):4828, 1998.
  33. Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM Journal on Applied Mathematics, 65(1):152–174, 2004.
  34. Boltzmann-type models with uncertain binary interactions. arXiv preprint arXiv:1709.02353, 2017.
  35. Novel type of phase transition in a system of self-driven particles. Physical review letters, 75(6):1226, 1995.
  36. Cédric Villani et al. Optimal transport: old and new, volume 338. Springer, 2009.
  37. The wiener–askey polynomial chaos for stochastic differential equations. SIAM journal on scientific computing, 24(2):619–644, 2002.

Summary

We haven't generated a summary for this paper yet.

Dice Question Streamline Icon: https://streamlinehq.com

Open Questions

We haven't generated a list of open questions mentioned in this paper yet.

Lightbulb Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Authors (2)

List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 1 tweet and received 1 like.

Upgrade to Pro to view all of the tweets about this paper: