Free Access
Volume 27, 2021
Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
Article Number S5
Number of page(s) 22
Published online 01 March 2021
  1. G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics. Multisc. Model. Simul. 11 (2013) 1–29. [Google Scholar]
  2. J.J. Alonso and J. Hicken, Introduction to multidisciplinary design optimization. In Vol. 222 of Aeronautics & Astronautics. Standford University (2012). [Google Scholar]
  3. N. Bellomo, A. Bellouquid and D. Knopoff, From the microscale to collective crowd dynamics. Multis. Model. Simul. 11 (2013) 943–963. [Google Scholar]
  4. C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. International Series in Pure and Applied Mathematics. McGraw-Hill (1978). [Google Scholar]
  5. Y. Bengio, P. Simard and P. Frasconi, Learning long-term dependencies with gradient descent is difficult. IEEE Trans. Neural Netw. 5 (1994) 157–166. [CrossRef] [PubMed] [Google Scholar]
  6. A.L. Bertozzi, J. Rosado, M.B. Short and L. Wang, Contagion shocks in one dimension. J. Stat. Phys. 158 (2015) 647–664. [Google Scholar]
  7. F. Bolley, J.A. Cañizo and J.A. Carrillo, Stochastic mean-field limit: non-Lipschitz forces and swarming. Math. Models Methods Appl. Sci. 21 (2011) 2179–2210. [Google Scholar]
  8. L. Bottou, Online learning and stochastic approximations. On-line Learn. Neural Netw. 17 (1998) 142. [Google Scholar]
  9. S. Bubeck, Convex optimization: algorithms and complexity. Found. Trends® Mach. Learn. 8 (2015) 231–357. [Google Scholar]
  10. J.A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: mean-field limit and Wasserstein distances. Collective dynamics from bacteria to crowds, volume 553 of CISM Courses and Lectures. Springer, Vienna (2014) 1–46. [Google Scholar]
  11. J.A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse, An analytical framework for consensus-based global optimization method. Math. Models Methods Appl. Sci. 28 (2018) 1037–1066. [Google Scholar]
  12. J.A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42 (2010) 218–236. [CrossRef] [Google Scholar]
  13. J.A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming. In Mathematical modeling of collective behavior in socio-economic and life sciences, Modelling and Simulation in Materials Science and Engineering. Birkhäuser Boston, Inc., Boston, MA (2010) 297–336. [Google Scholar]
  14. J.A. Carrillo, L. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty. Commun. Comput. Phys. 25 (2018) 508–531. [Google Scholar]
  15. F. Cucker and S. Smale, On the mathematics of emergence. Jpn. J. Math. 2 (2007) 197–227. [CrossRef] [Google Scholar]
  16. X. Dai and Y. Zhu, Towards theoretical understanding of large batch training in stochastic gradient descent. Preprint arXiv:1812.00542 (2018). [Google Scholar]
  17. A. Dembo and O. Zeitouni, Vol. 38 of Large deviations techniques and applications. Springer Science & Business Media (2009). [Google Scholar]
  18. R. Eberhart and J. Kennedy, Particle swarm optimization. In Proc. of IEEE International Conference on Neural Networks 4 (1995) 1942–1948. [Google Scholar]
  19. S.-Y. Ha, S. Jin and D. Kim, Convergence of a first-order consensus-based global optimization algorithm. Preprint arXiv:1910.08239 (2019). [Google Scholar]
  20. S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic Related Models 1 (2008) 415–435. [CrossRef] [Google Scholar]
  21. B. Hanin, Which neural net architectures give rise to exploding and vanishing gradients? In Adv. Neural Inf. Process. Syst. (2018) 582–591. [Google Scholar]
  22. M. Hauray and P.-E. Jabin, N-particles approximation of the Vlasov equations with singular potential. Arch. Ratl. Mech. Anal. 183 (2007) 489–524. [Google Scholar]
  23. J.H. Holland, Genetic algorithms. Sci. Am. 267 (1992) 66–73. [Google Scholar]
  24. R.A. Holley, S. Kusuoka and D.W. Stroock, Asymptotics of the spectral gap with applications to the theory of simulated annealing. J. Funct. Anal. 83 (1989) 333–347. [Google Scholar]
  25. L.C. Hsu, A theorem on the asymptotic behavior of a multiple integral. Duke Math. J. 15 (1948) 623–6323. [Google Scholar]
  26. C.-R. Hwang and S.-J. Sheu, Large-time behavior of perturbed diffusion Markov processes with applications to the second eigenvalue problem for Fokker-Planck operators and simulated annealing. Acta Appl. Math. 19 (1990) 253–295. [Google Scholar]
  27. T. Inglot and P. Majerski, Simple upper and lower bounds for the multivariate Laplace approximation. J. Approx. Theory 186 (2014) 1–11. [Google Scholar]
  28. P.-E. Jabin, A review of the mean field limits for Vlasov equations. Kinetic Related Models 7 (2014) 661–711. [Google Scholar]
  29. P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with W−1, kernels. Invent. Math. 214 (2018) 523–591. [Google Scholar]
  30. S. Jastrzebski, Z. Kenton, D. Arpit, N. Ballas, A. Fischer, Y. Bengio and A. Storkey, Three factors influencing minima in SGD. Preprint arXiv:1711.04623 (2017). [Google Scholar]
  31. S. Jin, L. Li and J.-G. Liu, Random batch methods (RBM) for interacting particle systems. J. Comput. Phys. 400 (2020) 108877. [Google Scholar]
  32. J. Kennedy, Swarm intelligence, handbook of nature-inspired and innovative computing. Springer (2006) 187–219. [Google Scholar]
  33. N.S. Keskar, D. Mudigere, J. Nocedal, M. Smelyanskiy and P.T.P. Tang, On large-batch training for deep learning: generalization gap and sharp minima. In International Conference on Learning Representations (2017). [Google Scholar]
  34. S. Kirkpatrick, C. Daniel Gelatt and M.P. Vecchi, Optimization by simulated annealing. Science 220 (1983) 671–680. [Google Scholar]
  35. T. Kolokolnikov, J.A. Carrillo, A. Bertozzi, R. Fetecau and M. Lewis, Emergent behaviour in multi-particle systems with non-local interactions [Editorial]. J. Phys. D 260 (2013) 1–4. [Google Scholar]
  36. J.P. McClure and R. Wong, Error bounds for multidimensional Laplace approximation. J. Approx. Theory 37 (1983) 372–390. [Google Scholar]
  37. P.J.M. van Laarhoven and E.H.L. Aarts, Simulated annealing: theory and applications. D. Reidel Publishing Co., Dordrecht (1987) 37. [Google Scholar]
  38. S. Liu, D. Papailiopoulos and D. Achlioptas, Bad global minima exist and SGD can reach them. Preprint arXiv:1906.02613 (2019). [Google Scholar]
  39. P.D. Miller, Applied asymptotic analysis. American Mathematical Society (2006). [Google Scholar]
  40. S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus. SIAM Rev. 56 (2014) 577–621. [Google Scholar]
  41. J.A. Nelder and R. Mead, A simplex method for function minimization. Comput. J. 7 (1965) 308–313. [Google Scholar]
  42. R. Pinnau, C. Totzeck, O. Tse and S. Martin, A consensus-based model for global optimization and its mean-field limit. Math. Models Methods Appl. Sci. 27 (2017) 183–204. [Google Scholar]
  43. H. Robbins and S. Monro, A stochastic approximation method. Ann. Math. Stat. (1951) 400–407. [Google Scholar]
  44. G. Toscani, Kinetic models of opinion formation. Commun. Math. Sci. 4 (2006) 481–496. [Google Scholar]
  45. C. Totzeck, R. Pinnau, S. Blauth and S. Schotthófer, A numerical comparison of consensus-based global optimization to other particle-based global optimization schemes. Proc. Appl. Math. Mech. 18 (2018) e201800291. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.