Open Access
Issue |
ESAIM: COCV
Volume 31, 2025
|
|
---|---|---|
Article Number | 20 | |
Number of page(s) | 46 | |
DOI | https://doi.org/10.1051/cocv/2025010 | |
Published online | 20 March 2025 |
- S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Related Models 1 (2008) 415–435. [Google Scholar]
- R. Carmona, F. Delarue, et al., Probabilistic Theory of Mean Field Games with Applications I–II. Springer (2018). [Google Scholar]
- 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: an Excursion Through Modeling, Analysis and Simulation (2014) 1–46. [Google Scholar]
- 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 (2010) 297–336. [Google Scholar]
- M. Fornasier and F. Solombrino, Mean-field optimal control. ESAIM: Control Optim. Calc. Var. 20 (2014) 1123–1152. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
- C. Orrieri, A. Porretta and G. Savaré, A variational approach to the mean field planning problem. J. Funct. Anal. 277 (2019) 1868–1957. [Google Scholar]
- A. Porretta, On the planning problem for the mean field games system. Dyn. Games Appl. 4 (2014) 231–256. [CrossRef] [MathSciNet] [Google Scholar]
- L. Ruthotto, S.J. Osher, W. Li, L. Nurbekyan and S.W. Fung, A machine learning framework for solving high-dimensional mean field game and mean field control problems. Proc. Natl. Acad. Sci. U.S.A. 117 (2020) 9183–9193. [Google Scholar]
- M. Burger, R. Pinnau, C. Totzeck and O. Tse, Mean-field optimal control and optimality conditions in the space of probability measures. SIAM J. Control Optim. 59 (2021) 977–1006. [CrossRef] [MathSciNet] [Google Scholar]
- B. Bonnet and H. Frankowska, Necessary optimality conditions for optimal control problems in wasserstein spaces. Appl. Math. Optim. 84 (2021) 1281–1330. [Google Scholar]
- G. Cavagnari, S. Lisini, C. Orrieri and G. Savaré, Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: equivalence and gamma-convergence. Jo. Diff. Equ. 322 (2022) 268–364. [Google Scholar]
- C. Jimenez, A. Marigonda and M. Quincampoix, Optimal control of multiagent systems in the wasserstein space. Calc. Var. Part. Diff. Equ. 59 (2020) 58. [Google Scholar]
- B. Bonnet and H. Frankowska, Semiconcavity and sensitivity analysis in mean-field optimal control and applications. J. Math. Pures Appl. 157 (2022) 282–345. [Google Scholar]
- M.S. Albergo and E. Vanden-Eijnden, Building normalizing flows with stochastic interpolants. arXiv preprint arXiv:2209.15571 (2023). [Google Scholar]
- I. Kobyzev, S.J. Prince and M.A. Brubaker, Normalizing flows: an introduction and review of current methods. IEEE Trans. Pattern Anal. Mach. Intell. 43 (2020) 3964–3979. [Google Scholar]
- J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375–393. [Google Scholar]
- Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: numerical methods for the planning problem. SIAM J. Control Optim. 50 (2012) 77–109. [Google Scholar]
- K. Elamvazhuthi and P. Grover, Optimal transport over nonlinear systems via infinitesimal generators on graphs. J. Computat. Dyn. 5 (2018) 1–32. [Google Scholar]
- K. Elamvazhuthi, S. Liu, W. Li and S. Osher, Dynamical optimal transport of nonlinear control-affine systems. J. Computat. Dyn. 10 (2023) 425–449. [Google Scholar]
- A. Agrachev and P. Lee, Optimal transportation under nonholonomic constraints. Trans. Am. Math. Soc. 361 (2009) 6019–6047. [Google Scholar]
- A. Hindawi, J.-B. Pomet and L. Rifford, Mass transportation with lq cost functions. Acta Appl. Math. 113 (2011) 215–229. [CrossRef] [MathSciNet] [Google Scholar]
- Y. Chen, T.T. Georgiou and M. Pavon, Optimal transport over a linear dynamical system. IEEE Trans. Automatic Control 62 (2016) 2137–2152. [Google Scholar]
- M. Fornasier, S. Lisini, C. Orrieri and G. Savaré, Mean-field optimal control as gamma-limit of finite agent controls. Eur. J. Appl. Math. 30 (2019) 1153–1186. [CrossRef] [Google Scholar]
- J.T. Beale and A. Majda, Vortex methods. I. Convergence in three dimensions. Math. Comp. 39 (1982) 1–27. [Google Scholar]
- J.T. Beale and A. Majda, Vortex methods. II. Higher order accuracy in two and three dimensions. Math. Comp. 39 (1982) 29–52. [Google Scholar]
- K. Craig and A.L. Bertozzi, A blob method for the aggregation equation. Math. Comp. 85 (2016) 1681–1717. [Google Scholar]
- J.A. Carrillo, K. Craig and F.S. Patacchini, A blob method for diffusion. Calc. Var. Part. Diff. Equ. 58 (2019) 1–53. [Google Scholar]
- K. Craig, K. Elamvazhuthi, M. Haberland and O. Turanova, A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling. Math. Computat. 92 (2023) 2575–2654. [Google Scholar]
- K. Craig, M. Jacobs and O. Turanova, Nonlocal approximation of slow and fast diffusion. J. Differ. Equ., 426 (2025) 782–852. [Google Scholar]
- J.A. Carrillo, A. Esposito and J.S.-H. Wu, Nonlocal approximation of nonlinear diffusion equations. Calc. Var. Partial Differ. Equ., 63 (2024) 100. [Google Scholar]
- P.L. Lions and S. Mas-Gallic, Une méthode particulaire déterministe pour des équations diffusives non linéaires. Comptes Rendus Acad. Sci. Ser. I Math. 332 (2001) 369–376. [Google Scholar]
- K. Oelschläger, Large systems of interacting particles and the porous medium equation. J. Diff. Equ. 88 (1990) 294–346. [Google Scholar]
- M. Burger and A. Esposito, Porous medium equation and cross-diffusion systems as limit of nonlocal interaction. Nonlinear Anal. 235 (2023) 113347. [Google Scholar]
- L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008). [Google Scholar]
- F. Santambrogio, Optimal Transport for Applied Mathematicians, Vol. 55. Birkäuser, NY (2015) 94. [Google Scholar]
- A. Figalli and F. Glaudo, An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows. EMS Textbooks in Mathematics (2021). [Google Scholar]
- L. Ambrosio, E. Brué and D. Semola, Lectures on Optimal Transport. Springer (2021). [Google Scholar]
- C. Villani, Topics in Optimal Transportation, Vol. 58. American Mathematical Society (2021). [Google Scholar]
- L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: in Metric Spaces and in the Space of Probability Measures. Springer Science & Business Media (2005). [Google Scholar]
- A. Bacciotti, Stability and Control of Linear Systems. Springer (2019). [Google Scholar]
- A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al., Pytorch: an imperative style, high-performance deep learning library. Adv. Neural Inform. Process. Syst. 32 (2019). [Google Scholar]
- R. Flamary, N. Courty, A. Gramfort, M.Z. Alaya, A. Boisbunon, S. Chambon, L. Chapel, A. Corenflos, K. Fatras, N. Fournier, L. Gautheron, N.T. Gayraud, H. Janati, A. Rakotomamonjy, I. Redko, A. Rolet, A. Schutz, V. Seguy, D.J. Sutherland, R. Tavenard, A. Tong and T. Vayer, Pot: Python optimal transport. J. Mach. Learn. Res. 22 (2021) 1–8. [Google Scholar]
- C.R. Harris, K.J. Millman, S.J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N.J. Smith, R. Kern, M. Picus, S. Hoyer, M.H. van Kerkwijk, M. Brett, A. Haldane, J.F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke and T.E. Oliphant, Array programming with NumPy. Nature 585 (2020) 357–362. [NASA ADS] [CrossRef] [Google Scholar]
- L. Ambrosio and G. Crippa, Continuity equations and ode flows with non-smooth velocity. Proc. Roy. Soc. Edinb. Sect. A Math. 144 (2014) 1191–1244. [CrossRef] [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.