Open Access
Volume 58, Number 3, May-June 2024
Page(s) 957 - 992
Published online 10 June 2024
  1. 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]
  2. J.-D. Benamou, Numerical resolution of an “unbalanced” mass transport problem. ESAIM: Math. Modell. Numer. Anal. – Modél. Math. Anal. Numér. 37 (2003) 851–868. [Google Scholar]
  3. 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]
  4. J.-D. Benamou and G. Carlier, Augmented lagrangian methods for transport optimization, mean field games and degenerate elliptic equations. J. Optim. Theory App. 167 (2015) 1–26. [Google Scholar]
  5. R. Bhatia, Matrix Analysis. Vol. 169. Springer Science & Business Media (2013). [Google Scholar]
  6. G. Bouchitté, Convex analysis and duality. Preprint arXiv:2004.09330 (2020). [Google Scholar]
  7. Y. Brenier, The initial value problem for the euler equations of incompressible fluids viewed as a concave maximization problem. Commun. Math. Phys. 364 (2018) 579–605. [Google Scholar]
  8. Y. Brenier and D. Vorotnikov, On optimal transport of matrix-valued measures. SIAM J. Math. Anal. 52 (2020) 2849–2873. [Google Scholar]
  9. S.C. Brenner, The Mathematical Theory of Finite Element Methods. Springer (2008). [Google Scholar]
  10. L.A. Caffarelli and R.J. McCann, Free boundaries in optimal transport and Monge-Ampere obstacle problems. Ann. Math. 171 (2010) 673–730. [Google Scholar]
  11. E.A. Carlen and J. Maas, An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker–Planck equation is gradient flow for the entropy. Commun. Math. Phys. 331 (2014) 887–926. [Google Scholar]
  12. E.A. Carlen and J. Maas, Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. J. Funct. Anal. 273 (2017) 1810–1869. [Google Scholar]
  13. J.A. Carrillo, K. Craig, L. Wang and C. Wei, Primal dual methods for Wasserstein gradient flows. Found. Comput. Math. 22 (2022) 389–443. [Google Scholar]
  14. Y. Chen, T.T. Georgiou and A. Tannenbaum, Matrix optimal mass transport: a quantum mechanical approach. IEEE Trans. Autom. Control 63 (2017) 2612–2619. [Google Scholar]
  15. Y. Chen, T.T. Georgiou and A. Tannenbaum, Interpolation of matrices and matrix-valued densities: the unbalanced case. Eur. J. Appl. Math. 30 (2019) 458–480. [Google Scholar]
  16. Y. Chen, W. Gangbo, T.T. Georgiou and A. Tannenbaum, On the matrix Monge–Kantorovich problem. Eur. J. Appl. Math. 31 (2020) 574–600. [Google Scholar]
  17. L. Chizat, G. Peyré, B. Schmitzer and F.-X. Vialard, An interpolating distance between optimal transport and Fisher–Rao metrics. Found. Comput. Math. 18 (2018) 1–44. [Google Scholar]
  18. L. Chizat, G. Peyré, B. Schmitzer and F.-X. Vialard, Scaling algorithms for unbalanced optimal transport problems. Math. Comput. 87 (2018) 2563–2609. [Google Scholar]
  19. L. Chizat, G. Peyré, B. Schmitzer and F.-X. Vialard, Unbalanced optimal transport: dynamic and Kantorovich formulations. J. Funct. Anal. 274 (2018) 3090–3123. [Google Scholar]
  20. A.J. Duran and P. Lopez-Rodriguez, The LpSpace of a positive definite matrix of measures and density of matrix polynomials inL1. J. Approximation Theory 90 (1997) 299–318. [Google Scholar]
  21. M. Erbar, M. Rumpf, B. Schmitzer and S. Simon, Computation of optimal transport on discrete metric measure spaces. Numer. Math. 144 (2020) 157–200. [Google Scholar]
  22. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press (2015). [Google Scholar]
  23. A. Figalli, The optimal partial transport problem. Arch. Ration. Mech. Anal. 195 (2010) 533–560. [Google Scholar]
  24. A. Figalli and N. Gigli, A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions. J. Math. App. 94 (2010) 107–130. [Google Scholar]
  25. G. Fu, S. Osher and W. Li, High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems. J. Comput. Phys. 491 (2023) 112375. [Google Scholar]
  26. W. Gangbo, W. Li, S. Osher and M. Puthawala, Unnormalized optimal transport. J. Comput. Phys. 399 (2019) 108940. [Google Scholar]
  27. Y. Gao, W. Li and J.-G. Liu, Master equations for finite state mean field games with nonlinear activations. Preprint arXiv:2212.05675 (2022). [Google Scholar]
  28. N. Gigli and J. Maas, Gromov–Hausdorff convergence of discrete transportation metrics. SIAM J. Math. Anal. 45 (2013) 879–899. [Google Scholar]
  29. P. Gladbach, E. Kopfer and J. Maas, Scaling limits of discrete optimal transport. SIAM J. Math. Anal. 52 (2020) 2759–2802. [Google Scholar]
  30. F. Golse, C. Mouhot and T. Paul, On the mean field and classical limits of quantum mechanics. Commun. Math. Phys. 343 (2016) 165–205. [Google Scholar]
  31. F. Golse and T. Paul, The Schrödinger equation in the mean-field and semiclassical regime. Arch. Ration. Mech. Anal. 223 (2017) 57–94. [Google Scholar]
  32. F. Golse and T. Paul, Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics. C. R. Math. 356 (2018) 177–197. [Google Scholar]
  33. L. Gross, Hypercontractivity and logarithmic sobolev inequalities for the Clifford-Dirichlet form. Duke Math. J. 42 (1975) 383–396. [Google Scholar]
  34. K. Guittet, On the time-continuous mass transport problem and its approximation by augmented Lagrangian techniques. SIAM J. Numer. Anal. 41 (2003) 382–399. [Google Scholar]
  35. R. Hug, E. Maitre and N. Papadakis, On the convergence of augmented lagrangian method for optimal transport between nonnegative densities. J. Math. Anal. App. 485 (2020) 123811. [Google Scholar]
  36. L.V. Kantorovich, On the translocation of masses. Dokl. Akad. Nauk. USSR (NS) 37 (1942) 199–201. [Google Scholar]
  37. M.J. Kastoryano and K. Temme, Quantum logarithmic Sobolev inequalities and rapid mixing. J. Math. Phys. 54 (2013) 052202. [Google Scholar]
  38. S. Kondratyev, L. Monsaingeon and D. Vorotnikov, A new optimal transport distance on the space of finite radon measures. Adv. Differ. Equ. 21 (2016) 1117–1164. [Google Scholar]
  39. H. Lavenant, Unconditional convergence for discretizations of dynamical optimal transport. Math. Comput. 90 (2021) 739–786. [Google Scholar]
  40. H. Lavenant, S. Claici, E. Chien and J. Solomon, Dynamical optimal transport on discrete surfaces. ACM Trans. Graphics (TOG) 37 (2018) 1–16. [Google Scholar]
  41. D. Le Bihan, Diffusion MRI: what water tells us about the brain. EMBO Mol. Med. 6 (2014) 569–573. [Google Scholar]
  42. W. Lee, R. Lai, W. Li and S. Osher, Generalized unnormalized optimal transport and its fast algorithms. J. Comput. Phys. 436 (2021) 110041. [Google Scholar]
  43. B. Li and J. Lu, Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics. J. Stat. Phys. 190 (2023) 161. [Google Scholar]
  44. B. Li and J. Zou, On a general matrix-valued unbalanced optimal transport problem. Preprint arXiv:2011.05845 (2023). [Google Scholar]
  45. W. Li, W. Lee and S. Osher, Computational mean-field information dynamics associated with reaction-diffusion equations. J. Comput. Phys. 466 (2022) 111409. [Google Scholar]
  46. M. Liero, A. Mielke and G. Savaré, Optimal transport in competition with reaction: the Hellinger–Kantorovich distance and geodesic curves. SIAM J. Math. Anal. 48 (2016) 2869–2911. [Google Scholar]
  47. M. Liero, A. Mielke and G. Savaré, Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures. Inventiones Math. 211 (2018) 969–1117. [Google Scholar]
  48. D. Lombardi and E. Maitre, Eulerian models and algorithms for unbalanced optimal transport. ESAIM: Math. Modell. Numer. Anal. – Modél. Math. Anal. Numér. 49 (2015) 1717–1744. [Google Scholar]
  49. J. Maas, Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261 (2011) 2250–2292. [Google Scholar]
  50. J. Maas, M. Rumpf, C. Schönlieb and S. Simon, A generalized model for optimal transport of images including dissipation and density modulation. ESAIM: Math. Modell. Numer. Anal. 49 (2015) 1745–1769. [Google Scholar]
  51. D. Mitrea, Distributions, Partial Differential Equations, and Harmonic Analysis. Springer (2013). [Google Scholar]
  52. G. Monge, Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris (1781). [Google Scholar]
  53. L. Monsaingeon and D. Vorotnikov, The schrödinger problem on the non-commutative Fisher-Rao space. Calculus Variations Part. Differ. Equ. 60 (2021) 14. [Google Scholar]
  54. A. Natale and G. Todeschi, Computation of optimal transport with finite volumes. ESAIM: Math. Modell. Numer. Anal. 55 (2021) 1847–1871. [Google Scholar]
  55. A. Natale and G. Todeschi, A mixed finite element discretization of dynamical optimal transport. J. Sci. Comput. 91 (2022) 38. [Google Scholar]
  56. N. Papadakis, G. Peyré and E. Oudet, Optimal transport with proximal splitting. SIAM J. Imaging Sci. 7 (2014) 212–238. [Google Scholar]
  57. G. Peyré, L. Chizat, F.-X. Vialard and J. Solomon, Quantum entropic regularization of matrix-valued optimal transport. Eur. J. Appl. Math. 30 (2019) 1079–1102. [Google Scholar]
  58. B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source. Arch. Ration. Mech. Anal. 211 (2014) 335–358. [Google Scholar]
  59. B. Piccoli and F. Rossi, On properties of the generalized Wasserstein distance. Arch. Ration. Mech. Anal. 222 (2016) 1339–1365. [Google Scholar]
  60. R.T. Powers and E. Størmer, Free states of the canonical anticommutation relations. Commun. Math. Phys. 16 (1970) 1–33. [Google Scholar]
  61. A.M. Rubinov, Abstract Convexity and Global Optimization. Vol. 44. Springer Science & Business Media (2013). [Google Scholar]
  62. W. Rudin, Real and Complex Analysis. Tata McGraw-hill Education (2006). [Google Scholar]
  63. E.K. Ryu, Y. Chen, W. Li and S. Osher, Vector and matrix optimal mass transport: theory, algorithm, and applications. SIAM J. Sci. Comput. 40 (2018) A3675–A3698. [Google Scholar]
  64. R. Teman, Numerical Analysis. Springer Science & Business Media (2012). [Google Scholar]
  65. C. Villani, Topics in Optimal Transportation. Number 58 in Graduate Studies in Mathematics. American Mathematical Society (2003). [Google Scholar]
  66. C. Villani, Optimal Transport: Old and New. Vol. 338. Springer Science & Business Media (2008). [Google Scholar]
  67. D. Vorotnikov, Partial differential equations with quadratic nonlinearities viewed as matrix-valued optimal ballistic transport problems. Arch. Ration. Mech. Anal. 243 (2022) 1653–1698. [Google Scholar]
  68. B.A. Wandell, Clarifying human white matter. Ann. Rev. Neurosci. 39 (2016) 103–128. [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.

Recommended for you