Open Access
Issue
ESAIM: M2AN
Volume 58, Number 3, May-June 2024
Page(s) 957 - 992
DOI https://doi.org/10.1051/m2an/2024024
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. [CrossRef] [EDP Sciences] [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. [CrossRef] [Google Scholar]
  8. Y. Brenier and D. Vorotnikov, On optimal transport of matrix-valued measures. SIAM J. Math. Anal. 52 (2020) 2849–2873. [CrossRef] [MathSciNet] [Google Scholar]
  9. S.C. Brenner, The Mathematical Theory of Finite Element Methods. Springer (2008). [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [Google Scholar]
  33. L. Gross, Hypercontractivity and logarithmic sobolev inequalities for the Clifford-Dirichlet form. Duke Math. J. 42 (1975) 383–396. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [PubMed] [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. [CrossRef] [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. [CrossRef] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [EDP Sciences] [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. [CrossRef] [EDP Sciences] [Google Scholar]
  51. D. Mitrea, Distributions, Partial Differential Equations, and Harmonic Analysis. Springer (2013). [CrossRef] [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. [CrossRef] [Google Scholar]
  54. A. Natale and G. Todeschi, Computation of optimal transport with finite volumes. ESAIM: Math. Modell. Numer. Anal. 55 (2021) 1847–1871. [CrossRef] [EDP Sciences] [Google Scholar]
  55. A. Natale and G. Todeschi, A mixed finite element discretization of dynamical optimal transport. J. Sci. Comput. 91 (2022) 38. [CrossRef] [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. [CrossRef] [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. [CrossRef] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [Google Scholar]
  68. B.A. Wandell, Clarifying human white matter. Ann. Rev. Neurosci. 39 (2016) 103–128. [CrossRef] [PubMed] [Google Scholar]

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