{Dòng chảy gradient Euclid, metric và Wasserstein: một cái nhìn tổng quan}

Ambrosio, L.: Movimenti minimizzanti. Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur. 113, 191–246 (1995) Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics, vol. 2062, pp. 1–155 Springer-Verlag (2013) Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Spaces of Probability Measures. Lectures in Mathematics. ETH Zurich, Birkhäuser (2005) Ambrosio, L., Gigli, N., Savaré, G.: Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam. 29, 969–986 (2013) Ambrosio, L., Gigli, N., Savaré, G.: Heat flow and calculus on metric measure spaces with Ricci curvature bounded below—the compact case. In: Brezzi, F., Colli Franzone, P., Gianazza, U.P., Gilardi, G. (eds.) Analysis and Numerics of Partial Differential Equations. “INDAM”, vol. 4, pp. 63–116. Springer, Berlin (2013) Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Inv. Math. 195(2), 289–391 (2014) Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163, 1405–1490 (2014) Ambrosio, L., Savaré, G.: Gradient flows of probability measures. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations: Evolutionary equations, vol. 3. Elsevier, Amsterdam (2007) Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford Lecture Series in Mathematics and its Applications, vol. 25. Oxford University Press, Oxford (2004) Aubin, J.-P.: Un théorème de compacité. C. R. Acad. Sci. Paris 256, 5042–5044 (1963) Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000) Benamou, J.-D., Carlier, G.: Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations. J. Opt. Theor. Appl. 167(1), 1–26 (2015) Benamou, J.-D., Carlier, G., Laborde, M.: An augmented Lagrangian approach to Wasserstein gradient flows and applications. ESAIM Proc. 54, 1–17 (2016) Benamou, J.-D., Carlier, G., Mérigot, Q., Oudet, É.: Discretization of functionals involving the Monge–Ampère operator. Numerische Mathematik 134(3), 611–636 (2016) Blanchet, A., Calvez, V., Carrillo, J.A.: Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model. SIAM J. Numer. Anal. 46(2), 691–721 (2008) Blanchet, A., Carrillo, J.-A., Kinderlehrer, D., Kowalczyk, M., Laurençot, P., Lisini, S.: A hybrid variational principle for the Keller–Segel system in \({\mathbb{R}}^2\). ESAIM M2AN 49(6), 1553–1576 (2015) Bouchitté, G., Buttazzo, G.: Characterization of optimal shapes and masses through Monge–Kantorovich equation. J. Eur. Math. Soc. 3(2), 139–168 (2001) Bouchitté, G., Buttazzo, G., Seppecher, P.: Shape optimization solutions via Monge–Kantorovich equation. C. R. Acad. Sci. Paris Sér. I Math 324(10), 1185–1191 (1997) Brenier, Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math 305(19), 805–808 (1987) Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44, 375–417 (1991) Brenier, Y.: Extended Monge–Kantorovich theory. In: Optimal transportation and Applications, 91–121, 2003, Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 2–8 (2001) Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les spaces de Hilbert. North-Holland Mathematics Studies, Amsterdam (1973) Burago, Yu., Gromov, M., Perelman G.: A. D. Alexandrov spaces with curvatures bounded below. Uspekhi Mat. Nauk47, 3–51 (1992); English translation: Russ. Math. Surv.47, 1–58 (1992) Buttazzo, G.: Semicontinuity, Relaxation, and Integral Representation in the Calculus of Variations. Longman Scientific and Technical, New York (1989) Cancès, C., Gallouët, T., Monsaingeon, L.: Incompressible immiscible multiphase flows in porous media: a variational approach https://arxiv.org/abs/1607.04009 (2016) (preprint) Carrillo, J.-A., Di Francesco, M., Figalli, A., Laurent, T., Slepčev, D.: Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math J. 156(2), 229–271 (2011) Carrillo, J.-A., McCann, R.J., Villani, C.: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoam. 19, 1–48 (2003) Carrillo, J.-A., McCann, R.J., Villani, C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179, 217–263 (2006) Carrillo, J.-A., Slepčev, D.: Example of a displacement convex functional of first order. Calc. Var. Partial Differ. Equ. 36(4), 547–564 (2009) Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999) Chizat, L., Peyré, G., Schmitzer, B., Vialard, F.-X.: Unbalanced optimal transport: geometry and Kantorovich formulation. arXiv preprint arXiv:1508.05216 (2015) Chizat, L., Peyré, G., Schmitzer, B., Vialard, F.-X.: An interpolating distance between optimal transport and Fisher–Rao. arXiv preprint arXiv:1506.06430 (2015) Choné, P., Le Meur, H.: Non-convergence result for conformal approximation of variational problems subject to a convexity constraint. Numer. Funct. Anal. Optim. 5–6(22), 529–547 (2001) Craig, K.: Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions, preprint available at arXiv:1512.07255 Daneri, S., Savaré, G.: Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Anal. 40, 1104–1122 (2008) De Giorgi, E.: New problems on minimizing movements. In: Baiocchi, C., Lions, J.L. (eds.) Boundary Value Problems for PDE and Applications, pp. 81–98. Masson, Paris (1993) Di Marino, S., Maury, B., Santambrogio, F.: Measure sweeping processes. J. Convex Anal. 23(1), 567–601 (2016) Di Marino, S., Mészáros, A.R.: Uniqueness issues for evolution equations with density constraints. Math. Models Methods Appl. Sci. 26(09), 1761–1783 (2016) Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47(2), 324–353 (1974) Ekeland, I., Moreno-Bromberg, S.: An algorithm for computing solutions of variational problems with global convexity constraints. Numerische Mathematik 115(1), 45–69 (2010) Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Society for Industrial and Applied Mathematics, Classics in Mathematics, Philadelphia (1999) Figalli, A., Gigli, N.: A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions. J. Math. Pures Appl. 94(2), 107–130 (2010) Fortin, M., Glowinski, R.: Augmented Lagrangian Methods Applications to the Numerical Solution of Boundary-Value Problems. North-Holland, Amsterdam (1983) Gallouët, T., Monsaingeon, L.: A JKO splitting scheme for Kantorovich-Fisher–Rao gradient flows, preprint available at arXiv:1602.04457 Gangbo, W.: An elementary proof of the polar factorization of vector-valued functions. Arch. Ration. Mech. Anal. 128, 381–399 (1994) Gangbo, W.: The Monge mass transfer problem and its applications. Contemp. Math. 226, 79–104 (1999) Gangbo, W., McCann, R.: The geometry of optimal transportation. Acta Math. 177, 113–161 (1996) Gigli, N.: On the Heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. Partial Differ. Equ. 39, 101–120 (2010) Gigli, N.: Propriétés géométriques et analytiques de certaines structures non lisses. Mémoire HDR, Univ. Nice-Sophia-Antipolis (2011) Gigli, N., Kuwada, K., Ohta, S.I.: Heat flow on Alexandrov spaces. Commun. Pure Appl. Math. LXVI, 307–331 (2013) Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5, 403–415 (1996) Hajłasz, P.: Sobolev spaces on metric-measure spaces. Contemp. Math. 338, 173–218 (2003) Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 688, 1–101 (2000) Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998) Kantorovich, L.: On the transfer of masses. Dokl. Acad. Nauk. USSR 37, 7–8 (1942) Keller, E.F., Segel, L.A.: Initiation of slide mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970) Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971) Kinderlehrer, D., Walkington, N.J.: Approximation of parabolic equations using the Wasserstein metric. ESAIM Math. Model. Numer. Anal. 33(04), 837–852 (1999) Kitagawa, J., Mérigot, Q., Thibert, B.: Convergence of a Newton algorithm for semi-discrete optimal transport. J. Eur. Math. Soc. (2016). Accepted 2017 Kondratyev, S., Monsaingeon, L., Vorotnikov, D.: A new optimal transport distance on the space of finite radon measures. arXiv preprint arXiv:1505.07746 (2015) Legendre, G., Turinici, G.: Second order in time schemes for gradient flows in Wasserstein and geodesic metric spaces https://hal.archives-ouvertes.fr/hal-01317769/document (2016) (preprint) Lévy, B.: A numerical algorithm for \(L^2\) semi-discrete optimal transport in 3D. ESAIM Math. Model. Numer. Anal 49(6), 1693–1715 (2015) Liero, M., Mielke, A., Savaré, G.: Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures. arXiv preprint arXiv:1508.0794 (2015) Liero, M., Mielke, A., Savaré, G.: Optimal transport in competition with reaction: the Hellinger–Kantorovich distance and geodesic curves. arXiv preprint arXiv:1509.00068 (2015) Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991 (2009) Matthes, D., McCann, R.J., Savaré, G.: A family of nonlinear fourth order equations of gradient flow type. Commun. Partial Differ. Equ. 34(10–12), 1352–1397 (2009) Maury, B., Roudneff-Chupin, A., Santambrogio, F.: A macroscopic crowd motion model of gradient flow type. Math. Models Methods Appl. Sci. 20(10), 1787–1821 (2010) Maury, B., Roudneff-Chupin, A., Santambrogio, F., Venel, J.: Handling congestion in crowd motion modeling. Net. Het. Media 6(3), 485–519 (2011) Maury, B., Venel, J.: A mathematical framework for a crowd motion model. C. R. Acad. Sci. Paris, Ser. I 346, 1245–1250 (2008) Maury, B., Venel, J.: A discrete contact model for crowd motion. ESAIM Math. Model. Numer. Anal. 45(1), 145–168 (2011) McCann, R.J.: Existence and uniqueness of monotone measure preserving maps. Duke Math. J. 80, 309–323 (1995) McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–159 (1997) McCann, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589–608 (2001) Mérigot, Q.: A multiscale approach to optimal transport. Comput. Gr. Forum 30(5), 1583–1592 (2011) Mérigot, Q., Oudet, É.: Handling convexity-like constraints in variational problems. SIAM J. Numer. Anal. 52(5), 2466–2487 (2014) Mészáros, A.R., Santambrogio, F.: Advection–diffusion equations with density constraints. Anal. PDE 9–3, 615–644 (2016) Monge, G.: Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris 666–704 (1781) Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2011) von Renesse, M.-K., Sturm, K.-T.: Entropic measure and Wasserstein diffusion. Ann. Probab. 37, 1114–1191 (2009) Rockafellar, R.T.: Convex Anal. Princeton University Press, Princeton (1970) Roudneff-Chupin, A.: Modélisation macroscopique de mouvements de foule, PhD Thesis, Université Paris-Sud (2011) Santambrogio, F.: Gradient flows in Wasserstein spaces and applications to crowd movement, Séminaire Laurent Schwartz no 27, École Polytechnique (2010) Santambrogio, F.: Flots de gradient dans les espaces métriques et leurs applications (d’après Ambrosio–Gigli–Savaré). In: Proceedings of the Bourbaki Seminar (2013) (in French) Santambrogio, F.: Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications no 87, Birkhäuser Basel (2015) Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16, 243–279 (2000) Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, vol. 44. Cambridge University Press, Cambridge (1993) Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196, 65–131 (2006) Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196, 133–177 (2006) Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford University Press, Oxford (2007) Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics. AMS, Providence (2003) Villani, C.: Optimal Transport: Old and New. Springer, Berlin (2008)