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Optimal transportation, modelling and numerical simulation

Published online by Cambridge University Press:  04 August 2021

Jean-David Benamou
Jean-David Benamou*
Affiliation:
INRIA Paris, Paris 12e, France E-mail: Jean-David.Benamou@inria.fr
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Abstract

We present an overviewof the basic theory, modern optimal transportation extensions and recent algorithmic advances. Selected modelling and numerical applications illustrate the impact of optimal transportation in numerical analysis.

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Research Article
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Acta Numerica , Volume 30 , May 2021 , pp. 249 - 325
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© The Author(s), 2021. Published by Cambridge University Press

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References

Achdou, Y., Cardaliaguet, P., Delarue, F., Porretta, A. and Santambrogio, F. (2020), Mean Field Games, Springer.CrossRefGoogle Scholar
Agueh, M. and Carlier, G. (2011), Barycenters in the Wasserstein space, SIAM J. Math. Anal. 43, 904–924.CrossRefGoogle Scholar
Alfonsi, A., Coyaud, R., Ehrlacher, V. and Lombardi, D. (2021), Approximation of optimal transport problems with marginal moments constraints, Math. Comp. 90, 689–737.CrossRefGoogle Scholar
Altschuler, J. M. and Boix-Adsera, E. (2020), Polynomial-time algorithms for multimarginal optimal transport problems with decomposable structure. Available at arXiv:2008.03006.Google Scholar
Ambrosio, L., Gigli, N. and Savare, G. (2005), Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser.Google Scholar
Andreev, R. (2017), Preconditioning the augmented Lagrangian method for instationary mean field games with diffusion, SIAM J. Sci. Comput. 39, A2763–A2783.CrossRefGoogle Scholar
Arnold, V. (1966), Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16, 319–361.CrossRefGoogle Scholar
Beiglböck, M. and Juillet, N. (2016), On a problem of optimal transport under marginal martingale constraints, Ann. Probab. 44, 42–106.CrossRefGoogle Scholar
Beiglböck, M., Henry-Labordère, P. and Touzi, N. (2017), Monotone martingale transport plans and Skorokhod embedding, Stochastic Process. Appl 127, 3005–3013.CrossRefGoogle Scholar
Benamou, J.-D. (2003), Numerical resolution of an ‘unbalanced’ mass transport problem, ESAIM Math. Model. Numer. Anal. 37, 851–868.CrossRefGoogle Scholar
Benamou, J.-D. and Brenier, Y. (2000), A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math. 84, 375–393.CrossRefGoogle Scholar
Benamou, J.-D. and Carlier, G. (2015), Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theory Appl. 167, 1–26.CrossRefGoogle Scholar
Benamou, J.-D. and Duval, V. (2019), Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem, Europ. J. Appl. Math. 30, 1041–1078.CrossRefGoogle Scholar
Benamou, J.-D. and Martinet, M. (2020), Capacity constrained entropic optimal transport, Sinkhorn saturated domain out-summation and vanishing temperature. Available at hal-02563022.Google Scholar
Benamou, J.-D., Carlier, G. and Hatchi, R. (2018), A numerical solution to Monge’s problem with a Finsler distance as cost, ESAIM Math. Model. Numer. Anal. 52, 2133–2148.CrossRefGoogle Scholar
Benamou, J.-D., Carlier, G. and Laborde, M. (2016a), An augmented Lagrangian approach to Wasserstein gradient flows and applications, ESAIM Proc. Surveys 54, 1–17.CrossRefGoogle Scholar
Benamou, J.-D., Carlier, G. and Nenna, L. (2016b), A numerical method to solve multi-marginal optimal transport problems with Coulomb cost, in Splitting Methods in Communication, Imaging, Science, and Engineering (Glowinski, R., Osher, S. J. and Yin, W., eds), Springer, pp. 577–601.CrossRefGoogle Scholar
Benamou, J.-D., Carlier, G. and Nenna, L. (2019a), Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm, Numer. Math 142, 33–54.CrossRefGoogle Scholar
Benamou, J.-D., Carlier, G., Cuturi, M., Nenna, L. and Peyré, G. (2015), Iterative Bregman projections for regularized transportation problems, SIAM J. Sci. Comput. 2, A1111– A1138.CrossRefGoogle Scholar
Benamou, J.-D., Carlier, G., Marino, S. Di and Nenna, L. (2019b), An entropy minimization approach to second-order variational mean-field games, Math. Models Methods Appl. Sci. 29, 1553–1583.CrossRefGoogle Scholar
Benamou, J.-D., Carlier, G., Mérigot, Q. and Oudet, E. (2016c), Discretization of functionals involving the Monge–Ampère operator, Numer . Math. 134, 611–636.Google Scholar
Benamou, J.-D., Collino, F. and Mirebeau, J.-M. (2016d), Monotone and consistent discretization of the Monge–Ampère operator, Math. Comp. 85, 2743–2775.CrossRefGoogle Scholar
Benamou, J.-D., Froese, B. D. and Oberman, A. M. (2014), Numerical solution of the optimal transportation problem using the Monge–Ampère equation, J. Comput. Phys. 260, 107–126.CrossRefGoogle Scholar
Benamou, J.-D., Gallouët, T. O. and Vialard, F.-X. (2019c), Second-order models for optimal transport and cubic splines on the Wasserstein space, Found. Comput. Math. 19, 1113– 1143.CrossRefGoogle Scholar
Benamou, J.-D., Ijzerman, W. L. and Rukhaia, G. (2020), An entropic optimal transport numerical approach to the reflector problem. Available at hal-02539799.CrossRefGoogle Scholar
Benmansour, F., Carlier, G., Peyré, G. and Santambrogio, F. (2009), Numerical approximation of continuous traffic congestion equilibria, Netw. Heterog. Media 4, 605–623.CrossRefGoogle Scholar
Berman, R. J. (2020), The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Ampère equations, Numer. Math. 145, 771–836.CrossRefGoogle Scholar
Bernot, M., Caselles, V. and Morel, J. M. (2008), Optimal Transportation Networks: Models and Theory, Vol. 1955 of Lecture Notes in Mathematics, Springer.Google Scholar
Bonneel, N., Peyré, G. and Cuturi, M. (2016), Wasserstein barycentric coordinates: Histogram regression using optimal transport, ACM Trans. Graphics 35, 71.CrossRefGoogle Scholar
Brenier, Y. (1989), The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc. 2, 225–255.CrossRefGoogle Scholar
Brenier, Y. (1991), Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44, 375–417.CrossRefGoogle Scholar
Brenier, Y. (2020), Examples of hidden convexity in nonlinear PDEs. Available at hal-02928398.Google Scholar
Brix, K., Hafizogullari, Y. and Platen, A. (2015), Solving the Monge–Ampère equations for the inverse reflector problem, Math. Models Methods Appl. Sci. 25, 803–837.CrossRefGoogle Scholar
Buttazzo, G., Pascale, L. De and Gori-Giorgi, P. (2012), Optimal-transport formulation of electronic density-functional theory, Phys. Rev. A 85, 062502.CrossRefGoogle Scholar
Caffarelli, L. (1992), The regularity of mappings with a convex potential, J. Amer. Math. Soc. 5, 99–104.CrossRefGoogle Scholar
Cancès, C., Gallouët, T. and Todeschi, G. (2020), A variational finite volume scheme for Wasserstein gradient flows, Numer. Math 146, 437–480.CrossRefGoogle Scholar
Carlier, G. (2001), A general existence result for the principal-agent problem with adverse selection, J. Math. Econ. 35, 129–150.CrossRefGoogle Scholar
Carlier, G. (2021), Classical and Modern Optimization, Imperial College Press. To appear.Google Scholar
Carlier, G., Eichinger, K. and Kroshnin, A. (2020), Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT. Available at hal-03084049.Google Scholar
Cavalletti, F. and Mondino, A. (2020), Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications. Available at arXiv:2004.08934.Google Scholar
Chambolle, A. and Pock, T. (2016), An introduction to continuous optimization for imaging, in Acta Numerica, Vol. 25, Cambridge University Press, pp. 161–319.Google Scholar
Chen, Y., Conforti, G. and Georgiou, T. T. (2018), Measure-valued spline curves: An optimal transport viewpoint, SIAM J. Math. Anal. 50, 5947–5968.CrossRefGoogle Scholar
Chizat, L., Peyré, G., Schmitzer, B. and Vialard, F.-X. (2018a), Scaling algorithms for unbalanced optimal transport problems, Math. Comput. 87, 2563–2609.CrossRefGoogle Scholar
Chizat, L., Peyré, G., Schmitzer, B. and Vialard, F.-X. (2018b), Unbalanced optimal transport: Dynamic and Kantorovich formulations, J. Funct. Anal. 274, 3090–3123.CrossRefGoogle Scholar
Chizat, L., Roussillon, P., Léger, F., Vialard, F.-X. and Peyré, G. (2020), Faster Wasserstein distance estimation with the Sinkhorn divergence. Available at arXiv:2006.08172.Google Scholar
Cominetti, R. and Martín, J. S. (1994), Asymptotic analysis of the exponential penalty trajectory in linear programming, Math. Program. 67, 169–187.CrossRefGoogle Scholar
Conforti, G. and Tamanini, L. (2021), A formula for the time derivative of the entropic cost and applications, J. Funct. Anal. 280, 108964.CrossRefGoogle Scholar
Cotar, C., Cotar, C., Friesecke, G., Friesecke, G., Pass, B. and Pass, B. (2015), Infinite-body optimal transport with Coulomb cost, Calc. Var. Partial Differential Equations 54, 717–742.CrossRefGoogle ScholarPubMed
Cullen, M. J. P. (2006), A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow, Imperial College Press.CrossRefGoogle Scholar
Cullen, M. J. P. and Purser, R. J. (1984), An extended Lagrangian theory of semigeostrophic frontogenesis, J. Atmos. Sci. 41, 1477–1497.2.0.CO;2>CrossRefGoogle Scholar
Cuturi, M. (2013), Sinkhorn distances: Lightspeed computation of optimal transport, in Advances in Neural Information Processing Systems 26 (NIPS 2013) (Burges, C. J. C. et al., eds), Curran Associates, pp. 2292–2300.Google Scholar
Dafni, G. D., McCann, R. J. and Stancu, A. (2013), Analysis and Geometry of Metric Measure Spaces: Lecture Notes of the 50th Séminaire de Mathématiques Supérieures (SMS), American Mathematical Society.CrossRefGoogle Scholar
Daneri, S. and Figalli, A. (2016), Variational Models for the Incompressible Euler Equations, Vol. 7 of AIMS on Applied Mathematics, American Institute of Material Sciences, pp. 1–48.Google Scholar
de Castro, P. M. M., Mérigot, Q. and Thibert, B. (2016), Far-field reflector problem and intersection of paraboloids, Numer. Math 134, 389–411.CrossRefGoogle Scholar
Di Marino, S. and Chizat, L. (2020), A tumor growth model of Hele-Shaw type as a gradient flow, ESAIM Control Optim. Calc. Var. 26, 103.CrossRefGoogle Scholar
Di Marino, S. and Gerolin, A. (2020), An optimal transport approach for the Schrödinger bridge problem and convergence of Sinkhorn algorithm, J. Sci. Comput. 85, 27.CrossRefGoogle Scholar
Di Marino, S., Gerolin, A. and Nenna, L. (2017), Optimal transportation theory with repulsive costs, in Topological Optimization and Optimal Transport: In the Applied Sciences, Vol. 17 of Radon Series on Computational and Applied Mathematics, De Gruyter, pp. 204–256.CrossRefGoogle Scholar
Ebin, D. G. and Marsden, J. (1970), Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92, 102–163.CrossRefGoogle Scholar
Ekeland, I. (2010), Notes on optimal transportation, Economic Theory 42, 437–459.CrossRefGoogle Scholar
Evans, L. C. (2001), Partial differential equations and Monge–Kantorovich mass transfer. Available at https://math.berkeley.edu/~evans/Monge-Kantorovich.survey.pdf.Google Scholar
Feydy, J. (2019), Geometric loss functions between sampled measures, images and volumes. Available at https://www.kernel-operations.io/geomloss/.Google Scholar
Feydy, J. (2020), Analyse de données géométriques, au delà des convolutions. PhD thesis, Mathématiques appliquées, Université Paris–Saclay.Google Scholar
Feydy, J., Séjourné, T., Vialard, F.-X., Amari, S.-i, Trouvé, A. and Peyré, G. (2019), Interpolating between optimal transport and MMD using Sinkhorn divergences, in Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS 2019) (Chaudhuri, K. and Sugiyama, M., eds), Vol. 89 of Proceedings of Machine Learning Research, PMLR, pp. 2681–2690.Google Scholar
Figalli, A. (2017), The Monge–Ampère Equation and its Applications, Zurich Lectures in Advanced Mathematics, European Mathematical Society.CrossRefGoogle Scholar
Figalli, A., Kim, Y.-H. and McCann, R. J. (2011), When is multidimensional screening a convex program?, 146, 454–478.CrossRefGoogle Scholar
Fortin, M. and Glowinski, R. (1985), Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, Vol. 15 of Studies in Mathematics and its Applications, North-Holland.Google Scholar
Friesecke, G. and Vögler, D. (2017), Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces. Available at arXiv:1801.00341.Google Scholar
Friesecke, G. and Vögler, D. (2018), Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces, SIAM J. Math. Anal. 50, 3996– 4019.CrossRefGoogle Scholar
Friesecke, G., Mendl, C. B., Pass, B., Cotar, C. and Klüppelberg, C. (2013), N-density representability and the optimal transport limit of the Hohenberg–Kohn functional, J. Chem. Phys. 139, 164109.CrossRefGoogle ScholarPubMed
Frisch, U., Matarrese, S., Mohayaee, R. and Sobolevski, A. (2002), A reconstruction of the initial conditions of the universe by optimal mass transportation, Nature 417, 260–262.CrossRefGoogle ScholarPubMed
Galichon, A. (2016), Optimal Transport Methods in Economics, first edition, Princeton University Press.CrossRefGoogle Scholar
Gallouët, T. O. and Mérigot, Q. (2018), A Lagrangian scheme à la Brenier for the incompressible Euler equations, Found. Comput. Math. 18, 835–865.CrossRefGoogle Scholar
Gangbo, W. and McCann, R. J. (1996), The geometry of optimal transportation, Acta Math. 177, 113–161.CrossRefGoogle Scholar
Gentil, I. (2020), The entropy, from Clausius to functional inequalities. Available at arXiv:2011.05206.Google Scholar
Ghoussoub, N., Kim, Y.-H. and Lim, T. (2019), Structure of optimal martingale transport plans in general dimensions, Ann. Probab. 47, 109–164.CrossRefGoogle Scholar
Glimm, T. and Oliker, V. (2003), Optical design of single reflector systems and the Monge– Kantorovich mass transfer problem, J. Math. Sci. 117, 4096–4108.CrossRefGoogle Scholar
Golse, F. and Paul, T. (2021), Quantum and semiquantum pseudometrics and applications. Available at arXiv:2102.05184.Google Scholar
Guéant, O. (2012), Mean field games equations with quadratic Hamiltonian: A specific approach, Math. Models Methods Appl. Sci. 22, 1250022.CrossRefGoogle Scholar
Guittet, K. (2003), On the time-continuous mass transport problem and its approximation by augmented Lagrangian techniques, SIAM J. Numer. Anal. 41, 382–399.CrossRefGoogle Scholar
Guo, I. and Loeper, G. (2018), Path dependent optimal transport and model calibration on exotic derivatives, SSRN Electron. J. doi:10.2139/ssrn.3302384.CrossRefGoogle Scholar
Haker, S., Zhu, L., Tannenbaum, A. and Angenent, S. (2004), Optimal mass transport for registration and warping, Internat. J. Comput. Vision 60, 225–240.CrossRefGoogle Scholar
Huesmann, M. and Trevisan, D. (2019), A Benamou–Brenier formulation of martingale optimal transport, Bernoulli 25, 2729–2757.CrossRefGoogle Scholar
Hug, R., Maitre, E. and Papadakis, N. (2020), On the convergence of augmented Lagrangian method for optimal transport between nonnegative densities, J. Math. Anal. Appl. 485, 123811.CrossRefGoogle Scholar
Jordan, R., Kinderlehrer, D. and Otto, F. (1998), The variational formulation of the Fokker– Planck equation, SIAM J. Math. Anal. 29, 1–17.CrossRefGoogle Scholar
Kitagawa, J., Mérigot, Q. and Thibert, B. (2019), Convergence of a Newton algorithm for semi-discrete optimal transport, J. Eur. Math. Soc. 21, 2603–2651.CrossRefGoogle Scholar
Kolouri, S., Park, S., Thorpe, M., Slepcev, D. and Rohde, G. K. (2017), Transport-based analysis, modeling, and learning from signal and data distributions, IEEE Signal Process. Magazine 34, 43–59.CrossRefGoogle Scholar
Kondratyev, S., Monsaingeon, L. and Vorotnikov, D. (2016), A new optimal transport distance on the space of finite Radon measures, Adv. Diff. Equations 21, 1117–1164.Google Scholar
Lasry, J.-M. and Lions, P.-L. (2007), Mean field games, Japan. J. Math. 2, 229–260.CrossRefGoogle Scholar
Lavenant, H. (2021), Unconditional convergence for discretizations of dynamical optimal transport, Math. Comp. 90, 739–786.CrossRefGoogle Scholar
Léonard, C. (2014), A survey of the Schrödinger problem and some of its connections with optimal transport, Discrete Contin. Dyn. Syst. 34, 1533–1574.CrossRefGoogle Scholar
Lévy, B. and Schwindt, E. L. (2018), Notions of optimal transport theory and how to implement them on a computer, Comput . Graph. 72, 135–148.CrossRefGoogle Scholar
Li, D., Lamoureux, M. P. and Liao, W. (2020), Application of an unbalanced optimal transport distance and a mixed L1/Wasserstein distance to full waveform inversion. Available at arXiv:2004.05237.Google Scholar
Liero, M., Mielke, A. and Savaré, G. (2016), Optimal transport in competition with reaction: The Hellinger–Kantorovich distance and geodesic curves, SIAM J. Math. Anal. 48, 2869–2911.CrossRefGoogle Scholar
Lions, J. L. (1971), Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer.CrossRefGoogle Scholar
Maas, J. (2011), Gradient flows of the entropy for finite Markov chains, J. Functional Analysis 261, 2250–2292.CrossRefGoogle Scholar
Mainini, E. (2012), A description of transport cost for signed measures, J. Math. Sci. 97, 837–855.CrossRefGoogle Scholar
Matthes, D. and Osberger, H. (2014), Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal. 48, 697–726.CrossRefGoogle Scholar
Maury, B., Roudneff-Chupin, A., Santambrogio, F. and Venel, J. (2011), Handling congestion in crowd motion modeling, Netw. Heterog. Media 6, 485.CrossRefGoogle Scholar
McCann, R. J. (1997), A convexity principle for interacting gases, Adv. Math 128, 153–179.CrossRefGoogle Scholar
Mellet, A., Perthame, B. and Quirós, F. (2017), A Hele-Shaw problem for tumor growth, J. Funct. Anal. 273, 3061–3093.CrossRefGoogle Scholar
Mémoli, F. (2011), Gromov–Wasserstein distances and the metric approach to object matching, Found. Comput. Math 11, 417–487.CrossRefGoogle Scholar
Mérigot, Q. (2011), A multiscale approach to optimal transport, Computer Graphics Forum 30, 1583–1592.CrossRefGoogle Scholar
Mérigot, Q. and Mirebeau, J.-M. (2016), Minimal geodesics along volume-preserving maps, through semidiscrete optimal transport, SIAM J. Numer. Anal. 54, 3465–3492.CrossRefGoogle Scholar
Mérigot, Q. and Thibert, B. (2020), Optimal transport: Discretization and algorithms. Available at hal-02494446.Google Scholar
Métivier, L., Brossier, R., Mérigot, Q. and Oudet, E. (2019), A graph space optimal transport distance as a generalization of L p distances: Application to a seismic imaging inverse problem, Inverse Problems 35, 085001.CrossRefGoogle Scholar
Métivier, L., Brossier, R., Mérigot, Q., Oudet, E. and Virieux, J. (2016), An optimal transport approach for seismic tomography: Application to 3D full waveform inversion, Inverse Problems 32, 115008.CrossRefGoogle Scholar
Natale, A. and Todeschi, G. (2020), A mixed finite element discretization of dynamical optimal transport. Available at hal-02501634.Google Scholar
Pal, S. (2019), On the difference between entropic cost and the optimal transport cost. Available at arXiv:1905.12206.Google Scholar
Papadakis, N., Peyré, G. and Oudet, E. (2014), Optimal transport with proximal splitting, SIAM J. Imaging Sci. 7, 212–238.CrossRefGoogle Scholar
Pass, B. (2015), Multi-marginal optimal transport: Theory and applications, ESAIM Math. Model. Numer. Anal. 49, 1771–1790.CrossRefGoogle Scholar
Pegon, P. (2017), Transport branché et structures fractales. PhD thesis, Mathématiques appliquées, Université Paris–Saclay (ComUE).Google Scholar
Peyré, G. (2015), Entropic approximation of Wasserstein gradient flows, SIAM J. Imaging Sci. 8, 2323–2351.CrossRefGoogle Scholar
Peyré, G. and Cuturi, M. (2019), Computational optimal transport, Found . Trends Mach. Learning 11, 355–607.CrossRefGoogle Scholar
Peyré, G., Cuturi, M. and Solomon, J. (2016), Gromov–Wasserstein averaging of kernel and distance matrices, in Proceedings of the 33rd International Conference on Machine Learning (ICML 2016) (Balcan, M. F. and Weinberger, K. Q., eds), Vol. 48 of Proceedings of Machine Learning Research, PMLR, pp. 2664–2672.Google Scholar
Rachev, S. T. and Rüschendorf, L. (2006), Mass Transportation Problems: Applications, Probability and its Applications, Springer.Google Scholar
Ramdas, A., Garcia, N. and Cuturi, M. (2017), On Wasserstein two sample testing and related families of nonparametric tests, Entropy 19, 47.CrossRefGoogle Scholar
Rochet, J.-C. and Chone, P. (1998), Ironing, sweeping, and multidimensional screening, Econometrica 66, 783–826.CrossRefGoogle Scholar
Rubinstein, J. and Wolansky, G. (2004), A variational principle in optics, J. Opt. Soc. Amer. A 21, 2164–2172.CrossRefGoogle ScholarPubMed
Salanié, B. and Galichon, A. (2012), Cupid’s invisible hand: Social surplus and identification in matching models. Available at hal-01053710.Google Scholar
Santambrogio, F. (2015), Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Vol. 87 of Progress in Nonlinear Differential Equations and Their Applications, Springer.CrossRefGoogle Scholar
Schmitzer, B. (2019), Stabilized sparse scaling algorithms for entropy regularized transport problems, SIAM J. Sci. Comput. 41, A1443–A1481.CrossRefGoogle Scholar
Steinerberger, S. (2020), On a Kantorovich–Rubinstein inequality. Available at arXiv:2010.12946.Google Scholar
Sturm, K.-T. (2020), The space of spaces: Curvature bounds and gradient flows on the space of metric measure spaces. Available at arXiv:1208.0434.Google Scholar
Symes, W. (1998), Mathematics of reflection seismology. Available at http://wwsorcas.com/book0/book0.pdf.Google Scholar
Vacher, A., Muzellec, B., Rudi, A., Bach, F. and Vialard, F.-X. (2021), A dimension-free computational upper-bound for smooth optimal transport estimation. Available at arXiv:2101.05380.Google Scholar
Vialard, F.-X. (2019), An elementary introduction to entropic regularization and proximal methods for numerical optimal transport. Available at hal-02303456.Google Scholar
Villani, C. (2003), Topics in Optimal Transportation, Graduate Studies in Mathematics, American Mathematical Society.Google Scholar
Villani, C. (2008), Optimal Transport: Old and New, Vol. 338 of Grundlehren der mathematischen Wissenschaften, Springer.Google Scholar
Wang, X.-J. (2004), On the design of a reflector antenna II, Calc. Var. Partial Differential Equations 20, 329–341.CrossRefGoogle Scholar
Yang, Y. and Engquist, B. (2018), Analysis of optimal transport and related misfit functions in full-waveform inversion, Geophys. 83, A7–A12.CrossRefGoogle Scholar