Open Access
Volume 28, 2022
Article Number 45
Number of page(s) 49
Published online 07 July 2022
  1. S. Alama, L. Bronsard and J.A. Montero, On the Ginzburg-Landau model of a superconducting ball in a uniform field. Annales de l'IHP Analyse non linéaire (2006) 237–267. [Google Scholar]
  2. D. Bao, C. Robles, Z. Shen et al., Zermelo navigation on Riemannian manifolds. J. Differ. Geometry 66 (2004) 377–435. [Google Scholar]
  3. M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Modern Birkhäuser Classics, Birkhauser, Basel (1997). [CrossRef] [Google Scholar]
  4. G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133–1148. [CrossRef] [MathSciNet] [Google Scholar]
  5. G. Barles and E. Rouy, A Strong Comparison Result for the Bellman equation arising in stochastic exit time control problems and its applications. Comm. Partial Differ. Equ. 23 (1998) 1995–2033. [CrossRef] [Google Scholar]
  6. G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271–283. [MathSciNet] [Google Scholar]
  7. J.-D. Benamou, G. Carlier and R. Hatchi, A numerical solution to Monge’s problem with a Finsler distance as cost. ESAIM: MMNA 52 (2018) 2133–2148. [CrossRef] [EDP Sciences] [Google Scholar]
  8. R.J. Berman, The Sinkhorn algorithm, parabolic optimal transport and geometric Monge-Ampère equations. Numer. Math. 145 (2020) 771–836. [CrossRef] [MathSciNet] [Google Scholar]
  9. F. Bonnans and S. Gaubert, Recherche opérationnelle. Aspects mathématiques et applications. Ellipse (2016). [Google Scholar]
  10. J. Bonnans, G. Bonnet and J.-M. Mirebeau, Monotone and second order consistent scheme for the two dimensional Pucci equation (2020). [Google Scholar]
  11. J.F. Bonnans, G. Bonnet and J.-M. Mirebeau, Second order monotone finite differences discretization of linear anisotropic differential operators. Math. Comput. 90 (2021) 2671–2703. [Google Scholar]
  12. O.P. Bruno and A. Prieto, Spatially dispersionless, unconditionally stable FC-AD solvers for variable-coefficient PDEs. J. Sci. Comput. 58 (2014) 331–366. [CrossRef] [MathSciNet] [Google Scholar]
  13. E. Casas, Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24 (1986) 1309–1318. [CrossRef] [MathSciNet] [Google Scholar]
  14. D. Chen, J.-M. Mirebeau and L.D. Cohen, Global minimum for a Finsler Elastica minimal path approach. Int. J. Comput. Vis. 122 (2017) 458–483. [CrossRef] [MathSciNet] [Google Scholar]
  15. Y. Chen, E. Fan and M. Yuen, The Hopf-Cole transformation, topological solitons and multiple fusion solutions for the n-dimensional Burgers system. Phys. Lett. A 380 (2016) 9–14. [CrossRef] [MathSciNet] [Google Scholar]
  16. X. Cheng and Z. Shen, Finsler geometry, An approach via Randers spaces (2012). [Google Scholar]
  17. L. Chizat, P. Roussillon, F. Léger, F.X. Vialard and G. Peyré, Faster wasserstein distance estimation with the Sinkhorn divergence. Adv. Neural Inf. Process. Syst. 33 (2020). [Google Scholar]
  18. L.D. Cohen, D. Chen and J.-M. Mirebeau, Finsler geodesics evolution model for region based active contours, in Proceedings of the British Machine Vision Conference (BMVC), edited by E.R.H. Richard C. Wilson and W.A.P. Smith. BMVA Press (2016) 22.1–22.12. [Google Scholar]
  19. M.B. Cohen, J. Kelner, R. Kyng, J. Peebles, R. Peng, A.B. Rao and A. Sidford, Solving directed laplacian systems in nearly- linear time through sparse LU factorizations, in 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), IEEE (2018) 898–909. [CrossRef] [Google Scholar]
  20. J.H. Conway and N.J.A. Sloane, Low-dimensional lattices. VI. Voronoi reduction of three-dimensional lattices. Proc. R. Soc. A 436 (1992) 55–68. [Google Scholar]
  21. M.G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. [CrossRef] [MathSciNet] [Google Scholar]
  22. K. Crane, M. Livesu, E. Puppo and Y. Qin, A Survey of Algorithms for Geodesic Paths and Distances. Preprint arXiv:2007.10430 (2020). [Google Scholar]
  23. K. Crane, C. Weischedel and M. Wardetzky, Geodesics in heat: a new approach to computing distance based on heat flow. ACM Trans. Graph. 32 (2013) 152:1–152:11. [Google Scholar]
  24. K. Crane, C. Weischedel and M. Wardetzky, The heat method for distance computation. Commun. ACM 60 (2017) 90–99. [CrossRef] [Google Scholar]
  25. M. Cuturi, Sinkhorn distances: lightspeed computation of optimal transport, in Proc. 26th International Conference on Neural Information Processing Systems — Volume 2 (2013) 2292–2300. [Google Scholar]
  26. C. De Lellis, F. Otto and M. Westdickenberg, Minimal entropy conditions for Burgers equation. Quart. Appl. Math. 62 (2004) 687–700. [CrossRef] [MathSciNet] [Google Scholar]
  27. R. Duits, S.P. Meesters, J.-M. Mirebeau and J.M. Portegies, Optimal paths for variants of the 2D and 3D Reeds-Shepp car with applications in image analysis. J. Math. Imag. Vision (2018) 1–33. [Google Scholar]
  28. A. Ern and J.-L. Guermond, Theory and practice of finite elements, vol. 159. Springer Science and Business Media (2013). [Google Scholar]
  29. J. Fehrenbach and J.-M. Mirebeau, Sparse non-negative stencils for anisotropic diffusion. J. Math. Imag. Vision 49 (2014) 123–147. [CrossRef] [Google Scholar]
  30. M. Feldman and R.J. McCann, Uniqueness and transport density in Monge’s mass transportation problem. Calc. Variat. Partial Differ. Equ. 15 (2002) 81–113. [CrossRef] [Google Scholar]
  31. E. Hopf, The partial differential equation ut+ uux= xx. Commun. Pure Appl. Math. 3 (1950) 201–230. [CrossRef] [Google Scholar]
  32. P. Houston, I. Muga, S. Roggendorf and K.G. Van Der Zee, The convection-diffusion-reaction equation in non-Hilbert Sobolev spaces: a direct proof of the inf-sup condition and stability of Galerkin’s method. Comput. Methods Appl. Math. 19 (2019) 503–522. [CrossRef] [MathSciNet] [Google Scholar]
  33. R. Kannan and Z.J. Wang, A high order spectral volume solution to the Burgers’ equation using the Hopf-Cole transformation. Int. J. Numer. Methods Fluids 69 (2012) 781–801. [CrossRef] [MathSciNet] [Google Scholar]
  34. H. Komiya, Elementary proof for Sion’s minimax theorem. Kodai Math. J. 11 (1988) 5–7. [CrossRef] [MathSciNet] [Google Scholar]
  35. F. Labelle and J.R. Shewchuk, Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation, in Proceedings of the nineteenth annual symposium on Computational geometry (2003) 191–200. [CrossRef] [Google Scholar]
  36. C. Léonard, A survey of the Schrodinger problem and some of its connections with optimal transport. Discr. Continu. Dyn. Syst. 34 (2014) 1533. [CrossRef] [Google Scholar]
  37. T.-T. Lu and S.-H. Shiou, Inverses of 2 x 2 block matrices. Comput. Math. Appl. 43 (2002) 119–129. [CrossRef] [MathSciNet] [Google Scholar]
  38. X.-N. Ma, N.S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem. Arch. Ratl. Mech. Anal. 177 (2005) 151–183. [CrossRef] [Google Scholar]
  39. J.-M. Mirebeau, Efficient fast marching with Finsler metrics. Numer. Math. 126 (2014) 515–557. [CrossRef] [MathSciNet] [Google Scholar]
  40. J.-M. Mirebeau, Minimal stencils for discretizations of anisotropic PDEs preserving causality or the maximum principle. SIAM J. Numer. Anal. 54 (2016) 1582–1611. [CrossRef] [MathSciNet] [Google Scholar]
  41. J.-M. Mirebeau, Fast-marching methods for curvature penalized shortest paths. J. Math. Imag. Vision 60 (2018) 784–815. [CrossRef] [Google Scholar]
  42. J.-M. Mirebeau, Riemannian fast-marching on cartesian grids, using voronoi’s first reduction of quadratic forms. SIAM J. Numer. Anal. 57 (2019) 2608–2655. [CrossRef] [MathSciNet] [Google Scholar]
  43. A.M. Oberman, Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44 (2006) 879–895. [CrossRef] [MathSciNet] [Google Scholar]
  44. S.-I. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds. Commun. Pure Appl. Math. 62 (2009) 1386–1433. [CrossRef] [Google Scholar]
  45. T. Ohwada, Cole-Hopf transformation as numerical tool for the Burgers equation. Appl. Comput. Math 8 (2009) 107–113. [MathSciNet] [Google Scholar]
  46. G. Randers, On an asymmetrical metric in the four-space of general relativity. Phys. Rev. 59 (1941) 195–199. [CrossRef] [MathSciNet] [Google Scholar]
  47. E. Selling, Uber die Binaren und Ternären Quadratischen Formen. J. Reine Angew. Math. 77 (1874) 143–229. [MathSciNet] [Google Scholar]
  48. J.A. Sethian and A.B. Vladimirsky, Ordered upwind methods for static Hamilton-Jacobi equations. Proc. Natl. Acad. Sci. USA 98 (2001) 11069–11074. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  49. R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Stat. 35 (1964) 876–879. [CrossRef] [Google Scholar]
  50. J. Solomon, F. de Goes, G. Peyré, M. Cuturi, A. Butscher, A. Nguyen, T. Du and L. Guibas, Convolutional Wasserstein distances: efficient optimal transportation on geometric domains. ACM Trans. Graph. 34 (2015) 66:1–66:11. [CrossRef] [Google Scholar]
  51. J. Solomon, R. Rustamov, L. Guibas and A. Butscher, Earth mover’s distances on discrete surfaces. ACM Trans. Graphics 33 (2014) 1–12. [CrossRef] [Google Scholar]
  52. S.R.S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20 (1967) 431–455. [Google Scholar]
  53. C. Villani, Optimal transport: old and new, vol. 338. Springer (2009). [CrossRef] [Google Scholar]
  54. F. Yang, L. Chai, D. Chen and L.D. Cohen, Geodesic via asymmetric heat diffusion based on Finsler metric, in Asian Conference on Computer Vision. Springer (2018) 371–386. [Google Scholar]
  55. F. Yang and L.D. Cohen, Geodesic distance and curves through isotropic and anisotropic heat equations on images and surfaces. J. Math. Imag. Vision 55 (2016) 210–228. [CrossRef] [Google Scholar]

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