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Volumn 34, Issue 3, 2012, Pages A1432-A1459

A numerical method for the elliptic monge-amperè equation with transport boundary conditions

Author keywords

Elliptic partial differential equations; Finite difference methods; Monge Amp ere equation; Optimal transport; Viscosity solutions

Indexed keywords

ASTROPHYSICS; BOUNDARY CONDITIONS; MESH GENERATION; NEWTON-RAPHSON METHOD; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; VISCOSITY;

EID: 84876121473     PISSN: 10648275     EISSN: 10957197     Source Type: Journal    
DOI: 10.1137/110822372     Document Type: Article
Times cited : (87)

References (48)
  • 1
    • 10044247612 scopus 로고    scopus 로고
    • Lecture notes on optimal transport problems
    • Funchal, 2000), Lecture Notes in Math. 1812, Springer, Berlin
    • L. Ambrosio, Lecture notes on optimal transport problems, in Mathematical Aspects of Evolving Interfaces (Funchal, 2000), Lecture Notes in Math. 1812, Springer, Berlin, 2003, pp. 1-52.
    • (2003) Mathematical Aspects of Evolving Interfaces , pp. 1-52
    • Ambrosio, L.1
  • 2
    • 0034407460 scopus 로고    scopus 로고
    • A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
    • J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375-393.
    • (2000) Numer. Math. , vol.84 , pp. 375-393
    • Benamou, J.-D.1    Brenier, Y.2
  • 5
    • 37449033900 scopus 로고    scopus 로고
    • A mixed formulation of the Monge-Kantorovich equations
    • J. W. Barrett and L. Prigozhin, A mixed formulation of the Monge-Kantorovich equations, M2AN Math. Model. Numer. Anal., 41 (2007), pp. 1041-1060.
    • (2007) M2AN Math. Model. Numer. Anal. , vol.41 , pp. 1041-1060
    • Barrett, J.W.1    Prigozhin, L.2
  • 6
    • 73949105127 scopus 로고    scopus 로고
    • Partial L1 Monge-Kantorovich problem: Variational formulation and numerical approximation
    • J. W. Barrett and L. Prigozhin, Partial L1 Monge-Kantorovich problem: Variational formulation and numerical approximation, Interfaces Free Bound., 11 (2009), pp. 201-238.
    • (2009) Interfaces Free Bound. , vol.11 , pp. 201-238
    • Barrett, J.W.1    Prigozhin, L.2
  • 7
    • 84974753170 scopus 로고
    • Convergence of approximation schemes for fully nonlinear second order equations
    • G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptot. Anal., 4 (1991), pp. 271-283.
    • (1991) Asymptot. Anal. , vol.4 , pp. 271-283
    • Barles, G.1    Souganidis, P.E.2
  • 8
    • 75749125433 scopus 로고    scopus 로고
    • Moving mesh generation using the parabolic Monge-Ampère equation
    • C. J. Budd and J. F. Williams, Moving mesh generation using the parabolic Monge-Ampère equation, SIAM J. Sci. Comput., 31 (2009), pp. 3438-3465.
    • (2009) SIAM J. Sci. Comput. , vol.31 , pp. 3438-3465
    • Budd, C.J.1    Williams, J.F.2
  • 9
    • 84990563964 scopus 로고
    • Boundary regularity of maps with convex potentials
    • L. A. Caffarelli, Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math., 45 (1992), pp. 1141-1151.
    • (1992) Comm. Pure Appl. Math. , vol.45 , pp. 1141-1151
    • Caffarelli, L.A.1
  • 10
    • 84968482174 scopus 로고
    • The regularity of mappings with a convex potential
    • L. A. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), pp. 99-104.
    • (1992) J. Amer. Math. Soc. , vol.5 , pp. 99-104
    • Caffarelli, L.A.1
  • 11
    • 0030366438 scopus 로고    scopus 로고
    • Boundary regularity of maps with convex potentials. II
    • L. A. Caffarelli, Boundary regularity of maps with convex potentials. II, Ann. of Math. (2), 144 (1996), pp. 453-496.
    • (1996) Ann. of Math. (2) , vol.144 , pp. 453-496
    • Caffarelli, L.A.1
  • 12
    • 84870649769 scopus 로고    scopus 로고
    • McConnell Brain Imaging Center
    • McConnell Brain Imaging Center, Brainweb: Simulated Brain Database, 2010, http://www. bic. mni. mcgill. ca/brainweb.
    • (2010) Brainweb: Simulated Brain Database
  • 13
    • 84967708673 scopus 로고
    • User's guide to viscosity solutions of second order partial differential equations
    • 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. (N. S.), 27 (1992), pp. 1-67.
    • (1992) Bull. Amer. Math. Soc. (N. S.) , vol.27 , pp. 1-67
    • Crandall, M.G.1    Ishii, H.2    Lions, P.-L.3
  • 14
    • 21544432712 scopus 로고    scopus 로고
    • Brainweb: Online interface to a 3D MRI simulated brain database
    • C. A. Cocosco, V. Kollokian, R. K.-S. Kwan, and A. C. Evans, Brainweb: Online interface to a 3D MRI simulated brain database, NeuroImage, 5 (1997), p. 425.
    • (1997) NeuroImage , vol.5 , pp. 425
    • Cocosco, C.A.1    Kollokian, V.2    Kwan, R.K.-S.3    Evans, A.C.4
  • 16
    • 53349146583 scopus 로고    scopus 로고
    • An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization
    • G. L. Delzanno, L. Chacon, J. M. Finn, Y. Chung, and G. Lapenta, An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization, J. Comput. Phys., 227 (2008), pp. 9841-9864.
    • (2008) J. Comput. Phys. , vol.227 , pp. 9841-9864
    • Delzanno, G.L.1    Chacon, L.2    Finn, J.M.3    Chung, Y.4    Lapenta, G.5
  • 17
    • 33646378906 scopus 로고    scopus 로고
    • An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions
    • E. J. Dean and R. Glowinski, An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions, Electron. Trans. Numer. Anal., 22 (2006), pp. 71-96.
    • (2006) Electron. Trans. Numer. Anal. , vol.22 , pp. 71-96
    • Dean, E.J.1    Glowinski, R.2
  • 18
    • 84962796427 scopus 로고    scopus 로고
    • On the numerical solution of the elliptic Monge-Ampère equation in dimension two: A least-squares approach
    • Springer, Dordrecht
    • E. J. Dean and R. Glowinski, On the numerical solution of the elliptic Monge-Ampère equation in dimension two: A least-squares approach, in Partial Differential Equations, Comput. Methods Appl. Sci. 16, Springer, Dordrecht, 2008, pp. 43-63.
    • (2008) Partial Differential Equations, Comput. Methods Appl. Sci. , vol.16 , pp. 43-63
    • Dean, E.J.1    Glowinski, R.2
  • 19
    • 0003073688 scopus 로고    scopus 로고
    • Partial differential equations and Monge-Kantorovich mass transfer
    • Int. Press, Boston, MA, 1999
    • L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, in Current Developments in Mathematics, 1997, Int. Press, Boston, MA, 1999, pp. 65-126.
    • (1997) Current Developments in Mathematics , pp. 65-126
    • Evans, L.C.1
  • 21
    • 0037118068 scopus 로고    scopus 로고
    • A reconstruction of the initial conditions of the universe by optimal mass transportation
    • U. Frisch, S. Matarrese, R. Mohayaee, and A. Sobolevski, A reconstruction of the initial conditions of the universe by optimal mass transportation, Nature, 417 (2002), pp. 260-262.
    • (2002) Nature , vol.417 , pp. 260-262
    • Frisch, U.1    Matarrese, S.2    Mohayaee, R.3    Sobolevski, A.4
  • 22
    • 77952098338 scopus 로고    scopus 로고
    • Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method
    • X. Feng and M. Neilan, Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method, SIAM J. Numer. Anal., 47 (2009), pp. 1226-1250.
    • (2009) SIAM J. Numer. Anal. , vol.47 , pp. 1226-1250
    • Feng, X.1    Neilan, M.2
  • 23
    • 59349099761 scopus 로고    scopus 로고
    • Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations
    • X. Feng and M. Neilan, Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations, J. Sci. Comput., 38 (2009), pp. 74-98.
    • (2009) J. Sci. Comput. , vol.38 , pp. 74-98
    • Feng, X.1    Neilan, M.2
  • 24
    • 80052702511 scopus 로고    scopus 로고
    • Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher
    • B. D. Froese and A. M. Oberman, Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher, SIAM J. Numer. Anal., 49 (2011), pp. 1692-1714.
    • (2011) SIAM J. Numer. Anal. , vol.49 , pp. 1692-1714
    • Froese, B.D.1    Oberman, A.M.2
  • 25
    • 78649322125 scopus 로고    scopus 로고
    • Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation
    • B. D. Froese and A. M. Oberman, Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation, J. Comput. Phys., 230 (2011), pp. 818-834.
    • (2011) J. Comput. Phys. , vol.230 , pp. 818-834
    • Froese, B.D.1    Oberman, A.M.2
  • 27
    • 11044234672 scopus 로고    scopus 로고
    • Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem: Nonlinear problems and function theory
    • T. Glimm and V. Oliker, Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem: Nonlinear problems and function theory, J. Math. Sci. (N. Y.), 117 (2003), pp. 4096-4108.
    • (2003) J. Math. Sci. (N. Y.) , vol.117 , pp. 4096-4108
    • Glimm, T.1    Oliker, V.2
  • 28
    • 11044226730 scopus 로고    scopus 로고
    • Optical design of two-reflector systems the Monge-Kantorovich mass transfer problem and Fermat's principle
    • T. Glimm and V. Oliker, Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle, Indiana Univ. Math. J., 53 (2004), pp. 1255-1277.
    • (2004) Indiana Univ. Math. J. , vol.53 , pp. 1255-1277
    • Glimm, T.1    Oliker, V.2
  • 30
    • 77249155146 scopus 로고    scopus 로고
    • An efficient numerical method for the solution of the L2 optimal mass transfer problem
    • E. Haber, T. Rehman, and A. Tannenbaum, An efficient numerical method for the solution of the L2 optimal mass transfer problem, SIAM J. Sci. Comput., 32 (2010), pp. 197-211.
    • (2010) SIAM J. Sci. Comput. , vol.32 , pp. 197-211
    • Haber, E.1    Rehman, T.2    Tannenbaum, A.3
  • 33
  • 34
    • 0001010507 scopus 로고
    • On a problem of Monge
    • L. V. Kantorovich, On a problem of Monge, Uspekhi Mat. Nauk., 3 (1948), pp. 225-226.
    • (1948) Uspekhi Mat. Nauk. , vol.3 , pp. 225-226
    • Kantorovich, L.V.1
  • 35
    • 85029011240 scopus 로고
    • Algorithms for the projection of a point onto an ellipsoid
    • Y. N. Kiselev, Algorithms for the projection of a point onto an ellipsoid, Liet. Mat. Rink., 34 (1994), pp. 174-196.
    • (1994) Liet. Mat. Rink. , vol.34 , pp. 174-196
    • Kiselev, Y.N.1
  • 36
    • 84972559573 scopus 로고
    • Neumann type boundary conditions for Hamilton-Jacobi equations
    • P. L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Math. J., 52 (1985), pp. 793-820.
    • (1985) Duke Math. J. , vol.52 , pp. 793-820
    • Lions, P.L.1
  • 37
    • 84990557141 scopus 로고
    • The Neumann problem for equations of Monge-Ampère type
    • P.-L. Lions, N. S. Trudinger, and J. I. E. Urbas, The Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math., 39 (1986), pp. 539-563.
    • (1986) Comm. Pure Appl. Math. , vol.39 , pp. 539-563
    • Lions, P.-L.1    Trudinger, N.S.2    Urbas, J.I.E.3
  • 38
    • 4644248996 scopus 로고    scopus 로고
    • Exact semi-geostrophic flows in an elliptical ocean basin
    • R. J. McCann and A. M. Oberman, Exact semi-geostrophic flows in an elliptical ocean basin, Nonlinearity, 17 (2004), pp. 1891-1922.
    • (2004) Nonlinearity , vol.17 , pp. 1891-1922
    • McCann, R.J.1    Oberman, A.M.2
  • 39
    • 85172006929 scopus 로고
    • On the approximation of linear elliptic differential equations by difference equations with positive coefficients
    • T. S. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Phys., 31 (1953), pp. 253-259.
    • (1953) J. Math. Phys. , vol.31 , pp. 253-259
    • Motzkin, T.S.1    Wasow, W.2
  • 40
    • 34247269678 scopus 로고    scopus 로고
    • Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems
    • 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), pp. 879-895.
    • (2006) SIAM J. Numer. Anal. , vol.44 , pp. 879-895
    • Oberman, A.M.1
  • 41
    • 47249122195 scopus 로고    scopus 로고
    • Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian
    • A. M. Oberman, Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), pp. 221-238.
    • (2008) Discrete Contin. Dyn. Syst. Ser. B , vol.10 , pp. 221-238
    • Oberman, A.M.1
  • 42
    • 34249964909 scopus 로고
    • On the numerical solution of the equation (2z/x2)(2z/y2) (2z/xy)2 = f and its discretizations
    • V. I. Oliker and L. D. Prussner, On the numerical solution of the equation (2z/x2)(2z/y2) (2z/xy)2 = f and its discretizations, I, Numer. Math., 54 (1988), pp. 271-293.
    • (1988) I, Numer. Math. , vol.54 , pp. 271-293
    • Oliker, V.I.1    Prussner, L.D.2
  • 43
    • 21644443786 scopus 로고
    • The Dirichlet problem for the multidimensional analogue of the Monge-Ampère equation
    • A. V. Pogorelov, The Dirichlet problem for the multidimensional analogue of the Monge-Ampère equation, Dokl. Akad. Nauk SSSR, 201 (1971), pp. 790-793.
    • (1971) Dokl. Akad. Nauk SSSR , vol.201 , pp. 790-793
    • Pogorelov, A.V.1
  • 44
    • 84972545670 scopus 로고
    • Characterization of the subdifferentials of convex functions
    • R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), pp. 497-510.
    • (1966) Pacific J. Math. , vol.17 , pp. 497-510
    • Rockafellar, R.T.1
  • 45
    • 66149192495 scopus 로고    scopus 로고
    • On the second boundary value problem for Monge-Ampère type equations and optimal transportation
    • N. S. Trudinger and X.-J. Wang, On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8 (2009), pp. 143-174.
    • (2009) Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , vol.8 , pp. 143-174
    • Trudinger, N.S.1    Wang, X.-J.2
  • 46
    • 0031446633 scopus 로고    scopus 로고
    • On the second boundary value problem for equations of Monge-Ampère type
    • J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math., 487 (1997), pp. 115-124.
    • (1997) J. Reine Angew. Math. , vol.487 , pp. 115-124
    • Urbas, J.1
  • 48
    • 1542342359 scopus 로고    scopus 로고
    • Topics in optimal transportation
    • AMS, Providence, RI
    • C. Villani, Topics in Optimal Transportation, Grad. Stud. Math. 58, AMS, Providence, RI, 2003.
    • (2003) Grad. Stud. Math. , vol.58
    • Villani, C.1


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