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Approximation of a nonlinear fractal energy functional on varying Hilbert spaces
Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature
Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA |
We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Ampère type. These surfaces need not be continuous up to the boundary of the domain and the Dirichlet boundary condition must be interpreted in a weak sense. As a consequence, sub-solutions do not always lie below super-solutions, standard comparison principles fail, and existing convergence theorems break down. By relying on a geometric interpretation of weak solutions, we prove a relaxed comparison principle that applies only in the interior of the domain. We provide a general framework for proving existence and stability results for consistent, monotone finite difference approximations and modify the Barles-Souganidis convergence framework to show convergence in the interior of the domain. We describe a convergent scheme for the prescribed Gaussian curvature equation and present several challenging examples to validate these results.
References:
[1] |
G. Alberti and L. Ambrosio,
A geometrical approach to monotone functions in $\mathbb{R}^n$, Math. Z., 230 (1999), 259-316.
|
[2] |
I. J. Bakelman, Generalized elliptic solutions of the Dirichlet problem for n-dimensional Monge-Ampère equations, In Nonlinear Functional Analysis and its Applications, volume 45 of P. Symp. Pure Math., pages 73-102. AMS, 1986. |
[3] |
I. J. Bakelman,
Convex Analysis and Nonlinear Geometric Elliptic Equations Springer Science & Business Media, 2012. |
[4] |
M. Bardi and I. Capuzzo-Dolcetta,
Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Springer Science & Business Media, 2008. |
[5] |
M. Bardi and P. Mannucci,
Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge-Ampère type, Forum Math., 25 (2013), 1291-1330.
|
[6] |
G. Barles and P. E. Souganidis,
Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.
|
[7] |
J.-D. Benamou, F. Collino and J.-M. Mirebeau,
Monotone and consistent discretization of the Monge-Ampere operator, Mathematics of computation, 85 (2016), 2743-2775.
|
[8] |
Z. Blocki,
On the Darboux equation, Zeszyty Naukowe Uniwersytetu Jagiello{\'n}skiego. Universitatis Iagellonicae Acta Mathematica, 1255 (2001), 87-90.
|
[9] |
J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, 2010.
![]() ![]() |
[10] |
S. C. Brenner, T. Gudi, M. Neilan and L.-Y. Sung,
$C^0$ penalty methods for the fully nonlinear Monge-Ampère equation, Math. Comp., 80 (2011), 1979-1995.
|
[11] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations i. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.
|
[12] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second order elliptic equations, Ⅲ: Functions of the eigenvalues of the Hessian, Acta Mathematica, 155 (1985), 261-301.
|
[13] |
Y. Chen and J. W. L. Wan, Monotone mixed narrow/wide stencil finite difference scheme for Monge-Ampère equation, https://arxiv.org/pdf/1608.00644.pdf, 2016. |
[14] |
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), 1-67.
|
[15] |
E. J. Dean and R. Glowinski,
Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type, Comput. Methods Appl. Mech. Engrg., 195 (2006), 1344-1386.
|
[16] |
M. Elsey and S. Esedoḡlu,
Analogue of the total variation denoising model in the context of geometry processing, Multiscale Model. Simul., 7 (2009), 1549-1573.
|
[17] |
X. Feng and M. Neilan,
Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations, J. Sci. Comput., 38 (2009), 74-98.
|
[18] |
J. M. Finn, G. L. Delzanno and L. Chacón, Grid generation and adaptation by Monge-Kantorovich optimization in two and three dimensions, In Proc. 17th Int. Meshing Roundtable, pages 551-568,2008. |
[19] |
B. D. Froese,
A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions, SIAM J. Sci. Comput., 34 (2012), A1432-A1459.
|
[20] |
B. D. Froese, Meshfree finite difference approximations for functions of the eigenvalues of the Hessian, Numer. Math. doi: 10.1007/s00211-017-0898-2, 2017. |
[21] |
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), 1692-1714.
|
[22] |
B. D. Froese and A. M. Oberman,
Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal., 51 (2013), 423-444.
|
[23] |
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, volume 224 of Grundlehren Math. Wiss, 2nd edition, Springer-Verlag, 1983.
![]() ![]() |
[24] |
C. E. Gutiérrez, The Monge-Ampère Equation, volume 44 of Progr. Nonlinear Differential Equations Appl., Springer Science & Business Media, 2001. |
[25] |
B. Hamfeldt and T. Salvador, Higher-order adaptive finite difference methods for fully nonlinear elliptic equations, J. Sci. Comput. in press. |
[26] |
Q. Han and J. -X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces volume 130, American Mathematical Society Providence, 2006. |
[27] |
H. Ishii and P.-L. Lions,
Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Diff. Eq., 83 (1990), 26-78.
|
[28] |
J. B. Kruskal,
Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping, Proc. Amer. Math. Soc., (1969), 697-703.
|
[29] |
P.-L. Lions,
Two remarks on Monge-Ampere equations, Ann. Mat. Pura Appl., 142 (1985), 263-275.
|
[30] |
G. Loeper and F. Rapetti,
Numerical solution of the Monge-Ampère equation by a Newton's algorithm, C. R. Math. Acad. Sci. Paris, 340 (2005), 319-324.
|
[31] |
J.-M. Mirebeau,
Discretization of the 3d Monge-Ampere operator, between wide stencils and power diagrams, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015), 1511-1523.
|
[32] |
A. Oberman,
The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694.
|
[33] |
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.
|
[34] |
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), 221-238.
|
[35] |
V. Oliker,
Embedding $S^n$ into $R^{n+1}$ with given integral Gauss curvature and optimal mass transport on $S^n$, Advances in Mathematics, 213 (2007), 600-620.
|
[36] |
V. I. Oliker and L. D. Prussner,
On the numerical solution of the equation $(\partial^2z/\partial x^2)(\partial^2z/\partial y^2)-(\partial^2z/\partial x\partial y)^2=f$ and its discretizations, I, Numer. Math., 54 (1988), 271-293.
|
[37] |
S. Osher and J. A. Sethian,
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.
|
[38] |
G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2006.
![]() ![]() |
[39] |
L.-P. Saumier, M. Agueh and B. Khouider,
An efficient numerical algorithm for the L2 optimal transport problem with periodic densities, IMA J. Appl. Math., 80 (2015), 135-157.
|
[40] |
M. Sulman, J. F. Williams and R. D. Russell,
Optimal mass transport for higher dimensional adaptive grid generation, J. Comput. Phys., 230 (2011), 3302-3330.
|
[41] |
N. S. Trudinger and J. I. E. Urbas,
The Dirichlet problem for the equation of prescribed Gauss curvature, Bull. Aust. Math. Soc., 28 (1983), 217-231.
|
[42] |
N. S. Trudinger and X. -J. Wang, The Monge-Ampère equation and its geometric applications, In Handbook of Geometric Analysis, volume 7 of Adv. Lect. Math., pages 467--524. Int. Press, 2008. |
[43] |
J. I. E. Urbas,
The generalized Dirichlet problem for equations of Monge-Ampere type, Annales de l'IHP Analyse non linéaire, 3 (1986), 209-228.
|
[44] |
C. Villani,
Topics in optimal transportation volume 58 of Graduate Studies in Mathematics AMS, Providence, RI, 2003. |
[45] |
H. Zhao,
A fast sweeping method for Eikonal equations, Math. Comp., 74 (2005), 603-627.
|
show all references
References:
[1] |
G. Alberti and L. Ambrosio,
A geometrical approach to monotone functions in $\mathbb{R}^n$, Math. Z., 230 (1999), 259-316.
|
[2] |
I. J. Bakelman, Generalized elliptic solutions of the Dirichlet problem for n-dimensional Monge-Ampère equations, In Nonlinear Functional Analysis and its Applications, volume 45 of P. Symp. Pure Math., pages 73-102. AMS, 1986. |
[3] |
I. J. Bakelman,
Convex Analysis and Nonlinear Geometric Elliptic Equations Springer Science & Business Media, 2012. |
[4] |
M. Bardi and I. Capuzzo-Dolcetta,
Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Springer Science & Business Media, 2008. |
[5] |
M. Bardi and P. Mannucci,
Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge-Ampère type, Forum Math., 25 (2013), 1291-1330.
|
[6] |
G. Barles and P. E. Souganidis,
Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.
|
[7] |
J.-D. Benamou, F. Collino and J.-M. Mirebeau,
Monotone and consistent discretization of the Monge-Ampere operator, Mathematics of computation, 85 (2016), 2743-2775.
|
[8] |
Z. Blocki,
On the Darboux equation, Zeszyty Naukowe Uniwersytetu Jagiello{\'n}skiego. Universitatis Iagellonicae Acta Mathematica, 1255 (2001), 87-90.
|
[9] |
J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, 2010.
![]() ![]() |
[10] |
S. C. Brenner, T. Gudi, M. Neilan and L.-Y. Sung,
$C^0$ penalty methods for the fully nonlinear Monge-Ampère equation, Math. Comp., 80 (2011), 1979-1995.
|
[11] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations i. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.
|
[12] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second order elliptic equations, Ⅲ: Functions of the eigenvalues of the Hessian, Acta Mathematica, 155 (1985), 261-301.
|
[13] |
Y. Chen and J. W. L. Wan, Monotone mixed narrow/wide stencil finite difference scheme for Monge-Ampère equation, https://arxiv.org/pdf/1608.00644.pdf, 2016. |
[14] |
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), 1-67.
|
[15] |
E. J. Dean and R. Glowinski,
Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type, Comput. Methods Appl. Mech. Engrg., 195 (2006), 1344-1386.
|
[16] |
M. Elsey and S. Esedoḡlu,
Analogue of the total variation denoising model in the context of geometry processing, Multiscale Model. Simul., 7 (2009), 1549-1573.
|
[17] |
X. Feng and M. Neilan,
Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations, J. Sci. Comput., 38 (2009), 74-98.
|
[18] |
J. M. Finn, G. L. Delzanno and L. Chacón, Grid generation and adaptation by Monge-Kantorovich optimization in two and three dimensions, In Proc. 17th Int. Meshing Roundtable, pages 551-568,2008. |
[19] |
B. D. Froese,
A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions, SIAM J. Sci. Comput., 34 (2012), A1432-A1459.
|
[20] |
B. D. Froese, Meshfree finite difference approximations for functions of the eigenvalues of the Hessian, Numer. Math. doi: 10.1007/s00211-017-0898-2, 2017. |
[21] |
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), 1692-1714.
|
[22] |
B. D. Froese and A. M. Oberman,
Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal., 51 (2013), 423-444.
|
[23] |
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, volume 224 of Grundlehren Math. Wiss, 2nd edition, Springer-Verlag, 1983.
![]() ![]() |
[24] |
C. E. Gutiérrez, The Monge-Ampère Equation, volume 44 of Progr. Nonlinear Differential Equations Appl., Springer Science & Business Media, 2001. |
[25] |
B. Hamfeldt and T. Salvador, Higher-order adaptive finite difference methods for fully nonlinear elliptic equations, J. Sci. Comput. in press. |
[26] |
Q. Han and J. -X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces volume 130, American Mathematical Society Providence, 2006. |
[27] |
H. Ishii and P.-L. Lions,
Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Diff. Eq., 83 (1990), 26-78.
|
[28] |
J. B. Kruskal,
Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping, Proc. Amer. Math. Soc., (1969), 697-703.
|
[29] |
P.-L. Lions,
Two remarks on Monge-Ampere equations, Ann. Mat. Pura Appl., 142 (1985), 263-275.
|
[30] |
G. Loeper and F. Rapetti,
Numerical solution of the Monge-Ampère equation by a Newton's algorithm, C. R. Math. Acad. Sci. Paris, 340 (2005), 319-324.
|
[31] |
J.-M. Mirebeau,
Discretization of the 3d Monge-Ampere operator, between wide stencils and power diagrams, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015), 1511-1523.
|
[32] |
A. Oberman,
The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694.
|
[33] |
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.
|
[34] |
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), 221-238.
|
[35] |
V. Oliker,
Embedding $S^n$ into $R^{n+1}$ with given integral Gauss curvature and optimal mass transport on $S^n$, Advances in Mathematics, 213 (2007), 600-620.
|
[36] |
V. I. Oliker and L. D. Prussner,
On the numerical solution of the equation $(\partial^2z/\partial x^2)(\partial^2z/\partial y^2)-(\partial^2z/\partial x\partial y)^2=f$ and its discretizations, I, Numer. Math., 54 (1988), 271-293.
|
[37] |
S. Osher and J. A. Sethian,
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.
|
[38] |
G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2006.
![]() ![]() |
[39] |
L.-P. Saumier, M. Agueh and B. Khouider,
An efficient numerical algorithm for the L2 optimal transport problem with periodic densities, IMA J. Appl. Math., 80 (2015), 135-157.
|
[40] |
M. Sulman, J. F. Williams and R. D. Russell,
Optimal mass transport for higher dimensional adaptive grid generation, J. Comput. Phys., 230 (2011), 3302-3330.
|
[41] |
N. S. Trudinger and J. I. E. Urbas,
The Dirichlet problem for the equation of prescribed Gauss curvature, Bull. Aust. Math. Soc., 28 (1983), 217-231.
|
[42] |
N. S. Trudinger and X. -J. Wang, The Monge-Ampère equation and its geometric applications, In Handbook of Geometric Analysis, volume 7 of Adv. Lect. Math., pages 467--524. Int. Press, 2008. |
[43] |
J. I. E. Urbas,
The generalized Dirichlet problem for equations of Monge-Ampere type, Annales de l'IHP Analyse non linéaire, 3 (1986), 209-228.
|
[44] |
C. Villani,
Topics in optimal transportation volume 58 of Graduate Studies in Mathematics AMS, Providence, RI, 2003. |
[45] |
H. Zhao,
A fast sweeping method for Eikonal equations, Math. Comp., 74 (2005), 603-627.
|




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