March  2018, 17(2): 671-707. doi: 10.3934/cpaa.2018036

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

Received  January 2017 Revised  August 2017 Published  March 2018

Fund Project: This work was partially supported by NSF DMS-1619807.

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.

Citation: Brittany Froese Hamfeldt. Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature. Communications on Pure and Applied Analysis, 2018, 17 (2) : 671-707. doi: 10.3934/cpaa.2018036
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. BenamouF. 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. BrennerT. GudiM. Neilan and L.-Y. Sung, $C^0$ penalty methods for the fully nonlinear Monge-Ampère equation, Math. Comp., 80 (2011), 1979-1995. 

[11]

L. CaffarelliL. 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. CaffarelliL. 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. CrandallH. 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. SaumierM. 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. SulmanJ. 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. BenamouF. 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. BrennerT. GudiM. Neilan and L.-Y. Sung, $C^0$ penalty methods for the fully nonlinear Monge-Ampère equation, Math. Comp., 80 (2011), 1979-1995. 

[11]

L. CaffarelliL. 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. CaffarelliL. 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. CrandallH. 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. SaumierM. 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. SulmanJ. 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. 

Figure 1.  (a) A viscosity solution with constant Gaussian curvature that does not achieve the Dirichlet boundary conditions and (b) a sub-solution that lies above this viscosity solution
Figure 2.  A finite difference stencil chosen from a point cloud (a) in the interior and (b) near the boundary
Figure 3.  Computational point cloud with $h=2^{-3}$
Figure 4.  Computed approximations ($h=2^{-7}$) to solutions that (a) are Lipschitz continuous (6.4.1), (b) have an unbounded gradient (6.4.2), and (c) do not achieve the Dirichlet data (6.4.3). (d) Error in discontinuous solution
Table 1.  Error in computed solutions
$C^{0, 1}$ $C^0$ Non-continuous
$h$ $\|u-u^h\|_\infty$ $\|u-u^h\|_\infty$ $\|u-u^h\|_\infty$ $\|u-u^h\|_\infty$
$2^{-3}$ $9.45\times10^{-2}$ $1.94\times10^{-1}$ $3.55\times10^{-1}$ $2.12\times10^{-1}$
$2^{-4}$ $9.27\times10^{-2}$ $1.61\times10^{-1}$ $3.33\times10^{-1}$ $1.83\times10^{-1}$
$2^{-5}$ $6.48\times10^{-2}$ $1.28\times10^{-1}$ $3.05\times10^{-1}$ $1.60\times10^{-1}$
$2^{-6}$ $6.41\times10^{-2}$ $1.09\times10^{-1}$ $2.90\times10^{-1}$ $1.33\times10^{-1}$
$2^{-7}$ $3.18\times10^{-2}$ $8.80\times10^{-2}$ $2.74\times10^{-1}$ $9.53\times10^{-2}$
$C^{0, 1}$ $C^0$ Non-continuous
$h$ $\|u-u^h\|_\infty$ $\|u-u^h\|_\infty$ $\|u-u^h\|_\infty$ $\|u-u^h\|_\infty$
$2^{-3}$ $9.45\times10^{-2}$ $1.94\times10^{-1}$ $3.55\times10^{-1}$ $2.12\times10^{-1}$
$2^{-4}$ $9.27\times10^{-2}$ $1.61\times10^{-1}$ $3.33\times10^{-1}$ $1.83\times10^{-1}$
$2^{-5}$ $6.48\times10^{-2}$ $1.28\times10^{-1}$ $3.05\times10^{-1}$ $1.60\times10^{-1}$
$2^{-6}$ $6.41\times10^{-2}$ $1.09\times10^{-1}$ $2.90\times10^{-1}$ $1.33\times10^{-1}$
$2^{-7}$ $3.18\times10^{-2}$ $8.80\times10^{-2}$ $2.74\times10^{-1}$ $9.53\times10^{-2}$
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