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

Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature

Brittany Froese Hamfeldt

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.

Keywords: Gaussian curvature, elliptic partial differential equations, Monge-Ampère equations, Gaussian curvature, viscosity solutions, finite difference methods.
Mathematics Subject Classification: Primary:35D40, 35J15, 35J25, 35J60, 65N06, 65N22;Secondary:35J66, 35J67, 35J70, 35J96, 53A99.
Citation: Brittany Froese Hamfeldt. Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature. Communications on Pure & 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.   Google Scholar

[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.  Google Scholar

[3]

I. J. Bakelman, Convex Analysis and Nonlinear Geometric Elliptic Equations Springer Science & Business Media, 2012.  Google Scholar

[4]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Springer Science & Business Media, 2008.  Google Scholar

[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.   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.   Google Scholar

[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.   Google Scholar

[8]

Z. Blocki, On the Darboux equation, Zeszyty Naukowe Uniwersytetu Jagiello{\'n}skiego. Universitatis Iagellonicae Acta Mathematica, 1255 (2001), 87-90.   Google Scholar

[9] J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, 2010.   Google Scholar
[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar
[24]

C. E. Gutiérrez, The Monge-Ampère Equation, volume 44 of Progr. Nonlinear Differential Equations Appl., Springer Science & Business Media, 2001. Google Scholar

[25]

B. Hamfeldt and T. Salvador, Higher-order adaptive finite difference methods for fully nonlinear elliptic equations, J. Sci. Comput. in press. Google Scholar

[26]

Q. Han and J. -X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces volume 130, American Mathematical Society Providence, 2006.  Google Scholar

[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.   Google Scholar

[28]

J. B. Kruskal, Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping, Proc. Amer. Math. Soc., (1969), 697-703.   Google Scholar

[29]

P.-L. Lions, Two remarks on Monge-Ampere equations, Ann. Mat. Pura Appl., 142 (1985), 263-275.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[32]

A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[38] G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2006.   Google Scholar
[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[44]

C. Villani, Topics in optimal transportation volume 58 of Graduate Studies in Mathematics AMS, Providence, RI, 2003.  Google Scholar

[45]

H. Zhao, A fast sweeping method for Eikonal equations, Math. Comp., 74 (2005), 603-627.   Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

I. J. Bakelman, Convex Analysis and Nonlinear Geometric Elliptic Equations Springer Science & Business Media, 2012.  Google Scholar

[4]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Springer Science & Business Media, 2008.  Google Scholar

[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.   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.   Google Scholar

[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.   Google Scholar

[8]

Z. Blocki, On the Darboux equation, Zeszyty Naukowe Uniwersytetu Jagiello{\'n}skiego. Universitatis Iagellonicae Acta Mathematica, 1255 (2001), 87-90.   Google Scholar

[9] J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, 2010.   Google Scholar
[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar
[24]

C. E. Gutiérrez, The Monge-Ampère Equation, volume 44 of Progr. Nonlinear Differential Equations Appl., Springer Science & Business Media, 2001. Google Scholar

[25]

B. Hamfeldt and T. Salvador, Higher-order adaptive finite difference methods for fully nonlinear elliptic equations, J. Sci. Comput. in press. Google Scholar

[26]

Q. Han and J. -X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces volume 130, American Mathematical Society Providence, 2006.  Google Scholar

[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.   Google Scholar

[28]

J. B. Kruskal, Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping, Proc. Amer. Math. Soc., (1969), 697-703.   Google Scholar

[29]

P.-L. Lions, Two remarks on Monge-Ampere equations, Ann. Mat. Pura Appl., 142 (1985), 263-275.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[32]

A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[38] G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2006.   Google Scholar
[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[44]

C. Villani, Topics in optimal transportation volume 58 of Graduate Studies in Mathematics AMS, Providence, RI, 2003.  Google Scholar

[45]

H. Zhao, A fast sweeping method for Eikonal equations, Math. Comp., 74 (2005), 603-627.   Google Scholar

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
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Figure 2.  A finite difference stencil chosen from a point cloud (a) in the interior and (b) near the boundary
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Figure 3.  Computational point cloud with $h=2^{-3}$
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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
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Table 1.  Error in computed solutions
$C^{0, 1}$ $C^0$ Non-continuous
$h$ $\|u-u^h\|_\infty$ $\|u-u^h\|_\infty$ $\|u-u^h\|_\infty$ $\|u-u^h\|_\infty$
$2^{-3}$ $9.45\times10^{-2}$ $1.94\times10^{-1}$ $3.55\times10^{-1}$ $2.12\times10^{-1}$
$2^{-4}$ $9.27\times10^{-2}$ $1.61\times10^{-1}$ $3.33\times10^{-1}$ $1.83\times10^{-1}$
$2^{-5}$ $6.48\times10^{-2}$ $1.28\times10^{-1}$ $3.05\times10^{-1}$ $1.60\times10^{-1}$
$2^{-6}$ $6.41\times10^{-2}$ $1.09\times10^{-1}$ $2.90\times10^{-1}$ $1.33\times10^{-1}$
$2^{-7}$ $3.18\times10^{-2}$ $8.80\times10^{-2}$ $2.74\times10^{-1}$ $9.53\times10^{-2}$
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$C^{0, 1}$ $C^0$ 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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