# Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge–Ampère Equation in Dimensions Two and Higher

## Abstract

The elliptic Monge–Ampère equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing, and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to the weak (viscosity) solution. In this article we build a wide stencil finite difference discretization for the Monge–Ampère equation. The scheme is monotone, so the Barles–Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation. Solutions of the scheme are found using a damped Newton's method. We prove convergence of Newton's method and provide a systematic method to determine a starting point for the Newton iteration. Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to nondifferentiable.

## References

1.
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.
2.
I. Bakelman, Convex Analysis and Nonlinear Geometric Elliptic Equations, Springer-Verlag, Berlin, 1994.
3.
G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptot. Anal., 4 (1991), pp. 271–283.
4.
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.
5.
J.-D. Benamou, B. D. Froese, and A. M. Oberman, Two numerical methods for the elliptic Monge-Ampère equation, ESAIM Math. Model. Numer. Anal., 4 (2010), pp. 737–758.
6.
D. P. Bertsekas, Convex Analysis and Optimization, Athena Scientific, Belmont, MA, 2003.
7.
J. F. Bonnans and H. Zidani, Consistency of generalized finite difference schemes for the stochastic HJB equation, SIAM J. Numer. Anal., 41 (2003), pp. 1008–1021.
8.
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.
9.
L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Commun. Pure Appl. Math., 37 (1984), pp. 369–402.
10.
L. A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2), 131 (1990), pp. 135–150.
11.
D. Cohen-Or, Space deformations, surface deformations and the opportunities in-between, J. Comput. Sci. Technol., 24 (2009), pp. 2–5.
12.
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.
13.
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.
14.
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.
15.
G. L. Delzanno, L. Chacón, 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.
16.
L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, in Current Developments in Mathematics 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, pp. 65–126.
17.
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.
18.
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.
19.
J. M. Finn, G. L. Delzanno, and L. Chacón, Grid generation and adaptation by Monge-Kantorovich optimization in two and three dimensions, in Proceedings of the 17th International Meshing Roundtable, (2008), pp. 551–568.
20.
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).
21.
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.
22.
B. D. Froese, A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions, SIAM J. Sci. Comput., to appear.
23.
T. Glimm and V. Oliker, Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem, J. Math. Sci. (N.Y.), 117 (2003), pp. 4096–4108.
24.
R. Glowinski, Numerical methods for fully nonlinear elliptic equations, in Proceedings of the 6th International Congress on Industrial and Applied Mathematics, R. Jeltsch and G. Wanner, eds., ICIAM 07, Invited Lectures, 2009, pp. 155–192.
25.
C. E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications 44, Birkhäuser Boston, Boston, MA, 2001.
26.
E. Haber, R. Horesh, and J. Modersitski, Numerical optimization for constrained image registration, preprint, 2010.
27.
S. Haker, A. Tannenbaum, and R. Kikinis, Mass preserving mappings and image registration, in MICCAI '01: Proceedings of the 4th International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer-Verlag, London, 2001, pp. 120–127.
28.
S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, Optimal mass transport for registration and warping, Internat. J. Comput. Vision, 60 (2004), pp. 225–240.
29.
Q. Han and J.-X. Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, Mathematical Surveys and Monographs 130, American Mathematical Society, Providence, RI, 2006.
30.
J. L. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Regional Conference Series in Mathematics 57, published for the Conference Board of the Mathematical Sciences, Washington, D.C., 1985.
31.
C. T. Kelley, Iterative methods for linear and nonlinear equations, Frontiers in Applied Mathematics 16, SIAM, Philadelphia, 1995.
32.
G. Loeper and F. Rapetti, Numerical solution of the Monge-Ampére equation by a Newton's algorithm, C. R. Math. Acad. Sci. Paris, 340 (2005), pp. 319–324.
33.
A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math., 99 (2004), pp. 365–379.
34.
A. M. Oberman, A convergent difference scheme for the infinity Laplacian: Construction of absolutely minimizing Lipschitz extensions, Math. Comp., 74 (2005), pp. 1217–1230.
35.
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.
36.
A. M. Oberman, Computing the convex envelope using a nonlinear partial differential equation, Math. Models Methods Appl. Sci., 18 (2008), pp. 759–780.
37.
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.
38.
A. M. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc. to appear, http://arxiv.org/abs/1007.0773.
39.
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), pp. 271–293.
40.
A. V. Pogorelov, Monge-Ampère equations of elliptic type, translated from the first Russian edition by L. F. Boron with the assistance of A. L. Rabenstein and R. C. Bollinger. P. Noordhoff Ltd., Groningen, 1964.
41.
A. V. Pogorelov, The Dirichlet problem for the multidimensional analogue of the Monge-Ampère equation, Dokl. Akad. Nauk SSSR, 201 (1971), pp. 790–793.
42.
G. Strang, Linear algebra and its applications, 2nd ed., Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1980.
43.
T. ur Rehman, E. Haber, G. Pryor, J. Melonakos, and A. Tannenbaum, 3D nonrigid registration via optimal mass transport on the GPU, Med. Image Anal., 13 (2009), pp. 931–940, 12.
44.
J. I. E. Urbas, The generalized Dirichlet problem for equations of Monge-Ampère type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), pp. 209–228.
45.
C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58, American Mathematical Society, Providence, RI, 2003.
46.
V. Zheligovsky, O. Podvigina, and U. Frisch, The Monge-Ampère equation: Various forms and numerical solution, J. Comput. Phys., 229 (2010), pp. 5043–5061.

## Information & Authors

### Information

#### Published In

SIAM Journal on Numerical Analysis
Pages: 1692 - 1714
ISSN (online): 1095-7170

#### History

Submitted: 22 July 2010
Accepted: 2 June 2011
Published online: 4 August 2011