article

A Least-Squares/Relaxation Method for the Numerical Solution of the Three-Dimensional Elliptic Monge---Ampère Equation

Authors:

Alexandre Caboussat,

Roland Glowinski,

Dimitrios GourzoulidisAuthors Info & Claims

Journal of Scientific Computing, Volume 77, Issue 1

Pages 53 - 78

https://doi.org/10.1007/s10915-018-0698-6

Published: 01 October 2018 Publication History

Abstract

In this article, we address the numerical solution of the Dirichlet problem for the three-dimensional elliptic Monge---Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a least-squares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and non-convex domains.

References

[1]

Aleksandrov, A.D.: Uniqueness conditions and estimates for the solution of the Dirichlet problem. Am. Math. Soc. Trans. 68, 89---119 (1968)

[2]

Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1998)

[3]

Awanou, G., Li, H.: Error analysis of a mixed finite element method for the Monge---Ampère equation. Int. J. Numer. Anal. Model. 11(4), 745---761 (2014)

[4]

Benamou, J.D., Froese, B.D., Oberman, A.M.: Two numerical methods for the elliptic Monge---Ampère equation. ESAIM Math. Model. Numer. Anal. 44(4), 737---758 (2010)

[5]

Benamou, J.D., Froese, B.D., Oberman, A.M.: Numerical solution of the optimal transportation problem using the Monge---Ampère equation. J. Comput. Phys. 49(4), 107---126 (2014)

Digital Library

[6]

Boehmer, K.: On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46(3), 1212---1249 (2008)

Digital Library

[7]

Botsaris, C.A.: A class of methods for unconstrained minimization based on stable numerical integration techniques. J. Math. Anal. Appl. 63(3), 729---749 (1978)

[8]

Brenner, S.C., Neilan, M.: Finite element approximations of the three dimensional Monge---Ampère equation. ESAIM Math. Model. Numer. Anal. 46(5), 979---1001 (2012)

[9]

Caboussat, A., Glowinski, R., Sorensen, D.C.: A least-squares method for the numerical solution of the Dirichlet problem for the elliptic Monge---Ampère equation in dimension two. ESAIM Control Optim. Calculus Var. 19(3), 780---810 (2013)

[10]

Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations I. Monge---Ampère equation. Commun. Pure Appl. Math. 37, 369---402 (2014)

[11]

Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence (1995)

[12]

Dean, E.J., Glowinski, R.: On the numerical solution of the elliptic Monge---Ampère equation in dimension two: a least-square approach. In: Glowinski, R., Neittaanmäki, P. (eds.) Partial Differ. Equ. Model. Numer. Simul. Comput. Methods Appl. Sci., vol. 16, pp. 43---63. Springer, Berlin (2008)

[13]

Engquist, B., Froese, B.D., Yang, Y.: Optimal transport for seismic full waveform inversion. Commun. Math. Sci. 14(8), 2309---2330 (2016)

[14]

Feng, X., Neilan, M.: Error analysis for mixed finite element approximations of the fully nonlinear Monge---Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 47(2), 1226---1250 (2009)

Digital Library

[15]

Feng, X., Neilan, M.: A modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations. SIAM J. Numer. Anal. 47(4), 2952---2981 (2009)

Digital Library

[16]

Feng, X., Neilan, M.: Vanishing moment method and moment solutions of second order fully nonlinear partial differential equations. J. Sci. Comput. 38(1), 74---98 (2009)

Digital Library

[17]

Feng, X., Neilan, M., Glowinski, R.: Recent developments in numerical methods for fully nonlinear 2nd order PDEs. SIAM Rev. 55, 1---64 (2013)

[18]

Feng, X., Neilan, M., Prohl, A.: Error analysis of finite element approximations of the inverse mean curvature flow arising from the general relativity. Numer. Math. 108, 93---119 (2007)

Digital Library

[19]

Froese, B.D., Oberman, A.M.: Convergent filtered schemes for the Monge---Ampère partial differential equation. SIAM J. Numer. Anal. 51(1), 423---444 (2013)

[20]

Geuzaine, C., Remacle, J.F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309---1331 (2009)

[21]

Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations. Springer, New York (1983)

[22]

Glowinski, R.: Finite element method for incompressible viscous flow. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. IX, pp. 3---1176. Elsevier, Amsterdam (2003)

[23]

Glowinski, R.: Numerical Methods for Nonlinear Variational Problems, 2nd edn. Springer, New York (2008)

[24]

Glowinski, R.: Numerical methods for fully nonlinear elliptic equations. In: Proceedings of the International Congress on Industrial and Applied Mathematics, 16---20 July 2007, Zürich. EMS Publishing House (2009)

[25]

Glowinski, R.: Variational Methods for the Numerical Solution of Nonlinear Elliptic Problem. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (2015)

Digital Library

[26]

Glowinski, R., Lions, J.L., He, J.W.: Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2008)

Digital Library

[27]

Glowinski, R., Lueng, T., Liu, H., Qian, J.: A novel method for the numerical solution of the Monge---Ampère equation. Private communication (2017)

[28]

Glowinski, R., Sorensen, D.C.: A quadratically constrained minimization problem arising from PDE of Monge---Ampère type. Numer. Algorithms 53(1), 53---66 (2009)

[29]

Gutiérrez, C.E.: The Monge---Ampère Equation. Birkhaüser, Boston (2001)

[30]

Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (1993)

Digital Library

[31]

Ishii, H., Lions, P.L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83(1), 26---78 (1990)

[32]

Kelley, C.: Iterative Methods for Linear and Nonlinear Equations. Society for Industrial and Applied Mathematics, Philadelphia (1995)

[33]

Kirk, B.S., Peterson, J.W., Stogner, R.H., Carey, G.F.: libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22(3---4), 237---254 (2006)

Digital Library

[34]

Liu, J., Froese, B.D., Oberman, A.M., Xiao, M.: A multigrid scheme for 3D Monge---Ampère equations. Int. J. Comput. Math. 94(9), 1850---1866 (2017)

Digital Library

[35]

Nochetto, R.H., Ntogkas, D., Zhang, W.: Two-scale method for the Monge---Ampère equation: convergence to the viscosity solution. arXiv:1706.06193 (2017)

[36]

Nochetto, R.H., Ntogkas, D., Zhang, W.: Two-scale method for the Monge---Ampère equation: pointwise error estimates. ArXiv:1706.09113 (2017)

[37]

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

[38]

Oberman, A.: Wide stencil finite difference schemes for the elliptic Monge---Ampère equations and functions of the eigenvalues of the Hessian. Discrete Continuous Dyn. Syst. B 10(1), 221---238 (2008)

[39]

Oliker, V.I., Prussner, L.D.: On the numerical solution of the equation $$z_{xx} z_{yy} - z_{xy}^2 = f$$zxxzyy-zxy2=f and its discretization. I. Numer. Math. 54, 271---293 (1988)

Digital Library

[40]

Picasso, M., Alauzet, F., Borouchaki, H., George, P.L.: A numerical study of some hessian recovery techniques on isotropic and anisotropic meshes. SIAM J. Sci. Comput. 33, 1058---1076 (2011)

Digital Library

[41]

Stojanovic, S.: Optimal momentum hedging via Monge---Ampère PDEs and a new paradigm for pricing options. SIAM J. Control Optim. 43, 1151---1173 (2004)

Digital Library

[42]

Tapia, R.A., Dennis, J.E., Schaefermeyer, J.P.: Inverse, shifted inverse, and Rayleigh quotient iteration as Newton's method. SIAM Rev. 60(1), 3---55 (2018)

[43]

Tychonoff, A.N.: The regularization of incorrectly posed problems. Dokl. Akad. Nauk. SSSR 153, 42---52 (1963)

Cited By

Wang PJin LLi ZYi L(2024)Spectral Collocation Method for Numerical Solution to the Fully Nonlinear Monge-Ampère EquationJournal of Scientific Computing10.1007/s10915-024-02617-y100:3Online publication date: 29-Jul-2024
https://dl.acm.org/doi/10.1007/s10915-024-02617-y
Lai MLee J(2023)Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère EquationJournal of Scientific Computing10.1007/s10915-023-02183-995:2Online publication date: 6-Apr-2023
https://dl.acm.org/doi/10.1007/s10915-023-02183-9
Caboussat AGlowinski RGourzoulidis D(2022)A Least-Squares Method for the Solution of the Non-smooth Prescribed Jacobian EquationJournal of Scientific Computing10.1007/s10915-022-01968-893:1Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s10915-022-01968-8
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A Least-Squares/Relaxation Method for the Numerical Solution of the Three-Dimensional Elliptic Monge---Ampère Equation

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Published In

cover image Journal of Scientific Computing

Journal of Scientific Computing Volume 77, Issue 1

October 2018

688 pages

ISSN:0885-7474

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Copyright © Copyright © 2018 Springer Science+Business Media, LLC, part of Springer Nature.

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Plenum Press

United States

Publication History

Published: 01 October 2018

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Cited By

Wang PJin LLi ZYi L(2024)Spectral Collocation Method for Numerical Solution to the Fully Nonlinear Monge-Ampère EquationJournal of Scientific Computing10.1007/s10915-024-02617-y100:3Online publication date: 29-Jul-2024
https://dl.acm.org/doi/10.1007/s10915-024-02617-y
Lai MLee J(2023)Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère EquationJournal of Scientific Computing10.1007/s10915-023-02183-995:2Online publication date: 6-Apr-2023
https://dl.acm.org/doi/10.1007/s10915-023-02183-9
Caboussat AGlowinski RGourzoulidis D(2022)A Least-Squares Method for the Solution of the Non-smooth Prescribed Jacobian EquationJournal of Scientific Computing10.1007/s10915-022-01968-893:1Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s10915-022-01968-8
Yadav NThije Boonkkamp JIjzerman W(2019)A Monge---Ampère Problem with Non-quadratic Cost Function to Compute Freeform Lens SurfacesJournal of Scientific Computing10.1007/s10915-019-00948-980:1(475-499)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.1007/s10915-019-00948-9
Glowinski RLiu HLeung SQian J(2019)A Finite Element/Operator-Splitting Method for the Numerical Solution of the Two Dimensional Elliptic Monge---Ampère EquationJournal of Scientific Computing10.1007/s10915-018-0839-y79:1(1-47)Online publication date: 1-Apr-2019
https://dl.acm.org/doi/10.1007/s10915-018-0839-y

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