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A Finite Element/Operator-Splitting Method for the Numerical Solution of the Two Dimensional Elliptic Monge–Ampère Equation

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Abstract

We discuss in this article a novel method for the numerical solution of the two-dimensional elliptic Monge–Ampère equation. Our methodology relies on the combination of a time-discretization by operator-splitting with a mixed finite element based space approximation where one employs the same finite-dimensional spaces to approximate the unknown function and its three second order derivatives. A key ingredient of our approach is the reformulation of the Monge–Ampère equation as a nonlinear elliptic equation in divergence form, involving the cofactor matrix of the Hessian of the unknown function. With the above elliptic equation we associate an initial value problem that we discretize by operator-splitting. To enforce the pointwise positivity of the approximate Hessian we employ a hard thresholding based projection method. As shown by our numerical experiments, the resulting methodology is robust and can handle a large variety of triangulations ranging from uniform on rectangles to unstructured on domains with curved boundaries. For those cases where the solution is smooth and isotropic enough, we suggest also a two-stage method to improve the computational efficiency, the second stage being reminiscent of a Newton-like method. The methodology discussed in this article is able to handle domains with curved boundaries and unstructured meshes, using piecewise affine continuous approximations, while preserving optimal, or nearly optimal, convergence orders for the approximation error.

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References

  1. Awanou, G.: Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: classical solutions. IMA J. Numer. Anal. 35(3), 1150–1166 (2014)

    Article  MATH  Google Scholar 

  2. Bakelman, I.J.: Convex Analysis and Nonlinear Geometric Elliptic Equations. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  3. Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  5. Brenner, S., Gudi, T., Neilan, M., Sung, L.: \({C}^0\) penalty methods for the fully nonlinear Monge–Ampère equation. Math. Comput. 80(276), 1979–1995 (2011)

    Article  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caboussat, A., Glowinski, R., Gourzoulidis, D.: A least-squares/relaxation method for the numerical solution of the three-dimensional elliptic Monge-Ampère equation. J. Sci. Comput. 77, 1–26 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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. Calc. Var. 19(3), 780–810 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caffarelli, L.A., Milman, M.: Monge–Ampère Equation: Applications to Geometry and Optimization: NSF-CBMS Conference on the Monge–Ampère Equation, Applications to Geometry and Optimization, July 9–13, 1997, Florida Atlantic University, Volume 226. American Mathematical Society, Providence (1999)

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

    Book  MATH  Google Scholar 

  11. Dean, E.J., Glowinski, R.: Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach. C. R. Math. Acad. Sci. Paris 336(9), 779–784 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dean, E.J., Glowinski, R.: Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least-squares approach. C. R. Math. Acad. Sci. Paris. 339(12), 887–892 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dean, E.J., Glowinski, R.: 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, 71–96 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Dean, E.J., Glowinski, R.: Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type. Comput. Methods Appl. Mech. Eng. 195(13), 1344–1386 (2006)

    Article  MATH  Google Scholar 

  15. Dean, E.J., Glowinski, R.: On the numerical solution of the elliptic Monge–Ampère equation in dimension two: a least-squares approach. In: Glowinski, R., Neittaanmaki, P. (eds.) Partial Differential Equations, pp. 43–63. Springer, Dordrecht (2008)

    Chapter  Google Scholar 

  16. Dean, E.J., Glowinski, R., Pan, T.W.: Operator-splitting methods and applications to the direct numerical simulation of particulate flow and to the solution of the elliptic Monge–Ampère equation. In: Cagnol, J., Zoésio, J.P. (eds.) Control Boundary Analysis, pp. 1–27. CRC, Boca Raton (2005)

    Google Scholar 

  17. D’Onofrio, L., Giannetti, F., Greco, L.: On weak Hessian determinants. In: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, vol. 16(3), pp. 159–169 (2005)

  18. Feng, X., Glowinski, R., Neilan, M.: Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Rev. 55(2), 205–267 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Feng, X., Neilan, M.: Mixed finite element methods for the fully nonlinear Monge–Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 47(2), 1226–1250 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Froese, B.D.: Meshfree finite difference approximations for functions of the eigenvalues of the Hessian. Numer. Math. 138(1), 75–99 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Froese, B.D., Oberman, A.M.: Convergent finite difference solvers for viscosity solutions of the elliptic Monge–Ampère equation in dimensions two and higher. SIAM J. Numer. Anal. 49(4), 1692–1714 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Froese, B.D., Oberman, A.M.: Fast finite difference solvers for singular solutions of the elliptic Monge–Ampère equation. J. Comput. Phys. 230(3), 818–834 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Froese, B.D.: A numerical method for the elliptic Monge–Ampère equation with transport boundary conditions. SIAM J. Sci. Comput. 34(3), A1432–A1459 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  26. Glowinski, R.: Numerical Nethods for Nonlinear Variational Problems. Springer, New York (1984). (2nd printing: 2008)

    Book  Google Scholar 

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

    Google Scholar 

  28. Glowinski, R.: Numerical methods for fully nonlinear elliptic equations. In: 6th International congress on industrial and applied mathermatics, ICIAM, vol. 7, pp. 155–192 (2009)

  29. Glowinski, R.: Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems. SIAM, Philadelphia (2015)

    Book  MATH  Google Scholar 

  30. Glowinski, R., Osher, S., Yin, W. (eds.): Splitting Methods in Communication, Imaging, Science, and Engineering. Springer, Berlin (2016)

    MATH  Google Scholar 

  31. Gutiérrez, C.E.: The Monge–Ampère Equation. Birkhaüser, Basel (2001)

    Book  MATH  Google Scholar 

  32. Hamfeldt, B.F., Salvador, T.: Higher-order adaptive finite difference methods for fully nonlinear elliptic equations. J. Sci. Comput. 75(3), 1282–1306 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lindsey, M., Rubinstein, Y.A.: Optimal transport via a Monge–Ampère optimization problem. SIAM J. Math. Anal. 49(4), 3073–3124 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Loeper, G., Rapetti, F.: Numerical solution of the Monge–Ampère equation by a Newton algorithm. C. R. Acad. Sci. Paris Ser. I 340, 319–324 (2005)

    Article  MATH  Google Scholar 

  35. Mohammadi, B.: Optimal transport, shape optimization and global minimization. C. R. Math. 344(9), 591–596 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nesterov, Y.: A method of solving a convex programming problem with convergence rate \({O}(1/k^2)\). Sov. Math. Dokl. 27(2), 372–376 (1983)

    MATH  Google Scholar 

  37. Nochetto, R., Ntogkas, D., Zhang, W.: Two-scale method for the Monge–Ampère equation: Convergence to the viscosity solution. In: Mathematics of Computation (2018). https://doi.org/10.1093/imanum/dry026

  38. Oberman, A.M.: 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(1), 221–238 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Oliker, V.I., Prussner, L.D.: On the numerical solution of the equation \(\frac{\partial ^2 z}{\partial x^2}\frac{\partial ^2 z}{\partial y^2}-\left(\frac{\partial ^2 z}{\partial x \partial y}\right)^2=f\) and its discretizations, i. Numer. Math. 54(3), 271–293 (1989)

    Article  Google Scholar 

  40. Persson, P.O., Strang, G.: A simple mesh generator in MATLAB. SIAM Rev. 46(2), 329–345 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964)

    Article  Google Scholar 

  42. Savin, O.: The obstacle problem for Monge-Ampère equation. Calc. Var. Partial Differ. Equ. 22(3), 303–320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Schaeffer, H., Hou, T.Y.: An accelerated method for nonlinear elliptic PDE. J. Sci. Comput. 69(2), 556–580 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. Strang, G., Fix, G.J.: An Analysis of The Finite Element Method, vol. 212. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  45. Su, W., Boyd, S., Candes, E.: A differential equation for modeling Nesterov’s accelerated gradient method: theory and insights. J. Mach. Learn. Res. 17, 1–43 (2016)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous reviewers of this article for most helpful comments and suggestions. The work of S. Leung is partially supported by the Hong Kong RGC Grants 16303114 and 16309316. The work of J. Qian is partially supported by NSF Grants 1522249 and 1614566.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX, 77204, USA

    Roland Glowinski

  2. Department of Mathematics, The Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Roland Glowinski

  3. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

    Hao Liu & Shingyu Leung

  4. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Jianliang Qian

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  1. Roland Glowinski
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  2. Hao Liu
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  4. Jianliang Qian
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Correspondence to Hao Liu.

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The original version of this article was revised: The article was originally published in SpringerLink with open access. With the author(s)’ decision to step back from Open Choice, the copyright of the article changed on October 2018 to ©Springer Science+Business Media, LLC, part of Springer Nature 2018.

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Glowinski, R., Liu, H., Leung, S. et al. A Finite Element/Operator-Splitting Method for the Numerical Solution of the Two Dimensional Elliptic Monge–Ampère Equation. J Sci Comput 79, 1–47 (2019). https://doi.org/10.1007/s10915-018-0839-y

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