Skip to main content
SpringerLink
Log in

A damped Newton algorithm for generated Jacobian equations

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

Generated Jacobian Equations have been introduced by Trudinger (Discrete Contin Dyn Syst A 34(4):1663–1681, 2014) as a generalization of Monge–Ampère equations arising in optimal transport. In this paper, we introduce and study a damped Newton algorithm for solving these equations in the semi-discrete setting, meaning that one of the two measures involved in the problem is finitely supported and the other one is absolutely continuous. We also present a numerical application of this algorithm to the near-field parallel reflector problem arising in non-imaging problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Abedin, F., Gutiérrez, C.E.: An iterative method for generated Jacobian equations. Calc. Var. Partial. Differ. Equ. 56(4), 101 (2017)

    Article  MathSciNet  Google Scholar 

  2. Berman, R.J.: Convergence rates for discretized Monge–Ampère equations and quantitative stability of optimal transport. Found. Comput. Math. 1–42 (2020)

  3. Boissonnat, J.-D., Wormser, C., Yvinec, M.: Curved voronoi diagrams. In: Effective Computational Geometry for Curves and Surfaces, Springer, pp. 67–116 (2007). https://hal.archives-ouvertes.fr/hal-00488446

  4. Caffarelli, L..A., Kochengin, S., Oliker, V..I.: On the numerical solution of the problem of reflector design with given far-field scattering data. Contemp. Math 226, 13–32 (1999)

    Article  MathSciNet  Google Scholar 

  5. Galichon, A., Kominers, S.D., Weber, S.: Costly concessions: an empirical framework for matching with imperfectly transferable utility. J. Polit. Econ. 127(6), 2875–2925 (2019)

    Article  Google Scholar 

  6. Guillen, N.: A primer on generated Jacobian equations: geometry, optics, economics. Not. Am. Math. Soc. 66, 1–10 (2019)

    Article  MathSciNet  Google Scholar 

  7. Guillen, N., Kitagawa, J.: Pointwise estimates and regularity in geometric optics and other generated Jacobian equations. Commun. Pure Appl. Math. 70(6), 1146–1220 (2017)

    Article  MathSciNet  Google Scholar 

  8. Gutiérrez, C.E., Tournier, F.: Regularity for the near field parallel refractor and reflector problems. Calc. Var. Partial. Differ. Equ. 54(1), 917–949 (2015)

    Article  MathSciNet  Google Scholar 

  9. Jiang, F., Trudinger, N.S.: On pogorelov estimates in optimal transportation and geometric optics. Bull. Math. Sci. 4(3), 407–431 (2014)

    Article  MathSciNet  Google Scholar 

  10. Kitagawa, J.: An iterative scheme for solving the optimal transportation problem. Calc. Var. Partial. Differ. Equ. 51(1–2), 243–263 (2014)

    Article  MathSciNet  Google Scholar 

  11. Kitagawa, J., Mérigot, Q., Thibert, B.: Convergence of a newton algorithm for semi-discrete optimal transport. J. Eur. Math. Soc. 21(9), 2603–2651 (2019)

    Article  MathSciNet  Google Scholar 

  12. Kochengin, S.A., Oliker, V.I.: Determination of reflector surfaces from near-field scattering data. Inverse Prob. 13(2), 363 (1997)

    Article  MathSciNet  Google Scholar 

  13. Machado, D.C.M.P., Mérigot, Q., Thibert, B.: Far-field reflector problem and intersection of paraboloids. Numer. Math. 134(2), 389–411 (2016)

    Article  MathSciNet  Google Scholar 

  14. Mérigot, Q., Meyron, J., Thibert, B.: An algorithm for optimal transport between a simplex soup and a point cloud. SIAM J. Imaging Sci. 11(2), 1363–1389 (2018)

    Article  MathSciNet  Google Scholar 

  15. Merigot, Q., Thibert, B.: Optimal transport: discretization and algorithms. In: Handbook of Numerical Analysis, vol. 22. Elsevier (2021) (to appear)

  16. Nöldeke, G., Samuelson, L.: The implementation duality. Econometrica 86(4), 1283–1324 (2018)

    Article  MathSciNet  Google Scholar 

  17. 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. Numerische Mathematik 54(3), 271–293 (1989)

    Article  Google Scholar 

  18. Oliker, V.: Mathematical aspects of design of beam shaping surfaces in geometrical optics. In: Kirkilionis, M., Krömker, S., Rannacher, R., Tomi, F. (eds.) Trends in Nonlinear Analysis, pp. 193–224. Springer (2003)

  19. Trudinger, N.S.: On the local theory of prescribed Jacobian equations. Discrete Contin. Dyn. Syst. A 34(4), 1663–1681 (2014)

    Article  MathSciNet  Google Scholar 

  20. Romijn, L.B., ten Thije Boonkkamp, J.H.M., Anthonissen, M.J.H., Ijzerman, W.L.: An iterative least-squares method for generated Jacobian equations in freeform optical design. SIAM J. Sci. Comput. 43(2), B298–B322 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We acknowledge the support of the French Agence Nationale de la Recherche through the project MAGA (ANR-16-CE40-0014).

Author information

Authors and Affiliations

  1. LJK, Grenoble INP, CNRS, Univ. Grenoble Alpes, 38000, Grenoble, France

    Anatole Gallouët & Boris Thibert

  2. Laboratoire de mathématiques d’Orsay, CNRS, Université Paris-Saclay, 91405, Orsay, France

    Quentin Mérigot

  3. Institut universitaire de France (IUF), Paris, France

    Quentin Mérigot

Authors
  1. Anatole Gallouët
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Quentin Mérigot
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Boris Thibert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatole Gallouët.

Additional information

Communicated by N.S. Trudinger.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gallouët, A., Mérigot, Q. & Thibert, B. A damped Newton algorithm for generated Jacobian equations. Calc. Var. 61, 49 (2022). https://doi.org/10.1007/s00526-021-02147-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-021-02147-7

Mathematics Subject Classification

Search

Navigation