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A penalty-regularization-operator splitting method for the numerical solution of a scalar Eikonal equation

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Abstract

In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.

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References

  1. Aronsson, G., Evans, L. C. and Wu, Y., Fast/slow diffusion and growing sandpiles, Journal of Differential Equations, 131, 1996, 304–335.

    Article  MathSciNet  MATH  Google Scholar 

  2. Barth, T. J. and Sethian, J. A., Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comput. Phys., 145(1), 1998, 1–40.

    Article  MathSciNet  MATH  Google Scholar 

  3. Caboussat, A. and Glowinski, R., A numerical method for a non-smooth advection-diffusion problem arising in sand mechanics, Commun. Pure Appl. Anal, 8(1), 2008, 161–178.

    Article  MathSciNet  Google Scholar 

  4. Caboussat, A. and Glowinski, R., Regularization methods for the divergence equation ▽ · u = f, J. Comput. Math., 30(4), 2012, 354–380.

    Article  MathSciNet  MATH  Google Scholar 

  5. Caboussat, A., Glowinski, R. and Pan, T. W., On the numerical solution of some Eikonal equations: An elliptic solver approach, to appear of Contemporary Applied Mathematics, Higher Education Press, Beijing and World Scientific, Singapore, 2013.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Caffarelli, L. and Crandall, M. G., Distance functions and almost global solutions of Eikonal equations, Comm. Partial Differential Equations, 3, 2010, 391–414.

    Article  MathSciNet  Google Scholar 

  8. Caffarelli, L. A. and Cabré, X., Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, Providence, RI,1995.

  9. Caffarelli, L. A. and Glowinski, R., Numerical solution of the Dirichlet problem for a Pucci equation in dimension two. Application to homogenization, J. Numer. Math., 16(3), 2008, 185–216.

    Article  MathSciNet  MATH  Google Scholar 

  10. Chan, T. and Shen, J., Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia, 2005.

    Book  Google Scholar 

  11. Crandall, M., Evans, L. and Lions, P. L., Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282, 1984, 487–502.

    Article  MathSciNet  MATH  Google Scholar 

  12. Dacorogna, B., Glowinski, R., Kuznetzov, Y. and Pan, T. W., On a conjuguate gradient/Newton/penalty method for the solution of obstacle problems. application to the solution of an Eikonal system with Dirichlet boundary conditions, Křížek, M., Neittaanmäki, P., Glowinski, R. and Korotov, S., editors, Conjugate Gradient Algorithms and Finite Element Methods, Springer-Verlag, Berlin, Heidelberg, 2004, 263–283.

    Chapter  Google Scholar 

  13. Dacorogna, B., Glowinski, R. and Pan, T. W., Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions, C. R. Acad. Sci. Paris, Sér. I, 336, 2003, 511–518.

    Article  MathSciNet  MATH  Google Scholar 

  14. Dacorogna, B. and Marcellini, P., Implicit Partial Differential Equations, Birkhaüser, Basel, 1999.

    Book  MATH  Google Scholar 

  15. Dacorogna, B., Marcellini, P. and Paolini, E., An explicit solution to a system of implicit differential equations, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 25, 2008, 163–171.

    Article  MathSciNet  MATH  Google Scholar 

  16. Dacorogna, B., Marcellini, P. and Paolini, E., Lipschitz-continuous local isometric immersions: Rigid maps and origami, Journal Math. Pures Appl., 90, 2008, 66–81.

    Article  MathSciNet  MATH  Google Scholar 

  17. Dacorogna, B., Marcellini, P. and Paolini, E., Origami and partial differential equations, Notices of the American Math. Soc., 57, 2010, 598–606.

    MathSciNet  MATH  Google Scholar 

  18. Dean, E. J. and Glowinski, R., Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type, Comp. Meth. Appl. Mech. Engrg., 195, 2006, 1344–1386.

    Article  MathSciNet  MATH  Google Scholar 

  19. Dean, E. J. and Glowinski, R., On the numerical solution of the elliptic Monge-Ampère equation in dimension two: A least-squares approach, in Glowinski, R. and Neittaanmäki, P., editors, Partial Differential Equations: Modeling and Numerical Simulation, 16, Computational Methods in Applied Sciences, Springer-Verlag, 2008, 43–63.

    Article  MathSciNet  Google Scholar 

  20. Dean, E. J., Glowinski, R. and Guidoboni, G., On the numerical simulation of Bingham visco-plastic flow: Old and new results, Journal of Non Newtonian Fluid Mechanics, 142, 2007, 36–62.

    Article  MATH  Google Scholar 

  21. Delbos, F., Gilbert, J. C., Glowinski, R. and Sinoquet, D., Constrained optimization in seismic reflection tomography: A Gauss-Newton augmented Lagrangian approach, Geophysical Journal International, 164, 2006, 670–684.

    Article  Google Scholar 

  22. Evans, L. C., Partial Differential Equations, 19, Graduate Texts in Mathematics, American Mathematical Society, 1998.

    Google Scholar 

  23. Flück, M., Hofer, T., Picasso, M., et al., Scientific computing for aluminum production, Int. J. Numer. Anal. and Modeling, 6(3), 2009, 489–504.

    MATH  Google Scholar 

  24. Glowinski, R., Finite Element Method for Incompressible Viscous Flow, IX, Handbook of Numerical Analysis (Ciarlet, P. G., Lions, J. L., eds), Elsevier, Amsterdam, 2003, 3–1176.

  25. Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, NY, 2nd edition, 2008.

    MATH  Google Scholar 

  26. Glowinski, R., Numerical methods for fully nonlinear elliptic equations, Invited Lectures, 6th Int. Congress on Industrial and Applied Mathematics, EMS, Zürich, Switzerland, 2009, 155–192.

    Google Scholar 

  27. Glowinski, R., Kuznetzov, Y. and Pan, T. W., A penalty/Newton/conjugate gradient method for the solution of obstacle problems, C. R. Acad. Sci. Paris, Sér. I, 336, 2003, 435–440.

    Article  MATH  Google Scholar 

  28. Glowinski, R., Lions, J. L. and Trémolières, R., Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications, North-Holland Publishing Co., Amsterdam, New York, 1981.

    Google Scholar 

  29. Gremaud, P. A. and Ide, N. R., Computation of nonclassical solutions to Hamilton-Jacobi problems, SIAM J. Sci. Comput., 21, 1999, 502–521.

    Article  MathSciNet  MATH  Google Scholar 

  30. Gremaud, P. A. and Kuster, C. M., Computational study of fast methods for the Eikonal equation, SIAM J. Sci. Comp., 27, 2006, 1803–1816.

    Article  MathSciNet  MATH  Google Scholar 

  31. Hysing, S. R. and Turek, S., The Eikonal equation: Numerical efficiency vs algorithmic complexity on quadrilateral grids, Proceedings of Algorithmy, 2005, 2, 2–31.

    Google Scholar 

  32. Kimmel, R. and Sethian, J. A., Computing geodesic paths on manifolds, Proceedings of National Academy of Sciences, 95(15), 1998, 8431–8435.

    Article  MathSciNet  MATH  Google Scholar 

  33. Majava, K., Glowinski, R. and Kärkkäinen, T., Solving a non-smooth eigenvalue problem using operatorsplitting methods, International Journal of Computer Mathematics, 84(6), 2007, 825–846.

    Article  MathSciNet  MATH  Google Scholar 

  34. Nečas, J., Introduction to the Theory of Nonlinear Elliptic Equations, JohnWiley & Sons, Ltd., Chichester, 1986.

    MATH  Google Scholar 

  35. Prigozhin, L., Sandpiles and rivers networks: Extended systems with nonlocal interactions, Phys. Rev. E, 49(2), 1994, 1161–1167.

    Article  MathSciNet  Google Scholar 

  36. Prigozhin, L., Variational model of sandpile growth, Euro. Journal of Applied Mathematics, 7, 1996, 225–235.

    MathSciNet  MATH  Google Scholar 

  37. Qin, F., Luo, Y., Olsen, K., et al., Finite-difference solution of the Eikonal equation along expanding wavefronts, Geophysics, 57(3), 1992, 478–487.

    Article  Google Scholar 

  38. Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, NJ, 1997.

    MATH  Google Scholar 

  39. Schlichting, H. and Gersten, K., Boundary Layer Theory, McGraw and Hill, 8th revised edition, Springer-Verlag, Berlin, 2000.

    Book  Google Scholar 

  40. Sethian, J. A., Fast marching methods, SIAM Rev., 41(2), 1999, 199–235.

    Article  MathSciNet  MATH  Google Scholar 

  41. Sethian, J. A. and Vladimirsky, A., Fast methods for the Eikonal and related Hamilton-Jacobi equations on unstructured meshes, Proceedings of National Academy of Sciences, 11, 2000, 5699–5703.

    Article  MathSciNet  Google Scholar 

  42. Zhao, H., A fast sweeping method for Eikonal equations, Math. Comp., 74, 2005, 603–627.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Haute Ecole de Gestion de Genève, HES-SO//University of Applied Sciences, Western Switzerland, Route de Drize 7, 1227, Carouge, Switzerland

    Alexandre Caboussat

  2. Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, Texas, 77204-3008, USA

    Roland Glowinski

Authors
  1. Alexandre Caboussat
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  2. Roland Glowinski
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Corresponding author

Correspondence to Alexandre Caboussat.

Additional information

In Honor of the Scientific Contributions of Professor Luc Tartar

This work was supported by the National Science Foundation (No.DMS-0913982).

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Caboussat, A., Glowinski, R. A penalty-regularization-operator splitting method for the numerical solution of a scalar Eikonal equation. Chin. Ann. Math. Ser. B 36, 659–688 (2015). https://doi.org/10.1007/s11401-015-0930-8

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  • DOI: https://doi.org/10.1007/s11401-015-0930-8

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