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Computing the Level Set Convex Hull

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Abstract

Quasiconvex (QC) functions are functions whose level sets are convex. The quasiconvex envelope (QCE) of a given function, g, is the maximal QC function below g. The QCE provides a level set representation for the convex hull of every level set of a given function. We present a nonlocal line solver for computing the QCE of a given function. The algorithm is based on solving the one dimensional problem on lines, which can be done by a fast marching or sweeping method. Convergence of the algorithm is established.

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References

  1. Avriel, M., Diewert, W.E., Schaible, S., Zang, I: Generalized Concavity. Springer (reprinted by SIAM classics) (1988)

  2. Arrow, K.J., Enthoven, A.C.: Quasi-concave programming. Econom. J. Econom. Soc. 29, 779–800 (1961)

  3. Barron, E.N., Goebel, R., Jensen, R.R.: Functions which are quasiconvex under linear perturbations. SIAM J. Optim. 22(3), 1089–1108 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barron, E.N., Goebel, R., Jensen, R.R.: The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete Contin. Dyn. Syst. Ser. B 17(6), 1693–1706 (2012)

  5. Barron, E., Goebel, R., Jensen, R.: Quasiconvex functions and nonlinear pdes. Trans. Am. Math. Soc. 365(8), 4229–4255 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barron, E.N., Jensen, R.R.: A uniqueness result for the quasiconvex operator and first order PDEs for convex envelopes. Annales de l’Institut Henri Poincare (C) Non Linear Anal. 31(2), 203–215 (2014)

  7. Carlier, G., Galichon, A.: Exponential convergence for a convexifying equation. ESAIM Control Optim. Calc. Var. 18(03), 611–620 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caffarelli, L.A., Spruck, J.: Convexity properties of solutions to some classical variational problems. Commun. Partial Differ. Equ. 7(11), 1337–1379 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Colesanti, A., Salani, P.: Quasi–concave envelope of a function and convexity of level sets of solutions to elliptic equations. Mathematische Nachrichten 258(1), 3–15 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Firey, W.J.: Shapes of worn stones. Mathematika 21(01), 1–11 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ishii, H., Mikami, T.: A level set approach to the wearing process of a nonconvex stone. Calc. Var. Partial Differ. Equ. 19(1), 53–93 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kawohl, B.: Rearrangements and convexity of level sets in PDE. Lect. Notes Math. 1150, 1–134 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. McAdams, A., Osher, S., Teran, J.: Crashing waves, awesome explosions, turbulent smoke, and beyond: applied mathematics and scientific computing in the visual effects industry. Not. AMS 57(5), 614–623 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Malladi, R., Sethian, J.A.: Image processing via level set curvature flow. Proc. Natl. Acad. Sci. 92(15), 7046–7050 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Oberman, A.M.: A convergent monotone difference scheme for motion of level sets by mean curvature. Numer. Math. 99(2), 365–379 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Oberman, A.M.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44(2), 879–895 (2006). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Oberman, A.M.: Computing the convex envelope using a nonlinear partial differential equation. Math. Models Methods Appl. Sci. 18(5), 759–780 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Oberman, A.M., Ruan, Y.: A partial differential equation for the rank one convex envelope. Submitted (2016)

  20. Osher, S., Sethian, J.A.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, vol. 3. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Tsai, Y.-H.R., Cheng, L.-T., Osher, S., Zhao, H.-K.: Fast sweeping algorithms for a class of Hamilton–Jacobi equations. SIAM J. Numer. Anal. 41(2), 673–694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Vese, L.: A method to convexify functions via curve evolution. Commun. Partial Differ. Equ. 24(9–10), 1573–1591 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhao, H.: A fast sweeping method for eikonal equations. Math. Comput. 74(250), 603–627 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montreal, Canada

    Bilal Abbasi & Adam M. Oberman

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  1. Bilal Abbasi
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  2. Adam M. Oberman
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Correspondence to Adam M. Oberman.

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Abbasi, B., Oberman, A.M. Computing the Level Set Convex Hull. J Sci Comput 75, 26–42 (2018). https://doi.org/10.1007/s10915-017-0522-8

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  • DOI: https://doi.org/10.1007/s10915-017-0522-8

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