Skip to main content
Springer Nature Link
Log in

Optimization in the Space of Smooth Shapes

  • Conference paper
  • First Online:
Geometric Science of Information (GSI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10589))

Included in the following conference series:

Abstract

The theory of shape optimization problems constrained by partial differential equations is connected with the differential-geometric structure of the space of smooth shapes.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)

    Book  MATH  Google Scholar 

  2. Bauer, M., Harms, P., Michor, P.M.: Sobolev metrics on shape space of surfaces. J. Geom. Mech. 3(4), 389–438 (2011)

    MATH  MathSciNet  Google Scholar 

  3. Berggren, M.: A unified discrete–continuous sensitivity analysis method for shape optimization. In: Fitzgibbon, W., Kuznetsov, Y., Neittaanmäki, P., Périaux, J., Pironneau, O. (eds.) Applied and Numerical Partial Differential Equations. Computational Methods in Applied Sciences, vol. 15, pp. 25–39. Springer, Dordrecht (2010). doi:10.1007/978-90-481-3239-3_4

    Chapter  Google Scholar 

  4. Céa, J.: Conception optimale ou identification de formes calcul rapide de la dérivée directionelle de la fonction coût. RAIRO Modelisation mathématique et analyse numérique 20(3), 371–402 (1986)

    Google Scholar 

  5. Delfour, M.C., Zolésio, J.-P.: Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. Advances in Design and Control, 2nd edn., vol. 22. SIAM (2001)

    Google Scholar 

  6. Gangl, P., Laurain, A., Meftahi, H., Sturm, K.: Shape optimization of an electric motor subject to nonlinear magnetostatics. SIAM J. Sci. Comput. 37(6), B1002–B1025 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ito, K., Kunisch, K., Peichl, G.H.: Variational approach to shape derivatives. ESAIM Control Optim. Calc. Var. 14(3), 517–539 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kriegl, A., Michor, P.W.: The Convient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. American Mathematical Society (1997)

    Google Scholar 

  9. Laurain, A., Sturm, K.: Domain expression of the shape derivative and application to electrical impedance tomography. Technical report No. 1863, Weierstraß-Institut für angewandte Analysis und Stochastik, Berlin (2013)

    Google Scholar 

  10. Michor, P.M., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005)

    MATH  MathSciNet  Google Scholar 

  11. Michor, P.M., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8(1), 1–48 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Michor, P.M., Mumford, D.: An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal. 23(1), 74–113 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mohammadi, B., Pironneau, O.: Applied Shape Optimization for Fluids. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

  14. Paganini, A.: Approximate shape gradients for interface problems. In: Pratelli, A., Leugering, G. (eds.) New Trends in Shape Optimization. ISNM, vol. 166, pp. 217–227. Springer, Cham (2015). doi:10.1007/978-3-319-17563-8_9

    Chapter  Google Scholar 

  15. Pantz, O.: Sensibilitè de l’èquation de la chaleur aux sauts de conductivitè. Comptes Rendus Mathematique de l’Académie des Sciences 341(5), 333–337 (2005)

    Google Scholar 

  16. Schmidt, S., Wadbro, E., Berggren, M.: Large-scale three-dimensional acoustic horn optimization. SIAM J. Sci. Comput. 38(6), B917–B940 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  17. Schulz, V.H., Siebenborn, M., Welker, K.: Structured inverse modeling in parabolic diffusion problems. SIAM J. Control Optim. 53(6), 3319–3338 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  18. Schulz, V.H., Siebenborn, M., Welker, K.: Efficient PDE constrained shape optimization based on Steklov-Poincaré type metrics. SIAM J. Optim. 26(4), 2800–2819 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  19. Siebenborn, M., Welker, K.: Computational aspects of multigrid methods for optimization in shape spaces. SIAM J. Sci. Comput. (2016). Submitted

    Google Scholar 

  20. Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization. Computational Mathematics, vol. 16. Springer, Heidelberg (1992)

    MATH  Google Scholar 

  21. Sturm, K.: Shape differentiability under non-linear PDE constraints. In: Pratelli, A., Leugering, G. (eds.) New Trends in Shape Optimization. ISNM, vol. 166, pp. 271–300. Springer, Cham (2015). doi:10.1007/978-3-319-17563-8_12

    Chapter  Google Scholar 

Download references

Acknowledgement

This work has been partly supported by the German Research Foundation within the priority program SPP 1962 under contract number Schu804/15-1.

Author information

Authors and Affiliations

  1. Department of Mathematics, Trier University, 54286, Trier, Germany

    Kathrin Welker

Authors
  1. Kathrin Welker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kathrin Welker .

Editor information

Editors and Affiliations

  1. Ecole Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land and Air Systems, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Welker, K. (2017). Optimization in the Space of Smooth Shapes. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-68445-1_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68444-4

  • Online ISBN: 978-3-319-68445-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics