PDE Constrained Shape Optimization as Optimization on Shape Manifolds

  • Volker H. Schulz
  • Martin Siebenborn
  • Kathrin WelkerEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)


The novel Riemannian view on shape optimization introduced in [14] is extended to a Lagrange–Newton as well as a quasi–Newton approach for PDE constrained shape optimization problems.


Shape optimization Riemannian manifold Newton method Quasi–Newton method Limited memory BFGS 


  1. 1.
    Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arian, E., Vatsa, V.N.: A preconditioning method for shape optimization governed by the Euler equations. Technical report, Institute for Computer Applications in Science and Engineering (ICASE), pp. 98–14 (1998)Google Scholar
  3. 3.
    Ameur, H.B., Burger, M., Hackl, B.: Level set methods for geometric inverse problems in linear elasticity. Inverse Prob. 20, 673–696 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Correa, R., Seeger, A.: Directional derivative of a minmax function. Nonlinear Anal. 9(1), 13–22 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Delfour, M.C., Zolésio, J.-P.: Shapes and Geometries: Analysis, Differential Calculus, and Optimization. Advances in Design and Control. SIAM Philadelphia, Philadelphia (2001)zbMATHGoogle Scholar
  6. 6.
    Gabay, D.: Minimizing a differentiable function over a differential manifold. J. Optim. Theor. Appl. 37(2), 177–219 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eppler, K., Harbrecht, H.: A regularized Newton method in electrical impedance tomography using shape Hessian information. Control Cybern. 34(1), 203–225 (2005)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Eppler, K., Schmidt, S., Schulz, V.H., Ilic, C.: Preconditioning the pressure tracking in fluid dynamics by shape Hessian information. J. Optim. Theory Appl. 141(3), 513–531 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8, 1–48 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mohammadi, B., Pironneau, O.: Applied Shape Optimization for Fluids, Numerical Mathematics and Scientific Computation. Clarendon Press, Oxford (2001)zbMATHGoogle Scholar
  11. 11.
    Nägel, A., Schulz, V.H., Siebenborn, M., Wittum, G.: Scalable shape optimization methods for structured inverse modeling in 3D diffusive processes. Comput. Vis. Sci. (2015). doi 10.1007/s00791-015-0248-9
  12. 12.
    Novruzi, A., Roche, J.R.: Newton’s method in shape optimisation: a three-dimensional case. BIT Numer. Math. 40, 102–120 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ring, W., Wirth, B.: Optimization methods on Riemannian manifolds and their application to shape space. SIAM J. Optim. 22, 596–627 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schulz, V.H.: A Riemannian view on shape optimization. Found. Comput. Math. 14, 483–501 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Schulz, V.H., Siebenborn, M., Welker, K.: Towards a Lagrange-Newton approach for PDE constrained shape optimization. In: Leugering, G., et al., (eds.) New Trends in Shape Optimization. International Series of Numerical Mathematics, vol. 166. Springer (2015). doi: 10.1007/978-3-319-17563-8
  16. 16.
    Schulz, V.H., Siebenborn, M., Welker, K.: Structured inverse modeling in parabolic diffusion problems. SICON (2014, submitted to). arXiv:1409.3464
  17. 17.
    Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, vol. 16. Springer, Heidelberg (1992)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Volker H. Schulz
    • 1
  • Martin Siebenborn
    • 1
  • Kathrin Welker
    • 1
    Email author
  1. 1.Department of MathematicsUniversity of TrierTrierGermany

Personalised recommendations