Abstract
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.
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References
Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)
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)
Ameur, H.B., Burger, M., Hackl, B.: Level set methods for geometric inverse problems in linear elasticity. Inverse Prob. 20, 673–696 (2004)
Correa, R., Seeger, A.: Directional derivative of a minmax function. Nonlinear Anal. 9(1), 13–22 (1985)
Delfour, M.C., Zolésio, J.-P.: Shapes and Geometries: Analysis, Differential Calculus, and Optimization. Advances in Design and Control. SIAM Philadelphia, Philadelphia (2001)
Gabay, D.: Minimizing a differentiable function over a differential manifold. J. Optim. Theor. Appl. 37(2), 177–219 (1982)
Eppler, K., Harbrecht, H.: A regularized Newton method in electrical impedance tomography using shape Hessian information. Control Cybern. 34(1), 203–225 (2005)
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)
Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8, 1–48 (2006)
Mohammadi, B., Pironneau, O.: Applied Shape Optimization for Fluids, Numerical Mathematics and Scientific Computation. Clarendon Press, Oxford (2001)
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). doi10.1007/s00791-015-0248-9
Novruzi, A., Roche, J.R.: Newton’s method in shape optimisation: a three-dimensional case. BIT Numer. Math. 40, 102–120 (2000)
Ring, W., Wirth, B.: Optimization methods on Riemannian manifolds and their application to shape space. SIAM J. Optim. 22, 596–627 (2012)
Schulz, V.H.: A Riemannian view on shape optimization. Found. Comput. Math. 14, 483–501 (2014)
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
Schulz, V.H., Siebenborn, M., Welker, K.: Structured inverse modeling in parabolic diffusion problems. SICON (2014, submitted to). arXiv:1409.3464
Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, vol. 16. Springer, Heidelberg (1992)
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Schulz, V.H., Siebenborn, M., Welker, K. (2015). PDE Constrained Shape Optimization as Optimization on Shape Manifolds. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_54
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