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401-3640-13L  Seminar in Applied Mathematics: Shape Calculus

SemesterHerbstsemester 2018
DozierendeR. Hiptmair
Periodizitäteinmalige Veranstaltung
KommentarNumber of participants limited to 10

KurzbeschreibungShape calculus studies the dependence of solutions of partial differential equations on deformations of the domain and/or interfaces. It is the foundation of gradient methods for shape optimization. The seminar will rely on several sections of monographs and research papers covering analytical and numerical aspects of shape calculus.
Lernziel* Understanding of concepts like shape derivative, shape gradient, shape Hessian, and adjoint problem.
* Ability to derive analytical formulas for shape gradients
* Knowledge about numerical methods for the computation of shape gradients.

1. The velocity method and Eulerian shape gradients: Main reference [SZ92, Sect. 2.8–2.11, 2.1, 2.18], covers the “velocity method”, the Hadamard structure theorem and formulas for shape gradients of particular functionals. Several other sections of [SZ92,Ch. 2] provide foundations and auxiliary results and should be browsed, too.

2. Material derivatives and shape derivatives, based on [SZ92, Sect. 2.25–2.32].

3. Shape calculus with exterior calculus, following [HL13] (without Sections 5 & 6). Based on classical vector analysis the formulas are also derived in [SZ92, Sects 2,19,2.20] and [DZ10, Ch. 9, Sect. 5]. Important background and supplementary information about the shape Hessian can be found in [DZ91, BZ97] and [DZ10, Ch. 9, Sect. 6].

4. Shape derivatives of solutions of PDEs using exterior calculus [HL17], see also [HL13,Sects. 5 & 6]. From the perspective of classical calculus the topic is partly covered in [SZ92, Sects. 3.1-3.2].

5. Shape gradients under PDE constraints according to [Pag16, Sect. 2.1] including a presentation of the adjoint method for differentiating constrained functionals [HPUU09, Sect. 1.6]. Related information can be found in [DZ10, Ch. 10, Sect. 2.5] and [SZ92, Sect. 3.3].

6. Approximation of shape gradients following [HPS14]. Comparison of discrete shape gradients based on volume and boundary formulas, see also [DZ10, Ch. 10, Sect. 2.5].

7. Optimal shape design based on boundary integral equations following [Epp00b], with some additional information provided in [Epp00a].

8. Convergence in elliptic shape optimization as discussed in [EHS07]. Relies on results reported in [Epp00b] and [DP00]. Discusses Ritz-Galerkin discretization of optimality conditions for normal displacement parameterization.

9. Shape optimization by pursuing diffeomorphisms according to [HP15], see also [Pag16,Ch. 3] for more details, and [PWF17] for extensions.

10. Distributed shape derivative via averaged adjoint method following [LS16].

[BZ97] Dorin Bucur and Jean-Paul Zolsio. Anatomy of the shape hessian via
lie brackets. Annali di Matematica Pura ed Applicata, 173:127–143, 1997.

[DP00] Marc Dambrine and Michel Pierre. About stability of equilibrium shapes. M2AN Math. Model. Numer. Anal., 34(4):811–834, 2000.

[DZ91] Michel C. Delfour and Jean-Paul Zolésio. Velocity method and Lagrangian formulation for the computation of the shape Hessian. SIAM J. Control Optim.,
29(6):1414–1442, 1991.

[DZ10] M.C. Delfour and J.-P. Zolésio. Shapes and Geometries, volume 22 of Advances in Design and Control. SIAM, Philadelphia, 2nd edition, 2010.

[EHS07] Karsten Eppler, Helmut Harbrecht, and Reinhold Schneider. On convergence in elliptic shape optimization. SIAM J. Control Optim., 46(1):61–83 2007.

[Epp00a] Karsten Eppler. Boundary integral representations of second derivatives in shape optimization. Discuss. Math. Differ. Incl. Control Optim., 20(1):63–78, 2000.
German-Polish Conference on Optimization—Methods and Applications (Żagań,

[Epp00b] Karsten Eppler. Optimal shape design for elliptic equations via BIE-methods. Int. J. Appl. Math. Comput. Sci., 10(3):487–516, 2000.

[HL13] Ralf Hiptmair and Jingzhi Li. Shape derivatives in differential forms I: an intrinsic perspective. Ann. Mat. Pura Appl. (4), 192(6):1077–1098, 2013.

[HL17] R. Hiptmair and J.-Z. Li. Shape derivatives in differential forms II: Application
to scattering problems. Report 2017-24, SAM, ETH Zürich, 2017. To appear in
Inverse Problems.

[HP15] Ralf Hiptmair and Alberto Paganini. Shape optimization by pursuing diffeomorphisms. Comput. Methods Appl. Math., 15(3):291–305, 2015.

[HPS14] R. Hiptmair, A. Paganini, and S. Sargheini. Comparison of approximate shape gradients. BIT Numerical Mathematics, 55:459–485, 2014.

[HPUU09] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE constraints, volume 23 of Mathematical Modelling: Theory and Applications. Springer, New York, 2009.

[LS16] Antoine Laurain and Kevin Sturm. Distributed shape derivative via averaged adjoint method and applications. ESAIM Math. Model. Numer. Anal., 50(4):1241–1267,2016.

[Pag16] A. Paganini. Numerical shape optimization with finite elements. Eth dissertation 23212, ETH Zurich, 2016.

[PWF17] A. Paganini, F. Wechsung, and P.E. Farell. Higher-order moving mesh methods for pde-constrained shape optimization. Preprint arXiv:1706.03117 [math.NA], arXiv, 2017.

[SZ92] J. Sokolowski and J.-P. Zolesio. Introduction to shape optimization, volume 16 of Springer Series in Computational Mathematics. Springer, Berlin, 1992.
Voraussetzungen / BesonderesKnowledge of analysis and functional analysis; knowledge of PDEs is an advantage and so is some familiarity with numerical methods for PDEs