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Second order directional shape derivatives of integrals on submanifolds
Universität Bayreuth, 95440 Bayreuth, Germany |
We compute first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. The result is a quadratic form in terms of one perturbation vector field that yields a second order quadratic model of the perturbed functional. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms.
References:
| [1] |
D. Bucur and J.-P. Zolésio,
Anatomy of the shape Hessian via Lie brackets, Ann. Mat. Pura Appl., 173 (1997), 127-143.
doi: 10.1007/BF01783465. |
| [2] |
M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 22 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2011.
doi: 10.1137/1.9780898719826. |
| [3] |
F. R. Desaint and J.-P. Zolésio,
Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151 (1997), 234-269.
doi: 10.1006/jfan.1997.3130. |
| [4] |
A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, 48 of Mathématiques & Applications (Berlin), Springer, Berlin, 2005.
doi: 10.1007/3-540-37689-5. |
| [5] |
M. Hintermüller and W. Ring,
A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., 64 (2003/04), 442-467.
doi: 10.1137/S0036139902403901. |
| [6] |
R. Hiptmair and J. Li,
Shape derivatives in differential forms I: An intrinsic perspective, Ann. Mat. Pura Appl., 192 (2013), 1077-1098.
doi: 10.1007/s10231-012-0259-9. |
| [7] |
S. Lang, Fundamentals of Differential Geometry, 191 of Graduate Texts in Mathematics., Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0541-8. |
| [8] |
F. Murat and J. Simon,
Etudes de problems d'optimal design, Lectures Notes in Computer Science, 41 (1976), 54-62.
|
| [9] |
A. Novruzi and M. Pierre,
Structure of shape derivatives, J. Evol. Equ., 2 (2002), 365-382.
doi: 10.1007/s00028-002-8093-y. |
| [10] |
V. H. Schulz,
A Riemannian view on shape optimization, Found. Comput. Math., 14 (2014), 483-501.
doi: 10.1007/s10208-014-9200-5. |
| [11] |
J. Simon, Second variation for domain optimization problems. control and estimation of distributed parameter systems, International Series of Numerical Mathematics, 91, Birkhäuser, Basel, 1989,361–378. |
| [12] |
J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Shape Sensitivity Analysis, 16 of Springer Series in Computational Mathematics., Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58106-9. |
| [13] |
M. Spivak, A comprehensive Introduction to Differential Geometry. Vol. III, Publish or Perish, Inc., Wilmington, Del., second edition, 1979. |
| [14] |
K. Sturm, Convergence of Newton's method in shape optimisation via approximate normal functions, Technical Report arXiv: 1608.02699, arXiv, 2016. |
| [15] |
K. Sturm,
A structure theorem for shape functions defined on submanifolds, Interfaces Free Bound., 18 (2016), 523-543.
doi: 10.4171/IFB/372. |
| [16] |
L. Younes, Shapes and Diffeomorphisms, 171 of Applied Mathematical Sciences., Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-12055-8. |
| [17] |
J.-P. Zolésio, Un résultat d'existence de vitesse convergente dans des problèmes d'identification de domaine, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), Aiii, A855–A858. |
| [18] |
J.-P. Zolésio, The material derivative (or speed) method for shape optimization, In Optimization of distributed parameter structures, Vol. II (Iowa City, Iowa, 1980), 50 of NATO Adv. Study Inst. Ser. E: Appl. Sci., Nijhoff, The Hague, 1981, 1089–1151. |
show all references
References:
| [1] |
D. Bucur and J.-P. Zolésio,
Anatomy of the shape Hessian via Lie brackets, Ann. Mat. Pura Appl., 173 (1997), 127-143.
doi: 10.1007/BF01783465. |
| [2] |
M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 22 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2011.
doi: 10.1137/1.9780898719826. |
| [3] |
F. R. Desaint and J.-P. Zolésio,
Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151 (1997), 234-269.
doi: 10.1006/jfan.1997.3130. |
| [4] |
A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, 48 of Mathématiques & Applications (Berlin), Springer, Berlin, 2005.
doi: 10.1007/3-540-37689-5. |
| [5] |
M. Hintermüller and W. Ring,
A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., 64 (2003/04), 442-467.
doi: 10.1137/S0036139902403901. |
| [6] |
R. Hiptmair and J. Li,
Shape derivatives in differential forms I: An intrinsic perspective, Ann. Mat. Pura Appl., 192 (2013), 1077-1098.
doi: 10.1007/s10231-012-0259-9. |
| [7] |
S. Lang, Fundamentals of Differential Geometry, 191 of Graduate Texts in Mathematics., Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0541-8. |
| [8] |
F. Murat and J. Simon,
Etudes de problems d'optimal design, Lectures Notes in Computer Science, 41 (1976), 54-62.
|
| [9] |
A. Novruzi and M. Pierre,
Structure of shape derivatives, J. Evol. Equ., 2 (2002), 365-382.
doi: 10.1007/s00028-002-8093-y. |
| [10] |
V. H. Schulz,
A Riemannian view on shape optimization, Found. Comput. Math., 14 (2014), 483-501.
doi: 10.1007/s10208-014-9200-5. |
| [11] |
J. Simon, Second variation for domain optimization problems. control and estimation of distributed parameter systems, International Series of Numerical Mathematics, 91, Birkhäuser, Basel, 1989,361–378. |
| [12] |
J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Shape Sensitivity Analysis, 16 of Springer Series in Computational Mathematics., Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58106-9. |
| [13] |
M. Spivak, A comprehensive Introduction to Differential Geometry. Vol. III, Publish or Perish, Inc., Wilmington, Del., second edition, 1979. |
| [14] |
K. Sturm, Convergence of Newton's method in shape optimisation via approximate normal functions, Technical Report arXiv: 1608.02699, arXiv, 2016. |
| [15] |
K. Sturm,
A structure theorem for shape functions defined on submanifolds, Interfaces Free Bound., 18 (2016), 523-543.
doi: 10.4171/IFB/372. |
| [16] |
L. Younes, Shapes and Diffeomorphisms, 171 of Applied Mathematical Sciences., Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-12055-8. |
| [17] |
J.-P. Zolésio, Un résultat d'existence de vitesse convergente dans des problèmes d'identification de domaine, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), Aiii, A855–A858. |
| [18] |
J.-P. Zolésio, The material derivative (or speed) method for shape optimization, In Optimization of distributed parameter structures, Vol. II (Iowa City, Iowa, 1980), 50 of NATO Adv. Study Inst. Ser. E: Appl. Sci., Nijhoff, The Hague, 1981, 1089–1151. |
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