zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Incorporating topological derivatives into level set methods. (English) Zbl 1044.65053
From the abstract: The aim of this paper is to investigate the use of topological derivatives in combination with the level set method for shape reconstruction and optimization problems. We propose a new approach generalizing the standard speed method, which is obtained by using a source term in the level set equation that depends on the topological derivative of the objective functional. The resulting approach can be interpreted as a generalized fixed-point iteration for the optimality system (with respect to topological and shape variations). Moreover, we apply the new approach for a simple model problem in shape reconstruction, where the topological derivative can be computed without additional effort. Finally, we present numerical tests related to this model problem, which demonstrate that the new method based on shape and topological derivative successfully reconstructs obstacles in situations where the standard level set approach fails.

65K10Optimization techniques (numerical methods)
49Q12Sensitivity analysis
65D99Numerical approximation
Full Text: DOI
[1] Adams, R. A.: Sobolev spaces. (1975) · Zbl 0314.46030
[2] Allaire, G.: Shape optimization by the homogenization method. (2002) · Zbl 0990.35001
[3] Allaire, G.; Jouve, F.; Toader, A. M.: A level-set method for shape optimization. CR acad. Sci. Paris ser. I 334, 1125-1130 (2002) · Zbl 1115.49306
[4] G. Allaire, F. Jouve, A.M. Toader, Structural optimization using sensitivity analysis and a level-set method, Preprint (CMAP École Polytechnique, Paris, 2003)
[5] H. Benameur, M. Burger, B. Hackl, Level set methods for geometric inverse problems in linear elasticity, Preprint, 2003
[6] Bendsøe, M. P.; Sigmund, O.: Topology optimization. (2002) · Zbl 1034.74041
[7] Bourdin, B.; Chambolle, A.: Design-dependent loads in topology optimization. ESAIM control optim. Calc. var. 9, 19-48 (2003) · Zbl 1066.49029
[8] Burger, M.: A level set method for inverse problems. Inverse problems 17, 1327-1356 (2001) · Zbl 0985.35106
[9] Burger, M.: A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces and free boundaries 5, 301-329 (2003) · Zbl 1081.35134
[10] Delfour, M. C.; Zolésio, J. P.: Shapes and geometries. Analysis, differential calculus, and optimization. (2001) · Zbl 1002.49029
[11] Dorn, O.; Miller, E. M.; Rappaport, C. M.: A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Inverse problems 16, 1119-1156 (2000) · Zbl 0983.35150
[12] O. Dorn, Shape reconstruction in 2D from limited-view multifrequency electromagnetic data, Preprint, 2000 · Zbl 0988.78012
[13] Garreau, S.; Guillaume, P.; Masmoudi, M.: The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control optim. 39, 1756-1778 (2001) · Zbl 0990.49028
[14] Giaquinta, M.: Introduction to regularity theory for nonlinear elliptic systems. (1993) · Zbl 0786.35001
[15] Guillaume, P.; Idris, K. Sid: The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control optim. 41, 1042-1072 (2002) · Zbl 1053.49031
[16] Hettlich, F.; Rundell, W.: Iterative methods for the reconstruction of an inverse potential problem. Inverse problems 12, 251-266 (1996) · Zbl 0858.35134
[17] M. Hintermüller, W. Ring, A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., in press · Zbl 1073.68095
[18] M. Hintermüller, W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional, J. Math. Imag. Vision 20 (2003) in press
[19] Ito, K.; Kunisch, K.; Li, Z.: Level-set function approach to an inverse interface problem. Inverse problems 17, 1225-1242 (2001) · Zbl 0986.35130
[20] Jiang, G. S.; Peng, D.: Weighted ENO-schemes for Hamilton--Jacobi equations. SIAM J. Sci. comput. 21, 2126-2143 (2000) · Zbl 0957.35014
[21] Litman, A.; Lesselier, D.; Santosa, F.: Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set. Inverse problems 14, 685-706 (1998) · Zbl 0912.35158
[22] Osher, S.; Santosa, F.: Level set methods for optimization problems involving geometry and constraints. I. frequencies of a two-density inhomogeneous drum. J. comput. Phys. 171, 272-288 (2001) · Zbl 1056.74061
[23] Osher, S.; Sethian, J. A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton--Jacobi formulations. J. comput. Phys. 79, 12-49 (1988) · Zbl 0659.65132
[24] Osher, S.; Fedkiw, R. P.: The level set method and dynamic implicit surfaces. (2002) · Zbl 1026.76001
[25] Santosa, F.: A level-set approach for inverse problems involving obstacles. ESAIM: control, optimisation calculus variations 1, 17-33 (1996) · Zbl 0870.49016
[26] A. Schumacher, Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungskriterien, PhD Thesis, Universität-Gesamthochschule-Siegen, 1995
[27] Sokolowski, J.; .Zochowski, A.: On the topological derivative in shape optimization. SIAM J. Control optim. 37, 1251-1272 (1999)
[28] Sokolowski, J.; .Zochowski, A.: Topological derivatives for elliptic problems. Inverse problems 15, 123-134 (1999)
[29] Sokolowski, J.; Zolesio, J. P.: Introduction to shape optimization. (1992) · Zbl 0487.49004
[30] Stolarska, M.; Chopp, D. L.; Moes, N.; Belytschko, T.: Modelling crack growth by level sets in the extended finite element method. Int. J. Numer. meth. Eng. 51, 943-960 (2001) · Zbl 1022.74049
[31] J.P. Zolesio, The material derivative (or speed) method for shape optimization, in: Optimization of Distributed Parameter Structures, vol. II, NATO Adv. Study Inst. Ser. E, Appl. Sci. 50 (1981) 1089--1151