Abstract
The new method is proposed for the numerical solution of a class of shape inverse problems. The size and the location of a small opening in the domain of integration of an elliptic equation is identified on the basis of an observation. The observation includes the finite number of shape functionals. The approximation of the shape functionals by using the so-called topological derivatives is used to perform the learning process of an artificial neural network. The results of computations for 2D examples show, that the method allows to determine an approximation of the global solution to the inverse problem, sufficiently closed to the exact solution. The proposed method can be extended to the problems with an opening of general shape and to the identification problems of small inclusions. However, the mathematical theory of the proposed approach still requires futher research. In particular, the proof of global convergence of the method is an open problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.A. Adams, Sobolev Spaces, Academic Press: New York, 1975.
A.R. Barron, “Universal approximation bounds for superpositions of a sigmoidal function,” IEEE Transactions on Information Theory, vol. 39, pp. 930–945, 1993.
D. Göhde, “Singuläre Störung von Randvertproblemen durch ein kleines Loch im Gebiet,” Zeitschrift fúr Analysis und ihre Anvendungen, vol. 4, no. 5, pp. 467–477, 1985.
M. Hagan and M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. on Neural Networks, vol. 5, no. 6, pp. 989–993, 1994.
S. Haykin, Neural Networks: A Comprehensive Foundation, 2nd edn. Prentice-Hall: New York, 1999.
Hecht-Nielsen, Neurocomputing, Addison-Wesley Publishing Company: Reading, MA, 1990.
A. Herwig, “Elliptische randvertprobleme zweiter Ordnung in Gebieten mit einer Fehlstelle,” Zeitschrift f¨ur Analysis und ihre Anvendungen, vol. 8, no. 2, pp. 153–161, 1989.
K. Hornik, M. Stinchcombe, and H. White, “Multilayer Feedforward Networks are Universal Approximators,” Neural Networks, vol. 3, pp. 551–560, 1990.
A.M. Il', “Matching of Asymptotic Expansions of Solutions of Boundary Value Problems,” Translations of Mathematical Monographs, vol. 102, AMS 1992.
T. Lewinski and J. Sokolowski, “Optimal Shells Formed on a Sphere. The Topological Derivative Method,” Rapport de Recherche No. 3495, INRIA-Lorraine, 1998.
T. Lewinski and J. Sokolowski, “Topological derivative for nucleation of non-circular voids,” in Differential Geometric Methods in the Control of Partial Differential Equations. 1999 AMS-IMS-SIAM Joint Summer Research Conference, University of Colorado, Boulder June 27-July 1, 1999, R. Gulliver, W. Littman, R. Triggiani (Eds.), Contemporary Mathematics, American Math. Soc. vol. 268, pp. 341–361, 2000.
T. Lewinski, J. Sokolowski, and A. Zochowski, “Justification of the bubble method for the compliance minimization problems of plates and spherical shells,” CD-Rom, 3rd World Congress of Structural and Multidisciplinary Optimization (WCSMO-3) Buffalo, Niagara Falls, New York, May 17-21, 1999.
W.G. Mazja, S.A. Nazarov, and B.A. Plamenevskii, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, 1, Berlin: Akademie-Verlag, 1991, 432 p. (English transl.: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, vol. 1, Basel: Birkh¨auser Verlag, 2000, 435 p.).
W.G. Mazja, S.A. Nazarov, and B.A. Plamenevski, Asymptotische Theorie elliptischer Randwertaufgaben, I, Akademie-Verlag: Berlin, 1991.
S.A. Nazarov and B.A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, De Gruyter Exposition in Mathematics 13, Walter de Gruyter, 1994.
S.A. Nazarov and J. Sokolowski, “Asymptotic analysis of shape functionals,” Les prépublications de l'Institut Élie Cartan 51, 2001.
J.R. Roche and J. Sokolowski, “Numerical methods for shape identification problems,” Special issue of Control and Cybernetics: Shape Optimization and Scientific Computations, vol. 5, pp. 867–894, 1996.
M. Schiffer and G. Szegö, “Virtual mass and polarization,” Transactions of the American Mathematical Society, vol. 67, pp. 130–205, 1949.
A. Shumacher, “Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungkriterien,” PhD Thesis, Universität-Gesamthochschule-Siegen, Siegen, 1995.
J. Sokolowski and A. Zochowski, “On topological derivative in shape optimization,” SIAM Journal on Control and Optimization, vol. 37, no. 4, pp. 1251–1272, 1999.
J. Sokolowski and A. Zochowski, “Topological derivative for optimal control problems,” Control and Cybernetics, vol. 28, no. 3, pp. 611–626, 1999.
J. Sokolowski and A. Zochowski, “Topological derivatives for elliptic problems,” Inverse Problems, vol. 15, no. 1, pp. 123–134, 1999.
J. Sokolowski and A. Zochowski, “Topological derivatives of shape functionals for elasticity systems,” Mechanics of Structures and Machines, vol. 29, pp. 333–351, 2001.
J. Sokolowski and A. Zochowski, “Optimality conditions for simultaneous topology and shape optimization,” Les prepublications de l'Institut lie Cartan 8, 2001.
J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Verlag: Berlin, 1992.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jackowska-Strumillo, L., Sokolowski, J., Żochowski, A. et al. On Numerical Solution of Shape Inverse Problems. Computational Optimization and Applications 23, 231–255 (2002). https://doi.org/10.1023/A:1020528902875
Issue Date:
DOI: https://doi.org/10.1023/A:1020528902875