The Topological Derivative Method and Artificial Neural Networks for Numerical Solution of Shape Inverse Problems

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(1)The Topological Derivative Method and Artificial Neural Networks for Numerical Solution of Shape Inverse Problems Lidia Jackowska-Strumillo, Jan Sokolowski, Antoni Zochowski. To cite this version: Lidia Jackowska-Strumillo, Jan Sokolowski, Antoni Zochowski. The Topological Derivative Method and Artificial Neural Networks for Numerical Solution of Shape Inverse Problems. [Research Report] RR-3739, INRIA. 1999, pp.18. �inria-00072925�. HAL Id: inria-00072925 https://hal.inria.fr/inria-00072925 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. The topological derivative method and artificial neural networks for numerical solution of shape inverse problems ˙ Lidia Jackowska-Strumiłło , Jan Sokołowski , Antoni Zochowski. No 3739 Juin 1999 ` THEME 4. ISSN 0249-6399. apport de recherche.

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(85) R Error for R=0.075. −3. x 10 6 4. R−Rsim. 2 0 −2 −4 −6 −8 1 0.8. 1 0.6. 0.8 0.6. 0.4 0.4. 0.2. 0.2 0. Y. 0. X. ®uvawyo 7 R. X Error for R=0.075. 0.08 0.06. X − Xsim. 0.04 0.02 0 −0.02 −0.04 −0.06 1 0.8. 1 0.6. 0.8 0.6. 0.4 0.4. 0.2 Y. ß)¯ Mú. "". ' . 0.2 0. 0. ®uvawyo R. X.

(86) Y Error for R=0.075. 0.08 0.06 0.04. Y−Ysim. 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 1 0.8. 1 0.6. 0.8 0.6. 0.4 0.4. 0.2. 0.2 0. Y. 0. X. ®uvawyo!!R. R Error for R=0.1. −3. x 10 8 6. R−Rsim. 4 2 0 −2 −4 −6 1 1. 0.8 0.8. 0.6 0.6 0.4 Y. 0.4 0.2. 0.2. ®uvawyo´R. X. ". . õ.

(87) X Error for R=0.1. 0.03 0.02 0.01. X−Xsim. 0 −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 1 1. 0.8 0.8. 0.6 0.6 0.4. 0.4 0.2. Y. 0.2. X. ®uvawyo´R. Y Error for R=0.1. 0.05 0.04 0.03. Y−Ysim. 0.02 0.01 0 −0.01 −0.02 −0.03 1 1. 0.8 0.8. 0.6 0.6 0.4 Y. ß)¯ Mú. "". ' . 0.4 0.2. 0.2. ®uvawyopR. X.

(88) R Error for R=0.18. −3. x 10 7 6. R−Rsim. 5 4 3 2 1 0 0.7 0.7. 0.6 0.65 0.6. 0.5. 0.55 0.5. 0.4. 0.45 0.4. 0.3. Y. 0.35. X. ®uvawyopaR. X Error for R=0.18. −3. x 10 5. X−Xsim. 0. −5. −10. −15 0.7 0.7. 0.6 0.65 0.6. 0.5. 0.55 0.5. 0.4 Y. 0.45 0.3. 0.4 0.35. . ®uvawyop R. X. ". . õ.

(89) Y Error for R=0.18. 0.015. 0.01. Y−Ysim. 0.005. 0. −0.005. −0.01 0.7 0.7. 0.6 0.65 0.6. 0.5. 0.55 0.5. 0.4 Y. 0.45 0.3. 0.4 0.35. ®uvawyopR. X. MâO,- 01 7',!v

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