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  4. Retrieving the Robin coefficient from single Cauchy data in elliptic systems

Retrieving the Robin coefficient from single Cauchy data in elliptic systems

MMC.
2022;
Volume 9, Number 3
: pp. 663–677
https://doi.org/10.23939/mmc2022.03.663
Received: February 16, 2022
Revised: July 04, 2022
Accepted: July 05, 2022

Mathematical Modeling and Computing, Vol. 9, No. 3, pp. 663–677 (2022)

Authors:
  1. A. El Madkouri
  2. A. Ellabib
1
Faculty of Sciences and Techniques, Department of Mathematics and Informatics LAMAI, Cadi Ayyad University, Marrakech, Morocco
2
Faculty of Sciences and Techniques, Department of Mathematics and Informatics LAMAI, Cadi Ayyad University, Marrakech, Morocco

The purpose of this work is to identify a Robin coefficient from available measurements on the accessible part of the boundary.  After recasting the inverse problem as an optimization problem, we study the issue of identifiability, stability, and identification.  For the reconstruction process, two regularized algorithms are designed, and the forward problem is approximated using the discontinuous dual reciprocity method.  The accuracy of the proposed approaches is tested in the case of noise–free and noisy data and the findings are very promising and encouraging.

boundary element method
discontinuous reciprocity approximation
Levenberg–Marquardt algorithm
quasi–Newton algorithms
Robin parameter reconstruction
  1. Chantasiriwan S.  Inverse heat conduction problem of determining time-dependent heat transfer coefficient.  International Journal of Heat and Mass Transfer. 42 (23), 4275–4285 (1999).
  2. Divo E., Kassab A. J., Kapat J. S., Chyu M.-K.  Retrieval of multi dimensional  heat  transfer  coefficient distributions using an inverse BEM-based regularized algorithm: numerical and experimental results.  Engineering Analysis with Boundary Elements. 29 (2), 150–160 (2005).
  3. Inglese G.  An inverse problem in corrosion detection.  Inverse Problems. 13 (4), 977–994 (1997).
  4. Alessandrini G., Piero L. D., Rondi L.  Stable determination of corrosion by a single electrostatic boundary measurement.  Inverse Problems. 19 (4), 973–984 (2003).
  5. Fang W., Cumberbatch E.  Inverse problems for metal oxide semiconductor field-effect transistor contact resistivity.  SIAM Journal on Applied Mathematics. 52 (3), 699–709 (1992).
  6. Beck J. V., Osman A. M.  Inverse problem for the estimation of time-and-space dependent heat transfer coefficients.  Journal of Thermophysics and Heat Transfer. 3 (2), 146–152 (1989).
  7. Kaup P. G., Santosa F.  Nondestructive evaluation of corrosion damage using electrostatic measurements.  Journal of Nondestructive Evaluation. 14 (3), 127–136 (1995).
  8. Chaabane S., Jaoua M.  Identification of Robin coefficients by the means of boundary measurements.  Inverse Problems. 15 (6), 1425–1438 (1999).
  9. Choulli M.  An inverse problem in corrosion detection: Stability estimates.  Journal of Inverse and Ill-Posed Problems. 12 (4), 349–367 (2004).
  10. Sinsich E.  Lipschitz stability for the inverse Robin problem.  Inverse Problems. 23 (3), 1311–1326 (2007).
  11. Alessandrini G., Piero L. D., Rondi L.  Stable determination of corrosion by a single electrostatic boundary measurement.  Inverse problems. 19 (4), 973–984 (2003).
  12. Chaabane S., Fellah I., Jaoua M., Leblond J.  Logarithmic stability estimates for a Robin coefficient in 2D Laplace inverse problems.  Inverse Problems. 20 (1), 47–59 (2004).
  13. Leblond J., Mahjoub M., Partington J. R.  Analytic extensions and Cauchy-type inverse problems on annular domains: stability results.  Journal of Inverse and Ill-Posed Problems. 14 (2), 189–204 (2006).
  14. Alessandrini G., Sincih E.  Detection of nonlinear corrosion by electrostatic measurements.  Applicable Analysis. 85 (1–3), 107–128 (2006).
  15. Kabanikhin S. I., Karchevsky A. L.  Optimizational method for solving the Cauchy problem for an elliptic equation.  Journal of Inverse and Ill-Posed Problems. 3 (1), 21–26 (1995).
  16. Slodiscka M., Vankeer R.  Determination of the convective transfer coefficient in elliptic problems from a nonstandard boundary condition.  In: J. Maryska, M. Tuma, J. Sembera (eds) Simulation, modelling, and numerical analysis, SIMONA 2000. 13–20 (2000).
  17. Fasino D., Inglese G.  Stability of the solutions of an inverse problem for Laplace's equation in a thin strip.  Numerical Functional Analysis and Optimization.  22 (5–6), 549–560 (2001).
  18. Ellabib A., El Madkouri A.  A Discontinuous Dual Reciprocity Method in Conjunction with a Regularized Levenberg–Marquardt Method for Source Term Recovery in Inhomogeneous Anisotropic Materials.  International Journal of Computational Methods. 17 (10), 2050002 (2020).
  19. El Madkouri A., Ellabib A.  A Preconditioned Krylov Subspace Iterative Methods for Inverse Source Problem by Virtue of a Regularizing LM-DRBEM. International Journal of Applied and Computational Mathematics. 6 (4), 94 (2020).
  20. El Madkouri A., Ellabib A.  Source term identification with discontinuous dual reciprocity approximation and quasi-Newton method from boundary observations.  Journal of Computational Mathematics. 39 (3), 311–332 (2020).
  21. Lions J. L., Magenes E.  Non-Homogeneous Boundary Value Problems and Applications.  Springer, Berlin (1972).
  22. Isakov V.  Inverse source problems. Vol. 34. American Mathematical Soc. (1990).
  23. Partridge P. W., Brebbia C. A., Wrobel L. C.  The Dual Reciprocity Boundary Element Method.  International Series on Computational Engineering.  Springer, Dordrecht (1991).
  24. Ahmedou Bamba S., Ellabib A., El Madkouri A.  Simulation of heat distribution in the human eye using discontinuous dual reciprocity boundary element method and non-overlapping domain decomposition approach.  Mathematical Modeling and Computing.  7 (1), 1–13 (2020).
  25. Ahmedou Bamba S., Ellabib A., El Madkouri A.  Numerical study of optimal control domain decomposition for nonlinear boundary heat in the human eye.  Journal of Mathematical Modeling. 8 (3), 219–240 (2020).
  26. Zhang Y., Zhu S.  On the choice of interpolation functions used in the dual-reciprocity boundary-element method.  Engineering Analysis with Boundary Elements. 13 (4), 387–396 (1994).
  27. Broyden C. G.  The convergence of a class of double-rank minimization algorithms 1. General considerations.  IMA Journal of Applied Mathematics. 6 (1), 76–90 (1970).
  28. Fletcher R.  A new approach to variable metric algorithms.  The Computer Journal. 13 (3), 317–322 (1970).
  29. Goldfarb D.  A family of variable-metric methods derived by variational means.  Mathematics of Computation. 24 (109), 23–26 (1970).
  30. Shanno D. F.  Conditioning of quasi-newton methods for function minimization.  Mathematics of Computation. 24 (111), 647–656 (1970).
  31. Hanke M.  A regularizing Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems.  Inverse problems. 13 (1), 79–95 (1997).
  32. Fletcher R.  Practical methods of optimization. John Wiley & Sons (2013).