Optimization method based on bio-inspired approach for solving a robin inverse problem
Resumo
In this paper, we are interested in solving an inverse problem of reconstructing an unknown Robin coefficient. It consists to identify the Robin parameter from some additional measurement in the accessible part of the boundary. This problem is knowing as an ill-posed problem in the Hadamard sense and require regularisation methods to be solved. To this end, we formulate the inverse problem into an optimization problem, where we consider the given data as a control, in addition to a regularizing term known as Thikhonov regularization. Then, we investigate the particle swarm optimization as a stochastic approach, in combination with a finite element method (FEM) to solve the optimization problem. The numerical results indicate that our approach yields accurate, convergent, and stable solutions for the considered Robin inverse problem in irregular 2D domain.
Downloads
Referências
Chantasiriwan, S. Inverse heat conduction problem of determining time-dependent heat transfer coefficient. International Journal Of Heat And Mass Transfer 42, 4275-4285, (1999).
Divo, E., Kassab, A., Kapat, J. & Chyu, M. Retrieval of multidimensional heat transfer coefficient distributions using an inverse BEM-based regularized algorithm: numerical and experimental results. Engineering Analysis With Boundary Elements 29, 150-160 (2005).
Inglese, G. An inverse problem in corrosion detection. Inverse Problems 13, 977-995, (1997).
Alessandrini, G., Del Piero, L., Rondi, L. & Others Stable determination of corrosion by a single electrostatic boundary measurement. Inverse Problems 19, 973-984 (2003).
Fang, W. & Cumberbatch, E. Inverse problems for metal oxide semiconductor field-effect transistor contact resistivity. SIAM Journal On Applied Mathematics 52, 699-709, (1992).
Beck, J. & Osman, A. Nonlinear inverse problem for the estimation of time-and-space-dependent heat-transfer coefficients. Journal Of Thermophysics And Heat Transfer 3, 146-152, (1989).
Chaabane, S. & Jaoua, M. Identification of Robin coefficients by the means of boundary measurements. Inverse Problems 15 1425, (1999).
Chaabane, S., Ferchichi, J. & Kunisch, K. Differentiability properties of the L1-tracking functional and application to the Robin inverse problem. Inverse Problems 20, 1083, (2004).
Alessandrini, G., Del Piero, L., Rondi, L. & Others Stable determination of corrosion by a single electrostatic boundary measurement. Inverse Problems 19, 973-984, (2003).
Hadamard, J. & Morse, P. Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Physics Today 6(8), 18 (1953).
Beck, J. & Murio, D. Combined function specification-regularization procedure for solution of inverse heat conduction problem. AIAA Journal 24, 180-185, (1986).
Avdonin, S., Kozlov, V., Maxwell, D. & Truffer, M. Iterative methods for solving a nonlinear boundary inverse problem in glaciology. Journal of Inverse and Ill-posed Problems 17(3), 239-258, (2009).
Alifanov, O. Solution of an inverse problem of heat conduction by iteration methods. Journal Of Engineering Physics. 26, 471-476, (1974).
Tikhonov, A. & Arsenin, V. Solutions of Ill-posed Problems: Andrey N. Tikhonov and Vasiliy Y. Arsenin. Translation Editor Fritz John, V. H. Winston and Sons (1977).
Hansen, P. Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank. SIAM Journal On Scientific And Statistical Computing 11, 503-518, (1990).
Fasino, D. & Inglese, G. An inverse Robin problem for Laplace’s equation: theoretical results and numerical methods. Inverse Problems 15, 41, (1999).
Chaabane, S., Elhechmi, C. & Jaoua, M. A stable recovery method for the Robin inverse problem. Mathematics And Computers In Simulation 66, 367-383, (2004).
Fang, W. & Lu, M. A fast collocation method for an inverse boundary value problem. International Journal For Numerical Methods In Engineering 59, 1563-1585, (2004).
Tajani, C., Jouilik, B. & Abouchabaka, J. Numerical identification of Robin coefficient by iterative method. Palestine Journal Of Mathematics 7, 64-72, (2018).
Fasino, D. & Inglese, G. Discrete methods in the study of an inverse problem for Laplace’s equation. IMA Journal of Numerical Analysis 19, 105-118, (1999).
Jin, B. Conjugate gradient method for the Robin inverse problem associated with the Laplace equation. International Journal For Numerical Methods In Engineering 71, 433-453, (2007).
Jouilik, B., Daoudi, J., Tajani, C. & Abouchabaka, J. Optimization approach based on genetic algorithm for a Robin coefficient problem. Palestine Journal Of Mathematics 11(3), 708–718, (2022).
Chaabane, S., Ferchichi, J. & Kunisch, K. Differentiability properties of the L1-tracking functional and application to the Robin inverse problem. Inverse Problems 20, 1083, (2004)
Chaabane, S., Elhechmi, C. & Jaoua, M. A stable recovery method for the Robin inverse problem. Mathematics And Computers In Simulation 66, 367-383, (2004).
Karaboga, D., Gorkemli, B., Ozturk, C. & Karaboga, N. A comprehensive survey: artificial bee colony (ABC) algorithm and applications. Artificial Intelligence Review 42, 21-57, (2014).
Socha, K. & Dorigo, M. Ant colony optimization for continuous domains. European Journal Of Operational Research 185, 1155-1173, (2008).
Kennedy, J. & Eberhart, R. Particle swarm optimization. Proceedings Of ICNN’95-international Conference On Neural Networks 4, 1942-1948, (1995).
Yang, X. & Hossein Gandomi, A. Bat algorithm: a novel approach for global engineering optimization. Engineering Computations 29, 464-483, (2012).
De Jong, K. Learning with genetic algorithms: An overview. Machine Learning 3, 121-138, (1988).
Vogel, C. Non-convergence of the L-curve regularization parameter selection method. Inverse Problems 12, 535, (1996).
Engl, H. Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Journal Of Optimization Theory And Applications 52, 209-215, (1987).
Jin, B., & Zou, J. Numerical estimation of piecewise constant Robin coefficient. SIAM journal on control and optimization 48, 1977-2002, (2009).
Jin, B., & Zou, J. Numerical estimation of the Robin coefficient in a stationary diffusion equation. IMA Journal of Numerical Analysis 3, 677-701, (2010).
Garcıia-Gonzalo, E. & Fern´andez-Martıinez, J. Convergence and stochastic stability analysis of particle swarm optimization variants with generic parameter distributions. Applied Mathematics And Computation 249, 286-302, (2014).
Pedersen, M. Good parameters for particle swarm optimization. Hvass Lab., Copenhagen, Denmark, Tech. Rep. HL1001, 1551-3203, (2010).
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
