bdbaa578-d02c-46d6-b4c6-ab7eb647658620231114090954842wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051919202291920222210.37394/23206.2023.22https://wseas.com/journals/mathematics/2023.phpJamalDaoudiSMAD Team, Department of Mathematics Polydisciplinary Faculty of Larache Abdelmalek Essaadi University MOROCCOChakirTajaniSMAD Team, Department of Mathematics Polydisciplinary Faculty of Larache Abdelmalek Essaadi University MOROCCOIn this paper, we are interested in solving the data completion problem for the Laplace equation. It consists to determine the missing data on the inaccessible part of the boundary from overspecified conditions in the accessible part. Knowing that this problem is severely ill-posed, we consider its formulation as an optimization problem using Tikhonov regularization. Then, we consider an optimization approach based on adapted Real Coded Genetic Algorithm (RCGA) to minimize the cost function and recover the missing data. The performed numerical simulations, with different domains, illustrate the accuracy and efficiency of the proposed method with an adequate regularization parameter, in addition to the good agreement between the numerical solutions and different noise level of the given data.111420231114202384285392https://wseas.com/journals/mathematics/2023/b865106-048(2023).pdf10.37394/23206.2023.22.92https://wseas.com/journals/mathematics/2023/b865106-048(2023).pdf10.5772/intechopen.106479Martinez-Luaces, V., Fernández-Plaza, J. & Rico, L., About the Notion of Inverse Problem in STEM Education, Active Learning-Research and Practice for STEAM and Social Sciences Education, IntechOpen, 2022. 10.1109/51.715491Gulrajani, R., The forward and inverse problems of electrocardiography, IEEE Engineering in Medicine and Biology Magazine, Vol.17, No.5, 1998, pp. 84-101. 10.1515/156939406777571012El Badia, A. & Farah, M., Identification of dipole sources in an elliptic equation from boundary measurements: application to the inverse EEG problem, Journal of Inverse & Ill-Posed Problems, Vol.14, No.4, 2006, pp. 331-353. Friedman, A. & Vogelius, M., Determining cracks by boundary measurements, Indiana University Mathematics Journal, Vol.38, No.2, 1989, pp.497-525. 10.1016/j.matcom.2008.11.019Yang, F., Yan, L. & Wei, T., Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method, Mathematics and Computers in Simulation, Vol.79, No.7, 2009, pp. 2148-2156. Hadamard, J., Lectures on Cauchy’s problem in linear partial differential equations, Yale university press, 1923. 10.2514/3.9240Beck, J. & Murio, D., Combined function specification-regularization procedure for solution of inverse heat conduction problem, AIAA Journal, Vol.24, No.1, 1986, pp. 180-185. 10.1515/jiip.2009.018Avdonin, 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, Vol.17, No.4, 2009, pp. 239-258. Tajani, C. & Abouchabaka, J., On the data completion problem for Laplace’s equation, Annals of the University of Craiova-Mathematics and Computer Science Series, Vol.45, No.1, 2018, pp. 11-36. A. N. Tikhonov, V. J. Arsenin, V., Solutions of ill-posed problems, Vh Winston, 1977. 10.1007/s10462-012-9328-0Karaboga, D., Gorkemli, B., Ozturk, C. & Karaboga, N., A comprehensive survey: artificial bee colony (ABC) algorithm and applications, Artificial Intelligence Review, Vol.42, 2014, pp. 21-57. 10.1109/icnn.1995.488968Kennedy, J. & Eberhart, R., Particle swarm optimization, Proceedings Of ICNN’95-international Conference On Neural Networks, Vol.4, 1995, pp. 1942-1948. 10.1016/j.ejor.2006.06.046Socha, K. & Dorigo, M., Ant colony optimization for continuous domains, European Journal of Operational Research, Vol.185, No.3, 2008, pp. 1155-1173. 10.1108/02644401211235834Yang, X. & Hossein Gandomi, A., Bat algorithm: a novel approach for global engineering optimization, Engineering Computations, Vol.29, No.5, 2012, pp. 464-483. Tominaga D., Koga N. & Okamoto M., Efficient numerical optimization algorithm based on genetic algorithm for inverse problem, Pro-ceedings of the 2nd Annual Conference on Genetic and Evolutionary Computation, 2000, pp. 251-258. 10.1061/(asce)0733-9496(2005)131:1(45)Mahinthakumar, G. & Sayeed, M., Hybrid genetic algorithm—local search methods for solving groundwater source identification inverse problems, Journal of Water Resources Planning and Management. Vol.131, No.1, 2005, pp. 45-57. 10.1080/10407799508928835Raudenskỳ, M., Woodbury, K., Kral, J. & Brezina, T., Genetic algorithm in solution of inverse heat conduction problems, Numerical Heat Transfer, Part B Fundamentals, Vol.28, No.3, 1995, pp. 293-306. 10.1007/s00466-003-0526-0Mera, N., Elliott, L. & Ingham, D., A multi-population genetic algorithm approach for solving ill-posed problems, Computational Mechanics, Vol.33, No.4, 2004, pp. 254-262. 10.1016/s0955-7997(03)00087-0Mera, N., Elliott, L. & Ingham, D. (2004). Numerical solution of a boundary detection problem using genetic algorithms, Engineering Analysis with Boundary Elements, Vol.28, No.4, 2004, pp. 405-411. Jouilik, B., Daoudi, J., Tajani, C. & Abouchabaka, J., Optimization approach based on genetic algorithm for a robin coefficient problem, Palestine Journal of Mathematics, Vol.11, No.3, 2022, pp. 708-718. 10.1016/j.neucom.2005.05.006Wang, X., He, Q., Chen, D. & Yeung, D., A genetic algorithm for solving the inverse problem of support vector machines, Neurocomputing, Vol.68, 2005, pp. 225-238. 10.1109/iembs.2006.259672Jiang, M., Xia, L. & Shou, G., The use of genetic algorithms for solving the inverse problem of electrocardiography, 2006 International Conference of the IEEE Engineering in Medicine and Biology Society, 2006, pp. 3907-3910. Mera, N., Elliott, L. & Ingham, D., Comparison of a genetic algorithm and a gradient based optimisation technique for the detection of subsurface inclusions, Acta Universitatis Apulensis. Mathematics-Informatics, Vol.4, 2002, pp. 123-134. 10.1109/cec.2004.1330888Suram, S., Bryden, K. & Ashlock, D., An evolutionary algorithm to estimate unknown heat flux in a one-dimensional inverse heat conduction problem, Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Vol.11, 2005, pp. 15. 10.2514/6.1995-1650Yamamoto, K. & Inoue, O., Applications of genetic algorithm to aerodynamic shape optimization, 12th Computational Fluid Dynamics Conference, 1995, pp. 1650. Jouilik, B., Daoudi, J., Tajani, C. & Abouchabaka, J., Numerical optimization algorithm based on genetic algorithm for a data completion problem, TWMS Journal of Applied and Engineering Mathematics, Vol.13, No.1, 2023, pp. 86-97. 10.1007/s10462-017-9605-zHussain, K., Mohd Salleh, M., Cheng, S. & Shi, Y., Metaheuristic research: a comprehensive survey, Artificial Intelligence Review, Vol.52, 2019, pp. 2191-2233. 10.2307/1551672Holland, J., Genetic algorithms, Scientific American (June 1992).[12] Intergovernmental Panel on Climate Change, Climate Change: the IPCC Assessment, eds. JT Houghton, GJ Jenkins and JJ Ephraums, Cambridge University Press, 1990. 10.1016/b978-0-08-050684-5.50016-1Wright, A., Genetic algorithms for real parameter optimization, Foundations of Genetic Algorithms, Vol.1, 1991, pp. 205-218. Herrea, F., Tackling real-coded genetic algorithms: operators and tools for behavioral analysis, Artificial Intelligence Review, Vol.12, No.4, 1998, pp. 265-319. 10.5120/ijca2017913370Kora, P. & Yadlapalli, P., Crossover operators in genetic algorithms: A review, International Journal of Computer Applications, Vol.162, No.10, 2017, pp. 34-36. 10.1016/j.amc.2006.10.047Deep, K. & Thakur, M., A new crossover operator for real coded genetic algorithms, Applied Mathematics and Computation, Vol.188, No.1, 2007, pp. 895-911.