Regularization Strategy for Estimating the Heat Source in Two-Dimensional Heat Transfer Processes
DOI:
https://doi.org/10.70567/mc.v42.ocsid8206Keywords:
BEMD, Heat transfer, Source estimation, RegularizationAbstract
In this work, a technique is studied for estimating the source term in a two-dimensional reaction– advection–diffusion–source equation. The identification of this term constitutes an ill-posed inverse problem in the sense of Hadamard, since the associated inverse operator is unbounded, which leads to high sensitivity to perturbations in the data. To address this difficulty, a regularization strategy based on the Bidimensional Empirical Mode Decomposition (BEMD) is proposed, aimed at mitigating the undesired variability induced by noise present in experimental or simulated measurements. A numerical example is presented to illustrate the effectiveness of the proposed methodology, and the obtained results are compared with those reported in the literature, where other classical regularization techniques are employed. In addition, a detailed analysis is carried out on the error distributions and the statistical properties of the approximation relative errors, in order to assess the robustness and stability of the developed approach.
References
Cortinovis A. y Kressner D. Low-rank approximation in the Frobenius norm by column and row subset selection. SIAM Journal on Matrix Analysis and Applications, 41(4):1651–1673, 2020. https://doi.org/10.1137/19M1281848.
Engl H.W., Hanke M., y Neubauer A. Regularization of inverse problems. volumen 375. Springer Science & Business Media, 1996. https://doi.org/10.1007/978-94-009-1740-8.
González L. y Suárez A. Improving approximate inverses based on Frobenius norm minimization. Applied Mathematics and Computation, 219(17):9363–9371, 2013. https://doi.org/10.1016/j.amc.2013.03.057.
Hadamard J. Lectures on Cauchy’s problem in linear partial differential equations, volumen 171. The Mathematical Gazette, 1924. https://doi.org/10.2307/3603014.
Huang N.E., Shen Z., Long S.R., Wu M.C., Shih H.H., Zheng Q., Yen N.C., Tung C.C., y Liu H.H. The empirical mode decomposition and the Hilbert spectrum for nonli-near and non-stationary time series analysis. Proceedings of the Royal Society of London. Series A: mathematical, physical and engineering sciences, 454(1971):903–995, 1998. https://doi.org/10.1098/rspa.1998.0193.
Kirsch A. et al. An introduction to the mathematical theory of inverse problems, volumen 120. Springer, 2011. https://doi.org/10.1007/978-3-030-63343-1.
Li X.X., Guo H.Z., Wan S.M., y Yang F. Inverse source identification by the modified regularization method on poisson equation. Journal of Applied Mathematics, 2012(1):971952, 2012. https://doi.org/10.1155/2012/971952.
Liu L., Liu B., Huang H., y Bovik A.C. No-reference image quality assessment based on spatial and spectral entropies. Signal processing: Image communication, 29(8):856–863, 2014. https://doi.org/10.1016/j.image.2014.06.006.
Morvidone M.A., Masci I., Rubio A.D., Kurtz M.L.A., Tasat D.R., y Piotrkowski R. Particle size and morphological evaluation of airborne urban dust particles by scanning electron microscopy and bidimensional empirical mode analysis. WSEAS Transactions on Environment and Development, 20:504–513, 2024. https://doi.org/10.37394/232015.2024.20.49.
Nunes J.C., Bouaoune Y., Delechelle E., Niang O., y Bunel P. Image analysis by bidimensional empirical mode decomposition. Image and vision computing, 21(12):1019–1026, 2003. https://doi.org/10.1016/S0262-8856(03)00094-5.
Painam R.K. y Manikandan S. BEMD based adaptive Lee filter for despeckling of SAR images. Advances in Space Research, 71(8):3140–3149, 2023. ISSN 0273-1177. https://doi.org/10.1016/j.asr.2022.12.009.
Rubio D., Sassano N., Morvidone M., y Piotrkowski R. Application of bidimensional empirical mode decomposition for particle identification and size determination. International Journal of Applied Mathematics, Computational Science and Systems Engineering, 6:186–192, 2024. https://doi.org/10.37394/232026.2024.6.16.
Sponring J. The entropy of scale-space. En Proceedings of 13th International Conference on Pattern Recognition, volumen 1, páginas 900–904. IEEE, 1996. https://doi.org/10.1109/ICPR.1996.546154.
Umbricht G.F. Identification of the source for full parabolic equations. Mathematical Modelling and Analysis, 26(3):339–357, 2021. ISSN 1392-6292/1648-3510. https://doi.org/10.3846/mma.2021.12700.
Umbricht G.F. y Rubio D. Regularization techniques for estimating the space-dependent source in an n-dimensional linear parabolic equation using space-dependent noisy data. Computers & Mathematics with Applications, 172:47–69, 2024. ISSN 0898-1221. https://doi.org/10.1016/j.camwa.2024.07.029.
Yang F. y Fu C.L. A mollification regularization method for the inverse spatial-dependent heat source problem. Journal of Computational and Applied Mathematics, 255:555–567, 2014. ISSN 0377-0427. https://doi.org/10.1016/j.cam.2013.06.012.
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