Abstract
In this chapter we present a method to predict the optimal value of the Tikhonov’s regularization parameter by solving simplified versions of the inverse problems considered. This can be of great benefit since methods such as the L-curve and the Fixed Point Iteration require the inverse problem to be solved several times in order to determine the optimal value for the regularization parameter. The main idea that supports the proposed approach is to solve the problem of interest using a low set of dimensions to represent the function to be estimated and, then, this solution is used to obtain an estimate for the regularization parameter of the complete model, based on the Fixed Point Iteration method. Tests are performed on three inverse heat transfer problems: estimation of the variable thermal conductivity of a biological tissue, estimation of the inlet temperature in parallel plates channel, and the estimation of the variable scattering albedo of a radiative transfer participating medium. The results obtained demonstrate the feasibility of the technique in three problems with potential practical applications in bioengineering, renewable energy and climate in alignment with the Sustainable Development Goals (SDG) 3, 4, 7, 9, 13, and 17 of the United Nations 2030 Agenda established in 2015.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aucejo, M., De Smet, O.: A generalized multiplicative regularization for input estimation. Mech. Syst. Signal Process. 157, 107637 (2021)
Bazán, F.S.V.: Fixed-point iterations in determining the Tikhonov regularization parameter. Inverse Probl. 24(3), 035001 (2008)
Beck, J.V., Arnold, K.J.: Parameter Estimation in Engineering and Science. Wiley, New York (1977)
Beck, J.V., Blackwell, B., Clair, C.R.S., Jr.: Inverse Heat Conduction: Ill-Posed Problems. Wiley, New York, NY (1985)
Bejan, A.: Convection Heat Transfer, 4th edn. Wiley, Hoboken, NJ (2013)
Benning, M., Burger, M.: Modern regularization methods for inverse problems 27, 1–111 (2018)
Bokar, J.C., Özisik, M.N.: An inverse analysis for estimating the time-varying inlet temperature in laminar flow inside a parallel plate duct. Int. J. Heat Mass Transf. 38(1), 39–45 (1995)
Bozzoli, F., Cattani, L., Rainieri, S., Bazán, F.S.V., Borges, L.S.: Estimation of the local heat-transfer coefficient in the laminar flow regime in coiled tubes by the Tikhonov regularisation method. Int. J. Heat Mass Transf. 72, 352–361 (2014)
Chalhoub, E.S., Campos Velho, H.F.: Estimation of the optical properties of seawater from measurements of exit radiance. J. Quant. Spectrosc. Radiat. Transf. 72, 551–565 (2002)
Chandrasekhar, S.: Radiative Transfer. Dover Publications, Inc., New York, NY (1960)
Cotta, R.M., Cotta, B.P., Naveira-Cotta, C.P., Cotta-Pereira, G.: Hybrid integral transforms analysis of the bioheat equation with variable properties. Int. J. Therm. Sci. 49(9), 1510–1516 (2010)
Elwassif, M.M., Kong, Q., Vazquez, M., Bikson, M.: Bio-heat transfer model of deep brain stimulation-induced temperature changes. J. Neural Eng. 3(4), 306 (2006)
Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34(4), 561–580 (1992)
Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadelphia (1998)
Hansen, P.C., O’Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14(6), 1487–1503 (1993)
Horvath, H., Alados Arboledas, L., Olmo, F.J., Jovanovic, O., Gangl, M., Kaller, W., Sánchez, C., Sauerzopf, H., Seidl, S.: Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing. J. Geophys. Res. Atmos. 107, 4386 (2002)
Jardim, L.C.S., Knupp, D.C., Sacco, W.F., Silva Neto, A.J.: Solution of a coupled conduction—radiation inverse heat transfer problem with the topographical global optimization method. In: Computational Intelligence in Emerging Technologies for Engineering Applications, pp. 53–71. Springer (2020)
Jiang, S.C., Ma, N., Li, H.J., Zhang, X.X.: Effects of thermal properties and geometrical dimensions on skin burn injuries. Burns 28(8), 713–717 (2002)
Ke, H., Tai, S., Wang, L.V.: Photoacoustic thermography of tissue. J. Biomed. Opt. 19(2), 026003 (2014)
Kengne, E., Lakhssassi, A., Vaillancourt, R.: Temperature distributions for regional hypothermia based on nonlinear bioheat equation of Pennes type: dermis and subcutaneous tissues. Appl. Math. 3(3) (2012)
Khanday, M.A., Nazir, K.: Mathematical and numerical analysis of thermal distribution in cancerous tissues under the local heat therapy. Int. J. Biomath. 10(7), 1750099 (2017)
Knupp, D.C., Canato, J.V.M., Silva Neto, A.J., Soeiro, F.J.C.P.: Radiative properties estimation and construction of confidence regions with a combination of the differential evolution algorithm and the likelihood method. Proc. Ser. Braz. Soc. Comput. Appl. Math. 5(1) (2017)
Knupp, D.C., Silva Neto, A.J.: Solution of the inverse radiative transfer problem of simultaneous identification of the optical thickness and space-dependent Albedo using Bayesian inference. Comput. Model. Eng. Sci. 96(5), 339–360 (2013)
Knupp, D.C., Silva Neto, A.J., Sacco, W.F.: Radiative properties estimation with the Luus-Jaakola and the particle collision algorithm. Comput. Model. Eng. Sci. (CMES) 54(2), 121 (2009)
Mellal, I., Oukaira, A., Kengene, E., Lakhssassi, A.: Thermal therapy modalities for cancer treatment: a review and future perspectives. Int. J. Appl. Sci. Res. Rev. 4(2), 14 (2017)
Momenroodaki, P., Haines, W., Fromandi, M., Popovic, Z.: Noninvasive internal body temperature tracking with near-field microwave radiometry. IEEE Trans. Microw. Theory Tech. 66(5), 2535–2545 (2017)
Morozov, V.A.: Regularization Methods for Solving Incorrectly Posed Problems. Springer, New York, NY (1984)
Moura Neto, F.D., Silva Neto, A.J.: Two equivalent approaches to obtain the gradient in algorithms for function estimation in heat conduction problems. In: Proceedings of the 34th National Heat Transfer Conference, Pittsburgh, PA (2000)
Moura Neto, F.D., Silva Neto, A.J.: An Introduction to Inverse Problems with Applications. Springer-Verlag, Berlin (2013)
Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308–313 (1965)
Ng, E.Y.K., Tan, H.M., Ooi, E.H.: Boundary element method with bioheat equation for skin burn injury. Burns 35(7), 987–997 (2009)
Oliva Soares, P., Silva Neto, A.J., Campos Velho, H.F., Soeiro, F.J.C.P.: A two step inverse problem for vertical temperature profile retrieval in cloudy atmosphere using artificial neural networks. In: Proceedings of the 22nd International Congress of Mechanical Engineering, Ribeirão Preto, Brazil, pp. 4364–4375 (2013)
Özen, Ş., Helhel, S., Cerezci, O.: Heat analysis of biological tissue exposed to microwave by using thermal wave model of bio-heat transfer (TWMBT). Burns 34(1), 45–49 (2008)
Özişik, M.N.: Radiative Transfer and Interactions with Conduction and Convection. Wiley, New Jersey (1973)
Özişik, M.N., Orlande, H.R.B.: Inverse Heat Transfer: Fundamentals and Applications. Taylor & Francis, New York (2000)
Pennes, H.H.: Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol. 1, 93–122 (1948)
Pradere, C., Joanicot, M., Batsale, J.C., Toutain, J., Gourdon, C.: Processing of temperature field in chemical microreactors with infrared thermography. Quant. InfraRed Thermogr. J. 3(1), 117–135 (2006)
Schena, E., Saccomandi, P., Fong, Y.: Laser ablation for cancer: past, present and future. J. Funct. Biomater. 8(2), 19 (2017)
Shah, R.K.E., London, A.L.: Laminar Flow Forced Convection in Ducts. Advances in Heat Transfer, vol. 1, p. 1. Academic Press, New York (1978)
Shah, J., Park, S., Aglyamov, S.R., Larson, T., Ma, L., Sokolov, K.V., Johnston, K., Milner, T., Emelianov, S.Y.: Photoacoustic imaging and temperature measurement for photothermal cancer therapy. J. Biomed. Opt. 13(3), 034024 (2008)
Silva Neto, C.A., Silva Neto, A.J.: Estimation of optical thickness, single scattering albedo and diffuse reflectivities with a minimization algorithm based on an interior points method. In: Proceedings of 17th International Congress of Mechanical Engineering, ABCM, São Paulo, SP, Brazil (2003)
Silva Neto, A.J., Özişik, M.N.: An inverse problem of simultaneous estimation of radiation phase function, albedo and optical thickness. J. Quant. Spectrosc. Radiat. Transf. 53(4), 397–409 (1995)
Stephany, S., Becceneri, J.C., Souto, R.P., Campos Velho, H.F., Silva Neto, A.J.: A pre-regularization scheme for the reconstruction of a spatial dependent scattering albedo using a hybrid ant colony optimization implementation. Appl. Math. Model. 34(3), 561–572 (2010)
Storn, R., Price, K.: Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997)
Tikhonov, A.N.: On the solution of ill-posed problems and the method of regularization. Dokl. Akad. Nauk SSSR 151(3), 501–504 (1963)
Wang, J., Silva Neto, A.J., Moura Neto, F.D., Su, J.: Function estimation with Alifanov’s iterative regularization method in linear and nonlinear heat conduction problems. Appl. Math. Model. 26(11), 1093–1111 (2002)
Wolfram Documentation Center. https://reference.wolfram.com/. Accessed 15 Mar 2021
Zhang, K., Li, W., Eide, H., Stamnes, K.: A bio-optical model suitable for use in forward and inverse coupled atmosphere-ocean radiative transfer models. J. Quant. Spectrosc. Radiat. Transf. 103, 411–423 (2007)
Acknowledgements
The authors acknowledge the financial support provided by CAPES—Coordination of Superior Level Staff Improvement (Finance Code 001), CNPq—National Council for Scientific and Technological Development, and FAPERJ—Carlos Chagas Filho Foundation for Supporting Research in the State of Rio de Janeiro. The authors acknowledge also the CAPES PrInt program grants (Process Code 88887.469279/2019-00 and 88887.311757/2018-00). This work has been also partially funded by the Spanish Ministry of Economy and Competitiveness with the support of the project TIN2017-86647-P (including funds from the European Regional Development Fund, ERDF).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
da Silva Jardim, L.C., Knupp, D.C., de Souza Monteiro de Barros, T.M., da Silva Abreu, L.A., Corona, C.C., Neto, A.J.S. (2022). Computational Intelligence and Tikhonov Regularization with Reduced Dimension Model: Applications in Health, Renewable Energy and Climate Heat Transfer Inverse Problems. In: Verdegay, J.L., Brito, J., Cruz, C. (eds) Computational Intelligence Methodologies Applied to Sustainable Development Goals. Studies in Computational Intelligence, vol 1036. Springer, Cham. https://doi.org/10.1007/978-3-030-97344-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-97344-5_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-97343-8
Online ISBN: 978-3-030-97344-5
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)