Abstract
In this short note deals with the nonlinear inverse problem of identifying a variable parameter in fourth-order partial differential equations using an equation error approach. These equations arise in several important applications such as car windscreen modeling, deformation of plates, etc. To counter the highly ill-posed nature of the considered inverse problem, a regularization must be performed. The main contribution of this work is to show that the equation error approach permits the use of \(H^1\) regularization whereas other optimization-based formulations commonly use \(H_2\) regularization. We give the existence and convergence results for the equation error formulation. An illustrative numerical example is given to show the feasibility of the approach.
Dedicated to Prof. Alemdar Hasanoglu (Hasanov) on his 60th birthday
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Acar, R.: Identification of the coefficient in elliptic equations. SIAM J. Control Optim. 31(5), 1221–1244 (1993)
Al-Jamal, M.F., Gockenbach, M.S.: Stability and error estimates for an equation error method for elliptic equations. Inverse Prob. 28(9), 095006-15 ( 2012)
Radu, V.: Application. In: Radu, V. (ed.) Stochastic Modeling of Thermal Fatigue Crack Growth. ACM, vol. 1, pp. 63–70. Springer, Heidelberg (2015)
Crossen, E., Gockenbach, M.S., Jadamba, B., Khan, A.A., Winkler, B.: An equation error approach for the elasticity imaging inverse problem for predicting tumor location. Comput. Math. Appl. 67(1), 122–135 (2014)
Gockenbach, M.S.: The output least-squares approach to estimating Lamé moduli. Inverse Prob. 23(6), 2437–2455 (2007)
Gockenbach, M.S., Jadamba, B., Khan, A.A.: Numerical estimation of discontinuous coefficients by the method of equation error. Int. J. Math. Comput. Sci. 1(3), 343–359 (2006)
Gockenbach, M.S., Jadamba, B., Khan, A.A.: Equation error approach for elliptic inverse problems with an application to the identification of Lamé parameters. Inverse Probl. Sci. Eng. 16(3), 349–367 (2008)
Gockenbach, M.S., Khan, A.A.: Identification of Lamé parameters in linear elasticity: a fixed point approach. J. Ind. Manag. Optim. 1(4), 487–497 (2005)
Gockenbach, M.S., Khan, A.A.: An abstract framework for elliptic inverse problems. I. An output least-squares approach. Math. Mech. Solids 12(3), 259–276 (2007)
Gockenbach, M.S., Khan, A.A.: An abstract framework for elliptic inverse problems. II. An augmented Lagrangian approach. Math. Mech. Solids 14(6), 517–539 (2009)
Hasanov, A.: Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate. Int. J. Non-Linear Mech. 42(5), 711–721 (2007). http://dx.doi.org/10.1016/j.ijnonlinmec.2007.02.011
Hasanov, A., Mamedov, A.: An inverse problem related to the determination of elastoplastic properties of a plate. Inverse Prob. 10(3), 601–615 (1994). http://stacks.iop.org/0266-5611/10/601
Jadamba, B., Khan, A.A., Rus, G., Sama, M., Winkler, B.: A new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location. SIAM J. Appl. Math. 74(5), 1486–1510 (2014)
Jadamba, B., Khan, A.A., Sama, M.: Inverse problems of parameter identification in partial differential equations. In: Mathematics in Science and Technology, World Scientific Publishing, Hackensack, NJ, 2011, pp. 228–258
Kärkkäinen, T.: An equation error method to recover diffusion from the distributed observation. Inverse Prob. 13(4), 1033–1051 (1997)
Kügler, P.: A parameter identification problem of mixed type related to the manufacture of car windshields. SIAM J. Appl. Math. 64(3), 858–877 (2004). (electronic)
White, L.W.: Estimation of elastic parameters in beams and certain plates: \(H^1\) regularization. J. Optim. Theory Appl. 60(2), 305–326 (1989)
White, L.W.: Estimation of elastic parameters in a nonlinear elliptic model of a plate. Appl. Math. Comput. 42(2, part II), 139–187 (1991)
Acknowledgments
The work of A.A. Khan is supported by a grant from the Simons Foundation (#210443 to Akhtar Khan). The work of B. Jadamba is supported by RITs COS Faculty Development Grant (FEAD) for 2014.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Caya, P., Jadamba, B., Khan, A.A., Raciti, F., Winkler, B. (2016). An Equation Error Approach for the Identification of Elastic Parameters in Beams and Plates with \(H_1\) Regularization. In: Kozubek, T., Blaheta, R., Å Ãstek, J., RozložnÃk, M., ÄŒermák, M. (eds) High Performance Computing in Science and Engineering. HPCSE 2015. Lecture Notes in Computer Science(), vol 9611. Springer, Cham. https://doi.org/10.1007/978-3-319-40361-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-40361-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-40360-1
Online ISBN: 978-3-319-40361-8
eBook Packages: Computer ScienceComputer Science (R0)