Abstract
The objective of this short note is to employ an equation error approach to identify a variable parameter in fourth-order partial differential equations. Existence and convergence results are given for the optimization problem emerging from the equation error formulation. Finite element based numerical experiments show the effectiveness of the proposed framework.
References
[1] ACAR, R.: Identification of the coefficient in the elliptic equations, SIAM J. Control Optim. 31 (1993), 1221-1244.10.1137/0331058Search in Google Scholar
[2] BANKS, H. T.-KUNISCH, K.: Estimation techniques for distributed parameter systems. In: System and Control: Foundations and Applications 1, Birkhauser, Boston, 1989.Search in Google Scholar
[3] GOCKENBACH, M. S.-KHAN, A. A.: Identification of Lam´e parameters in linear elasticity: a fixed point approach, J. Ind. Manag. Optim. 1 (2005), 487-497.10.3934/jimo.2005.1.487Search in Google Scholar
[4] GOCKENBACH M. S.-KHAN A. A.: Identification of Lam´e parameters in linear elasticity: a fixed point approach, J. Ind. Manag. Optim. 1 (2005), 487-497.10.3934/jimo.2005.1.487Search in Google Scholar
[5] GOCKENBACH, M. S.-KHAN, A. A.: An abstract framework for elliptic inverse problems. Part 1: an output least-squares approach, Math. Mech. Solids 12 (2007), 259-276.10.1177/1081286505055758Search in Google Scholar
[6] 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 (2006), 343-359.Search in Google Scholar
[7] GOCKENBACH, M. S.-JADAMBA, B.-KHAN, A. A.: Equation error approach for elliptic inverse problems with an application to the identification of Lam´e parameters, Inverse Probl. Sci. Eng. 16 (2008), 349-367.10.1080/17415970701602580Search in Google Scholar
[8] JADAMBA, B.-KHAN, A. A.-SAMA, M.: Inverse problems of parameter identification in partial differential equations. In: Mathematics in Science and Technology, World Sci. Publ., Hackensack, NJ, 2011, 228-258.Search in Google Scholar
[9] KÄRKKÄINEN, T.: An equation error method to recover diffusion from the distributed observation, Inverse Problems 13 (1997), 1033-1051.10.1088/0266-5611/13/4/009Search in Google Scholar
[10] KUGLER, P.: A parameter identification problem of mixed type related to the manufacture of car windshields, SIAM J. Appl. Math. 64 (2004), 858-877.10.1137/S0036139903423339Search in Google Scholar
[11] MARINOV, T. T.-VATSALA, A. S.: Inverse problem for coefficient identification in the Euler-Bernoulli equation, Comput. Math. Appl. 56 (2008), 400-410.10.1016/j.camwa.2007.11.048Search in Google Scholar
[12] MARINOV, T. T.-MARINOVA, R. S.: Coefficient identification in Euler-Bernoulli equation from over-posed data, J. Comput. Appl. Math. 235 (2010), 450-459.10.1016/j.cam.2010.05.048Search in Google Scholar
[13] VOGEL, C. R.: Computational methods for inverse problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.Search in Google Scholar
[14] WHITE, L. W.: Estimation of elastic parameters in beams and certain plates: H1 regularization. J. Optim. Theory Appl. 60 (1989), 305-326.Search in Google Scholar
[15] WHITE, L. W.: Stability of optimal output least squares estimators in certain beams and plate models, Appl. Anal. 39 (1990) 15-33. Search in Google Scholar
Mathematical Institute Slovak Academy of Sciences
