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Estimation of a Regularisation Parameter for a Robin Inverse Problem

  • Xi-Ming Fang (a1) (a2), Fu-Rong Lin (a1) and Chao Wang (a1)

Abstract

We consider the nonlinear and ill-posed inverse problem where the Robin coefficient in the Laplace equation is to be estimated using the measured data from the accessible part of the boundary. Two regularisation methods are considered — viz. L 2 and H 1 regularisation. The regularised problem is transformed to a nonlinear least squares problem; and a suitable regularisation parameter is chosen via the normalised cumulative periodogram (NCP) curve of the residual vector under the assumption of white noise, where information on the noise level is not required. Numerical results show that the proposed method is efficient and competitive.

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Corresponding author

*Corresponding author. Email address: chaowang.hk@gmail.com (C. Wang)

References

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[1] Alessandrini, G., Piero, L.D. and Rondi, L., Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Problems 19, 973984 (2003).
[2] Cakoni, F. and Kress, R., Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Problems and Imaging 1, 229245 (2007).
[3] Chaabane, S., Elhechmi, C. and Jaoua, M., A stable recovery method for the Robin inverse problem, Math. Comput. Simul. 66, 367383 (2004).
[4] Chaabane, S., Feki, I. and Mars, N., Numerical reconstruction of a piecewise constant Robin parameter in the two- or three-dimensional case, Inverse Problems 28 065016, 19pp (2012).
[5] Chaabane, S., Fellah, I., Jaoua, M. and Leblond, J., Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Problems 20, 4759 (2004).
[6] Chaabane, S. and Jaoua, M., Identification of Robin coefficients by the means of boundary measurements, Inverse Problems 15, 14251438 (1999).
[7] Chen, D., Maclachlan, S. and Kilmer, M., Iterative parameter-choice and multigrid methods for anisotropic diffusion denoising, SIAM J. Sci. Comput. 33, 29722994 (2011).
[8] Choulli, M., On the determination of an unknown boundary function in a parabolic equation, Inverse Problems 15, 659667 (1999).
[9] Dennis, J. and Schnabel, R., Numerical Methods for Unconstrained Optimisation and Nonlinear Equations, SIAM (1996).
[10] Fang, W. and Zeng, S., A direct solution of the Robin inverse problem, J. Integral Equ. Appl. 21, 545557 (2009).
[11] Fasino, D. and Inglese, G., An inverse Robin problem for Laplace's equation: Theoretical results and numerical methods, Inverse Problems 15, 4148 (1999).
[12] Fasino, D. and Inglese, D., Discrete methods in the study of an inverse problem for Laplace's equation, SIAM J. Numer. Anal. 19, 105118 (1999).
[13] Golub, G.H., Heath, M. and Wahba, G., Generalised cross-validation as a method for choosing a good ridge parameter, Technometrics 21, 215223 (1979).
[14] Hansen, P.C., Kilmer, M.E. and Kjeldsen, R.H., Exploiting residual information in the parameter choice for discrete ill-posed problems, BIT. 46, 4159 (2006).
[15] Hansen, P.C. and O’leary, D.P., The use of the L-curve in the regularisation of discrete ill-posed problems, SIAM J. Sci. Comput. 14, 14871503 (1993).
[16] Inglese, G., An inverse problem in corrosion detection, Inverse Problems 13, 977994 (1997).
[17] Jin, B., Conjugate gradient method for the Robin inverse problem associated with the Laplace equation, Int. J. Numer. Meth. Eng. 71, 433453 (2007).
[18] Jin, B. and Maass, P., Sparsity regularisation for parameter identification problems, Inverse Problems 28 (2012) 123001123071.
[19] Jin, B. and Zou, J., Inversion of Robin coefficient by a spectral stochastic finite element approach, J. Comp. Phys. 227, 32823306 (2008).
[20] Jin, B. and Zou, J., Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim. 48, 19772002 (2009).
[21] Kress, R., Linear Integral Equations, 2nd edition, Springer, New York (1999).
[22] Lin, F. and Fang, W., A linear integral equation approach to the Robin inverse problem, Inverse Problems 21, 17571772 (2005).
[23] Ma, Y. and Lin, F., Conjugate gradient method for estimation of Robin coefficients, East Asian J. Appl. Math. 4, 189204 (2014).
[24] Mojabi, P. and LoVetri, J., Adapting the normalised cumulative periodogram parameter-choice method to the Tikhonov regularisation of 2-D/TM electromagnetic inverse scattering using born iterative method, Prog. Electromagn. Res. M 1, 111138 (2008).
[25] Morozov, V.A., Nashed, Z. and Aries, A.B., Methods for Solving Incorrectly Posed Problems, Springer, New York (1984).
[26] Rust, B.W. and O’Leary, D.P., Residual periodograms for choosing regularisation parameters for ill-posed problems, Inverse Problems 24, 3400534034 (2008).

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