The application of domain derivative for heat conduction with mixed condition in shape reconstruction. (English) Zbl 1112.65092

The authors investigate the inverse problem of detecting a cavity within a potential field by measuring the Cauchy data on the exterior known boundary. Notably, the boundary conditions on the cavity are of mixed type. No uniqueness result is mentioned. The domain derivative follows closely references [F. Hettlich and W. Rundell, Inverse Probl. 12, No. 3, 251–266 (1996; Zbl 0858.35134); F. Hettlich, ibid. 11, No. 2, 371–382 (1995); erratum ibid. 14, 209–210 (1998; Zbl 0821.35147) and F. Hettlich and W. Rundell, ibid. 17, No. 5, 1465–1482 (2001; Zbl 0986.35129)]. Section 3 is quite incomplete, e.g. no justification how one chooses the regularization parameter \(\alpha\) and the numbers \(M\), \(Q\), \(\rho_1\), \(\rho_2\).


65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35R30 Inverse problems for PDEs
35K05 Heat equation
Full Text: DOI


[1] Kirsch, A., The domain derivative and two applications in inverse scattering theory, Inverse problems, 9, 81-96, (1993) · Zbl 0773.35085
[2] Colton, D.; Kress, R., Integral equation methods in scattering theory, (1983), Wiley Interscience New York · Zbl 0522.35001
[3] Chapko, R.; Kress, R.; Yoon, J.-R., On the numerical solution of an inverse boundary value problem for the heat equation, Inverse problems, 14, 853-867, (1998) · Zbl 0917.35157
[4] Haslinger, J.; Mäkinen, R.A.E., Introduction to shape optimization: theory, approximation, and computation, (2003), SIAM · Zbl 1020.74001
[5] Hettlich, F., Frechet derivatives in inverse obstacle scattering, Inverse problems, 11, 371-382, (1995) · Zbl 0821.35147
[6] Hettlich, F.; Rundell, W., Iterative methods for the reconstruction of an inverse potential problem, Inverse problems, 12, 251-266, (1996) · Zbl 0858.35134
[7] Hettlich, F., Erratum: frechet derivatives in inverse obstacle scattering, Inverse problems, 14, 209-210, (1998)
[8] Hettlich, F., The Landweber iteration applied to inverse conductive scattering problems, Inverse problems, 14, 931-947, (1998) · Zbl 0917.35160
[9] Hettlich, F.; Rundell, W., Identification of a discontinuous source in the heat equation, Inverse problems, 17, 1465-1482, (2001) · Zbl 0986.35129
[10] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer Berlin · Zbl 0691.35001
[11] Pironneau, O., Optimal shape design for elliptic systems, (1984), Springer Berlin · Zbl 0496.93029
[12] Delfour, M.C.; Zolésio, J.P., Shapes and geometries: analysis, differential calculus, and optimization, () · Zbl 0675.49010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.