no. 2
Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements
ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 2, pp. 227-240.

We recently derived a very general representation formula for the boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction (cf. Capdeboscq and Vogelius (2003)). In this paper we show how this representation formula may be used to obtain very accurate estimates for the size of the inhomogeneities in terms of multiple boundary measurements. As demonstrated by our computational experiments, these estimates are significantly better than previously known (single measurement) estimates, even for moderate volume fractions.

DOI : 10.1051/m2an:2003024
Classification : 35J20, 35B27, 35R30
Mots clés : conductivity inhomogeneities, volume estimates, low volume fraction 
@article{M2AN_2003__37_2_227_0,
     author = {Capdeboscq, Yves and Vogelius, Michael S.},
     title = {Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {227--240},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {2},
     year = {2003},
     doi = {10.1051/m2an:2003024},
     mrnumber = {1991198},
     zbl = {1137.35347},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2003024/}
}
TY  - JOUR
AU  - Capdeboscq, Yves
AU  - Vogelius, Michael S.
TI  - Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2003
SP  - 227
EP  - 240
VL  - 37
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2003024/
DO  - 10.1051/m2an:2003024
LA  - en
ID  - M2AN_2003__37_2_227_0
ER  - 
%0 Journal Article
%A Capdeboscq, Yves
%A Vogelius, Michael S.
%T Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2003
%P 227-240
%V 37
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2003024/
%R 10.1051/m2an:2003024
%G en
%F M2AN_2003__37_2_227_0
Capdeboscq, Yves; Vogelius, Michael S. Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 2, pp. 227-240. doi : 10.1051/m2an:2003024. http://www.numdam.org/articles/10.1051/m2an:2003024/

[1] G. Alessandrini, E. Rosset and J.K. Seo, Optimal size estimates for the inverse conductivity problem with one measurement. Proc. Amer. Math. Soc. 128 (2000) 53-64. | Zbl

[2] G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements. Preprint (2002). | MR | Zbl

[3] H. Ammari and J.K. Seo, A new formula for the reconstruction of conductivity inhomogeneities. Preprint (2002).

[4] H. Ammari, S. Moskow and M.S. Vogelius, Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM: Cont. Opt. Calc. Var. 9 (2003) 49-66. | Numdam | Zbl

[5] E. Beretta, E. Francini and M.S. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis. Preprint (2002). | MR | Zbl

[6] M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Problems 16 (2000) 1029-1042. | Zbl

[7] M. Brühl, M. Hanke and M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. 93 (2003) 635-654. | Zbl

[8] Y. Capdeboscq and M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. ESAIM: M2AN 37 (2003) 159-173. | Numdam | Zbl

[9] D.J. Cedio-Fengya, S. Moskow and M.S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Problems 14 (1998) 553-595. | Zbl

[10] A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement. Indiana Univ. Math. J. 38 (1989) 553-580. | Zbl

[11] S. He and V.G. Romanov, Identification of small flaws in conductors using magnetostatic measurements. Math. Comput. Simulation 50 (1999) 457-471.

[12] M. Ikehata and T. Ohe, A numerical method for finding the convex hull of polygonal cavities using the enclosure method. Inverse Problems 18 (2002) 111-124. | Zbl

[13] H. Kang, J.K. Seo and D. Sheen, The inverse conductivity problem with one measurement: stability and estimation of size. SIAM J. Math. Anal. 28 (1997) 1389-1405. | Zbl

[14] R.V. Kohn and G.W. Milton, On bounding the effective conductivity of anisotropic composites, in Homogenization and Effective Moduli of Materials and Media, J.L. Ericksen, D. Kinderlehrer, R. Kohn and J.-L. Lions Eds., Springer-Verlag, IMA Vol. Math. Appl. 1 (1986) 97-125. | Zbl

[15] O. Kwon, J.K. Seo and J.-R. Yoon, A real time algorithm for the location search of discontinuous conductivities with one measurement. Comm. Pure Appl. Math. 55 (2002) 1-29. | Zbl

[16] R. Lipton, Inequalities for electric and elastic polarization tensors with applications to random composites. J. Mech. Phys. Solids 41 (1993) 809-833. | MR | Zbl

Cité par Sources :