# Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements

Yves Capdeboscq; Michael S. Vogelius

- Volume: 37, Issue: 2, page 227-240
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topCapdeboscq, Yves, and Vogelius, Michael S.. "Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.2 (2003): 227-240. <http://eudml.org/doc/245120>.

@article{Capdeboscq2003,

abstract = {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.},

author = {Capdeboscq, Yves, Vogelius, Michael S.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {conductivity inhomogeneities; volume estimates; low volume fraction},

language = {eng},

number = {2},

pages = {227-240},

publisher = {EDP-Sciences},

title = {Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements},

url = {http://eudml.org/doc/245120},

volume = {37},

year = {2003},

}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2003

PB - EDP-Sciences

VL - 37

IS - 2

SP - 227

EP - 240

AB - 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.

LA - eng

KW - conductivity inhomogeneities; volume estimates; low volume fraction

UR - http://eudml.org/doc/245120

ER -

## References

top- [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. Zbl0944.35108
- [2] G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements. Preprint (2002). Zbl1010.35117MR1943396
- [3] H. Ammari and J.K. Seo, A new formula for the reconstruction of conductivity inhomogeneities. Preprint (2002). Zbl1040.78008
- [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. Zbl1075.78010
- [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). Zbl1089.78003MR2020923
- [6] M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Problems 16 (2000) 1029–1042. Zbl0955.35076
- [7] M. Brühl, M. Hanke and M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. 93 (2003) 635–654. Zbl1016.65079
- [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. Zbl1137.35346
- [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. Zbl0916.35132
- [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. Zbl0703.35165
- [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. Zbl0992.35118
- [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. Zbl0888.35131
- [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. Zbl0631.73012
- [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. Zbl1032.78005
- [16] R. Lipton, Inequalities for electric and elastic polarization tensors with applications to random composites. J. Mech. Phys. Solids 41 (1993) 809-833. Zbl0797.73046MR1214019

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.