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

Yves Capdeboscq; Michael S. Vogelius

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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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 37.2 (2010): 227-240. <http://eudml.org/doc/194160>.

@article{Capdeboscq2010,

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},

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

language = {eng},

month = {3},

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/194160},

volume = {37},

year = {2010},

}

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

DA - 2010/3//

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.; conductivity inhomogeneities; low volume fraction

UR - http://eudml.org/doc/194160

ER -

## References

top- 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.
- G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements. Preprint (2002).
- H. Ammari and J.K. Seo, A new formula for the reconstruction of conductivity inhomogeneities. Preprint (2002).
- 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.
- 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).
- M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Problems16 (2000) 1029-1042.
- M. Brühl, M. Hanke and M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math.93 (2003) 635-654.
- Y. Capdeboscq and M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. ESAIM: M2AN37 (2003) 159-173.
- 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 Problems14 (1998) 553-595.
- A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement. Indiana Univ. Math. J.38 (1989) 553-580.
- S. He and V.G. Romanov, Identification of small flaws in conductors using magnetostatic measurements. Math. Comput. Simulation50 (1999) 457-471.
- M. Ikehata and T. Ohe, A numerical method for finding the convex hull of polygonal cavities using the enclosure method. Inverse Problems18 (2002) 111-124.
- 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.
- 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.
- 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.
- R. Lipton, Inequalities for electric and elastic polarization tensors with applications to random composites. J. Mech. Phys. Solids41 (1993) 809-833.

## Citations in EuDML Documents

top- Yves Capdeboscq, Michael S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction
- Yves Capdeboscq, Michael S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction
- Roland Griesmaier, A general perturbation formula for electromagnetic fields in presence of low volume scatterers
- Roland Griesmaier, A general perturbation formula for electromagnetic fields in presence of low volume scatterers
- Stanislas Larnier, Mohamed Masmoudi, The extended adjoint method
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