Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T17:06:51.550Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal asymptotic estimates for the volume of internal inhomogeneitiesin terms of multiple boundary measurements

Published online by Cambridge University Press:  15 November 2003

Yves Capdeboscq  and
Michael S. Vogelius
Yves Capdeboscq
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. vogelius@hilbert.rutgers.edu.
Michael S. Vogelius
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. vogelius@hilbert.rutgers.edu.
Get access

Abstract

We recently derived a very general representation formulafor the boundary voltage perturbations caused by internalconductivity inhomogeneities of low volume fraction (cf. Capdeboscq and Vogelius (2003)). In this paper we show how thisrepresentation formula may be used to obtain veryaccurate estimates for the size of the inhomogeneitiesin 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.

Keywords

Conductivity inhomogeneitiesvolume estimateslow volume fraction.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 2 , March 2003 , pp. 227 - 240
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alessandrini, G., Rosset, E. and Seo, J.K., Optimal size estimates for the inverse conductivity problem with one measurement. Proc. Amer. Math. Soc. 128 (2000) 53-64. CrossRef
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).
Ammari, H., Moskow, S. and Vogelius, M.S., Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM: Cont. Opt. Calc. Var. 9 (2003) 49-66. CrossRef
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).
Brühl, M. and Hanke, M., Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Problems 16 (2000) 1029-1042. CrossRef
Brühl, M., Hanke, M. and Vogelius, M.S., A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. 93 (2003) 635-654. CrossRef
Capdeboscq, Y. and Vogelius, M.S., A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. ESAIM: M2AN 37 (2003) 159-173. CrossRef
Cedio-Fengya, D.J., Moskow, S. and Vogelius, M.S., Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Problems 14 (1998) 553-595. CrossRef
Friedman, A. and Isakov, V., On the uniqueness in the inverse conductivity problem with one measurement. Indiana Univ. Math. J. 38 (1989) 553-580.
He, S. and Romanov, V.G., Identification of small flaws in conductors using magnetostatic measurements. Math. Comput. Simulation 50 (1999) 457-471. CrossRef
Ikehata, M. and Ohe, T., A numerical method for finding the convex hull of polygonal cavities using the enclosure method. Inverse Problems 18 (2002) 111-124. CrossRef
Kang, H., Seo, J.K. and Sheen, D., The inverse conductivity problem with one measurement: stability and estimation of size. SIAM J. Math. Anal. 28 (1997) 1389-1405. CrossRef
Kohn, R.V. and Milton, G.W., 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. CrossRef
Kwon, O., Seo, J.K. and Yoon, J.-R., A real time algorithm for the location search of discontinuous conductivities with one measurement. Comm. Pure Appl. Math. 55 (2002) 1-29. CrossRef
Lipton, R., Inequalities for electric and elastic polarization tensors with applications to random composites. J. Mech. Phys. Solids 41 (1993) 809-833. CrossRef