A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction

Yves Capdeboscq; Michael S. Vogelius

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 1, page 159-173
  • ISSN: 0764-583X

Abstract

top
We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.

How to cite

top

Capdeboscq, Yves, and Vogelius, Michael S.. "A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction." ESAIM: Mathematical Modelling and Numerical Analysis 37.1 (2010): 159-173. <http://eudml.org/doc/194151>.

@article{Capdeboscq2010,
abstract = { We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities. },
author = {Capdeboscq, Yves, Vogelius, Michael S.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Voltage perturbations; conductivity inhomogeneities; low volume fraction.; voltage perturbations; low volume fraction},
language = {eng},
month = {3},
number = {1},
pages = {159-173},
publisher = {EDP Sciences},
title = {A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction},
url = {http://eudml.org/doc/194151},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Capdeboscq, Yves
AU - Vogelius, Michael S.
TI - A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 1
SP - 159
EP - 173
AB - We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.
LA - eng
KW - Voltage perturbations; conductivity inhomogeneities; low volume fraction.; voltage perturbations; low volume fraction
UR - http://eudml.org/doc/194151
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. H. Ammari and H. Kang, High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter. Preprint (2002).  Zbl1036.35050
  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 Control Optim. Calc. Var.9 (2003) 49-66.  Zbl1075.78010
  5. E. Beretta, A. Mukherjee and M.S. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of conductivity imperfections of small area. Z. Angew. Math. Phys.52 (2001) 543-572.  Zbl0974.78006
  6. 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).  Zbl1089.78003
  7. M. Brühl, M. Hanke and M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. (to appear).  Zbl1016.65079
  8. Y. Capdeboscq and M.S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements. ESAIM: M2AN (to appear).  Zbl1137.35347
  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 Problems14 (1998) 553-595.  Zbl0916.35132
  10. A. Friedman and M.S. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence. Arch. Ration. Mech. Anal.105 (1989) 299-326.  Zbl0684.35087
  11. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin, Heidelberg, New York (1983).  Zbl0562.35001
  12. 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
  13. 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
  14. F. Murat and L. Tartar, H-Convergence, in Topics in the Mathematical Modelling of Composite Materials, A. Cherkaev and R.V. Kohn Eds., Progress in Nonlinear Differential Equations and Their Applications, Vol. 31, pp. 21-43. Birkhäuser, Boston, Basel, Berlin (1997).  
  15. G.C. Papanicolaou, Diffusion in random media, Surveys in Applied Mathematics, Vol. 1, Chap. 3, J.B. Keller, D.W. Mclaughlin and G.C. Papanicolaou Eds., Plenum Press, New York (1995).  Zbl0846.60081

Citations in EuDML Documents

top
  1. Hoai-Minh Nguyen, Michael S. Vogelius, A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity
  2. Yves Capdeboscq, Michael S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements
  3. Yves Capdeboscq, Michael S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements
  4. Roland Griesmaier, A general perturbation formula for electromagnetic fields in presence of low volume scatterers
  5. Roland Griesmaier, A general perturbation formula for electromagnetic fields in presence of low volume scatterers
  6. Habib Ammari, Hyeonbae Kang, Sur le Problème de Conductivité Inverse
  7. Stanislas Larnier, Mohamed Masmoudi, The extended adjoint method
  8. Stanislas Larnier, Mohamed Masmoudi, The extended adjoint method

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.