The inverse conductivity problem with one measurement: Stability and estimation of size

Hyeonbae Kang, Jin Keun Seo, Dongwoo Sheen

Research output: Contribution to journalArticle

71 Citations (Scopus)

Abstract

We consider the inverse problem to the refraction problem div((1 + (k-1)χD)∇u) = 0 in Ω and ∂u/∂v = g on ∂Ω. The inverse problem is to determine the size and the location of an unknown object D from the boundary measurement ΛD(g) = u|∂Ω. The results of this paper are twofold: stability and estimation of size of D. We first obtain upper and lower bounds of the size of D by comparing ΛD(g) with the Dirichlet data corresponding to the harmonic equation with the same Neumann data g. We then obtain logarithmic stability in the case of the disks. In the course of deriving the stability, we are able to compute a positive lower bound (independent of D) of the gradient of the solution u to the refraction problem with the Neumann data g satisfying some mild conditions.

Original languageEnglish
Pages (from-to)1389-1405
Number of pages17
JournalSIAM Journal on Mathematical Analysis
Volume28
Issue number6
DOIs
Publication statusPublished - 1997 Jan 1

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Inverse Conductivity Problem
Refraction
Inverse problems
Inverse Problem
Dirichlet
Upper and Lower Bounds
Logarithmic
Harmonic
Lower bound
Gradient
Unknown

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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abstract = "We consider the inverse problem to the refraction problem div((1 + (k-1)χD)∇u) = 0 in Ω and ∂u/∂v = g on ∂Ω. The inverse problem is to determine the size and the location of an unknown object D from the boundary measurement ΛD(g) = u|∂Ω. The results of this paper are twofold: stability and estimation of size of D. We first obtain upper and lower bounds of the size of D by comparing ΛD(g) with the Dirichlet data corresponding to the harmonic equation with the same Neumann data g. We then obtain logarithmic stability in the case of the disks. In the course of deriving the stability, we are able to compute a positive lower bound (independent of D) of the gradient of the solution u to the refraction problem with the Neumann data g satisfying some mild conditions.",
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The inverse conductivity problem with one measurement : Stability and estimation of size. / Kang, Hyeonbae; Seo, Jin Keun; Sheen, Dongwoo.

In: SIAM Journal on Mathematical Analysis, Vol. 28, No. 6, 01.01.1997, p. 1389-1405.

Research output: Contribution to journalArticle

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