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 language | English |
|---|---|
| Pages (from-to) | 1389-1405 |
| Number of pages | 17 |
| Journal | SIAM Journal on Mathematical Analysis |
| Volume | 28 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - 1997 Jan 1 |
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All Science Journal Classification (ASJC) codes
- Analysis
- Computational Mathematics
- Applied Mathematics
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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 journal › Article
TY - JOUR
T1 - The inverse conductivity problem with one measurement
T2 - Stability and estimation of size
AU - Kang, Hyeonbae
AU - Seo, Jin Keun
AU - Sheen, Dongwoo
PY - 1997/1/1
Y1 - 1997/1/1
N2 - 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.
AB - 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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U2 - 10.1137/S0036141096299375
DO - 10.1137/S0036141096299375
M3 - Article
AN - SCOPUS:0001607563
VL - 28
SP - 1389
EP - 1405
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
SN - 0036-1410
IS - 6
ER -