著者

Ide T Isozaki H Nakata S Siltanen S

出版者

Institute of Physics

雑誌

Inverse problems (ISSN:02665611)

巻号頁・発行日

vol.26, no.3, pp.035001, 2010-03

被引用文献数

12 12

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Assume one is given a three-dimensional bounded domain with an unknown conductivity distribution inside. Further, suppose that the conductivity consists of a known background and unknown anomalous regions (inclusions) where conductivity values are unknown and different from the background. A method is introduced in Ide et al (2007 Commun. Pure Appl. Math. 60 1415–42) for locating inclusions approximately from noisy localized voltage-to-current measurements performed at the boundary of the body. The method is based on the use of complex geometrical optics solutions and hyperbolic geometry; numerical testing is presented in the aforementioned paper for the two-dimensional case. This work reports the results of computational implementation of the method in dimension three, where both the simulation of data and the computerized inversion algorithm are more complicated than in dimension two. Three new regularizing steps are added to the algorithm, resulting in significantly better robustness against noise. Numerical experiments are reported, suggesting that the approximate location of the inclusions can be reliably recovered from the data with a realistic level of measurement noise. Potential applications of the results include early diagnosis of breast cancer, underground contaminant detection and nondestructive testing.

著者

Gaitan P. Isozaki H. Poisson O. Siltanen S. Tamminen J. P.

出版者

Society for Industrial and Applied Mathematics

雑誌

SIAM journal on mathematical analysis (ISSN:00361410)

巻号頁・発行日

vol.45, no.3, pp.1675-1690, 2013-05

被引用文献数

5

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We consider an inverse boundary value problem for the heat equation on the interval $(0,1)$, where the heat conductivity $\gamma(t,x)$ is piecewise constant and the point of discontinuity depends on time: $\gamma(t,x) = k^2 \ (0 < x < s(t))$, $\gamma(t,x) = 1\ (s(t) < x < 1)$. First, we show that $k$ and $s(t)$ on the time interval $[0,T]$ are determined from a partial Dirichlet-to-Neumann map: $u(t,1) \to \partial_xu(t,1), \ 0 < t < T$, $u(t,x)$ being the solution to the heat equation such that $u(t,0)=0$, independently of the initial data $u(0,x)$. Second, we show that another partial Dirichlet-to-Neumann map: $u(t,0) \to \partial_xu(t,1), \ 0 < t < T$, $u(t,x)$ being the solution to the heat equation such that $u(t,1)=0$, restricts the pair $(k,s(t))$ to, at most, two cases on the time interval $[0,T]$, independently of the initial data $u(0,x)$.