scholarly journals Numerical solutions of boundary inverse problems for some elliptic partial differential equations

2009 ◽  
Author(s):  
Suxing Zeng
Keyword(s):  
Partial Differential Equations ◽  
Inverse Problems ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Boundary Inverse Problems

scholarly journals Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Kiwoon Kwon
Keyword(s):  
Inverse Problem ◽  
Partial Differential Equations ◽  
Medical Imaging ◽  
Inverse Problems ◽  
Differential Equations ◽  
Optical Tomography ◽  
Elliptic Partial Differential Equations ◽  
Unique Determination ◽  
Partial Differential ◽  
Uniqueness Results

Unique determination issues about inverse problems for elliptic partial differential equations in divergence form are summarized and discussed. The inverse problems include medical imaging problems including electrical impedance tomography (EIT), diffuse optical tomography (DOT), and inverse scattering problem (ISP) which is an elliptic inverse problem closely related with DOT and EIT. If the coefficient inside the divergence is isotropic, many uniqueness results are known. However, it is known that inverse problem with anisotropic coefficients has many possible coefficients giving the same measured data for the inverse problem. For anisotropic coefficient with anomaly with or without jumps from known or unknown background, nonuniqueness of the inverse problems is discussed and the relation to cloaking or illusion of the anomaly is explained. The uniqueness and nonuniqueness issues are discussed firstly for EIT and secondly for ISP in similar arguments. Arguing the relation between source-to-detector map and Dirichlet-to-Neumann map in DOT and the uniqueness and nonuniqueness of DOT are also explained.

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