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.

scholarly journals VARIATIVE SOLUTION OF THE COEFFICIENT INVERSE PROBLEM FOR THE HEAT EQUATIONS

2020 ◽  
Vol 72 (4) ◽  
pp. 23-27
Author(s):  
L. Yermekkyzy ◽  
Keyword(s):  
Inverse Problem ◽  
Partial Differential Equations ◽  
Inverse Problems ◽  
Differential Equations ◽  
Generalized Solution ◽  
Heat Equations ◽  
Coefficient Inverse Problem ◽  
Partial Differential ◽  
Additional Information ◽  
Coefficient Inverse Problems

One of the main types of inverse problems for partial differential equations are problems in which the coefficients of the equations or the quantities included in them must be determined using some additional information. Such problems are called coefficient inverse problems for partial differential equations. Coefficient inverse problems (identification problems) have become the subject of close study, especially in recent years. Interest in them is caused primarily by their important applied values. They find applications in solving problems of planning the development of oil fields (determining the filtration parameters of fields), in creating new types of measuring equipment, in solving problems of environmental monitoring, etc. The standard formulation of the coefficient inverse problem contains a functional (discrepancy), physics. When formulating the statements of inverse problems, the statements of direct problems are assumed to be known. The solution to the problem is sought from the condition of its minimum. Inverse problems for partial differential equations can be posed in variational form, i.e., as optimal control problems for the corresponding systems. A variational statement of one coefficient inverse problem for a one-dimensional heat equation is considered. By the solution of the boundary value problem for each fixed control coefficient we mean a generalized solution from the Sobolev space. The questions of correctness of the considered coefficient inverse problem in the variational setting are investigated.

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