A Note on the Transport Method for Hybrid Inverse Problems

There are several hybrid inverse problems for equations of the form \(\nabla \cdot D \nabla u - \sigma u = 0\) in which we want to obtain the coefficients \(D\) and \(\sigma\) on a domain \(\Omega\) when the solutions \(u\) are known. One approach is to use two solutions \(u_1\) and \(u_2\) to obtai...

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Bibliographic Details
Published in:arXiv.org 2019-12
Main Authors: Chung, Francis J, Hoskins, Jeremy G, Schotland, John C
Format: Article
Language:English
Subjects:
Coefficients
Fields (mathematics)
Inverse problems
Transport equations
Online Access:Get full text
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Summary:There are several hybrid inverse problems for equations of the form \(\nabla \cdot D \nabla u - \sigma u = 0\) in which we want to obtain the coefficients \(D\) and \(\sigma\) on a domain \(\Omega\) when the solutions \(u\) are known. One approach is to use two solutions \(u_1\) and \(u_2\) to obtain a transport equation for the coefficient \(D\), and then solve this equation inward from the boundary along the integral curves of a vector field \(X\) defined by \(u_1\) and \(u_2\). It follows from an argument of Guillaume Bal and Kui Ren that for any nontrivial choices of \(u_1\) and \(u_2\), this method suffices to recover the coefficients on a dense set in \(\Omega\). This short note presents an alternate proof of the same result from a dynamical systems point of view.
ISSN:2331-8422