Structural stability in aminimization problem and applications to conductivity imaging

M. Zuhair Nashed; Alexandru Tamasan

doi:10.3934/ipi.2011.5.219

Article Contents

2011, Volume 5, Issue 1: 219-236. Doi: 10.3934/ipi.2011.5.219

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Structural stability in a minimization problem and applications to conductivity imaging

M. Zuhair Nashed^1, and
Alexandru Tamasan^1,

1.
Department of Mathematics, University of Central Florida, Orlando, FL 32816

Received: June 30, 2009

Revised: June 30, 2010

Published: January 2011

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Abstract

We consider the problem of minimizing the functional $\int_\Omega a|\nabla u|dx$, with $u$ in some appropriate Banach space and prescribed trace $f$ on the boundary. For $a\in L^2(\Omega)$ and $u$ in the sample space $H^1(\Omega)$, this problem appeared recently in imaging the electrical conductivity of a body when some interior data are available. When $a\in C(\Omega)\cap L^\infty(\Omega)$, the functional has a natural interpretation, which suggests that one should consider the minimization problem in the sample space $BV(\Omega)$. We show the stability of the minimum value with respect to $a$, in a neighborhood of a particular coefficient. In both cases the method of proof provides some convergent minimizing procedures. We also consider the minimization problem for the non-degenerate functional $\int_\Omega a\max\{|\nabla u|,\delta\}dx$, for some $\delta>0$, and prove a stability result. Again, the method of proof constructs a minimizing sequence and we identify sufficient conditions for convergence. We apply the last result to the conductivity problem and show that, under an a posteriori smoothness condition, the method recovers the unknown conductivity.

Keywords:

Non-smooth optimization; bounded variation; degenerate elliptic equations; regularization; conductivity imaging.

Mathematics Subject Classification: Primary 49A45; Secondary 35R30.

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