Title

Structural Stability In A Minimization Problem And Applications To Conductivity Imaging

Keywords

Bounded variation; Conductivity imaging; Degenerate elliptic equations; Non-smooth optimization; Regularization

Abstract

We consider the problem of minimizing the functional ∫Ω a&pipe;∇u&pipe;dx with u in some appropriate Banach space and prescribed trace f on the boundary. For a ∈ L2(Ω) and u in the sample space H1(Ω), this problem appeared recently in imaging the electrical conductivity of a body when some interior data are available. When a ∈ C(Ω) ∩ L∞(Ω), the functional has a natural interpretation, which suggests that one should consider the minimization problem in the sample space BV(Ω). We show the stability of the minimum value with respect to a, in a neighborhood of a particular coeffcient. In both cases the method of proof provides some convergent minimizing procedures. We also consider the minimization problem for the non-degenerate functional ∫Ω a max{&pipe;∇u&pipe;,δ}dx, for some δ > 0, and prove a stability result. Again, the method of proof constructs a minimizing sequence and we identify suffcient 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. © 2011 American Institute of Mathematical Sciences.

Publication Date

2-1-2011

Publication Title

Inverse Problems and Imaging

Volume

5

Issue

1

Number of Pages

219-236

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.3934/ipi.2011.5.219

Socpus ID

79951752786 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/79951752786

This document is currently not available here.

Share

COinS