Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem
Bastian Gebauer Nuutti Hyvönen
In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.
keywords: inclusions. inverse elliptic boundary value problems Factorization method
Localized potentials in electrical impedance tomography
Bastian Gebauer
In this work we study localized electric potentials that have an arbitrarily high energy on some given subset of a domain and low energy on another. We show that such potentials exist for general $L^\infty_+$-conductivities in almost arbitrarily shaped subregions of a domain, as long as these regions are connected to the boundary and a unique continuation principle is satisfied. From this we deduce a simple, but new, theoretical identifiability result for the famous Calderón problem with partial data. We also show how to construct such potentials numerically and use a connection with the factorization method to derive a new non-iterative algorithm for the detection of inclusions in electrical impedance tomography.
keywords: Electrical impedance tomography factorization method. Calderon problem

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