Volume 16, Issue 8

We prove the instability of averages of the conductivity in the inverse boundary value problem of Calderón, also known as the inverse conductivity problem or EIT.

New representations for solutions and coefficients of evolutionary equations are presented in the paper. On the basis of these representations the theorems of solvability of inverse problems are obtained.

We consider the problem of determining the shape and location of cracks from Cauchy data on the boundary of semi-infinite domains modeling the reconstruction of cracks within a heat conducting medium from temperature and heat flux measurements. Our reconstructions are based on a pair of nonlinear integral equations for the unknown crack and the unknown flux jump on the crack that are linear with respect to the flux and nonlinear with respect to the crack. We propose two different iteration methods employing the following idea: Given an approximate reconstruction for the crack we first solve one of the equations for the flux and subsequently linearize the other equation for updating the crack. The foundations for this approach for solving the inverse problem in semi-infinite domains are provided and numerical experiments exhibit the feasibility of both methods and their stability with respect to noisy data.

A new numerical method for an inverse problem for an elliptic equation with unknown potential is proposed. In this problem the point source is running along a straight line and the source-dependent Dirichlet boundary condition is measured as the data for the inverse problem. A rigorous convergence analysis shows that this method converges globally, provided that the so-called tail function is approximated well. This approximation is verified in numerical experiments, so as the global convergence. Applications to medical imaging, imaging of targets on battlefields and to electrical impedance tomography are discussed.

In this article, we construct complex geometrical optics solutions with general phase functions for second order elliptic equations in two dimensions. We then use these special solutions to treat the inverse problem of reconstructing embedded inclusions by boundary measurements.

In the paper we consider linear ill-posed problems on sets of functions convex upwards or downwards along all lines that belong to a functions' domain and are parallel to coordinate axes. A regularizing algorithm is constructed such that an approximate solution tends to the exact one uniformly of some subsets of the domain. The algorithms to estimate an error of finite dimensional approximation and to find a lower and an upper functions that bound all approximation solutions are provided. As a model example, an inverse problem for a two-dimensional heat conduction equation is solved.