Petri Ola

Electromagnetic inverse problems and generalized Sommerfeld potentials

In this article we study the inverse boundary value problem for Maxwell’s equation; that is, we determine the scalar permeability, permissibility, and conductivity of a body from electromagnetic field measurements outside the body. In this paper, the problem is approached by using certain generalized Sommerfeld potentials that are shown to satisfy a vector-valued Schrodinger equation. By using these potentials, the inverse boundary value problem is reduced to the problem of determining the potential of this second-order operator. Further, it is shown that the electromagnetic material parameters can be determined by using electric or magnetic dipole point sources only.

The Inverse Conductivity Problem with an Imperfectly Known Boundary

We show how to eliminate the error caused by an incorrectly modeled boundary in electrical impedance tomography (EIT). In practical measurements, one usually lacks exact knowledge of the boundary. Because of this, the numerical reconstruction from the measured EIT data is done using a model domain that represents the best guess for the true domain. However, it has been noticed that an inaccurate model of the boundary causes severe errors for the reconstructions. We introduce a new algorithm to find a deformed image of the original isotropic conductivity based on the theory of Teichmuller spaces, and we implement it numerically.

RECOVERING SINGULARITIES FROM BACKSCATTERING IN TWO DIMENSIONS

We have shown that in two dimensions the leading singularities of the quantum mechanical scattering potential are determined by the backscattering data. We assume that the short range potential belongs to a suitable weighted Sobolev space, and by estimating the iterative terms in the Born-expansion we are able to show, that for example for Heaviside-type singularities across a smooth hypersurface, both the location and the size of the jump are recovered from backscattering. The main part of the proof consists in getting sharp enough estimates for the first non-linear Born-term. These estimates are proven using a recent characterization of W 1,p -functions due to P. Hajlasz, and a modification of the classical Triebels Maximal Inequality.

Inverse Boundary Value Problem for Maxwell Equations with Local Data

We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the boundary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov [4] for the Schrödinger equation to Maxwell equations.

Transmission Eigenvalues for Elliptic Operators

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-self-adjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.

TRANSMISSION EIGENVALUES FOR OPERATORS WITH CONSTANT COEFFICIENTS

In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order

Boundary integral equations for the scattering of electromagnetic waves by a homogeneous chiral obstacle

\ge 2

Electrical Impedance Tomography Problem With Inaccurately Known Boundary and Contact Impedances

with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.

The inverse conductivity problem with an imperfectly known boundary in three dimensions

The time‐harmonic scattering of electromagnetic waves by a homogeneous chiral media are considered herein. The scattering problem is reduced using Bohren’s decomposition to an equivalent boundary integral equation which is shown to be uniquely solvable except for a discrete set of values of electromagnetic parameters of the scatterer. It should be noted that the boundary integral operators appearing in this equation are the same as for a dielectric scatterer, and hence their mapping properties are well known.

On Absence and Existence of the Anomalous Localized Resonance without the Quasi-static Approximation

In electrical impedance tomography (EIT) the conductivity function inside a body is reconstructed based on current and voltage data on the boundary of the body. The traditional setting for the EIT problem assumes that the boundary of the body and the electrode-skin contact impedances are known a priori. However, in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances, and it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images. In this paper, we propose a novel reconstruction method in which the systematic errors caused by inaccurately known boundary and contact impedances are eliminated as part of the image reconstruction. The method is tested with real EIT data