Identification of an inclusion in multifrequency electric impedance tomography

The multifrequency electrical impedance tomography is considered to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation, we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous stability estimates. Our results in this paper are quite surprising in inverse conductivity problem since in general infinitely many input currents are needed to obtain the uniqueness in the determination of the conductivity.     

Determination of the coefficient in the stationary nonlinear equation of heat conduction

The article considers the problem of determining the solution-dependent coefficient of heat conductivity in a stationary nonlinear equation of heat conduction containing a parameter. Additional information for the determination of heat conductivity is provided by a function dependent on a parameter, which is obtained by solving a boundary-value problem. A uniqueness theorem is proved for the inverse problem.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 13–17, 1993.     

The Calderón problem for quasilinear elliptic equations

In this paper we show uniqueness of the conductivity for the quasilinear Calderón's inverse problem. The nonlinear conductivity depends, in a nonlinear fashion, of the potential itself and its gradient. Under some structural assumptions on the direct problem, a real-valued conductivity allowing a small analytic continuation to the complex plane induce a unique Dirichlet-to-Neumann (DN) map. The method of proof considers some complex-valued, linear test functions based on a point of the boundary of the domain, and a linearization of the DN map placed at these particular set of solutions.     

Problem of Determining the Anisotropic Conductivity in Electrodynamic Equations

Doklady Mathematics - For a system of electrodynamic equations, the inverse problem of determining an anisotropic conductivity is considered. It is supposed that the conductivity is described by a...     

The inverse scattering problem for a partially coated penetrable cavity with interior measurements

In this paper we consider the inverse scattering problem for a cavity that is bounded by a partially coated penetrable inhomogeneous medium of compact support and recover the shape of the cavity and the surface conductivity from a knowledge of measured scattered waves due to point sources located on a curve or surface inside the cavity. First, we prove that both the shape of the cavity and the surface conductivity on the coated part can be uniquely determined from a knowledge of the measured data. Next, we establish a linear sampling method for determining both the shape of the cavity and the surface conductivity. A central role in our justification is played by an eigenvalue problem which we call the exterior transmission eigenvalue problem. Finally, we present some numerical examples to illustrate the validity of our method.     

Numerical method for solving a two-dimensional electrical impedance tomography problem in the case of measurements on part of the outer boundary

The two-dimensional electrical impedance tomography problem is considered in the case of a piecewise constant electrical conductivity. The task is to determine the unknown boundary separating the regions with different conductivity values, which are known. Input information is the electric field measured on a portion of the outer boundary of the domain. A numerical method for solving the problem is proposed, and numerical results are presented.     

On the Main Aspects of the Inverse Conductivity Problem

We consider a nonlinear inverse problem for an elliptic partial differential equation known as the Calder{\''o}n problem or the inverse conductivity problem. Based on several results, we briefly summarize them to motivate this research field. We give a general view of the problem by reviewing the available results for $C^2$ conductivities. After reducing the original problem to the inverse problem for a Schr\"odinger equation, we apply complex geometrical optics solutions to show its uniqueness. After extending the ideas of the uniqueness proof result, we establish a stable dependence between the conductivity and the boundary measurements. By using the Carleman estimate, we discuss the partial data problem, which deals with measurements that are taken only in a part of the boundary.     

From restoration by topological gradient to medical image segmentation via an asymptotic expansion

The aim of this article is to present an application of the topological asymptotic expansion to the medical image segmentation problem. We first recall the classical variational of the image restoration problem, and its resolution by topological asymptotic analysis in which the identification of the diffusion coefficient can be seen as an inverse conductivity problem. The conductivity is set either to a small positive coefficient (on the edge set), or to its inverse (elsewhere). In this paper a technique based on a power series expansion of the solution to the image restoration problem with respect to this small coefficient is introduced. By considering the limit when this coefficient goes to zero, we obtain a segmented image, but some numerical issues do not allow a too small coefficient. The idea is to use the series expansion to approximate the asymptotic solution with several solutions corresponding to positive (larger than a threshold) conductivity coefficients via a quadrature formula. We illustrate this approach with some numerical results on medical images.     

Asymptotic and boundary integral formulae for the computational reconstruction of small conductivity imperfections

In this paper we investigate the electrostatic problem of determining conductivity profiles from the knowledge of boundary currents and voltages. We obtain an improved estimate for the voltage potential of a two-dimensional conductor having finitely many circular inclusions and piecewise constant conductivity profile. We derive an asymptotic expansion for the voltage potential in terms of the reference voltage potential and the location, size, and conductivity of the inhomogeneities. This representation is used to formulate the nonlinear least squares problem for estimating the location and size of the inhomogeneities. Required boundary data for the voltage potential are generated numerically by solving a system of integral equations. Computational experiments are presented to demonstrate the effectiveness of our identification procedure.     

Reconstruction of Less Regular Conductivities in the Plane

Abstract

We consider the inverse conductivity problem of how to reconstruct an isotropic electric conductivity distribution in a conductive body from static electric measurements on the boundary of the body. An exact algorithm for the reconstruction of a conductivity in a planer domain from the associated Dirichlet-to-Neumann map is given. We assume that the conductivity has essentially one derivative, and hence we improve earlier reconstruction results. The method relies on a reduction of the conductivity equation to a first order system, to which the ?¯-method of inverse scattering theory can be applied.     

Asymptotic and boundary integral formulae for the computational reconstruction of small conductivity imperfections

In this paper we investigate the electrostatic problem of determining conductivity profiles from the knowledge of boundary currents and voltages. We obtain an improved estimate for the voltage potential of a two-dimensional conductor having finitely many circular inclusions and piecewise constant conductivity profile. We derive an asymptotic expansion for the voltage potential in terms of the reference voltage potential and the location, size, and conductivity of the inhomogeneities. This representation is used to formulate the nonlinear least squares problem for estimating the location and size of the inhomogeneities. Required boundary data for the voltage potential are generated numerically by solving a system of integral equations. Computational experiments are presented to demonstrate the effectiveness of our identification procedure.     

On one problem of electrophysics with discontinuous nonlinearity

We consider a conductor heating problem in the following setting: a constant voltage and a constant temperature are maintained on the conductor surface, and the electric conductivity of the material experiences jumps when passing through certain temperatures. Earlier-obtained results for problems with a spectral parameter for equations of elliptic type with discontinuous nonlinearities are applied to this problem of electrophysics. We weaken the conditions imposed on the set of points of discontinuity of the nonlinearity (the electric conductivity of the conductor) occurring in the equation.     

Stable reconstruction of regular 1-Harmonic maps with a given trace at the boundary

We consider the numerical solvability of the Dirichlet problem for the 1-Laplacian in a planar domain endowed with a metric conformal with the Euclidean one. Provided that a regular solution exists, we present a globally convergent method to find it. The global convergence allows to show a local stability in the Dirichlet problem for the 1-Laplacian nearby regular solutions. Such problems occur in conductivity imaging, when knowledge of the magnitude of the current density field (generated by an imposed boundary voltage) is available inside. Numerical experiments illustrate the feasibility of the convergent algorithm in the context of the conductivity imaging problem.     

An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion

We rigorously derive an asymptotic expansion of the steady-state voltage potentials in the presence of a conductivity inclusion of small volume that is close to a planar surface. This new formula is motivated by the practically important inverse problem of imaging a conductivity inclusion near a planar interface.     

Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity

Studied in this paper is the Cauchy problem of the two-dimensional magnetohydrodynamics system with inhomogeneous density and electrical conductivity. It is shown that the 2-D incompressible inhomogeneous magnetohydrodynamics system with a constant viscosity is globally well-posed for a generic family of the variations of the initial data and an inhomogeneous electrical conductivity. Moreover, it is established that the system is globally well-posed in the critical spaces if the electrical conductivity is homogeneous.     

New exact multi-coated ellipsoidal inclusion model for anisotropic thermal conductivity of composite materials

The present study deals with a new micromechanical modeling of the thermal conductivity of multi-coated inclusion-reinforced composites. The proposed approach has been developed in the general frame of anisotropic thermal behavior per phase and arbitrary ellipsoidal inclusions. Based on the Green's function technique, a new formulation of the problem of multi-coated inclusion is proposed. This formulation consists in constructing a system of integral equations, each associated to the thermal conductivity of each coating and the reference medium. Thanks to the concept of interior- and exterior-point Eshelby's conduction tensors, the exact solution of the problem of multicoated inclusion is obtained. Analytical expressions of the intensity in each phase and the effective thermal conductivity of the composite, through homogenizations schemes such as Generalized self-consistent and Mori-Tanaka models are provided. Results of the present model are successfully compared with those issued from both analytical models and finite elements methods for composites with doubly coated inclusions. Moreover, the developed micromechanical model has been applied to a three phase composite materials in order to analyze combined effects of the aspect ratio and the volume fraction of the ellipsoidal inclusions, the anisotropy of the thermal conductivity of interphase, the thermal conductivity contrast between local phases on the predicted effective thermal conductivity.     

Asymptotic solution of the problem of nonstationary thermal conductivity of laminated anisotropic inhomogeneous shells

External asymptotic expansions of the solutions of the problem of nonstationary thermal conductivity of laminated anisotropic inhomogeneous shells under different boundary conditions on faces are constructed. We analyze the obtained two-dimensional resolving equations and investigate the asymptotic properties of the solutions of the problem of thermal conductivity. A physical justification of some features of the asymptotic expansion of temperature is presented.     

Investigation of boundary-value problem for slow flow of a sphere by viscous non-isothermal gas

We obtain a solution to a boundary-value problem of a flow of spherical form particle for stationary system of equations of viscous non-isothermal gaseous medium including the Stokes equation, heat conductivity equation, and state equation with account taken of dependence of viscosity, heat conductivity, and density of gaseous medium on temperature.     

Local Existence of Bounded Solutions to the Degenerate Stefan Problem with Joule's Heating

This paper deals with the degenerate Stefan problem with Joule's heating, which describes the combined effects of heat and electrical current Rows in a metal. The local existence of a bounded weak solution for the problem in proved. Also a degenerate thermistor problem with continuous conductivity is considered.     

Global existence of solutions for the 1-D radiative and reactive viscous gas dynamics

In this paper, we prove the existence of a global solution to an initial-boundary value problem for 1-D flows of the viscous heat-conducting radiative and reactive gases. The key point here is that the growth exponent of heat conductivity is allowed to be any nonnegative constant; in particular, constant heat conductivity is allowed.