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.     

Orthotropic conductivity reconstruction with virtual‐resistive network and Faraday's law

We obtain the existence and the uniqueness at the same time in the reconstruction of orthotropic conductivity in two‐space dimensions by using two sets of internal current densities and boundary conductivity. The curl‐free equation of Faraday's law is taken instead of the elliptic equation in a divergence form that is typically used in electrical impedance tomography. A reconstruction method based on layered bricks‐type virtual‐resistive network is developed to reconstruct orthotropic conductivity with up to 40% multiplicative noise. Copyright © 2015 John Wiley & Sons, Ltd.     

Sparsity reconstruction in electrical impedance tomography: An experimental evaluation

We investigate the potential of sparsity constraints in the electrical impedance tomography (EIT) inverse problem of inferring the distributed conductivity based on boundary potential measurements. In sparsity reconstruction, inhomogeneities of the conductivity are a priori assumed to be sparse with respect to a certain basis. This prior information is incorporated into a Tikhonov-type functional by including a sparsity-promoting ?¹-penalty term. The functional is minimized with an iterative soft shrinkage-type algorithm. In this paper, the feasibility of the sparsity reconstruction approach is evaluated by experimental data from water tank measurements. The reconstructions are computed both with sparsity constraints and with a more conventional smoothness regularization approach. The results verify that the adoption of ?¹-type constraints can enhance the quality of EIT reconstructions: in most of the test cases the reconstructions with sparsity constraints are both qualitatively and quantitatively more feasible than that with the smoothness constraint.     

The enclosure method for an inverse problem arising from a spot welding

A two dimensional version of a reconstruction problem of an unknown weld on the interface between two electric conductive plates is considered. It is assumed that the two plates have a same known isotropic homogeneous conductivity, and the line where the welding area is located is known. Under these assumptions, an explicit extraction formula of the location of the tips of the welding area on the line from a single set of an electric current density and the corresponding voltage potential on the boundary of the material formed by the plates is given. This result may have possibility of application to quality evaluation of spot welding fixation strength of a lamina. Copyright © 2015 John Wiley & Sons, Ltd.     

A level set method to reconstruct the discontinuity of the conductivity in EIT

In this paper, one level set method is applied to finding the interface of discontinuity of the conductivity in EIT(electrical impedance tomography) problem. By choosing one suitable velocity function, a level set reconstruction algorithm is proposed. The theoretical results for EIT problem and regularization are given. Finally the numerical examples demonstrate that the reconstruction algorithm is efficient and stable. The work was supported by the National Natural Science Foundation of China (Grant Nos. 10431030, 10771138), the Shanghai Natural Science Foundation (Grant No. 07JC14001), the National Basic Research Program (Grant No. 2005CB321701) and Ministry of Education of China and State Administration of Foreign Experts Affairs of China under an 111 project (Grant No. B08018).     

Retrieving the time-dependent thermal conductivity of an orthotropic rectangular conductor

The aim of this paper is to determine the thermal properties of an orthotropic planar structure characterized by the thermal conductivity tensor in the coordinate system of the main directions (Oxy) being diagonal. In particular, we consider retrieving the time-dependent thermal conductivity components of an orthotropic rectangular conductor from nonlocal overspecified heat flux conditions. Since only boundary measurements are considered, this inverse formulation belongs to the desirable approach of non-destructive testing of materials. The unique solvability of this inverse coefficient problem is proved based on the Schauder fixed point theorem and the theory of Volterra integral equations of the second kind. Furthermore, the numerical reconstruction based on a nonlinear least-squares minimization is performed using the MATLAB optimization toolbox routine lsqnonlin. Numerical results are presented and discussed in order to illustrate the performance of the inversion for orthotropic parameter identification.     

On computation of test dipoles for factorization method

In electrical impedance tomography, one tries to recover the spatial conductivity distribution inside a body from boundary measurements of current and voltage. In many important situations, the examined object has known background conductivity but is contaminated by inhomogeneities. The factorization method of Kirsch provides a tool for locating such inclusions. The computational attractiveness of the factorization technique relies heavily on efficient computation of Dirichlet boundary values of potentials created by dipole sources located inside the examined object and corresponding to the homogeneous Neumann boundary condition and to the known background conductivity. In certain simple situations, these test potentials can be written down explicitly or given with the help of suitable analytic maps, but, in general, they must be computed numerically. This work introduces an inexpensive algorithm for approximating the test potentials in the framework of real-life electrode measurements and analyzes how well this technique can be imbedded in the factorization method. The performance of the resulting fast reconstruction algorithm is tested in two spatial dimensions. The work of the second author was supported by the Academy of Finland (project 115013), the Finnish Funding Agency for Technology and Innovation (project 40084/06), the Finnish Cultural Foundation and the Finnish Foundation for Technology Promotion.     

The effective conductivity of a randomly inhomogeneous medium

Using a representation of the solution to the diffusion equation in a randomly inhomogeneous medium in the form of a Feynman path integral an explicit expression is obtained for the effective conductivity in a space of arbitrary dimension. A calculation of the path integral only turns out to be possible in the case of a large-scale limit. In particular, it is shown that in the three-dimensional case the expression for the effective conductivity does not admit of an expansion in terms of the conductivity variance. This indicates that the use of standard perturbation theory in the form of an expansion in terms of the conductivity fluctuations is incorrect.     

Single Measurement Detection Of a Discontinuous Conductivity

Let be a bounded region in R$sup:n$esup: 2. with C$sup:2$esup: boundary and conductivity ggr;Suppose that some region D CC may have been replaced with a material which has a differing C$sup:2$esup: conductivity profile. We show that by applying an appropriate current flux on and measuring the resulting potential on an open subset one can "detect" the presence of the region D, that is. the potentials induced on when D is present versus absent must differ. Moreover. if D and its conductivity are known to satisfy certain a priori restrictions, one can assert that the potentials induced on must differ by a fixed aount which does not depend on the domain D or its conductivity.     

Transverse electrical conductivity of a quantum collisional plasma in the Mermin approach

We derive formulas for the transverse electrical conductivity and the permittivity in a quantum collisional plasma using the kinetic equation for the density matrix in the relaxation approximation in the momentum space. We show that the derived formula becomes the classical formula when the Planck constant tends to zero and that when the electron collision rate tends to zero (i.e., the plasma becomes collisionless), the derived formulas become the previously obtained Lindhard formulas. We also show that when the wave number tends to zero, the quantum conductivity becomes classical. We compare the obtained conductivity with the conductivity obtained by Lindhard and with the classical conductivity     

A coupled complex boundary method for an inverse conductivity problem with one measurement

We recently proposed in [Cheng, XL et al. A novel coupled complex boundary method for inverse source problems Inverse Problem 2014 30 055002] a coupled complex boundary method (CCBM) for inverse source problems. In this paper, we apply the CCBM to inverse conductivity problems (ICPs) with one measurement. In the ICP, the diffusion coefficient q is to be determined from both Dirichlet and Neumann boundary data. With the CCBM, q is sought such that the imaginary part of the solution of a forward Robin boundary value problem vanishes in the problem domain. This brings in advantages on robustness and computation in reconstruction. Based on the complex forward problem, the Tikhonov regularization is used for a stable reconstruction. Some theoretical analysis is given on the optimization models. Several numerical examples are provided to show the feasibility and usefulness of the CCBM for the ICP. It is illustrated that as long as all the subdomains share some portion of the boundary, our CCBM-based Tikhonov regularization method can reconstruct the diffusion parameters stably and effectively.     

Existence and Regularity of Solutions of a Nonlinear Nonuniformly Elliptic System Arising from a Thermistor Problem

A thermistor is an electric circuit device made of ceramic material whose electric conductivity depends on the temperature. If the only heat source is the electric heating, the temperature and the electric potential satisfy a nonlinear elliptic system which is also degenerate if the electric conductivity is not uniformly bounded from above or away from zero. Under general boundary conditions, we establish existence and Hölder continuity of solutions of such a nonlinear nonuniformly elliptic system. When the elechic conductivity linearly depends on the temperature, we provide a non-uniqueness and non-existence example.     

A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity

We revisit the asymptotic formulas originally derived in [D.J. Cedio-Fengya, S. Moskow, M.S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998) 553–595; A. Friedman, M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Ration. Mech. Anal. 105 (1989) 299–326]. These formulas concern the perturbation in the voltage potential caused by the presence of diametrically small conductivity inhomogeneities. We significantly extend the validity of the previously derived formulas, by showing that they are asymptotically correct, uniformly with respect to the conductivity of the inhomogeneities. We also extend the earlier formulas by allowing the conductivities of the inhomogeneities to be completely arbitrary L^∞$L^{\infty }$, positive definite, symmetric matrix-valued functions. We briefly discuss the relevance of the uniform asymptotic validity, and the admission of arbitrary anisotropically conducting inhomogeneities, as far as applications of the perturbation formulas to “approximate cloaking” are concerned.     

On a reconstruction algorithm for the trajectory and control in a delay system

We discuss a problem of the dynamic reconstruction of unmeasured coordinates of the phase vector and unknown controls in nonlinear vector equations with delay. A regularizing algorithm is proposed for the reconstruction of both controls and unmeasured coordinates simultaneously with the processes. The algorithm is stable with respect to information noises and computational errors.     

Effect of discontinuity in density and conductivity on the hydromagnetic stability of a two-dimensional jet

The hydromagnetic stability of an incompressible, inviscid, and two-dimensional jet in the presence of a discontinuity in density and conductivity across the jet surfaces is studied against small and large wavelength disturbances. In the equilibrium state the magnetic field is uniform and aligned with the direction of jet velocity. Study of several different cases shows that discontinuity in density or conductivity or in the product of density and conductivity across the jet surfaces has a stabilizing effect on the jet.     

The Effective Conductivity of a Regular Composite Medium

The advantages of a Green's function method are exemplifiedby the analysis of steady potential flow through a homogeneousmedium containing a regular array of subdomains where a differentvalue of the conductivity prevails. It is shown, in particular,that the results previously obtained by Rayleigh for the squarearray of cylindrical obstacles with infinite conductivity arerecovered in a more efficient manner; and that the method issuitable for handling either widely dispersed or closely packedarrays.     

A variational proximal alternating linearized minimization in a given metric for limited-angle CT image reconstruction

Due to the restriction of computed tomography (CT) scanning environment, the acquired projection data may be incomplete for exact CT reconstruction. Though some convex optimization methods, such as total variation minimization based method, can be used for incomplete data reconstruction, the edge of reconstruction image may be partly distorted for limited-angle CT reconstruction. To promote the quality of reconstruction image for limited-angle CT imaging, in this paper, a nonconvex and nonsmooth optimization model was investigated. To solve the model, a variational proximal alternating linearized minimization (VPALM) method based on proximal mapping in a given metric was proposed. The proposed method can avoid computing the inverse of a huge system matrix thus can be used to deal with the larger-scale inverse problems. What’s more, we show that each bounded sequence generated by VPALM globally converges to a critical point based on the Kurdyka–Lojasiewicz property. Real data experiments are used to demonstrate the viability and effectiveness of VPALM method, and the results show that the proposed method outperforms two classical CT reconstruction methods.     

Reconstruction of a penetrable obstacle by complex spherical waves

We consider an inverse acoustic scattering problem for identifying a non-convex penetrable obstacle in three dimensions in a homogeneous medium. We apply the complex geometrical optics solutions with logarithmic phase, which is called complex spherical waves, to reconstruction problem. The reconstruction schemes will be demonstrated in the last section.     

MHD Couette flow of a viscous stratified fluid of large conductivity

The MHD Couette flow of a viscous stratified fluid of large electrical conductivity with suction and injection at the plane boundaries is studied when the plane boundaries are maintained at different temperatures. The Oseen type governing equations are formulated using the method suggested by Greenspan for stratified fluids. Introducing the similarity variables, the linearised equations are solved to obtain the velocity and temperature distributions. The results show that the behaviour of velocity and temperature in fluids of large conductivity is different from the behaviour of velocity and temperature for fluids of finite conductivity. The effect of the magnetic field on the load capacity is investigated for the case when the width of the channel is small.     

On the optimal reconstruction of a solution of the Poisson equation

We consider the problem of optimal reconstruction of a solution of the generalized Poisson equation in a bounded domain Q with homogeneous boundary conditions for the case in which the right-hand side of the equation is fuzzy. We assume that right-hand sides of the equations belong to generalized Sobolev classes and finitely many Fourier coefficients of the right-hand sides of the equations are known with some accuracy in the Euclidean metric. We find the optimal reconstruction error and construct a family of optimal reconstruction methods. The problem on the best choice of the coefficients to be measured is solved.