On the scattered field generated by a ball inhomogeneity of constant index

We consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a disk of radius $\epsilon$ and another one outside. We derive sharp estimates of the size of the scattered field caused by this disk inhomogeneity, for any frequencies and any contrast. We also provide a broadband estimate, that is, a uniform bound for the scattered field for any contrast, and any frequencies outside of a set which tend to zero with $\epsilon$.Comment: 37 pages, 3 figure

Elliptic regularity theory applied to time harmonic anisotropic Maxwell's equations with less than Lipschitz complex coefficients

The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with $C^{1,1}$ boundary. We assume that at least one of the material parameters is $W^{1,3+\delta}$ for some $\delta>0$. Using regularity theory for second order elliptic partial differential equations, we derive $W^{1,p}$ estimates and H\"older estimates for electric and magnetic fields up to the boundary. We also derive interior estimates in bi-anisotropic media.Comment: 19 page

Finite Element Approximation of Elliptic Homogenization Problems in Nondivergence-Form

We use uniform $W^{2,p}$ estimates to obtain corrector results for periodic homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme.Comment: 39 page

Combining Radon transform and Electrical Capacitance Tomography for a 2d+12d+1 imaging device

This paper describes a coplanar non invasive non destructive capacitive imaging device. We first introduce a mathematical model for its output, and discuss some of its theoretical capabilities. We show that the data obtained from this device can be interpreted as a weighted Radon transform of the electrical permittivity of the measured object near its surface. Image reconstructions from experimental data provide good surface resolution as well as short depth imaging, making the apparatus a $2d+1$ imager. The quality of the images leads us to expect that excellent results can be delivered by \emph{ad-hoc} optimized inversion formulas. There are also interesting, yet unexplored, theoretical questions on imaging that this sensor will allow to test

Numerical Computation of approximate Generalized Polarization Tensors

In this paper we describe a method to compute Generalized Polarization Tensors. These tensors are the coefficients appearing in the multipolar expansion of the steady state voltage perturbation caused by an inhomogeneity of constant conductivity. As an alternative to the integral equation approach, we propose an approximate semi-algebraic method which is easy to implement. This method has been integrated in a Myriapole, a matlab routine with a graphical interface which makes such computations available to non-numerical analysts

Foreign Object Detection and Quantification of Fat Content Using A Novel Multiplexing Electric Field Sensor

There is an ever growing need to ensure the quality of food and assess specific quality parameters in all the links of the food chain, ranging from processing, distribution and retail to preparing food. Various imaging and sensing technologies, including X-ray imaging, ultrasound, and near infrared reflectance spectroscopy have been applied to the problem. Cost and other constraints restrict the application of some of these technologies. In this study we test a novel Multiplexing Electric Field Sensor (MEFS), an approach that allows for a completely non-invasive and non-destructive testing approach. Our experiments demonstrate the reliable detection of certain foreign objects and provide evidence that this sensor technology has the capability of measuring fat content in minced meat. Given the fact that this technology can already be deployed at very low cost, low maintenance and in various different form factors, we conclude that this type of MEFS is an extremely promising technology for addressing specific food quality issues

On one dimensional inverse problems arising from polarimetric measurements of nematic liquid crystals

We revisit the problem of determining dielectric parameters in layered nematic liquid crystals from polarimetric measurements originally introduced by Lionheart & Newton. After a detailed analysis of the model, of the scales involved, and of natural obstacles to the reconstruction of more than one dielectric parameters, we produce two simple one-dimensional inverse problems which can be studied without any expertise in liquid crystals. We then confirm that very little can be recovered about the internal configuration of smooth dielectric parameters from these measurements, and give a uniqueness result for one of the two problem, when the unknown parameter satisfies a monotonicity property. In that case, the available data can be expressed in terms of Laplace and Hankel transforms.Comment: 23 pages 3 figure

An asymptotic representation formula for scattering by thin tubular structures and an application in inverse scattering

We consider the scattering of time-harmonic electromagnetic waves by a penetrable thin tubular scattering object in three-dimensional free space. We establish an asymptotic representation formula for the scattered wave away from the thin tubular scatterer as the radius of its cross-section tends to zero. The shape, the relative electric permeability and the relative magnetic permittivity of the scattering object enter this asymptotic representation formula by means of the center curve of the thin tubular scatterer and two electric and magnetic polarization tensors. We give an explicit characterization of these two three-dimensional polarization tensors in terms of the center curve and of the two two-dimensional polarization tensor for the cross-section of the scattering object. As an application we demonstrate how this formula may be used to evaluate the residual and the shape derivative in an efficient iterative reconstruction algorithm for an inverse scattering problem with thin tubular scattering objects. We present numerical results to illustrate our theoretical findings

An asymptotic representation formula for scattering by thin tubular structures and an application in inverse scattering

We consider the scattering of time-harmonic electromagnetic waves by a penetrable thin tubular scattering object in three-dimensional free space. We establish an asymptotic representation formula for the scattered wave away from the thin tubular scatterer as the radius of its cross-section tends to zero. The shape, the relative electric permeability and the relative magnetic permittivity of the scattering object enter this asymptotic representation formula by means of the center curve of the thin tubular scatterer and two electric and magnetic polarization tensors. We give an explicit characterization of these two three-dimensional polarization tensors in terms of the center curve and of the two two-dimensional polarization tensor for the cross-section of the scattering object. As an application we demonstrate how this formula may be used to evaluate the residual and the shape derivative in an efficient iterative reconstruction algorithm for an inverse scattering problem with thin tubular scattering objects. We present numerical results to illustrate our theoretical findings. Mathematics subject classifications (MSC2010): 35C20, (65N21, 78A46

Stability estimates for systems with small cross-diffusion

We discuss the analysis and stability of a family of cross-diffusion boundary value problems with nonlinear diffusion and drift terms. We assume that these systems are close, in a suitable sense, to a set of decoupled and linear problems. We focus on stability estimates, that is, continuous dependence of solutions with respect to the nonlinearities in the diffusion and in the drift terms. We establish well-posedness and stability estimates in an appropriate Banach space. Under additional assumptions we show that these estimates are time independent. These results apply to several problems from mathematical biology; they allow comparisons between the solutions of different models a priori. For specific cell motility models from the literature, we illustrate the limit of the stability estimates we have derived numerically, and we document the behaviour of the solutions for extremal values of the parameters