Generalized Polarization Tensors Research Articles

Inverse conductivity problem with one measurement: uniqueness of multi-layer structures

In this paper, we study the recovery of multi-layer structures in inverse conductivity problem by using one measurement. First, we define the concept of Generalized Polarization Tensors (GPTs) for multi-layered medium and show some important properties of the proposed GPTs. With the help of GPTs, we present the perturbation formula for general multi-layered medium. Then we derive the perturbed electric potential for multi-layer concentric disks structure in terms of the so-called generalized polarization matrix, whose dimension is the same as the number of the layers. By delicate analysis, we derive an algebraic identity involving the geometric and material configurations of multi-layer concentric disks. This enables us to reconstruct the multi-layer structures by using only one partial-order measurement.

Construction of inclusions with vanishing generalized polarization tensors by imperfect interfaces

AbstractWe investigate the problem of planar conductivity inclusion with imperfect interface conditions. We assume that the inclusion is simply connected. The presence of the inclusion causes a perturbation in the incident background field. This perturbation admits a multipole expansion of which coefficients we call as the generalized polarization tensors (GPTs), extending the previous terminology for inclusions with perfect interfaces. We derive explicit matrix expressions for the GPTs in terms of the incident field, material parameters, and geometry of the inclusion. As an application, we construct GPT‐vanishing structures of general shape that result in negligible perturbations for all uniform incident fields. The structure consists of a simply connected core with an imperfect interface. We provide numerical examples of GPT‐vanishing structures obtained by our proposed scheme.

Inverse Problem for a Planar Conductivity Inclusion

This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs.

Geometric multipole expansion and its application to semi-neutral inclusions of general shape

We consider the conductivity problem with a simply connected or multi-coated inclusion in two dimensions. The potential perturbation due to an inclusion admits a classical multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). The GPTs have been fundamental building blocks in conductivity inclusion problems. In this paper, we present a new concept of geometric multipole expansion and its expansion coefficients, named the Faber polynomial polarization tensors (FPTs), using the conformal mapping and the Faber polynomials associated with the inclusion. The proposed expansion leads us to a series solution method for a simply connected or multi-coated inclusion of general shape, while the classical expansion leads us to a series solution only for a single- or multilayer circular inclusion. We also provide matrix expressions for the FPTs using the Grunsky matrix of the inclusion. In particular, for the simply connected inclusion with extreme conductivity, the FPTs admit simple formulas in terms of the conformal mapping associated with the inclusion. As an application of the concept of the FPTs, we construct semi-neutral inclusions of general shape that show relatively negligible field perturbations for low-order polynomial loadings. These inclusions are of the multilayer structure whose material parameters are determined such that some coefficients of geometric multipole expansion vanish.

Localized Sensitivity Analysis at High-Curvature Boundary Points of Reconstructing Inclusions in Transmission Problems

In this paper, we are concerned with the recovery of the geometric shapes of inhomogeneous inclusions from the associated far field data in electrostatics and acoustic scattering. We present a local resolution analysis and show that the local shape around a boundary point with a mean curvature of high magnitude can be reconstructed more easily and stably. In proving this, we develop a novel mathematical scheme by analyzing the generalized polarization tensors (GPTs) and the scattering coefficients (SCs) coming from the associated scattered fields, which in turn boils down to the analysis of the layer potential operators that sit inside the GPTs and SCs via microlocal analysis. In a delicate and subtle manner, we decompose the reconstruction process into several steps, where all but one step depends on the global geometry, and one particular step depends on the mean curvature at the given boundary point. Then by a sensitivity analysis with respect to local perturbations of the curvature of the boundary surface, we establish the local resolution effects. Our study opens up a new field of mathematical analysis on wave superresolution imaging.

Polarization tensor in de Sitter gauge gravity

The gauge theory of the de Sitter group, [Formula: see text], in the ambient space formalism has been considered in this paper. This method is important to construction of the de Sitter super-conformal gravity and Quantum gravity. [Formula: see text] gauge vector fields are needed which correspond to [Formula: see text] generators of the de Sitter group. Using the gauge-invariant Lagrangian, the field equations of these vector fields have been obtained. The gauge vector field solutions are recalled. By using these solutions, the spin-[Formula: see text] gauge potentials has been constructed. There are two possibilities for presenting this tensor field: rank-[Formula: see text] symmetric and mixed symmetry rank-[Formula: see text] tensor fields. To preserve the conformal transformation, a spin-[Formula: see text] field must be represented by a mixed symmetry rank-[Formula: see text] tensor field, [Formula: see text]. This tensor field has been rewritten in terms of a generalized polarization tensor field and a de Sitter plane wave. This generalized polarization tensor field has been calculated as a combination of vector polarization, [Formula: see text], and tensor polarization of rank-2, [Formula: see text], which can be used in the gravitational wave consideration. There is a certain extent of arbitrariness in the choice of this tensor and we fix it in such a way that, in the limit, [Formula: see text], one obtains the polarization tensor in Minkowski spacetime. It has been shown that under some simple conditions, the spin-[Formula: see text] mixed symmetry rank-[Formula: see text] tensor field can be simultaneously transformed by unitary irreducible representation of de Sitter and conformal groups ([Formula: see text]).

Reconstruction of a small acoustic inclusion via time-dependent polarization tensors

Abstract This paper aims at introducing the concept of time-dependent polarization tensors (TDPTs) for the wave equation associated to a diametrically small acoustic inclusion, with constitutive parameters different from those of the background and size smaller than the operating wavelength. Firstly, the solution to the Helmholtz equation is considered, and a rigourous systematic derivation of a complete asymptotic expansion of the scattered field due to the presence of the inclusion is presented. Then, by applying the Fourier transform, the corresponding time-domain expansion is readily obtained after truncating the high frequencies. The new concept of TDPTs is shown to be promising for performing imaging. In particular, the optimization approach proposed by Ammari et al. (Ammari, H., Kang, H., Kim, & E. Lee, J.-Y. (2012)The generalized polarization tensors for resolved imaging. Part II: shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements. Math. Comp., 81, 839–860.) is extended to TDPTs. Numerical simulations are presented, showing that the TDPTs reconstructed from noisy measurements allow to image fine shape details of the inclusion.

Multi-scale Classification for Electrosensing

This paper introduces a premier and innovative (real-time) multi-scale method for target classification in electrosensing. The intent is that of mimicking the behavior of the weakly electric fish, which is able to retrieve much more information about the target by approaching it. The method is based on a family of transform-invariant shape descriptors computed from generalized polarization tensors (GPTs) reconstructed at multiple scales. The evidence provided by the different descriptors at each scale is fused using Dempster--Shafer theory. Numerical simulations show that the recognition algorithm we propose performs undoubtedly well and yields a robust classification.

Analytical shape recovery of a conductivity inclusion based on Faber polynomials

A conductivity inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from multistatic measurements. As a modification of GPTs, the Faber polynomial polarization tensors (FPTs) were recently introduced in two dimensions. In this study, we design two novel analytical non-iterative methods for recovering the shape of a simply connected inclusion from GPTs by employing the concept of FPTs. First, we derive an explicit expression for the coefficients of the exterior conformal mapping associated with an inclusion in a simple form in terms of GPTs, which allows us to accurately reconstruct the shape of an inclusion with extreme or near-extreme conductivity. Secondly, we provide an explicit asymptotic formula in terms of GPTs for the shape of an inclusion with arbitrary conductivity by considering the inclusion as a perturbation of its equivalent ellipse. With this formula, one can non-iteratively approximate an inclusion of general shape with arbitrary conductivity, including a straight or asymmetric shape. Numerical experiments demonstrate the validity of the proposed analytical approaches.

Reconstruction of Domains with Algebraic Boundaries from Generalized Polarization Tensors

This paper aims at showing the stability of the recovery of a smooth planar domain with a real algebraic boundary from a finite number of its generalized polarization tensors. It is a follow-up of ...

Electro-sensing of inhomogeneous targets

This paper addresses the electro-sensing problem for weakly electric fish in the case of inhomogeneous targets. It aims at providing a shape descriptor-based classification for inhomogeneous targets from measurements of the potentials on the skin of the fish. The approach is based on new invariants for the contracted generalized polarization tensors associated with inhomogeneous objects. The numerical simulations show that by comparing these invariants with those in a dictionary of precomputed homogeneous and inhomogeneous targets, one can successfully classify the inhomogeneous target.

Identification of an algebraic domain in two dimensions from a finite number of its generalized polarization tensors

This paper aims at studying how finitely many generalized polarization tensors of an algebraic domain can be used to determine its shape. Precisely, given a planar set with real algebraic boundary, it is shown that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors. The size of the matrix depends polynomially on the degree of the boundary of the algebraic domain. The density with respect to Hausdorff distance of algebraic domains among all bounded domains invites to extend via approximation our reconstruction procedure beyond its natural context. Based on this, a new algorithm for shape recognition/classification is proposed with some strong hints about its efficiency.

Dispersion characteristics of anisotropic unmagnetized ultra-relativistic transverse plasma wave with arbitrary electron degeneracy

Thermal momentum space anisotropy is ubiquitous in many astrophysical and laboratory plasma environments. Using Vlasov-Maxwell's model equations, a generalized polarization tensor for a collisionless ultra-relativistic unmagnetized electron plasma is derived. In particular, the tensor is obtained by considering anisotropy in the momentum space. The integral of moments of Fermi-Dirac distribution function in terms of Polylog functions is used for describing the border line plasma systems (TeTFe≈1) comprising arbitrary electron degeneracy, where Te and TFe, are thermal and Fermi temperatures, respectively. Furthermore, the effects of variation in thermal momentum space anisotropy on the electron equilibrium number density and the spectrum of electromagnetic waves are analyzed.

Corner Effects on the Perturbation of an Electric Potential

We consider the perturbation of an electric potential due to an insulating inclusion with corners. This perturbation is known to admit a multipole expansion whose coeffcients are linear combinations of generalized polarization tensors. We define new geometric factors of a simple planar domain in terms of a conformal mapping associated with the domain. The geometric factors share properties of the generalized polarization tensors and are the Fourier series coeffcients of a generalized external angle of the inclusion boundary. Since the generalized external angle contains the Dirac delta singularity at corner points, we can determine a criteria for the existence of corner points on the inclusion boundary in terms of the geometric factors. We illustrate and validate our results with numerical examples computed to a high degree of precision using integral equation techniques, the Nystrom discretization, and recursively compressed inverse preconditioning. (Less)

Reconstructing Fine Details of Small Objects by Using Plasmonic Spectroscopic Data. Part II: The Strong Interaction Regime

This paper is concerned with the inverse problem of reconstructing a small object from far field measurements by using the field interaction with a plasmonic particle which can be viewed as a passive sensor. It is a follow-up of the work [H. Ammari et al., Reconstructing fine details of small objects by using plasmonic spectroscopic data, SIAM J. Imag. Sci., to appear], where the intermediate interaction regime was considered. In that regime, it was shown that the presence of the target object induces small shifts to the resonant frequencies of the plasmonic particle. These shifts, which can be determined from the far field data, encodes the contracted generalized polarization tensors of the target object, from which one can perform reconstruction beyond the usual resolution limit. The main argument is based on perturbation theory. However, the same argument is no longer applicable in the strong interaction regime as considered in this paper due to the large shift induced by strong field interaction between the particles. We develop a novel technique based on conformal mapping theory to overcome this difficulty. The key is to design a conformal mapping which transforms the two particle system into a shell-core structure, in which the inner dielectric core corresponds to the target object. We show that a perturbation argument can be used to analyze the shift in the resonant frequencies due to the presence of the inner dielectric core. This shift also encodes information of the contracted polarization tensors of the core, from which one can reconstruct its shape, and hence the target object. Our theoretical findings are supplemented by a variety of numerical results based on an efficient optimal control algorithm. The results of this paper make the mathematical foundation for plasmonic sensing complete.

Reconstructing Fine Details of Small Objects by Using Plasmonic Spectroscopic Data

This paper is concerned with the inverse problem of reconstructing a small object from far-field measurements. The inverse problem is severely ill-posed because of the diffraction limit and low signal to noise ratio. We propose a novel methodology to solve this type of inverse problem based on an idea from plasmonic sensing. By using the field interaction with a known plasmonic particle, the fine detail information of the small object can be encoded into the shift of the resonant frequencies of the two particle system in the far field. In the intermediate interaction regime, we show that this information is exactly the generalized polarization tensors associated with the small object, from which one can perform the reconstruction. Our theoretical findings are supplemented by a variety of numerical results. The results in the paper also provide a mathematical foundation for plasmonic sensing.

Model based estimation of ellipsoidal object using artificial electric sense

In this article we address the issue of shape estimation using electric sense inspired by the active electric fish. These fish can perceive their environment by measuring the perturbations in a self-generated electric field caused by nearby objects. The approach proceeded in three stages. Firstly, the object was detected and its electric properties (insulator or conductor) identified. Secondly, the object was localized using the multiple signal classification algorithm, which was originally developed to localize a radio wave emitter using a network of antennas. Thirdly, the shape estimation relied on the concept of generalized polarization tensor, which enabled us to model the electric response of an object polarized by an ambient electric field. We describe the implementation of the approach through numerous experiments. The system was able to estimate shape with an average error of 16%, and opened the way toward further improvements. In particular, self-aligning the sensor with the ellipsoid through a reactive feedback makes the shape estimation errors drop to 10%.

Covariant and infrared-free graviton two-point function in de Sitter spacetime

In this paper, the two-point function of linearized gravitons on de Sitter (dS) space is presented. Technically, respecting the dS ambient space notation, the field equation is given by the coordinate-independent Casimir operators of the de Sitter group. Analogous to the quantization of the electromagnetic field in Minkowski space, the field equation admits gauge solutions. The notation allows us to exhibit the formalism of Gupta-Bleuler triplets for the present field in exactly the same manner as it occurs for the electromagnetic field. In this regard, centering on the spin-two part (the traceless part, ${\cal{K}}^t$), the field solution is written as a product of a generalized polarization tensor and a minimally coupled massless scalar field. Then, admitting a de Sitter-invariant vacuum through the so-called "Krein space quantization", the de Sitter fully covariant two-point function is calculated. This function is interestingly free of pathological large distance behavior (infrared divergence). Moreover, the spin-zero part (the pure-trace part; ${\cal{K}}^{pt}$) of the field is discussed in this paper. It is shown that the implications of the dS group unitary irreducible representations restrict the gauge-fixing parameter to the optimal value, which remarkably results in the pure-trace part be written in terms of a conformally coupled massless scalar field.

Numerical implementation for reconstruction of inhomogeneous conductivities via generalized polarization tensors

This paper deals with numerical methods for reconstruction of inhomogeneous conductivities. We use the concept of Generalized Polarization Tensors to do reconstruction. Basic resolution and stability analysis are presented. Least‐square norm methods with respect to Generalized Polarization Tensors are used for reconstruction of conductivities. Finally, reconstruction of three different types of conductivities in the plane is demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.

Construction of conformal mappings by generalized polarization tensors

We present a new systematic method to compute the Riemann mapping from the outside of the unit disc to the outside of a simply connected domain. We derive explicit relations between the coefficients of the Riemann mapping and the generalized polarization tensors associated with the domain. Because the generalized polarization tensors can be computed numerically, we are able to compute the coefficients of the Riemann mapping using these relations. Effectiveness of the method is validated by numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.