arXiv subjects

Habib Ammari

Publications and source records attributed to Habib Ammari.

194 records · Page 11Linked to original sources

arXiv2012

Tracking of a Mobile Target Using Generalized Polarization Tensors

In this paper we apply an extended Kalman filter to track both the location and the orientation of a mobile target from multistatic response measurements. We also analyze the effect of the limited-view aspect on the stability and the efficiency of our tracking approach. Our algorithm is based on the use of the generalized polarization tensors, which can be reconstructed from the multistatic response measurements by solving a linear system. The system has the remarkable property that low order generalized polarization tensors are not affected by the error caused by the instability of higher orders in the presence of measurement noise.

math.NA↗

arXiv2012

Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity

The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and it proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.

math.AP↗

arXiv2012

Boundary perturbations due to the presence of small linear cracks in an elastic body

In this paper, Neumann cracks in elastic bodies are considered. We establish a rigorous asymptotic expansion for the boundary perturbations of the displacement (and traction) vectors that are due to the presence of a small elastic linear crack. The formula reveals that the leading order term is ε^2 where εis the length of the crack, and the ε^3-term vanishes. We obtain an asymptotic expansion of the elastic potential energy as an immediate consequence of the boundary perturbation formula. The derivation is based on layer potential techniques. It is expected that the formula would lead to very effective direct approaches for locating a collection of small elastic cracks and estimating their sizes and orientations.

math.AP↗

arXiv2012

Spectral analysis of the Neumann-Poincaré operator and characterization of the gradient blow-up

When perfectly conducting or insulating inclusions are closely located, stress which is the gradient of the solution to the conductivity equation can be arbitrarily large as the distance between two inclusions tends to zero. It is important to precisely characterize the blow-up of the gradient. In this paper we show that the blow-up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue ${1}/{2}$ of a Neumann-Poincaré type operator defined on the boundaries of the inclusions. By comparing the singular function with the one corresponding to two disks osculating to the inclusions, we quantitatively characterize the blow-up of the gradient in terms of explicit functions.

math.AP↗

arXiv2012

Reconstruction of a piecewise smooth absorption coefficient by an acousto-optic process

The aim of this paper is to tackle the nonlinear optical reconstruction problem. Given a set of acousto-optic measurements, we develop a mathematical framework for the reconstruction problem in the case where the optical absorption distribution is supposed to be a perturbation of a piecewise constant function. Analyzing the acousto-optic measurements, we establish a new equation in the sense of distributions for the optical absorption coefficient. For doing so, we introduce a weak Helmholtz decomposition and interpret in a weak sense the cross-correlation measurements using the spherical Radon transform. We next show how to find an initial guess for the unknown coefficient and finally construct the true coefficient by providing a Landweber type iteration and proving that the resulting sequence converges to the solution of the system constituted by the optical diffusion equation and the new equation mentioned above. Our results in this paper generalize the acousto-optic process proposed in [Ammari, Garnier, Nguyen and Seppecher, 2012] for piecewise smooth optical absorption distributions.

math.AP↗

arXiv2012

Stability estimates for an inverse scattering problem at high frequencies

We consider an inverse scattering problem and its near-field approximation at high frequencies. We first prove, for both problems, Lipschitz stability results for determining the low-frequency component of the potential. Then we show that, in the case of a radial potential supported sufficiently near the boundary, infinite resolution can be achieved from measurements of the near-field operator in the monotone case.

math.AP↗

arXiv2012

Spectral theory of a Neumann-Poincare-type operator and analysis of cloaking due to anomalous localized resonance

The aim of this paper is to give a mathematical justification of cloaking due to anomalous localized resonance (CALR). We consider the dielectric problem with a source term in a structure with a layer of plasmonic material. Using layer potentials and symmetrization techniques, we give a necessary and sufficient condition on the fixed source term for electromagnetic power dissipation to blow up as the loss parameter of the plasmonic material goes to zero. This condition is written in terms of the Newtonian potential of the source term. In the case of concentric disks, we make the condition even more explicit. Using the condition, we are able to show that for any source supported outside a critical radius CALR does not take place, and for sources located inside the critical radius satisfying certain conditions CALR does take place as the loss parameter goes to zero.

math.AP↗

arXiv2012

Quantitative thermo-acoustic imaging: An exact reconstruction formula

This paper aims to mathematically advance the field of quantitative thermo-acoustic imaging. Given several electromagnetic data sets, we establish for the first time an analytical formula for reconstructing the absorption coefficient from thermal energy measurements. Since the formula involves derivatives of the given data up to the third order, it is unstable in the sense that small measurement noises may cause large errors. However, in the presence of measurement noise, the obtained formula, together with a noise regularization technique, provides a good initial guess for the true absorption coefficient. We finally correct the errors by deriving a reconstruction formula based on the least square solution of an optimal control problem and prove that this optimization step reduces the errors occurring and enhances the resolution.

math.AP↗

arXiv2011

Sharp estimates for the Neumann functions and applications to quantitative photo-acoustic imaging in inhomogeneous media

We obtain sharp $L^p$ and Hölder estimates for the Neumann function of the operator $\nabla \cdot γ\nabla - ik$ on a bounded domain. We also obtain quantitative description of its singularity. We then apply these estimates to quantitative photo-acoustic imaging in inhomogeneous media. The problem is to reconstruct the optical absorption coefficient of a diametrically small anomaly from the absorbed energy density.

math.AP↗

arXiv2011

Enhancement of near-cloaking. Part II: the Helmholtz equation

The aim of this paper is to extend the method of improving cloaking structures in the conductivity to scattering problems. We construct very effective near-cloaking structures for the scattering problem at a fixed frequency. These new structures are, before using the transformation optics, layered structures and are designed so that their first scattering coefficients vanish. Inside the cloaking region, any target has near-zero scattering cross section for a band of frequencies. We analytically show that our new construction significantly enhances the cloaking effect for the Helmholtz equation.

math.AP↗

arXiv2011

Enhancement of Near Cloaking Using Generalized Polarization Tensors Vanishing Structures. Part I: The Conductivity Problem

The aim of this paper is to provide an original method of constructing very effective near-cloaking structures for the conductivity problem. These new structures are such that their first Generalized Polarization Tensors vanish. We show that this in particular significantly enhances the cloaking effect. We then present some numerical examples of Generalized Polarization Tensors vanishing structures.

math-ph↗

arXiv2010

Layer Potential Techniques for the Narrow Escape Problem

The narrow escape problem consists of deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. We explicitly show the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems developed in \cite{book}, we also construct high-order asymptotic formulas for eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.

math.AP↗

arXiv2009

Progress on the Strong Eshelby's Conjecture and Extremal Structures for the Elastic Moment Tensor

We make progress towards proving the strong Eshelby's conjecture in three dimensions. We prove that if for a single nonzero uniform loading the strain inside inclusion is constant and further the eigenvalues of this strain are either all the same or all distinct, then the inclusion must be of ellipsoidal shape. As a consequence, we show that for two linearly independent loadings the strains inside the inclusions are uniform, then the inclusion must be of ellipsoidal shape. We then use this result to address a problem of determining the shape of an inclusion when the elastic moment tensor (elastic polarizability tensor) is extremal. We show that the shape of inclusions, for which the lower Hashin-Shtrikman bound either on the bulk part or on the shear part of the elastic moment tensor is attained, is an ellipse in two dimensions and an ellipsoid in three dimensions.

math.AP↗

arXiv2001

High-Order Terms in the Asymptotic Expansions of the Steady-State Voltage Potentials in the Presence of Conductivity Inhomogeneities of Small Diameter

We derive high-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of a finite number of diametrically small inhomogeneities with conductivities different from the background conductivity. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for inhomogeneities with Lipschitz boundaries and those with extreme conductivities.

math-ph↗