arXiv subjects

Mourad Sini

Publications and source records attributed to Mourad Sini.

At least 55 records · Page 3Linked to original sources

arXiv2022

On the Origin of Minnaert Resonances

It is well known that the presence, in a homogeneous acoustic medium, of a small inhomogeneity (of size $\varepsilon$), enjoying a high contrast of both its mass density and bulk modulus, amplifies the generated total fields. This amplification is more pronounced when the incident frequency is close to the Minnaert frequency $ω_{M}$. Here we explain the origin of such a phenomenon: at first we show that the scattering of an incident wave of frequency $ω$ is described by a self-adjoint $ω$-dependent Schrödinger operator with a singular $δ$-like potential supported at the inhomogeneity interface. Then we show that, in the low energy regime (corresponding in our setting to $\varepsilon\ll1$) such an operator has a non-trivial limit (i.e., it asymptotically differs from the Laplacian) if and only if $ω=ω_{M}$. The limit operator describing the non-trivial scattering process is explicitly determined and belongs to the class of point perturbations of the Laplacian. When the frequency of the incident wave approaches $ω_{M}$, the scattering process undergoes a transition between an asymptotically trivial behaviour and a non-trivial one.

math.AP↗

arXiv2021

The inverse source problem for the wave equation revisited: A new approach

The inverse problem of reconstructing a source term from boundary measurements, for the wave equation, is revisited. We propose a novel approach to recover the unknown source through measuring the wave fields after injecting small particles, enjoying a high contrast, into the medium. For this purpose, we first derive the asymptotic expansion of the wave field, based on the time-domain Lippmann-Schwinger equation. The dominant term in the asymptotic expansion is expressed as an infinite series in terms of the eigenvalues $\{λ_n\}_{n\in \mathbb{N}}$ of the Newtonian operator (for the pure Laplacian). Such expansions are useful under a certain scale between the size of the particles and their contrast. Second, we observe that the relevant eigenvalues appearing in the expansion have non-zero averaged eigenfunctions. We prove that the family $\{\sin(\frac{c_1}{\sqrt{λ_n}}\, t),\, \cos(\frac{c_1}{\sqrt{λ_n}}\, t)\}$, for those relevant eigenvalues, with $c_1$ as the contrast of the small particle, defines a Riesz basis (contrary to the family corresponding to the whole sequence of eigenvalues). Then, using the Riesz theory, we reconstruct the wave field, generated before injecting the particles, on the center of the particles. Finally, from these (internal values of these) last fields, we reconstruct the source term (by numerical differentiation for instance). A significant advantage of our approach is that we only need the measurements on $\{x\}\times(0, T)$ for a single point $x$ away from $Ω$, i.e., the support of the source, and large enough $T$.

math.AP↗

arXiv2021

Photo-acoustic inversion using plasmonic contrast agents: The full Maxwell model

We analyze the inversion of the photo-acoustic imaging modality using electromagnetic plasmonic nano-particles as contrast agents. We prove that the generated pressure, before and after injecting the plasmonic nano-particles, measured at a single point, located away from the targeted inhomogeneity to image, and at a given band of incident frequencies is enough to reconstruct the (eventually complex valued) permittivity. Indeed, from these measurements, we define an indicator function which depends on the used incident frequency and the time of measurement. This indicator function has differentiating behaviors in terms of both time and frequency. First, from the behavior in terms of time, we can estimate the arrival time of the pressure from which we can localize the injected nano-particle. Second, we show that this indicator function has maximum picks at incident frequencies close to the plasmonic resonances. This allows us to estimate these resonances from which we construct the permittivity. To justify these properties, we derive the dominant electric field created by the injected nano-particle when the incident frequency is close to plasmonic resonances. This is done for the full Maxwell system. To this end, we use a natural spectral decomposition of the vector space (L^2(D))^3 based on the spectra of the Newtonian and the Magnetization operators. As another key argument, we provide the singularity analysis of the Green's tensor of the Maxwell problem with varying permittivity. Such singularity is unusual if compared to the ones of the elliptic case (as the acoustic or elastic models). In addition, we show how the derived approximation of the electric fields propagates, as a source, in the induced pressure with the time.

math.AP↗

arXiv2021

The Effective Permittivity and Permeability Generated by a Cluster of Moderately Contrasting Nanoparticles

In a $3D$ bounded and $C^{1, α}$-smooth domain $Ω$, $α\in (0, 1)$, we distribute a cluster of nanoparticles enjoying moderately contrasting relative permittivity and permeability which can be anisotropic. We show that the effective permittivity and permeability generated by such cluster is explicitly characterized by the corresponding electric and magnetic polarization tensors of the fixed shape. The error of the approximation of the scattered fields corresponding to the cluster and the effective medium is inversely proportional to the dilution parameter $c_r:=\fracδ{a}$ where $a$ is the maximum diameter of the nanoparticles and $δ$ the minimum distance between them. The constant of the proportionality is given in terms of a priori bounds on the cluster of nanoparticles (i.e. upper and lower bounds on their permittivity and permeability parameters, upper bound on the dilution parameter $c_r$, the used incident frequency and the domain $Ω$). A key point of the analysis is to show that the Foldy-Lax field appearing in the meso-scale approximation, derived in \cite{AM}, is a discrete form of a (continuous) system of Lippmann-Schwinger equations with a related effective permittivity and permeability contrasts. To derive this, we prove that the Lippmann-Schwinger operator, for the Maxwell system, is invertible in the Hölder spaces. As a by-product, this shows a Hölder regularity property of the electromagnetic fields up to the boundary of the inhomogeneity.

math.AP↗

arXiv2020

An Introduction To The Mathematics Of The Imaging Modalities Using Small Scaled Contrast Agents

In the recent years, we witness a great interest in imaging, in a wide sense, using contrast agents. One of the reasons is that many imaging modalities, as the ones related to medical sciences, suffer from several shortcomings. The most serious one is the issue of instability. Indeed, it is, nowadays, a common certainty that classical inverse problems of recovering objects from remote measurements are, mostly, highly unstable. To recover the stability, it is advised to create, whenever possible, the missing contrasts in the targets to image. In this survey paper, we follow this direction and propose an approach how to analyze mathematically the effect of the injected agents on the different fields under consideration. These contrast agents are small-sized particles modeled with materials that enjoy high contrasts as compared to the ones of the background. These two properties allow them, under critical scales of size/contrast, to create local spots when excited from far. These local spots can be remotely recovered in stable ways. The accessible information on the target are encoded in theses spots. After stating a class of such imaging modalities that enter into this framework, as the acoustic imaging, photo-acoustic imaging, optical imaging and more, we provide detailed analysis for first two modalities where the contrast agents are micro-bubbles and nano-particles respectively.

physics.optics↗

arXiv2020

Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies

We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves). We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixed single backscattering direction. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region. 1. We show that, for each fixed used bubble, the contrasted far-fields reach their maximum value at, incident, frequencies close to the Minneart resonance (or the body-wave resonances depending on the types of bubbles we use). Hence, we can reconstruct this resonance from our data. The explicit dependence of the Minnaert resonance in terms of the background mass density of the background allows us to recover it, i.e. the mass density, in a straightforward way. 2. In addition, this measured contrasted far-fields allow us to recover the total field at the location points of the bubbles (i.e. the total field in the absence of the bubbles). A numerical differentiation argument, for instance, allows us to recover the bulk modulus of the targeted region as well.

math.AP↗

arXiv2020

Mathematical Analysis of the Photo-acoustic imaging modality using resonating dielectric nanoparticles: The 2D TM-model

We deal with the photoacoustic imaging modality using dielectric nanoparticles as contrast agents. Exciting the heterogeneous tissue, localized in a bounded domain $Ω$, with an electromagnetic wave, at a given incident frequency, creates heat in its surrounding which in turn generates an acoustic pressure wave (or fluctuations). The acoustic pressure can be measured in the accessible region $\partial Ω$ surrounding the tissue of interest. The goal is then to extract information about the optical properties (i.e. the permittivity and conductivity) of this tissue from these measurements. We describe two scenarios. In the first one, we inject single nanoparticles while in the second one we inject couples of closely spaced nanoparticles (i.e. dimers). This can be done using dielectric nanoparticles enjoying high contrasts of both its electric permittivity and conductivity. These results are possible using frequencies of incidence close to the resonances of the used dielectric nanoparticles. This allows us to solve the photoacoustic inverse problem with direct approximation formulas linking the measured pressure to the optical properties of the tissue. The results are justified in the 2D TM-model.

math.AP↗

arXiv2020

Analysis of the acoustic waves reflected by a cluster of small holes in the time-domain and the equivalent mass density

We study the time-domain acoustic scattering problem by a cluster of small holes (i.e. sound-soft obstacles). Based on the retarded boundary integral equation method, we derive the asymptotic expansion of the scattered field as the size of the holes goes to zero. Under certain geometrical constraints on the size and the minimum distance of the holes, we show that the scattered field is approximated by a linear combination of point-sources where the weights are given by the capacitance of each hole and the causal signals (of these point-sources) can be computed by solving a, retarded in time, linear algebraic system. A rigorous justification of the asymptotic expansion and the unique solvability of the linear algebraic system are shown under natural conditions on the cluster of holes. As an application of the asymptotic expansion, we derive, in the limit case when the holes are densely distributed and occupy a bounded domain, the equivalent effective acoustic medium (an equivalent mass density characterized by the capacitance of the holes) that generates, approximately, the same scattered field as the cluster of holes. Conversely, given a locally variable, smooth and positive mass density, satisfying a certain subharmonicity condition, we can design a perforated material with holes, having appropriate capacitances, that generates approximately the same acoustic field as the acoustic medium modelled by the given mass density (and constant speed of propagation). Finally, we numerically verify the asymptotic expansions by comparing the asymptotic approximations with the numerical solutions of the scattered fields via the finite element method.

math.AP↗

arXiv2019

Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles

This paper concerns the time-harmonic direct and inverse elastic scattering by an extended rigid elastic body surrounded by a finite number of point-like obstacles. We first justify the point-interaction model for the Lamé operator within the singular perturbation approach. For a general family of pointwise-supported singular perturbations, including anisotropic and non-local interactions, we derive an explicit representation of the scattered field. In the case of isotropic and local point-interactions, our result is consistent with the ones previously obtained by Foldy's formal method as well as by the renormalization technique. In the case of multiple scattering with pointwise and extended obstacles, we show that the scattered field consists of two parts: one is due to the diffusion by the extended scatterer and the other one is a linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one. As to the inverse problem, the factorization method by Kirsch is adapted to recover simultaneously the shape of an extended elastic body and the location of point-like scatterers in the case of isotropic and local interactions. The inverse problems using only one type of elastic waves (i.e. pressure or shear waves) are also investigated and numerical examples are present to confirm the inversion schemes.

math-ph↗

arXiv2019

Foldy-Lax approximation of the electromagnetic fields generated by anisotropic inhomogeneities in the mesoscale regime with complements for the perfectly conducting case

The Foldy-Lax (or the point-interaction) approximation of the electromagnetic fields generated by a cluster of small scaled inhomogeneities is derived in the mesoscale regime, i.e. when the minimum distance $δ$ between the particles is proportional to their maximum radi $a$ in the form $δ=c_r \; a$ with a positive constant $c_r$ that we call the dilution parameter. We consider two types of families of inhomogeneities. In the first one, the small particles are modeled by anisotropic electric permittivities and/or magnetic permeabilities with possibly complex values. In the second one, they are given as perfectly conductive inclusions. In both the cases, we provide the dominating field (the so-called Foldy-Lax field) with explicit error estimates in terms of the dilution parameter $c_r$. In the case of perfectly conductive inclusions, the results provided here improve sharply the ones derived recently in \cite{AB-SM:MMS2019}. Such approximations are key steps in different research areas as imaging and material sciences.

math.AP↗

arXiv2019

The equivalent media generated by bubbles of high contrasts: Volumetric metamaterials and metasurfaces

We deal with the point-interaction approximations for the acoustic wave fields generated by a cluster of highly contrasted bubbles for a wide range of densities and bulk moduli contrasts. We derive the equivalent fields when the cluster of bubbles is appropriately distributed (but not necessarily periodically) in a bounded domain $Ω$ of $\mathbb{R}^3$. We handle two situations. 1. In the first one, we distribute the bubbles to occupy a $3$ dimensional domain. For this case, we show that the equivalent speed of propagation changes sign when the medium is excited with frequencies smaller or larger than (but not necessarily close to) the Minnaert resonance. As a consequence, this medium behaves as a reflective or absorbing depending on whether the used frequency is smaller or larger than this resonance. In addition, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above a certain threshold, then the medium behaves as a 'wall', i.e. allowing no incident sound to penetrate. 2. In the second one, we distribute the bubbles to occupy a $2$ dimensional (open or closed) surface, not necessarily flat. For this case, we show that the equivalent medium is modeled by a Dirac potential supported on that surface. The sign of the surface potential changes for frequencies smaller or larger than the Minnaert resonance, i.e. it behaves as a smart metasurface reducing or amplifying the transmitted sound across it. As in the $3$D case, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above an appropriate threshold, then the surface allows no incident sound to be transmitted across the surface, i.e. it behaves as a white screen.

math.AP↗

arXiv2019

The point-interaction approximation for the fields generated by contrasted bubbles at arbitrary fixed frequencies

We deal with the linearized model of the acoustic wave propagation generated by small bubbles in the harmonic regime. We estimate the waves generated by a cluster of $M$ small bubbles, distributed in a bounded domain $Ω$, with relative densities having contrasts of the order $a^β, β>0, $ where $a$ models their relative maximum diameter, $a\ll 1$. We provide useful and natural conditions on the number $M$, the minimum distance and the contrasts parameter $β$ of the small bubbles under which the point interaction approximation (called also the Foldy-Lax approximation) is valid. With the regimes allowed by our conditions, we can deal with a general class of such materials. Applications of these expansions in material sciences and imaging are immediate. For instance, they are enough to derive and justify the effective media of the cluster of the bubbles for a class of gases with densities having contrasts of the order $a^β$, $β\in (\frac{3}{2}, 2)$ and in this case we can handle any fixed frequency. In the particular and important case $β=2$, we can handle any fixed frequency far or close (but distinct) from the corresponding Minnaert resonance. The cluster of the bubbles can be distributed to generate volumetric metamaterials but also low dimensional ones as metascreens and metawires.

math.AP↗

arXiv2019

Estimation of the heat conducted by a cluster of small cavities and characterization of the equivalent heat conduction

We estimate the heat conducted by a cluster of many small cavities. We show that the dominating heat is a sum, over the number of the cavities, of the heats generated by each cavity after interacting with each other. This interaction is described through densities computable as solutions of a close, and invertible, system of time domain integral equations of a second kind. As an application of these expansions, we derive the effective heat conductivity which generates approximately the same heat as the cluster of cavities, distributed in a 3D bounded domain, with explicit error estimates in terms of that cluster. At the analysis level, we use time domain integral equations. Doing that, we have two choices. First, we can favor the space variable by reducing the heat potentials to the ones related to the Laplace operator (avoiding Laplace transform). Second, we can favor the time variable by reducing the representation to the Abel integral operator. As the model under investigation has time-independent parameters, we follow here the first approach.

math.AP↗

arXiv2018

Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $Σ\subseteqΓ=\partialΩ$, where $Ω$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schrödinger operators with singular $δ$-type potential supported on the surface $Γ$ and of strength $α\in L^{p}(Γ)$, $p>2$.

math.AP↗

arXiv2018

Mathematical imaging using electric or magnetic nanoparticles as contrast agents

We analyse mathematically the imaging modality using electromagnetic nanoparticles as contrast agent. This method uses the electromagnetic fields, collected before and after injecting electromagnetic nanoparticles, to reconstruct the electrical permittivity. The particularity here is that these nanoparticles have high contrast electric or magnetic properties compared to the background media. First, we introduce the concept of electric (or magnetic) nanoparticles to describe the particles, of relative diameter $δ$ (relative to the size of the imaging domain), having relative electric permittivity (or relative magnetic permeability) of order $δ^{-α}$ with a certain $α>0$, as $0<δ<<1$. Examples of such material, used in the imaging community, are discussed. Second, we derive the asymptotic expansion of the electromagnetic fields due to such singular contrasts. We consider here the scalar electromagnetic model. Using these expansions, we extract the values of the total fields inside the domain of imaging from the scattered fields measured before and after injecting the nanoparticles. From these total fields, we derive the values of the electric permittivity at the expense of numerical differentiations.

math.AP↗

arXiv2018

The Foldy-Lax Approximation for the Full Electromagnetic Scattering by Small Conductive Bodies of Arbitrary Shapes

We deal with the electromagnetic waves propagation in the harmonic regime. We derive the Foldy-Lax approximation of the scattered fields generated by a cluster of small conductive inhomogeneities arbitrarily distributed in a bounded domain $Ω$ of $\mathbb{R}^3$. This approximation is valid under a sufficient but general condition on the number of such inhomogeneities $m$, their maximum radii $ε$ and the minimum distances between them $δ$, the form $$(\ln m)^{\frac{1}{3}}\fracεδ \leq C,$$ where $C$ is a constant depending only on the Lipschitz characters of the scaled inhomogeneities. In addition, we provide explicit error estimates of this approximation in terms of aforementioned parameters, $m, ε, δ$ but also the used frequencies $k$ under the Rayleigh regime. Both the far-fields and the near-fields (stated at a distance $δ$ to the cluster) are estimated. In particular, for a moderate number of small inhomogeneities $m$, the derived expansions are valid in the mesoscale regime where $δ\sim ε$. At the mathematical analysis level and based on integral equation methods, we prove a priori estimates of the densities in the $L^{2,Div}_t$ spaces instead of the usual $L^2$ spaces (which are not enough). A key point in such a proof is a derivation of a particular Helmholtz type decomposition of the densities. Those estimates allow to obtain the needed qualitative as well as quantitative estimates while refining the approximation. Finally, to prove the invertibility of the Foldy-Lax linear algebraic system, we reduce the coercivity inequality to the one related to the scalar Helmholtz model. As this linear algebraic system comes from the boundary conditions, such a reduction is not straightforward.

math.AP↗

arXiv2017

Characterization of the equivalent acoustic scattering for a cluster of an extremely large number of small holes

We deal with the time-harmonic acoustic waves scattered by a large number of small holes, of maximal radius $a, a<<1$, arbitrary (i.e. not necessarily periodically) distributed in a bounded part of a homogeneous background. We show that as their number $M$ grows following the law $M:=M(a):=O(a^{-s}), \; a<<1$, the collection of these holes has one of the following behaviors: 1. if $s<1$, then the scattered fields tend to vanish as $a$ tends to zero, i.e. the cluster is a soft one. 2. if $s=1$, then the cluster behaves as an equivalent medium modeled by a refraction index, supported in a given bounded domain $Ω$, which is described by certain geometry properties of the holes and their local distribution. The cluster is a moderate (or intermediate) one. 3. if $s>1$, then the cluster behaves as a totally reflecting extended body, modeled by a bounded and smooth domain $Ω$, i.e. the incident waves are totally reflected by the surface of this extended body. The cluster is a rigid one. These approximations are provided with explicit error estimates in terms of $a,\; a<<1$.

math-ph↗

arXiv2017

Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces

We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $Σ$ of) compact hypersurfaces $Γ=\partialΩ$, $Ω\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the free Laplacian on the whole space ${\mathbb{R}}^{n}$. Our results apply to all standard examples of boundary conditions, like Dirichlet, Neumann, Robin, $δ$ and $δ'$-type, either assigned on $Γ$ or on $Σ\subsetΓ$.

math-ph↗