Izar Azpiroz
email: izar.azpiroz@inria.fr

Hélène Barucq
email: helene.barucq@inria.fr

Julien Diaz
email: julien.diaz@inria.fr

Rabia Djellouli
email: ra-bia.djellouli@csun.edu

Full determination of elastic scatterer characteristics from FFP measurements

Keywords: Far-eld pattern, elasto-acoustic scattering, Fréchet derivative

A multi-stage computational strategy is proposed for determining all elastic scatterer characteristics including the shape, the material properties (Lamé coecients and density), and the location from the knowledge of far-eld pattern (FFP) measurements. The main feature of this new methodology is to utilize the well-known Tikhonov Newton algorithm in a multi-stage solution strategy. The adopted approach is critical for recognizing that the dierent nature and scales of the sought-after parameters as well as the frequency regime have dierent eect on the scattering observability. Numerical results obtained for illustrative two-dimensional elastic congurations highlight the performance of the designed solution methodology.

Reconstruction complète des caractéristiques d'un obstacle élastique à partir de mesure en champ lointain Résumé : Nous proposons une stratégie à plusieurs étapes pour déterminer toutes les caractéristiques physiques d'un diracteur élastique à partir de mesures acoustiques en champ lointain. Ces caractéristiques sont la forme, les paramètres de matériau (coecients de Lamé et densite), et la position. La principale caractéristique de cette nouvelle méthode est le couplage de l'algorithme de Tikhonov Newton avec une stratégie de résolution multi-étape. Cette approche est cruciale non seulement pour prendre en compte les diérences de natures et d'échelles des paramètres à reconstruire mais également les eets des diérents régimes de fréquence. Nous illustrons les performances de cette stratégie de résolution à l'aide de résultats numériques sur des congurations 2D.

Mots-clés : Champ lointain, diraction élasto-acoustique, dérivée de Fréchet

Introduction

Inverse scattering problems, where information about an unknown object such as a body, or an inhomogeneity in a material, or a potential, is to be recovered from measurements of waves of elds scattered by this object, are fundamental for exploring objects that are not accessible to in situ measurements. This class of problems is notoriously very dicult to investigate mathematically and/or numerically [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]. The challenge is due to the fact that these problems are not only nonlinear but also ill-posed in the sense of Hadamard [START_REF] Hadamard | Lectures on Cauchy's problem in linear partial dierential equations[END_REF]. In spite of their diculties, inverse scattering problems received a great deal of attention by mathematicians, scientists, and engineers, as attested by the prolicness of papers and conferences dedicated to this topic. This is due in part to their relevance to a wide range of important applications including seismology RR n°9247 [START_REF] Dahlen | Theoretical global seismology[END_REF][START_REF] Fichtner | Full seismic waveform modelling and inversion[END_REF][START_REF] Tarantola | Inverse problem theory and methods for model parameter estimation[END_REF][START_REF] Biondo | 3D seismic imaging[END_REF][START_REF] Virieux | An overview of full-waveform inversion in exploration geophysics[END_REF], radar and sonar [START_REF] Le | Principles of radar and sonar signal processing[END_REF], optics [START_REF] Walther | The ray and wave theory of lenses[END_REF], along with many other areas in science and medical imaging technology.

The inverse scattering problem considered in this paper consists in retrieving all the properties of an elastic scatterer from the knowledge of its far-eld patterns (FFP) corresponding to one or multiple incident plane waves (see Section 2). To the best of our knowledge, this "full" inverse scattering problem has not been previously studied neither mathematically nor numerically. This work constitutes the rst numerical attempt to simultaneously recover all the object properties: the shape of the scatterer, its material properties (the Lamé coecients and the density), and its location. All existing numerical methods have been designed and so far employed for only partial parameters reconstruction. More specically, extensive eorts have been devoted during the past ve decades to the determination of the shape of an unknown object from the knowledge of its corresponding FFP measurements and the nature of the object. Indeed, various computational procedures have been designed for this purpose (see, e.g., [START_REF] Colton | Recent developments in inverse acoustic scattering theory[END_REF][START_REF] Farhat | On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method[END_REF][START_REF] Cakoni | Qualitative methods in inverse scattering theory: An introduction[END_REF][START_REF] Ivanyshyn | Shape reconstruction of acoustic obstacles from the modulus of the far eld pattern[END_REF][START_REF] Djellouli | Inverse acoustic problems[END_REF][START_REF] Elschner | An optimization method in inverse acoustic scattering by an elastic obstacle[END_REF][START_REF] Monk | An inverse uidsolid interaction problem[END_REF][START_REF] Monk | Near eld sampling type methods for the inverse uidsolid interaction problem[END_REF][START_REF] Michael | Phaseless inverse scattering problems in three dimensions[END_REF][START_REF] Cakoni | The factorization method for a defective region in an anisotropic material[END_REF][START_REF] Cakoni | A qualitative approach to inverse scattering theory[END_REF][START_REF] Cakoni | Qualitative methods in inverse electromagnetic scattering theory: Inverse scattering for anisotropic media[END_REF][START_REF] Michael | Reconstruction procedures for two inverse scattering problems without the phase information[END_REF][START_REF] Qin | The linear sampling method for inhomogeneous medium and buried objects from far eld measurements[END_REF][START_REF] Barucq | Localization of small obstacles from back-scattered data at limited incident angles with full-waveform inversion[END_REF], and the references therein). The problem of nding the material parameters (Lamé coecients and density) of elastic scatterers has also been extensively studied and a wide range of numerical techniques have been developed to address it (see, e.g., [START_REF] Beylkin | Linearized inverse scattering problems in acoustics and elasticity[END_REF][START_REF] James E Warner | Inverse material identication in coupled acoustic-structure interaction using a modied error in constitutive equation functional[END_REF][START_REF] Hubmer | Lamé parameter estimation from static displacement eld measurements in the framework of nonlinear inverse problems[END_REF][START_REF] Lechleiter | Identifying Lamé parameters from time-dependent elastic wave measurements[END_REF] and the references therein). Likewise, various numerical approaches have been proposed to locate objects from the knowledge of their corresponding FFP measurements and the nature of the objects, as reported in [START_REF] Habib Ammari | A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a xed frequency[END_REF][START_REF] Ivanyshyn | Identication of sound-soft 3D obstacles from phaseless data[END_REF][START_REF] Fazli | An analytical approach to estimate the number of small scatterers in 2D inverse scattering problems[END_REF][START_REF] Griesmaier | A multifrequency MUSIC algorithm for locating small inhomogeneities in inverse scattering[END_REF][START_REF] Zhang | Recovering scattering obstacles by multi-frequency phaseless far-eld data[END_REF][START_REF] Zhang | Uniqueness results on phaseless inverse acoustic scattering with a reference ball[END_REF], among other references. To conclude, reference [START_REF] Bourgeois | On simultaneous identication of the shape and generalized impedance boundary condition in obstacle scattering[END_REF] is the only paper that we encountered in the literature that proposes a technique to reconstruct simultaneously the shape of an object along with the two parameters that characterize the impedance boundary condition adopted in the considered mathematical model. We have not come across of any work that solves the inverse scattering problem considered in Section 2 and delivers the full characteristics of elastic scatterers from their FFP measurements.

Given that, our goal is to propose a solution methodology to fully recover the properties of an object from the knowledge of its corresponding FFP measurements. The proposed approach is a multi-stage strategy in which a regularized Newton algorithm plays a central role (see Section 3). The idea of using regularized Newton-type methods for solving inverse scattering problems is not new. The novelty here is that it is employed in a multi-stage context in order to recognize that the sought-after parameters (shape, material parameters, location) have, depending on the frequency regime, dierent inuence on the scattering observability. The proposed strategy delivers rst the shape and material parameters and then the scatterer's location is obtained in the second stage. Note that the scale variability of the parameters is addressed here via a rescaling procedure applied to the forward problems (see Section 1.3), whereas an appropriate choice of the regularization matrix addresses the sensitivity to the sought-after parameters' inuence on the reconstruction (see Section 3.3).

The remainder of the paper is organized as follows. In Section 1, we specify the nomenclature and assumptions adopted in this paper, state the forward scattering problem, introduce the rescaling procedure employed to address the scale variability of the scatterer's parameters, and formulate the considered inverse scattering problem. Section 2 is devoted to the description of the designed solution methodology and its salient features. The main calculations required by the proposed inversion solver are described in Section 3. These are (a) the evaluation of the Jacobian matrix entries, (b) the solution of the forward problems that occur at each Newton iteration, and (c) the computation of the regularization matrix entries. A summary describing this new inversion algorithm along with its computational complexity are reported in Section 4. We present in Section 5 illustrative numerical parameters recovery for various scatterer's congurations including a disk, a non-convex octagon, and a mock-up submarine. The considered scatterers are made of steel and aluminum. These reconstructions have been performed using FFP measurements tainted with noise level ranging from 0% to 15%. The obtained results

Inria

highlight the performance eciency of the proposed multi-stage solution methodology. [START_REF] Patrick R Amestoy | A fully asynchronous multifrontal solver using distributed dynamic scheduling[END_REF] Preliminaries

We rst describe the notations and the assumptions we adopt throughout this paper. We then describe the mathematical formulation of the elasto-acoustic scattering problem and its nondimensionalized version. Last, we state the considered inverse problem.

Nomenclature and Assumptions

Throughout this chapter, we adopt the following nomenclature and assumptions:

• Ω s is a bounded domain of R 2 representing an elastic obstacle.

• Ω f is an innite domain representing the uid that surrounds Ω s , i.e., Ω f = R 2 \ Ω s .

• Γ is the boundary of Ω s and is assumed to be Lipschitz continuous.

• Σ is the exterior articial boundary. Σ is a circle.

• x = (x 1 , x 2 )
T is a point of R 2 and r = x 2 is the distance from the origin point to x (in m).

• S 1 is the unit circle in R 2 , i.e., S 1 = {x ∈ R 2 , x 2 = 1}.

• ∇ (resp. ∆) is the gradient (resp. Laplace) operator in R 2 .

• ν is the outward normal to the boundaries Γ and Σ, and ∂ ∂ν is the normal derivative operator.

• k is a positive number representing the wavenumber of the incident plane wave (in m -1 )

. k = 2π f c f , where f is the frequency of the propagating wave (in Hz) and c f is the sound speed in the uid (in m.s -1 ).

• ω represents the angular frequency of the propagating wave (in rad.s -1 ). ω = 2πf .

• d ∈ S 1 is a unit vector representing the direction of the incident plane wave.

• λ and µ are two positive numbers representing the Lamé coecients of the considered elastic object Ω s (in Pa).

• ρ s is a positive number representing the density of the elastic obstacle Ω s (in kg.m -3 ).

• M * denotes the adjoint matrix of M . M * is the complex conjugate of the transpose matrix of M , i.e, M * = M t .

RR n°9247

Ω s Γ Ω f = R n \ Ω s • Source p inc u p • Receiver
Figure 1: Schematic description of the problem statement.

Mathematical Formulation of the Problem

The scattering of time-harmonic acoustic waves by an elastic obstacle Ω s embedded in an innite homogeneous medium Ω f (see Fig. 1) can be formulated as the following boundary value problem (BVP):

(BV P )                                  ∇ • σ(u) + ω 2 ρ s u = 0 in Ω s (a) ∆p + k 2 p = 0 in Ω f (b) σ(u) • ν = -pν -gν on Γ (c) ω 2 ρ f u • ν = ∂p ∂ν + ∂g ∂ν on Γ (d) lim r→+∞ √ r ∂p ∂r -ikp = 0 (e) (1)
where the pair (u, p) represents the elasto-acoustic scattered eld vector; p is the scalar-valued uid pressure in Ω f , and u = (u x , u y ) t is the vector-valued displacement eld in Ω s ; g = p inc = e iω/c f x•d corresponds to the given incident plane wave. The stress tensor σ is related to the strain tensor by Hooke's law [START_REF] Miguel | Sound, structures, and their interaction[END_REF]. Note that this study is limited to the case of an isotropic medium, so that the stiness tensor is invariant under rotations and reections [START_REF] Miguel | Sound, structures, and their interaction[END_REF]. Consequently, we have

σ(u) = λ∇ • uI + 2µ (u), (2)
where is the strain tensor given by

(u) = 1 2 ∇u + (∇u) T . (3)
Hence, Ω s is assumed to be a homogeneous scatterer. Observe that the direct problem BVP

(1) contains the standard exterior Helmholtz problem given by equations (b) and (e), and the Navier equation given by (a) governing the equilibrium of an elastic scatterer. These equations are coupled via the transmission conditions given by (c) and (d). The rst one is a dynamic interface condition whereas the second one is a kinematic interface condition [START_REF] Miguel | Sound, structures, and their interaction[END_REF]. Furthermore, p ∞ , the far-eld pattern (FFP) corresponding to the scattered pressure eld p, can be expressed Inria using the following integral representation [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]:

p ∞ (x) = e iπ/4 √ 8πk Γ e -ik x•y ∂p ∂ν (y) - ∂e -ik x•y ∂ν p(y) dX; x ∈ S 1 . (4)
Note that BVP has been extensively studied. Mathematical results pertaining to existence, uniqueness, and regularity of the solution can be found in [START_REF] Miguel | Sound, structures, and their interaction[END_REF][START_REF] Barucq | On the existence and the uniqueness of the solution of a uidstructure interaction scattering problem[END_REF][START_REF] Hargé | Valeurs propres d'un corps élastique[END_REF][START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF][START_REF] Barucq | Mathematical determination of the Fréchet derivative with respect to the domain for a uid-structure scattering problem: Case of polygonal-shaped domains[END_REF][START_REF] Cj | Fluidsolid interaction: acoustic scattering by a smooth elastic obstacle[END_REF], among other references.

Nondimensionalization of the Forward Problem

The characteristic parameters of a given elastic scatterer Ω s are of dierent nature. Indeed, it has been reported in [START_REF] Azpiroz | Contribution to the Numerical Reconstruction in Inverse Elasto-Acoustic Scattering[END_REF][START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF] that the sensitivity of the FFP given by ( 4) to changes on the shape Γ is in general quite dierent to the ones on the material parameters λ, µ, and ρ s . Such a variability can have, depending on the frequency regime, a dramatic impact on the scattering observability. Furthermore, the magnitude of these parameters is also dierent. For example, in the numerical experiments reported in Section 5, the Lamé coecients λ and µ are of order 1GPa, the density is of order 10 4 kg m -3 , whereas the size of the obstacle is of order 10 -3 m. Such a dierence constitutes a serious numerical challenge when attempting to simultaneously recover these parameters from FFP measurements. Given that, there is undoubtedly a real advantage in rescaling the forward problem prior to solving the inverse obstacle problem introduced in Section 1.4. To this end, we proceed in a classical way by rescaling each of the parameters in BVP 1) by a characteristic unit of measure to be determined. Specically, we introduce the following notation:

v = v v 0 (5)
where v represents one of the parameters ω, c f , ρ s , λ, µ, u x , u y , p and v 0 is a characteristic unit of measure v to be determined. In addition, we use a dilation factor L for the position coordinates x = (x, y), i.e., x = 1 L x. Furthermore, we assume that u x,0 = u y,0 = u 0 . We then rewrite BVP [START_REF] Patrick R Amestoy | A fully asynchronous multifrontal solver using distributed dynamic scheduling[END_REF] in terms of the new dimensionless quantities v. We also set all the arising constants to be 1, i.e.,

C 0 x 2 0 ω 2 0 ρ s,0 = c 2 f,0 L 2 ω 2 0 = ω 2 0 u 0 ρ s,0 L p 0 = C 0 u 0 Lp 0 = ω 0 L c f,0 = 1. (6)
In doing so, we deduce that (û, p) also satises BVP(1) but with dimensionless quantities. Last, we must point out that in all numerical experiments reported in Section 5, we have selected:

• p 0 = 1Pa.

• ω 0 is the angular frequency in each considered numerical experiment.

• L = L (0) , where L (0) is the characteristic size of the initial obstacle Ω s (0) .

• C 0 = λ (0) +2µ (0) , where λ (0) and µ (0) are the Lamé coecients of the initial obstacle Ω s (0) .

The values of the remaining quantities ρ s,0 , c f,0 , u 0 are then deduced from the relations given by [START_REF] Azpiroz | Contribution to the Numerical Reconstruction in Inverse Elasto-Acoustic Scattering[END_REF].

RR n°9247

The Inverse Problem Formulation

As stated in the Introduction section, the main goal of this work is to fully characterize elastic scatterers from the knowledge of some corresponding FFP measurements. Therefore, we consider the following inverse obstacle problem (IOP):

Given one or several measured far eld patterns p ∞ (x) corresponding to one or several given directions d and wavenumbers k of incident plane waves, nd Γ the shape of the scatterer Ω s , its Lamé coecients (λ, µ), its density ρ s , and its location x c such that

F (Γ, λ, µ, ρ s , x c ) (x) = p ∞ (x); x ∈ S ⊆ S 1 (7)
where F is the far-eld operator that maps (Γ, λ, µ, ρ s , x c ) onto the far-eld pattern p ∞ . The tilde notation designates a measured quantity, i.e., a quantity that can be tainted with errors during the measurement process.

To the best of our knowledge, there were no attempts, reported in the literature, to investigate IOP [START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF], either mathematically or numerically. It is worth noting however that in the case of a rigid obstacle, i.e., Γ being the only unknown in IOP [START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF], in addition to an extensive numerical investigation (see, e.g., [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF][START_REF] Colton | Recent developments in inverse acoustic scattering theory[END_REF][START_REF] Mönch | A Newton method for solving the inverse scattering problem for a sound-hard obstacle[END_REF][START_REF] Kress | A quasi-Newton method in inverse obstacle scattering[END_REF][START_REF] Farhat | On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method[END_REF][START_REF] Ivanyshyn | Shape reconstruction of acoustic obstacles from the modulus of the far eld pattern[END_REF][START_REF] Colton | A simple method for solving inverse scattering problems in the resonance region[END_REF]), various results pertaining to the uniqueness have been established and can be found in [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF][START_REF] Ivanyshyn | Shape reconstruction of acoustic obstacles from the modulus of the far eld pattern[END_REF][START_REF] Kress | Inverse obstacle scattering with modulus of the far eld pattern as data[END_REF][START_REF] Ivanyshyn | Inverse scattering for surface impedance from phase-less far eld data[END_REF], among other references.

2

The Proposed Multi-Stage Solution Methodology

As stated earlier in the Introduction section, the main goal of this study is to propose an ecient computational procedure for solving IOP [START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF]. The method we have designed is a multi-stage strategy in which a regularized Newton algorithm plays a central role. The multi-stage feature of the method is adopted to recognize that the shape, the material parameters, and the location of the sought-after elastic object along with the frequency regime have dierent eects on the acoustic scattering observability. More specically:

• The shape of the scatterer and its material properties are of dierent nature and scales.

Their variations have dierent inuence on the FFP, as reported in [START_REF] Azpiroz | Contribution to the Numerical Reconstruction in Inverse Elasto-Acoustic Scattering[END_REF][START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF]. Hence, the proposed approach addresses the scale issue by considering the non dimensional formulation (see Section 1.3) when solving the forward scattering-type problems that occur at each Newton iteration. Furthermore, the dierent nature issue is addressed through the Tikhonov regularization procedure. Indeed, when retrieving simultaneously the shape and the material parameters, the proposed method uses dierent sets of regularization parameter values, as detailed in Section 3.3.

• It is well-known that, in general, Newton methods cannot determine the location of the sought-after object from the knowledge of the FFP intensity. This is due to the fact that the FFP intensity is invariant under translation of the scatterer [START_REF] Ivanyshyn | Inverse scattering for surface impedance from phase-less far eld data[END_REF][START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]. On the other hand, it is also well-known that, in practice, the measurements of the FFP intensity are more accurate than the measurements of the phase of the FFP. Hence, since accurate data are very important to ill-posed problems, the proposed method employs the FFP intensity measurements to rst recover the shape and the material parameters while keeping the location of the sought-after obstacle "frozen" at the origin of the plane. Once this stage is completed, the proposed algorithm uses the FFP measurements to locate the scatterer.

Inria

Next, we examine the algorithmic and computational considerations of the proposed solution methodology and report on its performance when applied to retrieving the parameters of various two-dimensional elastic objects including a disk, a non convex octagon, and a mockup submarine.

The Regularized Newton Method

Regularized iterative methods have been among the primary candidates since the mid-80s for solving both two -and three-dimensional scattering problems (see, e.g., [START_REF] Farhat | On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method[END_REF][START_REF] Kress | A quasi-Newton method in inverse obstacle scattering[END_REF][START_REF] Mönch | A Newton method for solving the inverse scattering problem for a sound-hard obstacle[END_REF][START_REF] Roger | Newton-Kantorovitch algorithm applied to an electromagnetic inverse problem[END_REF][START_REF] Pöppe | Newton-Kantorovich method applied to two-dimensional inverse scattering for an exterior Helmholtz problem[END_REF][START_REF] Tobocman | Inverse acoustic wave scattering in two dimensions from impenetrable targets[END_REF][START_REF] Sl | An ecient numerical method for exterior and interior inverse problems of Helmholtz equation[END_REF][START_REF] Kress | Newton method in inverse obstacle scattering inverse problem in engineering mechanics[END_REF][START_REF] Kress | Integral equation methods in inverse obstacle scattering[END_REF][START_REF] Gonzalo R Feijoo | Shape sensitivity calculations for exterior acoustics problems[END_REF][START_REF] Angell | The three dimensional inverse scattering problem for acoustic waves[END_REF][START_REF] Kristensson | Inverse problems for acoustic waves using the penalised likelihood method[END_REF]). The Newton iterative aspect of these methods addresses eectively the nonlinearity of the IOPs whereas the regularization component is incorporated to restore the stability, as dened by Hadamard in [START_REF] Hadamard | Lectures on Cauchy's problem in linear partial dierential equations[END_REF]. In what follows, we recall the main features of the regularized Newton method that we employ in the context of the proposed multi-stage solution methodology.

2.1.1 The parametrization.

We assume that the sought-after uid-structure interface Γ can be parametrized as follows:

Γ = Γ(s) ; s = [s 1 , ..., s Ns ] T ∈ R Ns (8)
where N s denotes the total number of shape parameters s j . The choice of the parameters depends on the a priori knowledge on the shape of the scatterer [START_REF] Farhat | On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method[END_REF][START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]. We set

q = s T , λ, µ, ρ s , x c T ∈ R Nq (9)
where x c = (x c , y c ) represents the location of the sought-after scatterer. N q is the total number of unknown parameters of IOP [START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF]. Hence, N q = N s + 5.

Since the far-eld pattern p ∞ of the scattered pressure eld p is measured at a nite number of given points xj ∈ S 1 , denoted by N x, we project IOP (7) onto a nite dimensional space of L 2 (S 1 ), which is the natural choice for the measurements' space, and transform it onto the following algebraic problem:

Find q ∈ R Nq such that F (q) (x j ) = p ∞ (x j ); j = 1, ..., N x (10)
Observe that the resulting inverse problem given by ( 10) is a discrete formulation of IOP [START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF]. It is a nonlinear algebraic system with N x equations and N q unknowns; N x being larger than N q when reconstructing with full aperture measurements.

The regularized Newton equation using the full data (FFP).

The solution of the nonlinear inverse problem IOP [START_REF] Barucq | On the existence and the uniqueness of the solution of a uidstructure interaction scattering problem[END_REF] by the Newton method incurs, at each Newton iteration n, the solution of the following linear algebraic system:

J F q (n) (x j ) δq (n) = p ∞ (x j ) -p (n) ∞ (x j ) ; j = 1, ..., N x (11)
where

• J F q (n) is the N x × N q Jacobian matrix of the operator F . J F q (n) (x l ) is the l th row vector whose entries are the Fréchet derivatives of the far-eld pattern with respect to the vector parameter coordinates q.

RR n°9247

• p

(n) ∞ (x l ) = F q (n) (x l )
is the far-eld pattern corresponding to the computed conguration Ω s(n) at iteration n.

Once the vector δq (n) ∈ R Nq is calculated, the value of the sought-after parameter vector is then updated as follows:

q (n+1) = q (n) + δq (n) (12)
Moreover, since in practice the number of observation points N x is greater than the number of the parameters N q , the linear system (11) is overdetermined. Therefore, it is solved in the leastsquares sense. Hence, the linear system given by ( 11) is replaced by the following corresponding normal equations:

J * F q (n) J F q (n) δq (n) = J * F q (n) p ∞ -p ∞ (n) (13)
where

• p ∞ is the FFP measurements vector, that is,

p ∞ = [ p ∞ (x 1 ), ..., p ∞ (x Nx )] T • p (n)
∞ is the FFP computed vector at iteration n, that is,

p (n) ∞ = p (n) ∞ (x 1 ), ..., p (n) ∞ (x Nx ) T .
Due to the ill-posedness nature of IOP [START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF], the discrete linear system ( 13) is severely illconditioned. This means that small perturbations in the data may lead to large changes in the solution. For this reason, a regularization procedure must be incorporated into [START_REF] Barucq | Localization of small obstacles from back-scattered data at limited incident angles with full-waveform inversion[END_REF] to address the lack of stability [START_REF] Werner Engl | Regularization of inverse problems[END_REF][START_REF] Farshatovich | Regularization of ill-posed problems by iteration methods[END_REF]. We propose a Tikhonov-based strategy to restore the stability to the linear system (13) [START_REF] Nikolaevich | Methods for solving ill-posed problems[END_REF][START_REF] Nikolaevich | Regularization of incorrectly posed problems[END_REF]. This consists in replacing (13) by the following regularized system:

J * F q (n) J F q (n) + R (n) δq (n) = J * F q (n) p ∞ -p ∞ (n) (14)
where R (n) is a N q × N q diagonal and denite positive matrix, called the regularization matrix.

The selection of the diagonal entries of R (n) is discussed in Section 3.3.

2.1.3

The regularized Newton equation using the reduced data (FFP intensity).

It is well-known that in practice the FFP intensity measurements are obtained with higher accuracy level than the corresponding phase measurements. In addition, accurate data are very important to ill-posed problems. Therefore, we propose to use the intensity of the measured far-eld pattern when retrieving the parameters s, λ, µ, and ρ s . Note that the intensity of the FFP is dened here as the square of the amplitude of the FFP:

U q (n) (x j ) = F q (n) (x j )F q (n) (x j ) =| p ∞ (n) (x j ) | 2 ; j = 1, ..., N x
Consequently, the computation of the vector δq (n) needed in [START_REF] Barucq | Mathematical determination of the Fréchet derivative with respect to the domain for a uid-structure scattering problem: Case of polygonal-shaped domains[END_REF] is obtained by solving the following regularized least-squares systems:

J * U q (n) J U q (n) + R (n) δq (n) = J * U q (n) U -U q (n) (15)
Inria where J U q (n) is the Jacobian matrix corresponding to the intensity operator U q (n) . It is given by:

J U q (n) = 2Re F (q (n) ) T J F (q (n) ) (16)
when Re designates the real part of a complex valued quantity and U denotes the measured intensity vector. It is dened by:

U = | p ∞ (x 1 ) | 2 , ..., | p ∞ (x Nx ) | 2 T
Similarly, U q (n) denotes the computed intensity vector at iteration n. It is given by:

U q (n) = U q (n) (x 1 ), ..., U q (n) (x Nx ) T .
We must point out that it is not possible in general to retrieve the location x c of the sought-after scatterer from the knowledge of its corresponding FFP intensity. This is due to the fact that the FFP intensity operator is invariant under translations of the scatterer Ω s (see, e.g., [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF][START_REF] Ivanyshyn | Inverse scattering for surface impedance from phase-less far eld data[END_REF][START_REF] Ivanyshyn | Identication of sound-soft 3D obstacles from phaseless data[END_REF] or Appendix D, pp. 207208 in [START_REF] Azpiroz | Contribution to the Numerical Reconstruction in Inverse Elasto-Acoustic Scattering[END_REF]). For this reason, the linear system (15) will be employed to recover the shape and the material parameters of Ω s only.

The Multi-Stage Solution Procedure

The proposed solution method is a multi-stage procedure in which the regularized Newton algorithm given by ( 14)-( 12) or ( 15)-( 12) is its corner stone. The proposed method recognizes that the sought-after parameters have dierent inuence on the scattering observability and can be described as follows:

Stage 0. We initiate the algorithm by selecting an arbitrary value of the initial vector parameter q (0) .

Stage 1. This stage is devoted to the simultaneous recovery of the shape and material parameters s T , λ, µ, ρ s T . This is accomplished by iterating the regularized Newton system [START_REF] Beylkin | Linearized inverse scattering problems in acoustics and elasticity[END_REF] and ( 12).

Stage 2. This stage is dedicated to the determination of the location x c of the elastic scatterer computed in Stage 1. This goal is achieved by iterating the regularized Newton system ( 14) and ( 12).

The Computational Requirements

We describe in this section the three critical steps of the algorithm: the evaluation of the Jacobians, the solution of the forward problem, and the computation of the regularization matrix.

Eective Evaluation of the Jacobians

As indicated in ( 14) and ( 15), a key step for employing the proposed regularized Newton algorithm is the computation, at each Newton iteration n, of the N x × N q entries of the Jacobian matrix J F (q (n) ). Such calculations must be performed eciently and with a high accuracy level to ensure stability, fast convergence, and cost eectiveness of the proposed computational procedure. To do this, we exploit the mathematical results pertaining to the dependence of the RR n°9247

operator F with respect to the domain [START_REF] Barucq | Fréchet dierentiability of the elasto-acoustic scattered eld with respect to Lipschitz domains[END_REF][START_REF] Barucq | Mathematical determination of the Fréchet derivative with respect to the domain for a uid-structure scattering problem: Case of polygonal-shaped domains[END_REF] and to the material parameters [START_REF] Azpiroz | Contribution to the Numerical Reconstruction in Inverse Elasto-Acoustic Scattering[END_REF][START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF]. The results established in [START_REF] Barucq | Fréchet dierentiability of the elasto-acoustic scattered eld with respect to Lipschitz domains[END_REF] state that the operator F is continuously dierentiable with respect to the domain Ω s . This result has been established assuming the domain Ω s to be Lipschitz continuous. Moreover, the result reported in [START_REF] Barucq | Mathematical determination of the Fréchet derivative with respect to the domain for a uid-structure scattering problem: Case of polygonal-shaped domains[END_REF] provides a characterization of the derivative of F with respect to the boundary of Ω s , that is of practical computational interest. The proof of the result assumes Ω s to be a Lipschitz continuous polygonal domain. Furthermore, the results established in [START_REF] Azpiroz | Contribution to the Numerical Reconstruction in Inverse Elasto-Acoustic Scattering[END_REF][START_REF] Azpiroz | Characterization of partial derivatives with respect to material parameters in a uid-solid interaction problem[END_REF] prove that F is also continuous dierentiable with respect to the material parameters and provide a characterization of its corresponding derivatives. More specically, let p

(n) l (resp. u (n) l
) be the derivative of the pressure eld p (resp. displacement eld u) in the direction of a considered parameter q l ; l = 1, ..., N q at iteration n, i.e., corresponding to the conguration Ω s (q (n) ). Then, the pair (p

(n) l , u (n) l
) is the solution of the following elasto-acoustic scattering-type problem:

(BV P )                                      ∇ • σ(u (n) l ) + ω 2 ρ s u (n) l = f (n) 1 in Ω s(n) (a) ∆p (n) l + k 2 p (n) l = f (n) 2 in Ω f (n) b (b) σ(u (n) l ) • ν + p (n) l ν = g (n) 1 on Γ (n) (c) ω 2 ρ f u (n) l • ν - ∂p (n) l ∂ν = g (n) 2 on Γ (n) (d) lim r→+∞ √ r ∂p (n) l ∂r -ikp (n) l = 0 (e) (17)
where f

(n) 1 , f (n) 2 , g (n) 1
and g

(n) 2

are functions whose expressions depend on the nature of the parameter q l . More specically:

• If q l is a shape or a position parameter, then f

(n) 1 = f (n) 2
= 0 and :

g (n) 1 = -h t j ∇σ(u (n) )ν -∇(p (n)T • h (n) j ν + σ(u (n) )[h j ] t ν + (p (n)T [h j ] t ν; g (n) 2 = -(ω 2 ρ f ∇u (n) -∇(∇(p (n)T ))h j • ν + (ω 2 ρ f u (n) -∇(p (n)T ) • [h j ] t ν.
where h

(n) l = ∂Γ ∂q l (q (n) ). (18)
• If q l is the Lamé coecient λ, then f

(n) 2 = g (n) 2
= 0 and we have:

f (n) 1 = -∇ • ∇ • u (n) I ; (19) g (n) 1 = -∇ • u (n) I • ν. (20)
• If q l is the Lamé coecient µ, then f

(n) 2 = g (n) 2
= 0 and we have:

f (n) 1 = -2 ∇ • (u (n) ); (21)
g (n) 1 = -2 (u (n) ) • ν. (22)
Inria

• If q l is the density coecient ρ, then f (n) 2 = 0; g (n) 1 = g (n) 2
= 0 and we have:

f (n) 1 = -ω 2 u (n) . ( 23
)
Once the derivatives p (n) l and u

(n) l are evaluated, the "far-eld pattern" p

(n) l,∞ , at iteration
n, is then post-processed using the integral representation given by (4) with p

(n) l and Γ (n) . Consequently, we deduce the entries of the Jacobian matrix J F (q (n) ) as follows:

J F (q (n) ) lj = p (n) l,∞ (x j ) j = 1, ..., N x; l = 1, ..., N q . ( 24
)
Hence, it follows that, at each regularized Newton iteration n, the Jacobian matrix J F (q (n) ) can be computed by solving N q direct elasto-acoustic scattering-type problems that dier only by additional terms in the transmission conditions and/or in the wave equations. This means that, at the algebraic level (that results from any nite element-type discretization [START_REF] Ciarlet | The nite element method for elliptic problems[END_REF]), the matrix entries of J F (q (n) ) are obtained by solving a single linear system with multiple right-hand sides.

Solving Eciently the Forward Problem

At each iteration n, the regularized Newton system (see ( 14) and ( 15)) calls for the solution of the elasto-acoustic scattering problem BVP(1) and BVP [START_REF] Bourgeois | On simultaneous identication of the shape and generalized impedance boundary condition in obstacle scattering[END_REF]. Although these problems are linear in nature, their numerical solutions are notoriously dicult due mainly to the following two reasons: (a) they are set in an unbounded (innite) domain, and (b) the scattered eld is highly oscillatory (for high frequency regime). To address the unboundness aspect of the problem, we propose to reformulate BVP(1) and BVP [START_REF] Bourgeois | On simultaneous identication of the shape and generalized impedance boundary condition in obstacle scattering[END_REF] in a bounded domain Ω f Σ , by surrounding the scatterer Ω s with an articial boundary Σ (see Fig. 2). The Sommerfeld condition (see (e) of BVP [START_REF] Patrick R Amestoy | A fully asynchronous multifrontal solver using distributed dynamic scheduling[END_REF] and BVP [START_REF] Bourgeois | On simultaneous identication of the shape and generalized impedance boundary condition in obstacle scattering[END_REF]) is then replaced by the following exterior boundary condition of rst order, in the sense of the pseudo-dierential framework adopted in [START_REF] Antoine | Baylissturkel-like radiation conditions on surfaces of arbitrary shape[END_REF]:

Ω s Ω f Γ p inc u p Σ
∂p ∂ν = ikp - κ 2 p on Σ. (25)

RR n°9247

where κ denotes the curvature of Σ. Since we have chosen Σ to be a circle of radius R, then κ = 1 R .

We have chosen this Robin-type boundary conditions for mainly the following two considerations:

(a) its computational simplicity and (b) its performance eciency. Indeed, the approximation of the boundary condition ( 25) by a nite element method introduces only additional masslike matrices dened on the exterior boundary Σ. Building these matrices is straightforward.

Moreover, when compared to a second-order absorbing boundary condition [START_REF] Bayliss | Radiation boundary conditions for wave-like equations[END_REF] or to a perfectly matched layer (PML) condition [START_REF] Chen | An adaptive uniaxial perfectly matched layer method for time-harmonic scattering problems[END_REF], it has been reported in [START_REF] Barucq | Long-term stable acoustic absorbing boundary conditions for regular-shaped surfaces[END_REF] that the rst-order absorbing boundary condition [START_REF] Colton | Recent developments in inverse acoustic scattering theory[END_REF] exhibits a comparable accuracy level, if not better, when employed in the low-and mid-frequency regime, i.e., the frequency band considered in this paper. Needless to say that one may use higher order absorbing boundary condition when performing these calculations in high frequency regime.

Once the direct scattering problems BVP (1) and BVP [START_REF] Bourgeois | On simultaneous identication of the shape and generalized impedance boundary condition in obstacle scattering[END_REF] are formulated in a bounded domain

Ω f
Σ , we solve them numerically using the interior penalty discontinuous Galerkin (IPDG) method introduced in [START_REF] Barucq | Ecient DG-like formulation equipped with curved boundary edges for solving elasto-acoustic scattering problems[END_REF]. The proposed IPDG method possesses two distinctive features. First, it employs high-order polynomial functions to ensure an accurate approximation of highly oscillating waves. Second, it is equipped with curved boundary edges to provide an accurate representation of the uid-structure interface Γ. The importance of an accurate representation of Γ has been demonstrated by the numerical results reported in [START_REF] Barucq | Ecient DG-like formulation equipped with curved boundary edges for solving elasto-acoustic scattering problems[END_REF]. These results indicate that (a) there is an improvement on the accuracy level by -at least-two orders of magnitude and (b) unlike when approximating Γ by a "broken" line, there are no more spurious internal resonances in the uid region.

The IPDG formulation can be expressed, at the algebraic level, as follows [START_REF] Barucq | Ecient DG-like formulation equipped with curved boundary edges for solving elasto-acoustic scattering problems[END_REF]:

A f + C B B * A s P U = F 1 F 2 (26)
where A f (resp. A s ) is a symmetric matrix given by:

A f = 1 ω 2 (K f -k 2 M f -J f + S f ), (27) resp.
A

s = K s -ω 2 ρ s M s -J s + S s , (28)
where

• K f (resp. K s ) is a block diagonal stiness matrix associated to the pressure (resp. the displacement) eld.

• M f (resp. M s ) is a block diagonal mass matrix associated to the pressure (resp. the displacement) eld.

• J f (resp. J s ) is a matrix that contains the jump terms. The entries of this matrix are dened over the interior edges in the uid (resp. in the solid).

• S f (resp. S s ) is a mass-like matrix dened over the interior edges in the uid (resp. in the solid) resulting from the penalty term.

C is the complex-valued damping matrix. It is a mass-like matrix, whose entries are all zeros except for the elements located at the exterior boundary Σ. The matrix A f + C is symmetric, but non-hermitian, and thus not positive-denite.

Inria

Note that the matrix A s is positive-denite up to the Jones frequencies [START_REF] Cj | Fluidsolid interaction: acoustic scattering by a smooth elastic obstacle[END_REF][START_REF] Barucq | On the existence and the uniqueness of the solution of a uidstructure interaction scattering problem[END_REF]. Note that the interior penalty term compensates the weak ellipticity of the equation operator. B is a mass-like boundary matrix whose entries are dened on the interface edges only, whereas F 1 and F 2 are the source vectors. The vector P ( resp. U ) is the uid pressure (resp. structural displacement) representation in the nite element basis.

The linear system given by ( 26) is thus composed of sparse matrices whose symmetry property allows for an optimized storage. This system is solved with the LU factorization procedure developed for sparse systems and incorporated in the open-source program suite, MUMPS [START_REF] Patrick R Amestoy | A fully asynchronous multifrontal solver using distributed dynamic scheduling[END_REF][START_REF] Patrick R Amestoy | Hybrid scheduling for the parallel solution of linear systems[END_REF].

Note that in the low-and mid-frequency regimes, the sparse implementation allows running the numerical experiments on a personal computer.

Last, as shown is Section 3.1, the Fréchet derivative of the elasto-acoustic scattered eld with respect to the parameter q j can be evaluated at each iteration by solving the same system of equations but with dierent right-hand sides. Then, the corresponding far-eld pattern, that is the Fréchet derivative of the FFP with respect to q j , is obtained by post-processing the solution using the integral representation given by ( 4) on the boundary Γ (n) .

Computation of the regularization matrix R (n)

It is well-known that the convergence of any regularized iterative algorithm strongly depends on the eectiveness of its regularization procedure. For this reason, various strategies have been developed for selecting "optimal" regularization values [START_REF] Alekseevich | On the solution of functional equations by the method of regularization[END_REF][START_REF] Alekseevich | Choice of parameter for the solution of functional equations by the regularization method[END_REF][START_REF] Wahba | Spline models for observational data[END_REF][START_REF] Per | The use of the L-curve in the regularization of discrete ill-posed problems[END_REF]. However, all these strategies have been designed based on theoretical considerations that unfortunately are limited to linear problems. Given that, we propose to employ a trial and error strategy for selecting, at each Newton iteration, the diagonal entries of the regularization matrix R (n) . Such a choice is a balance act between the stability of the algorithm and its accuracy. Note that the proposed "brute force" approach recognizes the dierent inuences of the sought-after parameters on the scattering observability. This is why the choice of the entries of R (n) depends on the algorithm stage.

In what follows, we describe how to select the regularization matrix R (n) in each stage of the proposed iterative algorithm.

3.3.1

The choice of R (n) in Stage I.

We rst recall that Stage I is dedicated to the simultaneous search of the shape parameter vector s ∈ R Ns and the material parameter triplet (λ, µ, ρ s ) ∈ R 3 . Therefore, the regularization matrix

R (n) is a (N s + 3) × (N s + 3) diagonal matrix whose diagonal entries R (n) ll are given by R (n) ll =      α (n) s ; 1 ≤ l ≤ N s α (n) p ; N s + 1 ≤ l ≤ N s + 2 α (n) ρs ; l = N s + 3 (29)
where the regularization parameters α

(n) s , α (n)
p and α

(n)

ρs are positive numbers. At each iteration n, the "optimal" value of the triplet (α

(n) s , α (n) p , α (n)
ρs ) is obtained by sweeping each of the three regularization parameters over a large interval of positive numbers and evaluating the corresponding residuals, i.e., the relative errors, in the euclidean norm in R N x , between the measured intensity vector U and the computed one U q (n) , and then selecting the value of the triplet that leads to the minimum residual (up to the noise level in the data). Note that we also monitor, at the same RR n°9247 time, the relative errors between two successive solutions q (n) and q (n+1) to determine whether or not the algorithm is stagnating, i.e., the algorithm is "trapped" at local minimum. Note that stagnation occurs when the successive relative error on the computed parameters is lower than a prescribed tolerance. In all the numerical experiments reported in Section 5, we have set this tolerance to be 0.1. Furthermore, we must point out that the proposed brute force approach for selecting, at each iteration, the "optimal" value of the regularization triplet is actually not a systematic three dimensional sweeping procedure that leads to a prohibitive computational cost.

The proposed strategy consists in tracking, the "optimal" value along a one-dimensional curve. Indeed, after an appropriate initialization, we proceed in three steps. In each step, we vary the values of one regularization parameter while the values of the two others remain "frozen". More specically, we proceed as follows:

• Initialization: selecting an initial guess. All three regularization parameters are chosen to be two order of magnitude greater than the largest diagonal entries of the normal system in [START_REF] Beylkin | Linearized inverse scattering problems in acoustics and elasticity[END_REF], i.e., we set:

α (0) s = α (0) p = α (0) ρs = 10 2 max 1≤l≤Ns+3 J * U q (0) J U q (0) ll (30)
•

Step 1: Varying α (n) s only. We start running the algorithm using the regularization matrix R (0) . At each iteration, we evaluate the relative residual and the successive relative error. If stagnation occurs and/or the relative residual increases, then the value of α (n) s is reduced while the values of the two others remain unchanged, i.e., α

A typical reduction factor of α We must point out that we have observed that only the sequence of the shape parameter vector s (n) tends to converge, in this step, to the sought-after parameter s.

• We have observed that at the end of this step, the sequence of material parameters (λ (n) , µ (n) ) tends to converge to the target parameter values (λ, µ).

• Step 3: Varying α (n) ρs only. We repeat exactly the same process described in Step 1 and

Step 2. The only dierence is that in this step, we decrease the value of α

(n) ρs while setting the value of α

(n) s (resp. α (n) p )
to be the last value selected in Step 1 (resp. Step 2).

Note that at the end of Step 3, the density sequence ρ (n) s tends to converge to the target value ρ s .

We must point out that in the event that the relative residual value does not reach the noise level, we repeat the full process starting with an initial guess for each parameter being the latest value selected at each corresponding step and using FFP measurements corresponding to a higher wavenumber k.

Inria 3.3.2 The Choice of R (n) in Stage II.
We recall that Stage II is dedicated to the determination of the location x c = (x c , y c ) using the regularized Newton equation given by ( 12)- [START_REF] Barucq | Localization of small obstacles from back-scattered data at limited incident angles with full-waveform inversion[END_REF]. Therefore, the regularization matrix R (n) is a 2 × 2 diagonal matrix given by R

(n) = α (n) c I ( 31
)
where I is the 2 × 2 identity matrix and α

(n) c
is a positive number representing the regularization parameter. Due to the small size of the linear system (13), the trial and error strategy for nding the optimal value of α

(n) c is simpler and "cheaper" than in Stage I.

Algorithm Summary and Computational Complexity

The description of the proposed solution methodology is summarized in this section along with the required calculations that drive the computational cost for its implementation.

Algorithm Summary

The proposed multi-stage solution methodology for solving the inverse problem IOP(7) can be summarized as follows.

Stage 0: Initialization. The proposed regularization Newton algorithm requires the following initial data:

A measured set of FFP, for one or several frequencies, at some observation points x1 , ..., xNx , i.e., the FFP vector

p ∞ = [ p ∞ (x 1 ), ..., p ∞ (x Nx )] T ∈ C N x
and its corresponding intensity

U = | p ∞ (x 1 ) | 2 , ..., | p ∞ (x Nx ) | 2 T
The tilde indicates that these measurements are tainted with a known noise level. An initial parameter vector q (0) = s (0) , λ (0) , µ (0) , ρ (0) s T ∈ R Ns+3 . The coordinates of the vector q (0) are selected arbitrarily, i.e., q (0) is a "blind" guess. Note that the initial location x c (0) of the scatterer is typically set to be the origin, that is, x c (0) = (0, 0) T .

We then construct the initial domain Ω

s and solve BVP(1) to deduce the FFP p (0) ∞ and its corresponding intensity U (0) .

Stage 1: Retrieving simultaneously the shape and material parameters q = [s, λ, µ, ρ s ] T . For a given frequency, this stage requires accomplishing successively the following tasks:

The construction, at each iteration n, of the Jacobian matrix J U (q (n) ) given by [START_REF] Biondo | 3D seismic imaging[END_REF].

The application of the regularized Newton algorithm given by ( 15) and ( 12) until convergence or stagnation is reached.

Note that the regularization matrix R (n) is selected according to the procedure described in Section 3.3.1. Moreover, at each iteration n, the intensity eld U (q (n) )

must be computed to monitor the algorithm convergence. If stagnation occurs, we RR n°9247 switch to a higher wavenumber k and repeat the procedure described in Stage 1 with an initial guess q (0) being the computed value q (n) at the stagnated iteration of the algorithm.

Stage 2: Recovering the location parameters, i.e., q = [x c , y c ] T

. The initial guess at this stage is q (0) = [0, 0] T and the scatterer Ω s corresponds to computed parameters

s (n) , λ (n) , µ (n) , ρ (n) s
T at convergence in Stage 1. This stage requires accomplishing successively the following tasks:

The construction, at each iteration n, of the Jacobian matrix J F (q

(n) ) dened in Section 2.1.2.
The application of the regularized Newton algorithm given by ( 14) and ( 12) until convergence or stagnation. Note that the regularization matrix R (n) is selected according to the procedure described in Section 3.3.2. Moreover, at each iteration n, the FFP p ∞ (q (n) ) must be computed to monitor the algorithm convergence.

Computational Complexity

The proposed multi-stage approach for solving IOP(7) mainly requires, for a given frequency, the following calculations.

Stage 0. Only one sparse linear system that results from the IPDG-discretization of BVP [START_REF] Azpiroz | Contribution to the Numerical Reconstruction in Inverse Elasto-Acoustic Scattering[END_REF] (see section 3.2) needs to be solved. This is accomplished using an LU factorization, as indicated in Section 3.2.

Stage 1. Two linear systems need to be solved:

-One sparse linear system with N s + 3 dierent right-hand sides in order to build the Jacobian matrix needed for the regularized Newton equation [START_REF] Beylkin | Linearized inverse scattering problems in acoustics and elasticity[END_REF].

-One small-size but full linear system ((N s + 3) × (N s + 3)) to determine the update δq (n) . This is accomplished, for each selected regularization matrix (see Section 3.3.1), using LU-factorization.

Stage 2. Similarly to Stage 2, two linear systems need to be solved:

-One sparse linear system with two dierent right-hand sides in order to build the Jacobian matrix needed for the regularized Newton equation ( 14).

-A 2×2 linear system to determine the update δq (n) . This is accomplished for each selected regularized parameter α c (see Section 3.3.2), using an analytical inversion.

Performance Assessment: Numerical Investigation

We investigate the convergence and the accuracy of the proposed multistage solution methodology, as well as its robustness to the noise level in the measurements. To this end, we consider We rst analyze the performance of the algorithm when the synthetic measurements are noisefree. We then investigate its sensitivity to the noise level. Last, we present results to illustrate the importance of using multiple frequencies to improve both the convergence and the accuracy, particularly when retrieving the material parameters.

5.1 Performance analysis in the absence of the noise.

We present in this section numerical results to assess the performance eciency of the proposed computational procedure when using noise-free synthetic FFP measurements. As stated earlier, these results have been obtained for three dierent scatterers: (a) a disk, (b) a non-convex octagon, and (c) a mockup-submarine.

Case of a disk-shaped scatterer.

As a proof of concept experiment, we consider here the simplest conguration: a disk-shaped domain made of steel. The synthetic FFP measurements are generated analytically using the Fourier series representation of the eld [START_REF] John J Bowman | Electromagnetic and acoustic scattering by simple shapes[END_REF][START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]. The FFP data have been measured at frequency f = 71.61kHz (i.e., ka = 3), which corresponds to a relatively low-medium frequency regime (see Fig. 3). The goal here is to determine the radius a, the material parameters (λ, µ, ρ), and the location center (x c , y c ) of a disk made of steel. The values of these six sought-after parameters are reported in the rst row of Table 1 and in Fig. 4.

Parameter In order to solve the considered six-parameter inverse problem (see ( 7)), we consider as an initial guess a disk whose characteristic parameter values are reported in the second row of Table 1 and in Fig. 4. The exterior boundary Σ (see [START_REF] Colton | Recent developments in inverse acoustic scattering theory[END_REF]) is a circle of radius 7.5 cm, which appears to be, given the considered frequency regime, far enough to neglect the eect of possible wave reections at Σ [START_REF] Barucq | Ecient DG-like formulation equipped with curved boundary edges for solving elasto-acoustic scattering problems[END_REF]. The resulting computational domain is discretized using an unstructured mesh of triangular-shaped elements with -at least-5 points per elements (see Fig. 5). Similarly to [START_REF] Barucq | Ecient DG-like formulation equipped with curved boundary edges for solving elasto-acoustic scattering problems[END_REF], we employ the IPDG solver using polynomial shape functions of fth order [START_REF] Ciarlet | The nite element method for elliptic problems[END_REF]. The results delivered by the proposed computational procedure are reported in Figs. 6789and Table 2. Recall that Stage I of the proposed algorithm uses the FFP intensity to deliver all the parameter values expect the location of the scatterer. The latter is retrieved in Stage II of the algorithm using the FFP. The following observations are noteworthy:

λ (GPa) µ (GPa) ρs (kg m -3 ) a (cm) x c (cm) y c (cm)
• The considered initial conguration (see Table 1) has been selected to ensure that the algorithm is "blind" to the sought-after parameter values. Indeed, the initial relative error on the radius (resp. the material parameters) is 50% (resp. 51%). The initial relative error on the location is about 100%. Consequently, the initial relative residual on the FFP intensity (resp. FFP) is over 80% (resp. 110%), as depicted in Fig. 3. Clearly, the proposed algorithm is applied outside the pre-asymptotic convergence region. and noise free measurements.

• Fig. 6 depicts the convergence performance of the proposed algorithm. It indicates that the convergence occurs after relatively a few iterations. More specically, the convergence in Stage I, which corresponds to the determination of the shape and material parameters from the FFP intensity measurements, is attained in 11 iterations. The relative residual on the FFP intensity drops from 80% to below 0.2%, as depicted in Fig. 3(b) and Fig. 7(c). Note that the relative residual on the FFP is however over 130% (see Fig. 7(a)).

Furthermore, the convergence in Stage II, which corresponds to the determination of the location from FFP measurements, is also achieved after 11 iterations. Indeed, the relative error on the FFP drops from 130% to below 0.1% (see Figs. • At convergence in Stage I the algorithm delivers the shape (resp. material) parameters with a relative error of about 0.07% (resp. 2%) (see Table 2 and Fig. 8(b)). Moreover, at convergence in Stage II, the algorithm determines the location of the scatterer with an accuracy level of about 0.01% (see Table 2 and Fig. 9(d)). Hence, the proposed algorithm retrieves the characteristic parameter values with a very high accuracy level.

RR n°9247

Parameter • The algorithm encountered stagnation phases during the iterative process in both stages, as reported in Figs. 89. Lowering the regularization parameter values, via the trial and error approach, enables to overcome this phenomenon. Indeed, this change allowed the computational procedure to keep iterating up to the tolerated residual level, i.e., the noise level.

λ (GPa) µ (GPa) ρs (kg m -3 ) a (cm) x c (cm) y c (cm)

Case of an octogonal-shaped domain.

In this experiment, we determine the characteristic parameter values of a non-convex octagon elastic scatterer Ω s immersed in water Ω f (see Fig. 10(a)). The sought-after scatterer Ω s is made of steel and its wet surface Γ is represented by the following piecewise linear parametrization:

Γ = (x c , y c ) + 8 j=1 (1 -t)X j + tX j+1 ; t ∈ [0, 1] (32)
where the vertices X j are given by: X j = s j cos θ j sin θ j ; j = 1, ..., 8.

(

) 33
Note that the vertices are uniformly distributed in the polar coordinate angle. The values of the shape parameters s j together with the center of the scatterer (x c , y c ), as well as the material parameters (λ, µ, ρ) are reported in Table 3.

Unlike the previous experiment pertaining to the disk-shaped scatterer, the synthetic FFP measurements cannot be generated analytically. Therefore, they were computed (see Fig. 11) using the IPDG solver introduced in [START_REF] Barucq | Ecient DG-like formulation equipped with curved boundary edges for solving elasto-acoustic scattering problems[END_REF]. However, in order to avoid the inverse crime [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF], we employed a mesh with a ner resolution than the one used during the inversion process (see Fig. 13 and Table 4).

In order to solve the considered thirteen-parameter inverse problem (see Eqs. ( 12), ( 14) and ( 15)), we considered as an initial guess a regular convex octagon elastic scatterer Ω (0) (see Fig. 10(b))

whose characteristic parameter values are reported in Table 3 and Fig. 12. The corresponding initial FFP is depicted in Fig. 11. The results of this inversion experiment are reported in Figs.

14-18 and in Table 5. The following observations are worth-while:

• The initial guess for the parameters values has been selected outside the pre-asymptotic convergence region, ensuring that the algorithm is "blind" to the sought-after thirteen values of the parameters, as reported in Table 3. Indeed, the initial relative errors in the shape parameters, Lamé coecients, density, and location of the scatterer, are 51.79%, 50.99%, 62.03%, 50%, respectively. These values lead to a computed initial FFP with a relative residual that exceeds 80% in the intensity (see Fig. • Fig. 14 illustrates the convergence performance of the algorithm. More specically, the determination of the shape and material parameters (in Stage I) is completed in about 18

iterations. The relative residual on the FFP intensity drops from above 80% to below 1%

(see Figure 15(c)). In Stage II, i.e., the determination of the location, the relative residual on the FFP drops from above 115% to below 1% in 13 iterations (see Figure 15(b)).

• At convergence in Stage I the algorithm delivers the shape parameters, the Lamé coecients, and the density with relative errors of 0.08%, 2% and 0.5%, respectively (see Table 5 and Figs. 16(e)-17(e)). At convergence, i.e., at the end of Stage II, the location of the scatterer is retrieved with a relative error of about 2% (see Fig. 17(g) and Table 5). Clearly, the proposed algorithm recovers the parameter values with an impressive accuracy level.

• Similarly to the disk experiment, stagnation phases occurred during the iterative process (see Figs. Relative Error (%) Target vs. Computed Values at convergence. f = 55.7kHz and noise free measurements.

using the following Fourier expansions:

Γ = (x ref , y ref ) + s 0 + M k=1 s 2k-1 cosθ + s 2k sinθ ; θ ∈ [0, 2π) (34)
The values of the shape parameters s j together with the center of the scatterer (x c , y c ), as well as the material parameters (λ, µ, ρ) are reported in Table 6 and Fig. 20.

(a) Target scatterer (b) Initial scatterer Similarly to the previous experiment pertaining to the non-convex octogonal scatterer, we have generated the synthetic FFP measurements points at frequency f = 55.7kHz (see Fig. 21) using the IPDG solver introduced in [START_REF] Barucq | Ecient DG-like formulation equipped with curved boundary edges for solving elasto-acoustic scattering problems[END_REF]. Here again, in order to avoid the inverse crime [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF], we employed a mesh ner than the one used during the inversion process, as indicated in Fig. 22 and Table 7.

To solve the resulting fourteen-parameter inverse problem (see Eqs. ( 12), ( 14) and ( 15)), we applied the proposed multistage algorithm starting from an initial conguration Ω (0) set to be a disk-shaped domain (see Fig. 19(b)). The characteristic parameter values of Ω (0) are reported in Table 6 and Fig. 20, and its corresponding FFP is depicted in Fig. 21. The results of this numerical experiment are reported in Target vs. Initial Guess.

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target conguration (see Table 6 and Figs. 20, 26(a)). The initial relative errors on the shape parameters, position parameter, Lamé coecients, and the density are 40.74%, 100%, 55.19%, and 85.18%, respectively. The initial relative residual on the FFP is of about 63% on the intensity (see Fig. 21(b)) and of about 120% on the eld (see Fig. 21(a)).

The convergence of the algorithm in this case is clearly demonstrated in Fig. 23. More specically, Stage I is completed in 24 iterations. The relative residual on the FFP intensity drops from 63% to 0.17% (see Fig. 24(c)). Furthermore, the algorithm converges in Stage II in 10 iterations. The relative residual on the FFP drops from 152% to 0.57% (see Figs. Similarly to the previous two experiments (the disk-and octogonal-shaped scatterers), the algorithm recovers the fourteen sought-after parameters with an impressive accuracy level.

Indeed, the relative errors on the computed shape parameters, Lamé coecients, and the density are 0.43%, 1.02%, and 2.73%, respectively (see Table 8 and Fig. 25). The center (x c , y c ) of submarine is recovered with a relative error of about 0.12% (see Table 8, Fig. 26).

The stagnation phenomenon also occurs here at several steps during the iteration process (see Figs. [START_REF] Colton | Recent developments in inverse acoustic scattering theory[END_REF][START_REF] Colton | A simple method for solving inverse scattering problems in the resonance region[END_REF]. This phenomenon is treated each time it happened by lowering the value of the regularization parameter via the trial and error procedure.

In all of the three experiments (disk, octagon, mockup-submarine), the proposed multistage strategy algorithm was able to successfully recover all the parameters from the knowledge of the FFP measurements corresponding to only one incident plane wave (i.e., one incident direction and one done frequency). This success is most likely due to the fact that the employed data are measured over the full aperture.

Region Table 7: Distribution of the triangles over the dierent computational domain regions: Mesh 1 vs. Mesh 2.

Γ s Γ f Ω s Ω f

Performance analysis in the presence of noise.

In what follows, we investigate the robustness and the accuracy of the proposed multistage solution methodology to the noise eect. To this end, we present illustrative numerical results corresponding to two congurations: a disk and an octagon.

5.2.1

Case of a disk-shaped domain.

We consider the same conguration adopted in the numerical experiment described in Section 5.1.1, i.e., a disk-shaped domain made of steel whose characteristic parameter values are reported in Table 1. The synthetic FFP measurements generated at frequency f = 71.61kHz (see Fig. 3) are however tainted with white noise to include possible errors in the measurements. Specically, we add to the FFP measured vector white noise of levels ranging from medium (5%), to high

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Parameter λ (GPa) µ (GPa) ρs (kg m -3 ) Target For all the considered white noise levels in the FFP data, the algorithm is initiated outside the pre-asymptotic convergence region (see Table 1). The initial conguration diers signicantly from the target one, leading to an initial relative residual over 80% for the intensity and over 110% for the full FFP.

Fig. 29 and Table 9 reveal that, in the three experiments, the convergence is achievable to the prescribed noise levels. Fig. 29 indicates that, in all three cases, the convergence is attained after at most 12 iterations in each stage.

• Figs. 30-33 and Table 9 indicate that, for all considered noise levels, the proposed solution strategy is able to retrieve all the parameters with a quite remarkable accuracy level. For example, even when the data are contaminated with a high noise level (10%), the proposed method recovers the shape and the location parameters with a relative error less than 5%.

In addition, the material parameters are obtained with a relative error of about 10% for the Lamé coecients and about 12% for the density (see Table 9). Furthermore, Figs. [START_REF] Farhat | On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method[END_REF][START_REF] Fazli | An analytical approach to estimate the number of small scatterers in 2D inverse scattering problems[END_REF] provide a comparison between the FFP corresponding to the target conguration and the one delivered by the proposed method at convergence. These gures also illustrate clearly the high accuracy level of the proposed inversion procedure.

Case of an octogonal-shaped scatterer.

We analyze here the noise eect on the determination of the characteristic parameter values corresponding to the non convex elastic scatterer described in Fig. 10(a) and Table 3. To this end, we consider the same numerical experiment set up presented in section 5.

• •

(c) Noise level 10%.

• •

(d) Noise level 15%.

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dierence is that we add to the measured FFP depicted in Fig. 11 white noise of three dierent levels: 5%, 10%, and 15%, as depicted in Figs. 34-35. The results reported in Figs. 36-40 and Table 10, clearly indicate that, similarly to the disk-shaped scatterer, the proposed computational strategy exhibits high performance eciency. Indeed, we observe that: • For all three noise levels, the proposed computational methodology is initiated outside the pre-asymptotic convergence region. Indeed, Fig. 36 shows that the initial residuals on the FFP intensity (resp. the full FFP), for all noise levels, exceed 80% (resp. 120%).

• Fig. 36 and Table 10 demonstrate the convergence of the proposed method. Indeed, the relative residuals reach the noise levels in -at most-18 iterations for each stage, regardless of the noise level.

• The sought-after parameters of the scatterer are retrieved with a high accuracy level, as reported in Table 10 and depicted in Figs. 37-38. • Figs. 39-40 also illustrate the high performance accuracy and robustness to the noise of the proposed method. These two gures show that the computed FFP at convergence is in agreement with the Target one.

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• •

(b) Noise level 5%.

• •

(c) Noise level 10%.

• •

(d) Noise level 15%.

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Figure 2 :

 2 Figure 2: Schematic description of the problem statement in a nite domain.

(n) s employed in the experiments reported in Section 5 is 1 / 10 .

 110 We continue iterating the algorithm and decreasing α (n) s until either convergence or stagnation is reached. At stagnation, we go to Step 2.

Step 2 :

 2 Varying α (n) p only. This step is similar to Step 1. The only dierence is that here the value of α (n) p varies whereas α (n) s remains frozen. Indeed, in this step, the value of α (n) ρs is still the initial value, i.e., α (n) ρs = α (0) ρs and the value of α (n) s is set to be the last selected value in Step 1. We iterate the algorithm and decrease the value of α (n) p at each stagnation or oscillation of the relative residual. We continue until convergence or stagnation are reached. At stagnation, we go to Step 3.

  synthetic FFP measurements corresponding to a single incident direction d = (1, 0) but for one or multiple frequencies. The measured data are obtained by sampling the full aperture of the FFP at 360 points. They are contaminated with white noise of levels ranging from 0% to 15% to mimic possible errors in the measurements. In what follows, we report on the determination of the values of the parameters characterizing three non-convex elastic scatterers: (a) a disk, (b) Inria a non-convex octagon, and (c) a mockup-submarine. Note that the rst two scatterers are made of steel, whereas the third one is made of aluminum.

Figure 3 :

 3 Figure 3: FFP for the disk-shaped scatterer experiment. Target vs. Initial Guess. f = 71.61kHz and noise free measurements.

Figure 4 :

 4 Figure 4: Material parameters for the disk-shaped scatterer experiment. Target vs. Initial Guess.

Figure 5 :

 5 Figure 5: Mesh resolution for the initial conguration of the computational domain for the disk shaped scatterer experiment.

Figure 6 :

 6 Figure 6: Convergence history in the case of a disk-shaped scatterer experiment. f = 71.61kHz

  7(a),(b)).

  11(b)) and 115% in the FFP (see Fig.11(a)). Full FFP at convergence in Stage I.

Figure 7 :

 7 Figure 7: FFP for the disk-shaped scatterer experiment. Target vs. Computed Values at convergence. f = 71.61kHz and noise free measurements.

Figure 8 :

 8 Figure 8: Material parameters for the disk-shaped scatterer experiment. Target vs. Computed Values at the occurred stagnation phases. f = 71.61kHz and noise free measurements.

Figure 9 :Figure 10 :

 910 Figure 9: Shape and location of the disk-shaped scatterer experiment. Target vs. Computed Values at the occurred stagnation phases. f = 71.61kHz and noise free measurements.

Figure 11 :

 11 Figure 11: FFP for the octogonal-shaped scatterer experiment. Target vs. Initial Guess. f = 55.7kHz and noise free measurements.

Figure 12 :

 12 Figure 12: Characteristic parameter values for octogonal-shaped scatterer experiment. Target vs. Initial Guess.

Figure 13 :

 13 Figure 13: Mesh resolution for the synthetic measurements (left) and for the inversion procedure at the initial computational domain conguration (right).

  16(a)-(e) and Figs. 17

5 . 1 . 3 Figure 14 :

 51314 Figure 14: Convergence history in the case of the octogonal-shape scatterer experiment. f = 55.7kHz and noise free measurements.

  Full FFP at convergence in Stage I.

Figure 15 :

 15 Figure 15: FFP for the octogonal-shaped satterer experiment. Target vs. Computed Values at convergence. f = 55.7kHz and noise free measurements.

Figure 16 :

 16 Figure 16: Material parameters for the octogonal-shaped scatterer experiment. Target vs. Computed Values at the occurred stagnation phases in Stage I. f = 55.7kHz and noise free measurements.

Figure 17 :Figure 18 :

 1718 Figure 17: Shape and location of the octogonal-shaped scatterer experiment. Target vs. Computed Values at the occurred stagnations. f = 55.7kHz and noise free measurements.RR n°9247

Figure 19 :

 19 Figure 19: Shape congurations for the mockup submarine experiments: Target vs. Initial Guess.

Figure 20 :

 20 Figure 20: Characteristic parameter values for the mockup submarine scatterer experiment.

  24(a),(b)).

  Number of triangles for Mesh 1 70 70 140 4920 Number of triangles for Mesh 2 52 52 105 2726

Figure 21 :Figure 22 :

 2122 Figure 21: FFP for the mockup submarine scatterer experiment. Target vs. Initial Guess. f = 55.7kHz and noise free measurements.

Figure 23 :

 23 Figure 23: Convergence history for the mockup submarine scatterer experiment. f = 55.7kHz and noise free experiments.

Figure 24 :

 24 Figure 24: FFP for the mockup submarine scatterer experiment. Target vs. Computed Values at convergence. f = 55.7kHz and noise free measurements.

Figure 25 :

 25 Figure 25: Material parameters for the mockup submarine scatterer experiment. Target vs. Computed Values at the occurred stagnation phases in Stage I. f = 55.7kHz and noise free measurements.

Figure 26 :

 26 Figure 26: Shape and location for the mockup submarine scatterer experiment. Target vs. Computed Values at the occurred stagnations. f = 55.7kHz and noise free measurements.

Figure 27 :

 27 Figure 27: FFP intensity for disk-shaped scatterer experiments. Target vs. Measured. f = 71.61kHz and noise levels in the measurements: 5%, 10%, 15%.

Figure 28 :

 28 Figure 28: Full FFP for disk-shaped scatterer experiments. Target vs. Measured. f = 71.61kHz and noise levels in the measurements: 5%, 10%, 15%.

  Noise level 15%.

Figure 29 :

 29 Figure 29: Convergence history for the disk-shaped scatterer experiments. Sensitivity to the noise level in the measurements f = 71.61kHz.

Figure 30 :

 30 Figure 30: Accuracy level on the material parameter values in the case of the disk-shaped scatterer experiments. Sensitivity to the noise level in the measurements. f = 71.61kHz.

Figure 31 :

 31 Figure 31: Accuracy level on the shape and location parameters in the case of the disk-shaped scatterer experiment. Sensitivity to the noise level in the measurements. f = 71.61kHz.

  Noise level 15%.

Figure 32 :

 32 Figure 32: Accuracy level on the FFP intensity for the disk-shaped scatterer experiments: Target vs. Computed Values at convergence. Sensitivity to the noise level in the measurements. f = 71.61kHz.

  Noise level 5%.

Figure 34 :

 34 Figure 34: FFP intensity for the octagon-shaped scatterer experiments: Target vs. Measured. f = 55.7kHz and noise levels in the measurement: 5%, 10%, 15%.

Figure 36 :

 36 Figure 36: Convergence history for the octogonal-shaped scatterer experiments. Sensitivity to the noise level in the measurements. f = 55.7kHz.

Figure 37 :

 37 Figure 37: Accuracy level on the material parameter values in the case of octagon-shaped scatterer experiments. Sensitivity to the noise level in the measurements. f = 55.7kHz.

Figure 38 :

 38 Figure 38: Accuracy level on the shape and location parameters in the case of the octagon-shaped scatterer experiment. Sensitivity to the noise level in the measurements. f = 55.7kHz.

Figure 43 :

 43 Figure 43: Imaginary FFP for the mockup submarine scatterer experiments: Target vs. Measured. f = 55.7kHz and noise levels in the measurements: 0%, 5%, 10%, 15%.

Figure 47 :

 47 Figure 47: Convergence history for the mockup submarine scatterer experiments. Sensitivity to the noise level in the measurements. Case of a two frequencies reconstruction: f 1 = 55.7kHz and f 2 = 159.15kHz.

Figure 49 :

 49 Figure 49: Accuracy level on the shape and location parameters' values for the mockup submarine scatterer experiments. Sensitivity to the noise level in the measurements. Case of a two frequencies reconstruction: f 1 = 55.7kHz and f 2 = 159.15kHz.

Table 1 :

 1 Characteristic parameter values for the disk-shaped scatterer experiment. Target vs.

	Target	115.40	76.90	7900	1.00	0.50	0.70
	Initial guess	50.00	50.00	3000	0.50	0.00	0.00
	Relative Error (%)	56.67	34.98	62.03	50	100	100
	Initial Guess.

Table 2 :

 2 Characteristic parameter values for the disk-shaped scatterer experiment. Target vs. Computed Values at convergence in Stage II. f = 71.61kHz and noise free measurements.

	Target	115.40	76.90	7900	1.00	0.50	0.70
	Computed	113.52	76.96	7757.616	0.99	0.49	0.70
	Relative Error	4.81	0.21	1.80	7.59 10 -2 5.71 10 -2 2.97 10 -5

Table 3 :

 3 Characteristic parameter values for the octogonal-shaped scatterer experiment. Target vs. Initial Guess.

	Region	Γ s	Γ f	Ω s	Ω f
	Number of triangles for Mesh 1 96 92 308 3862
	Number of triangles for Mesh 2 96 96 340 2556

Table 4 :

 4 Distribution of the triangles over the dierent computational domain regions: Mesh 1 vs. Mesh 2.

Table 5 :

 5 Characteristic parameter values of the octogonal-shaped scatterer experiment. Target vs. Computed Values at convergence. f = 55.7kHz and noise free measurements.

Table 8

 8

	and Figs. 23-26. The following observations are

Table 6 :

 6 Characteristic parameter values for the mockup submarine scatterer experiment. Target vs. Initial Guess.

Table 8 :

 8 Characteristic parameter values for the mockup submarine scatterer experiment. Target vs. Computed Values at convergence. f = 55.7kHz and noise free measurements.

(10%), and to very high (15%), as depicted in Figs.

27-28.

For each considered noisy FFP measured vector, we applied the proposed multistage algorithm from the same initial conguration (see Table

1

and Figs.

3-4

). The results of these three numerical experiments are reported in Figs.

29-33 and

Table 9. These results indicate the following.

Table 9 :

 9 Final relative residual and relative errors, sensitivity to the noise level in the measurements f = 71.61kHz for the disk-shaped scatterer experiments.

	Noise level	Relative Residual (%) Intensity Field	Relative Error (%) Shape Lamé Density Location
	0%	0.21	0.11	7.59 10 -2	1.36	1.80	0.01
	5%	6.06	5.07	0.36	3.43	1.38	0.48
	10%	11.75	9.95	0.57	5.06	3.41	0.42
	15%	17.36	14.88	0.60	6.97	9.85	0.64

Table 10 :

 10 Final relative residual and relative errors for the octogonal-shaped scatterer experiments, sensitivity to the noise level in the measurements. f = 55.7kHz.

	Noise level	Relative Residual (%) Intensity Field Shape Lamé Density Location Relative Error (%)
	0%	1.02	1.88	1.02	2.61	0.47	1.53
	5%	4.95	5.53	2.05	4.53	0.96	1.75
	10%	10.98	12.77	4.64	9.14	11.58	3.87
	15%	15.2	15.36	5.86	11.81	18.69	5.01

Table 11 :

 11 Relative Residual and Relative error for the mockup submarine scatterer experiments.Sensitivity to the noise level in the measurements. Case of a two frequencies reconstruction: f 1 = 55.7kHz and f 2 = 159.15kHz.

	Frequency Noise level	Relative Residual (%)		Relative Error (%)
	(kHz)		Intensity	Field Shape Lamé Density Location
		0%	0.17	0.57	0.43	4.96	1.15	0.12
	f 1 = 55.7	5%	5.12	5.23	1.60	17.55	4.95	0.53
		10%	9.30	11.34	3.79	14.19	14.33	2.29
		15%	15.29	17.82	2.34	17.15	11.36	4.75
		0%	1.42	1.09	0.50	0.90	0.84	0.21
	f 2 = 159.15	5%	4.05	6.46	0.53	0.89	0.90	0.21
		10%	12.76	13.05	3.75	2.52	0.68	1.66
		15%	13.01	14.62	2.33	6.74	8.37	0.91

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Recovery of the parameters with multiple-frequency measurements

We present here numerical results demonstrating that using FFP measurements corresponding to more than one frequency signicantly improves the performance accuracy of the proposed algorithm. The improvement in the accuracy is particularly noticeable when retrieving the material parameters with highly noisy data. We consider the thirteen parameters inverse problem introduced in section 5.1.3. Specically, the goal is to recover the parameter's values of the mockup submarine made of aluminum depicted in Fig. 19(a). These values are reported in Table 6 and Fig. 20. We initiate the algorithm from the same circular-shaped domain introduced in section 5.1.3 (see Fig. 19(b), Fig. 20 and Table 6). We employed in this numerical experiments two sets of FFP measurements. We rst apply the algorithm using FFP data corresponding to an angular frequency ω 1 = 0.35, i.e., f 1 = 55.7kHz, the frequency regime used in section 5.1.3.

When Stage II of the algorithm is completed, i.e., at convergence or stagnation, we re-apply the algorithm starting from the obtained nal domain conguration, but this time using FFP measurements corresponding to a higher frequency regime ω 2 = 1 (f 2 = 159.15kHz). Note that both sets of FFP measurements are tainted with white noise of level ranging from 0% to 15% (see Figs. [START_REF] Ivanyshyn | Inverse scattering for surface impedance from phase-less far eld data[END_REF][START_REF] Miguel | Sound, structures, and their interaction[END_REF][START_REF] Michael | Phaseless inverse scattering problems in three dimensions[END_REF]. The results of these numerical experiments are reported in Table 11 and Figs. 444546474849. These results suggest the following remarks:

• Similarly to all previous experiments, the algorithm, in this case, is initiated outside the pre-asymptotic convergence region. Indeed, the initial conguration, a circular-shaped domain, diers signicantly from the target mockup submarine, as indicated in Fig. 19 and Table 6.

• Fig. 47 reveals that the proposed algorithm has already converged and reached the noise level when using the FFP measurements corresponding to the rst frequency f 1 (see Figs.

44-46

). On the one hand, the accuracy on the shape parameters at convergence is quite satisfactory, as reported in Table 11 and depicted in Fig. 49. On the other hand, the recovery of the material parameters and location is less accurate but still acceptable, as reported in Table 11. Given the reasonable quality of the reconstruction, one may even consider to stop the experiment at this step.

• Using FFP measurements for higher frequency (f 2 = 159.15kHz) and re-initiating the algorithm from the computed conguration with the FFP data corresponding to the lower frequency (f 1 = 55.7kHz) enables to retrieve all the parameters with a quite remarkable accuracy, particularly for highly noisy measurements. Indeed, the accuracy on the material parameters is improved by a factor two to over an order of magnitude, depending on the noise level. Moreover, accuracy on the location is signicantly improved (up to a factor 5 in the case of 15% noise level), as reported in Table 11 and

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