M Aucejo

O De Smet

A space-frequency multiplicative regularization for force reconstruction problems

Keywords: Inverse problem, Force reconstruction, Multiplicative regularization, Mixed norms

Dynamic forces reconstruction from vibration data is an ill-posed inverse problem. A standard approach to stabilize the reconstruction consists in using some prior information on the quantities to identify. This is generally done by including in the formulation of the inverse problem a regularization term as an additive or a multiplicative constraint. In the present article, a space-frequency multiplicative regularization is developed to identify mechanical forces acting on a structure. The proposed regularization strategy takes advantage of one's prior knowledge of the nature and the location of excitation sources, as well as that of their spectral contents. Furthermore, it has the merit to be free from the preliminary definition of any regularization parameter. The validity of the proposed regularization procedure is assessed numerically and experimentally. It is more particularly pointed out that properly exploiting the space-frequency characteristics of the excitation field to identify can improve the quality of the force reconstruction.

Introduction

The characterization of dynamic forces acting on a structure remains one of the major industrial concerns to control broadband excitation sources or establish consistent excitation models for numerical simulation and design of complex structures. Unfortunately, the direct measurement of excitation forces is generally difficult or even impossible in practical situations. The basic idea to circumvent this practical limitation is to perform an indirect measurement from related accessible quantities such as displacement, velocity or acceleration fields.

Such techniques, referred to as force reconstruction problems, belong to the class of ill-posed inverse problems, meaning that the existence of a unique stable solution is not guaranteed. A possible solution to remedy this undesirable feature consists in including in the reconstruction problem some prior information on the forces to identify to constrain the space of admissible solutions. The mathematical transcription of this simple idea leads to express the inverse problem as a minimization problem, where prior information on the excitation forces is encoded in a regularization term. This regularization term can be incorporated in the formulation as an additive constraint, given rise to Tikhonov-like regularizations [START_REF] Tikhonov | Regularization of incorrectly posed problems[END_REF]. It should, however, be noted that a proper choice of the regularization term is crucial since it strongly conditions the quality of the reconstruction.

In general, two categories of reconstruction problem can arise in practical situations. The first one is related to the localization of excitation sources, while the second one consists in reconstructing the frequency spectrum or the time signal of prelocalized sources. Regarding the localization problem, the regularization term only reflects the spatial prior information on the sources to identify. It is often expressed as the p -norm of the desired solution vector. Such a norm is highly flexible to express one's prior knowledge on the nature of the forces to reconstruct [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF], since smooth solutions are obtained for p = 2 [START_REF] Tikhonov | Regularization of incorrectly posed problems[END_REF][START_REF] Boyd | Convex optimization[END_REF][START_REF] Renzi | Vibratory source identification by using the finite element model of a subdomain of a flexural beam[END_REF], while sparse excitation fields are promoted for p ≤ 1 [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF][START_REF] Fu | Efficient minimization methods of mixed l 2 -l 1 and l 1 -l 1 norms for image restoration[END_REF][START_REF] Chartrand | Nonconvex regularization for image segmentation[END_REF][START_REF] Grasmair | Non-convex sparse regularization[END_REF][START_REF] Chu | Robust bayesian superresolution approach via sparsity enforcing a priori for near-field aeroacoustic source imaging[END_REF]. It should be noticed that the spatial prior information is classically defined in a global manner, meaning that all the sources exciting the structure have the same attribute, i.e. localized, distributed or piecewise continuous. Such global regularization terms lead to poor reconstructions, if the actual excitation field combines sources of different nature, since the a priori has to reflect a compromise between contradictory distributions [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF]. To bypass this difficulty, a group regularization term, defined from p -norms, has been recently proposed to exploit local spatial prior information on both the nature and the location of excitation sources [START_REF] Aucejo | Bayesian structural source identification using local generalized gaussian priors[END_REF][START_REF] Aucejo | Bayesian source identification using local priors[END_REF]. In all the aforementioned methods developed in the frequency domain, the reconstruction problem is solved frequency by frequency, which is equivalent to suppose that the frequency spectrum of the identified sources is discontinuous. This lack of spectral continuity is inherent to these approaches and can induce potential inaccuracies in the reconstructed frequency spectrum if the sources are broadband [START_REF] Rezayat | Identification of dynamic forces using group-sparsity in frequency domain[END_REF]. On the other hand, for reconstructing the frequency spectrum or the time signal of prelocalized sources, the regularization term has to reflect prior information on the nature of the excitation signal. Since the sources are usually broadband, the force signal exhibits a certain continuity. That is why, the corresponding regularization term is generally constructed from the 2 -norm of the solution vector to identify [START_REF] Jacquelin | Force reconstruction: analysis and regularization of a deconvolution[END_REF][START_REF] Mao | Experimental study of hammer impact identification on a steel cantilever beam[END_REF][START_REF] Jang | Indirect measurement of the impulsive load to a nonlinear system from dynamic responses: Inverse problem formulation[END_REF][START_REF] Turco | Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives[END_REF]. Obviously, such identification methods are not suited for source localization and should fail when the locations of the potential sources does not match the actual ones.

Consequently, it appears that the vast majority of the methods proposed in the literature are generally unable to consistently tackle both localization and spectral/temporal reconstruction problems at the same time. To the best of our knowledge, only a few methods have been developed to address these issues. However, they are often limited to the reconstruction of point sources or to configurations where the spatial distribution of the sources and the nature of the excitation signals share the same characteristics such as the sparsity [START_REF] Jones | Detection of impact location and magnitude for isotropic plates using neural networks[END_REF][START_REF] Park | Monitoring Impact Events Using a System-Identification Method[END_REF][START_REF] Lourens | An augmented kalman filter for force identification in structural dynamics[END_REF][START_REF] Ginsberg | New approach for impact detection by finding sparse solution[END_REF][START_REF] Rezayat | Force identification using mixed and penalized optimization techniques[END_REF] or address the space-time (or space-frequency) reconstruction problem in a separated manner [START_REF] Turco | A strategy to identify excitating forces acting on structures[END_REF].

It is thus of primary interest to simultaneously exploit both the spatial and the spectral/temporal features of excitation sources to aid the reconstruction process in finding the best possible solution. These requirements are actually satisfied by regularization terms derived from a mixed p,q -norm. A mixed norm is a matrix norm defined for any matrix F by the following relation:

F p,q =   i j |F ij | p q p   1 q ∀(p, q) ∈ ]0, +∞[ 2 . (1)
Mixed 2,q -norms, for q ≤ 1, have revealed their suitability in signal and image recovery applications [START_REF] Kowalski | Sparse regression using mixed norms[END_REF][START_REF] Gramfort | Mixed-norm estimates for the m/eeg inverse problem using accelerated gradient methods[END_REF][START_REF] Zheng | Subspace weighted 2,1 minimization for sparse signal recovery[END_REF][START_REF] Wang | On recovery of block-sparse signals via mixed 2 / q (0 < q ≤ 1) norm minimization[END_REF]. In the context of force identification, Rezayat et al. first derive a Tikhonov-like regularization using a regularization term based on a mixed 2,1 -norm to reconstruct broadband point forces [START_REF] Rezayat | Identification of dynamic forces using group-sparsity in frequency domain[END_REF][START_REF] Rezayat | Force identification using mixed and penalized optimization techniques[END_REF].

In the present paper, an original regularization strategy is developed to solve both localization and spectral reconstruction problems within a unique framework. More precisely, the proposed approach first relies on the definition of a regularization term that properly reflects one's prior knowledge on the type (localized or distributed) of the excitation forces, as well as on the nature of the excitation signal. From a mathematical standpoint, this regularization term is constructed from the general mixed p,q -norm. Then, to derive the generic form of the identification problem, the proposed regularization term is included in the formulation as a multiplicative constraint. In doing so, a particular form of multiplicative regularization is obtained. This regularization strategy, originally developed by Van den Berg et al. [START_REF] Van Den Berg | Multiplicative regularization for constrast profile inversion[END_REF],

has several advantages compared to the more classical additive approaches, which explains its use in the present article. It has, in particular, the merit to be free from the preliminary definition of any regularization parameter. Accordingly, it is generally faster than the related Tikhonov-like regularization, but leads to similar reconstructed solutions [START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF]. To clearly highlight and explain the main features of the proposed regularization strategy, this article is divided into five parts. In section 2, the reconstruction model used for deriving the space-frequency regularization is detailed. Section 3 is devoted to the description of the generic formulation of the regularization problem. Its resolution is performed from an iterative procedure introduced in section 4.

Numerical and experimental validations of the space-frequency regularization are respectively proposed in sections 5 and 6. Obtained results point out the practical and potential interest in exploiting both spatial and spectral prior information for improving the quality of the force reconstruction. Finally, the last part of this paper introduces the theoretical basis of the possible extension of the space-frequency regularization to time domain applications.

Description of the reconstruction model

The definition of the space-frequency reconstruction problem requires the construction of a model describing the dynamic behavior of the structure and relating the measured vibration field to the excitation field to identify. To this end, let us consider the general situation where the studied structure is supposed linear and time-invariant. In this case, the dynamic behavior of the structure at a particular frequency f j is completely described by the transfer functions matrix H(f j ). Depending on the method used to derive this transfer function matrix, two reconstruction models can be defined. Indeed, if H(f j ) is measured, the reconstruction model is written [START_REF] Janssens | A pseudo-forces methodology to be used in characterization of structure-borne sound sources[END_REF][START_REF] Guillaume | An inverse method for the identification of localized excitation sources[END_REF][START_REF] Parloo | Force identification by means of in-operation modal models[END_REF]:

X(f j ) = H(f j ) F(f j ), ∀ j = 1, . . . , N, (2)
where X(f j ) is the measured vibration field and F(f j ) is the unknown excitation field at frequency f j and N is the number of studied frequencies.

On the other hand, if H(f j ) is numerically computed, the reconstruction model is then defined by the following relation [START_REF] Dobson | A review of the indirect calculation of excitation forces from measured structural response data[END_REF][START_REF] Leclere | Indirect measurement of main bearing loads in an operating diesel engine[END_REF]:

X(f j ) = H(f j ) F(f j ) + N(f j ) ∀ j = 1, . . . , N, (3)
where N(f j ) is the noise vector reflecting measurement and modeling errors.

To implement the proposed space-frequency regularization, one has to define a global model from the reconstruction model given by Eq. ( 2) or [START_REF] Boyd | Convex optimization[END_REF] at each studied frequency. Considering that the transfer function matrices are obtained from FEM, the global reconstruction model is written [START_REF] Rezayat | Force identification using mixed and penalized optimization techniques[END_REF]:

X = H F + N, (4)
where

H = diag [H(f 1 ), . . . , H(f j ), . . . , H(f N )]
is the global transfer functions matrix, X = X(f 1 ) T , . . . , X(f j ) T , . . . , X(f N ) T T is the global measured vibration field, F is the global unknown excitation field and N is the global noise vector.

Space-frequency multiplicative regularization

As already evoked in the introduction, a classical approach to stabilize an inverse problem consists in including in the formulation of the problem some prior information on the forces to identify. For this purpose, the inverse problem is generally expressed as a constrained minimization problem, namely:

min F F(X -H F) subject to R(F) ≤ β, (5)
where:

-F(X -H F) is the data-fidelity term which controls the a priori on the noise corrupting the data [START_REF] Green | Statistics of Images, the TV Algorithm of Rudin-Osher-Fatemi for Image Denoising and an Improved Denoising Algorithm[END_REF][START_REF] Nikolova | A variational approach to remove outliers and impulse noise[END_REF][START_REF] Le | A variational approach to reconstructing images corrupted by poisson noise[END_REF];

-R(F) is the regularization term that encodes prior information on the force distribution F [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF][START_REF] Chambolle | Total variation minimization and a class of binary mrf models[END_REF][START_REF] Hintermuller | Nonconvex TV q -models in image restoration: Analysis and a trust-region regularization based superlineraly con-vergennt solver[END_REF];

-β is some positive constant related to the solution variance.

For simplifying the resolution of the reconstruction problem, the constrained minimization problem is generally reformulated into an unconstrained form, in which the regularization term appears as an additive or a multiplicative constraint. The additive formulation actually corresponds to a Tikhonov-like regularization, that is:

F a = argmin F F(X -H F) + λ R(F), (6)
where λ is a Lagrange multiplier, a.k.a. the regularization parameter, which controls the trade-off between the data-fidelity and regularization terms.

The main issue of such a formulation is related to the choice of the regularization parameter. Generally, the optimal value of this parameter is automatically selected from adapted numerical procedures [START_REF] Golub | Generalized cross-validation as a method for choosing a good ridge parameter[END_REF][START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF]. However, these selection methods are often computationally intensive and time-consuming.

To circumvent this drawback, the multiplicative strategy can be used as an alternative [START_REF] Abubakar | Two -and three-dimensional algorithms for microwave imaging and inverse scattering[END_REF][START_REF] Abubakar | A multiplicative regularization approach for deblurring problems[END_REF][START_REF] Abubakar | An integral equation approach for 2.5-dimensional forward and inverse electromagnetic scattering[END_REF][START_REF] Li | Effective solution of nonlinear subsurface flow inverse problems in sparse bases[END_REF]. Formally, this approach simply writes:

F m = argmin F F(X -H F) • R(F). (7)
From a practical standpoint, this regularization method eliminates the need for the selection of the regularization parameter. As a result, the multiplicative strategy is significantly faster than the additive formulation. It should however be noted that both techniques lead to similar identified excitation fields [START_REF] Aucejo | Bayesian source identification using local priors[END_REF]. For this particular reason, the proposed space-frequency regularization method is based on a multiplicative approach. The choice of the data-fidelity and regularization terms is detailed in the following subsections.

Definition of the regularization term

An appropriate definition of the regularization term is all the more crucial since it conditions the quality of the reconstructed solution [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Boyd | Convex optimization[END_REF][START_REF] Aucejo | Bayesian source identification using local priors[END_REF][START_REF] Ginsberg | New approach for impact detection by finding sparse solution[END_REF].

In this paper, the regularization term is defined from the general mixed p,q -norm in order to properly exploit one's prior knowledge on the spatial and spectral characteristics of the forces to reconstruct. To highlight the main properties of the proposed regularization term and make its definition clearer, a step-by-step description is introduced. More precisely, it consists in explaining the construction of the space-frequency regularization term from the separate definition of frequency and spatial regularization terms.

Frequency regularization term

The frequency regularization term has to reflect prior information on the nature of the excitation signal. To this end, let us consider a reconstruction point i located on the structure. For this particular point, the frequency spectrum (a.k.a frequency group) is extracted from the global force vector F and is simply written:

F[i, :] = [F i (f 1 ), . . . , F i (f j ), . . . , F i (f N )] T . (8)
To take into account a wide class of excitation signals, the frequency regularization term can be expressed from a p -norm, that is:

R F[i, :] = F[i, :] p . (9)
Such a regularization term is well adapted to deal with broadband as well as sparse excitation signals. Indeed, when rotating machinery applications or milling processes are considered, the force signal is generally sparse, since only a few specific frequencies are excited [START_REF] Serviere | Blind source separation of noisy harmonic signals for rotating machine diagnosis[END_REF][START_REF] Huang | Vibration analysis in milling titanium alloy based on signal processing of cutting force[END_REF]. Such a sparsity pattern can be enforced using p ≤ 1 [START_REF] Chartrand | Nonconvex regularization for image segmentation[END_REF]. On the contrary, when the structure is excited by an impact or a shaker fed by a white noise signal, the force spectrum is necessarily broadband and continuous, i.e. rather smooth. In this situation, it is recommended to set p = 2 to promote the C 0 continuity of the force signal. To enforce a particular continuity pattern, such as C n or piecewise continuity, a differentiation operator L f has to be introduced to include in the formulation one's prior knowledge on the continuity of force spectrum [START_REF] Stephan | Total variation regularization for linear ill-posed inverse problems: Extensions and applications[END_REF][START_REF] Stickel | Data smoothing and numerical differentiation by a regularization method[END_REF]. As a result, a general form of frequency regularization term is:

R F[i, :] = L f F[i, :] p . (10)
Practically, the differentiation operator corresponds to the discretized form of a certain continuum derivative operator. For instance, to promote the C 1 continuity of the force spectrum, one has to set p = 2 and define a frequency differentiation operator L f of the form:

L f = 1 ∆f               -1.5 2 -0.5 -0.5 0 0.5 . . . . . . -0.5 0 0.5 0.5 -2 1.5               , ( 11
)
where ∆f is the frequency resolution.

Spatial regularization term

The spatial regularization term has to reflect prior information on the type of the excitation forces acting on a structure. To this end, let us consider the force vector at a particular frequency f j , namely:

F[:, j] = F(f j ) = [F 1 (f j ), . . . , F i (f j ), . . . , F M (f j )] T , ( 12
)
where M is the number of reconstruction points.

To derive a consistent spatial regularization term, it has to be flexible enough to properly reflect different priors. In this paper, we consider that only one type of sources excites the structure. Yet, the derivations presented in the next sections can be easily extended to the case where several types of sources act simultaneously on the structure. To this end, the concept of local regularization term has to be introduced [START_REF] Aucejo | Bayesian source identification using local priors[END_REF]. Here, the spatial regularization term is chosen such that:

R F[:, j] = L s F[:, j] q q , ( 13
)
where L s is the spatial differentiation operator that allows controlling the regularity of the solution.

As shown in Refs. [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Aucejo | Bayesian source identification using local priors[END_REF], the proposed regularization term is particularly well adapted for force reconstruction problems, since smooth solutions are promoted for q = 2 and L s = I M (I M : Identity matrix of dimension M ), while sparse solutions are favored for q ≤ 1 and L s = I M . For promoting piecewise continuous solutions, one can set q ≤ 1 and define L s as the discretized form of the n th -order differential operator. Thus, the solution is approximated by piecewise constant segments if n = 1 [see Eq. ( 11)], while the solution is approximated by piecewise polynomials of degree 1 if n = 2 [START_REF] Stephan | Total variation regularization for linear ill-posed inverse problems: Extensions and applications[END_REF].

Space-frequency regularization term

The role of the space-frequency regularization term is to reflect the experimenter's knowledge on both the spatial and spectral characteristics of the forces to identify. From the definition of the frequency and spatial regularization terms, it readily comes:

R(F) = L F q p,q = M i=1 (L F)[i, :] q p , ( 14
)
where L is the global space-frequency differentiation operator and the vector

(L F)[i]
is defined by analogy with Eq. ( 8). For implementation purposes, the space-frequency differentiation operator is expressed:

L = L s L f , (15)
where L s = I N ⊗ L s and L f are respectively the global spatial and frequency differentiation operator.

Such a regularization term introduces explicitly a coupling between the coefficients of the vector L F and allows promoting some structures observed in real signals [START_REF] Gramfort | Mixed-norm estimates for the m/eeg inverse problem using accelerated gradient methods[END_REF]. To illustrate this particular property of mixed norms, let us consider the simple case where L = I N M . Furthermore, if one represents the force vector F as a matrix, where the rows correspond to the force spectrum at a particular location and the columns to the force field at a specific frequency, it comes:

F =            F 1 (f 1 ) • • • F 1 (f j ) • • • F 1 (f N ) . . . . . . . . . F i (f 1 ) • • • F i (f j ) • • • F i (f N ) . . . . . . . . . F M (f 1 ) • • • F M (f j ) • • • F M (f N )            . ( 16
)
If we further assume that (p, q) = (2, 1), then the matrix F is supposed to be sparse along the lines (space) and full along the rows (frequency). In other words, one promotes the spatial sparsity of the excitation field (localized sources) and the continuity of its frequency spectrum (broadband source signal). Consequently, the present space-frequency regularization term is highly flexible, since it allows dealing with various force distributions and various excitation signals within a unique framework.

Definition of the data-fidelity term

The data-fidelity term is a measure of the difference between the measured vibration field X and the reconstructed vibration field H F. Consequently, it reflects prior information on the bias between measured and reconstructed vibration fields. In other words, it defines one's prior knowledge of the noise corrupting the data. A common assumption consists in considering that the vibration field is corrupted at each frequency by an additive Gaussian white noise. In this situation, the data-fidelity term is usually expressed as:

F(X -H F) = X -H F 2 2 (17)

Generic form of the space-frequency multiplicative regularization

The generic form of the proposed space-frequency multiplicative regularization is simply obtained by introducing Eqs. ( 14) and ( 17) into Eq. ( 7). In doing so, it readily comes:

F m = argmin F X -H F 2 2 • L F q p,q . ( 18
)

Resolution of the regularization problem

As already stated in section 3, the proposed formulation is flexible enough to properly incorporate prior information on the type of the sources acting on the structure, but also on the nature of the excitation signals. In return, the functional to minimize is generally non-convex, implying that the corresponding solution, if exists, has no closed-form expression. For this primary reason, the implementation of an iterative procedure is the keystone to obtain consistent regularized solutions. The requested algorithm should be easy to implement, sufficiently general to cope with a wide range of configurations depending on the values of p and q and should exhibit a linear or super-linear rate of convergence. All these requirements are met by the Iteratively Reweighted Least Squares algorithm [START_REF] Lawson | Contributions to the Theory of Linear Least Maximum Approximation[END_REF][START_REF] Rodriguez | An iteratively weighted norm algorithm for total variation regularization[END_REF][START_REF] Majumdar | Compressed sensing of color images[END_REF][START_REF] Daubechies | Iteratively re-weighted least squares minimization for sparse recovery[END_REF]. Such an iterative procedure has been successfully applied for solving force reconstruction problems at a particular frequency [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Aucejo | Bayesian source identification using local priors[END_REF][START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF]. In the next of this section, an adapted IRLS algorithm is derived to deal with the proposed space-frequency multiplicative regularization.

Basic principle

The proposed IRLS algorithm is a fixed-point algorithm adapted for solving minimization problems of the form of Eq. [START_REF] Park | Monitoring Impact Events Using a System-Identification Method[END_REF]. The underlying idea behind this algorithm is to define a fixed-point iteration having a unique and explicit solution. In doing so, it is expected to recover the solution of the original minimization problem when the convergence of the iterative process is reached. To derive the desired fixed-point iterate, the mixed p,q -norm to the power of q is recast into a squared weighted 2 -norm. In doing so, one obtains at iteration k+1:

F (k+1) m = argmin F X -H F 2 2 • W (k) 1/2 L F 2 2 , (19)
where

F (k+1) m
is the solution at iteration k+1 and

W (k)
is a global weighting diagonal matrix defined as a function of L F

m . In the present case, the coefficients of the global weighting matrix are defined such that:

W (k) I = W s (k) i • W f (k) i,j , (20)
where I = j + N (i -1) is a global index (for i = 1, . . . , M and j = 1, . . . , N )

and

W s (k) i and W f (k) i,j
are the weighting coefficients related to the space (s) and the frequency (f ) domains.

By setting

Y[i, :] = (L F (k) m )[i, :], the weighting coefficients W f (k) i,j and W s (k) i are written [see Appendix A]: W f (k) i,j = max p-2 , |Y i (f j )| p-2 (21)
and

W s (k) i = max 2 q p -1 , W f (k) i 1/2 Y[i, :] 2 2 q p -1 , (22)
where

W f (k) i = diag W f (k) i,1 , . . . , W f (k) i,N
and is a damping parameter that allows avoiding infinite weights. The damping parameter is automatically selected once for all at the beginning of the iterative process from the cumu-

lative histogram of L F (0)
m [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Rodriguez | Efficient minimization method for a generalized total variation functional[END_REF]. Actually, its value is calculated so that 5% of the values L F (0) m are less than or equal to .

To have a better insight on the regularization properties of the multiplicative strategy, the operational form of the previous minimization problem should be given. After some simple calculations, one finds the following explicit relation:

F (k+1) m = H H H + α (k+1) L H W (k) L -1 H H X, (23)
where α (k+1) is an adaptive regularization parameter, defined such that:

α (k+1) = X -H F (k) m 2 2 W (k) 1/2 L F (k) m 2 2 . ( 24
)
Practically, the adaptive regularization parameter automatically adjusts the amount of regularization throughout the iterative process, since its current value depends on the regularized solution computed at iteration k. This actually constitutes the definite advantage of the proposed multiplicative regularization over its additive counterpart, because updating the adaptive regularization parameter is almost costless.

Choice of the initial solution and stopping criterion

As any iterative procedure, the proposed IRLS algorithm requires the definition of a good initial guess and a reliable stopping criterion.

Choice of the initial solution

The choice of the initial solution is essential for a successful reconstruction. This choice is all the more crucial that the functional to minimize is generally non-convex. Actually, the initial solution has to satisfy several prerequisites to lead to relevant reconstructions. More precisely, it could be defined as a coarse solution of the minimization problem, easy to compute, but sufficiently close to the actual solution to ensure the convergence of the iterative process. Such requirements are fulfilled by a solution of the form:

F (0) m = H H H + α (0) L H L -1 H H X, (25)
where α (0) is a rough estimate of the converged value of the adaptive regularization parameter.

In Ref. [START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF], it has been shown that α (0) can be chosen from the following heuristic procedure:

1. Find estimates of the largest and the smallest singular values of

A = H L -1 H H L -1
, noted σ 1 and σ n respectively.

The estimate of the largest singular value is given by the upper bound of σ 1 , namely [START_REF] Horn | Topics in Matrix Analysis[END_REF]:

σ 1 (A) = A ∞ A 1 . ( 26
)
The estimation of the smallest singular value is obtained from σ 1 and an estimate κ of the condition number of A, namely:

σ n (A) = σ 1 (A) κ (A) . ( 27
)
2. Define a set S α 0 of possible values of α (0) ∈ [ σ n , σ 1 ] using a constant logarithmic spacing to take into account the decrease of the singular values.

3. Choose α (0) = median (S α 0 ).

Some comments should be made on the estimation procedure described above. This heuristic selection procedure has been developed with performance in mind. Indeed, classical automatic selection procedures, like the Generalized Cross Validation [START_REF] Golub | Generalized cross-validation as a method for choosing a good ridge parameter[END_REF] or the L-curve principle [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF], require the computation of the SVD of the system matrix, which can be time consuming when dealing with large reconstruction problems. To bypass this potential bottleneck, we have sought to develop a procedure close to the classical ones but avoiding the calculation of the SVD of the system matrix. That is why, estimates of the largest and smallest singular value are only calculated in step 1. Actually, all the performance gain are related to this calculation.

The second step is rather standard in the implementation of automatic selection procedures [see Ref. [START_REF] Hansen | Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems[END_REF] for instance]. The last step defines the initial adaptive regularization parameter as the median of the set S α 0 . It has been found experimentally that such an estimate allows generally obtaining a relevant initial solution for the iterative solver. However, because this estimation procedure is heuristic, it may sometimes fail to give a good starting point for the iterative process. In such a situation, it is always possible to choose α (0) as the regularization parameter picked by the L-curve or any other automatic selection procedure. Incidentally, the computational efficiency of the overall procedure is affected in proportion to the size of the transfer functions matrix.

Choice of the stopping criterion

Regarding the choice of the stopping criterion, a classical approach consists in stopping the iterative process when the relative variation of a certain quantity is less or equal to some tolerance. Actually, the proposed iterative procedure offers a natural definition of the stopping criterion, based on the relative variation δ of the adaptive regularization parameter. By definition, this convergence indicator is written:

δ = α (k+1) -α (k) α (k) . (28)
Experimentally, it has been found that setting the tolerance to 10 -8 allows obtaining consistent reconstruction, while preserving the time-performance of the IRLS algorithm [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF].

Generic resolution algorithm

A comprehensive overview of the proposed resolution algorithm is given in table 1.

Numerical validation

This numerical validation intends to investigate the practical interest of applying the proposed approach for solving force reconstruction problems.

Description of the numerical test case

The studied structure is a thin simply supported steel beam of length 1 m and cross-sectional area 3 × 10 -4 m 2 . The beam is excited by a point force of unit amplitude from 50 Hz to 500 Hz (frequency resolution: 1 Hz).

The coordinate of the point force, measured from the left end of the beam, Compute F from Eqs. ( 20), ( 21) and [START_REF] Turco | A strategy to identify excitating forces acting on structures[END_REF] Compute α (k+1) from Eq. ( 24)

Compute F (k+1) m from Eq. ( 23)

Update δ using Eq. ( 28)

k ← k + 1 end return F m ← F (k) m is x 0 = 0.6 m.
Practically, this excitation configuration allows not only simulating a broadband excitation such as a hammer impact, but also studying the influence of the proposed regularization term, since the excitation field spatially exhibits a sparse distribution of excitation sources.

To synthesize experimental vibration data, the global exact vibration displacement field X exact is first computed from a FE mesh of the beam made up with 20 plane beam elements, assuming that only transverse motions are measurable. Then, the exact displacement field is corrupted by an additive Gaussian white noise with a signal-to-noise ratio (SNR) equal to 30 dB. It should be added that a structural damping ratio of 0.01 has been introduced in the calculation to avoid infinite displacement amplitudes at resonance frequencies, whose the five first are given in Table 2. on the structure are of primary importance. The main interest in using such a numerical model is to allow the reconstruction of both external and reaction forces [START_REF] Renzi | Vibratory source identification by using the finite element model of a subdomain of a flexural beam[END_REF][START_REF] Aucejo | Bayesian source identification using local priors[END_REF].

Reference force vector

To measure the ability of the proposed space-frequency regularization in reconstructing relevant excitation fields, it is first necessary to define a reliable point of comparison. Such a reference force vector is derived from the global transfer functions matrix H and the global exact displacement X exact by using the following simple relation:

F ref = H -1 X exact . (29)
The typology of the sources acting on the structure is presented in Fig. 1.

More specifically, this figure proposes a spatial representation of the reference force vector at a resonance frequency of the structure (211 Hz), but also at a frequency lying outside resonance frequencies (150 Hz). It should be noted that the reference force vector exhibits, as expected, only point sources, namely the point force and the reaction forces at both ends of the beam.

For the sake of completeness, the exact displacement field at 150 Hz and 211 Hz is given in Fig. 2. This figure clearly show that the analysis displacement vibration field alone does not provide any information on the sought excitation field, which emphasizes the interest of developing an appropriate reconstruction strategy.

Naive reconstruction

When the displacement field is noisy, a naive approach is to solve the reconstruction problem in a least-squares sense, that is:

F naive = H + X, (30)
where

H + is the Moore-Penrose pseudoinverse of H.
Unfortunately, such a naive approach generally fails to locate and reconstruct the actual forces acting on a structure as illustrated in Figs. 3 and4.

It is especially noteworthy that the larger reconstruction errors of the point force spectrum are observed at resonance frequencies [see Fig. 4]. To justify this result, one has to keep in mind that for lightly damped structures the shape of the vibration response at resonance frequencies is close to that of the mode shapes. In this situation, the nearfield information is somewhat masked, which makes the reconstruction problem more involved. These preliminary results better explain the need for regularization methods and can serve as a basis to point out the main features of the proposed approach.

Reconstruction from the space-frequency regularization

To properly apply the proposed space-frequency regularization, one has to determine the parameters defining the regularization term. As already evoked in the introduction, consistent reconstructions can be obtained by properly exploiting the spatial and spectral characteristics of the force vector to identify. Regarding the spatial prior, one has to remind that the force vector is sparse [see Fig. 1]. Practically, this observation leads to define a parameter q promoting the sparsity of the solution. On the other is a particular case of the space-frequency approach, since it is obtained by defining N separate frequency groups for a reconstruction point i. In other words, this regularization strategy is written [START_REF] Aucejo | Bayesian source identification using local priors[END_REF]:

F m (f j ) = argmin F(j) X(f j ) -H(f j )F(f j ) 2 2 • F(f j ) q q , ∀ j = 1, . . . , N. (31)
The first step to assess the pertinence of the proposed SF regularization compared to the FbF strategy consists in appraising their ability in properly identifying the mechanical sources acting on the structure. The reconstructions proposed in Fig. 5 point out that both FbF and SF approaches provide a consistent identification of the reaction forces at boundaries. However, it is worth noting that the FbF regularization fails in localizing the point force at one of the resonance frequencies of the beam, which is not the case with the SF strategy [see Fig. 5b]. This observation is confirmed by the analysis of the spectrum of the point force reconstructed by the FbF or the SF approaches and presented in Fig. 6. Indeed, the force spectrum obtained using the FbF strategy exhibits large reconstruction errors around the resonance frequencies (94 Hz, 211 Hz and 376 Hz in the frequency range of interest).

On the contrary, the reconstruction error is limited to 1.6 dB at most over all the frequency range with the SF strategy. All these results tend to demonstrate that the proposed multiplicative regularization is well adapted to force reconstruction problems. Finally, one has to notice that the proposed regularization strategy is well adapted to solve large reconstruction problem, since the calculation of a SVD, that is generally used to compute the optimal regularization parameter, is avoided. This is all the more interesting than the calculation of the SVD can be time-consuming or even impossible for large-scale systems on a personal computer.

Influence of the choice of the norm parameters p and q

It can be argued that in real-life applications the proposed guidelines regarding the choice of the norm parameters p and q seem impractical, because the force distribution is unknown or there is no idea about the force to recover. We are prone to think that in real-life applications rough information about the force to recover is available. This information is provided by a careful analysis of the mechanical system. As a result, we think that a rough idea of the nature of the excitation signal and the type of the sources acting on the structure can be obtained, but this supposes that the user has some knowledge in mechanics.

Nevertheless, it is interesting to study the influence of an improper choice of p and q on the quality of the reconstructed solutions. To this end, two configurations are compared. The first one corresponds to the case where the the nature of the excitation signal is properly inferred, while the prior on the type of the source does not correspond to the actual spatial distribution.

This situation typically arises when (p, q) = (2, 2). The second configuration corresponds to the situation for which the type of the source is properly determined, while the prior on the nature of the excitation signal does not match the actual one. This situation is encountered when (p, q) = (1, 0.5).

The analysis of the spatial reconstructions at 150 Hz presented in Fig. 7 tends to show that an error on the choice of p seems less critical than an error on the choice of q. This observation is confirmed by the inspection of the spectrum of the reconstructed point force obtained from both configurations [see Fig. 8]. However, it should be noted that a poor choice of p lowered the overall quality of the reconstructed solution even if q is appropriately selected. This is particularly noticeable in Fig. 8b where the reconstruction error obtained for the second configuration with p = 1 is on the whole greater than that obtained with p = 2 [see Fig. 6b]. The proposed resolution algorithm being iterative, it could be interesting to determine how the noise corrupting the data affects its performances.

To this end, 10 3 samples of the measurement noise vector N, supposed to be a Gaussian white noise, are drawn for each of two following SNR: 30 dB and 25 dB. Then, the SF regularization is applied to each samples with (p, q) = (2, 0.5). This procedure allows quantifying the posterior uncertainty about the reconstructed solution by estimating the median and the 95% credible interval from the solution samples as well as monitoring the evolution of the number of iterations of the resolution algorithm.

As expected, Figs. 9 and 10 show that the posterior uncertainty about the reconstructed force vector is large at resonance frequencies and more generally increases as the SNR decreases, which is also associated to a loss of accuracy of the reconstruction. Regarding the number of iterations performed by the resolution algorithm to converge, the results are summarized in Table 3. It can be seen that the number of iterations increases as the SNR decreases, which is in line with our expectations. It is also worth noting that the related 95% credible interval around the median value is relatively sharp. Reference force spectrum and ( a ) 95% credible interval

Experimental validation

To confirm the conclusions drawn from the previous numerical experiment, the SF and FbF regularizations are applied to a real structure. The main objective of this experimental validation is to assess the performance of the proposed approach in operating conditions.

Description of the experimental set-up

The structure under test is a suspended (free) aluminum plate of 0.

Reconstruction of the excitation field

In the numerical validation, the proposed SF formulation has been compared with the related FbF regularization. A similar analysis process is proposed in this section. The analysis of the experimental set-up implies that the plate is only excited by a broadband point force. Consequently, it is necessary to define a regularization promoting the spatial sparsity of the excitation field as well as the continuity of the force spectrum. From the guidelines given in section 3.1, the tuning parameters of the SF regularization term are sets to (p, q) = (2, 0.5). To further compare the SF and FbF regularizations, it is interesting to see whether the reconstructed force spectrum far from the excitation point is close to 0 N. This expected result is confirmed in Fig. 13, where the reconstructed force spectrum at (x, y) = (0.109 m, 0.075 m) obtained from both regularization strategies are indeed close to 0 N. Although these results seem consistent with our expectations, it remains to verify the behavior of the SF and FbF approaches in the space domain. Indeed, the excitation field being reconstructed on a grid of 35× 29 points, corresponding to the measurement mesh, it is possible to study its spatial distribution. One can observe in Fig. 14 that both regularization techniques allows identifying relevant force distributions, since the point force is properly localized. More specifically, they perform equally well at non-resonance frequency (here 230 Hz), insofar as the point force amplitude is respectively estimated to 0.030 N and 0.036 N by the SF and FbF regularizations, while the measured amplitude is equal to 0.029 N. Furthermore, as already observed in the numerical validation and in Fig. 12, the main discrepancies can arises around the resonance frequencies. For instance, Fig. 14 points out that at 290 Hz, the SF and FbF regularizations behave rather differently, since the SF leads to a consistent reconstructed solution (identified: 0.015 N, actual: 0.014 N). On the contrary, the FbF strategy fails in properly estimating the point force amplitude (identified: 0.002 N). However, except this very local difference, both techniques provide similar results on whole frequency range [see Fig. 12]. In the end, the present experimental study shows that, in a particular real-world application, the SF multiplicative regularization leads to a reconstruction similar to that obtain with the related FbF multiplicative regularization. However, this observation is finally reassuring here, because the solution provided by the FbF strategy is already in good agreement with the reference one.

Extension to time domain applications

The proposed space-frequency regularization strategy can be easily adapted to derive a space-time regularization approach. To this end, only the global reconstruction model has to be modified. If the structure is time-invariant and initially at rest, the displacement field x k at instant t k is obtained from the discretized Duhamel's integral:

x k = ∆t k i=0 h k-i f i + n k , (32)
where ∆t = t k+1 -t k is the constant time step and h k and n k are respectively the impulse response matrix and the noise vector at instant t k .

The previous relation can be recast under the following matrix form:

X = H F + N, (33)
with:

H =         h 0 0 • • • 0 h 1 h 0 . . . . . . . . . . . . . . . 0 h Nt h Nt-1 • • • h 0         , X =         x 0 x 1 . . . x Nt         , F =         f 0 f 1 . . . f Nt         , N =         n 0 n 1 . . . n Nt         , (34)
where N t is the total number of time steps.

Consequently, using the previous mechanical model, the regularization procedure described in sections 3 and 4 can be directly implemented for time domain applications. It is also worth mentioning that the such a space-time regularization is a natural strategy for time domain applications, for which the reconstruction problem is classically established from a convolution model [START_REF] Jacquelin | Force reconstruction: analysis and regularization of a deconvolution[END_REF][START_REF] Qiao | Sparse deconvolution for large-scale ill-posed inverse problem of impact force reconstruction[END_REF]. We are prone to think that the extension and application of the proposed regularization strategy to force reconstruction problems in time domain is very promising, because of its ability to allow dissociate prior information on the spatial distribution and the time history of the force signals.

Conclusion

In the present study, the initial motivation was to propose a formulation of the force reconstruction problem able to fully exploit information available a priori on the type of the sources and the nature of the excitation signals.

To this end, a space-frequency multiplicative regularization has been introduced. This formulation is highly flexible, since it allows dealing with various force distributions and various excitation signals within a unique framework.

Practically, the regularization problem is solved from an adapted IRLS algorithm. The proposed experimental application tends to show that the SF multiplicative regularization gives results similar to those obtained with the related FbF multiplicative regularization. This apparently disappointing observation is actually reassuring, since, in the present case, the force signal identified by the FbF strategy is already in good agreement with the reference one. The potential benefit of the proposed SF strategy has been better highlight in the numerical validation, since it allows obtaining consistent reconstructions even at structural resonance frequencies, where the FbF approach generally fails. These proposed numerical and experimental applications consequently indicate that the SF regularization performs better than or, at least, similarly to the FbF regularization. Finally, a direct and natural promising extension of the proposed regularization has been introduced for time domain application. Its implementation and its performances will be the subject of a future publication.

is:

Y q p,q = M i=1 Y[i, :] p p q p = M i=1 W f (k) i 1/2 Y[i, :] 2 2 q p = M i=1 W f (k) i 1/2 Y[i, :] 2 2 q p -1 W f (k) i 1/2 Y[i, :] 2 2 = M i=1 W s (k) i W f (k) i 1/2
Y[i, :] where I = j + N (i -1).

Figure 1 :

 1 Figure 1: Reference force vector F ref [:, j] (a) at a non-resonance frequency (150 Hz) and (b) at a resonance frequency (211 Hz)

Figure 2 :

 2 Figure 2: Exact displacement vector X exact [:, j] (a) at a non-resonance frequency (150 Hz) and (b) at a resonance frequency (211 Hz)

Figure 3 :Figure 4 :

 34 Figure 3: Naive spatial reconstruction of the force vector F naive [j] (a) at a non-resonance frequency (150 Hz) and (b) at a resonance frequency (211 Hz) -(-) Reference and (--) Naive reconstruction

Figure 5 :

 5 Figure 5: Spatial reconstructions of the excitation field (a) at non-resonance frequency (150 Hz) and (b) at a resonance frequency (211 Hz) of the beam -(-) Reference, (--) SF regularization and (-• -) FbF regularization

Figure 6 :

 6 Figure 6: Spectrum of the reconstructed point force -(-) Reconstructed force spectrum and (--) Reference force spectrum

Figure 7 :Figure 8 :

 78 Figure 7: Spatial reconstructions of the excitation field at 150 Hz -(-) Reference, (--)Reconstructed excitation field for (p, q) = (2, 2) and (-• -) Reconstructed excitation field for (p, q) = (1, 0.5)

Figure 9 :

 9 Figure 9: Spatial reconstructions of the excitation field at 211 Hz for a SNR equal to (a) 30 dB and (b) 25 dB -Real part of the reconstructed force vector -(-) Reference, (--) Median of the samples and ( a ) 95% credible interval

Figure 10 :

 10 Figure 10: Spectrum of the reconstructed point force for a SNR equal to (a) 30 dB and (b) 25 dB -Real part of the reconstructed force vector -(-) Median of the samples, (--)

Figure 11 :

 11 Figure 11: Experimental set-up -(a) Suspended plate and (b) Excitation device

Figure 12

 12 Figure 12 presents a comparison of the force spectrum identified at ( x 0 , y 0 ) = (0.404 m, 0.256 m) by the SF and FbF regularizations with the reference signal measured by the force sensor between 150 Hz and 750 Hz (frequency resolution: 5 Hz). Contrary to what has been observed in the numerical validation, the SF and FbF regularizations provide similar results on the whole, but exhibit discrepancies at some resonance frequencies of the plate.

Figure 12 :Figure 13 :Figure 14 :

 121314 Figure 12: Reconstruction of the force spectrum at identified excitation point -(-) Reference measured by the force sensor, (--) SF regularization and (-.-) FbF regularization

4 ) 1 / 2 Y[i, :] 2 → 0

 41220 As stated previously, a damping parameter s should be introduced in order to avoid infinite weights when W f (k) i and q < p. For the sake of simplicity, the damping parameter is chosen such that f = s = , where is calculated from the cumulative histogram of L F

Table 1 :

 1 Generic resolution algorithm

Inputs: Global transfer functions matrix H, Global measured vibration field X, Global differentiation matrix L, Tolerance tol Output: Reconstructed force vector F m Initialization: Estimate α (0)

Table 2 :

 2 First five resonance frequencies

	ID	Value
	1	23.52 Hz
	2	94.11 Hz
	3 211.75 Hz
	4 376.45 Hz
	5 588.21 Hz
	Regarding finally the definition of the reconstruction model, the global
	transfer functions matrix H is computed from a FE model of the plate with
		21

Table 3 :

 3 Number of iteration with respect to the SNR

	SNR Median 95% CI
	30 dB	20	[19, 23]

[START_REF] Zheng | Subspace weighted 2,1 minimization for sparse signal recovery[END_REF][START_REF] Parloo | Force identification by means of in-operation modal models[END_REF]

Appendix A. Calculation of the weighting coefficients

The purpose of this appendix is to detail the calculation of the weighting coefficients W

]. From these definitions, the weighting coefficients are determined as follows:

The previous equation clearly shows that the weighting coefficients W f (k) i,j are theoretically defined by:

However, to avoid infinite weights when |Y i (f j )| → 0 and p < 2, a damping parameter f must be introduced.

The weighting coefficients W s (k) i are obtained in a similar fashion, that


Mathieu Aucejo
email: mathieu.aucejo@cnam.fr

Olivier De Smet

An iterated multiplicative regularization for force reconstruction problems

Keywords: Inverse problem, Force reconstruction, Multiplicative regularization, Iterated regularization

come

An iterated multiplicative regularization for force reconstruction problems

Introduction

Iterated Tikhonov regularization has been extensively studied by the inverse problem community. Classically, iterated Tikhonov regularization is defined as follows:

       F (0) = 0 F (k) = argmin F X -HF 2 2 + λ (k) F -F (k-1) 2 2 for k ≥ 1 , (1)
where in the context of force reconstruction X is the measured vibration field, H is the system matrix of the structure describing its dynamic behavior, F

is the force vector identified at iteration k and λ (k) is the regularization parameter at iteration k.

The previous formulation is referred to as non-stationary Tikhonov regularization [START_REF] Hanke | Nonstationary iterated Tikhonove regularization[END_REF][START_REF] Donatelli | On nondecreasing sequences of regularization parameters for nonstationary iterated tikhonov[END_REF][START_REF] Donatelli | Fast nonstationary preconditioned iterative methods for ill-posed problems, with application to image deblurring[END_REF][START_REF] Huang | On the choice of solution subspace for nonstationary iterated tikhonov regularization[END_REF][START_REF] Buccini | Regularizing preconditioners by non-stationary iterated Tikhonov with general penalty term[END_REF][START_REF] Buccini | Iterated tikhonov regularization with a general penalty term[END_REF], because the regularization parameter changes at each iteration. If the regularization parameter remains constant over the iterations, it is called stationary iterated Tikhonov regularization [START_REF] Buccini | Regularizing preconditioners by non-stationary iterated Tikhonov with general penalty term[END_REF][START_REF] Buccini | Iterated tikhonov regularization with a general penalty term[END_REF][START_REF] King | Approximation of generalized inverses by iterated regularization[END_REF][START_REF] Engl | On the choice of the Regularization Parameter for Iterated Tikhonov Regularization of Ill-Posed Problems[END_REF][START_REF] Gfrerer | An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates[END_REF][START_REF] Scherzer | Convergence rate of iterated tikhonov regularized solutions of nonlinear ill-posed problems[END_REF][START_REF] Jin | On the choice of the regularization parameter for ordinary and iterated regularization of nonlinear ill-posed problems[END_REF]. Practically, iterated Tikhonov regularization can be viewed as an iterative refinement of an initial solution obtained by applying the standard Tikhonov regularization [START_REF] Tikhonov | Regularization of incorrectly posed problems[END_REF]. Surprisingly, this regularization strategy seems to be seldom applied in mechanics despite its ability to compute improved identified solutions [START_REF] Cimetière | Solution of the cauchy problem using iterated tikhonov regularization[END_REF][START_REF] Jang | Numerical experiments on determination of spatially concentrated time-varying load on a beam: an iterative regularization method[END_REF].

In the context of force identification, iterated Tikhonov regularization has two major drawbacks. First, the regularization term promotes distributed solutions, which is not desirable when the structure is actually excited by localized or impulsive sources. This problem can be solved by implementing the iterated Tikhonov regularization in Banach spaces instead of Hilbert spaces [START_REF] Jin | Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces[END_REF][START_REF] Jin | Nonstationary iterated Tikhonov regularization in Banach spaces with uniformly convex penalty terms[END_REF][START_REF] Margotti | An inexct Newton regularization in Banach spaces based on the nonstationary iterated Tikhonov method[END_REF]. Practically, this means that the regularization term is express as the q -norm of the residual solution F -

F (k-1)
. Such a regulariza-tion term is highly flexible to express one's prior knowledge on the sources to identify, since smooth solutions are promoted for q = 2 [START_REF] Boyd | Convex optimization[END_REF][START_REF] Renzi | Vibratory source identification by using the finite element model of a subdomain of a flexural beam[END_REF], while sparse excitation fields are obtained with q ≤ 1 [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF][START_REF] Fu | Efficient minimization methods of mixed l 2 -l 1 and l 1 -l 1 norms for image restoration[END_REF][START_REF] Chartrand | Nonconvex regularization for image segmentation[END_REF][START_REF] Grasmair | Non-convex sparse regularization[END_REF][START_REF] Chu | Robust bayesian superresolution approach via sparsity enforcing a priori for near-field aeroacoustic source imaging[END_REF][START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Aucejo | Bayesian source identification using local priors[END_REF][START_REF] Qiao | Sparse deconvolution for the large-scale ill-posed inverse problem of impact force reconstruction[END_REF][START_REF] Li | A hierarchical bayesian method for vibration-based time domain force reconstruction problems[END_REF][START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF].

The second limiting factor is the estimation of the regularization parameter λ, that can be computed through automatic selection procedures such as the L-curve principle [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF] or the Generalized Cross-Validation [START_REF] Golub | Generalized cross-validation as a method for choosing a good ridge parameter[END_REF]. Indeed, when the reconstruction problem becomes large, this calculation can be very expensive, because they are usually based on the computation the Singular Value Decomposition (SVD) of the system matrix H. In this respect, the multiplicative regularization [START_REF] Abubakar | Two -and three-dimensional algorithms for microwave imaging and inverse scattering[END_REF][START_REF] Abubakar | A multiplicative regularization approach for deblurring problems[END_REF][START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF][START_REF] Aucejo | A space-frequency multiplicative regularization for force reconstruction problems[END_REF] is an interesting alternative, since it eliminates the need for the selection of the optimal regularization. However, as shown in the next of the paper, when this strategy is applied to very noisy data, as any Tikhonov-like regularization, identified solutions are generally disappointing. In such a situation an iterated approach is particularly well suited.

To combine the best of both worlds, an original iterated multiplicative regularization is proposed. In this paper, the main features of the proposed regularization strategy are set out, while its validity is assessed through a series numerical and real-world experiments. To clearly introduce the main features of the proposed strategy, this contribution is divided into four parts.

Before considering the core of the paper, the need for another regularization strategy is explained in section 2. Section 3 is devoted to the introduction of the proposed iterated multiplicative regularization. More precisely, a nonstationary version is introduced as well as its stationary counterpart. The comparison of both formulations is performed numerically in section 4 on two frequency domain applications. Obtained results demonstrate perhaps counterintuitively that proposed stationary formulation outperforms the associated non-stationary version regarding the quality of reconstructed solutions. The experimental validation presented in section 5 allows confirming the conclusions drawn in the numerical validation.

The need for another regularization strategy

As mentioned in the introduction, the iterated Tikhonov regularization given by Eq. ( 1) can lead to inaccurate reconstructions when the excitation field to identify is sparse. To illustrate this, let us consider the reconstruction in the frequency domain of a harmonic unit point force exciting a free-free steel beam at 350 Hz from a noisy measured vibration displacement field having a certain signal-to-noise ratio (SNR) [see section 4.1 for a detailed description of the test case].

Let us first consider the situation where the SNR is relatively high, e.g. 30 dB. In this case, it is clear from Fig. 1 that neither the standard Tikhonov regularization nor its stationary iterated version using the regularization parameter computed at the first iteration, namely λ (1) , from the L-curve principle give a satisfying identified solution. Indeed, the analysis of the result shows that the reconstructed excitation fields are too smooth compared to the target one. It should also be noted that the non-stationary iterated Tikhonov regularization does not provide a better identified solution. Indeed, for a traditional choice of λ (k) , namely [START_REF] Brill | Iterative solution of ill-posed problems: A survey[END_REF]:

λ (k) = λ (1) β k-1 , with β ∈ [0, 1], (2)
λ (k) decreases as k increases. As a result, if the regularization parameter λ (1) is optimally selected from the L-curve principle for instance, then the non-stationary iterated Tikhonov regularization leads to the least squares solution. As explained in the introduction, the previous formulation has two drawbacks, since it does not exploit one's prior knowledge of the sources to identify and it requires the calculation of a regularization parameter, which can be computationally intensive depending on the size of the reconstruction problem. To bypass these potential difficulties, a multiplicative regularization has been introduced [START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF][START_REF] Aucejo | A space-frequency multiplicative regularization for force reconstruction problems[END_REF]. Formally, it consists in solving the following minimization problem:

F = argmin F\{0} X -HF 2 2 • F q q , ( 3
)
where q is the norm parameter included in the interval ]0, +∞[, • q is the q -norm or quasi-norm. Practically, q is set to 2 if the solution is supposed to be distributed, while q can be chosen equal to or less than 1 if the solution is supposed to be localized [START_REF] Aucejo | Bayesian source identification using local priors[END_REF].

The application of the multiplicative regularization from the measured vibration field used previously gives, when q = 0.5, the reconstruction field presented in Figure 2. This figure shows that the reference excitation field and the reconstructed one are very close together when the measured data are little noisy (high SNR). Unfortunately, when the data are very noisy (low SNR), a degradation of the regularized solution accuracy is observed [see Figure 2].

In the light of these observations, it seems interesting to develop an iterated multiplicative regularization, combining the benefits of the multiplicative regularization and the iterated Tikhonov regularization. In particular, such an approach is potentially able to provide a refined regularized solution, while exploiting prior information on the sources to identify.

Iterated multiplicative regularization

This section aims at introducing the non-stationary iterated multiplicative regularization as well as its stationary counterpart. To render the pre- sentation more didactic, it is focused on the main concepts and features of the method.

Non-stationary version

By analogy with the standard iterated Tikhonov regularization, the nonstationary iterated multiplicative regularization (NSIMR) is expressed as:

         F (0) = 0 F (k) = argmin F\ F (k-1) X -HF 2 2 • F -F (k-1) q q for k ≥ 1 . (4)
Practically, the resolution of the minimization problem at iteration k ≥ 1 requires the implementation of an iterative procedure. In the present contribution, an Iteratively Reweighted Least Squares (IRLS) algorithm is implemented. Basically, it consists in iteratively computing the solution of the problem by recasting the q -norm into a weighted 2 -norm. As a result, at iteration k of the proposed iterated regularization, the estimated force vector

F (k,j)
at iteration j of the IRLS algorithm is explicitly given by [START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF]:

F (k,j) = H H H + α (k,j) W (k,j-1) -1 H H X + α (k,j) W (k,j-1) F (k-1) , (5)
where:

• W (k,j-1) is a diagonal weighting matrix depending on the norm parameter q and the residual solution at iteration j -

1, namely R (k,j-1) = F (k,j-1) -F (k-1)
[ [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF][START_REF] Aucejo | Bayesian source identification using local priors[END_REF][START_REF] Rodriguez | An iteratively weighted norm algorithm for total variation regularization[END_REF]. For the sake of completeness, it should be mentioned that the i th component of the weighting matrix at iterations outer iteration k and inner iteration j -

, is expressed as:

w (k,j-1) i = max (k) , r (k,j-1) i q-2 , (6)
where r

(k,j-1) i

is the i th of the residual solution R (k,j-1) and (k) is a small real positive number acting as a damping parameter. It allows avoiding infinite weights when r (k,j-1) i → 0 and q < 2. Its value is selected at the beginning of each outer iteration so that 5% of the values of R (k,0) are less than or equal to (k) .

• α (k,j) is the adaptive regularization parameter 1 , defined such that:

α (k,j) := X -H F (k,j-1) 2 2 W (k,j-1) 1/2 F (k,j-1) -F (k-1) 2 2 . ( 7
)
1 because its value is updated at each iteration of the IRLS algorithm.

8

When the IRLS algorithm has converged after N j iterations, the solu-

tion vector F (k)
, the weighting matrix W (k) and the adaptive regularization parameter α (k) are returned by the algorithm and are defined such that:

F (k) := F (k,N j ) , W (k) := W (k,N j ) and α (k) := α (k,N j ) . (8)
Consequently, the previous non-stationary iterated multiplicative algorithm is composed of a main (outer) iteration corresponding to the refinement stage and a nested (inner) iterative procedure to compute F

. Finally, to completely implement the IRLS algorithm, it remains to define the initial solution of each nested IRLS procedure as well as stopping criteria for both nested and outer loops.

The initial solution of the IRLS algorithm at outer iteration k > 1 is chosen as the solution of the following weighted Tikhonov regularization [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF]:

F (k,0) = argmin F X -HF 2 2 + α (k-1) W (k-1) 1/2 F -F (k-1) 2 2 = H H H + α (k-1) W (k-1) -1 H H X + α (k-1) W (k-1) F (k-1)
.

For the particular case k = 1, the initial solution is computed from the previous relation using W (0) = I (I: Identity matrix) and choosing α (0) from the heuristic selection procedure described in [START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF].

Regarding now, the definition of the stopping criteria, it should be noted that the proposed non-stationary regularization offers a natural definition of the stopping criteria based on the relative variation of the adaptive regularization parameter between two successive outer or inner iterations. Formally, the relative variations δ and γ of the adaptive regularization parameter for the inner and outer loops are respectively defined such that:

δ = α (k,j) -α (k,j-1) α (k,j-1) and γ = α (k) -α (k-1) α (k-1) . ( 10
)
As classically done in the literature, the iterative processes are stopped when the relative variations δ and γ are less than or equal to some tolerance. Experimentally, it has been found that setting the tolerances to 10 -8 and 10 -2 for the inner and outer loops respectively allows obtaining consistent reconstructions, while preserving the computational performances of the overall resolution process.

Stationary version

The stationary iterated multiplicative regularization is directly derived from the non-stationary version by fixing once for all the values of the weighting matrix W (k) and the adaptive regularization parameter α (k) to their values obtained after the first main iteration (i.e. k = 1) of the non-stationary algorithm, that is:

W (k) = W := W (1,N j ) and α (k) = α := α (1,N j ) . (11)
In other words, the stationary multiplicative regularization (SIMR) can be expressed as:

               F (0) = 0 F (1) , W, α = argmin F\{0} X -HF 2 2 • F q q for k = 1 F (k) = argmin F X -HF 2 2 + α W 1/2 F -F (k-1) 2 2 for k ≥ 2 . ( 12
)
At this stage, it is important to note that from k = 2, the explicit expression of the force vector

F (k)
is:

F (k) = H H H + α W -1 H H X + α W F (k-1) , for k ≥ 2. ( 13
)
Consequently, only the matrix

A = H H H + α W has to be inverted to com- pute F (k)
for any k ≥ 2. This allows deriving a computationally efficient algorithm, since if A is factorized using a LU decomposition for instance, the overall computation of

F (k)
for k ≥ 2 is almost as expensive as than the computation of F

.

The main differences with the non-stationary version concerns the definition of the stopping criteria. As shown in Eq. 12, the proposed stationary iterated multiplicative regularization first requires the resolution of a multiplicative regularization at iteration k = 1 from the IRLS procedure described in Ref. [START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF], defining thoroughly the initial solution and the stopping criterion as in section 3.1. Here, the question is when to stop the main iteration loop.

In the present case, because the adaptive regularization parameter remains constant after the first outer iteration, the stopping criterion is consequently related to the relative variation of the functional:

J( F (k) ) = X -H F (k) 2 2 + α W 1/2 F (k) -F (k-1) 2 2 ( 14
)
between two successive iterations. Practically, the algorithm is automatically stopped when a prescribed tolerance defined by the user is reached. Here, the tolerance is set to 10 -2 as for the non-stationary version.

Numerical validation

This numerical validation intends to investigate the differences between the two proposed iterated multiplicative regularizations through two frequencydomain applications. In the present numerical validations, only the reconstruction of the spatial distributions of sources associated to a harmonic point force exciting the studied structures at a particular frequency is considered. This is not a limitation of the method, since the force spectrum or history can be reconstructed from the proposed approach provided that the system matrix H is established accordingly. Finally, it is also worth noting that all the reconstructions are performed outside resonance frequencies. Indeed, at resonance frequencies, the inverse problem is known to be difficult. Because the system matrix is singular, the uniqueness of the solution is not ensured and the identified solution is generally not satisfying whatever the regularization approach used. It follows that the corresponding refined solution is

generally not satisfying at resonance frequencies as well.

1D structure

In the present application, the studied structure is a free-free steel beam with dimensions 1×0.03×0.01 m 3 excited by a unit point force at 350 Hz.

The coordinate of the point force, measured from the left end of the beam, is x 0 = 0.6 m.

Synthesis of the vibration field

In this example, the reconstruction is performed from the displacement field X measured at 350 Hz over the structure. To synthesize the measured vibration field, two numerical steps have been implemented. First, a finite element model of the beam made up with 20 plane beam elements has been used to compute the exact displacement field. Then, a Gaussian white noise with a prescribed SNR has been added to the exact data to simulate measurement errors, related to the transducers quality.

Reconstruction model

Let consider the practical situation where the vibration field X, measured over the surface of a structure, is caused by an unknown excitation field F. If the structure is linear, its dynamic behavior is completely determined by the transfer functions matrix H, relating the vibration field X to the unknown field F by the relation:

X = HF. (15)
In the present application, the FE model of the structure, used to compute the vibration field, has also been employed to compute the transfer functions matrix H by assuming that only bending motions are measurable. In other words, the transfer functions matrix have been dynamically condensed over the measurable dofs.

Application

To assess the validity of the proposed iterated multiplicative regularization strategy, it is compared to the ordinary multiplicative regularization (OMR). For this purpose, the choice of the tuning parameter q is crucial.

The analysis of the numerical test case shows that the beam is only excited by a point force. In this context, it is reasonable to set q = 0.5.

In section 2, it has been shown that, for this application example, the excitation field identified from OMR was in line with the target one, when the reconstruction is based on measured data with a high SNR value (30 dB in the present case). Consequently, both versions of the iterated multiplicative regularization are expected to provide reconstructions similar to that obtained with OMR. To confirm this intuition quantitatively, the relative error of the reconstructed solution is evaluated for each regularization strategy and given in Table 1. Formally, the relative error is given by:

E = F ref -F id 2 2 F ref 2 2 , ( 16
)
where F ref is the reference excitation field and F id is the force identified by OMR, NSMIR or SMIR. Furthermore, the computation time of each regularization strategy is also given in Table 1 to fairly compare their computational efficiency.

The results presented in Fig. 3 and Table 1 confirm our intuition, since the three approaches give similar reconstructions. It should however be noted that SMIR leads to a slightly better reconstruction for a computation time comparable to OMR, while NSMIR provides the same result as OMR for a computation time almost 30 times larger.

Perhaps more interestingly, for measured data with a low SNR level, the differences between each approach appear more clearly [see Fig. (--) OMR for q = 0.5, (-• -) NSIMR for q = 0.5 and (• • • ) SMIR for q = 0.5

F (k) ≈ F (k-1)
at each outer iteration.

This result can be explained by the fact that, in NSIMR, the regularization parameter is optimally adapted in order to find a solution minimizing the

residual solution F -F (k-1)
, which is not the case for SMIR. In addition, to have a global overview of the influence of the noise level on the solutions reconstructed from OMR and SMIR, a discussion is proposed in Appendix A.

As a side note, it should however be noted that, below 10 dB, OMR (and so NSMIR and SMIR) fails in providing consistent solutions as highlighted by the various simulations we have run 2 .

2 This is not the case of the related additive regularization described in Ref. [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF] which is based on automatic a posteriori selection procedures like the L-curve to determine the associated regularization parameter λ. Consequently, if needed, one can implement a stationary iterated additive regularization by simply computing at k = 1 the solution of the related additive regularization and replacing from k = 2 the adapted regularization parameter ᾱ by the regularization parameter λ obtained after the first outer iteration.

The latter iterated regularization is however computationally more expensive than that proposed in this paper [START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF]. (--) OMR for q = 0.5, (-• -) NSIMR for q = 0.5 and (• • • ) SMIR for q = 0.5

2D structure

In this second application, the studied structure is a free-free steel plate with dimensions 0.6 × 0.4 × 0.005 m 3 . The coordinates of the point force, measured from the lower left corner of the plate, are (x 0 , y 0 ) = (0.247 m, 0.218 m).

Synthesis of the vibration field

In this example, the reconstruction is performed from the velocity field X measured at 350 Hz over the structure. To properly simulate experimental measurements, the exact vibration displacement field X exact is first computed from a FE mesh of the plate made up with 187 shell elements, assuming that only bending motions are measurable. Then, a Gaussian white noise with a prescribed SNR is added to the exact data to simulate measurement errors.

In the next of this application, the generated exact data are corrupted so that the corresponding SNR is either equal to 30 dB or 13 dB.

Reconstruction model

As in section 4.1, the structure being linear, its dynamic behavior is governed by the transfer functions matrix H, relating the vibration field X to the unknown field F through Eq. 15. The FE model of the structure, used to compute the vibration field, has also been employed to compute the transfer functions matrix H by assuming that only normal velocities are measurable. In other words, the transfer functions matrix have been dynamically condensed over the measurable dofs.

Application

The analysis of the test case suggests that the target excitation field is very sparse. Consequently, it is reasonable to set q = 0.5. In doing so, all the necessary ingredients are available for reconstructing the excitation field at 350 Hz from OMR, NSIMR and SMIR.

As shown in Figs. 5 and6, all the regularization strategies gives excellent reconstructed excitation fields when the SNR is relatively high. Here again and for the same reasons as those pointed out in section 4.1, it is observed that OMR and NSMIR lead to the same reconstructed excitation field. Actually, the main differences are related to the computation time associated to each regularization strategy. Indeed, Table 3 shows that the computation time for OMR and SMIR are comparable, while that obtained for NSMIR is about 30 times larger. Here again, the situation is rather different in case of low SNR, since SMIR exhibits higher performances than OMR/NSMIR, regarding the re- 4]. Consequently, the application to a higher dimensional case confirms the observations made for the 1D case.

Experimental validation

The numerical validation has allowed drawing several interesting conclusions regarding the behavior of NSMIR and SMIR. First, NSMIR provides reconstructions similar to OMR. Second, SMIR significantly improves the quality of identified solutions in case of low SNR. To confirm all these conclusions, the analysis is extended to a real-world application, already used in Ref. [START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF] for validating OMR. Measurements of the vibration field were carried out with a scanning laser vibrometer on a grid of 35 × 29 points along x and y directions respectively using the force signal as phase reference. Regarding the reconstruction model, the structure under test being linear, it can be expressed as in Eq. ( 15). In the present application, the transfer function matrix H has been computed from a FE mesh designed to perfectly match the measurement mesh. As a result, it is composed of 952 shell elements, making the model theoretically valid up to 4500 Hz. Moreover, the calculation has been performed, considering that the bending motions as the only available data. In other words, the transfer functions matrix has been dynamically condensed over the measurable dofs.

Finally, it is worth noting that a global structural damping is used in the present experimental validation. Its value has been estimated from the modal damping ratios obtained from the measured FRFs.

Application

Before applying the different regularization strategies, the norm parameter q must be determined beforehand. The analysis of the experimental set-up suggests that the target excitation field is very sparse. Consequently, it is reasonable to set q = 0.5.

In the present experimental validation, the reconstruction is performed at 300 Hz, i.e. outside resonance frequencies. As shown in Fig. 10, the reconstructed excitation fields from OMR, NSMIR and SMIR are consistent with the target one, since the point force location is estimated at ( x 0 , y 0 ) = (0.404 m, 0.256 m). However, as observed in the numerical validation, SMIR allows obtaining a slightly better estimation of the point force amplitude for a computation time comparable to OMR, while NSMIR leads, here again, to a reconstruction similar to OMR, but for a larger computation time. In the present example, the improvement brought by SMIR is not spectacular, since the improvement is about 6% compared to OMR/NSMIR. However, this result is in line with our experience of laser vibrometry that allows obtaining rather clean measurements.

In the present experiment, SMIR provides only a slight improvement of

Conclusion

In the present contribution, an original iterated multiplicative regularization has been presented for identifying mechanical sources acting on a structure. The underlying idea behind this regularization strategy was to combine the advantages of the iterated Tikhonov regularization, which can be thought as an iterative refinement of an initial solution obtained using the standard Tikhonov regularization, and the multiplicative regularization that eliminates the need for the selection of the regularization parameter while taking into account one's prior knowledge on the sources to identify.

The resulting regularization strategy is aimed to be consequently flexible and computationally efficient. Practically, the iterated multiplicative regularization can be implemented in two different ways to give rise either to the non-stationary iterated multiplicative regularization (NSIMR) or to the stationary iterated multiplicative regularization (SIMR). Through numerical and experimental validations, it has been shown that SIMR allows improving the quality of the reconstructed solutions, while NSMIR gives regularized solutions similar to that obtained from the ordinary multiplicative regularization. This surprising result has been documented and explained in the paper.

However, the main conclusion of this paper is that SMIR can be used only when the measured data are very noisy (typically for SNR between 20 dB and 10 dB. In all other cases, the application of the ordinary multiplicative regularization is generally sufficient to obtain consistent reconstructions. A.6, the quality of the solutions identified from OMR significantly deteriorates as the SNR decreases. This is surprisingly not the case when the reconstructions are performed from SMIR, for which the relative error increases much more slowly than for OMR. In the light of these results, it is reasonable to say that OMR can be used when the noise level is greater than 20 dB. Below 20 dB, it seems necessary to employ SMIR to expect obtaining a better solution accuracy.

Figure 1 :

 1 Figure 1: Reconstruction of the excitation field at 350 Hz from a measured vibration field with high SNR (30 dB) -(-) Reference, (--) Standard Tikhonov regularization and (-• -) Stationary iterated Tikhonov regularization.

Figure 2 :

 2 Figure 2: Reconstruction of the excitation field at 350 Hz from a measured vibration field -(-) Reference, (--) Multiplicative regularization for q = 0.5 and SNR = 30 dB and (-• -) Multiplicative regularization for q = 0.5 and SNR = 13 dB.

4 and Ta- ble 2 ]Figure 3 :

 23 Figure 3: Spatial reconstruction of the excitation field at 350 Hz from a vibration field corrupted by an additive Gaussian white noise with a SNR of 30 dB -(-) Reference,

Figure 4 :

 4 Figure 4: Spatial reconstruction of the excitation field at 350 Hz from a vibration field corrupted by an additive Gaussian white noise with a SNR of 13 dB -(-) Reference,

Figure 5 :Figure 6 :

 56 Figure 5: Reconstructed excitation field at 350 Hz for SNR = 30 dB -(a) Reference, (b) OMR, (c) NSMIR and (d) SMIR

Figure 7 :Figure 8 : 5 Table 4 :5. 1 .Figure 9 :

 785419 Figure 7: Reconstructed excitation field at 350 Hz for SNR = 13 dB -(a) Reference, (b) OMR, (c) NSMIR and (d) SMIR

Figure 10 :

 10 Figure 10: Amplitude of the reconstructed excitation field at 300 Hz -(a) Excitation field obtained from OMR, (b) Excitation field obtained from NSMIR, (c) Excitation field obtained from SMIR and (d) Section view at y 0 = 0.256 m -(-) OMR, (--) NSMIR, (-• -) SMIR and (•) Location and amplitude of the measured force amplitude

  Appendix A. Influence of the noise level on the solutions identified from OMR and SMIR This appendix aims at giving to the reader a better insight into the performances of the proposed iterated regularization strategy with respect to the noise level of the vibration data. To this end, reconstructions are performed on the application test case presented in section 4.1 from OMR and SMIR only for six different SNR (30 dB, 25 dB, 20 dB, 15 dB and 13 dB). As shown in Fig. A.11 and Table

Figure A. 11 :

 11 Figure A.11: Spatial reconstruction of the excitation field at 350 Hz from a vibration field corrupted by an additive Gaussian white noise from (a) OMR and (b) SMIR -(-) Reference, (--) SNR = 30 dB, (-• -) SNR = 25 dB, (• • • ) SNR = 20 dB, (• • • •) SNR = 15 dB and (• • • •) SNR = 13 dB

Table 1 :

 1 Relative error and computation times for OMR, NSMIR and SMIR obtained

	from a reconstruction performed at 350 Hz from a vibration field corrupted by an additive
	Gaussian white noise with a SNR of 30 dB for a 1D structure
	Strategy Relative error (%) Computation time (ms)
	OMR	0.013	8
	NSMIR	0.013	218.5
	SMIR	0.010	11.2

Table 2 :

 2 Relative error and computation times for OMR, NSMIR and SMIR obtained from a reconstruction performed at 350 Hz from a vibration field corrupted by an additive

	Gaussian white noise with a SNR of 13 dB for a 1D structure
	Strategy Relative error (%) Computation time (ms)
	OMR	7.28	42.6
	NSMIR	7.28	167.7
	SMIR	0.47	44.5

Table 3 :

 3 Relative error and computation times for OMR, NSMIR and SMIR obtained from a reconstruction performed at 350 Hz from a vibration field corrupted by an additive Gaussian white noise with a SNR of 30 dB for a 2D structure

	Strategy Relative error (%) Computation time (ms)
	OMR	0.028	45.6
	NSMIR	0.028	1629.3
	SMIR	0.016	50.9

Table 5 :

 5 Computation times for OMR, NSMIR and SMIR obtained from a reconstruction

	performed at 300 Hz
	Strategy Computation time (s)
	OMR	8.89
	NSMIR	58.74
	SMIR	14.16
	the quality of the identified point force spectrum, since the results obtained
	from OMR are already in good agreement with the measured point force
	spectrum as demonstrated in Fig. 11 of Ref. [34].

Table A .

 A 6: Relative error (%) for OMR and SMIR for various noise levels

	Strategy 30 dB 25 dB 20 dB 15 dB 13 dB
	OMR	0.013	0.058	0.23	1.73	7.28
	SMIR	0.010	0.035	0.08	0.23	0.47