Mounir Sahli

Khaled Chetehouna
email: khaled.chetehouna@insa-cvl.fr

François Faubert

Chaimae Bariki

Nicolas Gascoin

Nadir Bellel

The characteristics of Magnesium-Lanthanum alloy obtained by an electrochemical process

Keywords: Mg-La alloy, Electrodeposition, aqueous solution, analysis techniques, coating

In this work, Magnesium-Lanthanum alloy was synthesized by an electrodeposition technique using an aqueous solution, based on Magnesium chloride hexahydrate and Lanthanum(III) Nitrate at a voltage of 4 Volts. A copper cathode plate is used for the deposition of the Mg-La alloy. The as-prepared powder was characterized by scanning electron microscope (SEM) to describe the morphology, energy dispersive spectroscopy (EDS) to determine the chemical composition, X-ray diffraction (XRD) and Fourier transform infrared (FTIR) spectra in order to define the chemical structure. Morphological description reveals the formation of heterogeneous chemical structures of very small size on the surface of the sample. EDS analysis indicates the presence of Mg, La and O as major elements and Cl as minor element in terms of weight percentages. X-ray results showed the existence of two distinct phases, Magnesium Hydroxide (Mg(OH) 2 ) and Lanthanum hydroxide (La(OH) 3 ), from the obtained deposit. FTIR analysis confirms the presence of the two phases identified in XRD diffractogram and it can be exhibited by clear peaks.

Introduction

Magnesium is the eighth most abundant element on the earth and constitutes about 2% of Earth's crust by weight. It is the third most copious element in seawater. It is characterized by various properties for instance a low density, a low cost, a very good malleability and ductility, a high strength/weight ratio and an environmentally friendly nature [START_REF] Zhou | Microstructural characteristics and mechanical properties of non-combustive Mg-9Al-Zn-Ca magnesium alloy friction stir welded joints[END_REF][START_REF] Yuan | Microstructural characteristics of joint region during diffusion-brazing of magnesium alloy and stainless steel using pure copper interlayer[END_REF][START_REF] Tharumarajah | Is there an environmental advantage of using magnesium components for lightweighting cars?[END_REF][START_REF] Hort | Magnesium alloys as implant materials -Principles of property design for Mg-RE alloys[END_REF][START_REF] Zhao | Reversibility of electrochemical magnesium deposition from tetrahydrofuran solutions containing pyrrolidinyl magnesium halide[END_REF]. The high electrochemical potential of magnesium (-2.4 Volts) and a substantial reactivity toward acids and their salts are their two chemical properties [START_REF] Rosalbino | Electrochemical behaviour assessment of novel Mg-rich Mg-Al-RE alloys (RE = Ce, Er)[END_REF][START_REF] Kim | Development of hight-driving potential and high-efficiency Mg-based sacrificial anodes for cathodic protection[END_REF]. As the lightest structural material, magnesium alloys have been used increasingly for structural compounds for several decades, particularly in the automotive, electronics and aerospace industries and expected to become one of the most promising lightweight materials in the 21st century [START_REF] Mehta | Investigation of wear properties of magnesium and aluminium alloys for automotive applications[END_REF][START_REF] Wang | Electrochemical behaviour of AZ61 magnesium alloy in dilute NaCL solutions[END_REF][START_REF] Alderliesten | The application of magnesium based Fibre Metal Laminates in aerospace structures[END_REF]. Recently, many investigations have been made in hydrogen storage materials and magnesium based alloys are considered to be the most promising [START_REF] Alderliesten | The application of magnesium based Fibre Metal Laminates in aerospace structures[END_REF][START_REF] Jurczyk | Hydrogen storage by Mg-based nanocomposites[END_REF][START_REF] Fan | Modifying microstructures and hydrogen storage properties of 85 mass% Mg-10 mass% Ni-5 mass% La alloy by ultra-high pressure[END_REF]. As benefits of the application of magnesium alloys, Cortés and Cantwell [START_REF] Gao | Phase structures and electrochemical properties of La 0.8-x Gd 0.2 Mg x Ni 3.1 Co 0.3 Al 0.1 hydrogen storage alloys[END_REF] mention improved electromagnetic protective capability and superior corrosion resistance. The addition of rareearth (RE) elements such as La, Y, Ce and Gd to Mg has received significant attention [START_REF] Cortés | The fracture properties of a fibre-metal laminate based on magnesium alloy[END_REF][START_REF] Stanford | The effect of rare earth elements on the behavior of magnesium-based alloys: Part 1 -Hot deformation behaviour[END_REF][START_REF] Gavras | Microstructure and property evaluation of high-pressure die-cast Mg-La-rare earth (Nd, Y or Gd) alloys[END_REF]. It has been also reported that Mg-RE alloys exhibit excellent mechanical properties. They are typified by higher specific strength and better creep resistance [START_REF] Kang | Thermodynamic evaluations and optimizations of binary Mg-light Rare Earth (La, Ce, Pr, Nd, Sm) systems[END_REF][START_REF] Mirza | A modified Johnson-Cook constitutive relationship for a rare-earth containing magnesium alloy[END_REF][START_REF] Bayani | Effect of rare earth elements addition on thermal fatigue behaviors of AZ91 magnesium alloy[END_REF]. To fabricate the Mg alloys, three methods have been explored: physical as PVD (Physical Vapour Deposition) [START_REF] Zhang | Microstructure and mechanical properties of as-cast and heat treated Mg-15Gd-3Y alloy[END_REF], mechanical as milling techniques [START_REF] Garcés | Texture of magnesium alloy films growth by physical vapour deposition (PVD)[END_REF] and chemical as CVD (Chemical Vapour Deposition) [START_REF] Yan | The influence of ball milling process on formation and electrochemical properties of amorphous MgNi hydrogen storage alloys[END_REF]. Besides these methods, metal alloys electrodeposition in aqueous ionic liquids are receiving increased interest [START_REF] Ch | Deposition of aluminium on magnesium by a CVD process[END_REF][START_REF] Chen | Electrodeposition of Li and electrochemical formation of Mg-Li alloys from the eutectic LiCl-KCl[END_REF][START_REF] Liu | A Electro-deposition process for fabrication of biomimetic super-hydrophobic surface and its corrosion resistance on magnesium[END_REF][START_REF] Liebenow | The electrodeposition of magnesium using solutions of organomagnesium halides, amidomagnesium halides and magnesium organoborates[END_REF][START_REF] Wang | Electrodeposition mechanism and characterization of Ni-Cu alloy coatings from a eutectic-based ionic liquid[END_REF]. It is the most economical method of prepared Mg-RE alloys and can be easily implemented on a large scale. At the same time, this method had other advantages: the phases of alloys could be controlled by electrochemical parameters and could be employed over a large temperature range. The aim of the work presented in this paper is to study the characteristics of the Magnesium-Lanthanum alloy elaborated by means electrodeposition technique. Section 2 describes the experimental methodology. It includes chemicals and samples preparation and characterization of Mg-La alloys. Section 3 examines the results obtained monitoring with discussions based to the ones found in literature. In the end, conclusions are drawn with some indicators to future analysis. . 6H 2 O were dissolved in 34 mL of ultra-pure water at room temperature. The obtained aqueous solutions with pH=6 were prepared for electrodeposition process. The electrochemical process consists in a cylindrical electrolysis chamber of an internal volume of 36.6 mL. The plexiglass vessel has as inner dimensions 1.8 cm radius and 3.6 cm height. A copper cathode plate and a tungsten thread anode, with a 2.2 cm gap, connected to a conventional potentiometric power source, were used to ensure the deposition of Mg-La alloy. The potential value selected to study its effect on the characteristics of the Mg-La alloys was 4 Volts. The deposition experiments were performed during 8h at ambient temperature (23 °C). The obtained deposit was scraped from the copper cathode plate, dried for 1h at 110 °C, milled manually in a marble mortar and subjected to further analyzes. Fig. 1 illustrates a photograph and a schematic overview of the defined experimental setup.

Experimental and procedures

Characterization of Mg-La alloy

The morphology and chemical composition of the obtained alloy were investigated by a scanning electron microscope (SEM, LEO 1455 VP, acceleration voltage of 20 kV) at a working distance of 15 mm equipped with an X-ray energy dispersive spectrometer (EDS, Inca X-sight, Oxford Instrument). The phase identification of the alloy were determined by means of X-ray diffractometer (XRD, Philips X' PERT) using Cu Kα radiation source (λ=1.54 Å, 40 kV, 30 mA) at room temperature. The scanning rate of 0.02°/s within the 2θ range going from 10° to 90° was used in order to increase counting statistics and to improve the signal/noise ratio. Thermogravimetric and differential scanning calorimetry analysis of the sample of the scraped deposit was performed by SETARAM apparatus under flowing Argon atmosphere at the heating rate of 10 °C/min in the temperature range of 25 °C to 600 °C. Fourier transform infrared (FTIR) spectroscopy of the alloys samples was carried out with a Nicolet 380 FT-IR spectrometer using the ATR technique. All spectra were collected with the resolution of 4 cm -1 in the range of 4000-500 cm -1 .

Results and discussion

As mentioned in the previous section, the characteristics of Mg-La alloy will be studied to 4 Volts. This section is structured as follows: a presentation of EDS and X-ray diffraction analyzes as well as the IR absorption spectroscopy studies.

EDS analysis of the as-deposited powders

Fig. 2 shows the surface morphology (SEM) of as-prepared Mg-La alloy. The image exhibits the formation of heterogeneous chemical structures and the different phases seem quite imbricated. Interestingly enough, the size of the aggregates found was in the order of nanometers to micrometers.

Fig. 2. SEM image of the as-deposited powder

The compositions of Mg-La alloy was examined by EDS, as shown in Table 1. The main elements detected on the surface of the coatings were Oxygen, Magnesium, Lanthanum and a little amount of Chlorine. The existence of O, Mg, La and Cl in the as-deposited powder can be justified by the electrolyte used in electrodeposition experiment. As we can see from this Table ,Mg Further to the above analyzes, the FTIR spectroscopy study is evident for the structural confirmation of the as-prepared powder. The FTIR transition spectrum of the as-deposited Mg-La alloy in the range of 4000-500 cm -1 for 4 Volts is revealed in Fig. 4. A sharp and strong peak at 3696 cm -1 , which has as transmittance value 75.67, can be ascribed to vibrations and stretching of OH bonds in Mg(OH) 2 [START_REF] Bocchetta | Template electrosynthesis of La(OH) 3 and Nd(OH) 3 nanowires using porous anodic alumina membranes[END_REF][START_REF] Xu | A TG-FTIR investigation into smoke suppression mechanism of magnesium hydroxide in asphalt combustion process[END_REF].

The band centred at 3606 cm -1 , corresponding at 79.37, is associated for stretching mode of OH -in Lanthanum (III) hydroxide [START_REF] Frost | Infrared emission spectroscopic study of brucite[END_REF][START_REF] Salavati-Niasari | Synthesis of lanthanum hydroxide and lanthanum oxide nanoparticles by sonochemical method[END_REF]. The bands at 3455 cm -1 and 1640 cm -1 , relating respectively to 79.24 and 81.70, are attributed to the hydroxyl groups in water as mentioned by Aghazadeh and Khosrow-pour [START_REF] Aghazadeh | A novel lanthanum hydroxide nanostructure prepared by cathodic electrodeposition[END_REF][START_REF] Khosrow-Pour | Large scale and uniform La(OH)3 nanorods prepared by template-free pulsed electrodeposition method[END_REF]. The two other peculiar bands at about 1434 cm -1 and 1048 cm -1 , characterized by the following values; 78.69 and 82.30, can be associated to asymmetric stretching vibrations of the carbonate group, which originate from the reaction of the as-prepared powders with CO 2 from air during the FTIR spectroscopy analysis [START_REF] Frost | Infrared emission spectroscopic study of brucite[END_REF][START_REF] Aghazadeh | A novel lanthanum hydroxide nanostructure prepared by cathodic electrodeposition[END_REF][START_REF] Wang | Characterizations of the thermal decomposition of brucite prepared by sol-gel technique for synthesis of nanocrystalline MgO[END_REF].

Conclusion

(1) The synthesis of Magnesium-Lanthanum powder due to an electrodeposition process at room temperature has been successfully treated.

(2) The different results on the behavior of the as-deposited powder are in agreement with those of literature Magnesium-based coatings.

(3) As future analyzes, a wide Mg-based would be studied with the different voltages and make a comparison in order to figure out the potential material for the hydrogen storage.

2. 1 . 3 ..6H 2 O

 132 Experimental Setup Magnesium chloride hexahydrate (MgCl 2 6H 2 O) and Lanthanum nitrate (La(NO 3 ) 6H 2 O) from Merck Group were employed as received. The ultra-pure water obtained by means of a MEDICA S/R/D (7/15) purification system was used to prepare the different solutions. A mass of 16.2 g of MgCl 2 and a mass of 4.3 g of La(NO 3 ) 3

1 .

 1 Illustrations of the electrodeposition experimental setup. (a) Photograph and (b) schematic overview.

  -La alloy in 4 Volts is characterized by high contents of Oxygen (40.7 %) and Lanthanum (31.5 %). The amounts of Magnesium and Chlorine are respectively 21.1 and 6.7 %. Table 1. Alloy element compositions in weight and atomic percentage ray diffraction analysis of the as-prepared powders The different phases of the as-deposited powder are discerned clearly in Fig. 3 through XRD diffractogram. According to the standard data JCPDS, two significant characteristic peaks are detected, which indicate the existence of Magnesium hydroxide (Mg(OH) 2 ) as a hexagonal structure with lattice constants of a = 3.148 Å and c = 4.787 Å [28] and Lanthanum(III) hydroxide (La(OH) 3 ) as a hexagonal structure with lattice constants a = 6.528 Å and c = 3.858 Å [29, 30]. As can be seen from this Figure, peaks at diffraction angle 2θ of 18.6°, 35.8°, 37.9°, 50.8°, 58.7° and 62.1° are related to Mg(OH) 2 whereas those at 2θ of 15.6°, 27.9°, 39.3°, 46.8°, 48.5° and 54.9° are associated to La(OH) 3 . The peak of Mg(OH) 2 can achieve a maximum of intensity value 917 and 207 as minimum. The two others distinct intensity values of the peak La(OH) 3 are respectively 895 and 185.

Fig. 3 .

 3 Fig. 3. X-ray diffraction pattern of the sample

Fig. 4 .

 4 Fig. 4. FTIR transition spectrum of the as-prepared powder


A Fast and Sparsity-Aware Generalization of SMART for Tomographic Particle Image Velocimetry Ioana Barbu and Cédric Herzet INRIA Rennes -Bretagne Atlantique Campus de Beaulieu, 35000 Rennes Email: cedric.herzet@inria.fr

Understanding the tridimensional (3D) motion of turbulent flows is one of the most challenging problems in fluid dynamics. Because the Navier-Stokes equations, describing the spatio-temporal evolution of a fluid, are intractable both at a theoretical and numerical level when dealing with turbulent flows, the idea has soon been proposed to resort to experimental measures. In this line of search, the most common approach is the so-called "Particle Image Velocimetry" (PIV) [START_REF] Adrian | Particle Image Velocimery[END_REF] which consists in seeding the fluid with passive particles 1 and accessing to motion measures by processing images of the spatiotemporal evolution of the latter. Recently, researchers have moved their attention to the tridimensional setup, where the 3D motion of a fluid must be inferred from the observation of a set of images captured at each time instant. The most-advanced experimental scheme addressing this problem is the so-called "Tomographic PIV" system, introduced by Elsinga et al. in [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF], [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF]. The work presented in this paper takes place in this particular applicative context.

We focus hereafter on an intermediate but important step arising in Tomographic PIV: the estimation of the 3D position of the particles from the set of collected images, i.e., the so-called "Volume reconstruction" problem [START_REF] Barbu | Tridimensional Estimation of Turbulent Fluid Velocity[END_REF]. Formally, the task consists in inverting a model of the form

y = Dx + noise, (1)
where y ∈ R m is a vector collecting the set of image pixels, x ∈ R n is a vector such that each element xj is associated to a possible (discretized) particle position in the volume under study: xj = 0 if and only if there is no particle at the corresponding location; finally, D ∈ R m×n is a "connection matrix" which relates the collected images to some particle configuration x. We refer the reader to [4, Chapter 2], for a detailed description of the model construction.

Some important facts about model (1): i) m n and the inversion of ( 1) is seriously ill-posed; ii) the nonzero elements of x corresponds to the energy scattered by the particles and are therefore nonnegative, x ≥ 0; iii) the number of possible positions n is usually much greater than the number of particles in the fluid, that is x 0 n; iv) m and n are typically large (e.g., m ∼ 10 6 , n ∼ 10 9 ).

The last item has motivated the TomoPIV community to focus on families of procedures exhibiting low computational and storage requirements. The most popular methodology in the current literature addressing the volume reconstruction problem is undeniably the so-called "Simultaneous Multiplicative Algebraic Reconstruction Technique" (SMART) [START_REF] Byrne | Iterative Image Reconstruction Algorithms Based on Cross-Entropy Minimization[END_REF] whose iterates can be expressed as

x (k+1) j = x (k) j m i=1 yi d T i x (k) γd ij ∀ j, (2)
where γ > 0 and di ∈ R n is the ith row of D. In the sequel, we will use the simple notation x (k+1) = SMART(x (k) ) to denote the 1 The particles therefore follow the fluid motion.

recursion defined in [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF]. SMART suffers from two main drawbacks: i) although it implicitly imposes the nonnegativity of the solution, it does not exploit the sparse nature of x; ii) it can have a very low speed of convergence, especially for highly-seeded fluids.

In this work, we show that SMART can be simply modified to circumvent these conundrums. We first notice that SMART can be interpreted as a nonlinear projected gradient (NPG) algorithm [START_REF] Beck | Mirror descent and nonlinear projected subgradient methods for convex optimization[END_REF], [START_REF] Petra | B-SMART: Bregman-Based First-Order Algorithms for Non-negative Compressed Sensing Problems[END_REF] applied to problem:

min x∈X KL(y, Dx), (3)
with X = R n + and where KL(•, •) denotes the Kullback-Leibler distance between two nonnegative vectors.

With this interpretation in mind, the sparsity of the sought solution can then simply be imposed by adding the additional constraint " x 1 ≤ τ " to problem (3). This is equivalent to contraining the optimization set to be of the form X = {x ∈ R n + |1 T x ≤ τ }. With this particular choice for X , it can be shown that the NPG iterates still obey a close-form recursion:

x (k+1) = x(k+1) if 1 T x(k+1) ≤ τ τ x(k+1) 1 T x(k+1) otherwise, (4)
where x(k+1) = SMART(x (k) ). In words, accounting for the sparsity of x in the reconstruction process only requires to add a normalization operation to the standard SMART recursion; the normalization occurs as soon as the inequality 1 T x(k+1) ≤ τ is not satisfied. We will refer to this sparsity-aware version of SMART as "SMART 1". The identification of SMART as an instance of NPG algorithm also allows us to propose an accelerated version of this type of procedure. In order to do so, we adopt the strategy proposed in [8, Section 5] which, with a simple modification of the standard NPG updates (inducing no inflation of the complexity order), achieves a rate of decrease of the cost function scaling as O(k -2 ) (versus O(k -1 ) for the standard NPG procedure). We will refer to this accelerated version of SMART (resp. SMART 1) as "A-SMART" (resp. "A-SMART 1").

Figures 1 and2 illustrate the performance achieved by the proposed variants of SMART. The abscissa represents the sparsity level of the sought vector x, expressed in terms of the "particles per pixel (ppp) ratio" (see [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF]), and the ordinate is the normalized correlation

Q x T x (k)
x x (k) , a figure of merit commonly used in the Tomographic PIV community. The numerical assessment emphasizes that: i) the accelerated version of SMART yields better results in terms of quality of reconstruction, despite solving the same convex problem -this is due to the higher rate of convergence of the latter; ii) constraining the sparsity within a noisy setting leads to an improvement of the quality of the estimated vector. , respectively, and then iteratively adapted following the Armijo rule. We set τ = x 0 (where x is the ground truth whose non-zero coefficients equal 1) in the "sparsity-aware" versions of (A-)SMART. Regardless of the ppp value, which translates here into sparsity levels growing from x 0 = 100 to x 0 = 300, SMART algorithms are outperformed by their accelerated versions. We also notice that, while A-SMART is outperformed by its '"sparsity-aware" counterpart, SMART and SMART 1 superimpose.

. Numerical assessment of the reconstruction quality of x (100) in a perturbed setting, where y i is affected by a Gaussian noise of zero mean and standard deviation equal to 0.1y i . The other parameters are identical to those used in Fig. 1. The performance of the algorithms follows the pattern observed within a non-perturbed setting (see Fig. 1), with the difference that SMART is outperformed by its '"sparsity-aware" counterpart.

  Figure 1. Numerical assessment of the reconstruction quality of x (100) in an ideal, noise-free scenario for a medium-scale setting (i.e., m = 6724, n = 99944). The results are averaged on 10 experiments. The relaxation parameter γ is initially chosen as 0.1, 0.05, 0.4 and 0.8 for SMART, SMART 1 , A-SMART and A-SMART 1

	Q
	0.98
	0.96
	0.94
	0.92
	0.9
	6 • 10 -2 8 • 10 -2 0.1	0.12 ppp	0.14	0.16
	SMART	A-SMART	SMART 1	A-SMART 1


Ioana Barbu
email: ioana.barbu@irstea.fr

Cédric Herzet
email: cedric.herzet@inria.fr

Accelerated, Sparsity-Aware Generalizations of Classical Algorithms for TomoPIV

We set up a generalized optimization framework for the classical Row Action Methods, otherwise customarily employed in the TomoPIV community. This scheme allows us to bypass known caveats of the algebraic programs. The so-enhance methods enable i) handling explicit constraints on the signal; ii) accelerating the rates of convergence. Comparative numerical simulations reveal superior performance of the latter with no inflation of the complexity.

Introduction

Understanding the tridimensional (3D) motion of turbulent flows is one of the most challenging problems in experimental fluid dynamics. Since its inception, the "Particle Image Velocimetry" (PIV) [START_REF] Meynart | Digital image processing for speckle flow velocimetry[END_REF] measurement technique has been the preferred modus operandi to tackle the turbulence from the experimental viewpoint. Recently, enhancements have been brought to the classical 2D scenario to enable 3D measurements. Within the so-called "Tomographic PIV (TomoPIV)" system, introduced by Elsinga et al. in [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF], the 3D motion of a fluid is inferred from the observation of a multiview set of images captured at each time frame. A crucial step in this development is the volumetric reconstruction of the particles' distribution at each time frame. Pragmatical motivations have led TomoPIV scientists to take an interest in algorithms belonging to the class of "Row Action Methods" [START_REF] Censor | Parallel Optimization: Theory, Algorithms and Applications[END_REF], which matched their expectations in terms of: i) data storage requirements; ii) computational time. This, on top of their implementation-friendly architecture and their ability to output satisfying solutions within a few iterations, has let the advances in the TomoPIV Volume Reconstruction Problem slowly come to a stall; moreover, other orthogonal solutions [START_REF] Barbu | Sparse models and pursuit algorithms for PIV tomography[END_REF][START_REF] Petra | On Sparsity Maximization in Tomographic Particle Image Reconstruction[END_REF] -believed to be computationally more expensive -have missed their transition from numerical to experimental assessment. However, Row Action Methods suffer from a certain number of caveats: i) their rates of convergence are usually linear [START_REF] Elfving | On Some Methods for Entropy Maximization and Matrix Scaling[END_REF]; ii) they do not explicitly exploit the physical knowledge on the sparse and noise-biased nature of the TomoPIV signal.

In summary, there is a need for i) algorithms satisfying the storage and computational time requirements of the physical system; ii) procedures able to account for some physical a priori on the signal and robust to noise affecting the observations. We address this problem in the remainder of the paper. In particular, we show that procedures classically appointed for the TomoPIV volume reconstruction problem can be recast within a general optimization framework, without any major change in their algorithmic architecture, nor the complexity thereof, inducing, among other perks: i) higher convergence rates; ii) the possibility to apply powerful convex-optimization tools to solve the problem.

The paper is organized as follows. Section 2 formalizes the observation model used in our subsequent derivations. In Sections 3 and 4, we lay-out an intuitive setting to show the progress of classical algebraic procedures towards more evolved optimization schemes. Section 5 brings numerical validation that the more elaborate schemes encompassing SMARTs and SIRTs should be the preferred method for tackling the TomoPIV Volume Reconstruction problem, leading us naturally to an anticipated conclusion in Section 6.

Model and Related Works

In a nutshell, we aim at modelling the interaction between the illuminated passive tracers flowing in the measurement region and the planes of the cameras placed around the volume of interest.

The key element conditioning the design of a realistic model comes from the physical properties of the seeders. The latter dramatically influences the projection model. In fact, the projection of a small particle on an image impacts a small vicinity of pixels; the latter form "airy spots" on an image instead of dot-like projections. A good approximation of this phenomenon was given by Champagnat et al. in [START_REF] Champagnat | Tomographic PIV: particles vs blobs[END_REF]. According to the said model, the image intensity y at a point u ∈ R 2 writes:

y(u) = ∑ p∈{1,...,n p } A(u -P(k p ))x(k p ), (1)
where n p counts the number of particles captured in the region of interest, k p ∈ R 3 registers for the location of the p th tracer in the volume and functions x : R 3 → R + , P : R 3 → R 2 and A : R 2 → R + stand for the volume intensity, projection and the Point Spread Function (PSF) -which defines the formation of the airy spots -, respectively. We can easily arrive at a discrete analogue of (1), namely:

y = Ax + noise, (2)
where y ∈ R m is a vector collecting the set of image pixels at points u belonging to the bidimensional image grids; x ∈ R n is a vector such that each element x j is associated to a discretized particle position (i.e., a voxel) in the visualized volume: x j = 0 if and only if there is no particle at the corresponding location; finally, A ∈ R m×n is the "projection matrix" which relates the collected images to some particle configuration x. We refer the reader to [4, Chapter 2], for a detailed description of our model construction, which relies on an approximation of the PSF. The system (2) has a series of notable properties -i) m n and the inversion of ( 2) is seriously ill-posed; ii) the nonzero elements of x correspond to the energy scattered by the particles and are therefore nonnegative, i.e., x ≥ 0; iii) the number of seeded particles is far less numerous than the number of voxels, i.e.,1 x 0 n; iv) m and n are typically large (e.g., m ∼ 10 6 , n ∼ 10 9 )some of which may condition the choice of the method to invert the system.

Classical Algorithms

It is precisely the high dimensionality of the system (2) that motivated the TomoPIV community to resort to Row-Action Methods, interesting for their low computational and storage requirements. The latter encompass the well-known "Simultaneous Multiplicative Algebraic Reconstruction Technique (SMART)" [START_REF] Byrne | Iterative Image Reconstruction Algorithms Based on Cross-Entropy Minimization[END_REF] and the "Simultaneous Iterative Reconstruction Technique (SIRT)" [START_REF] Andersen | Simultaneous Algebraic Reconstruction Technique (SART): a superior implementation of the ART algorithm[END_REF][START_REF] Cimmino | Calcolo approsimato per le soluzioni dei sistemi di equazioni lineari[END_REF].

The SIRTs include, inter alia, known procedures such as "Cimmino" [START_REF] Cimmino | Calcolo approsimato per le soluzioni dei sistemi di equazioni lineari[END_REF] or "SART" [START_REF] Andersen | Simultaneous Algebraic Reconstruction Technique (SART): a superior implementation of the ART algorithm[END_REF]. In its comprehensive form, a solution of ( 2) is sought as a fixed point of the following recursion

x (k+1) = x (k) + α (k) WA T Γ(y -Ax (k) ), (3)
where α (k) > 0 and W, Γ are some diagonal positive-definite matrices. In the sequel, we will use the simple notation x (k+1) = SIRT(x (k) ) to denote the recursion defined by [START_REF] Auslender | Interior gradient and proximal methods for convex and conic optimization[END_REF]. Within the TomoPIV context, Cimmino and SART have been advocated in [START_REF] Petra | Extended and Constrained Cimmino-type Algorithms with Applications in Tomographic Image Reconstruciton[END_REF] and [START_REF] Petra | Enhancing Sparsity by Constraining Strategies: Constrained SIRT versus Spectral Projected Gradient Methods[END_REF], respectively.

The most popular procedure for TomoPIV is the "Simultaneous Multiplicative Algebraic Reconstruction Technique" (SMART) [START_REF] Byrne | Iterative Image Reconstruction Algorithms Based on Cross-Entropy Minimization[END_REF]. The iterates can be expressed as

x (k+1) j = x (k) j m ∏ i=1 y i a T i x (k) γa i j ∀ j, (4)
where γ > 0 and a i ∈ R n is the ith row of A. Henceforth, we will use the simple notation x (k+1) = SMART(x (k) ) to denote the recursion defined in [START_REF] Barbu | Tridimensional Estimation of Turbulent Fluid Velocity[END_REF]. SMART was introduced into the TomoPIV setting in [START_REF] Atkinson | An efficient simultaneous reconstruction technique for tomographic particle image velocimetry[END_REF], before being broadly relayed by most contributors.

Although conceptually interesting, SIRT and SMART do not allow to account for other known trademarks of the x -more specifically, its sparsity -and they can have a very low speed of convergence, especially for highly-seeded fluids.

Towards an Optimization Framework

In this section, we expose our proposed methodology. We start by introducing some useful notations. Then, we show that by simply modifying classical procedures with regard to an optimization framework we can circumvent known shortcomings of the former.

Notations

We define the indicator function I X (x) of a set X as

I X (x) = 0 if x ∈ X , +∞ otherwise.
The proximal operator of a convex proper function f :

R n → R ∪ ∞ is defined as prox f (v) = arg min x∈X f(x) + 1 2 x -v 2 2 .

SIRT and SMART as Particular Cases of Optimization Programs

We first notice that SIRT can be interpreted as a proximal gradient method (PGM) [START_REF] Parikh | Proximal Algorithms[END_REF], which is an algorithmic variant of the well-know gradient projected method. In fact, the SIRT iteration depicted by Equation (3) corresponds to the PGM step when the latter procedure is applied to problem:

min x 1 2 y -Ax 2 2 + λr(x), (5)
provided that W = I, Γ = I and r(x) is constant function, while λ > 0 is some tuning parameter.

We can enforce the sparsity and the nonnegativity of the sought solution by an appropriate choice of r(x). For example, let r(x) = λ x 1 + I R n + (x), the PGM recursion reads as

x (k+1) = prox λ x 1 +I R n + (SIRT(x (k) )), (6)
where the proximal operator prox λ x 1 +I R n + (•) takes the simple form:

prox λ x 1 +I R n + (v) i = v i -λ if v i ≥ λ 0 otherwise. ( 7
)
Plugging Equation ( 7) into Equation ( 6) translates to the projection on the positive orthant of the soft thresholded current estimate. Simply put, enforcing the positivity and the sparsity of the solution comes down to computing a SIRT iteration, followed by a simple thresholding operation of the solution (which induces no inflation of the complexity cost).

In an equivalent manner, we recognize that SMART can be interpreted as a nonlinear projected gradient (NPG) algorithm [START_REF] Beck | Mirror descent and nonlinear projected subgradient methods for convex optimization[END_REF][START_REF] Petra | B-SMART: Bregman-Based First-Order Algorithms for Non-negative Compressed Sensing Problems[END_REF] which is also a variant of the gradient projected method -applied to problem:

min x∈X KL(y, Ax), (8)
with X = R n + and where KL(•, •) denotes the Kullback-Leibler distance between two nonnegative vectors.

The sparsity of the sought solution can then simply be imposed by adding the additional constraint " x 1 ≤ τ" to problem [START_REF] Byrne | Iterative Image Reconstruction Algorithms Based on Cross-Entropy Minimization[END_REF]. This is equivalent to contraining the optimization set to be of the form X = {x ∈ R n + |1 T x ≤ τ}. With this particular choice for X , it can be shown that the NPG iterates still obey a closed-form recursion:

x (k+1) = x(k+1) if 1 T x(k+1) ≤ τ τ x(k+1) 1 T x(k+1) otherwise, (9)
where x(k+1) = SMART(x (k) ). Altogether, the "sparsity-aware" SMART iteration is updated conditionally: following the ubiquitous Equation ( 4), if the current estimate is sparse enough and by the SMART solution normalized by the 1 -norm of the estimate, otherwise. Seemingly, no significant complexity cost is induced by this update.

The identification of SIRT and SMART as instances of PGM and NPG algorithms, respectively, allows us to propose an accelerated version of this type of procedures. In order to do so, we adopt the strategies proposed in [START_REF] Beck | A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems[END_REF] (respectively [3, Section 5]) which achieves a rate of decrease of the cost function scaling as O(k -2 ) (versus O(k -1 ) for the standard procedures) by applying the PGM update x (k+1) = SIRT(z (k+1) ) (respectively, the NPG update x(k+1) = SMART(z (k+1) )) to a proper interpolation point

z (k+1) = x (k) + ζ (k) (x (k) -x (k-1) ), (10)
with ζ (k) ∈ [0, 1]. We notice that the operation [START_REF] Champagnat | Tomographic PIV: particles vs blobs[END_REF] induces no inflation of the complexity order.

Numerical Validation

We carry out a comparative numerical assessment of the classical methods versus their enhanced counterparts. Nomenclature-wise, sparsity-aware variants of SIRT and SMART will respectively append " 1 " to their label (see equations ( 6) and ( 9), respectively). We will prefix the accelerated versions of the classical algorithms (and of their sparsity-enhanced companions) by "A-" in our figure legend.

We simulate a medium-scale perturbed setting (i.e., m = 6724, n = 99944), where y i is affected by a Gaussian noise of zero mean and standard deviation equal to 0.1y i . The results are averaged on 10 experiments and we allow for 100 iterations. The relaxation parameter γ (see Equation ( 4)) is initially chosen as 0.1, 0.05, 0.4 and 0.8 for SMART, A-SMART, SMART 1 and A-SMART 1 , respectively, while λ = 0.01 (see Equation ( 5)). The so-chosen initial values of λ is then iteratively adapted following the Armijo rule. We set τ = x 0 (where x is the ground truth whose non-zero coefficients equal 0) -see Equation [START_REF] Censor | Parallel Optimization: Theory, Algorithms and Applications[END_REF]. As suggested in [START_REF] Parikh | Proximal Algorithms[END_REF], we update ζ (k) = k k+3 ; for specifics of this action relate to Equation [START_REF] Champagnat | Tomographic PIV: particles vs blobs[END_REF]. particles per pixel (ppp) 100) in terms of the normalized correlation against the "particles per pixel (ppp)" ratio.

x T x (k) x x (k) SMART A-SMART SMART 1 A-SMART 1 SIRT A-SIRT SIRT 1 A-SIRT 1
Figure 1 illustrates the performance achieved by the proposed methods. The abscissa represents the sparsity level of the sought vector x, expressed in terms of the "particles per pixel (ppp) ratio" (see [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF]), and the ordinate is the normalized correlation x T x (k) x x (k) , a figure of merit commonly used in the Tomographic PIV community. We notice in particular that i) standard algorithms are outperformed by their accelerated versions, despite solving the same convex problem -this is due to the higher convergence rates of the latter; ii) generally, the unregularized algorithms are outperformed by their "sparsity-aware" counterparts -this is achieved by accounting for an additional information on the signal when solving the inverse TomoPIV problem.

Conclusion

This paper campaigns for the understanding of some classical Row Action Methods from the optimization viewpoint. The advantage of tackling the TomoPIV problem from this angle is twofold. First, the algorithmic structure barely suffers any change as the proposed scheme yields closed-form updates which do not affect the complexity cost, nor the parallelizable architecture of SMART. Second, the newly-found mathematical properties allow for a remarkable flexibility to obtain accelerated, sparsity-aware enhancements and merit further extensive study.

Figure 1 :

 1 Figure 1: Numerical assessment of the reconstruction quality of x (100) in terms of the normalized correlation against the "particles per pixel (ppp)" ratio.

Here the notation x 0 refers to the " 0 -norm" which returns the number of nonzero elements in its argument.


Ioana Barbu
email: ioana.barbu@irstea.fr

Cédric Herzet
email: cedric.herzet@inria.fr

A New Approach for Volume Reconstruction in TomoPIV with the Alternating Directions Method of Multipliers

We adapt and import into the TomoPIV scenery a fast algorithm for solving the volume reconstruction problem. Our approach is based on the reformulation of the volume reconstruction task as a constrained optimization problem and the resort to the "Alternating Directions Method of Multipliers" (ADMM). This approach is shown to be an interesting alternative to the "row-action" methods, classically used by the Tomo-PIV community. In particular, our problem formulation allows to: i) take explicitly into account the level of the noise affecting the data; ii) account for both the nonnegativity and the sparsity of the solution. Experiments on a numerical TomoPIV benchmark show that the proposed framework is a serious contender for the state-of-the-art.

Introduction

The Tomographic PIV (TomoPIV) has been considered, since its debut by Elsinga et al. [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF], the experimental solution to the understanding of turbulent fluid velocity. Its methodology deals with inferring the tridimensional (3D) motion from the observation of a multiview set of images captured at each time frame. A crucial step in this development is the volumetric reconstruction of the particles' distribution at each time frame and this problem has thus been studied in many works of literature, see [START_REF] Barbu | Tridimensional Estimation of Turbulent Fluid Velocity[END_REF]Chapter 3] for a review. The most popular of these contributions belong to the class of the so-called "row-action method" [START_REF] Censor | Row-Action Methods for Huge and Sparse Systems and Their Applications[END_REF]. These procedures accomplish the volume reconstruction task with low computational and storage requirements; unfortunately, they also often suffer from a low convergence rate to the desired solution. As a result, while a few iterations of these procedures may lead to pretty poor reconstruction performance, letting them running up to convergence may lead to an unacceptable computational burden.

New enhanced tools have thus to be employed to deal with this conundrum. We exhibit, in our companion paper [START_REF] Barbu | Accelerated, Sparsity-Aware Generalizations of Classical Algorithms for TomoPIV[END_REF], ways to accelerate convergence rates of standard methods with no inflation of the computational cost. In the current work, we focus our attention on an alternative approach to the volume reconstruction problem by resorting to the powerful "Alternating Direction Method of Multipliers" (ADMM) [START_REF] Eckstein | On the Douglas-Rachford Splitting Method and the Proximal Point Algorithm for Maximal Monotone Operators[END_REF], an optimization procedure for large-scale problems which has recently become popular in the machine-learning community. Our motivation is backed up by the numerous advantages of the method, namely i) accounting for both the nonnegativity and the sparsity of the solution ii) explicitly handling the noise levels affecting the observations iii) possessing -in certain variants of the proposed paradigm -the same complexity per iteration as the "row action" methods, but at faster convergence rates than the latter.

The paper is organized as follows. Section 2 briefly presents the model underlying the volume reconstruction problem. In Section 3, we formalize the reconstruction task as a convex optimization problem and apply the ADMM technique to this particular setup. Section 4 reports convincing numerical results and Section 5 concludes with a few remarks.

Problem Formulation

The estimation of the 3D positions of the particles from the set of the collected images consists in inverting a model of the form

y = Ax + noise, (1)
where y ∈ R m is a vector collecting the set of image pixels; x ∈ R n is a vector such that each element x j is associated to a discretized particle position (i.e., a voxel) in the visualized volume: x j = 0 if and only if there is no particle at the corresponding location; finally, A ∈ R m×n is the "projection matrix" which relates the collected images to some particle configuration x. We refer the reader to our companion paper [START_REF] Barbu | Accelerated, Sparsity-Aware Generalizations of Classical Algorithms for TomoPIV[END_REF] for a flavour of the reasoning justifying Equation ( 1) and to [2, Chapter 2] for a detailed description of the model construction.

When n > m and A is full rank, (1) has an infinity of solutions. In order to distinguish between them, we can exploit some information available on the TomoPIV system: i) the elements of x corresponds to intensity and should therefore be nonnegative; ii) the sought vector x is typically sparse, that is, it contains much more zero than nonzero elements. Constraining the solution to be nonnegative is often an easy task. Accouting for sparsity is usually more tricky and many works have focussed on this problem in the last decades [START_REF] Barbu | Sparse models and pursuit algorithms for PIV tomography[END_REF][START_REF] Cornic | Computationally efficient sparse algorithms for tomographic PIV Reconstruction[END_REF][START_REF] Petra | TomoPIV meets Compressed Sensing[END_REF]. A standard technique proposed in this literature to enforce sparsity consists in looking for the solution of (1) with the minimal 1 norm. We adopt this approach hereafter. Our volume reconstruction problem then takes the form of the following optimization problem

min x≥0 x 1 such that y -Ax 2 ≤ ε, (2)
where ε > 0 is a parameter accounting for the level of the noise corrupting the collected data. The resulting optimization problem is convex, but nonsmooth. This, on top of the high dimensionality of the system (typically, m ∼ 10 6 , n ∼ 10 9 ), incites us to consider the volume reconstruction problem within the ADMM framework.

TomoPIV meets ADMM

In this section, we expose our methodology based on ADMM to solve the volume reconstruction problem. We start by introducing some useful notations. Then, we depict the general ADMM framework. Finally, we aim attention at the ADMM iterates to address the problem (2) and show its pertinence within our context.

Notations and Useful Computations

We define the indicator function I X (x) of a set X as

I X (x) = 0 if x ∈ X , +∞ otherwise.
The projection operator of a point v on a convex set X is defined as

Π X (v) = arg min x∈X x -v 2 .
The proximal operator of a convex proper function f :

R n → R ∪ ∞ is defined as prox f (v) = arg min x∈X f(x) + 1 2 x -v 2 2 .
3.2 Quick Tutorial on ADMM ADMM focusses on the following type of optimization problems

min x,z f(x) + g(z) such that Dx + z = 0, (3)
where f : R m → R, g : R n → R are closed, proper and convex functions. The ADMM is an iterative procedure searching for a minimizer of (3) via the following recursion

x (k+1) = arg min x f(x) + ρ 2 Dx + z (k) + u (k) 2 2 z (k+1) = arg min z g(z) + ρ 2 Dx (k+1) + z + u (k) 2 2 u (k+1) = u (k) + Dx (k+1) + z (k+1) , (4)
where ρ > 0. We refer the reader to the very good tutorial on ADMM [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF] for an explanation of the rationale behind this type of methodology.

ADMM has recently caught the attention of the signal-processing community for several reasons. First, the conditions on f and g in (3) are mild (they are not required to be differentiable and can take on infinite values); therefore, problem (3) encompasses a large number of optimization problems as particular cases. Second, the ADMM recursion (4) converges to a solution of (3) under very general conditions, see [5, section 3.2]. Third, the algorithmic scheme has been shown empirically to converge to modest accuracy to a solution of (3) in only a few tens of iterations. Finally, the optimization problems involved in the updates of x (k+1) and z (k+1) admit very fast solution or closed-form expression in many setups. These two last features make ADMM very appealing in large-scale problems where modest accuracy is often sufficient but complexity load is of utmost importance.

ADMM Applied to the TomoPIV Problem

Our ensuing developments are inspired from the "C-SALSA" algorithm introduced in [START_REF] Alfonso | An Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems[END_REF]. In order to take the algorithmic cue from the upper-evoked scheme, we express the problem (2) along the lines of that expressed by Equation (3):

min x,z 1 ,z 2 I R n + (z 1 ) + z 1 1 + I B(y,ε) (z 2 ) such that I n A x - z 1 z 2 = 0, (5)
where B(y,ε) = {v ∈ R m | yv 2 ≤ ε} is the 2 ball of radius ε centered on y.

Particularizing the ADMM update rules (4) to the targeted problem [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF], we obtain the following recursions:

x (k+1) = arg min x x -z (k) 1 -u (k) 1 2 2 + ρ 2 Ax -z (k) 2 -u (k) 2 2 2    z (k+1) 1 = prox ρ -1 z 1 1 +I R n + (x (k+1) -u (k) 1 ) z (k+1) 2 = Π B(y,ε) (Ax (k+1) -u (k) 2 ) u (k+1) 1 = u (k) 1 -x (k+1) + z (k+1) 1 u (k+1) 2 = u (k) 2 -Ax (k+1) + z (k+1) 2 .
We can make the following remarks on the complexity of these update rules. The "x-update" is tantamount to solving a least-square problem. Several options are possible to carry out this task. First, one can rely on standard optimality conditions to characterize the solution. The identification of the latter requires to solve a linear system leading to a complexity scaling (at best) as O(mn + n 2 ). Another route to solve the least-square problem consists in resorting to iterative search techniques such as, for example, the conjugate-gradient method (CGM). This approach leads to a complexity scaling as O(mn) per iteration.

The z-update requires to apply some simple proximal and projection operators. The update of z 1 relies on the proximal operator of the 1 -norm defined by

prox ρ -1 z 1 1 +I R n + (v) i = v i -ρ -1 if v i ≥ ρ -1 0 otherwise, (6)
which is a simple element-wise thresholding operation. The complexity of (6) thus scales as O(n). The update of z 2 requires the projection of Ax (k+1)u

(k)
2 onto the 2 ball B(y,ε) and has a simple closed-form expression, see [1, section 3]:

Π B(y,ε) (v) = y + ε v-y v-y 2 if v -y 2 > ε v -y if v -y 2 ≤ ε (7)
This operation can thus be performed with a complexity scaling as O(n). The updates of u 1 and u 2 only involve simple vector additions.

As a consequence, the overall complexity of the ADMM recursion is dominated by the x-update, that we can solve both analytically and iteratively. The two approaches will be considered in our simulation in the next section. To rehash some of the information provided earlier, the analytical resolution of the x-update scales (at best) as O(mn + n 2 ), whereas its iterative counterpart has a complexity scaling as O(mn) per iteration. We note that a complexity scaling as O(mn) is similar to the row-action methods, widely used in the TomoPIV community.

On the other hand, the complexity O(mn + n 2 ) is prohibitive for typical dimensions of the TomoPIV systems. However, this approach will serve as a point of comparison in our simulations.

Numerical Validation

Figure 1 depicts comparative numerical assessment of our newly advanced methods against state-of-the-art SMART [START_REF] Byrne | Iterative Image Reconstruction Algorithms Based on Cross-Entropy Minimization[END_REF] and its accelerated counterpart -that we have coined A-SMART -introduced in our companion paper [START_REF] Barbu | Accelerated, Sparsity-Aware Generalizations of Classical Algorithms for TomoPIV[END_REF]. Nomenclature-wise, positivity-enforcement will be suggested by appending "+" to algorithms' labels -(A-)SMARTs excluded, which implicitly constraints non-negativity -, while the " 1 " hallmark will depict the sparsity constraint. The approximated variant of ADMM will be refered to as ADMM-CGM and will follow the similar paradigm concerning the appended labels.

We simulate a medium-scale perturbed setting (i.e., m = 6724, n = 99944), where y i is affected by a Gaussian noise of zero mean and standard deviation equal to 0.1y i . The results are averaged on 10 experiments and the number of iterations counts 50. We notice in particular that state-of-the-art SMART is outperformed by the newly generalized (A-)SMART( 1 ) schemes -see [START_REF] Barbu | Accelerated, Sparsity-Aware Generalizations of Classical Algorithms for TomoPIV[END_REF] -and by the here-introduced ADMM procedures and their approximated variants ADMM-CGM. This reinforces our statement that the latter should be resolutely invited -along accelerated versions of SMART -into the experimental TomoPIV scene. x T x (k) x x (k) Figure 1: Numerical assessment of the reconstruction quality of x (80) in terms of the normalized correlation against the "particles per pixel (ppp)" ratio.

Conclusion

This paper advocates a new class of algorithms to undertake the TomoPIV Volume Reconstruction Problem. The latter make for an interesting alternative to the classical schemes employed for the TomoPIV Reconstruction Problem in terms of quality of reconstruction versus complexity and speed.