Izaak Neri

Norbert Kern

Andrea Parmeggiani

The Totally Asymmetric Simple Exclusion Process on Networks

Keywords: numbers: 89.75.-k, 64.60.-i, 05.60.Cd, 02.50.-r

We study the totally asymmetric simple exclusion process (TASEP) on complex networks, as a paradigmatic model for transport subject to excluded volume interactions. Building on TASEP phenomenology on a single segment and borrowing ideas from random networks we investigate the effect of connectivity on transport. In particular, we argue that the presence of disorder in the topology of vertices crucially modifies the transport features of a network: irregular networks involve homogeneous segments and have a bimodal distribution of edge densities, whereas regular networks are dominated by shocks leading to a unimodal density distribution. The proposed numerical approach of solving for mean-field transport on networks provides a general framework for studying TASEP on large networks, and is expected to generalize to other transport processes.

Delivering matter, energy or information is a crucial requirement for the functioning of any complex system, ranging from the sub-cellular level of biological organisms to globespanning man-made structures. Transport is often organized along line-like pathways, which are in turn interconnected to form a network structure. In this perspective, diffusion along networks has been studied extensively (see, e.g., [START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF][START_REF] López | [END_REF]). On the other hand, interactions play a fundamental role in the transport properties of many systems: intracellular traffic of molecular motors on the cytoskeleton, pedestrian traffic on an ensemble of paths and traffic of information packages on the internet are but a few prominent examples. They lead to emergent collective phenomena, such as the appearance of jams.

The totally asymmetric simple exclusion process (TASEP) is a paradigmatic model for one-dimensional non-equilibrium transport subject to excluded volume interactions: entities ("particles") hop in a given direction, but cannot occupy the same place [START_REF] Schütz | Phase Transitions and Critical Phenomena[END_REF]. TASEP was initially introduced as a model for the kinetics of RNA polymerization by ribosomes [START_REF] Macdonald | [END_REF], but has since then received much general interest, including from fundamental statistical physics [START_REF]Nonequilibrium Statistical Mechanics in One Dimension[END_REF] and mathematics [START_REF] Liggett | Stochastic Interacting Systems: Contact, Voter and Exclusion Processes[END_REF]. Numerous generalizations have been developed and applied to various areas, such as the collective motion of motor proteins along cytoskeletal filaments, vehicular traffic, etc. [START_REF] Frey | [END_REF][START_REF] Chowdhury | Traffic and Granular Flow '07[END_REF].

The collective behavior of exclusive transport on a network, however, is not well understood at this stage. A body of numerical work on TASEP-like models on networks, often with complex details, exists in the context of traffic [START_REF] Chowdhury | Traffic and Granular Flow '07[END_REF], but less so on biological transport [START_REF] Greulich | [END_REF]. In terms of a paradigmatic analysis based on TASEP, knowledge is still limited to either simple topologies with at most two junctions [10,11], or involves structureless links, e.g. in tree-like networks [12].

In this letter we address the question of how the topology of a network affects its TASEP transport characteristics. Combining concepts from the area of complex networks [START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF] with mean field (MF) methods for TASEP in the presence of junctions [11] we construct the global behavior from that of single segments. This allows us to rationalize many features of transport on large-scale random networks in terms of theoretical arguments, and furthermore leads to an algorithm to numerically solve the MF problem of TASEP transport on a large scale network. In particular we argue that irregularity, i.e. randomness in the vertex degrees, strongly modifies the transport properties of a network.

TASEP on a network. -We generalize the TASEP transport rules [START_REF] Schütz | Phase Transitions and Critical Phenomena[END_REF][START_REF] Macdonald | [END_REF] to a closed network of N S directed segments and N V vertices or junctions. The segments consist of L sites along which particles perform unidirectional random sequential hops subject to hard-core on-site exclusion. At the junctions, particles from k in v incoming segments compete for occupying the same vertex site v, while a particle can leave the junction through one of k out v outgoing segments with equal probability. Figure 1 serves as an illustration for a network with M=30 junctions. We write ρ v for the average occupancy

∂ρ v ∂t = v →v J (v ,v) - v←v J (v,v ) , (1)
the sums run over vertices v identifying incoming segments (v , v) and over vertices v for outgoing segments (v, v ).

We briefly review the behavior of an isolated segment linked to reservoirs, which we will build on. Its average density ρ and current J are known to be homogeneous, provided that segments are at least of moderate size, such that boundary effects remain small. Both are set by the entry rate α and exit rate β [13]:

ρ(α, β) =    α α ≤ β, α < 1/2 (LD) 1 -β β ≤ α, β < 1/2 (HD) 1/2 α, β ≥ 1/2 (MC) (2)
with the current given by the parabolic current-density relation

J(α, β) = ρ(α, β) 1 -ρ(α, β) . (3)
These three homogeneous phases are high density (HD), low density (LD) and maximal current (MC), but for α = β a non-homogeneous shock phase (SP) arises, for which LD and HD regions coexist on the same segment, separated by a diffusing domain wall [14].

Mean-field theory and algorithm. -We extend the MF analysis of [11] to large scale networks to establish an algorithm operating on the junction occupancies ρ v as the only variables. To do so, we neglect correlations between neighbouring sites, as is usually done in MF [13]. Then the entry (exit) rate α (β) for a segment are effective rates, which depend only on the occupancies of the adjacent vertices [11]:

α = ρ v /k out v and β = 1 -ρ v . (4)
The entry rate α is reduced according to the out-degree k out v , since particles on vertex v are distributed uniformly over all outgoing edges. Assuming homogeneous segments, the effective rates in ( 2) and ( 3) can be substituted into the continuity equation ( 1) to yield a closed set of N V equations in the vertex occupancies:

∂ρ v ∂t = v →v J ρ v k out v , 1-ρ v - v ←v J ρ v k out v , 1-ρ v , (5)
where the sums run over all vertices v (v ) which are upstream (downstream) from v. The microscopic dynamics lack particle-hole symmetry at the junctions, as is reflected in Eq. ( 5) by the factor 1/k out v in the entry rate. The numerical MF algorithm consists of iteratively finding the stationary solution to (5), thereby achieving considerable computational advantage upon simulations since we only need to update the N V junction occupancies ρ v . In the following we will study transport on random networks as model systems, complementing MF solutions of Eqs. ( 5) by explicit simulations. Throughout, we exploit the fact that macroscopic observables are self-averaging on these ensembles, as we have verified directly by analyzing different network instances. TASEP on a Bethe network. -As an example of closed random graphs with regular topology we consider the directed Bethe network, drawn from the c-regular ensemble [START_REF] Bollobás | Random graphs[END_REF], in which all vertices have identical connectivity c = k in = k out . The undirected Bethe network is well known in statistical mechanics from the study of, for example, spin models on graphs [START_REF] Baxter | Chapter 4 in Exactly solved models in statistical mechanics[END_REF]. Figure 2 shows the average segment current J as a function of the overall density ρ. The standard result for the current J(ρ) = ρ(1-ρ) is recovered for c =1, where the Bethe network reduces to a simple ring. As connectivity is increased, the overall current is reduced: this may appear counter-intuitive, but reflects the fact that vertices progressively become bottlenecks and block the flow of particles. More precisely, the current parabola is truncated at intermediate densities by a plateau-like region, which widens but lowers with connectivity c. To interpret these results, we first give an explanation based on the phenomenology of TASEP on a line. To this end, we point out that an analytical solution to Eqs. ( 5) can be given for the Bethe network, since all N V equations become identical due to equal vertex connectivities. Therefore the solution requires identical occupancies ρ v for all vertices, from which segment currents and densities follow: all vertices, and therefore all segments, are equivalent. The transition from the LD to the HD phase appears when the effective rates (4) are equal (α = β), i.e. at ρ v = c/(c + 1). Using Eqs. [START_REF] López | [END_REF][START_REF] Schütz | Phase Transitions and Critical Phenomena[END_REF][START_REF] Macdonald | [END_REF] this leads to distinct regimes for the current, yielding the truncated parabola for the MF current-density relation:

J (ρ) = c (c+1) 2 for ρ * < ρ < 1 -ρ * ρ(1 -ρ) otherwise , (6)
with LD phases (for ρ < ρ * ), and HD phases (for ρ > 1 -ρ * ), where ρ * = 1/(c + 1). The plateau can be rationalized in terms of domain wall phenomenology [14], recalling that α = β indicates a SP with a diffusing domain wall between LD and HD zones which coexist on the same segment. Since both zones have complementary densities (ρ LD = 1 -ρ HD ), the current is not affected as further particles are accomodated by growing the HD zones at the expense of the LD zones [14,15]. These MF arguments capture the essential transport features well, as is shown by the simulation data on Fig. 2 (and this remains true also for triangular and square lattices). Deviations arise, however, on the current plateau (where MF underestimates the current) and close to the transitions (where explicit simulations furthermore reveal the particlehole asymmetry, which increases with connectivity c). The numerical MF algorithm does not provide a solution on the plateau, since the assumption of homogeneous segments does not hold, but it otherwise reproduces the theoretical results with great precision. In summary, despite its random nature, TASEP transport through a Bethe network may be understood in terms of a single effective vertex, similar to the Ising model on a Bethe lattice [START_REF] Baxter | Chapter 4 in Exactly solved models in statistical mechanics[END_REF].

TASEP on a Poissonian network. -In order to explore the effect of irregularity, i.e. non-uniform vertex connectivity, we study TASEP on the Poissonian ensemble, closely related to the Erdös-Rényi ensemble [START_REF] Bollobás | Random graphs[END_REF]: any two vertices are connected with probability c/N V , yielding an average connectivity c. In order to avoid artefacts we consider transport on the strongly connected component (SCC) of the Poissonian network, in which each vertex can be reached from all other vertices. We find the SCC using an algorithm developed by Tarjan [START_REF] Tarjan | [END_REF]. For illustration we refer to Fig. 1, which shows a particular realization of a Poissonian ensemble with c = 1, the SCC highlighted in bold.

We first comment on the transport characteristic J(ρ) of TASEP transport on Poisson networks. Figure 3 shows the current J (averaged over all segments) as a function of density ρ, for various connectivities c. The MF results are in excellent agreement with simulations, thereby validating the MF algorithm for disordered networks. The comparison to Bethe networks with identical connectivities c shows that both networks carry the same current J(ρ) at very low and very high densities. However, they behave very differently at intermediate densities. We observe that (i) currents in Poissonian networks are significantly lower than in the corresponding Bethe networks, (ii) even on the MF level, the current J(ρ) no longer possesses particle-hole symmetry (ρ ↔ 1-ρ), and (iii) the density at which the highest current is achieved lies below half-filling, and progressively reduces with connectivity c. The most striking difference, however, is the absence of a plateau in J(ρ) for the Poissonian network. In contrast to Bethe networks, this suggests that no SP segments involving domain walls arise over any extended density range.

A finer understanding is obtained by analysing how the transport features of individual segments are distributed across the network. Consider first the distribution of segment densities, shown for a connectivity c = 10 and an overall density ρ = 0.3, see Fig. 4(a). It is bimodal with peaks corresponding to segments carrying either a very low or a very high density. We are therefore dealing with two sub-networks of segments either in LD or HD phase, in contrast to the Bethe scenario 6) is shown for the same c (dotted lines). Results are not ensemble-averaged: sample to sample fluctuations are of the order of the symbol sizes, but mean field and simulation results for a given topology coincide as in the example.

where all segments are in SP at intermediate densities (single peak in red). Interestingly, the distribution of the segment currents remains unimodal for Poisson networks, Fig. 4(b), thus putting a similar load on all segments. The bimodal density distribution is also the key to understanding how the network adjusts to higher overall densities, by successively switching individual segments from a LD to a HD phase. The inset in Fig. 4(a), obtained for ρ = 0.7, shows that the typical densities of HD/LD segments are not significantly modified, whereas the proportion of the HD network grows at the expense of its LD counterpart as further particles are added. This is further documented by the fraction n HD (n LD ) of edges in HD (LD) phases, Fig. 4(c). For the Poissonian network n HD is roughly equal to the overall density ρ, thereby confirming that the change from LD to HD in irregular networks occurs progressively. This linear behavior of n HD is furthermore very robust with respect to variations of average connectivity c (data not shown), implying that this picture remains valid for all connectivities c.

Transport on highly connected networks. -Significant insight into TASEP transport on general random networks can be gained from analyzing the (MF) high-connectivity limit. Ultimately all vertices constitute bottlenecks, taking their occupancy close to saturation. Therefore we expand the MF Eqs.

(5), posing ρ v = 1 -r v /c, with r v ∼ O(1). The distribution of individual segment densities ρ s then becomes [19]

W S (ρ s ) = k p deg (k) k c dr W V (r) δ (ρ s -ρ ± ) (7)
where p deg (k)k/c represents the degree distribution of vertices when selecting a random segment, W V (r) is the distribution of r, which follows from the MF Eqs. [START_REF]Nonequilibrium Statistical Mechanics in One Dimension[END_REF]. The high (low) density value ρ + (ρ -) are set by the vertex saturation parameter r/c relative to the vertex degree k, as

ρ ± = ρ -= 1/k if r/c > 1/k ρ + = 1 -r/c if r/c < 1/k . (8)
This shows that the bimodal distribution W S in the segment densities, with a fraction of segments in LD and a fraction in HD, is a general feature for complex irregular networks. This is particularly well illustrated in the strong connectivity limit c → ∞, where W S (ρ s ) reduces to

W S (ρ s ) = (1 -ρ) δ(ρ s ) + ρ δ(1 -ρ s ) . ( 9
)
The prefactors of the δ functions explain the linear behavior of n HD , and our above interpretation for Poissonian networks hence generalizes to general irregular random networks.

Conclusions and

Outlook. -TASEP transport on closed random networks has been shown to lead to very different scenarios for regular and irregular topologies. Despite the minimal character of TASEP, there may be direct implications: the presence of bimodality, which we have shown to be robust, in biological tracer experiments would make our findings directly useful for their interpretation. But our results also raise interesting questions, such as the interplay of biological transport and crowding, and their possible regulation by the cytoskeletal network.

An important result of our study is that MF arguments lead to very good predictions for the global transport properties of closed networks, and provides a framework for their interpretation. Moreover, our numerical MF method gives access to system sizes currently beyond the reach of simulations. Generalizations of the approach to open random networks and other network topologies seem straightforward. In addition, we may also expect it to generalize to other transport processes, as long as the behavior of an individual segment is known and boundary-controlled, i.e. determined by the occupancy of the junctions it connects to.

IN thanks Francesco Turci for many fruitful discussions.

  x

FIG. 1 .

 1 FIG. 1. (color online). An example of a Poissonian network of M = 30 junctions with c = 1 and its strongly connected component (bold). Every segment consists of L sites on which particles undergo TASEP dynamics. At the junctions the particles choose one of k out v outlets.

5 FIG. 2 .

 52 FIG. 2. (color online). The average segment current J as a function of the overall density ρ on a Bethe network of degree c superposing numerical mean-field predictions (closed symbols), explicit simulations (NV = 80 junctions, L = 100 sites per edge, open symbols) and a one-vertex analytical result Eq. (6) (solid lines). The dashed lines delimit the different phases for c = 2.

FIG. 3 .

 3 FIG. 3. (color online). The current-density characteristic of the strongly connected component of Poissonian graphs of given average connectivity c. Simulations (markers) with NV = 200 junctions and segments of L = 100 sites are compared to mean field results (solid lines) for the identical network. For comparison, the Bethe result Eq. (6) is shown for the same c (dotted lines). Results are not ensemble-averaged: sample to sample fluctuations are of the order of the symbol sizes, but mean field and simulation results for a given topology coincide as in the example.

FIG. 4 .

 4 FIG. 4. (color online). Distribution of segment properties for networks of connectivity c = 10, comparing a Poisson network (black) to a Bethe network (dashed red). Symbols show simulation results, lines are mean-field predictions. (a) Distribution of segment densities WS(ρs) for an overall density ρ = 0.3 (inset: ρ = 0.7). Convergence of the density peak to a delta peak is very slow in simulations due to collective fluctuations in the shock phase, closed markers correspond to fourfold longer runs than open ones. (b) Distribution of segment currents W (Js) for ρ = 0.3. (c) Fraction of segments in high (nHD) and low (nLD) density segments as a function of the overall density ρ.

We thank A.C. Callan-Jones for a critical reading of the manuscript and acknowledge support from ANR-09-BLAN-0395-02 and from the University of Montpellier 2.


Amel Ben Abda
email: amel.benabda@enit.rnu.tn

Jacques Henry
email: jacques.henry@inria.fr

Fadhel Jday
email: fadheldj@yahoo.com

Boundary data Completion: The method of boundary value problem factorization

We consider the following data completion problem for the Laplace equation in the cylindrical domain:

smooth bounded open set and a > 0), limited by the faces Γ 0 = {0}×O and Γ a = {a}×O. The Neumann and Dirichlet boundary conditions are given on Γ 0 while no condition is given on Γ a . The completion data problem consists in recovering a boundary condition on Γ a . This problem has been known since Hadamard [12] to be ill-posed. The problem is set as an optimal control problem with a regularized cost function. To obtain directly an approximation of the missing data on Γ a we use the method of factorization of elliptic boundary value problems. This method allows to factorize a boundary value problem in the product of two parabolic problems. Here it is applied to the optimality system (i.e. jointly on the state and adjoint state equations). ‡ Let us recall that the Sobolev space H 1 2 00 (O) is defined in Theorem 11.7, p 72 of [19] as the 1 2interpolated space of H 1 0 (O) and L 2 (O).

Introduction

In this work we are concerned by solving the Cauchy problem with data given on a part of the boundary in the framework of Laplace's equation. This problem is known to be ill-posed since Hadamard [START_REF] Hadamard | Lectures on Cauchy Problem in Linear Partial Equation[END_REF]. There are many approaches for solving such an inverse problem, the more popular ones being those of optimal control techniques [START_REF] Blum | Numerical simulation and optimal control in Plasma Physics with Application to Tokamaks[END_REF], [START_REF] Lions | Controle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF] or approximate control technique [START_REF] Bodart | Existence of approximate controls for a semilinear equation[END_REF] or analytic approximation of functions [START_REF] Leblond | Analytic extensions and Cauchy-type inverse problems on annular domains: stability results[END_REF]. The quasi-reversibily method was introduced by Lattès and Lions [START_REF] Lattès | Méthode de Quasi-réversibilité et Applications[END_REF] in the late 60's and it allows to regularize a large class of ill-posed problems. After [START_REF] Klibanov | A computational quasi-reversibility method for Cauchy problems for Laplace's equation[END_REF], in recent papers [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation[END_REF], [START_REF] Bourgeois | Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation[END_REF], [START_REF] Bourgeois | The method of quasi-reversibility to solve the Cauchy problems for elliptic partial differential equations[END_REF], [START_REF] Bourgeois | Conditional stability for ill-posed elliptic Cauchy problems : the case of C1,1 domains (part I)[END_REF], L. Bourgeois revisited the method in the framework of Cauchy's problem for Laplace's equation and studied its stability. In this method the ill-posed secondorder PDE problem is replaced by a fourth-order well posed problem depending on a small regularizing parameter. Notice that in almost all these studies the authors use some regularizing techniques. For example we would like to point out the paper [START_REF] Takeuchi | Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation[END_REF] where the authors explored a discretized Tikhonov regularization for a Cauchy problem corresponding to an elliptic PDE by reproducing kernel Hilbert space.

In this paper, we consider the following problem:

(P 0 )          ∆u = 0 in Ω, u = 0 on Σ, u = T on Γ 0 , ∂u ∂n = Φ on Γ 0 ,
where Ω is a cylinder in R n , Ω =]0, a[×O, O ⊂ R n-1 is a bounded domain, a being a strictly positive real number. Let Σ =]0, a[×∂O be the lateral boundary of Ω and Γ 0 = {0} × O and Γ a = {a} × O be the faces. To stress the particular role of the coordinate along the axis of the cylinder we denote it by x while y ∈ R n-1 denotes the coordinates in the section, perpendicular to the axis. This cylindrical geometry and the Laplace operator make the presentation of the factorization method easy, but the method can be generalized to regular non cylindrical domains and to more general self adjoint second order elliptic operators.

On Γ 0 one has the Cauchy data (Φ, T ) ∈ (H

1 2 00 (O)) ×H 1 2
00 (O) ‡ whereas no boundary condition is available on Γ a . The aim of this paper is the recovering of these missing boundary data exploiting the over specified one on the side Γ 0 . The existence of a solution of this problem is not insured for arbitrary Cauchy data (Φ, T ). This problem is treated in [START_REF] Andrieux | Solving Cauchy problems by minimizing an energylike functional[END_REF], [START_REF] Chakib | Convergence analysis for finite element approximation to an inverse Cauchy problem[END_REF] in a general domain Ω having a boundary : ∂Ω = Γ 0 ∪ Γ 1 . Andrieux et al. [START_REF] Andrieux | Solving Cauchy problems by minimizing an energylike functional[END_REF] rephrase the missing boundary data recovery problem as a control one with two states, each one satisfying one boundary condition on Γ 0 , the control being the overdetermined boundary conditions on Γ a . This method is an iterative one and therefore requires a resolution of the forward problems for each new data (φ, T ). In this work we use the same formulation as in [START_REF] Andrieux | Solving Cauchy problems by minimizing an energylike functional[END_REF] in terms of optimal control but we add a regularization term to insure the existence and uniqueness of the optimality system [START_REF] Lions | Controle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF]. We make use of the factorization method which transforms the elliptic boundary value problem into two parabolic ones. It allows the direct evaluation of the missing boundary data for all Cauchy data (Φ, T ).

The boundary data recovering as an optimal control problem

If ϕ, t on Γ a match the boundary data we will have an overdetermined boundary problem given by:

             ∆u = 0 in Ω, u = 0 on Σ, ∂u ∂n = Φ, u = T on Γ 0 , u = t, ∂u ∂n = ϕ on Γ a .
We consider now, the following optimal control problem: for all (η, τ ) ∈ (H

1 2 00 (O)) × H 1 2
00 (O), the state (u 1 , u 2 ) is defined by :

       ∆u 1 = 0 in Ω, u 1 = 0 on Σ, u 1 = T on Γ 0 , ∂u 1 ∂n = η on Γ a (1)
       ∆u 2 = 0 in Ω, u 2 = 0 on Σ, ∂u 2 ∂n = Φ on Γ 0 , u 2 = τ on Γ a . (2)
If (φ, T ) are compatible then u 1 = u 2 when (η, τ ) = (ϕ, t). We introduce the cost function E(η, τ ) defined by:

E(η, τ ) = Ω (∇u 1 -∇u 2 ) 2 dx dy. (3)
for all (η, τ ) ∈ U ad = (H

1 2 00 (O)) × H 1 2
00 (O). When it exists, the optimal control is given by: (ϕ, t) = arg min{E(η, τ ); (η, τ ) ∈ U ad }.

The spaces (H

1 2 00 (O)) and H 1 2
00 (O) are equipped, respectively, with the following inner product (see [START_REF] Lions | Problèmes aux limites Non homogènes et Applications[END_REF]):

(u, v) (H 1 2 00 (O)) = Γa (-∆ Γa ) -1 4 u(-∆ Γa ) -1 4 v dy = Γa u(-∆ Γa ) -1 2 v dy, for all u, v ∈ (H 1 2

(O)) , and

(f, g)

H 1 2 00 (O) = Γa (-∆ Γa ) 1 4 f (-∆ Γa ) 1 4 g dy = Γa f (-∆ Γa ) 1 2 g dy, for all f, g ∈ H 1 2
00 (O).

Factorization of boundary value problems

The method was presented in [START_REF] Henry | Factorization of second order elliptic boundary value problems by dynamic programming[END_REF]. We consider the problem of Poisson in the open cylindrical domain Ω defined by:

(P 1 )            -∆u = f in Ω, u = 0 on Σ, u = g on Γ a , - ∂u ∂x = u 0 on Γ 0 .
We apply the method of invariant embedding developped by R. Bellman in [START_REF] Bellman | Dynamic Programming[END_REF]. In control theory this method gives the optimal feedback : one embeds the problem of control in a family of similar problems defined between the present and the final time.

In our work this invariant embeding is spatial: we embed the problem (P 1 ) in a family of similar problems (P s,k ) defined on Ω s =]0, s[×O. For every problem we impose a boundary condition of Dirichlet u| s Γs = k Γ s = {s} × O.

(P s,k )            -∆u s = f in Ω s , u s = 0 on Σ s , u s = k on Γ s , - ∂u s ∂x = u 0 on Γ 0 .
Let u s = u s (k) be the solution of (P s,k ). It's clear that (P 1 ) is exactly (P s,k ) with s = a and k = g. The calculation is now similar to the one developped by Lions in [START_REF] Lions | Controle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF] to find the optimal feedback for a problem of optimal control for a parabolic equation. For every s ∈]0, a] we define the Dirichlet-Neumann mapping (D-N), P (s) by

P (s)k = ∂γ s ∂x | |Γs , where γ s is the solution of            -∆γ s = 0 in Ω s , γ s = 0 on Σ s , γ s = k on Γ s , - ∂γ s ∂x = 0 on Γ 0 .
Let w be defined by w(s) = ∂β s ∂x | Γs , where β s is the solution of

           -∆β s = f in Ω s , β s = 0 on Σ s , β s = 0 on Γ s , - ∂β s ∂x = u 0 on Γ 0 .
and w(0) = -u 0 .

The linearity of (P s,k ) yields u s = β s + γ s . So by the well-posedness of the problem on γ and the continuity of the trace mapping

∂u s ∂x | Γs = P (s)k + w. (4)
It is shown in ( [START_REF] Henry | Factorization of second order elliptic boundary value problems by dynamic programming[END_REF]) that for every s ∈]0, a], P (s) :

H 1 2 00 (O) -→ H 1 2
00 (O) is a coercive, self-adjoint, linear continuous operator and w(s) = ∂βs ∂x |Γs ∈ H 1 2 00 (O) . We consider for x ≤ s the restriction of u s (k) at Ω x . It is solution of the problem (P x,u s (x,k) ).

By the same calculation as previously we have the relation:

∂u s ∂x (x, k) = P (x)u s (x, k) + w(x). (5)
On a formal way, we take the derivative of (5) with respect to x (stressing the particular role plaid by x we denote it by d dx ) and using the equation satisfied by u s ,

d 2 u s dx 2 = -∆ y u s -f = dP dx u s + P du s dx + dw dx
Subtituting du s dx from (5) and for x = s we find

(∆ y + dP dx (s) + P 2 (s))k + P (s)w(s) + dw(s) dx + f (s) = 0.
In this calculation k and s are arbitrary and we deduce:

dP dx + P 2 = -∆ y , for x ∈]0, a]. (6)
For the residual term dw dx

+ P w = -f, for x ∈]0, a]. (7)
One can extend by continuity this argument to x = 0. Writing the relation (5) at x = 0, and by using the fact that u(0) is arbitrary, we get the initial conditions for P and w

P (0) = 0, w(0) = -u 0 . (8)
Adding the condition on Γ a for u, we summarize the result: an equivalent formulation of (P 1 ) is

           dP dx + P 2 = -∆ y , P (0) = 0, dw dx + P w = 0, w(0) = -u 0 , - du dx + P u = -w, u(a) = g. (9)
The system ( 9) is uncoupled because one can integrate the two first equations in x from 0 to a to find P and w, then u is obtained by backward integration. In a formal way, having P , one can factorize (P 1 ) in

-( d dx + P )( d dx -P )u = f. ( 10
)

The factorization method as a tool to the boundary data recovering

In this section we show that applying the factorization method to the states u 1 , u 2 , we can express the cost function explicitly in terms of the controls τ , η. We embed the control problem in a family of similar problems defined on Ω s . For (1) we impose a Neumann boundary condition on Γ s :

du s 1 dx | Γs = α,
       ∆u s 1 = 0 in Ω s , u s 1 = 0 on Σ s , u s 1 = T on Γ 0 , ∂u s 1 ∂n = α on Γ s , (11)
       ∆u s 2 = 0 in Ω s , u s 2 = 0 on Σ s , ∂u s 2 ∂n = Φ on Γ 0 , u s 2 = β on Γ s . (12)
We consider the Neumann-Dirichlet(N-D) mapping for

u 1 : α -→ u s 1 | Γs and the Dirichlet-Neumann (D-N) for u 2 : β -→ du s 2 dx | Γs . From section 3 we have du s 2 dx | Γs = P (s)β + w 2 (s), with P (s) ∈ L(H 1 2 00 (O), (H 1 2
00 (O)) ), P (0) = 0 and w 2 (0) = -Φ. The invariant embeding of section 3 leads to

dP dx + P 2 = -∆ y , P (0) = 0, ( 13
)
dw 2 dx + P w 2 = 0, w 2 (0) = -Φ. (14)
Then u 2 is found by backward integration of

du 2 dx = P u 2 + w 2 , u 2 (a) = τ (15)
We can treat the (N-D) operator Q in a similar way, for u 1 satisfying (11),

u s 1 (s) = Q(s)α + w 1 (s), Q(s) ∈ L((H 1 2 0,0 (O)) , H 1 2
0,0 (O)). Considering the invariant embeding for x ≤ s we get

u s 1 (x) = Q(s) du s 1 dx (x) + w 1 (x).
Taking the derivative with respect to x du s

1 dx = dQ dx du s 1 dx + Q d 2 u s 1 dx 2 + dw 1 dx .
Substituting u s 1 , using [START_REF] Chakib | Convergence analysis for finite element approximation to an inverse Cauchy problem[END_REF], taking x = s and using the fact that α is arbitrary we get dQ dx -Q∆ y Q = I, Q(0) = 0, ( 16)

dw 1 dx -Q∆ y w 1 (x) = 0, w 1 (0) = T. (17)
After solving these equations we find u 1 by backward integration of the implicit differential equation

u 1 = Q du 1 dx + w 1 , u 1 (a) = Q(a)η + w 1 (a). (18)

Solving the optimal control problem

Let us show now that the energy functional E can be expressed directly in terms of the control variable η and τ using the operators P and Q. So there is no need of introducing an adjoint state to derive the optimality condition. Saying it in another way, the classical decoupling of the optimality system between state and adjoint state using a Riccati equation is done here at the same time as the factorization of the state equation.

From now on we will denote P = P (a) and Q = Q(a). Let A be the matrix operator

A = Q -QP -P Q P .
Let us denote [η, τ ] the row vector with component η and τ . Let us remark that in relation to the ill-posedness of the Cauchy problem, the operator A is not coercive.

Proposition 1 The cost functional E can be written equivalently

E(η, τ ) = C + [η, τ ]A[η, τ ] -2 Γa (w 1 P τ + w 2 Qη) dy,
where C is a constant not depending on η and τ . If a minimum (ϕ, t) of E is reached it satisfies

A[ϕ, t] = [Qw 2 , P w 1 ] . (19)
Proof Let α and β be the solutions of

           ∆α = 0 in Ω, α = 0 on Σ, α = T on Γ 0 , ∂α ∂n = 0 on Γ a , (20)
           ∆β = 0 in Ω, β = 0 on Σ, ∂β ∂n = Φ on Γ 0 , β = 0 on Γ a . (21)
Using Green's formula and ( 1), ( 2)

E(η, τ ) = Ω (∇u 1 -∇u 2 ) 2 dx dy = Γ 0 ( ∂u 1 ∂n - ∂u 2 ∂n )(u 1 -u 2 ) dy + Γa ( ∂u 1 ∂n - ∂u 2 ∂n )(u 1 -u 2 ) dy = Γ 0 ( ∂u 1 ∂n -Φ)(T -u 2 ) dy + Γa ( ∂u 1 ∂n - ∂u 2 ∂n )(u 1 -u 2 ) dy = Γ 0 (-ΦT + ∂u 1 ∂n T + Φu 2 - ∂u 1 ∂n u 2 ) dy + Γa (η - ∂u 2 ∂n )(u 1 -τ ) dy.
Using the full Green's formula for the term in

∂u 1 ∂n u 2 E(η, τ ) = Γ 0 (-2ΦT + ∂u 1 ∂n α + ∂β ∂n u 2 ) dy + Γa (ητ -u 1 ∂u 2 ∂n ) dy + + Γa (η - ∂u 2 ∂n )(u 1 -τ ) dy,
and for the terms in ∂u 1 ∂n α and ∂β ∂n u 2

E(η, τ ) = -2 Γ 0 ΦT dy + Γ 0 (T ∂α ∂n + βΦ) dy - Γa (ηα + ∂β ∂n τ ) dy + Γa (ητ -u 1 ∂u 2 ∂n ) dy + Γa (η - ∂u 2 ∂n )(u 1 -τ ) dy = -2 Γ 0 ΦT dy + Γ 0 (T ∂α ∂n + βΦ) dy + Γa (η(u 1 -α) + ( ∂u 2 ∂n - ∂β ∂n )τ ) dy -2 Γa u 1 ∂u 2 ∂n dy
From ( 15), [START_REF] Leblond | Analytic extensions and Cauchy-type inverse problems on annular domains: stability results[END_REF] we have

u 1 (a) = Q(a) ∂u 1 ∂n (a) + w 1 (a) = Q(a)η + w 1 (a) ∂u 2 ∂n (a) = P (a)u 2 (a) + w 2 (a) = P (a)τ + w 2 (a).
Then from the definition of α and β we have

α(a) = w 1 (a), ∂β ∂n (a) = w 2 (a). ( 22
) Then E(η, τ ) = -2 Γ 0 ΦT dy + Γ 0 (T ∂α ∂n + βΦ) dy + Γa (ηQη + τ P τ ) dy -2 Γa (Qη + w 1 )(P τ + w 2 ) dy
The relation at the minimum results from the quadratic form of E, its convexity and the selfadjointness of P and Q.

Proposition 2 Let w 1 and w 2 be given by ( 17) and ( 14). The relation [START_REF] Lions | Problèmes aux limites Non homogènes et Applications[END_REF] at the minimum of E is equivalent to

(I -QP )t = Qw 2 + w 1 and (I -P Q)ϕ = P w 1 + w 2 . ( 23
)
The compatibility of the data (Φ, T ) with respect to the Cauchy problem, is given by one of the three following assertions that are equivalent (i) (Qw 2 , P w 1 ) ∈ (A),

(ii) Qw 2 + w 1 ∈ (I -QP ), (iii) w 2 + P w 1 ∈ (I -P Q).

Proof

The relation [START_REF] Lions | Problèmes aux limites Non homogènes et Applications[END_REF] reads

Qϕ -QP t = Qw 2 P t -P Qϕ = P w 1 (24)
By the coerciveness of P and the Lax-Milgram Theorem, P is an isomorphism from H

In the same way the first equation (24) yields

ϕ = P t + w 2 . (26)
Then ( 23) is obtained by the combination Q(26)+( 25) and ( 26)+P (25). Reciprocally, assume (23) is satisfied. Let us show that I -QP is injective. Suppose there exists τ such that (I -QP )τ = 0. This means that considering the homogeneous problems (Φ = T = 0) (1), (2) we get

u 1 (η)| Γa = u 2 | Γa , for η = ∂u 2 ∂n | Γa .
Then by Holmgren's Theorem, u 1 = u 2 and by the boundary conditions at Γ 0 and Holmgren's Theorem once again u 1 = u 2 = 0, and so τ = 0. Similarly I -P Q is injective From (23) the combination -P (23

) 1 + (23) 2 yields (I -P Q)(ϕ -P t) = (I -P Q)w 2 .
And so we recover (26) and in the same way we can recover (25). Then ( 19) is obtained by a linear combination of ( 26) and(25).

Let us now turn to the compatibility condition for the data (Φ, T ). If (Qw 2 , P w 1 ) ∈ (A) then there exists (ϕ, t) ∈ (H

u 1 (a) = u 2 (a).
Similarly from the first relation ( 24)

∂u 1 ∂n | Γa = ∂u 2 ∂n | Γa .
This implies that (ϕ, t) is the solution of the inverse problem of data completion and then (Φ, T ) are compatible.

Reciprocally if (Φ, T ) are compatible then the inverse problem has a unique solution (ϕ, t) that is the minimum of the function E. Then Proposition 1 gives

A[ϕ, t] = [Qw 2 , P w 1 ] .
Let us show that assertion 2) implies 3). Let Qw 2 + w 1 ∈ (I -QP ) and t such that t -QP t = Qw 2 + w 1 . Then t -w 1 = QP t + Qw 2 . Let us set ϕ = P t + w 2 . We have

Qϕ = QP t + Qw 2 = t -w 1 .
Multiplying by P P Qϕ = P t -P w 1 = ϕ -w 2 -P w 1 , that is

w 2 + P w 1 ∈ (I -P Q).
By a similar calculation 3) implies 2).

From the previous calculation if (24) is satisfied (23) is also and so 1) implies 2) and 3). Also we already showed that reciprocally 2) and 3) imply 1).

Link with the Steklov-Poincaré operator approach

In this part we present a link between the previous approach and the calculation of [START_REF] Azaiez | El Fekih On Cauchy's problem: II. Completion, regularization and approximation Inverse Problems[END_REF] using the Steklov-Poincaré operator. For all (Φ, T ) ∈ (H

1 2 00 (Γ a )) × H 1 2
00 (Γ a ) and for all τ ∈ H 1 2 00 (Γ a ), we consider the problem (1), ( 2), where we control u 1 with a Dirichlet condition u 1 = τ on Γ a in order to be in the same framework than [START_REF] Azaiez | El Fekih On Cauchy's problem: II. Completion, regularization and approximation Inverse Problems[END_REF], that is

u 1 now satisfies      ∆u 1 = 0 in Ω, u 1 = 0 on Σ, u 1 = T on Γ 0 , u 1 = τ on Γ a (27)
We denote the solutions of ( 27), ( 2) by u 1 (τ, T ) and u 2 (τ, Φ) to recall the dependencies with respect to the control and the data. We denote u 1 (τ, 0) (resp u 2 (τ, 0)) by u 1 (τ ) (resp u 2 (τ )) and u 1 (0, T ) (resp u 2 (0, Φ) by u 0 1 (τ ) (resp u 0 2 (τ )). We introduce now the bilinear forms s(., .), s D (., .) and s N (., .) in H

s(τ, δ) = s D (τ, δ) -s N (τ, δ) = Ω ∇u 1 (τ )∇u 1 (δ)dx dy - Ω ∇u 2 (τ )∇u 2 (δ)dx dy, l(δ) = - Ω ∇u 0 1 (T )∇u 1 (δ)dx dy + Ω ∇u 0 2 (Φ)∇u 2 (δ)dx dy -(Φ, u 2 (δ)) L 2 (Γ 0 ) .
The boundary data recovery amounts to finding t ∈ H 1 2 00 (Γ a ) such that ∇u 1 (t) = ∇u 2 (t) on the face Γ a . This is equivalent to finding t ∈ H

1 2 00 (Γ a ) verifying s(t, δ) = l(δ) for all δ ∈ H 1 2 00 (Γ a ).
Proposition 3 The bilinear forms s, s D and s N and the linear form l are related to the operators solution of the Riccati equations ( 13), ( 16) and the remainder functions w 1 and w 2 by s N (τ, δ)

= (P τ, δ), s D (τ, δ) = (Q -1 τ, δ), s(τ, δ) = ((Q -1 -P )τ, δ), l(δ) = (Q -1 w 1 + w 2 , δ), for all τ, δ ∈ H 1 2
00 (Γ a ). Proof By Green's formula

s D (τ, δ) = Ω ∇u 1 (τ )∇u 1 (δ) dx dy = Γa ∂u 1 ∂n (τ )δ dy = Γa Q -1 τ δ dy s N (τ, δ) = Ω ∇u 2 (τ )∇u 2 (δ) dx dy = Γa ∂u 2 ∂n (τ )δ dy = Γa P τ δ dy s(τ, δ) = s D (τ, δ) -s N (τ, δ) = Γa (Q -1 -P )τ δ dy l(δ) = - Ω ∇u 0 1 (T )∇u 1 (δ) dx dy + Ω ∇u 0 2 (Φ)∇u 2 (δ) dx dy -(Φ, u 2 (δ)) L 2 (Γ 0 ) = - Γ 0 T ∂u 1 ∂n (δ) dy - Γ 0 Φu 2 (δ) dy.
From ( 20), ( 21), using the full Green's formula and the relation ( 22) one gets

l(δ) = - Γ 0 α ∂u 1 ∂n (δ) dy - Γ 0 ∂β ∂n u 2 (δ) dy = Γa w 1 ∇u 1 (δ) dy + Γa w 2 δ dy.
We shall denote by S(x) the operator S(x) = Q -1 (x) -P (x) and so s(τ, δ) = (Sτ, δ).

Remark 1 The definition of u 1 given by ( 27) is the same than the one of (1) if τ and η are linked by τ = Q(a)η + w 1 .

In this case the second relation ( 24) is automatically satisfied which proves that the two formulations are equivalent.

Remark 2 These relations may be particularly usefull if one wants to study the sensitivity of the data completion problem to the size a of the domain because the derivatives of P , Q, w 1 and w 2 with respect to a are given by ( 13), ( 16), ( 17), ( 14).

Generalization

In this section we extend the calculations of section 4 with two controls η, τ to the case where the two states u 1 , u 2 are controlled through Robin conditions on Γ a . We consider the following two mixed well posed problems for both η and τ in H

1 2 00 (Γ a ) , r 1 > 0 and r 2 > 0.        ∆u 1 = 0 in Ω, u 1 = 0 on Σ, u 1 = T on Γ 0 , ∂u 1 ∂n + r 1 u 1 = η + r 1 τ on Γ a (28)        ∆u 2 = 0 in Ω, u 2 = 0 on Σ, ∂u 2 ∂n =Φ on Γ 0 , ∂u 2 ∂n + r 2 u 2 = η + r 2 τ on Γ a (29)
Here again, the next step is to build an error functional on the two fields by

E r 1 r 2 (η, τ ) = Ω (∇u 1 -∇u 2 ) 2 dx dy,
with, u 1 and u 2 being the solution of ( 28) and ( 29). If the minimum exists we define (ϕ, t) = arg min E r 1 r 2 (η, τ ).

Indeed, u 1 and u 2 are obviously equal when the pair (Φ, T ) is compatible and one gets on Γ a

u 1 = u 2 = t, ∂u 1 ∂n = ∂u 2 ∂n = ϕ.
Let us set

Q 1 = (I + r 1 Q) -1 , P 1 = (P + r 2 I) -1 , X 1 = QQ 2 1 + P P 2 1 -2QP Q 1 P 1 , X 2 = r 2 1 QQ 2 1 + r 2 2 P P 2 1 -2r 1 r 2 QP Q 1 P 1 , X 3 = r 1 QQ 2 1 + r 2 P P 2 1 -(r 1 + r 2 )QP Q 1 P 1 , X 4 = r 1 Q 1 (Q 1 -I -r 1 QQ 1 ) -2r 2 P P 1 Q 1 , X 5 = r 2 P 1 (r 2 P 1 -I -P P 1 ) -2r 1 r 2 QQ 1 P 1 X 6 = Q 1 (Q 1 -I) -R 1 Q 2 1 Q -2P P 1 Q 1 X 7 = P 1 (r 2 P 1 -I -P P 1 ) -2r 2 P 1 QQ 1 .

Then setting

A r 1 r 2 = X 1 X 3 X 3 X 2 ,
by the same calculations as in section 4 we get

E r 1 r 2 (η, τ ) = C + [η, τ ]A r 1 r 2 [η, τ ] + Γa ((X 4 w 1 + X 5 w 2 )τ + (X 6 w 1 + X 7 w 2 )η) dy.
With this formulation we can recover various situations: taking r 1 = 0, r 2 = +∞ we recover the formulation (1), ( 2) and

E r 1 r 2 (η, τ ) = E(η, τ ).
Taking r 1 = r 2 = +∞ we have

u 1 | Γa = u 2 | Γa = τ,
so this is the framework of [START_REF] Azaiez | El Fekih On Cauchy's problem: II. Completion, regularization and approximation Inverse Problems[END_REF] recalled in section 5. Then

E r 1 r 2 (η, τ ) = ((Q -1 -P )τ, τ ) -2(Q -1 w 1 + w 2 , τ ) + C.
The case r 1 = r 2 < +∞ corresponds to a Steklov-Poincaré approach as in [START_REF] Azaiez | El Fekih On Cauchy's problem: II. Completion, regularization and approximation Inverse Problems[END_REF] but with a Robin condition control. In the domain decomposition method it is known to be numerically more efficient.

Explicit Formulae

Determination of the interfacial operators

In this section we take full advantage of the cylindrical geometry of Ω to derive explicit formulae for the Dirichlet-Neumann P (a) and Neumann-Dirichlet Q(a) operators at the face Γ a .

Proposition 4 Let U (x) be the C 0 semi-group with generator -2(-∆ y ) 1 2 . For cylindrical domains Ω the solution of ( 13) is explicitly given by

P (x) = (-∆ y ) 1 2 (I -U (x))(I + U (x)) -1 . ( 30
)
The solution Q(x) of ( 16) is then given by

Q(x) = (-∆ y ) -1 P (x).
The operators P and Q commute and R is given by

R(x) = I -P (x)Q(x) = I -Q(x)P (x) = 4U (x)(U (x) + I) -2 .

Proof

The formula (30) is proved in [START_REF] Bouarroudj | On a Direct Study of an Operator Riccati Equation Appearing in Boundary Value Problems Factorization[END_REF] by using a Yosida regularization. From this formula, P being a function of -∆ y commutes with it and any of its power. Let us multiply ( 13) by (-∆ y ) -1 . We deduce that (-∆ y ) -1 P is a solution of ( 16), which has a unique solution so that

Q = (-∆ y ) -1 P = (-∆ y ) -1 2 (I -U (x))(I + U (x)) -1 .
Then R = I -P Q is given by

R(x) = I -(-∆ y ) -1 Q 2 (x) = 4U (x)(U (x) + I) -2 .
Similarly S(x) = Q -1 (x) -P (x) is given by

S(x) = 4(-∆ y ) 1 2 U (x)(I -U 2 (x)) -1 .

Behaviour with respect to the length of the cylinder

We use here the explicit formulae obtained in section [START_REF] Bourgeois | Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation[END_REF] to study the dependence of the ill-posedness of the problem with respect to the length of the cylinder. If the data (Φ, T ) are compatible the solution (ϕ, t) of the problem (P 0 ) is given by

(S(a) -1 w 2 (a) + R(a) -1 w 1 (a), R(a) -1 w 2 (a) -∆ y S(a) -1 w 1 (a)).
Let {α n , n ∈ N}, α n > 0 be the spectrum of (-∆ y )

1 2 . The sequence (α n ) increases to infinity. Then the spectrum of R(a) is given by

{ 4e -2αna (1 + e -2αna ) 2 , n ∈ N},
that of S(a) is given by {α n 4e -2αna 1 -e -4αna , n ∈ N}, and that of (-∆ y ) -1 S(a) by

{ 4e -2αna α n (1 -e -4αna )
, n ∈ N}.

One notices that these sequences decrease exponentially to 0, when n goes to infinity, and that illustrates the ill-posedness of the inverse problem (P 0 ). We also notice that if the length of the cylinder increases, the speed of convergence of the sequences to zero increases. So these formulae allow to quantify the ill-posedness of the inverse problem according to the length of the cylinder (i.e. the distance between the surface where the data are overdetermined and the one where they are missing).

Regularizing the data completion problem

First let us recall the regularization process performed in [START_REF] Azaiez | El Fekih On Cauchy's problem: II. Completion, regularization and approximation Inverse Problems[END_REF] :

s(t, δ) + εs D (t, δ) = l(δ) for every δ ∈ H 1 2
00 (Γ a ), which we write with our notations

S(a)t + εQ(a) -1 t = w 2 + Q(a) -1 w 1 , (31) or equivalently
(R(a) + εI)t = Q(a)w 2 + w 1 . (32)
In a classical way, the parameter ε is fixed according to the noise on the measurements on Γ 0 . It is clear that we can only estimate the modes of t corresponding to the eigenvalues of S(a) larger than those of the regularizing term εQ(a) -1 . We shall define them as the computable modes Let us define the parameter d by

d = 2 -ε -2 √ 1 -ε ε .
Proposition 5 The computable modes for the regularization (31) are those verifying

α n ≤ 1 2a ln( 1 d
).

Proof

We denote by S ε the regularization of S,

S ε = S + εQ(a) -1 .
Let (λ n ) n , (resp (λ ε n ) n ) be the eigenvalues of S (resp of S ε ). We have

λ ε n = 4α n e -2αna 1 -e -4αna + εα n 1 + e -2αna 1 -e -2αna .
The nth mode is computable if

λ n > εα n 1 + e -2αna
1 -e -2αna , that is 4e -2αna (1 + e -2αna ) 2 > ε. Setting X = e -2αna we must have

εX 2 + 2(ε -2)X + ε < 0.
The zeroes of this polynomial are X 1 = d and

X 2 = 2-ε+2 √ 1-ε ε . So we must have α n ≤ 1 2a ln( 1 d ). ( 33
)
Remark 3 Condition (33) shows how the number of computable modes depends on the regularization parameter ε and that it decreases as a increases.

We propose now a new regularization of the optimal control problem by considering the cost function

E ε (η, τ ) = Ω (∇u 1 -∇u 2 ) 2 dx dy + 2ε Ω ∇u 1 ∇u 2 dx dy (34)
By calculations similar to Proposition 1 one can show that

Ω ∇u 1 ∇u 2 dx dy = Γ 0
ΦT dy + Γa (Qη + w 1 )(P τ + w 2 ) dy.

Defining

A ε = Q -(1 -ε)QP -(1 -ε)P Q P ,
the solution of the regularized optimal control problem is given by

A ε [ϕ, t] = (1 -ε)[Qw 2 , P w 1 ]
which is well posed for ε > 0 sufficiently small. After elimination, the regularized version of R is

R ε = (I -(1 -ε) 2 P Q).
Using the former calculations in the cylindrical case its spectrum is

{2(1 + (1 -ε) 2 ) e -2αna (1 + e -2αna ) 2 + (2ε -ε 2 ) 1 + e -4αna (1 + e -2αna ) 2 , n ∈ N}. Setting d = ε + 1 - √ 1 + 2ε ε ,
and neglecting the terms in ε 2 we have the same result as in proposition 5:

α n ≤ 1 2a ln( 1 d ), (35)
but as d > d , more modes could be computable by this regularization. Furthermore the influence of this regularization is less on the first modes and for small a than the previous one. This is desirable as these situations correspond to less ill-posed problems.

Numerical results

In this section we present numerical experiments on the factorization based method we propose for the data completion problem. We call this method "the direct method" because it needs the resolution of three differential equations as opposed to the method of [START_REF] Andrieux | Solving Cauchy problems by minimizing an energylike functional[END_REF] which we call iterative because it uses an iterative algorithm of minimization.

We consider problem (P 0 ) on Ω = ]0, a[ × ]0, π[. The Cauchy data on Γ 0 are Φ = sin(y) et T = sin(y). Then the harmonic function u(x, y) = e x sin(y) is the solution of (P 0 ) and the missing data to be recovered on Γ a are ϕ th = ∂u ∂n (a, y) = e a sin(y) et t th = u(a, y) = e a sin(y).

For the numerical experiments with the direct method, a finite difference method is used. The grid points are : a ij = (ih, jk) for i = 1...n and j = 1...n -1, n ∈ N, h = a n and k = π n . There is no boundary value problem to solve. Instead we have to solve an initial value problem for the operator Riccati equations ( 6), [START_REF] Klibanov | Carleman Estimates for Coefficient Inverse Problems and Numerical Applications[END_REF]. Let us first consider a semi-discretization with respect to the coordinate y by a finite difference method. The Dirichlet to Neumann operator P becomes a (n -1) × (n -1) matrix P i at x = ih (i ∈ [0, n]) that connects the values u i,. of the solution at the gridpoints on section x = ih to the approximation of ∂u ∂x on the same section. This matrix satisfies a matrix Riccati equation in x. After a discretization in the x direction (i ∈ N), this equation is solved by an explicit Euler scheme respecting stability conditions. Namely one has P i = P i-1 (I -hP i-1 ) -h∆ y,k , P 0 = 0, where ∆ y,k is the three diagonal matrix associated with the three points approximation to the second derivative. A similar scheme is applied to the Neumann-Dirichlet operator

Q i = Q i-1 (I -h∆ y,k Q i-1 ) + hI, Q 0 = 0.
The method has been implemented in a Scilab program.

Comparison with the iterative method

In this section we compare the iterative method of [START_REF] Andrieux | Solving Cauchy problems by minimizing an energylike functional[END_REF] with a regularizing term with ours.

In both cases the data completion problem is set as a control problem as formulated in 1, 2, 3.

For the iterative method we solve the state equation by a P 1 finite element method on a regular triangular meshing with the same gridpoints using the FreeFEM software. The minimization uses a conjugate gradient method.

We denote t d , t i the Dirichlet condition on the section Γ a computed respectively by the direct method and the iterative method. Let err d and err i be respectively the relative errors with respect to the theoretical solution t th in L 2 norm of respectively t d , t i . Let time 1 be the computing time for the operators P , Q and R, time 2 be the computing time for the functions w 1 , w 2 and for solving and ( 19) and time 3 be the computing time for the iterative method.

For this computation we took ε = 10 As we do not use the same software for the direct and iterative methods we cannot compare directly the computing times. Nevertheless Scilab being an interpreted language it is known to have slower execution times which favours the direct method for small size problems. These results are in accordance with known facts for the comparison between direct and iterative methods for solving numerically linear systems. The complexity for factorizing a (n, n) matrix is O(n 3 ). In our framework the factorization is obtained by solving the Riccati equations and so this increase of complexity is in accordance with the increase of time 1 . On the other hand this computation is independent of the data (Φ, T ) which means that once P , Q and R have been computed for an extra set of data only a computation of duration time 2 is needed. As time 2 remains small this gives an advantage to the direct method if a large set of data has to be treated.

Comparison of regularizations

In section 7.3 we presented a new way of regularizing the data completion problem. In this section we compare it with the one of [START_REF] Andrieux | Solving Cauchy problems by minimizing an energylike functional[END_REF]. Let err dr2 be the relative error between t th and t d when the classical regularization of [START_REF] Andrieux | Solving Cauchy problems by minimizing an energylike functional[END_REF] is used.

We set ε = 10 -2 and n = 50. We study the effect of the length of the cylinder a. As it was stated in section 7.3 our regularization adapts itself to the case of small lengthes where the problem is less ill-posed.

Stability

In this section we study the stability of the data completion problem to noise on the data. Let err dn the relative error on t d when a noise is added to the data. The level of the noise is 10 -1 . We set n = 100 and a = π 2 .

err dn err d ε = 10 -1 0.4 0.4 ε = 10 -2 4.10 -2 7.10 -2 ε = 10 -4 0.19 10 -2 ε = 10 -6

2.75 10 -2

Conclusion

The factorization method allows an explicit recovering of the boundary missing data when the Cauchy problem is rephrased into an optimal control one. In this formulation the operators P and Q do not depend on the data. The main feature of the proposed method is that if we have a collection of boundary data to recover, that is a collection of Cauchy data (Φ, T ) one has only to solve the initial value problems involving w 1 and w 2 . Furthermore the proposed method permits to perform analytical computations which throw new insights into the missing data recovery problem. In particular this method allowed us to propose a new way of regularizing the problem that is self adaptative for less ill-posed problems.

In our future work we will try to extend the method to more general geometries than the cylinder and to general second order self adjoint operators, and study the effect of small perturbations of the surfaces of measurements or of completion.

  and for (2) a Dirichlet boundary condition : u 2 | s Γs = β. The originality of the approach lies in the choice of the embedding: the controls τ , η do not show up in the problems set in Ω s

  a )) . Then the second equation (24) yields t = Qϕ + w 1 .

P w 1 = P u 1

 11 a ) solution of the system (24). Let us define u 1 and u 2 as previously with u 2 | Γa = t and ∂u 1 ∂n | Γa = ϕ. Then the second relation (24) gives ∂u 2 ∂n -w 2 = P Q ∂u 1 ∂n + Then P (a)u 1 (a) = P (a)u 2 (a), and by the coerciveness of P and Lax Milgram Theorem:

2 00 2 00

 22 a ) and the linear form l in H 1 (Γ a ) defined for all τ, δ ∈ H 1 (Γ a ) by :

  -8 and a = π 2 .

		time 1 (sec.) time 2 (sec.) err d	time 3 (sec.)	err i
	n=100	2.9	0.135	10 -2	10.53	9.10 -3
	n=200	38.9	0.89	5.10 -3	47.80	4.10 -3
	n=300	198.2	1.88	3.10 -3	124.3	2.10 -3

Acknowledgments

The authors acknowledge the financial support of INRIA associated team Enée. The LAMSIN researchers' work is supported an a regular basis by the Tunisian Ministry for Higher Education, Scientific Research and Technology.


Jacques Henry

Bento Louro

Maria Do

Céu Soares

Factorization of linear elliptic boundary value problems in non cylindrical domains

We present a method of factorization for linear elliptic boundary value problems considered in non cylindrical domains. We associate a control problem to the boundary value problem which regularizes it. The technique of change of variables is used to study this problem.

Introduction

In [START_REF] Angel | Dynamic programming and partial differential equations[END_REF] Angel and Bellman proposed a method based on spatial invariant embedding for transforming a second-order elliptic boundary value problem in a rectangle, in a system of first-order decoupled initial Email addresses: jacques.henry@inria.fr (Jacques Henry), bjl@fct.unl.pt (Bento Louro), mcs@fct.unl.pt (Maria do Céu Soares).

value problems which can be solved by a two-step process. In [START_REF] Henry | Factorization of second order elliptic boundary value problems by dynamic programming[END_REF], Henry and Ramos gave a complete justification for this transformation, in the case of the Poisson equation in an n-dimensional cylindrical domain. The invariant embedding was performed using the coordinate along the axis of the cylinder. The Neumann to Dirichlet (NtD) operator on a section of the cylinder was shown to satisfy a Riccati equation. The method is similar to the one used by [START_REF] Lions | Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF], for deriving the optimal feedback for optimal control problems of parabolic equations. Its justification is based on a Galerkin method. A shorter proof was given in [START_REF] Bouarroudj | On a direct study of an operator Riccati equation appearing in boundary value problems factorization[END_REF]. In this study, we generalize this method to non cylindrical domains. The relationship with a control problem is used to regularize the Riccati equation. We use a change of variables in order to set the problem in a cylindrical domain. It was shown, in the case of the cylinder, that the LU block factorization of matrix of the problem discretized by finite differences, can be viewed as a discretization of the factorized version of the boundary value problem. Other discretizations lead to new numerical schemes.

Elliptic boundary value problem in a non cylindrical domain

We denote the elements of R N by (x 1 , x 2 , . . . , x N ) = (x, y), where x = x 1 and y = (x 2 , . . . , x N ) to stress the particular role of x 1 . For each x ∈ [0, a], let O x be a bounded open set in R N -1 . We shall call a quasi-cylinder with respect to x, a set Ω defined in R N by Ω = ∪ 0<x<a (x, O x ), where y ∈ R N -1 is the coordinate in the section. We make the following regularity assumption on Ω: as in [START_REF] Da Prato | An optimal control problem for a parabolic equation in non cylindrical domains[END_REF], we assume that each O x has a C 2 boundary and that there exist

C 2 diffeomorphims T x in R N -1 , T x (O a ) = O x ,
where T x denotes the flow associated with the speed field -∂ ∂x T x (y) = V (x, T x (y)), continuous with respect to x ∈ [0, a], y ∈ R N -1 and verifying T a (y) = y. We also consider Σ = ∪ 0<x<a (x, ∂O x ) to be the "lateral boundary" of the domain. Further, Γ 0 = {0} × O 0 and Γ a = {a} × O a are the "faces" of the domain.

We consider the Poisson problem

(P 0 )            -∆u = - ∂ 2 u ∂x 2 -∆ y u = f, in Ω, u |Σ = 0, - ∂u ∂x | Γ0 = 0, u |Γ a = u 1 ,
where f ∈ L 2 (Ω) and

u 1 ∈ H 1/2
00 (O a ) (see [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] for the definition of this space). This problem has a unique solution in

L 2 (0, a; H 2 (O x ) H 1 0 (O x )) H 1 (0, a; L 2 (O x )), which is the space of functions verifying a 0 u 2 H 2 (Ox) dx < ∞, u | ∂Ox = 0 and a 0 u 2 L 2 (Ox) + ∂u ∂x 2 L 2 (Ox) dx < ∞.
As in [START_REF] Henry | Factorization of second order elliptic boundary value problems by dynamic programming[END_REF], we want to factorize this problem by invariant embedding in the family of similar problems

2 (P s,h )    -∆u s = f in Ω s = ∪ 0<x<s (x, O x ), u s | Σs = 0, - ∂u s ∂x | Γ0 = 0, u s | Γs = h, (1)
where h ∈ H In Theorem 5.3 we arrive to a Riccati equation satisfied by this operator, after a change of variables.

Optimal control framework and regularization

We can formulate (P 0 ) as an optimal control problem:

(OC 0 )      ∂u ∂x = v in Ω, u| Σ = 0, u| Γa = u 1 , inf v J (v) = 1 2 Ω (|∇ y u -∇ y u d | 2 + |v| 2 )dx dy,
where u stands for the state, v for the control and u d is given almost everywhere by

   -∆ y u d (x) = f (x) in O x , u d | ∂Ox = 0.
This is an ill-posed problem, since J not defined for every v ∈ L 2 (Ω).

In order to overcome this difficulty, we make a parabolic regularization of (OC 0 ):

(OC ε )      ∂u ε ∂x + √ ε∆ y u ε = v in Ω, u ε | Σ = 0, u ε | Γa = u 1 , inf v J (v) = 1 2 Ω (|∇ y u ε -∇ y u d | 2 + |v| 2 )dx dy.
We can prove (see [START_REF] Lions | Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF]) that this control problem has a unique solution, u ε , in L 2 (0, a; H 1 0 (O x )). The adjoint state p ε is given by:

   - ∂p ε ∂x + √ ε∆ y p ε = -∆ y u ε -f , in Ω, p ε | Σ = 0, p ε | Γ0 = 0,
and v ε = -p ε is the optimality condition. The problem (OC ε ) corresponds to the regularization of (P 0 ) with a 4th-order operator in y:

(P ε )            - ∂ 2 u ε ∂x 2 -∆ y u ε + ε∆ 2 y u ε = f, in Ω, u ε | Σ = 0, - ∂u ε ∂x | Γ0 = 0, u ε|Γ a = u 1 , ( ∂u ε ∂x + √ ε∆ y u ε )| Σ = 0.

Change of variables

The derivation with respect to s of the invariant embedding (1), as done in [START_REF] Henry | Factorization of second order elliptic boundary value problems by dynamic programming[END_REF], is not obvious as P acts on a space of functions depending on s. To avoid this problem, we are going to make a change of coordinates, following [START_REF] Da Prato | An optimal control problem for a parabolic equation in non cylindrical domains[END_REF]. Using the flow T x , the quasi-cylindrical domain Ω can be mapped isomorphically from the cylinder Q =]0, a[×O a , by the change of variables (x, Y ) → (x, y) = (x, T x (Y )), Y ∈ O a . Let DT x be the jacobian of the transformation T x and J

x = det(DT x ). Set A(x)z = -J -1 x ∇ Y .(J x (DT x ) -T (DT x ) -1 ∇ Y z) and B(x)z = -(DT x ) -T ∇ Y z.V • T x , where z(x, Y ) = u(x, T x (Y )) = u • T x (Y ).
Being in the framework of [START_REF] Da Prato | An optimal control problem for a parabolic equation in non cylindrical domains[END_REF], by the change of variables the regularized problem (OC ε ) becomes

           ∂z ε ∂x - √ εA(x)z ε + B(x)z ε = v • T x in Q, z ε | Σ = 0, z ε | Γa = u 1 , inf v J (v) = 1 2 Q ((DT -1 x (∇ Y z ε -∇ Y z d )) 2 + |v • T x | 2 )J x dx dY , (2)
with z d = u d • T x and Σ =]0, a[×∂O a . Now the adjoint state is given by

   - ∂q ε ∂x - √ εA (x)q ε + B (x)q ε = A(z ε -z d ), in Q, q ε | Γ0 = 0, q ε | Σ = 0,
where Γ0 = {0} × O a and the optimality condition becomes v • T x = -q ε . The invariant embedding with respect to the subdomains

Q s =]0, s[×O a furnishes a linear operator Pε (s) ∈ L(L 2 (O a ), L 2 (O a )),
such that the traces on {s} × O a satisfy -q ε = Pε

z ε + r ε , so that z ε verifies ∂z ε ∂x - √ εA(x)z ε + B(x)z ε = Pε (x)z ε + r ε .
Then we can prove, as in [START_REF] Da Prato | An optimal control problem for a parabolic equation in non cylindrical domains[END_REF], that Pε satisfies the well posed Riccati equation in the cylinder Q, in the following sense

( d Pε dx ϕ, ψ) Jx +( √ εA(x)ϕ, Pε (x)ψ) Jx + ( √ ε Pε (x)ϕ, A(x)ψ) Jx -(B(x)ϕ, Pε (x)ψ) Jx -( Pε (x)ϕ, B(x)ψ) Jx + ( Pε (x)ϕ, Pε (x)ψ) Jx = (A(x)ϕ, ψ) Jx , ∀ϕ, ψ ∈ H 1 0 (O a ),
with Pε (0) = 0.

Convergence results

We can prove the following theorem: Theorem 5.1 As ε -→ 0, for x = s and a fixed z ε

(s) = h ∈ H 1 0 (O a ), we have z ε → z, in H 1 (Q s ), and √ εAz ε → 0, in L 2 (Q s ).
Then, using (2), we can prove that

1 / 2 00

 12 (O s ), Γ s = {s} × O s and s ∈]0, a[. By linearity of (P 0 ), for every s ∈]0, a[ we define the Dirichlet to Neumann map through P (s)h + w = ∂u ∂x | Γs .

Theorem 5 . 2

 52 As ε -→ 0, ( ∂z ε ∂x + B(x)z ε )| x=s → ( ∂z ∂x + B(x)z)| x=s in L 2 (O a ). This implies that, for h ∈ H 1 0 (O a ), Pε h -→ P h, in L 2 (O a ). As a consequence, we obtain the desired result: Theorem 5.3 The Dirichlet-Neumann operator P satisfies ( d P dx ϕ, ψ) Jx -(B(x)ϕ, P (x)ψ) Jx -( P (x)ϕ, B(x)ψ) Jx + +( P (x)ϕ, P (x)ψ) Jx = (A(x)ϕ, ψ) Jx , ∀ϕ, ψ ∈ H 1 0 (O a ).

( 3 )

 3 Problem (P 0 ) is now equivalent to the factorized formulation, with P given by (3),dr dx -B r + P r = -Az d , r(0) = 0, dz dx + B(x)z -P z = r, z(a) = u 1 , in Q.

Acknowledgements

This work was partially supported by Financiamento Base 2010 ISFL-1-297 and project PTDC/MAT/ 109973/2009 from FCT/MCTES/PT.


Gaëlle Ferré
email: gaelle.ferre@univ-nantes.fr

MULTIMODAL ANALYSIS OF DISCOURSE MARKERS 'DONC', 'ALORS' AND 'EN FAIT' IN CONVERSATIONAL FRENCH

Keywords: Discourse markers, spoken French, prosody, gesture

This paper, which is based on a multimodal corpus of conversational French, aims at describing some prosodic and gestural characteristics of the three discourse markers 'donc' ('so'), 'alors' ('so' or 'then') and 'en fait' ('in fact'). The pragmatic functions of 'donc' and 'alors' have been deeply analyzed in a few studies , Vlemings [12] and Zénone [13] and [14]) but none of them described the phonetic characteristics of the markers. The present paper also proposes the association of 'en fait' with 'donc' and 'alors' since it is met in similar contexts in discourse.

INTRODUCTION

In the field of pragmatics, Mosegaard-Hansen [START_REF] Mosegaard-Hansen | Alors and donc in spoken French: A reanalysis[END_REF], Vlemings [START_REF] Vlemings | The discourse use of French donc in imperative sentences[END_REF] and Zénone [START_REF] Zénone | Marqueurs de consécution : le cas de donc[END_REF] and [START_REF] Zénone | La consécution sans contradiction : donc, par conséquent, alors, ainsi, aussi (première partie)[END_REF], among others, have described the functions of French discourse markers (DMs. For a definition of Discourse Marker, see [START_REF] Fraser | What are discourse markers?[END_REF] and [START_REF] Schiffrin | Discourse Markers[END_REF]) 'donc' ('so') and/or 'alors' ('so' or 'then') in context. English DM 'so' has also been the object of some works by Bolden [START_REF] Bolden | Little Words That Matter: Discourse Markers ''So'' and ''Oh'' and the Doing of Other-Attentiveness in Social Interaction[END_REF] and [START_REF] Bolden | Implementing incipient actions: The discourse marker 'so' in English conversation[END_REF], and in some respect Local [START_REF] Local | Phonetic Detail and the Organisation of Talk-in-Interaction[END_REF]. Looking at the data, we decided to add 'en fait' ('in fact') to the analysis since the DM is met in very similar contexts, although the three markers do not have the same meaning and can be considered as having a prototypical function.

Apart from Local's study [START_REF] Local | Phonetic Detail and the Organisation of Talk-in-Interaction[END_REF] which served as a methodological framework for the present paper, none of the other studies presented above described any phonetic characteristic of the DMs.

To make up for this lack of description, some prosodic and gestural characteristics of the DMs will be presented according to 5 of their possible functions in discourse (presented in section 2) which are productive enough to allow statistical treatment of the data. This paper is part of a larger study which could not be published in its whole here for lack of space.

PRAGMATIC FUNCTIONS OF DMs

Three of the functions that are described in the pragmatics literature mentioned in the introduction are the following:

• Conclusion (Concl): the topic is coming to an end. A function described by Local [7] for 'stand-alone so', except that in French the marker is most of the time followed by 'euh' ('uh') in this function that may be assumed by either 'donc' or 'alors' to a lesser degree.

Ex: ça fait référence à un truc (0.790 1 ) donc euh (2.64) ['it refers to something so uh']

•

Inference (Inf): what is said can be inferred from preceding discourse. A function that may be assumed by the 3 DMs.

Ex: y avait une espèce d'écran géant, donc un matériel d'enfer ['there was a sort of huge screen so (it was) incredible equipment']

•

Resumption (Res): marker used to resume the topic after an interruption or a parenthesis. A function that may be assumed by the 3 DMs. In our own pragmatic analysis of the corpus, we thought that two other functions should be added:

• Auto-correction (AC): the speaker changes a word he/she has just mentioned or starts anew after abandoning a previous syntactic construction. A function that may be assumed by the 3 DMs.

Ex: et ils sont là relativement souvent parce qu'ils ont un en fait c'est une pépinière ['and they are here quite often since they have a in fact it's a tree nursery']

•

Punctuation (P): used at the end of a Turn Contructional Unit (TCU: an utterance that is complete from the syntactic, prosodic and semantic point of view) as a punctuation mark, much as French DM 'quoi'. This function is mostly always assumed by 'en fait'.

Ex: y a que des tout petits patelins où y a rien quoi en fait, où y a un bistrot qui fait tout ['there are only very small villages without a single shop really in fact, with only one pub where you can buy everything']

As mentioned in the introduction, the markers do not have exactly the same meaning although they may be used with similar functions in discourse. Briefly put, the core meaning of 'donc' is one of consequence, whereas 'alors' is a temporal marker and 'en fait' introduces some unexpected discourse item (or a discourse item presented as such by the speaker) and may be understood as 'contrary to expectation'. The meaning of the markers may however be modified by the presence of other adjacent DMs in clusters, but also by the use of the marker in different contexts, so that they become more or less multifunctional: 'alors' is the most multifunctional DM, whereas 'en fait' is the least multifunctional of the three markers, 'donc' being intermediary. This has an impact on the phonetic realization of the DMs and on the presence of hand gestures.

In terms of pronunciation, we found that the most multifunctional markers show a much wider range of pronunciations. For instance, 'donc' is pronounced on a continuum ranging from total absence of nasalization (and plosive elision) to a spreading of nasalization over the two plosives (and insertion of epenthetic [ŋ] in between the nasal vowel and the final plosive which may undergo a voicing assimilation too). On the contrary, 'en fait', the least multifunctional marker, is pronounced with less variety.

We also noticed that the presence of co-verbal hand gestures is much higher with a marker like 'alors' than it is with 'en fait', and we believe that this is also due to the multifunctionality of DM 'alors'.

CORPUS AND ANNOTATIONS

This paper is based on a three-hour multimodal corpus of non-elicited conversational French between 6 speakers (CID Corpus described in [START_REF] Bertrand | Le CID -Corpus of Interactional Data -Annotation et Exploitation Multimodale de Parole Conversationnelle[END_REF]) 2 . The dialogues were transcribed and annotations were made in prosody using Praat [START_REF] Boersma | Praat: doing phonetics by computer (Version 5.1.05[END_REF] and in gesture using Elan [START_REF] Elan | Elan Linguistic Annotator[END_REF]. These annotations are described in Table 1 below. Once again, many other parameters could have been taken into account for this study which does not mean to be exhaustive. Prosodic parameters were based on perception, including degree of stress. Primary stress falls on a syllable that is fully pronounced and is realized with a marked peak in intensity and/or F0, whereas secondary stress falls on a syllable that is fully pronounced but not uttered with any marked peak in intensity and F0. No stress has been assigned to a syllable that is reduced phonemically and doesn't have any marked F0/intensity peak. The Intonational Phrase (IP) can be described as a superior prosodic constituent (Selkirk [11]), with an F0 contour that does not show any break, and which includes a major prosodic prominence called "nuclear stress". The corpus yielded a total number of 718 occurrences of the 3 DMs, among which there were 124 'alors', 248 'en fait' and 346 'donc'. And the number of occurrences were the following in each of the 5 functions under study here:

RESULTS

For each function assumed by the markers, two types of statistics were done on the data to draw out its main characteristics: a proportion and an Anova test.

Auto-correction

When the DMs have a function of auto-correction, they have the following prosodic characteristics:

• They carry a primary stress (Prop.Test: Xsquared = 11.2, df = 1, p-value < 0.01),

  Ex: et donc je cherchais c'était dans les urgences vasculaires et tout ça (1.04) et donc je voyais des panneaux qui m'indiquaient un truc ['and so I was looking for, I was in the Emergency Stroke Services and stuff, and so I could see something on signboards']

Table 1 :

 1 Prosodic and gestural annotations.

	Prosody	Gesture
	Degree of stress (0, 1, 2)	Presence	of	co-verbal
	F0 contour (Flat, Falling,	gestures 3
	Rising) on the whole DM	Gesture onset
	Pitch range (high, mid, low) Head movements
	Voice quality	Gaze
	Pause (before, after)
	Position of DM in IP
	Position of DM in TCU
	Marker duration

Table 2 :

 2 Number of DMs for each pragmatic function.

	Function	Donc	Alors	En fait	Total
	Conc	58	2	0	60
	AC	18	12	21	51
	P	1	0	74	75
	Inf	115	46	3	164
	Res	141	21	27	189
	TOTAL	333	81	125	539

Silent pause in seconds.

Supported by the French National Research Agency (Project number: ANR BLAN08-2_349062), the OTIM project is referenced on the following webpage: http://aune.lpl.univ-aix.fr/~otim/.

Gestures were annotated according to McNeill[START_REF] Mcneill | Hand and Mind: What Gestures Reveal about Thought[END_REF]; their typology and the annotation process have been fully described in Bertrand et al.[START_REF] Bertrand | Le CID -Corpus of Interactional Data -Annotation et Exploitation Multimodale de Parole Conversationnelle[END_REF].

• Their F0 contour is rising (Prop.Test: Xsquared = 7.6, df = 1, p-value < 0.01),

• They come in initial position in the IP (Prop.Test: X-squared = 9.1, df = 1, p-value < 0.01), and in medial position in the TCU (Prop.Test: X-squared = 22.9, df = 1, p-value < 0.01),

• When the DM is 'en fait', it is shorter than its average duration (One-way F-Test: X-squared = 1.5, df = 31, p-value < 0.05).

The gestural characteristics of the DMs are:

Hand gestures (when present) are initiated during the production of the DM (Prop.Test: Xsquared = 6.3, df = 1, p-value < 0.05).

Inference

When the DMs have a function of inference, they have the following prosodic characteristics:

• They either carry a secondary stress (Prop.Test: X-squared = 4.8, df = 1, p-value < 0.05) or are unstressed (Prop.Test, X-squared = 4.9, df = 1, p-value < 0.05),

• Their F0 contour is flat (Prop.Test: Xsquared = 17.5, df = 1, p-value < 0.01),

• They come in initial position in the IP (Prop.Test: X-squared = 37.1, df = 1, p-value < 0.01) but not necessarily in the TCU.

The gestural characteristics of the DMs are:

•

Gaze is turned towards the other participant (Prop.Test: X-squared = 13.7, df = 1, pvalue < 0.01),

• They are accompanied with 'emblems' (as defined by McNeill [START_REF] Mcneill | Hand and Mind: What Gestures Reveal about Thought[END_REF]) -speakers count the two parts of the inference on their fingers -showing that two discourse units are linked to each other (Prop.Test: X-squared = 4.6, df = 1, p-value < 0.05).

Topic resumption

When the DMs have a function of topic resumption, they have the following prosodic characteristics:

• They come in medial position in the IP (Prop.Test: X-squared = 4.4, df = 1, p-value < 0.05) but not necessarily in the TCU. This is due to the fact that many DMs playing this function are preceded and followed by other DMs like 'et' ('and') or 'euh' ('uh'),

•

The DMs that aren't preceded by another DM are preceded by a silent pause (Prop.Test: Xsquared = 17.6, df = 1, p-value < 0.01),

• When the DM is 'alors', it is shorter in duration than the average duration for this marker (One-way F-Test: X-squared = 1.2, df = 20, p-Value < 0.01).

The gestural characteristics of the DMs are:

•

Gaze is turned away from the other participant (Prop.Test: X-squared = 15.4, df = 1, pvalue < 0.01) therefore hindering the other participant from taking the turn.

Conclusion

When the DMs have a function of conclusion, they have the following prosodic characteristics:

• They have a flat F0 contour (Prop.Test: Xsquared = 17.4, df = 1, p-value < 0.01),

• They are uttered in the lower range of the speaker's pitch (Prop.Test: X-squared = 28.8, df = 1, p-value < 0.01),

• They come in initial position in the IP (Prop.Test: X-squared = 25.1125, df = 1, p-value = 5.408e-07) and in the TCU (Prop.Test: X-squared = 39.8, df = 1, p-value < 0.01),

• They are preceded by a silent pause (Prop.Test: X-squared = 11.7, df = 1, p-value < 0.01) although there isn't necessarily a pause after them (typically because they are immediately followed by the filled pause 'euh'),

• They are uttered with either a creaky (Prop.Test: X-squared = 7.2, df = 1, p-value < 0.01) or a quiet voice (Prop.Test: X-squared = 7.4, df = 1, p-value < 0.01),

• They are longer than their average duration (One-way F-Test: X-squared = 1.3, df = 57, pvalue < 0.01).

The gestural characteristics of the DMs are: •

Hand gestures (when present) do not have their onset during the production of the DMs (Prop.Test: X-squared = 5.8, df = 1, p-value < 0.05),

• They are typically accompanied with adaptors (self-touching / grooming gestures) (Prop.Test: X-squared = 11.6, df = 1, p-value < 0.01). What is relevant here is that adaptors are not co-verbal gestures and are massively produced when the participant is listening to the interlocutor. Therefore, the co-occurrence of adaptors and markers playing a function of conclusion is congruent.

Punctuation

When the DMs have a function of punctuation, they have the following prosodic characteristics:

• They carry a secondary stress (Prop.Test: X-squared = 11.1, df = 1, p-value < 0.01),

• They are uttered in the lower range of the speaker's pitch (Prop.test: X-squared = 4.6, df = 1, p-value < 0.05),

• They come in final position in the IP (Prop.Test: X-squared = 86.7, df = 1, p-value < 0.01) and in the TCU (Prop.Test: X-squared = 161.1, df = 1, p-value < 0.01),

• They are uttered with a partial devoicing (Prop.Test: X-squared = 13.2, df = 1, p-value < 0.01) or a quiet voice (Prop.Test: X-squared = 9.9, df = 1, p-value < 0.01),

• They are followed by a silent pause (Prop.Test: X-squared = 35.8, df = 1, p-value < 0.01) although there is no pause before them (Prop.Test: X-squared = 26.1, df = 1, p-value < 0.01).

The gestural characteristics of the DMs are:

•

Gaze is turned towards the other participant, thus indicating that he/she can take the turn at this point (Prop.Test: X-squared = 6.5, df = 1, p-value < 0.05),

• Hand gestures (when present) are not initiated during the utterance of the DM (Prop.Test: X-squared = 6.8, df = 1, p-value < 0.01).

CONCLUSION

The three French discourse markers 'donc', 'alors' and 'en fait' have been examined in this paper in a multimodal perspective which offers a complement to existing studies in pragmatics.

We have seen that five functions that the DMs assume in discourse can be distinguished thanks to the degree of stress of the DM, its pitch range and contour, length and voice quality, but also thanks to the position of the DM in the Intonation Phrase and the Turn Constructional Unit.

At the level of gestures, we note that although head movements were annotated, no regularity could be observed that would lead to statistically significant results. Functions are rather distinguished thanks to gaze direction (away from other participant in the case of topic resumption and towards participant in the case of inference and punctuation) and mostly hand gesture onset (a gesture is not initiated during a marker that has a function of conclusion or punctuation). Gesture type is relevant only for the inference and conclusion functions (DMs produced together with emblems and adaptors).


Xavier Délen
email: xavier.delen@institutoptique.fr

Igor Martial

Julien Didierjean

Nicolas Aubry

Damien Sangla

François Balembois

Patrick Georges

34 W continuous wave Nd:YAG single crystal fiber laser emitting at 946 nm

Taking advantage of the pump beam confinement and the excellent thermal management offered by Nd:YAG single crystal fibers, we demonstrated a maximum output power of 34 W at 946 nm for an incident pump power of 86 W. A high slope efficiency of 53 % was obtained. There was no rollover observed in the efficiency curves and the maximum output power was only limited by the available pump power.

Recent developments of laser sources using the 4 F 3/2 -4 I 9/2 transition of Nd:YAG to obtain emission at 946 nm, have been mostly driven by the demand of high power blue laser sources. Frequency doubled 946 nm lasers have numerous potential applications such as biological and medical diagnostics, trace gas detection, Raman spectroscopy. UV light generation which can be obtained by frequency quadrupling is also of great interest for many applications such as ozone remote sensing or writing of fiber Bragg grating.

, the development of high brightness diode-pumped sources has lead to the design of multiwatt continuous-wave lasers. The highest cw power ever achieved at 946 nm are in the range of the few ten's of watts : 15 W in a bulk end-pumped

configuration [START_REF] Zhou | Continuous-wave, 15.2 W diode-end-pumped Nd : YAG laser operating at 946 nm[END_REF], 25 W with a thin disk laser [START_REF] Jiancun | 25-W Diode-Pumped Continuous-Wave Quasi-Three-Level Nd:YAG Thin Disk Laser[END_REF] and 105 W with a planar waveguide [START_REF] Mackenzie | Power-scaling Nd:YAG's quasi-four-level transition[END_REF]. These output power values are far below what can be obtained with Nd:YAG for 1064 nm or 1338 nm four level transitions producing over hundred watts [START_REF] Zhang | Diode-side-pumped Nd:YAG laser at 1338 nm[END_REF][START_REF] Wilhelm | Design and comparison of composite rod crystals for power scaling of diode end-pumped Nd:YAG lasers[END_REF]. Indeed, the design of a high power 946 nm Nd:YAG laser is difficult for several reasons.

It is a quasi three level transition and the emission cross section is about one order of magnitude lower than at 1064 nm. It implies working with high pump intensities in order to obtain efficient laser operation. It is therefore necessary to focus the pump beam on a small diameter. Using typical high power fiber coupled laser diodes (for example with core diameter of 400 µm and a numerical aperture of 0.22), the pump beam Rayleigh length is relatively short and the crystal length is then limited to a few millimeters. To ensure a good overlap between pump and laser beams and a large overall absorption, a high doping concentration has to be used, typically in the order of 1 %. This leads limitation of the performance at 946 nm in multiple ways. Firstly, non radiative effects as fluorescence quenching and energy transfer upconversion occcur at this doping level. These processes do not only limit the population inversion but also induce an extra heat load which contributes to the crystal temperature increase. Secondly, the high absorption coefficient leads to high local temperature increase, particularly on the pumped face of the crystal. The temperature gradients induce signal beam deformations and mechanical stress which can even lead to the crystal fracture. Moreover, crystal properties such as thermal conductivity and thermooptic coefficient also evolve in a disadvantageous way with temperature inducing higher thermal lensing [START_REF] Aggarwal | Measurement of thermo-optic properties of Y3Al5O12, Lu3Al5O12, YAIO(3), LiYF4, LiLuF4[END_REF][START_REF] Zagumennyi | The Thermal Conductivity of YAG and YSAG Laser Crystals[END_REF].

Finally, the population of the lower level increases with temperature and induces more reabsorption losses and shifts the oscillation threshold.

To keep both high pump intensity and low temperature increase different pump geometries have been used.

Bonding of undoped caps to a laser crystal is one way to limit thermal effects caused by the deformation of the crystal faces. This technique was tested in a longitudinally-pumped, 1.1-at.% Nd:YAG composite crystal, but significant degradation of laser performance at 946 nm was observed for pump powers in excess of 30 W (M 2 =13 at maximum pump power) [START_REF] Zhou | Continuous-wave, 15.2 W diode-end-pumped Nd : YAG laser operating at 946 nm[END_REF]. The thin disk configuration represents a more efficient way to limit the temperature while allowing high pump intensity via multi-pass pumping. It has been used with the quasi three level transition in Nd:YAG but the low emission cross section requires to work with small pump beam diameters to have enough gain. It results in thermal effects which limit the power scaling [START_REF] Jiancun | 25-W Diode-Pumped Continuous-Wave Quasi-Three-Level Nd:YAG Thin Disk Laser[END_REF][START_REF] Pavel | Multipass pumped Nd-based thin-disk lasers: continuous-wave laser operation at 1.06 and 0.9 mu m with intracavity frequency doubling[END_REF]. Another way is optical guiding: pump confinement allows the use of Nd:YAG crystals with lower doping concentration and to reduce thermal effects.

Pump guiding has been successfully used in a double clad 20 mm long planar waveguide with a 0.6% Nd 3+ doping concentration [START_REF] Mackenzie | Power-scaling Nd:YAG's quasi-four-level transition[END_REF]. Even if the output power was the highest with 105 W, the beam quality was very limited in the non-guided direction. For a 35 W output power, the M² beam quality factor was given to be respectively 2.8 and 56 in the guided and non-guided direction [START_REF] Mackenzie | An efficient high-power 946 nm Nd:YAG planar waveguide laser[END_REF].

Nd:YAG single crystal fibers use the same concept of pump guiding but with lower doping concentration (0.2 %) and longer gain medium (50 mm) [START_REF] Didierjean | High-power laser with Nd : YAG single-crystal fiber grown by the micro-pullingdown technique[END_REF]. As opposed to the planar waveguide, the signal beam is in free propagation in the single crystal fiber, and its profile is defined by a cavity with two external mirrors for mode filtering. The concept of single crystal fiber has already been used successfully for a quasi-three level laser: the Yb:YAG with an output power of 50 W at 1030 nm [START_REF] Sangla | High power laser operation with crystal fibers[END_REF]. Consequently, this laser crystal geometry seems to have a good potential for the development of high power quasi-three level Nd:YAG lasers at 946 nm.

In this paper, we report, for the first time to our knowledge, laser operation at 946 nm with a Nd:YAG single crystal fiber. By Finite Element Analysis (FEA), we firstly estimate the potential of this geometry compared to standard bulk crystals in terms of temperature. Then we present experimental results : temperature measurement by infrared imaging and performance of the Nd:YAG single crystal fiber at 946 nm.

Compared to a classical Nd:YAG crystal, single crystal fibers have two specificities : the low doping concentration (0.2 %) associated with a long length (50 mm) and the small section (1mm diameter). In order to understand the interest of this geometry, we carried out a comparison with a set of FEA temperature modelization for different Nd:YAG bulk crystals (1% or 0.2% doping rate with a square 3x3 mm section) and for a Nd:YAG single crystal fiber (0.2 % doping rate and 1 mm diameter). For the simulation, no attention was paid on the crystal lengths as we calculated the temperature of the pumped face assuming a radial heat flow. In order to simplify the calculations, we supposed also a perfect contact between the Nd:YAG and its mount, no saturation of absorption and a heat source coming only from the quantum defect between 808 nm and 946 nm (14.6%). The pump power was set at 86 W over a spot of 500 µm diameter: those conditions corresponded to the experimental setup described in the following. Figure 1 shows the temperature rise maps which were obtained as the results of the simulations.

Lowering the doping concentration from 1% to 0.2% distributes the pumping heat load over a longer length and induces a significant decrease of the maximal temperature rises by over three times (from 194°C to 51°C). The small diameter of the single crystal fiber also induces a temperature reduction by over a factor of two (from 51°C to 19°C) by bringing the heat sink closer to the heat source. Consequently, the single crystal fiber geometry could be very useful to reduce the temperature increase and the associated gradient. However, the reduction of the transverse section led also to the reduction of the exchange surface with the mount. Consequently, the heat transfer coefficient between the mount and the single crystal fiber need to be as high as possible in order to take all the benefit of this geometry. This key point was experimentally investigated on a 0.2% Nd:YAG single crystal fiber with a diameter of 1 mm integrated in a water cooled TARANIS module provided by Fibercryst. The thermal contact between the module and the heat sink is ensured by a carbon foil. The water cooling temperature was 25°C during the experiment. The measurements were carried out on the pumped face with a thermal camera and a ZnSe beam splitter, in a setup similar to the one used by Chénais et al. [START_REF] Chénais | Direct and absolute temperature mapping and heat transfer measurements in diode-end-pumped Yb:YAG[END_REF]. Figure 2 shows the temperature increase map of the crystal for a pump power of 86 W. The temperature difference between the edge of the single crystal fiber and the crystal mount was found to be about 5°C. This very low difference shows the high quality of the thermal contact between the single crystal fiber and the cooling mount (copper block). It reveals also a maximal temperature increase of 31°C. This value was higher than the one calculated by FEA analysis. It is due to the higher quantum defect between the pump photons and fluorescence photons which is centered around 1134 nm whereas the laser emits at 946 nm. However, the temperature raise was kept reasonably small even at this high pumping power of 86 W predicting good performance for 946 nm laser operation. As shown on figure 3, a simple linear cavity made of two concave mirrors was used for the experiment. The dichroïc meniscus M 1 had a radius of curvature of 100 mm, was highly reflective at 946 nm and had a low reflectivity at both 808 nm and 1064 nm in order to prevent parasitic oscillation. For the output coupler, we used concave mirrors with a radius of curvature of 100 mm, a high transmission at 1064 nm (T>80%) and transmission of 2%, 5%, 10% and 15% at 946 nm. The facets of the single crystal fiber were anti-reflection coated for a broadband spectral range between 600 nm and 1100 nm. The Nd:YAG crystal was 0.2% doped, 50 mm long and had a diameter of 1 mm. The water cooling temperature was set at 14°C during the experiment. The pump beam was delivered by a laser diode module coupled in a 400 µm core diameter fiber with a numerical aperture of 0.22.

The output of the fiber was imaged inside the single crystal fiber using two doublets of focal length 60 mm and 80 mm. The maximum nominal output power of the diode was 110 W and the maximum incident pump power on the crystal was 86 W. The lasing wavelength was monitored with an optical spectrum analyzer and only laser emission at 946 nm has been observed. The output power versus the incident pump power is shown on figure 4 for the different transmissions of the output coupler. The threshold was found for an incident pump power between 15 W and 20 W. The maximal slope efficiency was obtained for an output coupler with a 10% transmission and reached 53%. The maximum output power was 34 W for an incident pump power of 86 W which gives an optical conversion efficiency of 39.5%. At this power level, the M 2 beam quality factor was below 5 in both directions. In addition, no rollover was observed in the efficiency curves showing the limited impact of the temperature rise and thermal lensing provided by the 1 mm diameter single crystal fiber together a very efficient thermal management.

In conclusion, we have demonstrated that high power Nd:YAG 946 nm laser design is possible using single crystal fiber configuration. It offers good thermal management and provides pump energy confinement. A total output power of 34 W was obtained with high slope efficiency of 53% and with a relatively good beam quality. To the best of our knowledge, it is the highest output power ever reported for an end-pumped Nd:YAG laser emitting at 946 nm with a signal beam in free propagation in the laser cavity. It is also the highest brightness for a 946 nm laser source at this level of power.

Further power scaling can be considered since no roll over has been observed in the efficiency curves. The total output power was only limited by the available pump power. In particular, the full potential of the single crystal fibers could be used in a dual end pump configuration.

Xavier Délen acknowledges the DGA (French Ministry of Defense) for the partial funding of his PhD and Fibercryst for the loan of the TARANIS-Nd module.

Fig. 1 .

 1 Fig. 1. Temperature rise maps calculated using finite elements analysis.

Fig. 2 .

 2 Fig. 2. Temperature increase map of the input face of a 0.2% doped Nd:YAG single crystal fiber under 86 W diode pumping over 500 µm at 808 nm.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Experimental setup.

Fig. 1 .

 1 Fig. 1. Temperature map calculated using finite elements analysis.

Fig. 2 .

 2 Fig. 2. Temperature increase map of the input face of a 0.2% doped Nd:YAG single crystal fiber under 86 W diode pumping over 500 µm at 808 nm.

Fig. 3 .

 3 Fig. 3. Experimental setup.

Fig. 4 .

 4 Fig. 4. Laser output power versus incident pump power for different output couplers.