Serge Dumont
email: serge.dumont@u-picardie.fr

Eric Lebon
email: lebon@lmgc.univ-montp2.fr

WAVELET-GALERKIN METHOD FOR PLANE ELASTOSTATICS

Keywords: Elastostatics, wavelet transform, llctitious domains, augmented Lagrangian. 1

This paper is devoted to a new algorithm to solve elastostatics problems on any shape domain. Our approach is based on a wavelet-Galerkin method. Due to the form of the wavelet bases, the general domain is included in a square Ectitious domain, and the initial problem is extended over this square. The boundary conditions are imposed by an augmented Lagrangian method. Numerical examples are presented. Comparisons are done with analytic solutions and with Enite element computations.

INTRODUCTION

Solving hard problems in computational structure mechanics, requires conceiving accurate, robust and high performance algorithms. Classically, this kind of problem is solved by finite element methods, but, it may be very difficult to generate the grid, for example when the geometry of the solid is complex or in the study of localized phenomena. Moreover, the localization of high gradient zones is not easy, and usually needs stress tensor computation.

We propose a~ original alternative method, based on wavelet transform. The advantages of this wavelet-Galerkin method are, in particular: localization, adaptativity and the fact that there is no creation of a grid. This kind of method has already been used by the authors to solve periodic homogenization problems and permits us studying very inhomogeneous materials [START_REF] Dumont | Wavelet-Galerkin method for heterogeneous media[END_REF], [START_REF] Dumont | Homogeneisation periodique et Elasticite[END_REF].

On the other hand, there is no fast wavelet transform on general domains. We have therefore been interested in a fictitious domain method consisting of including the general domain in a larger square or rectangle.

This method is already used with the finite element method or with the finite difference method when the shape of the domain is too complex.

The paper is organized as follows: in section 2, we introduce the method that is used to solve a bidimensional elastostatics problem on any shape domain with the wavelet transform, and we give mathematical results. Section 3 is a brief summary of wavelets' theory and of the wavelet-Galerkin method. Section 4 is devoted to the boundary conditions and the way we take them into consideration. In section 5, we give the numerical algorithms that we use to solve the linear system. Finally, section 6 contains some numerical examples whose validity is shown by comparing the results with analytical solutions and with solutions given in the literature.

MATHEMATICAL FORMULATION OF THE PROBLEM

Some Functional Spaces

In this first section, we define some functional spaces which will be used in the following.

We consider an open bounded domain w in JR. 2 • We suppose that the boundary Bw of w can be written as aw = awD u awF where IBwDI > 0 (IBwvl = faw ds).

D

The displacements on w are in the space, [START_REF] Attouch | Viscosity solutions of minimization problems[END_REF] and the associated contraints are in the space,

L = ( L~ym(w)) 4 = { a;j, i, j = 1, 2, 3 : a;j E £ 2 (w), a;j = aji a.e.} (2)
We will also consider the square domain n = (0, 1) x (0, 1), and the functional spaces,

v = (H~(n)f = {v E L;(n) with the scalar product, 2 '"" 8u; 8u; ] d + L..J----X. k= 1

QXk QXk

• On the parts aw D and aw F of the boundary aw' we define the trace:

(4) ( 1/2 ) 2 ( 1/2 ) 2 'Yo : V >---t H 00 (8w 0 ) 'YF : V >---t H 00 (8wF) . (5)
• On awD, let us define,

( 1/2( )2 M1 = Hoo awD)
and its dual:

for the duality product: (w,J.ti), = fawD WJ.li ds.

M 1 has the scalar product,

(u,v) 8 w = { uv ds, D law D and the norm: llwiiM, = inf{llvllv : v E H~(!1) "'f 0 (v) = w} . • On awF, let us define, .M2 = (H~6 2 (8wF)/
and its dual:

for the duality product: (~t2, g)

2 = faw F 1t2Y ds. (6) (7)
The scalar product on M 2 , and the norms on M 2 and M~ are defined as those on M 1 and M~.

The Elasticity Problem

In this part, we consider a bidimensional elastic material with rigid tensor IK, occupying the open bounded domain w in JR 2 • We want to find the displacement u solution of the equilibrium problem:

Problem 1. j, u 0 , and g be given, find u and u such that:

div u + f = 0 in w u= uo u.n = g u = IKe( u) On QWD On QWF (behaviour law) (8)
where n is the unit vector of the outward normal to 8w, e(u) is the deformation tensor associated to u, e(u) = grad,(u) i.e. e;j(u) = !(ui,j + Uj,i), i,j = 1 or 2.

Classically, iff E (L 2 (w)r, uo E M 1 , gEM~ and l8w 0 1 > 0, this elasticity problem has an unique solution (u, u) with u E L, u E H [START_REF] Fichera | Existence theorems in elasticity[END_REF].

The Fictitious Domains

We want to solve this problem using the wavelet transform. Due to the form of the wavelets, it is necessary to introduce an associated method. Then, we include the domain w in a larger square domain n, and we extend the problem (Pw) on 0: Problem 2. j, u 0 , and g be given, find u and iT such that:

(Pn) div iJ + j = 0 ii=uo a.n=g iJ=lKe(ii)
u n -periodic where j is an extension of f on n. Remark. The condition ii 0-periodic is not necessary but in the following only periodic wavelets are used.

From a given function f, the computation of the periodic extention f off may be done by different ways according to the properties of f and w (by periodizing f or extending f by a constant, etc.). n

Fig.1: The fictitious domains

If i E ( L 2 (f2) r, Uo E Ml, g E M!z and J8wD I > 0, this problem has an unique solution (iJ,u) (with u E V) equal to the solution (a,u) of (Pw) on w [START_REF] Khalfaoui | Contact unilateral et ondelettes: formulation du probleme[END_REF], [START_REF] Fichera | Existence theorems in elasticity[END_REF]. This solution can be written as a problem of minimization under constraint: Problem 3. Let J, uo and g be given. Find u such that,

(Pm) u E Argmin ¢ { u = u 0 a.n=g (10)
where [START_REF] Dumont | Elasticite et ondelettes : Formulations mathematiques et mise en oeuvre numerique, 2nd Colloque de Calcul des structures[END_REF] 2 Jn ln 2 Jn ln is the potential energy of the system.

tj>(v) = ~ r u(v): e(v) dx-r iv dx = ~ r JKe(v) : e(v) dx-r iv dx
Ifu 0 E (L 2 (8wv)) 2 , g E (L 2 (8wF)) 2 , j E (£ 2 (!1)) 2
, this problem has an unique solution (u, u) with u E H [START_REF] Dumont | Homogeneisation periodique et Elasticite[END_REF], [START_REF] Khalfaoui | Contact unilateral et ondelettes: formulation du probleme[END_REF]. •

The Augmented Lagrangian

In order to solve this problem, we have chosen to use the augmented Lagrangian method.

The interests of this method are that the displacement solution of problem (Pr) is exact for all r (contrary to a penalty method for example), and the term depending on r acts as a preconditioner of a simply Lagrangian method. This augmented Lagrangian method consists in adding a new variable A; for each constraint i, called the Lagrangian multiplier. With those variables, we build a function called Lagrangian such that the solution is a saddle point of the Lagrangian (minimum for the displacement, maximum for the multipliers).

More precisely, we define the augmented Lagrangian of the problem of minimization under constraint (Pm) by,

Lr(v, Jtl, /t2) = tf>(v) + (Jtl, v-uo) 1 + (Jt2, u( v).n-g) 2 + ~llv-uoll;,, (12)
and we have the theorem.

Theorem 1. [START_REF] Brezzi | On the existence, uniqueness and approximation of saddle point problems arising from Lagmnge multipliers[END_REF] The point (u, A1, .A2) in V x M; x M2 solution of {Pm) is a saddle point of Lr. In other words, (u, .A1, .A2) is the unique solution of the problem:

(

Remark. The augmented Lagrangian is a partial augmented Lagrangian, only the first constraint is concerned.

The Variational Problem

To find the saddle point solution of problem (Pr ), we take the derivative of the augmented Lagrangian equal to zero. In other words, we have,

for all (v, Jtl, Jt2) in V x M; x M2. ( 14
)
Then, we have to solve the following linear problem.

Problem 4. Let j, U 0 , and g be given. Find (u, AI, A 2 ) in V X M1 X M2 such that, with

(P/v) { ar(u, v) + bi(v, AI)+ b2(v, A2) = h(v) bi ( u, JJI) = ei (JJI) b2( u, /J2) = e2 (JJ2) \fv E V VJJI E M1 "'JJ2 E M2 { ar(u, v) =In <7(u): e(v) dx + r(u, v) 8 ..,D; h(v) =In fv dx + r(uo, v) 8 ..,v bi(v,JJI) = (v,JJI) 1 ;ei(JJI) = (uo,JJI) 1 b2(v,JJ2) = (JJ2,<7(v).n).;e2(JJ2) = (iJ2,9)
Using the LBB theorem [START_REF] Babuska | The finite element method with Lagmngian multipliers[END_REF], [START_REF] Brezzi | On the existence, uniqueness and approximation of saddle point problems arising from Lagmnge multipliers[END_REF], [START_REF] Dumont | Elasticite et ondelettes : Formulations mathematiques et mise en oeuvre numerique, 2nd Colloque de Calcul des structures[END_REF], [START_REF] Khalfaoui | Contact unilateral et ondelettes: formulation du probleme[END_REF], it is easy to show the existence and unicity of the triplet (u,AI,A 2 ) in V X M1 X M2.

The problem (P/v) is similar to the classical mixed formulations as presented by Babuska in [START_REF] Babuska | The finite element method with Lagmngian multipliers[END_REF].

THE WAVELET-GALERKIN METHOD 3.1. One Dimensional Daubechies Wavelets

In this section, we present a brief summary of one dimensional Daubechies wavelets [START_REF] Daubecides | Orthonormal bases of compactly supported wavelets[END_REF].

Let M be a positive integer. There exists a sequence of real numbers {h(n)}.~o, ... ,•M-1>

a function <p called the scaled function and a function 1/J, orthogonal to the first one called the associated wavelet, such that,

2M-I <p(x) = v'2 L h(n) <p(2x-n) Vx E JR, ( 17
) n=O 2M-I 1/J(x) = v'2 L g(n) <p(2x-n) Vx E JR, ( 18
) n=O where g(n) = ( -1)n h(2M-1-n), n = 0, ... , 2M-1.
The support of <p and 1/J are included in the closed and compact interval [0, 2M-1].

Let

\fx E iR, \fj,k E Z (19)
The functions ' P,. and 1/J,. verify,

We define Vj the closure of the space generated by {'P;•' k E Z}. For all j E Z, this subspace Vj verifies the following conditions,

n Vj = {0} L2(1R.) = u Vj. ( 21
)
jEZ jEZ

The set of spaces {Vj, j E Z} is called a multiresolution analysis of L 2 (1R.).

Periodic Wavelets

Due to the form of the problem, it is necessary to define periodic wavelets and a multiresolution analysis of (£~(!1)) 2

• First, we define the periodic scaled function ¢;• by, ¢;.(x)=2il 2 L'f'(2i(x+n)-k) j ::::::

0 j k = 0, ... , 2j -1, (22)
nEZ and the same for the periodic associated wavelet 1/J. Then, we can define the periodic scale function on JR. 2 by,

The closure spaces l'iP generated by { <P1 • ; k,, k 2 = 0, ... , 2i -1} is a multiresolution where dis the direction (d = 1, 2). The dimension of a space V; is equal to 2 2 i+ 1 •

Remark. If we denote by {W;,j 2::: 0} the subspace of (£~(0))

Elastostatics Discretization

In this section, we discretize the problem (Pfv) using a Galer kin method written in the wavelet basis defined above. We first need to compute the projection into a space Vj of the plane elasticity operator,

a(u,v) = LJK e(u): e(v) dx forall u,vEV (31)
Then we obtain the proposition below.

Proposition 1. The projection of the plane elasticity operotor into the wavelet basis with

scale equal to j is given by a matrix lC of order 2 2 i+l defined by [IC(i,,i,)(e,,e,)] = [/Ci,.di,!EA;

where /Ci,! is an "elementary matrix" of order 2, where with K = (p,q)

.6. = (a,/3) J = (j., j,)

For the proof, see [START_REF] Dumont | Representation of plane elastostatics operators in Daubechies wavelets[END_REF].

[ F" F,

IC• _ u + 33 !,! - 21 12
F,, + F 33 (36) if f3 = 2 then (m., m 2 ) = (0, 1), tl.

1 = (numJ; ~ = (n,m 2 ), and f(n) is the derivative of order n. Thus, we have to compute the scalar product of two wavelets and their derivatives. The algorithm to compute these terms is based on theorems given by G. Beylkin [START_REF] Beylkin | On the Representation of Opemtors in Bases of Compactly Supported Wavelets[END_REF], [START_REF] Beylkin | Fast wavelet Transform and Numerical Algorithms I[END_REF] and Dahmen-Micchelli [START_REF] Dahmen | Using the refinement equation for evaluating integmls of wavelets[END_REF]. These terms are computed via filter banks as eigenvectors of a low order matrix.

In the case of several materials, see [START_REF] Dumont | Representation of plane elastostatics operators in Daubechies wavelets[END_REF] for the computation of ~::.!. As before, these terms are computed via filter banks as eigenmatrices of a low order tensor.

WAVELETS AND BOUNDARY CONDITIONS

We now have to compute the function and operators defined in ( 16) and especially the boundary terms. This computation is based on the works of Wells-Zhou [START_REF] Wells | Wavelet solutions for the Dirichlet problem[END_REF]. The main idea here will be the following: We can represent the function and the measure defined on the boundary in terms of wavelet series defined in a square region and the coding is independent of the geometry of the domain w.

First, assume that w is a bounded domain in JR 2 whose boundary ow can be written as, ow= {x E IR 2 : F(x) = 0}

(37)

for some Lipschitz function F. Then, the unit normal vector n along the boundary ow can be written as,

'i!F n = IV Fl. ( 38
)
The idea is now to compute an integral defined on ow from an integral defined on IR 2 • Let assume we can extend f (defined on ow) to JR 2 , then we have,

r f ds = -r f llowll Jaw JJR.2 (39)
where llowll is the boundary measure defined by,

II ow II = -'ilx~ .n ( 40
)
(X~ is the characteristic function of w: X~ (x) = 1 if x E w, = 0 otherwise). For more details, see [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]. The functions X~ are decomposed in the wavelet basis and so it is not difficult to compute coefficients of the measure ll8wll are obtained, the computation of the terms fiR.' f Y'xw .n are similar to the computation of the terms of the stiffness matrix [START_REF] Wells | Wavelet solutions for the Dirichlet problem[END_REF].

Remark. The wavelets used in the discretizations of the measure ll8wjj, u 0 and g are the same of those used in the discretization of f and the matrix K..

NUMERICAL ALGORITHMS 5.1. The Uzawa Method

The Uzawa algorithm is an iterative method to solve a linear system like [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]. It consists of: first, finding the wavelet coefficients of the displacement associated with given mutipliers coefficients by solving a linear system; then, reactualizing the multipliers coefficients with those of the displacement; and finally, going to the first step while the residuous is greater than a chosen threshold e. In our example, this algorithm can be summarized by the following diagram:

, 0 L.
(1) 'Y E JR. ' be given;

(2) minimization on v, which leads to solve (k 2: 1):

Find uk ( E JR.N; ) such that:

(3) Actualization of the multiplier:

(4) If 1Juk+ 1 -ukll N. 2: e, go to (2).

'2 (R. J )

Remark. The parameter Pk will be constant equal to p, and we can show that if 0 < p:::; 2r/IIPJII 2 where P 2 is the projection from H into M~, the method converges to the solution [START_REF] Fortin | Methode de Lagrangien augmente -Applications a la resolution numerique de problemes aux limites[END_REF].

On each step, we solve the linear system with the conjugate gradient method, described below.

The Conjugate Gradient

The conjugate gradient method is an iterative method to solve symmetric linear system. This method can be described with the following schema:

We initiate with, { xo be chosen;

Po = ro = b -Axo ( 41)
Fork= 0, 1, ...

-l!r.ll' .

-(Ap• ,N)'

(42)

Remark. The algorithm needs 2cN operations for each step, where N is the order of the matrix A, and c the average number of not vanishing coefficients by line.

The conjugate gradient converges in at most N iterations if the matrix's order is equal

to N, and the rate of the convergence depends on the condition number of the matrix A.

From the second step onward of the Uzawa method, the conjugate gradient initialization is made with the displacement coefficients solution of the previous step.

In order to evaluate the condition number of matrix K and the convergence of the conjugate gradient method, we have chosen not to use a preconditioner (diagonal for example, as it is done in works of Dahmen [START_REF] Dahmen | Multilevel preconditioning[END_REF]).

NUMERICAL RESULTS

Boundary Integrals

We present in this section the computation of perimeters for different geometries: a circle, a diamond, and a square. These results enable us to validate the computations of the terms on the boundary given by equations (38), (39), ( 40).

The geometries are, -a circle defined by (x-0.5) 2 + (y-0.5) 2 < R;

-a diamond defined by lx-0.51 -IY-0.51 < R;

-a square defined by Sup(lx-0.51, IY-0.51) < R.

with the value of the radius ranging from R = 0.2 to R = 0.45. For the computations, the different geometries are included in a squared box with the edge equal to 1. In all the following, we will use the Daubechies wavelets of order 3 at the level j equal to 5 (2048 degrees of freedom). The results are given in table 1.

The results obtained for the circle are quite precise. We observe on the one hand that we need to choose a step sufficiently small to correctly describe the circle and on the other hand it is necessary to chose a sufficiently large box to avoid problems on the edges of the circle. The maximum of the precision is obtained for a ratio disc radius-square edge equal to 0.4.

A Numerical Example

We present the example of a circle with radius equal to 0.25 included in a squared box with edge equal to 1. The problem is given by the following equations, { -div(JKe( u)) = 0 u=an E = 72GPa, v = 0.32 (Aluminium).

We present in figures 2 to 4 the numerical results. It is well known that the Uzawa algorithm has only a linear convergence [START_REF] Fortin | Methode de Lagrangien augmente -Applications a la resolution numerique de problemes aux limites[END_REF].

The linear system is solved by a preconditioned (SSOR) conjugate gradient method. We show in the figure 3 the computed displacement with Daubechies wavelets of order 3.

CONCLUSION

We have developped an original algorithm to solve elastostatics problems: a wavelet-Galerkin method coupled with augmented Lagrangian. Numerical results show the efficiency of the method and the difficulty in optimizing the different parameters. The method seems to be less efficient near the boundary than a finite elements computation, but adaptativity inside the domain w can be obtained very easily. Other ad vantages of the wavelets are the. lack of mesh and that the structure has only to be defined by a function characteristic of the boundary. This algorithm is promising and will permit us to treat more complex problems (real structures, non linear problems, ... ).

2 2. 6 .

 26 The Unicity of the Solution[START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]

F 12 F 21 ] 3 = 1 or 2 io 6 J

 1221326 ~ j., j, ~ j (direction of the derivative); scale of each wavelet (or scale function). terms of the matrix IC, if each ll\pq is a constant function (as in the case of only one material), equation ( 33) gives us ( ditto for A-:-',•-) 1 then (n., n,) = (1, 0); if a= 2 then (n., n 2 ) = (0, 1); if f3 = 1 then (m., m 2 ) = (1, 0);

Fig. 2 :

 2 Fig.2: Section of the computed displacement u 1 (x 1 , 0.5)

in w on ow with a= 0. 1 (Fig. 3 :

 13 Fig. 3: Residuum along the Uzawa iterations

Fig. 4 :Figure 4

 44 Fig. 4: Computed displacement

Fig. 5 :

 5 Fig. 5: Section of the displacement

Table 2

 2 Number of iterations and error (r = 10 6 , p = 5.10\t:uzawa = 10-4 )

	Ray	Uzawa	e.G. iterations	Error iterations
	R=0.2	31	241		uo-2
	R=0.3	29	152		5.10-3
	R=0.4	20	197		4.10-3
	.....
	"-20	IZI. !50	IZI. 7:i	J... IlliZI

'ilx~ and n with the coefficients A defined in (35), then 'ilx~.n. Finally, when the wavelet

Results are less precise for the two other cases due to the nondifferentiability of the boundary. These results are similar to those obtained in [START_REF] Wells | Wavelet solutions for the Dirichlet problem[END_REF].

An Analytical Example

We present the following example,

where w is a disc with the radius ranging from R = 0.2 to R = 0.4, included in a squared box with the edge equal to 1. The solution u (in n) of this problem is, Table 2 and figure 2 show the efficiency and the precision of the method.


Pegah Alizadeh

Peggy Cellier

Thierry Charnois
email: thierry.charnois@lipn.univ-paris13.fr

Bruno Crémilleux

Albrecht Zimmermann
email: albrecht.zimmermann@unicaen.frpeggy.cellier@irisa.fr

An Experimental Approach For Information Extraction in Multi-Party Dialogue Discourse

Keywords: Information Extraction, Dialogue Texts, Topic Modeling, Term Weighting

In this paper, we address the task of information extraction for transcript of meetings. Meeting documents are not usually well structured and are lacking formatting and punctuations. In addition, the information are distributed over multiple sentences. We experimentally investigate the usefulness of numerical statistics and topic modeling methods on a real data set containing multi-part dialogue texts. Such information extraction can be used for different tasks, of which we consider two: contrasting thematically related but distinct meetings from each other, and contrasting meetings involving the same participants from those involving other. In addition to demonstrating the difference between counting and topic modeling results, we also evaluate our experiments with respect to the gold standards provided for the data set.

Introduction

Many organizations (and people) spend large amount of their professional time on (or in) meetings. When the meetings are finished, they should be analyzed w.r.t. different aspects such as preparing meeting summaries, lists of envisioned projects, decisions, problems, action points etc. Preparing an automated approach for analyzing meetings would therefore be expected to be a boon for both meeting participants and non-meeting actors, e.g. managers, auditors etc. For the sake of capturing as rich a data set as possible, meetings can be recorded and stored in audio and/or video form. Thanks to new technologies such as Speechto-Text 1 , all registered dialogue during the meetings can be transformed into textual transcripts. In general, speech recognition or natural language processing on dialogue based conversations is difficult because they consist of multi-part interactions with extreme variability. On the other hand, the transcript of meetings -output of speech recognition methods -usually includes unstructured word streams, weak punctuation, formatting or capitalizations.

In this paper we experimentally evaluate how to extract information and topics from a meeting based corpus, namely AMI [START_REF] Carletta | Unleashing the killer corpus: experiences in creating the multieverything AMI meeting corpus[END_REF][START_REF]Augmented multi-party interaction[END_REF] (See Section 3). We consider this kind of information as the first stepping stone towards summarizing a meeting. It is not obvious which approaches are well-suited to this question but the research literature on extracting representative terms is vast. We chose to evaluate representatives of two different paradigms: a counting statistic, tf-idf often used in information retrieval and event detection, and a topic modeling approach, NMF .

Term extraction from meetings is obviously not a means to itself but a stepping stone towards more complex tasks. The questions we ask are therefore:

1. Can we identify the terms specific for an individual meeting in comparison to thematically related but distinct meetings? 2. Can we identify which meetings should be grouped together, e.g. by identifying common names of participants? 3. To what degree do extracted sets of words agree with the provided goldstandard summarizations?

We contribute to answer these questions by showing that tf-idf approaches perform well on some tasks (e.g. meeting characterisations and speakers identification) within a meeting acted out by a single group speaking about the same scenario. Whereas topic modeling is better for summarization and theme extraction within a set of thematically related meetings acted out by different groups of participants. In the other words, the two approaches complement each other.

In the following section, we give an overview of the related work, and in Section 3 we describe the corpus used in the paper. We describe and define the approaches we have evaluated in Section 4. Section 5 describes the experimental evaluation and discusses the results for the three settings described above in detail. In the last section, we conclude and outline perspectives.

Background and Related Work

In this paper we are interested in spoken multi-party human meetings and our goal is to extract information such as as summary, important events, decisions taken during the meeting, identified problems, proposed solutions and decided action points, etc. We are also interested in knowing which topics where discussed in a particular meeting as well as having making the discussion of particular topics accessible.

A meeting analyzer system called CALO has been proposed in [START_REF] Tur | The calo meeting assistant system[END_REF][START_REF] Tur | The CALO meeting speech recognition and understanding system[END_REF] that first transcribes meeting speech to text automatically, and then analyses the meeting in several phases: dialogue act segmentation and tagging, topic segmentation and identification, problem and decision detection, and summarization. With respect to this paper's scope, they identify topic and segment meetings using a generative topic model derived from Latent Dirichlet Allocation (LDA) method to learn the topic model automatically [START_REF] Purver | Unsupervised topic modelling for multi-party spoken discourse[END_REF]. Using an unsupervised learning approach, they produce the meeting segmentations simultaneously to learn what (meeting topic) and when (meeting segmentation) people talk about during meetings. An alternative approach for segmenting meetings consists of tracking changes in lexical distributions: [START_REF] Galley | Discourse segmentation of multi-party conversation[END_REF] and [START_REF] Georgescul | Exploiting structural meetingspecific features for topic segmentation[END_REF] translate the lexical distributions into a discriminative classifier and apply it on meeting transcripts.

Action items and decision extraction from meetings is another concern of this paper. A structural approach for detecting items and decisions has been proposed in [START_REF] Tur | The CALO meeting speech recognition and understanding system[END_REF][START_REF] Tur | The calo meeting assistant system[END_REF]. They classify meeting utterances w.r.t their roles in the process: task definition, agreement, and acceptance of responsibility. Next, they detect actions [START_REF] Purver | Detecting and summarizing action items in multi-party dialogue[END_REF] and decisions [START_REF] Fernández | Modelling and detecting decisions in multi-party dialogue[END_REF] from the role patterns. Another approach in the literature extracts important words (related to problems) using classifiers or sequence models using a lexical approach [START_REF] Fernández | Identifying relevant phrases to summarize decisions in spoken meetings[END_REF].

Tur et al. [START_REF] Tur | The calo meeting assistant system[END_REF][START_REF] Tur | The CALO meeting speech recognition and understanding system[END_REF] extract different shortened version of meetings according to different evaluation measures. They introduce a method that computes oracles of summaries and selects the one with maximum performance according to "ROUGE" [START_REF] Riedhammer | Packing the Meeting Summarization Knapsack[END_REF].

An alternative view consists of considering concepts such as decisions, problem identifications, and dialogue acts as events occurring during a meeting. Event detection is a well-established [START_REF] Kleinberg | Bursty and hierarchical structure in streams[END_REF] and active [START_REF] Gurin | Discovering news events that move markets[END_REF] research field, which in recent years has focused on social media platforms, particularly Twitter [START_REF] He | Analyzing feature trajectories for event detection[END_REF][START_REF] Sayyadi | Event detection and tracking in social streams[END_REF][START_REF] Ramage | Characterizing microblogs with topic models[END_REF][START_REF] Weng | Event detection in twitter[END_REF][START_REF] Lau | On-line trend analysis with topic models:#twitter trends detection topic model online[END_REF]. Compared to meeting dialogue, Twitter is an interesting case because it shares characteristics with it, such as weak punctuation or nonstandard use of words. Topic modeling has been used for event detection [START_REF] Ramage | Characterizing microblogs with topic models[END_REF][START_REF] Lau | On-line trend analysis with topic models:#twitter trends detection topic model online[END_REF]. Yet other approaches focus on statistical properties of the terms that occur in different documents, identifying "bursty" terms [START_REF] He | Analyzing feature trajectories for event detection[END_REF][START_REF] Weng | Event detection in twitter[END_REF], i.e. terms that during a given period occur much more often than before or after, and/or clustering them [START_REF] Sayyadi | Event detection and tracking in social streams[END_REF] to identify coherent topics.

AMI in Focus

The AMI meeting corpus contains 100 meeting hours captured using many synchronised recording devices2 . All meeting participants, both native or nonnative speakers, speak in English. In total, there are 171 meetings divided into two groups: scenario-based meeting and non-scenario based meeting. A scenario-based meeting simulates the discussions of team of four participants that are developing a remote control from start to prototype. For each simulation, the design process is divided into four parts which are held during a single day [START_REF] Carletta | Unleashing the killer corpus: experiences in creating the multieverything AMI meeting corpus[END_REF]. The "non-scenario based" meetings include real recorded meetings about designing various systems, as well as a small subset of scenarios that are different from the remote control one.

The AMI documents have been transcribed orthographically correct (while maintaining contractions) with annotated subsets of meetings including name entities, dialogues acts, abstractive and extractive summaries [START_REF] Carletta | Unleashing the killer corpus: experiences in creating the multieverything AMI meeting corpus[END_REF]. Abstractive summaries are texts of about 200 words for each meeting, consisting of free text giving a general abstract of the meeting plus specific explanations of the decisions, problems or issues encountered during the meeting. Extractive ones identify the parts of meeting related to the abstractive summary. Note that each sentence in an abstractive summary can be referred to several dialogue acts in the corresponding extractive summary, and each dialogue act can refer to several sentences in the abstractive summary.

Extracting Information from Meeting Transcripts

The aim of the paper is to extract information that can be used to derive a meeting summary of a meeting including several participants. We are interested in an approach allowing us to extract the "important" words from a document, which can then be used in further steps, such as summarizing the meeting.

We calculate a weight for each word in a meeting document, where the weight expresses the relative word importance for the meeting with respect to a set of comparison meeting documents. Our hypothesis is that in any meeting words with higher weights are more likely to provide relevant information about and characteristics of the meeting. In this section, we present the two approaches we have evaluated: Term Frequency-Inverse Document Frequency (tf-idf ), a statistical measure of a word's relative importance, as a reference method, and Nonnegative Matrix Factorization, a topic modeling approach that groups words into sets and assigns weights (probabilities) to each topic.

Term Frequency-Inverse Document Frequency

Term Frequency-Inverse Document Frequency (tf-idf ) is a measure that gives a higher weight to a term (a word) that appears frequently in a particular document but infrequently in the entire corpus. The term frequency (tf(w, d)) is simply the number of times that a word w appears in a document d. Inverse document frequency (idf) is the logarithmically-scaled proportion of documents in the corpus in which the word w appears. More formally:

idf(w, D) = log N 1 + |d ∈ D s.t. w ∈ d| (1)
where N is the total number of documents in corpus and the denominator is the (laplace-corrected) number of documents d in which a word w appears in corpus D. Thus, the tf-idf is:

tf-idf(w, d, D) = tf(w, d)idf(w, d)
As an example for meeting documents, a stop-word such as "and" may appear often in a document, but not have a high tf-idf value because it is repeated in almost every meeting. However, a word such as "lunch" has a high weight, because it appears frequently in a particular document but less often in the entire corpus. tf-idf is, for instance, the measure used to quantify terms' importance in [START_REF] Sayyadi | Event detection and tracking in social streams[END_REF], and a slight modification is used in [START_REF] He | Analyzing feature trajectories for event detection[END_REF].

Topic Modeling: Non-negative Matrix Factorization

There exist various topic modeling methods allowing us to identify the topics present in text documents. Among the considerable number of proposals for probabilistic methods such as Latent Dirichlet Allocation (LDA) or nonprobabilistic methods such as Non-negative Matrix Factorization (NMF ), we focus on the NMF in this paper [START_REF] Lee | Learning the parts of objects by nonnegative matrix factorization[END_REF]. Indeed, preliminary experiments (not given in this paper) showed that algebraic approaches perform better than probabilistic methods on our meeting documents.

In an NMF context, we assume that the matrix A ∈ R m×n represents m unique words in a corpus of n meetings (documents). The goal is to decompose the A matrix into two non-negative matrices such that A ∼ WH. The columns in W ∈ R m×k are the topics and the rows are the words represented in the topics. Each item in matrix identifies a non-negative weight for a word in relation to a topic. The matrix H ∈ R k×n indicates to what degree a meeting is related to the k of topics.

Preliminary Experiments with AMI Corpus

To evaluate the effectiveness of the two methods in treating meeting data, we perform several different experiments. In a first experiment, we assess whether tf-idf can be used to identify the characteristics of particular meetings when compared to other meetings in the same context. In a second experiment, we evaluate whether tf-idf and NMF can be used to help in grouping meetings. Finally, we assess the agreement of extracted sets of words with the two types of existing summaries (extractive and abstractive) for the meetings in the AMI corpus.

AMI Corpus Preparation

We evaluate the effectiveness of our proposed techniques on the AMI data set [START_REF]Augmented multi-party interaction[END_REF][START_REF] Carletta | Unleashing the killer corpus: experiences in creating the multieverything AMI meeting corpus[END_REF] containing multi-party dialogues3 . The AMI data set delivers its meeting documents in different divisions such as divided meeting in terms of words, time, speakers and etc. We generated the plain text files manually using these segmentations and divisions which are available online 4 .

To study the scenario-based meetings, we first classify meeting files w.r.t their subject. According to [START_REF] Carletta | Unleashing the killer corpus: experiences in creating the multieverything AMI meeting corpus[END_REF], each 3 to 5 meetings documents are acted out by a single group speaking about the same scenario, divided into several parts.

For this reason, we regroup the 138 scenario-based meetings into 34 "effective" meeting blocks. On the other hand, there exist 33 non-scenario based meetings as well that will be studied in our future works.

Characterizing individual meetings

As we have described before, the underlying motivation for this work is to help summarize meetings, in particular identifying decisions taken, action items defined, problems identified, and responsibilities assigned. The scenario-based AMI meetings are intended to simulate the trajectory of a particular project (designing a remote control) from start to finish, condensed into a single day. All 34 groups acting out this scenario follow the same script but will, of course, vary in the concrete implementation of the script. However, if there is something inherent in each stage of the project, i.e. each individual meeting of the same block, we would expect those characteristics to show up in the terms extracted by tf-idf .

To evaluate our hypothesis, we use the following experimental setup: we treat a given block of meetings, i.e. meetings acted out by the same group, as the corpus and identify for each individual meeting in this block the twenty highestscoring words according to tf-idf . For each of these words, we then count how often it occurred in the set of words derived from the same stage, e.g. we gather the sets of words derived from the first meeting of each block (M 1 ) and count duplicates. Table 1 shows the results for each stage, after thresholding word frequency at 4, i.e. 10%, rounded up 5 . We remind the reader that we did not use stemming for those texts. The number at the beginning of each cell therefore reports the modulo stemming, e.g. "cats" counts towards "cat", the number in parentheses the actual count of the word. There are 34 different groups that acted out the scenarios, which means that, for instance, the term "animal" was used in 73.5% of all first meetings of this scenario.

Table 1 (including 10% of twenty highest scoring words for the whole 34 meeting blocks) demonstrates most of the words are appeared in the first stage or the last one while the two stages in the middle have a small share. For instance in M 1 stage, "animal", "cat" and "favourite" have repeated 29, 19 and 12 times respectively. But in the M 2 stage less words appear, for instance "age", "lunch" and "teletext" have appeared each for 9 times. We see that the first and last meetings of a block have more related words than the second and third meetings. This shows that the script aligns the different groups quite closely in the beginning and in the end, but that they diverge in the middle two meetings. The first meeting seems always to be a brain-storm related to animals, and the fourth meeting about how to evaluate the success of the project. In the case of the third meeting, we can speculate that the discussion turned around what materials to use, and the presence of "lunch" as a highly-ranked word in the second meeting reminds us that this meeting took place at the end of the morning.

Grouping meetings

Earlier, we have contrasted individual meetings against other meetings within a particular process. In the real world it's not clear how to group meetings, however. In the best case scenario, whoever starts the meeting recording would indicate what project or issue the meeting is related to but we cannot rely on this information being available. A slightly weaker assumption relies on participants stating their names. For example, if Ada, Billy, Christine and Dolores work together on one project, and Elise, Frank, Gerd and Heather on a second one, we would assume that certain combinations of names identify groups of meetings belonging together. We therefore change the underlying document for calculating the tf-idf score, merging all meetings of a single block into one document, and treating the other merged blocks as comparison documents. The second column from the left of Table 2 shows examples of what happens when we contrast the 3-5 meetings enacted by a particular group against all other meetings in the corpus and as we expected, several of the extracted terms are names 6 . When contrasting individual sessions in a block against the other ones in the same block, as in the preceding section, names weren't highly ranked, which is to be expected given that the entire scenario is acted out by the same group of participants.

This indicates that one can contrast meetings against the entire available corpus and group them with other meetings showing the same combination of names, if they have not been grouped together by the person in charge of recording. The results need improvement, however: while we know that each scenario was acted out by four people, the average number of names recovered by tf-idf is only 2.44, with a standard deviation of 1.44. Main contributors to this standard deviation are blocks 25, 27, and 31, neither of which gave rise to a single name. It is possible that participants in those blocks did not state their names (often), a situation that might arise in real-world situations, especially if a certain group of people has already been working together for a while.

In this context, we would also like to point out an interesting observation in our experiments: for the block grouping M 52 , M 53 , M 54 , M 55 , one of the highestranking words is "shoulda". This is a non-formal contraction of "should have" and the fact that it is tagged as having one of the 20th highest scores for that block (actually the 8th-highest) indicates that certain participants' manner of speech might be enough to group meetings. Other contractions that we have extracted include "you'll" and "ain't". We intend to explore this phenomenon in the future.

As a comparison approach to using tf-idf on merged meetings, we also evaluated NMF . We use NMF to assign a topic to each scenario-based meeting block. As has been mentioned earlier in this section, there are 34 scenario based meeting blocks in total. For this reason we use NMF topic modeling with exactly 34 topics to classify the scenario-based meeting blocks, i.e. each block of meetings is assigned to exactly one topic.

To implement the NMF method, we used the gensim package [START_REF] Řehůřek | Software Framework for Topic Modelling with Large Corpora[END_REF]. After classification, each meeting should belong to a topic class. As for tf-idf , each topic is represented by the 20 most important candidate words according to the induced model. Referring to Table 2, the extracted words for five meeting blocks 1 to 5 is given in the last column 7 . While we can recover names using tf-idf , this is not reliably the case when using NMF -the average number of recovered names is 0.88, with a standard deviation of 0.91. This means that the information derived the tf-idf score and the topic model are complementary. Interestingly enough, when looking at meeting block 4 in Table 2, we find "david" and "jordan" in the NMF results, who refer to the same person in the acting group.

Quality of extracted sets of words

In order to evaluate the extracted information of the meetings, we compare them with the abstractive and extractive summaries as follows.

Given a set of words S extracted from a meeting M , we define the accuracy S w.r.t the abstractive summary (extractive summary) of M , s abs (M ) (s ext (M )), as:

acc abs (S, M ) = |S ∩ s abs (M )| |S| (acc ext (S, M ) = |S ∩ s ext (M )| |S| ), i.e.,
number of words of S that appear in the abstractive (extractive) summary divided by the number of words in S. We match words modulo stemming, for instance, if S contains "painting" and s abs /s ext a similar word such as "paint", the word is considered to have appeared.

We assess the quality of the extracted sets of words in two ways: Figures 1 2 show how sets of words extracted as described in Section 5.2 perform w.r.t. to abstractive and extractive summaries. In overall, abstractive accuracy results are always less than 0.6 but in many cases they vary between 0.0 to 0.4. It is noticeable that for almost half of the blocks, the abstractive accuracy for the first meeting is lowest. As we saw in Table 1, the brain-storming in the first meeting involved concrete animal names that are not necessarily presented in the abstractive meeting.

Most extractive accuracies are above 0.6 and noticeable cases have the accuracy more than 0.8. As expected extractive accuracy is higher than the abstractive one. This is because our accuracy measure computes the percentage of appearing words in each summary. Since the extractive summary includes more words as a subset of the meeting documents, the accuracy value for extractive summaries is far higher than the abstractive one. In addition, abstractive summaries by definition abstract from the actual contents of the meeting.

The second assessment uses the sets of words extracted according to the description in Section 5.3, and accuracies are calculated w.r.t. the merged summaries of all meetings in a block. The upper graph in Figure 3 reports the accuracies of sets of words extracted from each meeting block using tf-idf . Similar to the preceding results, extractive accuracy is higher than the abstractive one. According to the graph, the extractive accuracy is more than 0.6 while the abstractive accuracy is less than 0.4 for all the scenario-based meeting blocks.

The lower graph in the figure shows abstractive and extractive accuracies for sets of words extracted by NMF . By comparing this graph with the tf-idf results, we see that NMF gives better results for both summary types. This is the flip side of NMF 's inability to pick out names -instead of picking up particularities of the group -NMF learns topics describing the meeting block, matching summaries better. Since topic modeling methods compute the probability with which each text is related to each topic, Figure 4 shows for each meeting block how probable it is that it concerns a particular topics. The color bar in the right side of the figure shows the probabilities i.e. a point with a color closer to red is more highly probable. For instance, meeting 1 concerns topic 5 with a probability of more than 0.9 while, it is about topics 8, 25, 31 and 3 with a very low probability (less than 0.1). It shows that there are no ambiguities -each meeting block has one topic assigned to it with very high probability (in excess of 0.9), and all others are at less than 0.1 probability.

Conclusions and Perspectives

In this article, we tested representatives of two paradigms for term extraction from documents -numerical statistics and topic modeling -to extract important information from texts in the context of recorded and transcribed multiparty dialogues, more specifically the AMI data set. We evaluated three settings: characterizing individual meetings in comparison to other thematically related meetings, identifying names of participants tying together related meetings, and matching extracted sets of words to provided gold-standard summaries.

In terms of the first setting, tf-idf performs rather well, consistently characterizing first and last meetings of a scenario, and showing good extractive accuracies (and to a lesser degree abstractive ones), i.e. comparisons of the extracted sets to gold-standard summaries. When it comes to identifying names, tf-idf shows better performance than NMF but they are arguably not good enough to reliably group meetings, explicit tagging before recording is therefore probably required.

Topic modeling via NMF , on the other hand shows better performance w.r.t the provided gold standard (abstractive and extractive summaries) when we attempt to characterize a full scenario, i.e. a block of thematically related meetings acted out by one group.

A conclusion to be drawn from our results is therefore that the two approaches complement each other, providing results that are useful for different aspects of the larger task of characterizing and summarizing meetings. To continue in this direction, automatically grouping meetings seems to be the most important issue to improve on because working on related meetings is foundational for other tasks, i.e. characterizing different types of meetings and/or identifying trends that help in understanding the progression of a particular project. A second question is how to arrive at summaries from terms sets -the process of extracting sentences related to extracted terms, and translating those into abstractive summaries in turn, is not an obvious one but needs to be tackled.

Fig. 1 .

 1 Fig. 1. Abstractive accuracy for sets of words extracted from individual meetings, compared to the other meetings in the same block, using tf-idf.

Fig. 2 .

 2 Fig. 2. Extractive accuracy for sets of words extracted from individual meetings, compared to the other meetings in the same block, using tf-idf.

Fig. 3 .

 3 Fig. 3. The upper graph (lower graph) indicates the abstractive and extractive accuracies for sets of words extracted from scenario-based meeting blocks using tf-idf (NMF ).

Fig. 4 .

 4 Fig. 4. The x axis indexes the 34 scenario-based meeting blocks while the Y axis indicates the list of 34 topics. The figure shows the 5 highest-ranked topics for each meeting block. The heat map on the right hand side maps probabilities to the colors used in the plot.

Table 1 .

 1 The most often repeated high-scoring words (according to tf-idf ) for individual meetings in scenario-based blocks

	M1	M2	M3	M4
	25: animal	9: age, lunch, teletext	8: solar, wood	17: criteria
	19: cat (15)	8: percent, young (4)	6: titanium	13: seven
	12: favourite	6: pay	5: spongy, concepts	12: evaluation
	11: tool (5), dog (7)	5: settings, zap,	4: dark, sample, dou-	9: sample
		seventy, users	blecurved, materials,
			sensor, banana, circuit,
			cases, vegetables, fruit
	7: training, draw	4: set, messages,		8: special, false
		group, mode, infrared,
		recognition, speech
	6: rabbit, profit, fish			6: evaluate, process
	4: friendly, bird, tail,			5: prototype, leader-
	whiteboard, width, charac-			ship, creativity, under
	teristics, elephant, morning
				4: scale, single, fifteen, team,
				average, budget, curve

Table 2 .

 2 Example of words extracted using tf-idf t (left) and NMF (right) for the first five scenario-base meeting blocks

Tool example: http://www.vocapia.com

Available here: http://groups.inf.ed.ac.uk/ami/download/

http://groups.inf.ed.ac.uk/ami/download/

https://github.com/pegahani/AMI-prep

For full results, we refer the readers to: https://github.com/pegahani/Event_ detection/blob/master/result/result_4_4.txt

For full results, we refer the readers to: https://github.com/pegahani/Event_ detection/blob/master/result/result_4_block_scen.txt

For more results you can visit: https://github.com/pegahani/Event_detection/ blob/master/result/Topic_modeling_nmf_block_34_topics.txt

Acknowledgement

This work is supported by the FUI 22 (REUs project) and the ANR (French Research National Agency) funded project NARECA ANR-13-CORD-0015.