R Émi

Christof Sparber
email: sparber@uic.edu

NONLINEAR DYNAMICS OF SEMICLASSICAL COHERENT STATES IN PERIODIC POTENTIALS

We consider nonlinear Schrödinger equations with either local or nonlocal nonlinearities. In addition, we include periodic potentials as used, for example, in matter wave experiments in optical lattices. By considering the corresponding semiclassical scaling regime, we construct asymptotic solutions, which are concentrated both in space and in frequency around the effective semiclassical phase-space flow induced by Bloch's spectral problem. The dynamics of these generalized coherent states is governed by a nonlinear Schrödinger model with effective mass. In the case of nonlocal nonlinearities we establish a novel averaging type result in the critical case.

INTRODUCTION

Coherent states have been originally introduced in quantum mechanics to describe wave packets minimizing the uncertainty principle. This property makes coherent states highly attractive for the study of semiclassical asymptotics, see, e.g., [START_REF] Hagedorn | Semiclassical quantum mechanics. I. The → 0 limit for coherent states[END_REF][START_REF] Faou | Computing semiclassical quantum dynamics with Hagedorn wavepackets[END_REF][START_REF] Paul | Semi-classical methods with emphasis on coherent states[END_REF]. Indeed, it can be shown that for Schrödinger equations with sub-quadratic potentials, coherent states retain their shape, providing minimum uncertainty at all time in the quadratic case [START_REF] Combescure | Quadratic quantum Hamiltonians revisited[END_REF], and up to Ehrenfest time in general [START_REF] Bouzouina | Uniform semiclassical estimates for the propagation of quantum observables[END_REF]. Recently, extensions to weakly nonlinear situations have been studied in [START_REF]Nonlinear coherent states and Ehrenfest time for Schrödinger equations[END_REF][START_REF] Carles | A nonlinear adiabatic theorem for coherent states[END_REF]. In addition, the semiclassical dynamics of coherent states under the influence of (highly oscillatory) periodic potentials has been investigated by the authors in [START_REF] Carles | Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials[END_REF]. In the present work we combine the effects coming from periodic and nonlinear potentials.

To this end, we consider nonlinear Schrödinger equations which, after scaling into dimensionless coordinates, appear in the following semiclassical form:

(1.1)

iε∂ t ψ ε + ε 2 2 ∆ψ ε = V per x ε ψ ε + ε α f (|ψ ε | 2 )ψ ε , ψ ε |t=0 = ψ ε 0 ,
where t ∈ R, x ∈ R d , and d ∈ N denotes the spatial dimension (usually d = 3). Moreover, ε ∈ (0, 1] denotes a (small) semiclassical parameter, i.e. a dimensionless rescaled Planck's constant. The factor ε α measures the (asymptotic) strength of the nonlinearity: the larger the α > 0, the weaker the nonlinear effects. In the following, we shall allow for two different types of gauge invariant nonlinearities:

• Local nonlinearities: f (|ψ ε | 2 ) = λ|ψ ε | 2σ
, with σ ∈ N and λ ∈ R, allowing for focusing (attractive) and defocusing (repulsive) situations. • Nonlocal nonlinearities of convolution type:

f (|ψ ε | 2 ) = K * |ψ ε | 2 , with K(x) ∈
R a given interaction kernel. Finally, the term V per (x/ε) denotes a highly oscillatory periodic potential. More precisely, let Γ ≃ Z d be some regular lattice, then we assume that for all y ∈ R d : V per (y + γ) = V per (y) with γ ∈ Γ. In addition, we shall assume V per ∈ C ∞ (R d ). Equation (1.1) describes the propagation of waves on macroscopic length-and time-scales, i.e. over many periods of the periodic potential. The parameter ε ≪ 1 consequently describes the ratio between microscopic (quantum mechanical) and the macroscopic scales.

Nonlinear Schrödinger equations with periodic potentials arise in various physical contexts: A by now classical example is the mean-field description of electrons propagating in a crystalline solid [START_REF] Guillot | Semiclassical asymptotics in solid state physics[END_REF][START_REF] Spohn | Long time asymptotics for quantum particles in a periodic potential[END_REF] under the additional influence of a self-consistent electric field. The latter is usually modeled by means of a nonlocal Hartree nonlinearity [START_REF] Bechouche | Semiclassical limit for the Schrödinger-Poisson equation in a crystal[END_REF] for a semiclassical study via Wigner measures. Another situation in which (1.1) applies is the description of Bose-Einstein condensates in so-called optical lattices, cf. [START_REF] Choi | Bose-Einstein condensates in an optical lattice[END_REF]. In the regime of dilute gases, such condensates can be modeled by the Gross-Pitaevskii equation with cubic nonlinearity σ = 1, cf. [START_REF] Pitaevskii | International Series of Monographs on Physics[END_REF]. Note however, that other nonlinearities also arise, see, for example, [START_REF] Berloff | Nonlocal nonlinear Schroedinger equations as models of superfluidity[END_REF] where a nonlocal term is used for the description of superfluid Helium. In addition, strong magnetic confinement allows for the experimental realization of quasi-one-dimensional (cigar-shaped) condensates, or quasi two-dimensional condensates, motivating the fact that we consider (1.1) in general dimensions d ∈ N, cf. [START_REF] Kolomeisky | Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation[END_REF]. A third example for the appearances of (1.1) stems from the description of wave packets propagating within nonlinear photonic crystals [START_REF]Nonlinear Photonic Crystals[END_REF] and where the nonlinear response of the media is modeled via a Kerr nonlinearity σ = 1. In this case, the underlying assumption in the derivation of (1.1) is the existence of a preferred direction of propagation, implying that the appropriate model is stated in dimension

f (|ψ ε | 2 ) = |ψ ε | 2 * 1/| • |, see, e.g.,
d = 1.
In all of these situations, the joint effects of nonlinearity, periodicity and dispersion (or, quantum pressure), can lead to the existence of stable localized states conserving the form upon propagation and collisions. Gap solitons, discrete breathers and compactons are examples of such states. Here we shall present another possibility, which will arise from semiclassical description via coherent states. Even though the nonlinearity in our case is weaker than in the above mentioned situations (due to the fact that α > 0), the obtained asymptotic solutions will nonetheless experience nonlinear effects in leading order, provided α is of critical size (to be made precise later on). The latter will depend on the precise form of the nonlinearity.

To present our results, we recall the classical Bloch eigenvalue problem [START_REF] Wilcox | Theory of Bloch waves[END_REF]:

(1.2) H(k)χ m (•, k) = E m (k)χ m (•, k), m ∈ N.
Denoting by Y the centered fundamental domain of Γ, E m (k) ∈ R and χ m (•, k) denote, respectively, the m-th eigenvalue/eigenvector pair of

(1.3) H(k) = 1 2 (-i∇ y + k) 2 + V per (y) , y ∈ Y, parametrized by the crystal momentum k ∈ Y * ≃ T d .
We shall assume that at t = 0,

(1.4) ψ ε (0, x) ∼ ε→0 ε -d/4 u 0 x -q 0 √ ε χ m x ε , p 0 e ip0•(x-q0)/ε ,
where u 0 denotes some smooth and rapidly decaying profile. In other words, the initial data ψ ε 0 can be approximated by a highly oscillatory Bloch eigenfunction χ m modulated by a (generalized) coherent state, i.e. a wave function which is localized both in space and in frequency.

Remark 1.1. In particular, the choice u 0 (z) = exp(-|z| 2 /2) yields a classical coherent state, i.e. ground state of the harmonic oscillator potential, as modulation. The states (1.4) are more general, though, since we can allow for any u 0 ∈ S(R d ), the Schwartz space of rapidly decaying, smooth functions. We also remark that the same class of initial data have recently been considered in [START_REF] Carles | Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials[END_REF], where the situation of linear Schrödinger equations with combined periodic and slowly varying external potentials has been considered (see B for more details).

Provided such initial data, we shall show that the solution of (1.1) can be approximately (in a sense to be made precise) described by the following semiclassical wave packet:

(1.5) ψ ε (t, x) ∼ ε→0 ϕ ε (t, x) := ε -d/4 u t, x -q(t) √ ε χ m x ε , p 0 e iφm(t,x)/ε
where q(t) = q 0 + t∇ k E m (p 0 ) describes the macroscopic shift of the centre of mass and the highly oscillatory phase function φ m is

(1.6) φ m (t, x) = p 0 • (x -q 0 ) -tE m (p 0 ).
To this end, we need to give sense to the group velocity ∇ k E m (k) and thus, we have to assume from now on, that:

(1.7) E m (k) is a simple eigenvalue in the vicinity of k = p 0 .

(Of course, p 0 ∈ R d has to be understood modulo Γ * in this case.) In other words, we have to avoid that two Bloch bands cross at p 0 , that is E m (p 0 ) = E n (p 0 ), for m = n. It is known that at such crossing points E m (k) is no longer differentiable, causing the above asymptotic description (which is based on an adiabatic decoupling of the slow and fast degrees of freedom) to break down.

Remark 1.2. Clearly, the non-crossing condition given above restricts our choice for the initial wave vectors p 0 ∈ R d . It is known however that the set of band crossings has Lebesgue measure zero. For example, in the case d = 1, band crossings can only occur at k = 0 or at the boundary of the Brillouin zone.

So far, we have not said what determines the profile u = u(t, z) appearing in (1.5). Its time-evolution depends on the strength of the nonlinearity, i.e. on the size of α > 0 in the case of local nonlinearities (the situation for nonlocal ones will be described later on). We shall find that the critical size is α c = 1 + dσ/2 and when α = α c , u solves the following homogenized nonlinear Schrödinger equation

(1.8) i∂ t u + 1 2 div z (∇ 2 k E m (p 0 )) • ∇ z u = λ m |u| 2σ u, u |t=0 = u 0 , with effective coupling constant λ m = λ Y |χ m (y, p 0 )| 2σ+2 dy.
Note that the dispersive properties of (1.8) are determined by an effective mass matrix ∇ 2 k E m (p 0 ) ∈ R d×d , which itself depends on the choice of the initial wave vector (cf. [START_REF] Duan | Effective mass approach for a Bose-Einstein condensate in an optical lattice[END_REF] for a recent study).

In the next section, we derive this effective mass equation from multi-scale expansion. A rigorous stability result is then proved in Section 3. The case of nonlocal nonlinearities is treated in Section 4. As we shall see, in situations where the kernel K is homogeneous, the critical value α c depends on the degree of homogeneity (like in the case of a local nonlinearity), and the analogue of (1.8) is an envelope equation with a nonlocal nonlinearity. On the other hand, if the kernel K is a smooth function, then α c = 1, and in sharp contrast with the other situations studied in this paper, the analogue of (1.8) for α = α c is found to be a linear equation, see Section 4.2.

MULTI-SCALE EXPANSION

2.1. The hierarchy of equations. Except for the treatment of the nonlinear term, we resume the strategy followed in [START_REF] Carles | Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials[END_REF]. We seek the solution ψ ε of (1.1) in the form (2.1)

ψ ε (t, x) = ε -d/4 U ε t, x -q(t) √ ε , x ε e iφm(t,x)/ε ,
where the phase φ m (t, x) is given by (1.6) and the function

U ε = U ε (t, z, y) admits an asymptotic expansion (2.2) U ε (t, z, y) ∼ ε→0 j∈N ε j/2 U j (t, z, y)
with smooth profiles U j which, in addition, are assumed to be Γ-periodic with respect to y. For ψ ε given by the ansatz (2.1), we compute

iε∂ t ψ ε + ε 2 2 ∆ψ ε -V per x ε ψ ε = ε -d/4 e iφm/ε 2 j=0 ε j/2 b ε j (t, z, y) (z,y)= x-q(t) √ ε , x ε with b ε 0 = -∂ t φ m U ε + 1 2 ∆ y U ε - 1 2 |p 0 | 2 U ε + ip 0 • ∇ y U ε -V per (y)U ε , b ε 1 = -i q(t) • ∇ z U ε + (∇ y • ∇ z ) U ε + ip 0 • ∇ z U ε , b ε 2 = i∂ t U ε + 1 2 ∆ z U ε .
Using (1.6) and the fact that

q(t) = ∇ k E m (p 0 ), we can rewrite b ε 0 , b ε 1 as b ε 0 = (E m (p 0 ) -H (p 0 ))U ε , b ε 1 = i (p 0 -∇ k E m (p 0 )) • ∇ z U ε + (∇ y • ∇ z ) U ε ,
where H(p 0 ) is the Bloch Hamiltonian (1.3) evaluated at k = p 0 (again, this has to be understood modulo Γ * ). Introducing the following linear operators:

L 0 = E m (p 0 ) -H (p 0 ) , L 1 = i (p 0 -∇ k E m (p 0 )) • ∇ z + ∇ y • ∇ z , L 2 = i∂ t + 1 2 ∆ z ,
and expanding U ε in powers of ε ∈ (0, 1], we consequently need to solve the following hierarchy of equations:

(2.3)      L 0 U 0 = 0, L 0 U 1 + L 1 U 0 = 0, L 0 U 2 + L 1 U 1 + L 2 U 0 = F (U 0 ),
where, for α c = 1 + dσ/2, we find:

F (U 0 ) = 0 if α > α c , λ|U 0 | 2σ U 0 if α = α c .
In the next subsection, we shall focus on the resolution of (2.3).

The effective mass equation.

Given the form of L 0 , the equation

L 0 U 0 = 0 implies (2.4) U 0 (t, z, y) = u(t, z)χ m (y, p 0 ) .
By Fredholm's alternative, a necessary and sufficient condition to solve the equation

L 0 U 1 + L 1 U 0 = 0, is that L 1 U 0 is orthogonal to ker L 0 , that is: (2.5) χ m , L 1 U 0 L 2 (Y ) = 0.
Given the expression of L 1 and the formula (2.4), we compute

L 1 U 0 = i (p 0 -∇ k E m (p 0 )) • ∇ z u(t, z)χ m (y, p 0 ) + ∇ y χ m (y, p 0 ) • ∇ z u(t, z).
Now, we make use of the algebraic identities derived in Section A. In view of (A.2), we infer that (2.5) is automatically fulfilled. We thus obtain

U 1 (t, z, y) = u 1 (t, z)χ m (y, p 0 ) + u ⊥ 1 (t, z, y), where u ⊥
1 , the part of U 1 which is orthogonal to ker L 0 , is obtained by inverting an elliptic operator:

u ⊥ 1 = -L -1 0 L 1 U 0 .
Note that the formula for L 1 U 0 can also be written as

L 1 U 0 = -i∇ k (E m (p 0 ) -H (p 0 )) χ m (y, p 0 ) • ∇ z u(t, z).
Taking into account (A.1), this yields:

u ⊥ 1 (t, z, y) = -i∇ k χ m (y, p 0 ) • ∇ z u(t, z).
At this stage, we shall, for simplicity, choose u 1 = 0, in which case U 1 becomes simply a function of u:

(2.6) U 1 (t, z, y) = -i∇ k χ m (y, p 0 ) • ∇ z u(t, z).
As a next step in the formal analysis, we must solve

L 0 U 2 + L 1 U 1 + L 2 U 0 = F (U 0 )
. By the same argument as before, we require

(2.7) χ m , L 1 U 1 + L 2 U 0 -F (U 0 ) L 2 (Y ) = 0.
With the expression (2.6), we compute

L 1 U 1 = d j,ℓ=1 (p 0 -∇ k E m (p 0 )) j ∂ k ℓ χ m (y, p 0 ) -i∂ 2 yj k ℓ χ m (y, p 0 ) ∂ 2 zjz ℓ u,
and we also have

L 2 U 0 = i∂ t + 1 2 ∆ z u(t, z) χ m (y, p 0 ) .
We consequently infer

χ m , L 1 U 1 + L 2 U 0 L 2 (Y ) = i∂ t u + 1 2 ∆ z u - d j,ℓ=1 χ m , ∂ kj E m (p 0 ) ∂ k ℓ χ m + i∂ 2 yjk ℓ χ m L 2 (Y ) ∂ 2 zj z ℓ u.
In the case α = α c , by making the last sum symmetric with respect to j and ℓ, and using (A.3), we finally obtain the nonlinear Schrödinger equation (1.8) with effective mass tensor

M = ∇ 2 k E m (p 0 ) and coupling constant λ m := λ χ m , |χ m | 2σ χ m L 2 (Y ) = λ Y |χ m (y, p 0 )| 2σ+2 dy.
In addition, we can write

(2.8) U 2 (t, z, y) = u 2 (t, z)χ m (y, p 0 ) + u ⊥ 2 (t, z, y), where u ⊥ 2 = -L -1 0 (L 1 U 1 + L 2 U 0 )
. We shall also impose u 2 ≡ 0 and thus

U 2 = u ⊥ 2 .
Assume for the moment that we can solve (1.8). Then, we have the following result, which establishes some basic regularity properties of our multi-scale expansion (where we denote by H k the usual L 2 (R d ) based Sobolev space). Lemma 2.1. Suppose (1.7) holds true and let u ∈ C([0, T ]; H k ), be a solution of (1.8) up to some T > 0.

Then U j ∈ C([0, T ]; H k-j z × W ∞,∞ (Y )), for j = 0, 1, 2.
Proof. First note that (y, k) → χ m (y, k) is smooth and bounded together with all its derivatives, provided (1.7) holds true. Having this in mind, the proof follows directly from the construction of {U j } j=0,1,2 as solutions to the system (2.3).

Remark 2.2. Note that in the case α > α c nonlinear effects are absent at leading order since we obtain, instead of (1.8), a linear effective mass equation:

(2.9)

i∂ t u lin + 1 2 div z ∇ 2 k E m (p 0 ) • ∇ z u lin = 0, u |t=0 = u 0 .
This type of equation has been derived in [START_REF] Piatnitski | Homogenization of the Schrödinger equation and effective mass theorems[END_REF][START_REF] Sparber | Effective mass theorems for nonlinear Schrödinger equations[END_REF], using a different asymptotic scaling.

MAIN RESULTS

In this section we shall make the computations given above rigorous and prove a nonlinear stability result. As a first step we need to guarantee the existence of a smooth solution to (1.8), at least locally in-time.

Construction of an approximate solution.

The dispersion relation of (1.8) is given by a real-valued symmetric matrix. Standard techniques (see, e.g., [START_REF] Tao | Nonlinear dispersive equations[END_REF]) yield the existence of a unique local solution, provided that the initial datum is sufficiently smooth: Lemma 3.1. Let u 0 ∈ H k with k > d/2. There exists T c ∈ (0, +∞] and a unique maximal solution u ∈ C([0, T c );

H k ) to (1.8), such that u(t, •) L 2 = u 0 L 2 . The solution is maximal in the sense that if T c < ∞, then lim t→Tc u(t, •) H k = +∞.
The solution u(t, •) may not exist for all times, even if λ 0, i.e. even if the nonlinearity in the original equation (1.1) is defocusing. However, we can claim T c = ∞ in either of the following cases (see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]):

• ∇ 2 k E m (p 0 ) is positive definite and λ m 0, or • ∇ 2 k E m (p 0
) is negative definite and λ m 0. On the contrary, if for instance ∇ 2 k E m (p 0 ) is positive definite and λ m < 0 (focusing nonlinearity), finite time blow up (that is, T c < ∞) may occur, see, e.g., [START_REF] Tao | Nonlinear dispersive equations[END_REF][START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]. This is the case typically if the initial datum is "too large": for any fixed profile u 0 ∈ S(R d ), if one considers u |t=0 = Λu 0 , there exists Λ 0 > 0 such that for all Λ Λ 0 , T c < ∞. Note that in other situations, where the signature of

∇ 2 k E m (p 0 ) is non-trivial (hence d 2)
, the issue of global existence vs. finite time blow-up is an open question. Remark 3.2. Clearly, for α > α c these issues do not occur, since the leading order profile u lin solves the linear equation (2.9) and hence exists for all times, T c = +∞.

ψ ε app (t, x) := ε -d/4 U 0 + √ εU 1 + εU 2 t, x -q(t) √ ε ,
x ε e iφm(t,x)/ε , which satisfies the original equation (1.1) up to some remainder terms:

iε∂ t + ε 2 2 ∆ -V per ψ ε app -λε α |ψ ε app | 2σ ψ ε app = e iφm/ε ε d/4 ε 3/2 (r ε 1 + r ε 2 ) (t, z, y) (z,y)= x-q(t) √ ε , x ε -λR ε (t, x).
The remainder terms are given by

r ε 1 (t, z, y) = L 2 U 1 (t, z, y), r ε 2 (t, z, y) = L 1 U 2 (t, z, y) + √ εL 2 U 2 (t, z, y),
and

R ε = ε α |ψ ε app | 2σ ψ ε app if α > α c , while if α = α c = 1 + dσ/2, R ε = ε 1+dσ/2 |ψ ε app | 2σ ψ ε app - ε ε d/4 e iφm/ε U 0 t, x -q(t) √ ε , x ε 2σ U 0 t, x -q(t) √ ε , x ε .
This, together with the regularity result established in Lemma 2.1 then directly yields the following proposition.

Proposition 3.3. Assume (1.7) and let α α c = 1 + dσ/2. Then, we can find ψ ε app such that:

1. For all T ∈ [0, T c ), ψ ε app has a coherent state structure on [0, T ], in the sense that there exists C independent of ε ∈ (0, 1] such that, with ϕ ε defined in (1.5),

for α = α c , sup t∈[0,T ] ψ ε app (t, •) -ϕ ε (t, •) L 2 (R d ) C √ ε, for α > α c , sup t∈[0,T ] ψ ε app (t, •) -ϕ ε lin (t, •) L 2 (R d ) C √ ε,
where ϕ ε lin is the approximate solution constructed from u lin , solving (2.9). 2. The function solves (1.1) up to a small error:

iε∂ t ψ ε app + ε 2 2 ∆ψ ε app = V per x ε ψ ε app + λε α |ψ ε app | 2σ ψ ε app + εw ε ,
where the remainder term w ε satisfies: for all T > 0, with T < T c in the case α = α c , there exists C > 0 independent of ε > 0 such that

sup t∈[0,T ] w ε (t, •) L 2 (R d ) C ε min(α-αc,1/2) if α > α c , √ ε if α = α c .
Note that because of the factor ε in front of the time derivative, it is natural to represent a small error term as ε times as small term.

3.2. Nonlinear stability. It remains to prove nonlinear stability of the approximate solution constructed above. For the sake of simplicity we shall do so only for α = α c and d = 1. The (physically less interesting) case α > α c can be proved analogously and a possible generalization to higher dimensions is indicated in Remark 3.6 below. For

ε 0 > 0, set f ε H 1 ε := sup 0<ε ε0 f ε L 2 + ε∂ x f ε L 2 ,
which is equivalent to the usual H 1 -norm for every (fixed) ε > 0. The approach that we present is similar to the one followed in [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF]: First, we need to construct a more accurate approximate solution than the one stated in Proposition 3.3. Taking the asymptotic expansion presented in Section 2 one step further, we can gain a factor √ ε in Proposition 3.3. More precisely, we can construct ψ ε app such that:

(3.2) sup t∈[0,T ] ψ ε app (t, •) -ψ ε app (t, •) H 1 ε C √ ε,
and

iε∂ t ψ ε app + ε 2 2 ∆ ψ ε app = V per x ε ψ ε app + λε 1+σ/2 | ψ ε app | 2σ ψ ε app + ε w ε ,
where the additional factor √ ε is reflected in the error estimate

(3.3) sup t∈[0,T ] w ε (t, •) H 1 ε Cε.
Note that in this case the corrector U 1 is not the same for ψ ε app , since unlike what we have done in §2.2, we can no longer assume u 1 = 0. Rather, u 1 now solves an evolution equation, which is essentially (1.8) linearized about u, with a non-trivial source term (see [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF] for more details). Therefore, the estimate (3.2) must be expected to be sharp in general.

Having constructed such an improved approximation ψ ε app we can state the following nonlinear stability result: Theorem 3.4. Let d = 1, α = 1 + σ/2, σ ∈ N, and Assumption (1.7) hold. In addition, suppose that the initial data satisfy:

(3.4) ψ ε 0 -ψ ε app|t=0 L 2 (R) = O(ε), ε∂ x ψ ε 0 -ψ ε app|t=0 L 2 (R) = O(1).
Let T ∈ [0, T c ). Then, there exists ε 0 = ε 0 (T ) such that for ε ∈ (0, ε 0 ], the solution of (1.1) exists on [0, T ]. Moreover, there exists C independent of ε ∈ (0, ε 0 ] such that

sup t∈[0,T ] ψ ε (t, •) -ϕ ε (t, •) L 2 (R) C √ ε.
where ϕ ε is defined in (1.5).

Proof. The scheme of the proof is the same as the proof of Theorem 4.5 in [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF], so we shall only give the main steps. Fix T < T c and let η ε = ψ ε -ψ ε app be the error between the exact and the approximate solution. It satisfies

iε∂ t η ε + ε 2 2 ∆η ε = V per x ε η ε + λε 1+σ/2 |ψ ε | 2σ ψ ε -| ψ ε app | 2σ ψ ε app -ε w ε , with η ε |t=0 L 2 = O(ε), ε∂ x η ε |t=0 L 2 = O(1)
by assumption. From [START_REF] Rauch | Lectures on geometric optics[END_REF], we have:

Lemma 3.5 (Moser's lemma). Let R > 0, s ∈ N and F (z) = |z| 2σ z, σ ∈ N. Then there exists C = C(R, s, σ) such that if v ε satisfies (ε∂ x ) β v ε L ∞ (R)
R, 0 β s,

and δ ε satisfies δ ε L ∞ (R) R, then 0 β s (ε∂ x ) β (F (v ε + δ ε ) -F (v ε )) L 2 C 0 β s (ε∂ x ) β δ ε L 2
.

We apply this lemma with v ε = ε 1/4 ψ ε app , and s = 0, s = 1 successively: there exists R > 0 independent of ε ∈ (0, 1] such that

sup t∈[0,T ] β=0,1 (ε∂ x ) β v ε (t) L ∞ (R)
R.

Set δ ε = ε 1/4 η ε . By assumption and the Gagliardo-Nirenberg inequality,

(3.5) δ ε |t=0 L ∞ = ε 1/4 η ε |t=0 L ∞ ε 1/4 √ 2ε -1/2 η ε |t=0 1/2 L 2 ε∂ x η ε |t=0 1/2 L 2 Cε 1/4 .
As long as δ ε (t) L ∞ R, energy estimates and Moser's lemma with s = 0 yield

η ε (t) L 2 η ε (0) L 2 + C t 0 η ε (s) L 2 ds + t 0 w ε (s) L 2 ds,
where we have used the homogeneity of F . By Gronwall's Lemma, for t T :

η ε (t) L 2 C(T ) η ε (0) L 2 + t 0 w ε (s) L 2 ds Cε.
Applying the operator ε∂ x to the equation satisfied by η ε , we infer similarly

ε∂ x η ε (t) L 2 ε∂ x η ε (0) L 2 + C t 0 η ε (s) H 1 ε ds + t 0 ε∂ x w ε (s) L 2 ds + 1 ε ∂ y V per L ∞ t 0 η ε (s) L 2 ds,
where the last term stems from the relation

[ε∂ x , V per (x/ε)] = ∂ y V per (x/ε) ∈ L ∞ , since
V per is smooth and periodic. Thus,

ε∂ x η ε (t) L 2 C + C t 0 ε∂ x η ε (s) L 2 ds + Ct.
Gronwall's lemma now yields ε∂ x η ε (t) L 2 C(T ). In view of the Gagliardo-Nirenberg inequality,

δ ε (t) L ∞ = ε 1/4 η ε (t) L ∞ √ 2ε -1/4 η ε 1/2 L 2 ε∂ x η ε 1/2 L 2 C(T )ε 1/4 .
For ε sufficiently small (depending of T ), δ ε (t) L ∞ R for all t ∈ [0, T ], and the result follows from a bootstrap argument.

The above theorem shows nonlinear stability of the approximate solution up to times of order O(1), i.e. independent of ε, provided that the initial data are well-prepared, in the sense given in (3.4). Essentially this means that ψ ε 0 contains not only U 0 , but also U 1 associated to ψ ε app . We shall not insist further on this aspect, which is probably a technical artifact, and remark that in the linear case a stronger result is valid, see [START_REF] Carles | Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials[END_REF] where stability is proved up to the so-called Ehrenfest time O(ln 1/ε), and no well-preparedness as in (3.4) is needed (an initial error O(ε r ) for some r > 0 suffices).

Remark 3.6. If x ∈ R d with d
2, the proof can be easily adapted, provided an even better approximate solution is constructed. The reason is that, instead of (3.5), one needs to rely on the following Gagliardo-Nirenberg inequality

δ ε L ∞ (R d ) Cε -d/2 δ ε 1-d/(2s) L 2 (R d ) |ε∇| s δ ε d/(2s) L 2 (R d ) , for s > d/2.
Thus, in order to account for the singular factor ε -d/2 , one is forced to construct an approximate solution ψ ε app to a sufficiently high order in ε (see [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF] for more details).

THE CASE OF NONLOCAL NONLINEARITIES

In this section we shall show how to perform the same asymptotic analysis as before in the case of nonlocal nonlinearities. In other words, we consider (4.1)

iε∂ t ψ ε + ε 2 2 ∆ψ ε = V per x ε ψ ε + ε α (K * |ψ ε | 2 )ψ ε
with K(x) ∈ R some given interaction kernel. In the following we shall focus on two particular choices of interaction kernels K which are physically relevant.

4.1. Homogeneous kernels. In this subsection we shall consider functions of the form

K(x) = λ|x| µ , λ ∈ R, with µ ∈ R \ {0} such that -min(2, d) < µ 2.
For example, the choice µ = -1 in d = 3 corresponds to the classical Hartree nonlinearity, modeling a self-consistent, repulsive (λ > 0) Coulomb interaction. The case µ > 0 has been recently studied in [START_REF] Masaki | Analysis of Hartree equation with an interaction growing at the spatial infinity[END_REF]. Like in the case of local nonlinearities, the critical exponent α c depends on the homogeneity µ, namely α c = 1 -µ/2. This can be seen as follows: We plug the ansatz

ϕ ε (t, x) = ε -d/4 u t, x -q(t) √ ε χ m x ε , p 0 e iφm(t,x)/ε into the convolution term ε 1-µ/2 |x| µ * |ψ ε | 2 . This yields ε 1-µ/2-d/2 R d |x -ξ| µ u t, ξ -q(t) √ ε 2 χ m ξ ε , p 0 2 dξ.
We want this term to be of order O(ε) in our asymptotic expansion, to mimic the approach presented in §2. In this case, it will consequently appear within b ε 2 , leading to the effective mass equation. In order to show that this is indeed the case, we rewrite the initial convolution as

ε 1-µ/2-d/2 R d |ξ| µ u t, x -ξ -q(t) √ ε 2 χ m x -ξ ε , p 0 2 dξ,
and use the substitution z = (x -q(t))/ √ ε in the envelope u, and y = x/ε in χ m :

ε 1-µ/2-d/2 R d |ξ| µ u t, z - ξ √ ε 2 χ m y - ξ ε , p 0 2 dξ.
Setting ζ = ξ/ √ ε, this can be written as

ε |ζ| µ |u (t, z -ζ)| 2 χ m y - ζ √ ε , p 0 2 dζ.
Then, the following averaging result can be proved:

Proposition 4.1. Let (1.7) hold true and assume that ζ → |ζ| µ |u (t, z -ζ)| 2 is in L 1 (R d ).
Then, for all k ∈ Y * , it holds This result can be seen as a variant of the two-scale convergence results introduced in [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF][START_REF] Allaire | Homogenization and two-scale convergence[END_REF], and used in [START_REF] Piatnitski | Homogenization of the Schrödinger equation and effective mass theorems[END_REF]. The main difference here is the convolution structure.

R d |ζ| µ |u (t, z -ζ)| 2 χ m y - ζ √ ε , k 2 dζ -→ ε→0 R d |ζ| µ |u (t, z -ζ)| 2 dζ. In addition, if ζ → |ζ| µ |u (t, z -ζ)| 2 is in W 1,1 (R d ),
Proof. We decompose y → |χ m (y, k)| 2 into its generalized Fourier series (recall that Γ ≃ Z d ) and write

|ζ| µ |u (t, z -ζ)| 2 χ m y - ζ √ ε 2 dζ = γ∈Γ |ζ| µ |u (t, z -ζ)| 2 c γ e iγ•(y-ζ/ √ ε) dζ = γ∈Γ c γ e iγ•y |ζ| µ |u (t, z -ζ)| 2 e -iγ•ζ/ √ ε dζ.
By Riemann-Lebesgue lemma, for each term with γ = 0, the limit, as ε → 0, is zero.

Then only the term corresponding to γ = 0 remains, with

c 0 (k) = Y |χ m (y, k)| 2 dy = 1,
since the eigenfunctions χ m (•, k) form an orthonormal basis of L 2 (Y ). By the Dominated Convergence Theorem, we can exchange the sum over γ ∈ Γ and the limit ε → 0 in the above computation provided that ζ → |ζ| µ |u (t, z -ζ)| 2 is in L 1 (with an error o(1)).

In the case where the function is in W Assuming that u is sufficiently smooth and decaying, we can use the above averaging result and perform the same asymptotic expansion as given in Section 2 to arrive at the effective nonlinear Schrödinger equation (4.2)

i∂ t u + 1 2 div z (∇ 2 k E m (p 0 )) • ∇ z u = λ(|z| µ * |u| 2 )u, u |t=0 = u 0 .
For µ < 0, existence of a smooth solution u ∈ C([0, T c ), H k ), locally in time, can be proved along the same lines as in [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and hence, a result analogous to the one stated in Proposition 3.3 is straightforward. In the case µ > 0 one can follow the arguments of [START_REF] Masaki | Analysis of Hartree equation with an interaction growing at the spatial infinity[END_REF], using a functional framework which is more intricate, however (the Sobolev spaces H k are not sufficient but have to be intersected with weighted L 2 spaces), and we shall not do so here. In a similar spirit, stability in the sense of Theorem 3.4 follows from an adaptation of Lemma 3.5, which we leave out.

4.2. Smooth kernels. If in (4.1) the interaction kernel K(x) is a given smooth function, bounded as well as its derivative, then α c = 1 (corresponding formally to the case µ = 0). Such a situation appears for example in [START_REF] Berloff | Nonlocal nonlinear Schroedinger equations as models of superfluidity[END_REF], where

K(x) = a 1 + a 2 |x| 2 + a 3 |x| 4 e -A 2 |x| 2 + a 4 e -B 2 |x| 2 ,
with constants a 1 , a 2 , a 3 , a 4 ∈ R, A, B > 0. Resuming the above computations in this context, we find:

K * |ψ ε | 2 = ε -d/2 R d K(ξ) u t, x -ξ -q(t) √ ε 2 χ m x -ξ ε , p 0 2 dξ = R d K(ζ √ ε) |u (t, z -ζ)| 2 χ m y - ζ √ ε , p 0 2 dζ -→ ε→0 K(0) R d |u (t, z -ζ)| 2 dζ = K(0) u(t) 2 L 2 = K(0) u 0 2 L 2 ,
due to mass conservation, along with an O( √ ε) convergence rate under suitable assumptions. In particular, this shows that, as ε → 0, the nonlinear effects become negligible. Indeed, in this case the envelope equation becomes

i∂ t u + 1 2 div z (∇ 2 k E m (p 0 )) • ∇ z u = K(0) u 0 2 L 2 u, u |t=0 = u 0 .
The right hand side involves a constant potential term, which can be gauged away via v(t, x) = u(t, x)e itK(0) u0 2 L 2 . The remaining amplitude v(t, x) then solves a free Schrödinger equation with effective mass tensor ∇ 2 k E m (p 0 ).

APPENDIX A. SOME USEFUL ALGEBRAIC IDENTITIES

For the derivation of the effective mass equation (1.8) we shall rely on several algebraic identities, which can be derived from Bloch's spectral problem (for more details see, e.g., [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF]): First, taking the gradient w.r.t. to k of (1.2), we have

(A.1) ∇ k (H(k) -E m ) χ m + (H(k) -E m ) ∇ k χ m = 0
and, by taking the in L 2 (Y )-scalar product with χ m , we obtain

∇ k E m = χ m , ∇ k H(k)χ m L 2 (Y ) + χ m , (H(k) -E m ) ∇ k χ m L 2 (Y ) .
Since H(k) is self-adjoint, the last term is zero, thanks to (1.2). We infer

(A.2) ∇ k E m (k) = χ m , (-i∇ y + k) χ m L 2 (Y ) .
Differentiating (A.1) again, we have, for all j, ℓ ∈ {1, . . . , d}:

APPENDIX B. ADDING AN ADDITIONAL, SLOWLY VARYING POTENTIAL

As a possible extension of our study, one might want to consider the case where the wave function is not only under the influence of the nonlinearity and the periodic potential, but also add an additional slowly varying external potential V (t, x), i.e.

iε∂ t ψ ε + 1 2 ∆ψ ε = V per x ε ψ ε + V (t, x)ψ ε + ε α f (|ψ ε | 2 )ψ ε .
At least formally, this can be done by combining our analysis with the results given in [START_REF] Carles | Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials[END_REF]: To this end, we define the semi-classical band Hamiltonian h sc m (k, x) = E m (k) + V (t, x), (k, x) ∈ Y * × R d , and denote the corresponding semiclassical phase space trajectories by (B.1) q(t) = ∇ k E m (p(t)) , q(0) = q 0 , ṗ(t) = -∇ x V (t, q(t)) , p(0) = p 0 .

This system is the analogue of the classical Hamiltonian phase space flow, in the presence of an additional periodic potential V per . In order to make sure that the system (B.1) is well-defined, it is sufficient to assume that E m (p(t)) is a simple eigenvalue (|E m (p(t)) -E n (k)| = 0 for all n = m, t ∈ R, k ∈ Y * ); see e.g. [START_REF] Carles | Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials[END_REF], where examples of such situations are given. The approximate solution under the form of a coherent state within the m-th Bloch band is then given by: ϕ ε (t, x) = ε -d/4 u t,

x -q(t) √ ε χ m x ε , p(t) e iΦm(t,x)/ε with q(t), p(t) obtained from (B.1). The highly oscillatory phase takes the form Φ m (t, x) = S m (t) + p(t) • (x -q(t)), where S m (t) ∈ R is the (purely time-dependent) semi-classical action S m (t) = t 0 p(s) • ∇E m (p(s)) -h sc m (p(s), q(s)) ds.

Note that Φ m simplifies to (1.6) in the case where V (t, x) = 0. In this case, the governing equation for the leading profile u(t, z) is found to be a nonlinear Schrödinger equation with time-dependent quadratic potential, time-dependent effective mass ∇ 2 k E m (p(t)) and coupling constant λ m (t), see [START_REF] Carles | Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials[END_REF] for more details. These features make it difficult to give sufficient conditions under which where the solution u(t, z) is global, i.e. T c = ∞. Indeed, the signature of ∇ 2 k E m (p(t)) may change, and the existence of Strichartz estimates for the linear part is a non-trivial issue. Moreover, λ m (t) may also change sign, making the analysis even more delicate.

Lemma 3 .

 3 1 provides the existence of a local-in-time solution u of the effective mass equation. In view of the multi-scale expansion given in Section 2, we can thus define an approximate solution by(3.1)

  then the above convergence holds with an error of order O( √ ε).

∂ 2

 2 kj k ℓ (H(k) -E m ) χ m + ∂ kj (H(k) -E m ) ∂ k ℓ χ m + ∂ k ℓ (H(k) -E m ) ∂ kj χ m + (H(k) -E m ) ∂ 2 kj k ℓ χ m = 0.Taking the scalar product with χ m , we have:(A.3) ∂ 2 kj k ℓ E m (k) = δ jℓ + -i∂ yj + k j ∂ k ℓ χ m + (-i∂ y ℓ + k ℓ ) ∂ kj χ m , χ m L 2 (Y ) -∂ k ℓ E m ∂ kj χ m + ∂ kj E m ∂ k ℓ χ m , χ m L 2 (Y ) .

  1,1 we obtain an error O( √ ε). The reason is that the coefficients (c γ ) γ∈Γ decrease rapidly for large |γ|, since y → |χ m (y, k)| 2 is smooth, provided Assumption (1.7) holds true and thus we can perform an integration by parts, and use dominated convergence again.

Mathematics Subject Classification. 81Q20, 35A35, 35Q40, 81Q05. R. Carles is supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01). C. Sparber has been supported by the Royal Society through his University Research Fellowship.1


Eric Cl Avier

Pierre Heroux
email: pierre.heroux@univ-rouen.fr

J Oel Gardes

Eric Trupin
email: eric.trupin@univ-rouen.fr

Ground-Truth Production and Benchmarking Scenarios Creation W ith DocM ining

I n this paper we pres ent the DocMining platform and its application to ground-truth datas ets production and page s egmentation evaluation. DocMining is a highly modular framework dedicated to document interpretation where document proces s ing tas k s are modeliz ed with s cenarios . We pres ent here two s cenarios which us e PDF documents , found on the web or produced from XML files , as bas is of the ground-truth datas et.

Introduction

Al gorithm perf ormance eval uation has become a maj or chal l enge f or document anal ysis systems. I n order to choose the right al gorithm according to the domain or to tune al gorithm parameters, users must have eval uation scenarios at their disposal . But ef f icient perf ormance eval uation can onl y be achieved with a representative ground-truth dataset. Theref ore users must have the possibil ity to create, access or modif y ground-truth datasets too.

M any approaches and tool s have been proposed f or benchmarking page segmentation al gorithm and producing ground-truth datasets. Former architectures and environment can be f ound in [START_REF] Kanungo | TRUEVIZ: a groundtruth/metadata editing and visualizing toolkit for OCR[END_REF] [START_REF] Mao | Software architecture of PSET: a page segmentation evaluation toolkit[END_REF] [START_REF] Phelps | The Multivalent Browser : A Platform for New Ideas[END_REF] . The ground-truth datasets are usual l y def ined with image regions f eatures l ike l ocation, l abel , reading order, contained text. But inf ormation needed f or a ground-truth dataset depends more on the user' s eval uation intention than on a f ormal def inition. Theref ore it must be possibl e to upgrade a ground-truth dataset according to the evol ving needs f or eval uation.

Perf ormance eval uation criteria, we f ound in previous works, are various and ref l ect those needs. Noticeabl e criteria are overl ap ratio, regions al ignment, spl it/ merge errors.

This paper describes the DocM ining pl atf orm and its appl ication to ground-truth dataset production and perf ormance eval uation.

This paper is organiz ed as f ol l ow : f irst, we present the architecture of the DocM ining pl atf orm and its maj or components. This pl atf orm is aimed at providing a f ramework f or document interpretation, but its modul ar architecture al l ows mul ti-purpose appl ications based on scenarios. Then we present an appl ication of the architecture where scenarios are designed f or groundtruthing and benchmarking page segmentation al gorithms.

The DocM ining pl atf orm

The DocM ining proj ect is supported by The DocM ining consortium, incl uding f our academic partners, PSI Lab ( Rouen, France) , Proj ect Qgar ( LORI A, Nancy, France) , L3i Lab ( La Rochel l e, France) , DI UF Lab ( Fribourg, Switz erl and) , and one industrial partner, GRI Lab f rom France Tel ecom R&D ( Lannion, France) . DocM ining is a mul ti purpose pl atf orm and is characteriz ed by three maj or aspects.

At f irst, the DocM ining architecture rel ies on a document-centered approach. Document processings communicate through the document itsel f ; such an approach avoids the probl ems of data scattering usual l y met in cl assical document processing chains.

Second, the DocM ining f ramework is based on a pl ug-in oriented architecture. Devel opers can convenientl y add new processings, making thus the pl atf orm easil y upgradeabl e. Document visual iz ation and manipul ation tool s are al so designed according to this approach, so that a user is abl e to f ul l y customiz e the interactions with the document structure.

Third, the pl atf orm handl es scenario-based operations. Running a scenario col l ects users' experience, which becomes part of the scenario itsel f . The scenario may then be transf ormed into a new processing corresponding to a higher-l evel granul arity.

So the DocM ining architecture is real l y modul ar because a user can create his own obj ects, integrate his own processings into the pl atf orm, design his own interf aces, def ine and run his own scenarios. I n this way, the pl atf orm may be used f or various interesting purposes such as benchmarking scenario creation, knowl edge base creation, parameters tuning, etc.

Architecture overview

The platform is based on a Java/XML architecture and relies on four major components:

The PSI Library, deriving from different research works at PSI laboratory, proposes processing chains using statistical and/or structural approaches [START_REF] Delalandre | Symbols Recognition by Global-Local Structural Approaches, Based on the Scenarios Use, and with a XML Representation of Data[END_REF]. It contains a classification tools library and a XML data management library.

The Qgar software system [START_REF] Ph | A complete system for the analysis of architectural drawings[END_REF] is developed by the same-named project at LORIA (www.qgar.org). It is aimed at the design of document analysis applications.

The XMillum (for XML Illuminator) platform [START_REF] Hitz | An architecture for editing document recognition results using XML[END_REF] is developped by the Software, Image & Document Engineering team, at the Departement of Computer Science of the University of Fribourg. It is a general and extensible tool for the edition and visualization of all kinds of document recognition data, that are transformed into XML using XSLT stylesheets. Display and editing functionalities are delegated to plugins.

The ImTrAc package, developed by the GRI Lab at FranceTelecom R&D Lannion, provides a process engine to control processing execution and a scenario engine to control scenario execution, as well as tools for processing integration and scenario creation. An overview of the platform architecture is given in figure 1.

The document structure

The DocMining architecture is based on a document centered approach. A document is represented by an XML tree built according to an XML schema. Basic elements are graphical objects defined by their type (Binary Image, Connected Component, Text Block, etc), their source (the document they are extracted from), and their location in the image. W e did not try to build a complete predefined taxonomy of possible types: The users of the platform can define their own graphical object types when necessary. A graphical object includes intrinsic data describing the way the object is physically represented. The XML schema we have defined for that is based on basic data types such as Freeman Chain, Feature Vector, etc., but, just like previously, a user can define its own data types if necessary. However a document is more than a simple description in terms of graphical objects and data. Its structure also contains information (name, parameters, etc) about processings which have been applied to the document and which have provided the objects.

As shown in figure 1, objects included in the document structure are visualized with XMillum. XSLT stylesheets define what objects may be visualized, how they are visualized, and how events involving objects (e.g. mouse clicks) are handled. Each object is associated to a Java class, which performs the rendering.

Interaction between processings

As shown in figure 1, a processing has no direct access to the document structure and cannot modify it if a so-called contract, defined according to an XML schema, has not been established with the document. The contract describes the processing behavior: the way the processing modifies the XML document structure (by adding, removing or updating nodes), the kind of graphical objects it produces, and parameters that do not require access to the document structure. The objects a processing may modify or access are defined by specifying the " trigger" node (the node that enables the execution of the processing) and the " updated" nodes (the nodes which are modified by the processing).

Scenarios

In order to achieve interpretation tasks, users can interactively build scenarios, which are defined as structured combinations of document processings. There are two ways of creating a scenario. The first way is based on the contracts of the processes. As objects inputs and outputs are specified for all processings in the corresponding contracts, it is possible to determine which processings can feed a given process and then to combine processings. The other way relies on a XMillum component that we have specifically developed for the DocMining platform. It provides means to interact with the ImTrAc processing engine and to visualize the structure of the document. For each object of a document, the ImTrAc engine is able to supply the list of processings that may be applied. Once the user has chosen a processing, the engine supplies its parameters list so as to be able to launch the corresponding process. W hen the process terminates, the document structure is updated and the user can then interact with the newly created objects.

Each user action on the document is recorded in a scenario, which may be applied later to another document. Each step of a scenario acts as a trigger and includes an XPath expression describing the way the required objects have to be extracted from the document.

Page Segmentation evaluation

3.1. Obtaining the document base PDF (Adobe Portable Document Format) documents serve as basis for our ground-truth dataset. Indeed, the PDF format is widely used in many applications (newspaper, advertising, slides, … ) and PDF documents can be easily found on the web. Moreover search engines (like google) allow to refine a search according to the document format. So it is very easy to build a PDF document base where many domains are represented.

A ground-truth dataset can also be built with newly created PDF documents. Figure 2 shows different ways to create a PDF representation of an XML document [START_REF] Carlisle | De XML à PDF via xmltex, XSLT et PassiveTeX[END_REF]. Each of the tools is freely available for download. The input XML document may be or may contain an instance of a widely used DTD such as DocBook, TEI, MathML, etc. Stylesheets (DSSSL or XML) are given for some of theses DTD. These can be modified so that several PDF documents with different formatting attributes may be created from a unique XML document. With those different approaches, it is possible to build a document base which contains "real life" documents obtained through an internet search and "problem specific" documents built from an XML source.

Building the ground-truthing scenario

The main drawback of the PDF format is that it is based on a pure display approach, structural and logical information is not directly accessible, those information must be computed from the low level objects contained in the PDF document.

The ground-truth dataset is obtained through a three steps scenario:

-select the PDF documents.

-extract the physical structure from their PDF representation. -save the generated ground-truth structure. For each scenario step we have defined and developed a processing observing our contract approach. For example, figure 3 shows the contract we defined for the extraction of the physical structure of a PDF document. Contract noticeable elements are bold typed:

-handled_object : the object which is processed. Here the trigger node and the updated node are the same (see 2.3). The document must contain a PdfDoc element to launch the processing. -process_config : the parameters of the processing -produced_object : the processing produces three kind of objects (text lines, words and images) with an unknown cardinality (indicated by the list attribute). -<process_property class_name="PdfSeg"> <service name="nodeAdd"> <handled_object> <object_doc type="PdfDoc"/> <process_config> <param type="ParamIn" name="ExtractImage" support="Data" param_value="0" info="if 1 extract the images"/> <param type="ParamIn" name="RemoveWord" support="Data" param_value="1" info="if 1 remove word textpieces (make the document lighter)"/> </process_config> <produced_object> <object_doc type="TextLine" list='yes'/> <object_doc type="Word" list='yes'/> <object_doc type="Image" list='yes'/> </produced_object> </handled_object> </service> </process_property> figure 3 : contract of the PDF segmentation processing Structure extraction from the PDF is done by using the PDF parsing API provided in the Multivalent package [START_REF] Phelps | The Multivalent Browser : A Platform for New Ideas[END_REF] a java platform dedicated to the visualization of various formats documents

The resulting structure is then marshalled into an XML file observing our XML schema. Therefore, this file contains the ground-truth information of the document Figure 4 shows an excerpt of the ground truth structure and the figure 5 its visualization with XMillum. <object_doc object_id="56" type="TextLine"> <object_pos h="11" w="127" x="14" y="660"/> <object_data> <ascii_data>daz, et créé ce week-end à </ascii_data> </object_data> </object_doc> <object_doc object_id="59" type="TextLine"> <object_pos h="11" w="127" x="14" y="670"/> <object_data> <ascii_data>Bonnefontaine dans le cadre des </ascii_data> </object_data> </object_doc> <object_doc object_id="67" type="TextLine"> Finally, our ground-truth dataset contains information concerning text lines, words (content and location) and images contained in the document. Thus we obtain a partial ground-truth that is sufficient for the first tests.

Building the benchmarking scenario

The benchmarking scenario is composed by three steps :

-Transformation of the PDF document into an image. -Physical structure extraction using a page segmentation processing. -Structure matching evaluation by extracting the corresponding ground-truth in the ground-truth dataset.

Transformation of the PDF document in an image is done by a processing that encapsulates a ghostscript command. At the present time, we have two segmentation algorithms, one based on a classical top down approach and the other one based on an hybrid approach [START_REF] Parodi | An efficient pre-processing of mixed-content document images for OCR systems[END_REF]. Both algorithms produce a resulting XML structure which is matched with the ground-truth dataset to measure the regions overlap ratio. Ground-truth information is extracted from the dataset by using Xpath expressions which allows to select the desired corresponding structure.

Figure 6 shows the contract we defined for this node matching step. The major parameters are bold typed :

-The parameter named KnowledgeBase refers to the ground-truth file. -The parameter named XpathSelector refers to an Xpath expression used to extract the desired objects from the ground-truth file. -The parameter SegmentedObjects which is another Xpath expression, refers to the graphical objects obtained after the segmentation step. The flexibility of Xpath expressions allows the user to select exactly what he needs, he can modify those selection expressions by choosing another kind of object (words for example) or by adding constraints (for example small areas may be filtered)

Node matching itself is done with Yanikoglu's method based on the ON pixels contained in a zone [START_REF] Yanikoglu | Pink panther: a complete environment for ground-truthing and benchmarking document page segmentation[END_REF]. In order to ignore insignificant differences between the ground-truth regions and the segmented ones, only the black pixels content of the areas are taken into account <process_property class_name="NodeMatch"> <service name="nodeAdd"> <handled_object> <object_doc type="BinaryImage"/> <process_config> <param type="ParamIn" name="KnowledgeBase" support="Data" param_value="gd_base.xml" info="tree base"/> <param type="ParamIn" name="XPathSelector" support="Data" param_value= "//object_doc[@type='TextLine']"/> <param type="ParamIn" name="SegmentedObjects" support="ObjectDoc" param_value= "object_doc[@type='TextLine']"/> </process_config> <produced_object> <object_data> <feature name= "NodeMatching"/> </object_data> </produced_object> </handled_object> </service> </process_property> figure 6 : Contract of the Node Matching process

Conclusion and future works

Although many page segmentation evaluation problems are not yet addressed in this paper (blocks labeling, reading order, errors evaluation), we think that the DocMining architecture is well suited to tackle many aspects of ground-truthing and benchmarking. Indeed, DocMining's strong modularity can help building groundtruthing benchmarking scenarios according to users needs Editing and visualization tools for manipulating groundtruth datasets may be added as well. Moreover, processing modularity allows user to design their own performance evaluation algorithm.

Therefore, the platform architecture allows future works to include errors evaluation processings and tools for adding new features to the datasets (labels, reading order).

In this paper, the solution we use to generate groundtruth and benchmarking scenarios for page segmentation is based on PDF documents. As shown on figure 2, PDF documents can be produced from XML data. The markup of an XML document often describes its logical structure. Therefore, a ground-truth dataset may be built with XML documents representing the logical structure and their associated PDF version corresponding to the physical structure.

Figure 1 :

 1 Figure 1 : DocMining platform architecture

figure 2 :

 2 figure 2 : different ways for producing a PDF document from an XML source

figure 4 :

 4 figure 4 : ground-truth structure extracted from a PDF File