Chetra Mang
email: chetra.mang@irt-systemx.fr

Julien Cortial
email: julien.cortial@safrangroup.com

Chiara Nardoni
email: chiara.nardoni@irt-systemx.fr

Felipe Bordeu
email: felipe.bordeu@safrangroup.com

A deterministic approach for shape and topology optimization under uncertain loads

Keywords: shape and topology optimization, level set, random uncertainties, deterministic approach

The present work aims at handling uncertain loads in shape and topology optimization. More specifically, we minimize objective functions combining mean values and variances of standard cost functions and assume that uncertainties are small and generated by a finite number N of random variables. A deterministic approach that relies on a second-order Taylor expansion of the cost function has been proposed by Allaire & Dapogny. 1 That method requires a computational effort comparable to the one for an N-load problem. This work presents a general framework to handle uncertainties on arbitrary static load cases where perturbations on both surface forces and body forces are considered. We demonstrate the effectiveness of the approach in the context of level-set-based topology optimization for the robust compliance minimization on three-dimensional test cases.

Introduction

Shape and topology optimization is routinely used in industry in the design process in mechanical parts. Most if not all commercial grade tools assume that exact functional specifications are available whereas in practice these can only be determined approximately, especially during early stages of design, when topology optimization provides the most benefits. Moreover, the feasibility and optimality of the solution can be very dependent on a slight change on the actual requirements. [START_REF] Allaire | A deterministic approximation method in shape optimisation under random uncertainties[END_REF] Unless uncertainty has been taken into account, the chosen design may turn out to be suboptimal, or even worse, defective. [START_REF] Pons-Prats | Robust shape optimisation in aeronautics[END_REF] Therefore, the development of procedures that improve the robustness of the solution should increase the added value of shape and topology optimization in the industrial design process.

Several strategies have been proposed to account for the effects of small uncertainties. On the one hand, when no statistical information is available, a worst-case design approach can be followed : The idea is simply to minimize the maximal value of the objective function for any possible perturbation. The computation is made tractable by assuming that it is possible to linearize the criterion with respect to the perturbation. [START_REF] Allaire | A linearized approach to worst case design in parametric and geometric shape optimisation[END_REF] While easy to implement, this strategy can lead to structures with poor nominal performances. On the other hand, when the user has some knowledge on the statistical distribution of the perturbation, a probabilistic description can be introduced. Then it becomes possible to consider the mean value and the variance of the objective fonction in the optimization problem, leading to solutions with better performance than in the previous case.

In this work, the latter approach is followed. The paper is organized as follows. First, the theoretical framework, mostly based on Allaire & Dapogny's work, [START_REF] Allaire | A deterministic approximation method in shape optimisation under random uncertainties[END_REF] allows a general treatment of uncertain loads. Then, the special case of the robust compliance is derived and further developed. Finally, the method is implemented in the context of level-set-based topology optimization and illustrated on a three-dimensional test-case.

Shape and topology optimisation via level set

Shape and topology optimisation

The goal of shape and topology optimization is to find a shape that is a solution of a problem of the form:

min Ω∈O ad J (Ω), (1)
where J (Ω) is a cost function that depends on the domain Ω ∈ R d and O ad is the set of admissible shapes.

Relying on Hadamard's notion of shape derivative, we evaluate the cost function sensitivity with respect to a certain class of domain perturbations in order to implement an iterative continuous optimization algorithm. A variation of a domain Ω is given by:

Ω θ = (I + θ )(Ω), (2)
where θ ∈ W 1,∞ (R d , R d ) and for θ sufficiently small,

(I + θ ) is diffeomorphism in R d . A function J (Ω) admits a shape derivative if the mapping θ → J (Ω θ ) from W 1,∞ (R d , R d ) to R is Fréchet differentiable at θ = 0.
The Fréchet derivative (or shape derivative) of J (Ω) is denoted by θ → J (Ω)(θ ) and defined as:

J (Ω θ ) = J (Ω) + J (Ω)(θ ) + o(θ ), (3)
where

|o(θ )| ||θ || W 1,∞ (R 3 ,R 3 ) θ →0 -→ 0 and J (Ω) is a continuous linear form on W 1,∞ (R d , R d ).
The shape derivative depends on the normal trace of the perturbation θ on Ω. [START_REF] Henrot | Variation et optimisation de formes: une analyse géométrique[END_REF] In many relevant cases, it can be written as:

5 ∀θ ∈ W 1,∞ (R 3 , R 3 ), J (Ω)(θ ) = ∂ Ω v Ω (s)θ n Ω ds, (4)
where n Ω is outward unit normal to the boundary ∂ Ω and v Ω (s) depends on J (Ω).

Level set method

The level set method introduced by Osher and Sethian 6 is used for the numerical implementation of the shape and topology optimization problem. In this framework, a shape Ω ⊂ R d is represented by the negative subdomain of an auxiliary level set function φ : R d → R :

∀x ∈ R d ,    φ (x) < 0 if x ∈ Ω φ (x) = 0 if x ∈ ∂ Ω φ (x) > 0 if x ∈ Ω c (5)
The motion of a domain Ω(t) for t ∈ [0, T ] according to a normal velocity field V (x,t) translates into an Hamilton-Jacobi equation for the associated level set function φ (t, .):

∂ φ ∂t +V |∇φ | = 0,t ∈ (0, T ), x ∈ R d , ( 6
)
where t is a pseudo-time, the upper bound T is analogous to the step size of a gradient descent and the advection field V derives from the shape derivative D Ω of the objective function J (Ω). Instead of taking the direction of steepest descent V = -D Ω , a regularisation and an extension process to compute V from D Ω is implemented. [START_REF] Allaire | Structural optimization using shape sensitivity analysis and a level set method[END_REF] 3 Random perturbations on mechanical loads in shape and topology optimization

∈ L 2 (Ω) d and g ∈ L 2 (Γ N ) d with Γ N ⊂ ∂ Ω and Γ 0 = ∂ Ω\ (Γ N ∪ Γ D ).
The mechanical problem obeys the system of equations :

       -div(σ (u Ω )) = f in Ω u Ω = 0 on Γ D σ (u Ω )n = g on Γ N σ (u Ω )n = 0 on Γ 0 (7)
We introduce small perturbations f to the body forces

f = f 0 + f such that f 0 ∈ L 2 (Ω) d , f ∈ L 2 (Ω) d and ĝ to the surface forces g = g 0 + ĝ such that g 0 ∈ L 2 (Γ N ) d , ĝ ∈ L 2 (Γ N ) d .
A second order Taylor expansion of a cost function

C (Ω, f , g) = Ω j( f , u Ω )dx + Γ N k(g, u Ω )ds (8) around f 0 ∈ L 2 (Ω) d and g 0 ∈ L 2 (Γ N ) d yields : C (Ω, f 0 + f , g 0 + ĝ) = Ω j( f 0 + f , u Ω ) dx + Γ N k(g 0 + ĝ, u Ω ) ds = Ω j( f 0 , u Ω ) dx + Ω ( f j( f 0 , u Ω ). f + u Ω j( f 0 , u Ω ).u 1 Ω ( f )) dx + 1 2 Ω ( 2 f j( f 0 , u Ω )( f , f ) + 2 f u j( f 0 , u Ω )( f , u 1 Ω ( f )) + 2 f j( f 0 , u Ω )(u 1 Ω ( f ), u 1 Ω ( f ))) dx + R f ( f ) Γ N k(g 0 , u Ω ) ds + Γ N ( g k(g 0 , u Ω ). ĝ + u Ω k(g 0 , u Ω ).u 1 Ω ( ĝ)) ds + 1 2 Γ N ( 2 g k(g 0 , u Ω )( ĝ, ĝ) + 2 g u j(g 0 , u Ω )( ĝ, u 1 Ω ( ĝ)) + 2 g j(g 0 , u Ω )(u 1 Ω ( ĝ), u 1 Ω ( ĝ))) ds + R g ( ĝ) (9) where R f ( f ) = Ω 1 0 ∂ 3 C ∂ f d ( f , f , f ) dt dx (10)
and

R g ( ĝ) = Γ N 1 0 ∂ 3 C ∂ g d ( ĝ, ĝ, ĝ) dt ds (11) R( f , ĝ) = R f ( f )+R g ( ĝ)
is the residual of the expansion. We assume that the perturbations are random and accordingly we set f ≡ f (x, ω), ĝ ≡ ĝ(x, ω), for x ∈ Ω and ω ∈ O, where (O, F , P) is a probability space.

Mean and variance of the cost function

Mean of the cost function

Let :

M (Ω) = O C (Ω, f (., ω), g(., ω)) P(dω) (12) denote the mean of the cost function C (Ω, f (., ω), g(., ω)) It is assumed that f ∈ L 2 (O, L 2 (Ω) d ) (respectively ĝ) is a finite sum of deterministic functions f i ∈ L 2 (Ω) d , i = 1, ..., N (respectively g i ) weighted by independant Gaussian random variables ξ f i (respectively ξ g i ): f (x, ω) = Σ N i=1 f i (x)ξ f i (ω)χ f i (13) ĝ(x, ω) = Σ N i=1 g i (x)ξ g i (ω)χ g i (14)
where χ f i : Ω → {0, 1} (respectively χ g i ) is an indicator function that identifies the region affected by the perturbation f i (respectively g i ).

Then it comes:

M (Ω) = Ω j( f 0 , u Ω )χ f 0 dx + 1 2 Ω (Σ N i=1 2 f j( f 0 , u Ω )( f i , f i ))χ f i dx +Σ N i=1 Ω ( f u j( f 0 , u Ω )( f i , u 1 Ω,i ))χ f i dx + 1 2 Ω (Σ N i=1 2 u j( f 0 , u Ω )(u 1 Ω,i , u 1 Ω,i ))χ f i dx Γ N k(g 0 , u Ω )χ g 0 ds + 1 2 Γ N (Σ N i=1 2 g k(g 0 , u Ω )(g i , g i ))χ g i ds +Σ N i=1 Γ N ( g u k(g 0 , u Ω )(g i , u 1 Ω,i ))χ g i ds + 1 2 Γ N (Σ N i=1 2 u k(g 0 , u Ω )(u 1 Ω,i , u 1 Ω,i ))χ g i ds (15) such that u 1 Ω,i := u 1 Ω ( f i , g i ) is the unique solution in H 1 Γ D (Ω) d of the problem :    -div(σ (u 1 Ω,i )) = f i χ f i dans Ω u 1 Ω,i = 0 sur Γ D σ (u 1 Ω,i )n = g i χ g i sur Γ N (16)
Variance of the cost function

The variance V (Ω) of the cost function C (Ω) is defined as:

V (Ω) = O (C (Ω, f (., ω)) -M (Ω)) 2 P(dω) (17)
Following Allaire & Dapogny, [START_REF] Allaire | A deterministic approximation method in shape optimisation under random uncertainties[END_REF] we obtain the expression:

Ṽ (Ω) = Σ N i=1 (a 2 Ω,i + b 2 Γ N ,i ) (18)
where

a Ω,i := Ω ( f j( f 0 , u Ω ). f i + u j( f 0 , u Ω ).u 1 Ω,i )χ f i dx and b Γ N ,i := Γ N ( g k(g 0 , u Ω ).g i + u k(g 0 , u Ω ).u 1 Ω,i )χ g i dx

Shape derivatives of mean and variance of the cost function

Shape derivative of the mean of the cost function

According to Theorem 14 in Allaire & Dapogny, 1 the Fréchet derivative of the mean of the cost function is given by :

∀θ ∈ O ad , M (Ω)(θ ) = Γ N ( j( f 0 , u Ω )χ f 0 + 1 2 Σ N i=1 2 f j( f 0 , u Ω )( f i , f i )χ f i +Σ N i=1 f u j( f 0 , u Ω )( f i , u 1 Ω,i )χ f i + 1 2 Σ N i=1 2 u j( f 0 , u Ω )(u 1 Ω,i , (u 1 Ω,i )χ f i + σ (u Ω ) : e(p 0 Ω )χ f 0 +Σ N i=1 σ (u 1 Ω,i ) : e(p 1 Ω,i )χ f i -f 0 .p 0 Ω χ f 0 -Σ N i=1 f i .p 1 Ω,i χ f i )θ n ds + Γ N ( ∂ ∂ n k(g 0 , u Ω )χ g 0 +Hk(g 0 , u Ω )χ g 0 + 1 2 Σ N i=1 2
g k(g 0 , u Ω )(g i , g i )χ g i +Σ N i=1 g u k(g 0 , u Ω )(g i , u 1 Ω,i )χ g i + 1

2 Σ N i=1 2 u k(g 0 , u Ω )(u 1 Ω,i , (u 1 Ω,i )χ g i -∂ ∂ n k(g 0 , p 0 Ω )χ g 0 -Hk(g 0 , p 0 Ω )χ g 0 )θ n ds, (19) p 0 Ω and p 1 Ω,i for i = 1, ..., N are the (N + 1) adjoint states, defined as the unique solutions of H 1 Γ D (Ω) d of the following variational problems :

3. 1

 1 A model problem Let us consider a linear elastic solid domain Ω ⊂ R d . The Dirichlet boundary condition is imposed on Γ D ⊂ ∂ Ω. Body and surface forces are respectively denoted by f

Figure 2 :

 2 Figure 2: Optimal shape at iteration 120 for (m 1 = 0, m 2 = 0) (top), (m 1 = 0.005, m 2 = 0.0006) (middle) and (m 1 = 0.008, m 2 = 0.001) (bottom).

Figure 3 :

 3 Figure 3: Convergence histories for compliance (upper) and volume (lower) for (m 1 = 0, m 2 = 0) (top), (m 1 = 0.005, m 2 = 0.0006) (middle) and (m 1 = 0.008, m 2 = 0.001) (bottom).

EUROGEN 2019

September 12-14, 2019, Guimarães, Portugal ∀v ∈ H 1 Γ D (Ω) [START_REF] Allaire | A linearized approach to worst case design in parametric and geometric shape optimisation[END_REF] , Ω (σ (p 0 Ω ) : e(v)) dx = -Ω ( u j( f 0 , u Ω )v) dx - [START_REF] Allaire | A deterministic approximation method in shape optimisation under random uncertainties[END_REF] 2 Ω (Σ N i=1

Shape derivative of variance of the cost function

The Fréchet derivative of the variance of the cost function is given by :

4 Application for robust compliance

The so-called compliance function is defined as :

It is a special case of (8) with j( f , u Ω ) = f .u Ω and k(g, u Ω ) = g.u Ω . The expressions for the mean and the variance of the cost function as well as the associated shape derivatives are then :

where

Compliance minimization with uncertain loads

In the following example, we consider the volume-constrained minimization of the weighted sum

of the approximate mean value and standard deviation of the compliance (23) where δ 0.

Let us consider the test case of a three-dimensional chair. The design domain is a subset of a 1.2 × 1 × 2 box, as shown on Figure 1. At the bottom of the chair, four small height cylidrical non-designed zones are clamped. The seat and the backrest are also non-design zones, denoted respectively by χ 1 and χ 2 .

The vertical nominal force applied on the seat surface is equal to f 10 = -0.025e 3 , and the horizontal nominal force applied on the backrest surface is equal to f 20 = -0.003e 1 . We introduce the perturbations f11 = m 1 ξ 1 e 1 on the seat surface, and f21 = m 2 ξ 2 e 3 on the backrest surface, where ξ 1 and ξ 2 denote two independant identically distributed Gaussian random variables with distribution N (µ, s).

We can write the applied surface forces as :

In the example below, the numerical values for the mean and the variance are set to µ = 1. and s = 0.01. The optimized shapes obtained at iteration 120 for (m 1 , m 2 ) ∈ {(0, 0), (0.005, 0.0006), (0.008, 0.001)} are displayed on Figure 2. This result clearly shows that the optimized solution changes significantly when uncertainties are taken into account. The convergence histories of compliance and volume for the three cases are shown on Figure 3.


Jean-Baptiste Pomet

Claude Samson

Claude Samson Exponential

Exponential stabilization of nonholonomic systems in power form

In: proceedings of held in Rio de Janeiro (Brasil), September 1994.

IFAC Symposium on Robust Control

  Fig. 2: Convergence of x, y, and theta


Annamaria Barbagallo

AND MICHEL TH ÉRA Dedicated to Professor Octavian-Emil Ernst

Michel Théra

Nearness Set-Valued

Antonino Maugeri

SET-VALUED ORTHOGONALITY AND NEARNESS

Keywords:

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In articles ( [START_REF] Campanato | A Cordes type condition for nonlinear nonvariational systems[END_REF][START_REF] Campanato | On the condition of nearness between operators[END_REF]) published at the end of the eighties, S. Campanato introduced and studied the notion of nearness between two functions defined on a set S and taking values in a real normed vector space (X, • ). More precisely, a function f : S → X is near function g : S → X in the sense of Campanato, iff there exist two real constants α > 0 and κ ∈ [0, 1) such that [START_REF] Alonso | On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces[END_REF] (f (s 1 ) -αg(s 1 ))

-(f (s 2 ) -αg(s 2 )) ≤ κ f (s 1 ) -f (s 2 )
for every s 1 , s 2 ∈ S. When X is a Banach space, S. Campanato proved [4, Theorem 1 & 2] that if f is bijective and f is near g with constants α and κ, then g is necessarily a bijection. Moreover, the Lipschitz modulus of the bijective function g • f -1 : X → X is less than or equal to α 1-κ , namely

g(f -1 (u)) -g(f -1 (v)) ≤ α 1 -κ u -v ∀u, v ∈ X.
When S = X, f is the identity operator and g is a linear function, Campanato results boils down to the well-known Neumann's Lemma [START_REF] Hansen | Functional analysis. Entering Hilbert space[END_REF]Lemma 5.1.6], see also [START_REF] Barbagallo | Orthogonality in locally convex spaces: Two nonlinear generalizations of Neumann's lemma[END_REF] for more details and extensions:

Lemma 1.1 (Neumann's lemma). Let X be a real Banach space, and A be a bounded linear operator on X, such that

(2) ∃ α > 0, 0 ≤ κ < 1 : Id -α A ≤ κ Id ,
where • is the operator norm defined by

A := inf{c ≥ 0 : A(x) ≤ c x , ∀x ∈ X}.
Then A is invertible, and

A -1 ≤ α 1-κ .
A strongly related notion to the Campanato nearness is the Birkoff-James orthogonality in a locally convex space (see, [9, Definition 1.1]). Given a vector space Y equipped with a family of semi-norms we say that the vector

u ∈ Y is Birkhoff-James orthogonal to v ∈ Y on (Y, P) iff (3) p ι (u) ≤ p ι (u -t v) ∀t ∈ R, ι ∈ I.
Of course, when Y is a normed space, P contains only one element, namely the norm of Y , and we retrieve the original Birkhoff's definition (see [START_REF] Birkhoff | Orthogonality in linear metric spaces[END_REF]; the reader is also referred to the very complete survey [START_REF] Alonso | On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces[END_REF]). An abstract notion of the Campanato nearness can be defined as follows: given two vectors u and v in a locally convex space (Y, P), we say that u is Campanato near v iff there exist two real constants > 0 and κ ∈ [0, 1) such that (4)

p ι (u -α v) ≤ κ p ι (u) ∀ι ∈ I.
The original Campanato nearness is obtained for the particular case of the locally convex space (X S , P), where X S denotes the set of all functions from S to X, and the set P contains all semi-norms of the form

(5) p s1,s2 : X S → R, p s1,s2 (f ) := f (s 1 ) -f (s 2 ) , ∀f ∈ X S ,
for all the points s 1 , s 2 ∈ S such that s 1 = s 2 . As no confusion risks to occur, we will drop, in the remaining part of this note, the wording "Birkhoff-James" and "Campanato", and simply speak of orthogonality and nearness.

The main object of this note is to extend the definitions of orthogonality and nearness from the original case of single-valued functions, to the case of set-valued mappings. As many operators of interest in non-smooth optimisation are set-valued -like the subdifferential of a convex function, to pick an example out of many -a correct definition of nearness covering the set-valued case should be a valuable tool in establishing set-valued generalizations of the Neumann and Campanato results. This note is organized as follows. Section 2 addresses several very simple attempts to define set-valued orthogonality and nearness. Their coherence and possible uses is discussed in the light of several examples.

As a consequence of this analysis, we propose in Section 3 a new definition of orthogonality and nearness in the set-valued setting.

Three attempts of defining set-valued orthogonality and nearness

This section is devoted to the study of three attempts to define orthogonality and nearness for set-valued mappings. The first one is based on the notion of Pompeiu-Hausdorff distance in a metric space X, while the two others are stated in terms of the selections of the two involved set-valued mappings. The limitations of these three attempted definitions are highlighted by some elementary examples.

2.1.

Distance-based nearness : first attempt of a definition. Let us consider the Pompeiu-Hausdorff distance, defined between two subsets of X by the following formula:

d(A, B) := max(sup v∈B inf u∈A u -v , sup u∈A inf v∈B u -v ) ∀A, B ⊂ X.
Let us recall that by a set-valued mapping (multifunction, correspondance, pointto-set, in some other terminologies) F , we mean a function between S and the set P(X) of all the subsets (possibly empty) of X. Throughout the paper, we will use the notation F : S ⇒ X.

A simple transposition to the set-valued setting of the original definition of orthogonality and nearness leads us to the following tentative definition.

Definition 1. Given two set-valued mappings F, G : S ⇒ X, we say that F is orthogonal to G iff d(F (s 1 ) -tG(s 1 ), F (s 1 ) -tG(s 2 ))) ≥ d(F (s 1 ), F (s 2 )) ∀t ∈ R, s 1 , s 2 ∈ S,
and F is near G iff there are two real constants α > 0 and κ ∈ [0, 1) such that

d(F (s 1 ) -αG(s 1 ), F (s 1 ) -αG(s 2 ))) ≤ κ d(F (s 1 ), F (s 2 )), ∀s 1 , s 2 ∈ S.
However, within the setting of this definition, there are set-valued mappings which are not near themselves, as proved by the following result.

Proposition 2.1. Let S be a set containing at least two points, and X be a real normed vector space containing at least one non-null vector. Then, there is a setvalued mapping F : S ⇒ X which is not near itself, according to Definition 1.

Proof of Proposition 2.1. Let s 1 and s 2 two different elements from S, and consider the set-valued mapping F : S ⇒ X given by

F (s) =    B X if s = s 1 {0} if s = s 1
, where B X is the closed unit ball in X.

Let us pick a positive real number t. Since X contains at least a non-null element, the Pompeiu-Hausdorff distance between the sets t B X and {0} amounts to t. Accordingly, the Pompeiu-Hausdorff distance between the two sets

F (s 1 ) -αF (s 1 ) = (1 + α) B X , F (s 2 ) -αF (s 2 ) = {0}
equals to 1 + α, and we deduce that

d(F (s 1 ) -αG(s 1 ), F (s 2 ) -αG(s 2 )) = 1 + α > κ = κ d(F (s 1 ), F (s 2 ))
for any two real constants α > 0 and κ ∈ [0, 1).

Let us remark that the existence of set-valued applications which are not near themselves is still achieved even when the Pompeiu-Hausdorff distance is replaced by one of the numerous distances between sets available in the mathematical literature.

It is easy to verify that, if δ : 2 X × 2 X → R + is any of the distances described in the survey paper ( [START_REF] Conci | Distances between set -a survey[END_REF]), then the function e : R + → R + , e(s) := δ({0}, s B X ∀s ≥ 0 in increasing. Accordingly, Proposition 2.1 holds true even if the Pompeiu-Hausdorff distance is replaced in Definition 1 by another distance between sets. It appears thus that no valid definition of the set-valued orthogonality and nearness can be achieved by using distance functions.

2.2.

Selections-based nearness: second attempt of a definition. Given F : S ⇒ X, we call selection of F any function σ F : S → X such that σ F (s) ∈ F (s) for any s ∈ S.

Our second very simple attempt to define a correct set-valued orthogonality and nearness is based on the analysis of all the selections of the two set-valued mappings involved.

Definition 2. Let F, G : S ⇒ X. Then F is orthogonal to (respectively near) G iff any selection of F is orthogonal to (respectively near) any selection of G.

Once again, this definition leads to the existence of set-valued mappings which are not near themselves. Proposition 2.2. Let S be a set containing at least two points, and X be a real normed vector space containing at least one non-null vector. Then, there is a setvalued mapping F : S ⇒ X which is not near itself, according to Definition 2.

Proof of Proposition 2.2. Let us prove that the constant set-valued mapping F : S ⇒ X, F (s) := B X ∀s ∈ S, is not near itself according to Definition 2.

As the vector space X contains non-null vectors, it follows that there is u, a vector of norm equal to 1. Let s 1 and s 2 two different elements of S. The functions

σ 1 : S → X σ 1 (s) :=    u, if s = s 1 -u, if s = s 1 , and
σ 2 : S → X σ 2 (s) :=    u, if s = s 1 -u, if s = s 1 ,
are thus two selections of the set-valued mapping F . But

(σ 1 (s 1 ) -α σ 2 (s 1 )) -(σ 1 (s 2 ) -α σ 2 (s 2 )) = 2(1 + α) > 2κ = κ σ 1 (s 1 ) -σ 1 (s 2 )
for any two real constants α > 0 and κ ∈ [0, 1).

2.3.

Selections-based nearness: third attempt of a definition. The first two attempts to define set-valued nearness were too restrictive. In both cases, setvalued mappings not being near themselves have been identified. Accordingly, in this subsection we address a selection-based definition which is considerably broader than Definitions 1 and 2. Definition 3. A set-valued mapping F is orthogonal to (respectively near) G iff any selection of F is orthogonal (respectively near) to at least one selection of G, and for any selection of G, there is at least one selection of F which is orthogonal to ( respectively near) it.

It is now obvious that any set-valued mapping is near itself, in the sense of Definition 3, so the main difficulty plaguing Definitions 1 and 2 is now removed. However, Definition 3 is not satisfactory since, unlike the single-valued case, basic algebraic properties are not inherited from F to G in the case when F is near G, as proved by the following result. Proposition 2.3. Let S := {s 1 , s 2 } be a set containing two points, and X be a real normed vector space containing at least one non-null vector. Then, there is a set-valued mapping F : S ⇒ X which is injective, in the sense that F (s 1 ) = F (s 2 ), and which is near a constant set-valued mapping G, according to Definition 3.

Proof of Proposition 2.3. Let u be a non-null vector from X, and define Each of the two set-valued mappings F and G has exactly four selections:

F : S ⇒ X, F (
σ F,1 (s 1 ) := -u, σ F,1 (s 2 ) := 0, σ F,2 (s 1 ) := -u, σ F,2 (s 2 ) := u σ F,3 (s 1 ) := u, σ F,1 (s 2 ) := 0, σ F,4 (s 1 ) := u, σ F,2 (s 2 ) := u are the selections of F , while the selections of G are σ G,1 (s 1 ) := 0, σ G,1 (s 2 ) := 0, σ G,2 (s 1 ) := 0, σ F,2 (s 2 ) := u σ G,3 (s 1 ) := u, σ F,1 (s 2 ) := 0, σ G,4 (s 1 ) := u, σ F,2 (s 2 ) := u. It is straight-forward to prove that σ F,1 is near σ G,2 , that σ F,2 is near σ G,2 , that σ F,3 is near σ G,3 , that σ F,4 is near σ G,1
, and that σ F,4 is near σ G,4 . So the set-valued mappings F and G satisfy conditions of Definition 3. Yet, F is injective, while G is constant.

Nearness for set-valued functions

In view of the detailed analysis achieved in Section 2, we conclude the correct definition of set-valued orthogonality and nearness should be broader than Definition 2, but more restrictive than Definition 3. In order to attain this objective, let us introduce the following notations. Given F, G : X ⇒ X, we say that a binary relation R on X is (F,G)-compatible if ∀s ∈ S, ∀u ∈ F (s), ∃v ∈ G(s) s.t. u R v, and ∀s ∈ S, ∀v ∈ G(s), ∃u ∈ F (s) s.t. u R v.

Moreover, a selection σ G of the set-valued mapping G is said R-compatible with one selection σ F de F , if σ F (s) R σ G (s) for any s ∈ S.

We are now ready to state the main notion of this note. It is now easy to see that any set-valued function F is near itself (just take for R the identity relation, x R y ⇔ x = y, remark that R is (F, F )-compatible, and notice that any selection σ F of F is R-compatible only with itself).

In a future article, we will investigate the properties of the set-valued mapping F which are inherited by any set-valued mapping G such that F is near G.

  s 1 ) := {-u, u}, F (s 2 ) := {0, u} and G : S ⇒ X, G(s 1 ) = G(s 2 ) := {0, u}.

Definition 4 .

 4 A set-valued mapping F is said to be orthogonal (respectively near) to G iff there exists a (F, G)-compatible binary relation R on X such that any selection σ F of F is orthogonal (respectively near) to any R-compatible selection σ G of G.


EIFER Samuel Thiriot
email: samuel.thiriot@res-ear.ch

Impact of the Interaction Network on the Dynamics of Word-of-Mouth with Information Seeking

Word-of-Mouth refers to the dynamics of interpersonal communication occurring during the diffusion of innovations (novel practices, ideas or products). According to field studies, word-of-mouth is made of both information seeking and proactive communication: individuals first become aware of the existence of an innovation, then start actively seeking out for the expert knowledge required to evaluate the innovation; when they hold the expert knowledge, they might start promoting it pro-actively. Successful diffusion of innovation requires the individuals to hold both awareness and expert knowledge, so they can evaluate the innovation and use it properly. A computational model "USA/IPK" was recently proposed to study the role and impact of information seeking on the dynamics of word-of-mouth. We propose here an analysis of the impact of the network of interaction on the dynamics of this model. We compare the dynamics of the model over networks generated with different algorithms with the original dynamics. The results demonstrate the dynamics of the model are similar across tested networks, with the noticeable exception of the efficiency of the diffusion which varies between networks having similar densities and sizes.

Introduction

Word-of-mouth: Evidence on Information Seeking

When individuals discuss an innovation (a novel product, practice, idea) [START_REF] Arndt | Role of product-related conversations in the diffusion of a new product[END_REF], they spread the word about its existence and qualities. More people become aware of a product through word-of-mouth than traditional advertisement [START_REF] Jagdish | Word of Mouth in Low Risk Innovations[END_REF]. Most consumers attribute a higher importance to interpersonal influence than other sources [START_REF] Norman | The Mathematical Theory of Epidemics[END_REF]. As a consequence, word-of-mouth is consistently said to determine the success or failure of innovations [START_REF] Arndt | Role of product-related conversations in the diffusion of a new product[END_REF] and facilitate the diffusion of products [START_REF] Norman | The Mathematical Theory of Epidemics[END_REF].

Word-of-mouth was often reduced to an epidemic process in which individuals "contaminate" each other with information [9]. Yet interpersonal communication about innovations or products [2] does not only include the proactive emission of information, but also communications initiated by people who seek out information about an innovation [7,[START_REF] Arndt | Role of product-related conversations in the diffusion of a new product[END_REF]. When an individual discovers the existence of the innovation, that is when he receives awareness knowledge from advertisement or another individual, he might engage (or not) in information seeking depending on his characteristics [7,[START_REF] Arndt | Role of product-related conversations in the diffusion of a new product[END_REF]. Expert knowledge covers "how to" use the innovation (know-how knowledge) [START_REF] Arndt | Role of product-related conversations in the diffusion of a new product[END_REF], "why" the innovation works (principles-knowledge) [START_REF] Arndt | Role of product-related conversations in the diffusion of a new product[END_REF], product category or product-class knowledge or brand knowledge . This expert knowledge might be gathered from individuals who hold it prior to the diffusion of the innovation because of their education or training, because they read specialized press, had experience with another product of same brand, category or class, or because they received this information from another individual. Once they hold the expert knowledge, people might engage into pro-actively passing the word around about the innovation, for instance because they are willing to help others [7] or because they are satisfied or dissatisfied after adoption [START_REF] Thiriot | Customer Satisfaction and Word of Mouth[END_REF].

Information seeking stands as a step required for most individuals to be able to decide to adopt or reject a product [START_REF] Norman | The Mathematical Theory of Epidemics[END_REF]. In the case of the diffusion of disruptive innovations such as vaccination or contraceptives [START_REF] Arndt | Role of product-related conversations in the diffusion of a new product[END_REF], information seeking is even seen as a mandatory step for individuals to adopt the innovation, as it enables people to understand why it works and how to use it. For instance, parents do not accept the vaccination of their children without gathering more knowledge first [26]. Even if innovations might be adopted without expert knowledge, the misuse of the innovation may later cause its discontinuance [START_REF] Arndt | Role of product-related conversations in the diffusion of a new product[END_REF]. As a consequence, a company or organization promoting an innovation attempts to maximize the proportion of the population which is not only aware, but also holds expertise on the innovation [START_REF] Arndt | Role of product-related conversations in the diffusion of a new product[END_REF].

Word-of-mouth: Computational Models

Numerous computational or mathematical models [19,20,16] were designed to understand the diffusion of information and innovations [12,11,17], assess the potential diffusion of products [4], and recommend strategies to accelerate or maximize this diffusion [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF]. The three main types of models related to information diffusion are based on information cascades, social influence and social learning [17,[START_REF] Girvan | Community structure in social and biological networks[END_REF]. The models developed in these last two categories describe the flow of influence within the population, without explicitly representing the flow of information, and can not be used to study the impact of information seeking. Marketing models based on information cascades [9, [START_REF] Goldenberg | Talk of the Network: A Complex Systems Look at the Underlying Process of Word-of-Mouth[END_REF][START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF] rely on an analogy with epidemic models [9,6] such as the SIR model [15,3]: every individual is either in state Susceptible (no information), Infective (informed and pro-actively passing the information to others) or Recovered (informed but passive). A Susceptible individual becomes Infective (S → I) when he meets an Infective individual. After a given time, Infective agents become Recovered (I → R). When enough individuals are passing the word around, information cascades appear in the simulations, as observed in reality [17]. In this case, the cumulated curve of the proportion of people informed in time follows the traditional S-shaped curve. Unfortunately, because they only include the information passing behaviour without any information seeking, these models only capture part of the dynamics of word-of-mouth.

A computational model was recently proposed to explore the dynamics of word-of-mouth with information seeking [START_REF] Chatterjee | The Innovation Diffusion Process in a Heterogeneous Population: A Micromodeling Approach[END_REF]. In this model built as an evolution of the SIR model, two dimensions of knowledge are distinguished: awareness knowledge which refers to knowing the innovation exists, and expert knowledge which allows the actual understanding of the innovation. The agents representing the individuals are associated with states of information for awareness and expertise, and are associated with traits determining their behaviour regarding information (e.g. curious agents will seek out information when they become aware). In a typical simulation, the population is initialized as unaware of the innovation. A given proportion of agents k is initialized with the expert knowledge, which they are supposed to know because of their education, training or experience with another similar innovation. A low proportion of expertise k in the population is said to represent a disruptive innovation; a higher proportion of expertise represents an incremental innovation that most people can understand as soon as it is advertised. During N steps of the simulation, an advertisement campaign dispatches the awareness knowledge to a small proportion of the population. Then the interpersonal interactions in the model drive the propagation of knowledge. Agents who became aware of the innovation, and have the trait curious, start to seek out for expert knowledge around them; by doing so, they propagate awareness around them; when they discover the expert knowledge, they stop seeking out information; the individuals having trait enthusiastic start promoting the innovation, whilst the others become passive. A typical simulation exhibits two S-shaped curves: the higher is the curve of awareness. The lower one is the curve of people who hold both awareness and expert knowledge. This model enables to experiment on the dynamics of information seeking: what are the relative impacts of curious people and enthusiastic ones? On which conditions can we enhance the retrieval of expertise in the population? Published results highlight a surprisingly high efficiency of information seeking, which enables most of the population to achieve gathering an expert knowledge held by as few as 0.1% of the population. Simulations experiments also suggest three different regimes for different proportions of initial knowledge, suggesting different communication strategies for disruptive and incremental innovations.

Research question

Each agent in a simulation of the USA/IPK model is located over an interaction network which determines with which agents it interacts at each time step. The dynamics of the model published so far [START_REF] Chatterjee | The Innovation Diffusion Process in a Heterogeneous Population: A Micromodeling Approach[END_REF] were only simulated over Watts-Strogatz networks. Yet existing knowledge on real social networks suggests they might have different characteristics than WS ones, including a skewed distribution of degrees, a strong modularity or a core-periphery structure. As the structure of interactions usually plays a central role on the dynamics of a model, we question here the impact of the networks of interactions on the dynamics of the IPK/USA model: do different networks lead to different qualitative dynamics? In the next section 2, we describe an experimental protocol to tackle these questions. After a detailed description of the USA/IPK model, we select several network generators reproducing statistical properties observed in real networks. We also detail the implementation of these experiments to facilitate replication. In the Results section 3 p. 6, we depict the results of these computations, and compare the qualitative dynamics of the model in its space of parameters with the published ones. We then discuss (4 p.11) the implications of these findings on the methodology to compare computational models over networks, the practical findings for the USA/IPK model, and open novel questions arising from these findings.

Experimental protocol

Selected Random Network Generators

As for an epidemic model and other models in which the dynamics depends on cascades, some statistical properties would obviously impact the dynamics of the model. If the density of the network increase, each agent seeking out expert knowledge is more likely to find it out, and agents pro actively transmitting information would have more impact. In the same way, networks having different sizes, diameters or average path lengths might bias the results because the increased steps required for information to flow in the entire population. In order to avoid these trivial effects of networks on the dynamics, we select network generators which are all compliant with the small-world effect (low density, short average path length) and all have a high clustering (or transitivity rate). We then set the parameters of these random network generators so that they generate networks having close statistical properties for density, average path length, count of vertices and diameter. We retain for these experiments three network generators that are able to reach similar statistical properties.

[WS] The famous Watts and Strogatz β-model [START_REF] Gilly | A Dyadic Study of Interpersonal Information Search[END_REF] requires as parameters N the size of the network, nei the neighbourhood of the original lattice and p rewire the rewiring probability. This algorithm starts with a regular lattice of N nodes in which nodes are connected with their nei neighbours (thus having 2.nei degree). It then rewires each link with probability p rewire . We define here n = 1000, nei = 5, p = 0.055; this leads to networks having a density of 0.01, average path length 4.4 ± 0.3 and a clustering rate of 0.47 ± 0.02.

[FF] The Forest Fire model was proposed by Leskovec [18] as an algorithm that creates networks having most of the properties observed in real networks, including communities, skewed distribution of degrees, a core-periphery structure. The network is grown step by step, each new node A being attached to m old nodes. Moreover, each time a new link is created between A and B, A explores the outgoing and incoming neighbours of B. A create links with outgoing nodes of B with a forward probability p, and also creates links with incoming nodes of B with probability p.r, with r the backward burning ratio. As this step is ran recursively, A is said to "burn" all the possible links. Here FF is set up with n = 1000, f w.prob = 0.37, bw.f actor = 0.9, ambs = 1. The resulting networks have a density of 0.01 ± 0.005, an average path length of 4.13 ± 0.2, and a clustering rate of 0.26 ± 0.2.

[SII] We also test a simple model1 that creates networks composed of several communities (sets of nodes having a strong density). This model, later named SII for Simple Interconnected Islands, starts by creating n islands of identical size size. Each island is a random graph in which links exist with probability p.in. Each island is connected with all the other islands with n.inter links, each being created between two nodes randomly picked from each island. Density and transitivity in SSI networks can easily be tuned by varying the n, size and p.in parameters, while the average path length may be tuned with n.inter. Its distribution of degree is nearly a Poisson-like one (as each island is a random network). This average path length remains low, because all the islands are interconnected. For this study we use n = 24 islands, size = 42 nodes per island, p.in = 0.235 of wiring probability within islands, n.inter = 1 link between each pair of island. The resulting networks count 1008 vertices, have a density of 0.0101 ± 0.002, an average path length of 4.36 ± 0.02, and a clustering rate of 0.21 ± 0.1.

Space of parameters for the USA/IPK model

In order to introduce information seeking, the authors of the USA/IPK distinguish awareness knowledge which refers to the existence of the innovation of interest, and expert knowledge which refers to the information required to understand, or use in the right way, the innovation. On the awareness dimension, the agent is either Unaware (knows the innovation exists), Seeking (discovered the innovation exists and seeks out for expert knowledge) or Aware (knows the innovation exists but does not actively search for information). On the expertise dimension, the agent is either Ignorant (does not holds the expert knowledge), Proactive (holds the expert knowlege and shares it around him) or Knowledgeable (passively holds the expert knowledge, without sharing it pro-actively).

The authors introduced personality variables which are randomly initialized at the beginning of the population based on ratios part provided as model parameters and constant during the simulation. "Curious" individuals refer to agents which, when they receive awareness, shift to the "Seeking" state; non curious agents would transition to "aware" instead. "Enthusiastic" agents refer to agents which, when they received the expert knowledge, promote it pro actively around them; non enthusiastic agents would transition directly to Knowledgeable instead. "Supporter " agents are those who, when they discover awareness when they are already holding expert knowledge, start promoting the information. We redirect the interested reader to the complete description of the model [START_REF] Chatterjee | The Innovation Diffusion Process in a Heterogeneous Population: A Micromodeling Approach[END_REF].

For each network generator, we explore the same space of parameters of the model as in the original study to facilitate comparisons: the proportion of curious and enthusiastic people is explored in [0 : 1] by steps of 0.05. The proportions of supporters include 0.0, 0.1 and 0.5. The proportions of initial knowledge k include k = 0.01 (1%, disruptive innovation), k = 0.1 (incremental innovation), k = 0.5 (standard product). For each point of the space of parameters, 10 simulations are started with a different random seed. We measure the final proportion of the population holding both the awareness and the expert knowledge.

The interest of the dynamics of the initial study can not be reduced to one unique epidemic threshold; the interesting elements were mostly qualitative, as the observation of three different regimes depending on the proportion of initial expertise, or the asymmetry in the role of information seeking and proactive communication. As a consequence, we drive an extensive exploration of the space of parameters and interpret the dynamics of the model in this space as in the original study. We will come back on every qualitative finding of the original study and compare the result with different networks.

Implementation

These experiments require the chaining of the generation of complex networks and the agent-based simulation over this network. For each simulation experiment, a different random seed is defined, a network is generated, its statistical properties are analyzed, then the network is loaded by the simulation model and the simulation is ran until the end of the diffusion dynamics. In order to facilitate the exploration of this experimental design, facilitate reproducibility and avoid manipulation errors, we use the OpenMole scientific software [21] which enables the description of computation workflows in which tasks are chained with each other. The generation of the networks is done in R using the igraph package [5]. The network is then written to a file in graphml format. The simulation is based on the original sourcecode of the USA/IPK model as found in the github repository2 . Its execution is driven in Netlogo [25] v6. The OpenMole software drive the exploration of the space of parameters, the transmission of parameters and data between the generation of network and the simulation, and drives the computation in parallel. Results were analyzed in R. The source code required to reproduce this exploration is shared in the same github repository.

Experimental results

Figures 1,2 and 3 depict the simulation results for initial proportions of expertise of respectively 1%, 10% and 50%. These figures represent the results of 276676 simulations.

In the original experiment, authors noted the high efficiency of word-ofmouth to gather the expert knowledge scattered in the population. The same result is obtained here on WS or SII: when there is 1% of expertise in the population (Fig. 1), having 30% of curious and 30% of enthusiasts leads to more than 90% of success. On FF however, the efficiency is noticeably lower: diffusion to the entire population is very rare. We can reproduce the original results on WS, but we also identified a counterexample with the FF networks. We conclude this efficiency depends on the structure of the network of interactions and can not be told to be systematic.

Authors highlighted threshold effects in the original study: a small increase in the proportion of curious or enthusiasts leads to a significant shift in the proportion of informed people. These effects are also visible here. On every network when k = 0.01 (Fig. 1), a small increase of the proportion of curious between 0.0 to 0.05 leads to diffusion rates as different as 0 or 100%.

Original results highlighted three distinct regimes of the model depending on the proportion of initial expertise k, with information seeking leading the dynamics when k is low, symmetric roles of information seeking and supporters with k higher, and first importance of supporters when k is very high (k >= 0.5). Thee distinct regimes also appear in our experiments depending on the value of k. For k = 0.01 (Fig. 1), there are vertical dark areas of the left side of the figures which traduce an absence of diffusion when there are no curious individuals in the population. In this part of the space of parameters, information seeking thus stands as a mandatory step for the communication to start. For k = 0.1 (Fig. 2), the same effect is visible when there are no supporters (left figures), less visible with a few supporters, and not visible when there are many supporters (right figures). In this regime, curious and enthusiast agents play a similar role: at least one of them is required for the diffusion to start. For k = 0.5 (Fig. 3), the success of diffusion is less sensitive to the proportion of curious agents; the supporter parameter here stands as a key element along with the proportion of enthusiasts; it would mean that when expert knowledge is widely available in the population, information seeking is less important than proactive communication. We confirm with these results the existence of three regimes depending on the proportion of initial expertise k on WS and SII networks.

The original study points out the strong asymmetry in the impact of the proportions of curious and proactive agents: while diffusion can occur with only a few curious and no enthusiastic, it is not possible with only a few enthusiastic and no curious. This effect has a potential strong impact for policies, as it suggests it is more important to create campaigns of information that make people seek out for information than make people spread the word when they are knowledgeable. We also observe this result here: a minimal proportion of curious being required on any network and any proportion of initial expertise when there are no supporters. No other parameters plays this role in the dynamics. This suggests, in line with field observations, that information seeking is a mandatory step for diffusion of information.

In the original study, the impact of supporters (that is individuals who spread the word when they discover awareness after holding the expert knowledge) only have an important impact when the initial amount of knowledge is high. We observe the same impact here: the proportion of supporters is essential for k = 0.5, visible for k = 0.1, and barely noticeable for k = 0.01. This observation stands on all the networks. The original study also highlighted how it is surprisingly easier to retrieve expertise when there is fewer initial expert knowledge in the population. This finding was counter-intuitive: if more expert knowledge is disseminated in the population, then any agent seeking out information is likely to find it; the more initial expert knowledge, the more efficient the diffusion result should be. We find here the same results as in the initial study: with k = 0.5, the final diffusion appears lower than with k = 0.1 on every network. As in the initial study, this observation is explained by the diffusion process: if there are few individuals holding expertise, the people initially seeking out will create long D R A F T chains of information seeking; when one agent of this chain gathers expertise, they will all collect it back (denoted "information gathering chain" in the initial study). If most people are initially knowledgeable however, as soon as an agent seeks out for information, he will find out expertise, and stop seeking out; he will not have raised the attention of many others during the process. Our experiments observe this phenomenon on every network. The comparison of the dynamics of the model over different networks mostly underlines the overall difference in the efficiency of the diffusion over networks, with diffusion being more successful over SII and WS, and far less successful over FF. As these networks have similar density and the same order of magnitude of average path length, these characteristics are unlikely to explain those differences. Other statistical indicators of the dynamics of the model also do not explain why the diffusion would be more difficult over FF networks. The duration of the diffusion is longer over WS than FF and SII, but those last two share the same distribution of duration. Observations of the dynamics of diffusion over the network suggests the core-periphery structure of FF might explain this phenomenon: the initial expert knowledge scattered in the periphery is more difficult to retrieve from the network (less dense areas). Moreover, despite the FF networks having an average path length being slightly lower than the ones of SII and WS networks in our settings, their diameter remains considerably higher, with an average of 14 instead of 7 for WS and SSI. This longer diameters makes the information less accessible in the network.

Discussion

The experimental protocol deployed for these computational studies enabled us to successfully run hundreds of thousands of simulation, each being run over a different network whom statistical properties were measured. The use of scientific workflows to run these computations appears as a relevant scheme to analyze the impact of networks on simulated dynamics ; the combination of the OpenMole software to coordinate the exploration of the parameters, and of R/igraph to generate and analyze the networks, stands as an efficient and reliable solution.

Regarding the USA/IPK model under study, our simulation experiments confirmed the qualitative dynamics of the model are similar over different networks: importance of information seeking to trigger diffusion of innovation, asymetry between information search and proactive transmission, difficult diffusion when too much expertise is available. Any recommendation for policies based on the initial model would still stand after our computational study. Note that these experiments can not prove that the dynamics would be the same on any network; they just increase the likelihood of this statement. However, our results demonstrated the efficiency of diffusion can be significantly different over different natures of networks (as demonstrated by the results over FF networks compared to WS and SII). As for any other multi-agent model using a network to describe the structure of interactions, this raises the question of which network should be used to investigate real-world dynamics.

Fig. 1 .

 1 Fig. 1. Final proportion of the population holding both awareness and expertise when 1% of the population initially holds the expertise. (top) WS (middle) FF (bottom) SII

Fig. 2 .

 2 Fig. 2. Final proportion of the population holding both awareness and expertise when 10% of the population initially holds the expertise. (top) WS (middle) FF (bottom) SII

Fig. 3 .

 3 Fig. 3. Final proportion of the population holding both awareness and expertise when 50% of the population initially holds the expertise. (top) WS (middle) FF (bottom) SII

This algorithm is highly similar to the one proposed by Newman and Girvan [8] as a test bed for community detection, in which two parameters drive the probability of existence of links respectively intra and inter communities. Our model facilitates the present experiments due to the guaranteed connectedness of the network.

https://github.com/samthiriot/model-wom-USA-IPK

D R A F T


Emmanuel Hache
email: emmanuel.hache@ifpen.frfrederic.lantz@ifpen.fr

Frédéric Lantz

Oil price volatility: An Econometric Analysis of the WTI Market

Keywords: Oil Prices, Futures Markets, Markov Switching Regime models # 79. I. ABADA, O. MASSOL

La collection "Les cahiers de l'économie" a pour objectif de présenter des travaux réalisés à IFP Energies nouvelles et à IFP School, travaux de recherche ou notes de synthèse en économie, finance et gestion. La forme peut être encore provisoire, afin de susciter des échanges de points de vue sur les sujets abordés. Les opinions émises dans les textes publiés dans cette collection doivent être considérées comme propres à leurs auteurs et ne reflètent pas nécessairement le point de vue d' IFP Energies nouvelles ou d' IFP School.

Pour toute information sur le contenu, prière de contacter directement l'auteur. Pour toute information complémentaire, prière de contacter

Introduction

An extensive literature exists on the respective roles of macroeconomic factors on the oil prices trend. The impact of the short-term variation of stocks [START_REF] Fattouh | Basis Variation and the Role of Inventories: Evidence from the Crude Oil Market, Oxford Energy Studies[END_REF][START_REF] Pierru | Speculation without Oil Stockpiling as a Signature: A Dynamic perspective[END_REF], the monetary policies and by extension the changes in exchange rates (Audigé & all, 2010;[START_REF] Hamilton | Understanding crude oil prices[END_REF][START_REF] Mignon | Les liens entre les fluctuations du prix du pétrole et du taux de change du dollar[END_REF] or interest rates [START_REF] Hamilton | Understanding crude oil prices[END_REF], or the cyclical nature of the petroleum industry [START_REF] Smith | World Oil: Market or Mayhem ?[END_REF][START_REF] Fattouh | Oil Market Dynamics through the Lens of the 2002-2009 Price Cycle[END_REF][START_REF] Lescaroux | The petroleum market: The ongoing oil price "shock" and the next "countershock[END_REF] are commonly put forward to explain prices movements. The speculation factor is "by definition" difficult to measure. Some authors [START_REF] Krugman | The Oil Nonbubble[END_REF][START_REF] Smith | World Oil: Market or Mayhem ?[END_REF][START_REF] Hamilton | Understanding crude oil prices[END_REF] reject the hypothesis that speculation has a role to play in the markets. Nevertheless, the introduction of new rules by the Commodity Futures Trading Commission (CFTC) at the end of 2000 and the implementation of expansionary monetary policies after September 11 have triggered a strong growth in the transaction volumes and an increase in the short-term volatility of the oil prices [START_REF] Chevalier | Rapport du groupe de travail sur la volatilité des prix du pétrole[END_REF][START_REF] Medlock | Who is in the Oil Futures Markets and How Has it Changed ?[END_REF] in the financial markets, such as the New York Mercantile Exchange (NYMEX).

In this context, we focus on interactions that may exist between a physical crude oil price and the level of activity in financial oil markets. The data studied are relative to the price of crude oil on the North American market: the spot price for WTI (Cushing, Oklahoma), the futures prices on the NYMEX, the transaction volumes and open interest in the same market. These information are all in the public domain, and were drawn from the U.S. Energy Information Administration and from the weekly market business reports of the CFTC.

In the first section, we briefly define the major changes introduced by the new regulations of the CFTC in 2000 and the consequences in term of transaction volumes. In the second section, we identify the interactions between the physical and the financial crude oil prices and the impact of the short-run trader's behavior by using econometric analysis. The main conclusions are summarised in the last section.

A new deal in the financial markets after 2000

Following the introduction at the end of December 2000 of the law modernising commodities markets, the Commodity Futures Modernization Act1(CFMA), two major changes have been observed. On the one hand, by studying available data from January 1993 to January 2009, we observed a marked rise in transaction volumes. Measured in batches of 1 000 barrels (a standard financial contract for WTI on the NYMEX), these transactions have risen, for twomonth term contracts, from around 52 000 in 1993 to 136 000 in 2008, i.e. multiplied by a factor of two and a half, with a peak of 165 000 in 20072. On the other hand, the share of non-commercial3 players increased from around 20% before 2001, to over 50% on average since 2006. In addition, their share in the volume of global transactions reached almost 60% at the beginning of the third quarter of 2008, a period during which crude prices reached record levels. According to [START_REF] Medlock | Who is in the Oil Futures Markets and How Has it Changed ?[END_REF], during the 1990s we could observe ten active contracts on NYMEX, representing in barrel equivalents (1 contract = 1 000 barrels) over 150 million barrels per day, or more than twice the global demand for crude oil at that time. In recent years, this figure has changed to almost seven, with around 600 million barrels being exchanged through financial contracts. During the previous two decades and especially in the initial phase of construction of the commodities markets, the main objective of the different derivatives marketplaces was to attract and concentrate the liquidity required for commercial 1 For more information, see the CFTC website at http://www .cftc.gov/lawandregulation/index.htm 2 During the same period of time, consumption of petroleum products only increased by 12% in the United States, and by 20% world-wide. 3 Thanks to the obligatory declarations that must be made by the various traders on NYMEX in order to operate in the financial markets, it is possible to determine the volumes for each trader, and to make so-called "open" positions (open interest) more comprehensible. Until September 2009, the CFTC classified the various parties into three categories of traders (See the breakdown up to 2009 produced by the CFTC at http://www.cftc.gov/marketreports/commitmentsoftraders/cot_about.html#P16_3370) so-called "commercial" traders (Commercial traders are those who are active in the petroleum supply chain (producers, stockholders and refiners), and who are in the market to achieve arbitrage between a physical position and a financial one. Noncommercial traders act in the market without any physical counterpart for their deals), "non-commercial" traders, and "others", with this last category corresponding to small volumes of transactions which it is impossible to attribute to one or other of the first two. traders to achieve hedging activities. Nevertheless, the rise in transaction volumes has been accompanied by a concentration of traders' liquidity on the shortest maturity contracts exchanged in the commodities markets. This factor has been observed and studied in the past [START_REF] Lautier | Term structure models of commodity prices: a review[END_REF], but it seems to be reinforced since 2000. For WTI futures prices, we observed between 1993 and 2008 a decrease in transaction volumes as contract terms grew longer, and a virtual absence of liquidity for long-term contracts (Figure 2). In fact, the inadequate information available at any given moment t on contracts whose maturity period is greater than one year does not give traders the incentives to trade in the market. In consequence, the liquidity for distant contracts -a maturity greater than 4 months -decreases sharply. We have split our sample into two sub-periods, the first one from January 1993 to December 2003, and the second from January 2004 to February 2009. This segmentation does not help to highlight the introduction of the CFMA (December 2000) as a catalyst for new trader behaviour. Nevertheless, it enables us to focus both on the actual increase and on the starting point of the acceleration in transaction volumes in the markets observed from January 2004. This breakdown has also demonstrated very different decrease profiles of the transaction volumes between the two chosen sub-periods. The profile is thus distinctly more pronounced over the more recent period (Figure 4), which highlight that the share of transaction volumes dealing with short-term contracts has risen strongly since 2004.

Methodology

The econometric analysis covers the relationship between the spot price and the future price of WTI. As transaction volumes have risen in particular for the shortest terms, we have focused on the relationship between the spot price (spott) and the price for two-month forward contracts (p2t)4.

The data cover the period from 4th April 1999 to 26th January 2009. The sample thus contains 511 weekly observations.

As an initial stage, we carried out unit root tests on each of the series studied. However, the period covered by the analysis was marked by contrasting price movements, which may show up as possible breaks. This led us to implement the testing procedure suggested by [START_REF] Perron | Further evidence on breaking trend functions in macroeconomic variables[END_REF], which grew out of the work of Perron and Vogelsang (1992a,b) as well as that of [START_REF] Zivot | Further evidence on the great crash, the oil price shock and the unit root test hypothesis[END_REF]. In these tests, the null hypothesis is that the temporal series is characterised by the presence of a unit root and a constant, which may be null, with the presence of a break. We distinguished between an instantaneous effect (denoted by AO for "Additive Outlier") and an effect with a transition affecting the constant (c) or both the constant and the slope (c,s) in the Dickey-Fuller regression (respectively denoted by IO(c) and IO(c,s) for "Innovational Outlier").

Subsequently, we tested for the existence of a long-term equilibrium (a cointegration relationship) between the future price and the spot price. As with the tests for unit root, we tested for the presence of a potential break during the period under study with a Gregory and Hansen test (1996a,b). Furthermore, we tested for changing variability of the error term (an autoregressive conditional heteroscedasticity -ARCH test) in the estimated relationship.

Estimating the short-term dynamic between the two prices is more difficult, and manifests several changes that are shown up by the structural break tests. Nevertheless, econometric estimation over several sub-periods shows similar empirical results for non-adjacent subperiods.

Thus we tried to build a short-term model allowing for two distinct states. To do so, we estimated a Markov chain model allowing for changes in the short run dynamic. We briefly describe the Markov Switching Vector Error Correction (MSVEC) model below.

A Markov chain is a random process {Yt,t=0,1,2... } which takes its values from a finite statespace E. This sequence has the Markov property, according to which, when Yt is known, it is possible to ignore the past when predicting the future. The change from one state to another is governed by a transition function.

In our case, for every sequence of random variable (Yt), we suppose that this follows a

Markov process with two states E={E1,E2}, corresponding to the two presumed states of price movements. Around the long-term equilibrium, we therefore allow two short-term dynamics with a probability of switching from one state to the other.

Consequently, the model can be expressed as:

(1) where zt is the Markov state, which is a member of E={E1,E2}, , µ(zt)is the term vector, which is constant for each state zt, is the matrix of coefficients for the delayed terms for delays j=1,...p, is the matrix of coefficients for the cointegration model for each of the states , and ut designates the residuals.

Constructing the model also supplies the probability of switching from one state to the other, hence we can deduce the unconditional probability p* of being in state E2. Having constructed the model (1), we then sought to explain the probability of being in each of the states through the use of indicators linked to transaction volumes and to traders' positions.

Empirical results

The two series spot t and p2 t are first-order integrated, I(1) as is demonstrated by the unit root tests (Augmented Dickey-Fuller, Phillips-Perron, and Kwiatkowski-Phillips-Schmidt-Shin)

shown in Table 1. Note: The superscripts "*" "**" "***" indicate the degree of significance associated with the 10%, 5% and 1% quantiles.

The cointegration test enables us to identify an equilibrium relationship between the spot price and the forward price (Table 3). Nevertheless, we performed a test of cointegration with a structural break on the long term relationship between the prices including a constant (Table 4). This led us to envisage a possible break, for both the constant and the variable at the same time, at the end of 2004 (Dec. 28-2004). The other break date found in Model (c) is above the highest 85% of observations. Incidentally, we evaluated the cointegration model using a regression method (Engle and Granger), and carried out an ARCH test (F(1,503) = 7.9347, prob(F)=0.00504).

This enables acceptance of the alternative hypothesis of change in the variance of the error terms around the equilibrium relationship. Finally, we decided to keep the equilibrium between the spot price and the forward price over the whole sample, considering that the short-term dynamic should change over different sub-periods. Note: The superscripts "*" "**" "***" indicate the degree of significance associated with the 10%, 5% and 1% quantiles.

a: The number of lags is determined from the AIC criterion

The Markov Switching Vectorial Error Correction Model (1) has been estimated according to the long term equilibrium (2). From the unit root test with structural break, we decide to introduce a dummy variable (du04) to improve the parameters estimation (du04t=0 until Dec 7-2004 and du04t=1 after).

Two different states are clearly identified through the MS-VECM estimation as observed in table 5. of the observations from the period under study (and 9.7% for p*>0.5).

The probability to move from state 1 to state 2 is low (0.03), whilst there is more important probability (0.22) to come back from state 2 to state 1. In the second state, the lagged residual of the long term equilibrium has an important impact on the short term dynamic of prices (-0.588) and the variances of the residuals are largest for both spot and future price variations than they are in the first state. Thus , this second state can therefore be regarded as a crisis state.

On the contrary, the first state (Standard state) , which is more commonly observed over the sample, is characterized by a moderate effect of the lagged long term residuals (-0.139), which indicates that the spot price and 2-month forward price will return steadily to their long-term equilibrium. The estimated coefficients are lowest, in absolute value, in state E1 than in state E2 which denotes a smoothest evolution over this standard state. Units: 1000 barrels Consequently, we estimated the probability of being in the state E 2 according to a set of variables related to the transactions on the future markets by using a logit model. For this purpose, the following explanatory variables were selected :

-the ratio of the transactions for the first forward contract V_1 t to the one month lagged open interest OI t-4 (4 weeks before).

-the volume of transactions V_1 t

The following adjustment was performed on the sub-sample from January 2007 to January

Conclusion

Recent events in the crude oil markets, such as the sharp rise in prices between January and July 2008, to almost $147 per barrel and the collapse a few weeks later to under $35 per barrel, have left many analysts and researchers puzzled by the underlying explanations for determination of prices. The analysis carried out of the crude oil market through our work on spot and forward prices for West Texas Intermediate (WTI) leads us to the following conclusions. The increased volatility of WTI prices since the beginning of the 2000s, which has recently been studied (Chevallier, 2010;[START_REF] Fattouh | Oil Market Dynamics through the Lens of the 2002-2009 Price Cycle[END_REF], is a significant clue in understanding market prices volatility. Furthermore, this development should be linked to that observed in the major commodities markets (e.g. NYMEX), that is, a marked rise in transaction volumes dealing with futures contracts, in particular on the shortest maturities (2 to 4 months), and more especially since 2004. From the cointegration test, we could not reject the hypothesis of a long term equilibrium between spot and future prices. Nevertheless, two distinct states (so-called "standard" and "crisis" states) should be distinguished for the short term dynamic. Subsequently, we cannot reject the hypothesis that variations in the positions of non-commercial players in the financial markets for crude oil may affect the probability that the standard state will prevail. The behaviour of non-commercial players may thus play a destabilising role in petroleum markets. Additionally, our probit model shows clearly the importance of changes in the amount of open interest in the switch between states.

The increase in the volume of transactions on financial trading floors should nevertheless be kept in perspective. In fact, as we discussed in the first section, the main objective of the Commodities Exchange is to attract and to concentrate the liquidity for hedging and trading purpose. The first contract for heavy fuel oil in the NYMEX was launched in 1978 and was abandoned because of inadequate liquidity's volume. Furthermore, the transactions volume figures must be handled with care, for at least two reasons. The strategy of non-commercial players is partly based on managing price differentials over a certain period of time (calendar spread), between different commodities or by-products (intra or inter market spread), these activities create a high degree of fluidity for these contracts. It enables the commercial player to be able to achieve a physical arbitrage on time and enables also many non-commercial players to close their positions before the expiration of the contract. On Commodity Exchange we observe a feedback effect between the degree of liquidity, the probability of closing a position with a very low transaction cost (The Exchange often requires an added-margin if the position is open near the expiration date) and the volume of transactions. Keeping the market liquid needs the non-commercial players to "feed" the market.

Because the oil future market has been characterized by a growing spread between the number of transactions and the oil demand in volume term when it raises up, the uncertainties on the oil demand leads to an increasing price volatility associated to an inflating number of transactions. During such instable period, the number of future contracts sold during the previous periods appears too high related to the real economic activity that contributes to increase the price volatility.

In this context, it would seem particularly appropriate to assess whether there might exist, at least theoretically, an optimal level of liquidity in the financial markets, with a view to regulating this. This liquidity index could depend on the number of participants, the volume of transactions, the degree of concentration of seller and buyer, the historical volatility and the observed price of the underlying commodity during a determined period of time.

We could also focus on margins requirement issues. Indeed the initial margin and the maintenance margin (which represents around 75 % of the initial margin in average) are in practice based on historical or implied volatility of the contracts (based on the underlying commodity). Their level are set by the Exchange and are subject to change depending on the market conditions but not on a daily basis (or intra-day basis) 5 . The implementation of a daily basis change on Margin Requirement could send a clear message to the players in term of market limitation and also a valuable information about market risk.

Figure 1 :

 1 Figure 1: Spot price for WTI crude oil (US dollars per barrel)

Figure 2 :

 2 Figure 2: Average volume of transactions of future prices for WTI by term

Figure 3 :

 3 Figure 3: Price of crude oil and financial versus physical ratio in the WTI market

Figure 4 :

 4 Figure 4: Average transaction volumes as a function of the term in the price of WTI (NYMEX)

Figure 5 :

 5 Figure 5: Unconditional Probability of state 2 (Crisis state)

  2009, a period marked by acceleration in the rise of crude oil prices: that the variable has a significant influence on the switch to state E 2 .The abrupt increase inthe ratio between the short-term transaction volumes and the open interest drives the switch in the crisis state. The change in the open interest relative to the observed demand for crude oil during this period explains the change in the explanatory variables and the state change. The ratio between open interest and demand, as well as the price of crude, both increase progressivelythis dual phenomenon is explained by the low short-term price elasticity of demand. When demand falters, position-closing shows up as a sharp rise in the relative ratios of volumes to open interest and the switch to the crisis state.

Table 1 :

 1 Unit root tests

	Variables	ln(spot)	∆ln(spot)	ln(p2t)	∆ln(p2t)
	ADF (c,t)	-1.702	-25.455***	-1.354	-24.436***
	number of lags	0	0	0	0
	ADF (c)	-1.790	-25.382***	-1.825	-24.361***
	number of lags	3	0	0	0
	ADF (none)	0.458	-25.384***	0.511	-24.353***
	number of lags	1	0	0	0
	PP (c,t)	-1.565	-25.388***	-1.413	-24.373***
	number of lags	7	7	8
	PP (c)	-1.879	-25.295***	-1.820	-24.295***
	number of lags	6	7	7	8
	PP (none)	0.422	-25.292***	0.524	-24.287***
	number of lags	6	7	7	8
	KPSS (c)	2.526***	0.226	2.564***	0.234
	number of lags	17	6	17	7
	However, the unit root test with structural breaks (AO and IO(c) versions) reveals such breaks
	for respectively May and July 2008 when prices reach their maximum values. The IO(c,s) test
	reveals different break dates for spot t and p2 t , respectively in March at the beginning of the
	sharp rise, and in December as prices plummeted (table 2). Nevertheless, the structural breaks
	should be included in the interval [15%; 85%], i.e., the sub-period [sept. 26 -2000; July 31-

Note: The superscripts "*" "**" "***" indicate the degree of significance associated with the 10%, 5% and 1% quantiles. 2007] to be interpreted, which does not match with the dates highlighted by the tests.

Table 2 : Tests for unit root with break (Perron)

 2

	Variables	ln(spott)	∆ln(spott)	ln(p2t)	∆ln(p2t)
	IO(c)	-3.491	-7.664***	-3.241	-8.011***
	Break date	Dec. 07-2004	July 08-2008	Dec. 07-2004	July 08-2008
	IO(c,s)	-3.172	-7.453***	-2.679	-7.970***
	Break date	June 13-2000	March 25-2008	May 15-2001	Dec. 30-2008
	AO	-3.526	-7.225***	-3.466	-7.623***
	Break date	Jan. 13-2009	May 13-2008	Jan. 13-2009	May 06-2008

Table 3 : Test of cointegration between ln(spot t ) and ln(p2 t )

 3

	H0 :	Intrinsic value	Statistical	Threshold value	Statistical	Threshold value Test
	order = r		Test λmax	Test λmax 5%	Test Trace	Trace 5%
	r=0	0.0368	19.068	11.225	19.415	12.321
	r≤1	0.0006	0.347	4.130	0.347	4.129
	Thus the relationship may be written:
	ln(p2t)=1.000787 ln(spott)		(2)
	(0.00158)
	(): Standard deviation

Table 4 : Gregory and Hansen test

 4

	Model	ADF	Date	k a
	c	-6.261***	25-11-2008	2
	c,s	-6.042***	28-12-2004	2

Table 5 : MS-VECM model of the dynamic between spot prices and 2-month forward prices

 5 Source: calculations by the authorsThe first state can be observed over the greater part of the sample, while the second state corresponds to specific events: September 11, 2001, the start of the Iraq conflict in 2003, the winter peak observed in 2006, and movements observed during spring and early summer 2008. Considering the observations for which p* is greater than 0.9, this state represents 6%

				State				State 2
			Coeff Std Dev. T-Student	Sign.	Coeff	Std Dev. T-Student	Sign.
		Prob(E2/E1)	0.030	0.011	2.644 **
		Prob(E1/E2)					0.224	0.078	2.654 ***
	∆(ln(p_spot(t)) = du04					-0.048	0.029	-1.630 *
		∆(ln(p_spot(t-1))	-0.107	0.049	-2.177 **	0.241	0.224	1.074
		∆(ln(p_2(t-1))					-0.607	0.336	-1.803 *
		εLT(t-1)	-0.114	0.024	-4.825 ***	-0.546	0.255	-2.134 **
	∆(ln(p_2(t)) =	∆(ln(p_spot(t-1))	0.103	0.061	1.692 *	0.484	0.248	1.949 *
		∆(ln(p_2(t-1))	-0.217	0.042	-5.117 ***	-0.529	0.330	-1.600 *
	Residuals	Var(uspot)	0.002	0.000	12.323 ***	0.010	0.003	3.290 ***
		Var(up_2)	0.002	0.000	12.273 ***	0.007	0.002	3.069 ***
		Cov(uspot,up_2 )	0.002	0.000	12.440 ***	0.007	0.002	3.471 ***

Table 7 : Average amount of open interest associated with short-term states

 7

		Commercial	Non commercial	Others
	p*≤0,5
	(state 1)	-14040	19721	-5680
	p*>0,5
	(state 2)	-4324	8587	-4263
	Whole sample	-13097	18640	-5543

It's also possible to study the relationship between the spot price and the one-month forward contract (the shortest maturity), but the authors considered the very close proximity of quotation values between these two "terms" could potentially affect the econometric results.

Acknowledgement

Many people have provided helpful comments and suggestions on the analysis, and we are pleased to take this opportunity to thank them. We are especially grateful to Arash FARNOOSH & Sandrine ROL. All remaining errors are ours. The views expressed herein are strictly those of the authors and are not to be construed as representing those of the IFP Energies Nouvelles.

27/05/2001 27/08/2001 27/11/2001 27/02/2002 27/05/2002 27/08/2002 27/11/2002 27/02/2003 27/05/2003 27/08/2003 27/11/2003 27/02/2004 27/05/2004 27/08/2004 27/11/2004 27/02/2005 27/05/2005 27/08/2005 27/11/2005 27/02/2006 27/05/2006 27/08/2006 27/11/2006 27/02/2007 27/05/2007 27/08/2007 27/11/2007 27/02/2008 27/05/2008 27/08/2008 27/11/2008

Source: calculations by the authors Into our model we subsequently introduced data concerning the transaction volumes observed depending on the probability of being in one or other of the two short-term states. To this end, we subdivided the observations according to whether the probability p* was above or below 0.5 (Table 6). It would appear that average transaction volumes for 1-month contracts rise sharply in line with p* whereas average volumes associated with other terms remain approximately stable. The "Cahiers de l'Économie" Series

The "Cahiers de l'économie" Series of occasional papers was launched in 1990 with the aim to enable scholars, researchers and practitioners to share important ideas with a broad audience of stakeholders including, academics, government departments, regulators, policy organisations and energy companies.

All these papers are available upon request at IFP School. All the papers issued after 2004 can be downloaded at: www.ifpen.fr

The list of issued occasional papers includes:


Eduard Curcȃ
email: curca@math.univ-lyon1.fr.

The divergence equation with L ∞ source

Keywords: divergence equation, Riesz products, function spaces. MSC 2010 classification: 42B37, 42B05 2

come

The divergence equation with L ∞ source

Introduction

In this paper, we improve the following result of Preiss ([4]) and McMullen (Theorem 2.1, in [START_REF] Mcmullen | Lipschitz maps and nets in Euclidean space[END_REF]):

Theorem 1 There exists g ∈ L ∞ (T 2 ) with zero integral, such that there are no f 1 ,

f 2 ∈ W 1,∞ (T 2 ) with g = ∂ 1 f 1 + ∂ 2 f 2 .
The proof in [START_REF] Preiss | Additional regularity for Lipschitz solutions of PDE[END_REF] is "geometric", the one in [START_REF] Mcmullen | Lipschitz maps and nets in Euclidean space[END_REF] relies essentially on Ornstein's non-inequality ( [START_REF] Ornstein | A non-inequality for differential operators in the L1 norm[END_REF]).

Note that, in the above statement, the conditions on f 1 , f 2 are isotropic, i.e., we require ∂ l f j ∈ L ∞ (T 2 ) for all l, j = 1, 2. In what follows, we will prove that, under some mild regularity assumptions on f 1 , f 2 , the above requirements can be weakened to anisotropic conditions. Namely, it is enough to impose ∂ 2 f 2 ∈ L ∞ (T 2 ). In order to state this more precisely, we introduce the following spaces of distributions.

Suppose λ : N → (0, ∞) is a decreasing function such that λ (k) → 0 when k → ∞. To such a function we associate the Banach space of those distributions whose Fourier transform decays at the rate at least λ. More precisely, consider the space

S λ T 2 :=    f ∈ D (T 2 ) sup n∈Z 2 f (n) λ (|n|) < ∞    ,
endowed with the norm given by

f S λ := sup n∈Z 2 f (n) λ (|n|) , f ∈ S λ T 2 .
To mention only few examples, we note that, for any

m ∈ N * , W m,1 (T 2 ) → S λ (T 2 ), with λ (|n|) = 1/ (1 + |n|) m and, if s > 0, the fractional Sobolev space H s (T 2 ) is embedded in S λ (T 2 ) for λ (|n|) = 1/ (1 + |n|) s .
With this notation, we can formulate our result.

Theorem 2 Suppose λ : N → (0, ∞) is decreasing to 0. There exists g ∈ L ∞ (T 2 ) such that there are no f 0 , f 1 , f 2 ∈ S λ (T 2 ) with ∂ 2 f 2 ∈ L ∞ (T 2 ) and g = f 0 + ∂ 1 f 1 + ∂ 2 f 2 .
We can easily observe that Theorem 2 implies Theorem 1. Indeed, if

f 1 , f 2 ∈ W 1,∞ (T 2 ) then ∂ 2 f 2 ∈ L ∞ (T 2
) and, as we mentioned above, we have

f 1 , f 2 ∈ S λ (T 2 ) for λ (|n|) = 1/ (1 + |n|). Also, even the weaker regularity condition f 0 , f 1 , f 2 ∈ H ε (T 2 ), ∂ 2 f 2 ∈ L ∞ (T 2 ) (ε > 0,
a small fixed number) rules out the existence of a solution. Intuitively, f ∈ S λ (T 2 ), with λ slowly decaying, means that f is "slightly better" than L 1 . The above result asserts that solutions with such regularity satisfying

∂ 2 f 2 ∈ L ∞ (T 2 ) need not exist.
Finally, we discuss the most important aspect, which is the proof of Theorem 2. Our proof completely avoids the use of Ornstein's non-inequality. It is an adaptation of the Riesz products based proof, given by Wojciechowski in [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF], of the fact that there exist L 1 functions which are not divergences of W 1,1 vector fields. We follow the general structure of his proof making the needed modifications in order to handle the L ∞ case. While the proof in [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF] relies on a relatively difficult lemma (Lemma 1, in [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF]), in our case, the role of this lemma will be played by Lemma 3 below, which is elementary and easy. Another aspect of our proof is the presence of the function λ. This allows us to quantify the regularity that we impose to the solution and to improve the result described by Theorem 1. The approach based on Ornstein's non-inequality does not seem to be suited for obtaining this improvement.

We also mention that the proof given below of Theorem 2 is self-contained and elementary.

Lemma 3 Suppose z 1 ,..., z N are some complex numbers. Then, there exist σ 1 ,...,

σ N ∈ {0, 1} such that N k=1 σ k z k ≥ 1 π N k=1 |z k | .
Proof. We follow [START_REF] Rudin | Real and Complex Analysis[END_REF]. View z 1 ,..., z N as vectors in R 2 . For a given θ ∈ [0, 2π], let r θ := (cos θ, sin θ). If H θ is the half-plane given by

H θ := z ∈ R 2 | z, r θ ≥ 0 ,
we have

1 2π 2π 0 N k=1,z k ∈H θ z k dθ ≥ 1 2π 2π 0 N j=1 z j , r θ + dθ = N j=1 1 2π 2π 0 z j , r θ + dθ,
and we easily see that, for all j,

1 2π 2π 0 z j , r θ + dθ = |z j | 1 2π 2π 0 (cos θ) + dθ = 1 π |z j | .
Using the above inequality, we complete the proof of Lemma 3 via a mean value argument.

We will also need few facts concerning the trigonometric polynomials. Fix a finite sequence (a k ) k=1,N in Z 2 . For each finite sequence (α 1 , ..., α N ) of complex numbers we have the following expansion rule:

N k=1 (1 + α k cos t, a k ) = 1 + N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0   ε j =0 α j 2   e i t,ε 1 a 1 +...+ε k a k .
Suppose, moreover, that (a k ) k=1,N is component-wise lacunary, i.e., there exists a constant

M > 3 such that |a k+1 (1)| / |a k (1)| > M and |a k+1 (2)| / |a k (2)| > M for all 1 ≤ k ≤ N -1.
Then, all the expressions ε 1 a 1 + ... + ε k a k in the above formula are distinct and nonzero. Hence, if α 1 , ..., α N and β 1 , ..., β N are complex numbers, by using the above formula and the relation between convolution and the Fourier transform, we obtain

N k=1 (1 + α k cos •, a k ) * N k=1 (1 + β k cos •, a k ) = N k=1 1 + α k β k 2 cos •, a k . ( 1
)
We will also use the following standard algebraic identity:

N k=1 (1 + c k ) = 1 + N k=1 c k k-1 j=1 (1 + c j ) (2)
for any complex numbers c 1 ,..., c N .

Proof of Theorem 2. Suppose that the assertion of Theorem 2 is false and fix a function λ as in the statement. Then, by the open mapping principle, there exists a constant C > 0 such that for any g ∈ L ∞ (T 2 ) there exist distributions

f 0 , f 1 , f 2 ∈ S λ (T 2 ), satisfying g = f 0 + ∂ 1 f 1 + ∂ 2 f 2 , with the properties that ∂ 2 f 2 ∈ L ∞ (T 2
) and

f 0 S λ + f 1 S λ + f 2 S λ + ∂ 2 f 2 L ∞ ≤ C g L ∞ . ( 3
)
Let N be a large positive integer such that ln N > 25πC and consider the functions on T 2

g N (t) :

= N k=1 1 + i k cos t, a k and G N (t) := N k=1 (1 + cos t, a k ) ,
where the finite sequence (a k ) k=1,N in (N * ) 2 is defined below. Using Lemma 3, applied to the sequence of complex numbers

z k := 1 k k-1 j=1 1 + i 2j for k = 1, ..., N ,
(here and after the product over an empty set is by convention equal to 1), we can find a sequence σ 1 ,..., σ N ∈ {0, 1} such that

N k=1 σ k k k-1 j=1 1 + i 2j ≥ 1 π N k=1 1 k k-1 j=1 1 + 1 4j 2 1/2 ≥ 1 π N k=1 1 k ≥ 1 π ln N . ( 4
)
Now we impose the sequence (a k ) k=1,N to satisfy the following properties:

(i) (a k ) k=1,N is component-wise lacunary; (ii) If σ k = 1, then a k (1) + 1≤j≤k-1 ε j a j (1) λ a k (2) + 1≤j≤k-1 ε j a j (2) < 1 4 N for all ε 1 , ..., ε k-1 ∈ {-1, 0, 1} ; (iii) If σ k = 0, then a k (2) + 1≤j≤k-1 ε j a j (2) λ a k (1) + 1≤j≤k-1 ε j a j (1) < 1 4 N for all ε 1 , ..., ε k-1 ∈ {-1, 0, 1} .
(By convention the sum over an empty set is equal to 0.) Such a sequence can be easily constructed by induction on k: if a 1 ,..., a k-1 are chosen, then we choose a k (2) much larger than a k (1), or a k (1) much larger than a k (2), depending on whether σ k = 1 or σ k = 0 respectively. Since λ is decreasing to 0, we can satisfy in this way the conditions (ii), respectively (iii). Also, the condition (i) can be easily satisfied.

We now return to the proof of Theorem 2. Note that

g N L ∞ = N k=1 1 + 1 k 2 1/2
≤ e π 2 /12 < 3, and also G N ≥ 0 and

G N L 1 = 1. ( 5
)
Using ( 1) and ( 2), we get

G N * g N (t) = N k=1 1 + i 2k cos t, a k = 1 + N k=1 i 2k cos t, a k k-1 j=1 1 + i 2j cos t, a j . ( 6
)
Consider the sets

A : = N k=1 σ k =1 {ε 1 a 1 + ... + ε k a k | ε 1 , ..., ε k ∈ {-1, 0, 1} , ε k = 0} , B : = N k=1 σ k =0 {ε 1 a 1 + ... + ε k a k | ε 1 , ..., ε k ∈ {-1, 0, 1} , ε k = 0} .
Since the sequence (a k ) k=1,N is component-wise lacunary, we have

({0} × Z) ∩ (A ∪ B) = ∅, (Z× {0}) ∩ (A ∪ B) = ∅ and A ∩ B = ∅, while clearly |A ∪ B| ≤ 3 N . In particular, |A| ≤ 3 N , |B| ≤ 3 N .
Using now ( 6) and ( 4), we obtain

|P A G N * g N (0)| = N k=1 iσ k 2k k-1 j=1 1 + i 2j ≥ 1 2π ln N , (7)
where P A is the linear operator on trigonometric polynomials, satisfying P A e i t,n = e i t,n if n ∈ A and P A e i t,n = 0 otherwise. On the other hand, according to our assumption and (5), we can find

f 0 , f 1 , f 2 ∈ S λ (T 2 ), satisfying g N = f 0 + ∂ 1 f 1 + ∂ 2 f 2 , with the properties that ∂ 2 f 2 ∈ L ∞ (T 2 ) and f 0 S λ + f 1 S λ + f 2 S λ + ∂ 2 f 2 L ∞ ≤ 3C.
Let us note that

P A G N * g N = P A G N * f 0 + P A G N * ∂ 1 f 1 + P A G N * ∂ 2 f 2 . ( 8
)
We next estimate each term on the right hand side of (8).

For the second term, we have:

P A G N * ∂ 1 f 1 L ∞ = G N * P A ∂ 1 f 1 L ∞ ≤ G N L 1 P A ∂ 1 f 1 L ∞ = P A ∂ 1 f 1 L ∞ ≤ |A| max n∈A ∂ 1 f 1 (n) = |A| max n∈A |n(1)| f 1 (n) ≤ |A| max n∈A |n(1)| λ (|n|) f 1 S λ ≤ |A| max n∈A |n(1)| λ (|n (2)|) f 1 S λ ≤ 3 N 4 -N 3C < 3C,
where we have used (ii).

For the third term, we have:

P A G N * ∂ 2 f 2 L ∞ = G N * ∂ 2 f 2 -P B G N * ∂ 2 f 2 L ∞ ≤ G N * ∂ 2 f 2 L ∞ + P B G N * ∂ 2 f 2 L ∞ ≤ G N L 1 ∂ 2 f 2 L ∞ + G N L 1 P B ∂ 2 f 2 L ∞ = ∂ 2 f 2 L ∞ + P B ∂ 2 f 2 L ∞ ≤ 3C + |B| max n∈B ∂ 2 f 2 (n) = 3C + |B| max n∈B |n(2)| f 2 (n) ≤ 3C + 3 N max n∈B |n(2)| λ (|n|) f 2 S λ ≤ 3C + 3 N max n∈B |n(2)| λ (|n (1)|) f 2 S λ ≤ 3C + 3 N 4 -N 3C < 6C,
where we have used the identity G N = P A G N + P B G N and (iii). Finally, the first term is easier to handle. We have:

P A G N * f 0 L ∞ = G N * P A f 0 L ∞ ≤ P A f 0 L ∞ ≤ |A| max n∈A f 0 (n) ≤ |A| max n∈A λ (|n|) f 0 S λ ≤ |A| max n∈A |n(1)| λ (|n(2)|) f 0 S λ ≤ 3 N 4 -N 3C < 3C.
These estimates together with (8) give us

P A G N * g N L ∞ ≤ 3C + 6C + 3C = 12C,
which contradicts (7), since ln N > 25πC.

Remark. (1) Similarly, a closer look to the proof in [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF] gives the following analogue of Theorem 2 in the case of L 1 .

Theorem 4 Suppose λ : N → (0, ∞) is decreasing to 0. There exists g ∈ L 1 (T 2 ) such that there are no f

0 , f 1 , f 2 ∈ S λ (T 2 ) with ∂ 2 f 2 ∈ L 1 (T 2 ) and g = f 0 + ∂ 1 f 1 + ∂ 2 f 2 .
(2) The d-dimensional case, with d ≥ 3, can be easily obtained from Theorem 2. More precisely, we have

Theorem 5 Let d ≥ 2. Suppose λ : N → (0, ∞) is decreasing to 0. There exists g ∈ L ∞ (T d ) such that there are no f 0 , f 1 , f 2 ,..., f d ∈ D T d with f 0 , f 1 , f 2 ∈ S λ T d , ∂ 2 f 2 ∈ L ∞ (T d ) and g = f 0 + ∂ 1 f 1 + ∂ 2 f 2 + ... + ∂ d f d . Indeed, consider a g ∈ C ∞ (T 2 ) and ψ ∈ C ∞ (T d-2
) such that 0 ≤ ψ ≤ 1 and T d-2 ψ = 1. If the above result were not true, we could find f

0 , f 1 , f 2 ,..., f d ∈ D T d such that g ⊗ ψ = f 0 + ∂ 1 f 1 + ∂ 2 f 2 + ... + ∂ d f d and f 0 S λ( T d ) + f 1 S λ( T d ) + f 2 S λ( T d ) + ∂ 2 f 2 L ∞ (T d ) ≤ C g L ∞ (T 2 ) .
Without loss of generality, we can suppose that f 0 , f 1 , f 2 ,..., f d are smooth. Integrating this equation in the last d -2 coordinates, we reduce the problem to the 2-dimensional case: f j (t, τ ) dτ , for j = 0, 1, 2,

g = f 0 + ∂ 1 f 1 + ∂ 2 f
satisfy f 0 S λ (T 2 ) + f 1 S λ (T 2 ) + f 2 S λ (T 2 ) + ∂ 2 f 2 L ∞ (T 2 ) ≤ C g L ∞ (T 2 ) .
Here, we have used the fact that, for all n ∈ Z 2 , f j (n ) = f j (n , 0) ≤ λ (|(n , 0)|) f j S λ( T d ) = λ (|n |) f j S λ( T d ) .

(3) Using Lemma 3, and adapting the technique in [START_REF] Kazaniecki | Ornstein's non-inequalities: Riesz product approach[END_REF], we can obtain similar anisotropic Ornstein type inequalities adapted to the L ∞ case. We give below an example. For any ε > 0, there exists a trigonometric polynomial f on T 2 , depending on ε, such that

ε ∂ 3 1 ∂ 2 2 f L ∞ ≥ ∂ 4 1 f L ∞ + ∂ 2 1 ∂ 4 2 f L ∞ + ∂ 1 ∂ 6 2 f L ∞ + ∂ 8 2 f L ∞ .

2

 2 wheref j (t) := T d-2

Acknowledgements

The author thanks Petru Mironescu for useful discussions and suggestions. He was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).