Romuald Elie
email: elie@ensae.frkharroubi@ensae.fr

Idris Kharroubi Ceremade

Probabilistic Representation and Approximation for Coupled Systems of Variational Inequalities *

Keywords: BSDE with jumps, variational inequalities, viscosity solutions, Monte Carlo simulations, switching problems. MSC Classification (2000): 93E20, 60H10, 60H30, 35K85, 49L25

Our study is dedicated to the probabilistic representation and numerical approximation of solutions of coupled systems of variational inequalities. We interpret the unique viscosity solution of a coupled system of variational inequalities as the solution of a one-dimensional constrained BSDE with jumps. This new representation allows for the introduction of a natural probabilistic numerical scheme for the resolution of these systems.

Introduction

Pardoux and [START_REF] Pardoux | Backward SDEs and Quasilinear Parabolic Partial Differential Equations[END_REF] developed the theory of backward stochastic differential equations, providing a probabilistic representation of solutions of quasi-linear parabolic PDEs. Coupling the diffusion process with a pure jump process, [START_REF] Pardoux | Probabilistic interpretation of a system of semilinear parabolic PDEs[END_REF] The focus of this note is to extend this type of Feynman-Kac representation to the more general class of coupled systems of quasilinear variational inequalities arising, for example, in optimal impulse or switching problems. We will typically consider systems of PDE of the form

- ∂v i ∂t -L i v i -f (i, ., (v k ) 1≤k≤m , σ(i, .) ⊤ D x v i ) ∧ min 1≤j≤m
h(i, j, ., v i , v j , σ(i, .) ⊤ D x v i ) = 0, (1.1) on I × [0, T ) × R d , with terminal condition v i (T, .) = g(i, .) on I × R d , (

where, for any i ∈ I := {1, . . . , m}, L i is a linear second order local operator

L i v i (t, x) := b(i, x) • D x v i (t, x) + 1 2 tr(σσ ⊤ (i, x)D 2 x v i (t, x)) , (1.3)
and b, σ, f , h and g are Lipschitz continuous functions. As observed by [START_REF] Bouchard | A stochastic target formulation for optimal switching problems in finite horizon[END_REF], this PDE appears in the resolution of optimal switching problems as well as stochastic target problems with jumps. The major difficulty arises from the coupling between all the components (v i ) i≤m of the solution and the use of different linear operators at each line. When m is large, the numerical resolution of (1.1)-(1.2) by classical PDE approximation methods is very tricky and highly computational. We intend to provide here a probabilistic representation to (1.1)-(1.2) leading to an efficient probabilistic numerical scheme. When b and σ are independent of the regime i ∈ I and the constraint functions are of the form h : (i, j, ., y i , y j , .) → y iy jc i,j , Hu and Tang (2007) interpret the vector solution to (1.1)-(1.2) as a multi-dimensional BSDE with terminal condition and oblique reflections. The challenging derivation of a convergent numerical approximation for this type of BSDE is of great interest and is currently under study. The approach of this paper relies instead on a recent reinterpretation of obliquely multi-dimensional reflected BSDEs in terms of one-dimensional constrained BSDEs with jumps, as introduced in Elie and [START_REF] Elie | Constrained Backward SDEs with Jumps: Application to Optimal Switching[END_REF]. The idea is to consider, as in [START_REF] Pardoux | Probabilistic interpretation of a system of semilinear parabolic PDEs[END_REF], a random regime driven by a pure jump transmutation process, allowing to retrieve simultaneously some information concerning all the components of the solution.

Given a d-dimensional Brownian motion W and an independent Poisson measure µ on R + ×I, we consider, for any initial condition e := (t, i, x) ∈ [0, T ]×I ×R d =: E, the unique I ×R d -valued solution (I e s , X e s ) of the SDE:

I s = i + s t I (j -I r-)µ(dr, dj) X s = x + s t b(I r , X r )dr + s t σ(I r , X r ) • dW r , t ≤ s ≤ T . (1.4)
Formally, given a smooth solution (v i ) i∈I to (1.1)-(1.2), the process Y := v I e (., X e . ) satisfies

Y t = g(I e T , X e T ) + T t f(I e s , X e s , Y s + U s , Z s )ds + K T -K t - T t Z s • dW s - T t I U s (j)µ(ds, dj) (1.5) on [0, T ],
where we denote

Z s := σ ⊤ (I e s-, X e s )D x v I e s-(s, X e s ), U s (.) := v . (s, X e s ) -v I e s-(s, X e s ), and
K s := s 0 [- ∂v I e u ∂t -L I e u v I e u -f(I e u , ., (v k ) 1≤k≤m , σ ⊤ (I e u , .)D x v I e u )](u, X e u )du. Since v satisfies (1.
1), we expect the following constraint to be satisfied:

h(I e s-, j, X e s , Y s-, Y s-+ U s (j), Z s ) ≥ 0 , j ∈ I, t ≤ s ≤ T . (1.6)
The BSDE (1.5) combined with constraint (1.6) falls into the class of constrained BSDEs with jumps and admits a unique minimal solution under mild conditions on the coefficients. We reinterpret the Y -component of the solution as the unique viscosity solution to the coupled system of variational inequalities (1.1)- (1.2). This new Feynman-Kac representation is meaningful to the BSDE literature since:

• It extends the results of [START_REF] Kharroubi | Backward SDEs with constrained jumps and Quasi-Variational Inequalities[END_REF] to more general constraints and driver functions depending on U . This allows for a strong coupling between the dynamics of the value function components and gives a minimality condition in some particular cases. (2006).This leads to a convergent numerical scheme based on time discretization, Monte Carlo simulations and projections.

The rest of the paper is organized as follows. In Section 2, we discuss existence, uniqueness and penalization, and give a minimality condition for constrained BSDEs with jumps (1.5)-(1.6). Section 3 presents the viscosity properties and the numerical approximation is detailed in the last section.

Notation. Throughout this paper, we are given a finite horizon T and a probability space (Ω, G, P) endowed with a d-dimensional standard Brownian motion W = (W t ) t≥0 , and an independent Poisson random measure µ on R + × I, with intensity measure λ(di)dt for some positive finite measure λ on I := {1, . . . , m}. We denote E := [0, T ] × I × R d . For a smooth function ϕ : [0, T ] × R d × I → R, ∂ϕ ∂t , D x ϕ and D 2 x ϕ denote resp. the derivative of ϕ w.r.t. t, the gradient and the Hessian matrix of ϕ w.r.t. x. The dependence in ω ∈ Ω is omitted when it is obvious.

Constrained Forward Backward SDEs with jumps

We present in this section the constrained forward backward SDEs with jumps and recall the existence and uniqueness results of [START_REF] Elie | Constrained Backward SDEs with Jumps: Application to Optimal Switching[END_REF]. We discuss the correspondence between the value function associated to Y and the U component of the solution. Under additional regularity of the value function, we provide a Skorohod type minimality condition for the considered BSDE.

Existence and uniqueness of a minimal solution via penalization

As discussed above, the forward process is a transmutation-diffusion process composed of a pure jump process I and a diffusion without jump X whose dynamics depends on I. For any initial condition e := (t, i, x) ∈ E, (I e , X e ) is the unique solution to (1.4) starting from (i, x) at time t.

For any initial condition e ∈ E, a solution to the constrained BSDE with jumps is a quadruplet

(Y e , Z e , U e , K e ) ∈ S 2 × L 2 W × L 2 μ × A 2 satisfying (1.5)-(1.6), where • S 2 is the set of real valued G-adapted càdlàg processes Y on [0, T ] s.t. Y S 2 := E sup 0≤r≤T |Y r | 2 1 2 < ∞, • L p W is the set of progressive R d -valued processes Z s.t. Z L p W := E T 0 |Z r | p dr 1 p < ∞, p ≥ 1, • L p μ is the set of P ⊗ σ(I) measurable maps U : Ω × [0, T ] × I → R s.t. U L 2 μ := E T 0 I |U s (j)| 2 λ(dj)ds 1 p < ∞, p ≥ 1,
• A 2 is the closed subset of S 2 composed by nondecreasing processes K with K 0 = 0.

Furthermore, (Y, Z, U, K) is referred to as the minimal solution to (1.5)-(1.6) whenever we have Y ≤ Y ′ a.s., for any other solution (Y ′ , Z ′ , U ′ , K ′ ). In order to ensure existence and uniqueness of a minimal solution to (1.5)-(1.6) for any initial condition, we make the following assumptions.

(H0) The following holds:

(i) There exists a constant L s.t.

|f (i, x, (u j ) j∈I , z) -f (i, x, (u ′ j ) j∈I , z ′ )| ≤ L|(z, (u j ) j∈I ) -(z ′ , (u ′ j ) j∈I )| , |h(i, j, x, y, u j , z, j) -h(i, j, x, y ′ , u ′ j , z ′ )| ≤ L|(y, z, u j ) -(y, z ′ , u ′ j )| , for all (x, i, j, y, z, u, y ′ , z ′ , u ′ ) ∈ R d × I 2 × [R × R d × R I ] 2 , and |f (i, x, (u j ) j∈I , z)| + |h(i, j, x, y, u j , z)| ≤ L 1 + |(y, z, (u j ) j∈I )| , for all (x, i, j, y, z, (u i ) i∈I ) ∈ R d × I 2 × R × R d × R I .
(ii) The function h(i, j, x, y, ., z) is non-increasing for all (i, x, y, z, j)

∈ I × R d × R × R d × I.
(iii) There exist two constants C 1 ≥ C 2 > -1 and a measurable map γ :

I × R d × R × R d × [R I ] 2 × I → [C 2 , C 1 ] such that, for any (i, x, y, z, u, u ′ ) ∈ I × R d × R × R d × [R I ] 2 , f (i, x, y + u, z) -f (i, x, y + u ′ , z ′ ) ≤ I (u j -u ′ j )γ(i, x, y, z, u, u ′ , j)λ(dj) .
(H1) For any e = (t, i, x) ∈ E, there exists a quadruple ( Ỹ e , Ze , Ũ e , Ke )

∈ S 2 × L 2 W × L 2 μ × A 2 solution to (1.5)-(1.6), with Ỹ e t = ṽI e t (t, X e t )
, for some deterministic function ṽ satisfying

|ṽ i (t, x)| ≤ C(1 + |x|) on E.
We provide in Remark 3.2 a more tractable sufficient condition under which (H1) holds.

The construction of the minimal solution is done by penalization. For any initial condition e ∈ E and n ∈ N, we introduce (Y e,n , Z e,n , U e,n ) solution to the following penalized BSDE

Y t = g(I e T , X e T ) + T t f(I e s , X e s , Y s + U s , Z s )ds - T t I U s (j)µ(ds, dj) - T t Z s • dW s + n T t I [h(I e s-, j, X e s , Y s-, Y s-+ U s (j), Z s )] -λ(dj)ds , 0 ≤ t ≤ T . ( 2
Y e,n -Y e L 2 W + Z e,n -Z e L p W + U e,n -U e L p μ -→ 0, n → ∞, 1 ≤ p < 2 .
Proof. This result is a direct application of Theorem We denote by (v n ) n∈N the sequence of deterministic functions defined by v n : e ∈ E → Y e,n t and we shall use indifferently the notation v n (t, i, x) or v n i (t, x), for (t, i, x) ∈ E. Under (H0)-(H1), we know from Proposition 2.1 that v is the pointwise limit of (v n ) n∈N . Proof. Fix e ∈ E. Since (v i ) i∈I is continuous, the process Y e inherits the quasi-left continuity of (I e , X e ). Combining (2.3) and Proposition 2.1 leads to max j∈I U e (j) -U e,n (j)

Representation of U and the minimality condition

S 2 -→ n→∞ 0.

We deduce from (2.2) and Lemma 5. 3 Link with coupled systems of variational inequalities

In this section, we interpret the minimal solution of (1.

Viscosity properties of the penalized BSDE

The penalized parabolic integral partial differential equation (IPDE) associated to (2.1) is naturally defined for each n ∈ N by

   - ∂ϕ i ∂t -L i ϕ i -f (i, ., (ϕ j ) j∈I , σ ⊤ (i, .)D x ϕ i ) -n I h i, j, ., ϕ i , ϕ j , σ ⊤ i D x ϕ i -λ(dj) = 0 on [0, T ) × R d × I, and v i (T, .) = g(i, .) on I × R d , (3.1)
where L is the m-dimensional Dynkin operator associated to X, defined in (1.

(i, t, x) ∈ I × [0, T ) × R d and ϕ ∈ C 1,2 ([0, T ] × R d ) such that (t, x) is a global minimum (resp. max.imum) of (v n i -ϕ), we have - ∂ϕ ∂t -L i ϕ -f (i, ., (v n j ) j∈I , σ ⊤ (i, .)D x ϕ) -n I [h(i, j, ., v n i , v n j , σ ⊤ (i, .)D x ϕ)] -λ(dj) (t, x) ≥ (resp. ≤ ) 0.
Proof. Fix n ∈ N. The continuity of v n follows from similar arguments as in the proof of Lemma 2.1 in [START_REF] Pardoux | Probabilistic interpretation of a system of semilinear parabolic PDEs[END_REF]. According to the representations detailed in the proof of Proposition 2.2, the viscosity property of v n fits in the framework of Theorem 4.1 in Pardoux et al. (1997), up to the comparison theorem for BSDE, which is replaced by Theorem 2.5 in Royer (2006). 2

Viscosity properties of the constrained BSDE with jumps

Formally, passing to the limit in (3.1) when n goes to infinity, we expect v to be a solution of (1.1) on [0, T ) × R d × I. As for the boundary condition, we cannot expect to have v(T -, .) = g, and we shall consider the relaxed boundary condition given by min v ig(i, .) , min

j∈I h i, j, ., v i , v j , σ ⊤ (i, .)D x v i (T -, x) = 0 on I × R d . (3.2)
Remark 3.1. In the particular case where the driver function f is independent of (y, z, u) and the constraint function is given by h : (i, j, x, y, y + v, z) → -c i,jv with c a given cost function, we retrieve the system of variational inequalities associated to switching problems min -

∂v i ∂t -L i v i -f (i, .), min j∈I [v i -v j -c i,j ] = 0 , on [0, T ) × R d × I , (3.3)
min v i -g(i, .), min j∈I v i -v j -c i,j (T -, .) = 0 , on R d × I . (3.4) Thus, if (3.4) satisfies a comparison theorem, v(T -, .
) is the smallest function greater than g satisfying (3.4). In particular, we retrieve the terminal condition v(T -, .) = g proposed by [START_REF] Hu | Multi-dimensional BSDE with oblique Reflection and optimal switching[END_REF] when the terminal condition g satisfies the cost constraint.

In order to define viscosity solutions of (1.1)-(3.2), we introduce, for any locally bounded vector function (u i ) i∈I on [0, T ] × R d its lower semicontinuous and upper semicontinuous (lsc and usc for short) envelopes u * and u * defined for (t, x)

∈ [0, T ] × R d by u * (t, x) = lim inf (t ′ ,x ′ )→(t,x),t ′ <T u(t ′ , x ′ ), and u * (t, x) = lim sup (t ′ ,x ′ )→(t,x),t ′ <T u(t ′ , x ′ ) . Definition 3.1. A vector function (u i ) i∈I , lsc (resp. usc) on [0, T ) × R d , is called a viscosity supersolution (resp. subsolution) to (1.1)-(3.2) if, for each (i, t, x) ∈ I × [0, T ] × R d and ϕ ∈ C 1,2 ([0, T ] × R d ) such that (t, x) is a global minimum (resp. maximum) of (u i -ϕ), we have, if t < T , min - ∂ϕ ∂t -L i ϕ -f (i, ., (u j ) j∈I , σ ⊤ (i, .)D x ϕ), min j∈I h(i, j, ., u i , u j , σ ⊤ (i, .)D x ϕ) (t, x) ≥ (resp. ≤) 0, if t = T , min u i -g(i, .
), min j∈I h(i, j, ., u i , u j , σ ⊤ (i, .)D x ϕ) (T, x) ≥ ( resp. ≤) 0 . Combining Fatou's lemma with standard estimates on X and linear growth conditions on g and ṽ, see (H1), we get that sup t∈[0,T ] |v i (t, x)| 2 ≤ C(1+|x| 2 ) with C > 0. Thus, v is locally bounded. We observe that the viscosity property of v in the interior of the domain is based on the same arguments as the one presented in the proof of Theorem 4.1 of [START_REF] Kharroubi | Backward SDEs with constrained jumps and Quasi-Variational Inequalities[END_REF]. The only difference comes from the more general form of the coefficients f and h. This is not a relevant issue here since they are continuous. In order to alleviate the presentation of the paper, we choose to omit it here and only prove the viscosity property (3.2) on the maturity boundary. (i) Let us first consider the supersolution property of v * to (3.2). Let (i,

A locally bounded vector function

(u i ) i∈I on [0, T )× R d is
x 0 ) ∈ I × R d and ϕ ∈ C 1,2 ([0, T ] × R d ) such that (T, x 0 ) is a null global minimum of ([v * ] i -ϕ).
Passing to the limit of the viscosity properties of the penalized BSDE, we get

min j∈I h(i, j, x, [v * ] i , [v * ] j , σ ⊤ (i, .)D x ϕ)](T, x 0 ) ≥ 0 .
Furthermore v n (T, .) = g, n ∈ N, so that the monotonic property of the sequence of continuous functions (v n ) n∈N gives v * (T, .) ≥ g. Therefore v * is a viscosity supersolution of (3.2).

(ii) We now turn to the subsolution property of v * . We argue by contradiction and suppose the existence of (i,

x 0 ) ∈ I × R d and ϕ ∈ C 1,2 ([0, T ] × R d ) such that 0 = (v * i -ϕ)(T, x 0 ) = max [0,T ]×R d (v * i -ϕ) , (3.5)
and min ϕg(i, .) , min j∈I h(i, j, ., ϕ, v * j , σ ⊤ (i, .)D x ϕ) (T, x 0 ) =: 2ε > 0. The regularity of v * , ϕ and D x ϕ as well as the monotonic property of h lead to the existence of an open neighborhood O of (T, x 0 ) ∈ [0, T ] × R d , and Υ, r > 0 such that for all (t, x, η, η ′ ) ∈ O × (-Υ, Υ) × B(0, r), we get min ϕηg(i, .) ,

min j∈I h(i, j, ., ϕ -η, v * j , σ ⊤ (i, .)[D x ϕ + η ′ ]) (t, x) ≥ ε . (3.6)
We introduce a sequence (

t k , x k ) k valued in [0, T )×R d satisfying (t k , x k ) → (T, x 0 ) and v i (t k , x k ) → v * i (T, x 0 ). Let us choose δ > 0 such that [t k , T ] × B(x k , δ) ⊂ O
for k large enough, and introduce the modified test function ϕ k given by

ϕ k (t, x) := ϕ(t, x) + ζ |x -x k | 2 δ 2 + C k φ x -x k δ + √ T -t , where 0 < ζ < Υ ∧ δr, φ is a regular function in C 2 (R d ) such that φ| B(0,1) ≡ 0, φ| B(0,1) c > 0, lim |x|→∞ φ(x) 1+|x| = ∞, and C k > 0 is a constant to be determined later. Since (v * -ϕ k )(t, x) ≤ (v * -ϕ)(t, x) -ζ |x-x k | 2 δ 2 for (t, x) ∈ [t k , T ] × R d , we deduce from (3.5) that (v * -ϕ k )(t, x) ≤ -ζ, for (t, x) ∈ [t k , T ] × ∂B(x k , δ).
Choosing C k large enough, the particular form of the function φ leads to

(v * i -ϕ k )(t, x) ≤ - ζ 2 , for (t, x) ∈ B(x k , δ) c × [t k , T ] . (3.7)
Thanks to the √ Tt term in the modified test function ϕ k , we deduce that

- ∂ϕ k ∂t -L i ϕ k -f i, ., (v * j + [ϕ k -η -v * i ]1 j=i ) j∈I , σ ⊤ (i, .)D x ϕ k (t, x) ≥ 0 , (3.8)
for any (t, x, η) ∈ [t k , T )×B(x k , δ)×(-Υ+ζ, Υ) and k large enough. We now choose η < Υ∧ ζ 2 ∧ε and introduce the stopping time θ k := inf s ≥ t k ; X e k s / ∈ B(x k , δ) or I e k s = I e k s-∧ T , where e k := (t k , i, x k ). Let us finally consider the process

(Y k , Z k , U k , K k ) given on [t k , θ k ] by                    Y k s := ϕ k (s, X e k s ) -η 1 s∈[t k ,θ k ) + v I e k s (θ k , X e k θ k )1 s=θ k , Z k s := σ ⊤ (I e k s-, X e k s )D x ϕ k (s, X e k s ), U k s := v * j (s, X e k s ) -[ϕ k (s, X e k s ) -η] 1 j =I e k s-j∈I , K k s := - s t k ∂ϕ k ∂t + L I e k r ϕ k + f(I e k r , ., (v * j + [ϕ k -η -v * I e k r ]1 j=I e k r ) j∈I , Z k r ) (r, X e k r )dr - s t k I (ϕ k -η -v * j )(r, X e k r )µ(dr, dj) + ϕ k -η -v I e k θ k (θ k , X e k θ k )1 s=θ k .
One easily checks from (3.6)

-(3.7)-(3.8) that (Y k , Z k , U k , K k ) is solution to Y s = v I e k θ k (θ k , X e k θ k ) + θ k s f(I e k r , X e k r , Y r + U r , Z r )dr - θ k s Z r • dW r - θ k s I U r (j)µ(dr, dj) + K θ k -K r on [t k , θ k ],
together with the constraint h(I e k r-, j, X e k r , Y r-, Y r-+ U r (j), Z r ) ≥ 0 a.e., j ∈ I. Since (Y e k , Z e k , U e k , K e k ) is a minimal solution to this constrained BSDE with jumps, we deduce

ϕ k (t k , x k ) -η = ϕ(t k , x k ) + T -t k -η ≥ v i (t k , x k ) ,
for all k large enough.

Letting k go to infinity, this contradicts (3.5) and concludes the proof. 2

Remark 3.2. The main drawback of this representation is the necessity of Assumption (H1).

Following similar arguments as in the proof of Proposition 6.3 in [START_REF] Kharroubi | Backward SDEs with constrained jumps and Quasi-Variational Inequalities[END_REF], observe that it is satisfied whenever there exists a Lipschitz function

(w i ) i∈I ∈ [C 2 (R d )] I supersolution to (3.
2) satisfying a linear growth condition, and there exists a constant C > 0 such that

L i w i + f (., (w j ) j∈I , σ ⊤ i Dw i ) ≤ C on R d , i ∈ I.

A comparison argument

In this section, we provide sufficient conditions characterizing the value function v as the unique viscosity solution of (1.1)-(3.2). This gives in particular the continuity of v, leading to the strong convergence by penalization and the minimality condition, presented in Section 2. The proof relies as usual on a comparison argument, which holds under the following additional assumptions.

(H2) The following holds: (i) For any i ∈ I, f (i, .) is convex in ((y j ) j∈I , z) and increasing in u i .

(ii) For any i, j ∈ I, h(i, j, .) is concave in (y i , y j , z) and decreasing in y i .

(iii) There exists a nonnegative vector function (Λ i ) i∈I ∈ [C 2 (R d )] I and a positive constant ρ such that, for all i ∈ I, Λ i ≥ g i , lim |x|→∞

Λ i (x)
1+|x| = ∞ and we have :

L i Λ i + f (i, ., (Λ j ) j∈I , σ ⊤ (i, .)D x Λ i ) ≤ ρΛ i and min j∈I h(i, j, ., Λ i , Λ j , σ ⊤ (i, .)D x Λ i ) > 0 .
An approximation of Y n at time 0 is computed recursively following the backward scheme for k

= T /h -1, • • • , 0 :          Z n,h t k := 1 h E t k Y n,h t k+1 (W t k+1 -W t k ) U n,h t k (i) := 1 h E t k Y n,h t k+1 μ((t k ,t k+1 ]×{i}) λ(i) , i ∈ I Y n,h t k := E t k Y n,h t k+1 + t k+1 t k f n (I s , X h t k , Y n,h t k+1 , Z n,h t k , U n,h t k )ds (4.1)
where E t k denotes the conditional expectation with respect to G t k . Following the arguments of Section 2.5 in [START_REF] Bouchard | Discrete-time approximation of decoupled forward-backward SDE with jumps[END_REF] and identifying (Y n,h , Z n,h , U n,h ) as a process constant on each interval (t k , t k+1 ], we verify the convergence of this discrete-time approximation :

Y n -Y n,h S 2 + Z n -Z n,h L 2 W + U n -U n,h L 2 μ -→ h→∞ 0, n ∈ N. (4.2)
Step 3. Approximation of the conditional expectations. The last step consists in estimating the conditional expectation operators E t k arising in (4.1). We adopt here the approach of Longstaff-Schwarz generalized in [START_REF] Gobet | Rate of convergence of empirical regression method for solving generalized BSDE[END_REF] relying on least square regressions.

Fix N ∈ N and simulate N independent copies of the Brownian increments (W j t k+1 -W j t k ) 0≤k≤T /h and the poisson measure (μ j ((t k , t k+1 ] × I) 0≤k≤T /h . For each simulation j ≤ N , define I N j and X h,N j as the trajectories of I and X h . By induction, one can easily verify the Markov property of the process (Y n,h , Z n,h , U n,h ) defined in (4.1):

Y n,h t i = c n,h k (I t i , X h t i ), Z n,h t i = a n,h k (I t i , X h t i ), U n,h t i = b n,h k (I t i , X h t i ),
for some deterministic functions (a n,h k , b n,h k , c n,h k ) k≤n . The idea is to approximate these functions using Ordinary Least Square (OLS) estimators. Given L ∈ N, we introduce a collection of basis functions (a L l , b L l , c L l ) 1≤l≤L of R × R d × R d . For each trajectory j ≤ N , define the associated terminal value given by Y n,h,L,N j,tn := g I N k,tn (X h,N j,tn ). Now we define recursively (Z n,h,L,N j,t k , U n,h,L,N j,t k ), backward in time for k = T /h -1, • • • , 0, by computing the OLS approximations as follows:

(α 1 , • • • , αL ) := arg min α 1 ,••• ,α L 1 N N j=1 1 h Y n,h,L,N j,t k+1 [W j t k+1 -W j t k ] - L l=1
α l a L l (I N j,t k , X h,N j,t k ) αl a L l (I N j,t k , X h,N j,t k ) and U n,h,L,N j,t k (i) := L l=1 βl (j)b L l (I N j,t k , X h,N j,t k ) , i ∈ I. γ l c L l (I N j,t k , X h,N j,t k ) 2 in order to deduce the OLS approximation Y n,h,L,N j,t k := L l=1 γl c L l (I N j,t k , X h,N j,t k ).

It remains to introduce (γ

We refer to [START_REF] Gobet | Rate of convergence of empirical regression method for solving generalized BSDE[END_REF] for the control of the statistical error due to the approximation of the conditional expectation operators by OLS projections, and, by extension,

Y n,h -Y n,hL,N S 2 + Z n,h -Z n,h,L,N L 2 W + U n,h -U n,h,L,N L 2 μ -→ N,L→∞ 0 , n ∈ N , h > 0. (4.
3)

The convergence of the algorithm follows from (2.2), (4.2) and (4.3). The derivation of a convergence rate requires precisions on the influence of n on the discretization and statistical errors, as well as a control of the penalization error. This challenging point is left to further research.

  extend this representation to systems of coupled semilinear PDEs with different linear differential operators on each line. Introducing restrictions on the domain of the backward process, El Karoui et al. (1997) cover the class of variational inequalities. Constraining instead the jump part of the solution, Kharroubi et al. (2010) consider quasilinear variational inequalities.

Proposition 2 . 2 . 2

 222 Let (H0)-(H1) holds. For any e ∈ E and stopping time θ valued in [t, T ], we have Y e θ = v I e θ (θ, X e θ ), and the process U represents asU e s (j) = v j (s, X e s )v I e s-(s, X e s ) , j ∈ I, t ≤ s ≤ T . (2.3) Proof. According to Proposition 2.1, we simply need to provide similar representations for the penalized BSDE (2.1). Fix e ∈ E. For any stopping time θ valued in [t, T ], uniqueness of solution of (2.1) and the Markov property of (I e , X e ) directly give to Y e,n θ = v n I e θ (θ, X e θ ). Denoting Ũ e,n s (j) := v n j (s, X e s )v n I e s-(s, X e s ), for j ∈ I and 0 ≤ s ≤ T , we deduce from (2.1) that I Ũ e,n s (j)µ(ds, dj) = Y e,n s -Y e,n s-= I Ũ e,n s (j)µ(ds, dj) , 0 ≤ s ≤ T . Therefore E T 0 I (U e,n s (j) -U e,n s (j)) 2 λ(dj)ds = 0 and the proof is complete.Under an extra regularity assumption on the function v satisfied under Assumption (H2) below, the previous representation leads to a Skorohod type minimality condition for (1.5)-(1.6).Corollary 2.1. Let (H0)-(H1) holds. Suppose (v i ) i∈I is continuous and the function h does not depend on z. Then, for any e ∈ E, the minimal solution (Y e , Z e , U e , K e ) satisfies

  5)-(1.6) as the unique viscosity solution of the PDE (1.1)-(1.2), thus generalizing the representation derived in Kharroubi et al. (2010), Pardoux et al. (1997) and Peng and Xu (2007).

Proposition 3 . 1 .

 31 3). Since the penalized BSDE falls into the class of BSDE with jumps studied by[START_REF] Pardoux | Probabilistic interpretation of a system of semilinear parabolic PDEs[END_REF], we deduce the following Feynman-Kac representation result. Under (H0)-(H1), the functions (v n ) n are continuous viscosity solutions of (3.1). Indeed, for any n ∈ N, v n (T, .) = g and, for any

2 ,

 2 ( β, • • • , βL )(i) := arg min β 1 ,••• ,β L

•

  It generelizes the conclusions of[START_REF] Peng | Constrained BSDE and viscosity solutions of variational inequalities[END_REF] derived in the no-jump case.

	• It offers a PDE representation to reflected BSDEs with interconnected obstacles introduced in Hamadène and Zhang (2008) since they relate directly to constrained BDSE with jumps,
	see Elie and Kharroubi (2009).
	• It generalizes the use of diffusion-transmutation process in Pardoux et al. (1997) to systems of variational inequalities.
	This representation leads to a natural probabilistic algorithm for the resolution of (1.1)-(1.2).
	The constrained BSDE with jumps is replaced by a penalized BSDE with jumps, which is ap-
	proximated by the discrete-time scheme studied in Bouchard and Elie (2008) and Gobet et al.

  Theorem 2.1. Suppose (H0)-(H1) holds. For any e := (t, i, x) ∈ E, there exists a unique quadruple (Y e , Z e , U e , K e ) ∈ S 2 × L 2 W × L 2 μ × A 2 minimal solution to (1.5)-(1.6) with K e predictable, and v i : (t, x) → Y t,i,x t defines a deterministic map from E into R. Moreover (Y e , Z e , U e ) is the limit of the (Y e,n , Z e,n , U e,n ) n∈N in the following sense

.1) Under (H0), we get from

[START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF]

existence and uniqueness of a solution of (2.1). We introduce K e,n := . 0 I [h(I e s-, j, X e s , Y e,n s-, Y e,n s-+ U e,n s (j), Z e,n s )] -λ(dj)ds, for any (e, n) ∈ E × N.

  2.1 in Elie and Kharroubi (2009). 2 Under additional regularity on Y e , we can improve the previous convergence up to p = 2. Proposition 2.1. If (H0)-(H1) holds, (Y e,n ) n∈N converges increasingly to Y e , for any e ∈ E. Additionally, if the process Y e is quasi-left continuous in time, we have Y e -Y e,n S 2 + Z e -Z e,n Fix e ∈ E and observe from Proposition 2.1 in Elie and Kharroubi (2009) that Y e,n converges increasingly to Y e . Since µ is a Poisson measure, the process Y e,n is quasi-left continuous. If Y e has the same regularity, the predictable projections of Y e and Y e,n are simply given by (Y e t-) We deduce from the weak version of Dini's theorem, see Dellacherie and Meyer (1980) p. 202, that Y e,n converges uniformly to Y e on [0, T ], and the dominated convergence theorem gives us Y e -Y e,n S 2 -→

		L 2 W	+ U e -U e,n	L 2 μ + K e -K e,n	S 2 -→
	with standard estimates of the form		n→∞	0. Combined
	Z e,n+p -Z e,n 2 L 2 W	+ U e,n+p -U e,n 2 L 2

n→∞ 0 , e ∈ E . (2.2) Proof. t and (Y e,n t-) t . This leads to Y e t-= lim n→∞ Y e,n t-. μ + K e,n+p -K e,n 2 S 2 ≤ C Y e,n+p -Y e,n 2 S 2 , this implies that the sequences (Z n ), (U n ) and (K n ) are Cauchy and hence convergent. 2 Remark 2.1. Under the additional Assumption (H2) below, (v i ) i∈I is interpreted as the unique viscosity solution to (1.1)-(1.2), see Theorem 3.2. In this case, (v i ) i∈I is continuous, Y t = v It (t, X t ) is quasi-left continuous and Proposition 2.1 holds.

  8 in Gegoux-Petit and Pardoux (1995), which also holds for càglàd functions, that

	t	T	min j∈I	h(I e s-, j, X e s , Y e,n s-, Y e,n s-+ U e,n s (j)) dK e,n s	-→ n→∞	t	T	min j∈I	h(I e s-, j, X e s , Y e s-, Y e s-+ U e s (j)) dK e s .

Since

T t min j∈I h(I e s-, j, X e s , Y e,n s-, Y e,n s-+ U e,n s (j)) dK e,n s ≤ 0 and (1.6) holds, we get (2.4). 2

  Proof. First, following the proof of Lemma 3.3 and Remark 3.2 in Kharroubi et al. (2010), standard estimates on the penalized BSDE (2.1) lead to

	if u E sup t∈[0,T ]	|Y e,n t | 2 ≤ C 1 + E |g(I e T , X e T )| 2 +	t	T	called a viscosity solution to (1.1)-(3.2) s | 2 ds + sup |X e s∈[0,T ] |ṽ I e s (s, X e s )|

* and u * are respectively viscosity supersolution and subsolution to (1.1)-(3.2). Theorem 3.1. Under (H0)-(H1), the function v is a (discontinuous) viscosity solution to (1.1)-(3.2). 2 , e ∈ E .

  1 , • • • , γL ) the minimizer of the mean square error

	1 N	N j=1	Y n,h,L,N j,t k+1	+	t k+1 t k f

n (I N j,s , X h,N j,t k , Y n,h,L,N t k+1 , Z n,h,L,N t k , U n,h,L,N t k )ds -L l=1

An example where (H2) holds is given for the case of optimal switching in [START_REF] Bouchard | A stochastic target formulation for optimal switching problems in finite horizon[END_REF].

Remark 3.3. As in [START_REF] Bouchard | A stochastic target formulation for optimal switching problems in finite horizon[END_REF], (iii) allows us to construct a nice strict supersolution of (1.1) allowing to control solutions of (1.1)-(3.2) by convex perturbations. Following the approach of [START_REF] Kharroubi | Backward SDEs with constrained jumps and Quasi-Variational Inequalities[END_REF], the general form of f and h forces us to add the extra convexity assumptions (i) and (ii). Theorem 3.2. Let (H0)-(H1)-(H2) holds. Then, for any U lsc (resp. V usc) viscosity supersolution (resp. subsolution) of (1.1)

In particular, v is continuous and it is the unique viscosity solution of (1.1)-(3.2) satisfying a linear growth condition.

We omit the proof of this comparison theorem which is a natural extension of Theorem

Numerical issues

The numerical resolution of systems of variational inequalities of the form (1.1)-(1.2) usually relies on the use of iterated free boundary. We first solve the system without boundary condition and consider recursively the system constrained by the boundary condition coming from the previous iteration. In a switching problem, we constrain the solution associated to n + 1 possible switches by the obstacle built from the solution where only n switches are allowed. Such a numerical approach is computationally demanding. We present here a natural convergent algorithm based on the approximation of the solution to the corresponding constrained BSDE with jumps (1.5)-(1.6). We combine a penalization procedure with the discrete-time scheme studied by [START_REF] Bouchard | Discrete-time approximation of decoupled forward-backward SDE with jumps[END_REF] and the statistical estimation projection presented in [START_REF] Gobet | Rate of convergence of empirical regression method for solving generalized BSDE[END_REF]. Thanks to the previous Feynman-Kac representation, this gives rise to a convergent probabilistic algorithm solving coupled systems of variational inequalities.

We fix an initial condition e ∈ E and omit it in the expressions for ease of presentation. Suppose that (H0)-(H1)-(H2) holds. The algorithm is divided in three steps.

Step 1. Approximation by penalization. We first approach the constrained BSDE with jumps (1.5)-(1.6) by its penalized version (2.1) characterized by a driver f n := fn[h] -as in Section 2.1. We deduce from Proposition 2.1 that the penalization error converges to 0 as n goes to infinity, see (2.2).

Step 2. Time discretization. Observe that the pure jump process I can be simulated perfectly and denote by (τ l ) l its jump times on [0, T ]. We introduce the Euler time scheme approximation X h of the forward process X defined on the concatenation (s l ) l of the regular time grid {t k := kh, k = 1, . . . , T /h} with the jumps (τ l ) of I: X h 0 = X 0 and X h s l+1 := X h s l + b(I s l , X h s l )(s l+1s l ) + σ(I s l , X h s l )[W s l+1 -W s l ].

We deduce an approximation Y n,h T of Y n T at maturity given by g I T (X h T ). The penalized BSDE (2.1) can now be discretized by an extension of the scheme exposed in [START_REF] Bouchard | Discrete-time approximation of decoupled forward-backward SDE with jumps[END_REF]


J-L Chenot
email: jean-loup.chenot@mines-paristech.fr

L Fourment

R Ducloux

E Wey

FINITE ELEMENT MODELLING OF FORGING AND OTHER METAL FORMING PROCESSES

Keywords: Metal, Plasticity, Finite Element, Numerical Simulation, Industrial Forming Processes

The fundamental mechanical formulation is recalled for simulation of metal forming processes. The basic principles of 3-dimensional finite element discretization and of time integration are summarized. Several important numerical developments for efficient computation of large plastic deformation are briefly described. Various fields of applications to real processes are reviewed. Illustrative examples are mentioned to show the utilization of the commercial computer code Forge3 in industry, as a very flexible tool to design metal forming sequences.

HISTORICAL INTRODUCTION

The first attempts to simulate large plastic deformations started in the 70's with two dimensional approximations, but applications were scarce due to the limited performance of computers [START_REF] Zienkiewicz | Elastosolution of engineering problems: initial stress, finite element approach[END_REF][START_REF] Lee | New solutions to rigid plastic deformation problems using a matrix method[END_REF][START_REF] Cornfield | Theoretical prediction of plastic flow in hot rolling including the effect of various temperature distribution[END_REF]. Several threedimensional computations were performed at the research level in the 80's [START_REF] Surdon | Finite element calculations of three-dimensional hot forging[END_REF], but it is only in the 90's that real industrial examples could be run [START_REF] Soyris | Forging of a connecting rod: 3-D finite element calculation[END_REF][START_REF] Chenot | Recent Progresses in Finite Element Simulation of the forging process[END_REF] and at the end of 90 that it begun to be used in industry. Since 2000 many improvements have made computations faster and more accurate and now the possibility of process optimization can be achieved.

MECHANICAL BASIS

For a much more detailed account of the theoretical background, the reader is referred to [START_REF] Wagoner | Metal forming analysis[END_REF].

CONSTITUTIVE EQUATIONS

The mechanical law that is often used in metal forming is isotropic elastic viscoplastic with an additive decomposition of the strain rate and a power law: (2)

e p ε = ε + ε 1 1 1 (1)
1 1 1/ ( ) / - ′ = - ε σ ε σ ε σ ε σ 1 p m K R K σ (3)
Where e ε 1 is the elastic strain rate, p ε 1 the plastic or visco plastic component, d J /dt is for the Jauman derivative, λ and µ are the Lamé coefficients, σ' is deviatoric stress tensor, σ is the usual equivalent stress, K is the consistency and mis the strain rate sensitivity. More general material behaviour can be used:

-Temperature dependant laws, -Anisotropic equation (Hill), -Anisotropic kinematic hardening is defined in term of the backstress X and the fourth rank tensor H:

X H X ( -) ( -) ′ ′ σ = σ σ (4)
The evolution of X is governed by the hardening function R and the parameter β:

2 3 R / ( / ) Χ = β ∂ ∂ε ε 1 1 (5)

CONTACT AND FRICTION

Several important conditions must be taken into account:

unilateral contact with friction, between the work-piece and rigid tools, modelled by e. g. a generalized Coulomb law:

v v n τ = -µ -σ ∆ ∆ ( ) / (6)
Where n σ is the normal stress and v ∆ the velocity difference between tool and part.

contact between deformable bodies by a penalty approach.

INTEGRAL FORMULATON

For an incompressible, or quasi incompressible flow, it is desirable to utilize a mixed formulation. In the domain Ω of the part, this formulation is written for any virtual velocity and pressure fields p v*, * as:

0 dS * v - *)dV pdiv(v - dV * : ' c = 1 1 1 Ω ∂ Ω Ω τ ε σ 1 (7) 0 * ) ) ( ( = + -1 Ω dV p p v div 1 κ (8)

THERMAL COUPLING

During heating of the preform, followed by hot forming, cooling and heat treatments of the part, it is necessary to introduce thermal and mechanical coupling. The classical heat equation for deformable bodies is written simply:

m 1 w dT 1c div(kgrad(T)) f ( 3 ) dt + = + ε 1 (9)
where the last term is a fraction f w of the viscoplastic heat dissipation. The constitutive law depends on temperature. For example we have the equations:

( ) n 0 0 0 1 K K 2 2 exp( / T), m m m T = + β = + (9)
The radiation condition on the free surface s ∂Ω is written:

4 4 r r 0 T k (T T ) n ∂ - = ε σ - ∂ (9)
where: r ε is the emissivity parameter, r σ the Stefan constant and o T the outside temperature.

NUMERICAL METHODS

FINITE ELEMENT DISCRETIZATION

To-day a satisfactory compromise is based on a mixed velocity and pressure formulation using tetrahedral elements, and a bubble function for the velocity field, in order to stabilize the solution for incompressible or quasi incompressible materials. The velocity field is discretized with shape functions N n , in term of nodal velocity vectors V n :

2 = n n n ) ( N 1 V v (9)
Using isoparametric elements the mapping between the physical space with coordinates x and the reference space, with coordinate ξ ξ ξ ξ is: [START_REF] Bousseta | Adaptive remeshing based on a posteriori error estimation for forging simulation[END_REF] The strain rate tensor can be computed with the help of the conventional B descritized linear operator:

2 = n n n ) ( N 1 X x
n n n 2 V B = 2 1 ( 11
)
The discretized mixed integral formulation for the mechanical problem is expressed in term of the deviatoric stress increment ∆σ' and pressure increment ∆p:

c U n n n n n R ( ) dV (p p) trace(B ) dV ( ) ( v/ v )N dS 0 Ω Ω ∂Ω ′ ′ = σ + ∆σ - + ∆ + µ -σ ∆ ∆ = 1 1 1 : Β : Β : Β : Β (12)
And the discretized mass conservation:

P m m R ( t div(v) p)M dV 0 Ω = κ ∆ + ∆ = 1 (13)

TIME DISCRETIZATION

The most widely used integration method for the velocity formulation still remains the simple Euler one step scheme. It is similar to the incremental displacement approach where the displacement is written:

u v t ∆ = ∆ .
It generally observed that a two levels Runge and Kutta method leads to a considerable improvement of the solution, which is particularly desirable when the workpiece undergoes very large rotations, such as in ring rolling.

CONTACT AND MULTI BODY COUPLING

Contact between tools and part or between several deformable materials can be imposed at the node level or with a surface approach. The so-called master and slave method is popular, but in order to avoid penetration between antagonists, there is a need for a more symmetrical formulation when the materials have similar hardness (see [START_REF] Fourment | A Quasi-symmetric formulation for contact between deformable bodies[END_REF]). When the contact zone is reduced with respect to the component size, more accurate formulations are required and significant improvements can be brought by smoothing the discretized surface of the obstacle by using higher order interpolations [START_REF] Hachani | Improved contact treatment for metal forming processes with reduced contact area[END_REF].

MESHING, REMESHING AND ADAPTIVITY

Automatic dynamic remeshing during the simulation of the whole forming process is almost always necessary, as elements undergo very high strain which could be at the origin of degeneracy. Before this catastrophic event, decrease of element quality must be evaluated and a remeshing module must be launched periodically to recover a satisfactory element quality. The global mesh can be completely regenerated, using a Delaunay or any front tracing method, but the method of iterative improvement of the mesh, with a possible local change of element structure and connectivity, seems to be much more effective.

In order to avoid the necessity for the user to perform several computations, with different meshes to check the accuracy, an error estimation can be developed using for example the generalization of the method proposed by Zienckiewicz and Zhu. Then, if the rate of convergence of the computation is known, the local mesh refinement necessary to achieve a prescribed tolerance can be computed, and the meshing modules are improved to be able to respect the refinement when generating the new mesh (see [START_REF] Bousseta | Adaptive remeshing based on a posteriori error estimation for forging simulation[END_REF]), or the achieve the best accuracy for a given maximum nodes number.

ITERATIVE AND PARALLEL SOLVING

For industrial, complicated applications with short delays, the computing time can be decreased dramatically using several or several tens of processors. This requires to use an iterative solver and to define a partition of the domain, each sub domain being associated with a processor. But the parallelization is made more complex due to remeshing and the remeshing process itself must be parallelized.

BIMESH METHOD

In order to reduce substantially the very long computational time required for the simulation of incremental processes such as cogging or ring rolling, a Bimesh method was developed [START_REF] Ramadan | A two mesh method for speeding-up incremental processes such as cogging[END_REF]. It consists in using different finite element meshes for the resolution of the different physical problems: a main fine mesh to store the results and to carry out the thermal computations, and an intermediate coarser mesh for the mechanical calculations. It makes it possible to take advantage of the localised deformations that characterize incremental processes to coarsen the mechanical mesh and consequently reduce the computational time.

MULTI GRID METHODS

In the multi grid method the principle is different: several grids of different refinement are defined, but the objective is to compute the solution on the finest one. This is achieved iteratively by using different operators: smoothing, restriction and extension, which allow us to compute the solution faster than with conventional iterative methods. The effectiveness of the method was demonstrated on forging in [START_REF] Rey | A node-nested Galerkin multigrid method for metal forging simulation[END_REF].

APPLICATIONS

COMPLEX FORGING

The computer code Forge 3 is now designed to control numerical accuracy with the possibility of error estimation and adaptive refinement [START_REF] Soyris | Forging of a connecting rod: 3-D finite element calculation[END_REF][START_REF] Bousseta | Adaptive remeshing based on a posteriori error estimation for forging simulation[END_REF]. A mechanical and coupled thermal analysis takes into account the elasticity of the tools, the heat conduction and possibly the elastic yielding of the forging press. It is also possible to simulate several operations, including heat treatments, by transferring automatically the results from one sequence to the next one.

DRAWING, ROLLING

In such processes, the deformation zone is localized. When an incremental approach is preferred in order to determine the beginning of the process and the free surface when the steady state is reached, an ALE method is used to avoid complete remeshing at every step. Applications are shown in [START_REF] Fourment | Application of the Arbitrary Lagrangian Eulerian formulation to the numerical simulation of stationary forming processes with dominant tangential material motion[END_REF].

RING ROLLING

Ring rolling is a complex process which is considered as incremental and which exhibits very large rotations. Several approaches were tested to avoid excessive numerical inaccuracy during the large number of rotations of the work-piece, including use of second order time integration and more recently the use of a polar coordinates and an improved scheme for local contact [START_REF] Hachani | Improved contact treatment for metal forming processes with reduced contact area[END_REF].

FLOW FORMING

Flow Forming is a 3-dimensional process for producing axisymmetric tubular parts. The preform is attached to a rotating mandrel, rollers apply a localised action on the wall of the part, the material of which is moved along the axial direction. Using a Lagrange -Euler development, the process can be simulated with Forge3 in order to predict possible geometric defects and the final microstructure [START_REF] Massoni | 3-D finite element modelling of flow forming and incremental forming[END_REF].

MECHANICAL ASSEMBLY

The typical mechanical assembly process is riveting and it involves for example the interaction of two tools and three deformable bodies: the rivet and two sheets. In this case a robust multi body contact algorithm must be used (see for example [START_REF] Wagoner | Metal forming analysis[END_REF][START_REF] Fourment | A Quasi-symmetric formulation for contact between deformable bodies[END_REF]) in order to simulate accurately the fastening stage. Moreover if one is interested in testing the strength of the joint it is compulsory to introduce the weakening due to damage, not only during a tension test but during the riveting process itself [START_REF] Bouchard | Numerical modeling of self-pierce riveting -From riveting process modelling down to structural analysis[END_REF].

MACHINING

Machining is a complex process that can be analyzed at difference scales. When the formation of the chip and the tool duration are the main objective, a numerical model can be utilized to predict the distribution of strain, strain rate and temperature in the vicinity of the tip. This approach will allow us to predict the possible formation of an adiabatic shear band (see for example [START_REF] Fourment | An adaptive A.L.E. finite element formulation for the 3d simulation of adiabatic shear band formation during high speed machining[END_REF]) and ultimately to optimize the tool geometry.

PROCESS OPTIMIZATION

Most often process optimization is performed in industry by trial and error, using the experience of the engineers or of the technicians. Due to the progress in computer technology, and with improved software, now available in Forge3, it is possible to achieve process optimization, for example in forging, using only process initial data and material behaviour. The optimization is developed in order to design the best tool shape and to prevent defect during forming, as it shown in [START_REF] Ducloux | Optimization of a range of 2D and 3D bulk forming processes by a meta-model assisted evolution strategy[END_REF][START_REF] Ejday | Meta-model assisted Multi-Objective Optimization -Application to non-steady 3D metal forming processes[END_REF].

We have briefly shown that with the Forge3 simulation code, a very large variety of metal forming processes can be treated and optimized. New applications appear each year and the computer code is developed continuously to include new modules for existing or new processes and also to improve the accuracy of the results and to accelerate the computations. It is expected that in the very near future, the software will allow the user to predict final local properties in the work-piece and evaluate the in service durability of the parts and components.


S Cavret
email: cavret@isara.fr

N Laurent

B Videmann

M Mazallon

S Lecoeur

Assessment of deoxynivalenol (DON) adsorbents and characterisation of 1 their efficacy using complementary in vitro tests. 2 3

Keywords: mycotoxins, deoxynivalenol, adsorption, intestinal transfer bentonite, sepiolite, Saccharomyces cerevisiae

Deoxynivalenol (DON) is a prevalent and resistant mycotoxin found in cereals and their 11 related products. Adsorbents appear to provide an opportunity to decrease DON absorption in 12 animals, but it is very difficult to evaluate their real efficacy due to their specificity. It is often 13 senseless to extrapolate results obtained with one mycotoxin to another one and even to 14 extrapolate results obtained in vitro in buffer to in vivo situation. We have carried out 15 experiments to characterize the properties of potential DON adsorbents. Tests in buffer pH 7 16 first allowed us to focus on six adsorbents: activated charcoal, cholestyramin, saccharomyces 17 cerevisiae mannans, algae β-glycan, fungus β-glycan and leguminous plant. The use of 18 equilibrium sorption models suggested non-saturated phenomenon and involved variable 19 mechanisms according to the specific material. Then, tests with Caco-2 cell model showed a 20 high reduction of DON cytotoxicity on proliferative intestinal cells and of DON absorption by 21 differentiated intestinal cells, when adsorbent was added (except for cholestyramin).

22 Otherwise, values were not always in accordance with those obtained in buffer. Our work 23 allowed us to point out five potential DON adsorbents and to propose a complementary in 24 vitro test allowing the enhancement of the determination of adsorbent properties.

Introduction 6

Fusariotoxins are secondary metabolites from certain species of the Fusarium genus, 7 present in soil which probably produces the most important quantity of toxins in temperate 8 Northern hemisphere (North America, Europe or Asia) (Placinta 1999). Among fusariotoxins, Our purpose was first to determine those adsorbents that effectively bind DON with 8 adsorption/desorption tests. After a screening of 18 potential adsorbents, 6 were selected and 9 their properties for DON characterised. Then, in order to focus on effects at the intestinal 10 level, intestinal cells were used to better describe consequences of the presence of adsorbents 11 in response to 2 questions. On the one hand, our experiments aimed at evaluating an eventual 12 protector role of adsorbents against DON cytotoxicity on proliferative intestinal cells. On the 13 other hand, we studied the adsorbent impact on DON transfer through intestinal epithelium in 14 order to specify an eventual effect on DON bioavailability at intestinal level. These experiments were carried out on the enterocyte model Caco-2. Cultures of Caco-2 cell 16 monolayers, isolated from a human colon carcinoma, are commonly used as a model system 17 to study intestinal uptake and transport processes. Differentiated post-confluent Caco-2 cells 18 exhibit well-developed microvilli and, grown on semi-permeable supports, form tight The Caco-2 cell monolayers were rinsed twice with phosphate buffered saline and pre-16 incubated for 30 min at 37°C in Hank's balanced salt solution (HBSS). Apical (AP) and 17 basolateral (BL) chamber volumes were maintained respectively at 1.5 and 2 ml.

18

Transepithelial transport was assayed at day 21 after seeding by adding DON (10 µM) to the 19 apical compartment containing or not adsorbent material (1 mg/ml) and measured for 8 hours.

20

DON was diluted in ethanol so that final volume of ethanol added to the donor compartment 21 did not exceed 0.5 %. At each sampling time, 200 µl of transport medium were withdrawn 22 from the receiver basolateral compartment and replaced by fresh buffer. Four repetitions were 23 carried out.

24

The same experiments were repeated pre-incubating DON and adsorbent 1 h before combined and evaporated to dryness under a stream of nitrogen at room temperature. 200 µl 14 of 1,1-bis(p-chlorophenyl)-2,2-dichloroethylene (DDE 20 ng/ml in hexane) was added to the residue as an internal standard. Samples were analyzed by GC-ECD (Hewlett Packard 5890 16 set II) using a capillary column Rtx-5 (length: 60m, internal diameter: 0.25mm, film 17 thickness: 0.25µm). GC conditions: carrier gas linear velocity, 1.0 ml/min; injector 18 temperature, 250°C; splitless, 0.75 min; detector temperature ( 63 Ni-ECD), 300°C; column 19 oven temperature, hold at 50°C (1 min), programmed 50-250°C at 30°C/min, 270-290°C at 20 15°C/min, hold at 290°C (12 min), 290-300°C at 30°C/min and hold at 300°C (3 min). The 21 detection limit and the quantification limit were 0.01 µM and 0.03 µM respectively.

14

than long bladed material (47.3±7.3% and 52.8±7% against 32.3±6.5% and 39.8±8.4%).

DON adsorption at pH 7 and pH 3 (Tables 2 and3). 4 and5). Isotherm adsorption models are usually used to believed that adsorption is significant as soon as 1/n is lower than 1. The Langmuir model 12 supposes only one surface site type, an instantaneous and reversible binding, then a dynamic 13 equilibrium, a limited number of fixation sites and an adsorption that reaches a steady state.

14

The lower b is, the higher the adsorbent affinity (Hussein et al., 2004). Kl corresponds to the energy needed for binding.

16

Adsorbent effects on DON cytotoxicity on proliferative Caco-2 cells. 17 On the one hand, results obtained with an adsorbent alone showed no significant 18 cytotoxic effects on proliferative intestinal cells, except with cholestyramin where cell 19 viability decreased by 31 ± 3.5% after a 24 h exposure (Figure 2). On the other hand, after 20 DON exposure, cell viability slightly decreased by 15±1% for a DON concentration of 1µM 21 and was very significantly diminished for higher concentrations (respectively 35±1 and 22 44±2% for 10 and 50µM) (Figure 3). With the exception of cholestyramin, every adsorbent 23 material succeeded in maintaining at least a higher cell viability than the control, for every Even in wells exposed to cholestyramin, TEER values reached 400 ± 10 Ω.cm 2 at the 4 beginning of the transfer experiment and 450 ± 13 Ω.cm 2 after. These measurements indicated 5 no diminution of epithelium integrity during transfer experiments, assessing confluent and 6 viable cells. Cholestyramin no longer prevented DON transfer after as little as 1 h. The 7 results showed that the presence of another adsorbent in apical medium leads to a decrease of 8 DON transfer in basal medium (Figure 4). The pre-incubation of activated charcoal, 9 cholestyramin or mannans with DON did not influence DON transfer in intestinal cells 10 (Figure 5). However, incubating leguminous plant, algae or fungus β-glycans with DON 1 h 11 before cell exposure led to a higher inhibition of DON transepithelial transfer.

12 After 1 h exposure, DON transfer (without adsorbent) was about 5% of the total 13 amount transferred in 6 h. Addition of an adsorbent after 1 h exposure limited DON transfer 14 but it needed 3 h to reach levels similar to those obtained when DON and adsorbent were 15 added together. The presence of cholestyramin in apical medium 1 h after DON 16 contamination had no consequence on the toxin transfer in epithelium. 3

We tested different β-glycans and it appeared that bound type (1-3 or 1-6) or ramification

4
might not be sufficient criteria to characterise their adsorption properties. Among the glycans 5 tested, algae and fungus ones showed a better and quite efficient adsorption of DON. pH 6 appeared to have an impact on these 2 β-glycans properties as they presented a low efficacy 7 but a high adsorption capacity at pH 7, and the inverse at pH 3.

8 Chlorophyllin showed a highly significant effect on DON but its prohibited use in

9

  deoxynivalenol (DON) is the most frequently detected (Placinta et al. 1999; Bennett and 10 Klich 2003; Mac Donald et al. 2004) and is commonly found in cereals and their related 11 products, and cereal components constituting an important part of animal feed (Miraglia 12 1996; Bennett and Klich 2003; Sudakin 2003). Studies measuring plasmatic concentration of 13 DON after oral administration demonstrated that the toxin is effectively absorbed by the 14 organism (Prelusky et al. 1986; Prelusky et al. 1988; Azcona-Olivera et al. 1995; Meky et al. 15 2003). In monogastric animals, and particularly in pigs, this toxin often causes a decrease of 16 alimentary uptake, digestive perturbations, lesions, and sometimes even the animal's death 17 (Larsen et al. 2004; Eriksen and Pettersson 2004; Sklan et al. 2003). Economic loss can have 18 an important impact on breeding and the question of a potential transfer of the toxin or some 19 residue to animal products has not yet been solved. 20 In spite of preventative efforts, crop contamination is often inevitable (Birzele 2000; 21 Edwards 2004; Schrödter 2004). Furthermore, DON appears very resistant and it is currently 22 really difficult to decontaminate a crop (Aziz et al. 1997; McKenzie et al. 1997; Yumbe-et al. 2003; Sabater-Vilar et al. 2007) but it would be of great interest 1 to test adsorbent properties in the presence of intestinal cells in order to focus on what 2 happens at the intestinal level. Indeed, the greater part of DON appears to be absorbed at the 3 level of the first intestinal segments (Eriksen et al. 2003; Danicke et al. 2004). The use of a 4 model simulating pig gastrointestinal tract confirmed that DON absorption principally 5 occurred at the small intestinal level, and specified jejunum preponderance (Avantaggiato et 6 al. 2004).

  7

  the difference between the initial and final concentrations of DON in 12 the testing solution. 13 Adsorption and desorption experiments.

143

  Screening allowed us to determine 6 potential DON adsorbents. Previously 15 described experiments were repeated with buffer solution (pH 7 and 3) containing increasing 16 concentrations of DON (1, 2, 6, 8 10 and 12 µM). Residues of samples obtained at pH 3 were 17 re-suspended in phosphate buffer pH 7, homogenised, centrifuged and supernatants were 18 analysed to highlight an eventual desorption phenomenon. Use of equilibrium sorption models. 20 Sorption data was tested against the linear, Freundlich and Langmuir adsorption 21 isotherm models. 22 Linear isotherm equation was: a =DON amount adsorbed at equilibrium (µg/ml); C e =DON concentration in buffer 1 at equilibrium (µg / ml); a, b=constants determined from the slope and intercept calculated 2 using linear regression (Microsoft Excel function, Excel:PC 2003).Plotting log C a against log C e , we used the linear equation below testing the Freundlich 4 model: 5 log C a = log Kf + (1/n) log C e 6 where: C a =DON amount adsorbed at equilibrium (µg / ml); C e =DON concentration in buffer 7 at equilibrium (µg / ml); Kf, n= Freundlich constants determined from the slope and intercept 8 calculated using linear regression (Microsoft Excel function, Excel:PC 2003). 9 The linear form of the Langmuir model as shown below was used: 10 1/C a = 1/(Kl x b x C e ) + (1/b) 11 where: C a =DON amount adsorbed at equilibrium (µg / ml); C e =DON concentration in buffer 12 at equilibrium (µg / ml); Kl, b= Langmuir constants determined from the slope and intercept 13 calculated using linear regression (Microsoft Excel function, Excel:PC 2003).

  tests on Caco-2 cells 16 Cell culture. 17 The Caco-2 cell line was obtained from American Tissue Culture Collection (ATCC 18 N°HTB37) and was used between passage 24 and 34. The cells were maintained in 19 Dulbecco's Modified Eagle's Medium containing 10 % heat-inactivated foetal bovine serum, 20 1 % non-essential amino acids, 5 % penicillin (100 U/ml) and streptomycin (100 mg/ml) in a 21 humidified atmosphere of 95 % air and 5 % CO 2 at 37°C. Cells were split every 7 days at a 22 density of 10 6 cells/flask. For transport studies, cells were harvested with trypsin-EDTA and 23 seeded on polycarbonate membrane inserts (0.2 µm pore diameter, 25 mm diameter; NUNC, cells were used for experiments at day 21 post-seeding, when morphological 1 differentiation was reached (Pinto et al. 1983). Cell culture media and reagents came from 2 Invitrogen (Cergy Pontoise, France). Foetal bovine serum came from Perbio Science 3 (Brebières, France). Culture flasks, and plates came from Falcon (VWR international, provided by Sigma Aldrich (St Quentin Fallavier, France) ("In vitro 7 toxicology assay kit, MTT based"). Cells were seeded on 96-well plates at a density of 4 x 10 4 8 cells/well. MTT tests were carried out on proliferative cells at day 3 after seeding: cultures 9 were incubated 24h with DON (0, 1, 10 and 50 µM) either with or without adsorbent material 10 (1 mg/ml), and treated according to the manufacturer's recommendations. Acidified formazan 11 crystals formation was measured at 570 nm, with a reference wavelength of 690 nm. The 12 amount of dye produced was proportional to the number of metabolically active live cells. 6 13 repetitions were performed. 14 Transport studies.

15

 15

  analysis.24Results of adsorption tests in aqueous buffer were analysed by comparison of means. mean flux of 3 independent experiments ± S.D. Statistical analysis was 2 performed using one-way analysis of variance (ANOVA), and a p-value of < 0.05 was 3 considered significant.

7

  Six adsorbents significantly decreased DON concentration in phosphate buffer, 8 regardless of the level of DON concentration (0.3 and 0.6 mg DON/g adsorbent): activated 9 charcoal (respectively 99.9±0.1% and 98.4±0.4%), cholestyramin (97.8 ± 1.2 % and 68.6 ± 15 10 %), saccharomyces cerevisiae mannans (84.6±1% and 88.1±4.7%), algae β-glycan 11 (81.1±2.3% and 78.8±2%), fungus β-glycan (80.0±1.9% and 85.6±0.7%) and chlorophyllin 12 (85.1±3.1% and 89.6±0.4 %). Leguminous plant samples presented such heterogeneity that 13 obtained values could show dramatic variations. Short bladed material showed better results

16

  Whatever the pH and concentration, activated charcoal led to a decrease of more than 17 95% of DON concentration in solutions. Cholestyramin reduced DON levels in buffer by 18 about 60% at pH 7. The amount of DON adsorbed by Saccharomyces cerevisiae mannans 19 decreased when the DON concentration increased at pH 7. This phenomenon was not 20 observed at pH 3. DON levels in buffers decreased by between 70 and 90% in the presence of 21 algae or fungus β-glycans. DON adsorption by short-bladed leguminous plants remained 22 constant at about 50% at pH 7 but this material hardly decreased DON concentration at pH 3. 23 Desorption tests never demonstrated a DON desorption phenomenon when residues 24 obtained at pH 3 were re-suspended at pH 7 (data not shown). Linear regression of the linear,

23

  describe pesticide, radionuclide, ZEA adsorption(Avom et al. 2001; Hussein et al. 2004, Ramos et al. 1996). Here, they allowed for a better characterisation of the binding reactions 4 involved. They provided a relation between DON concentration in buffer and the amount of 5 DON absorbed per mg of adsorbent for a given temperature (37°C). The linear model is 6 available for highly diluted medium and stationary experimental conditions. It is based on the 7 hypothesis of unlimited adsorption capacity and constant affinity. The Freundlich model does 8 not need any specific knowledge about adsorbent material and supposes an unlimited number 9 of adsorption sites. Kf constant obtained is proportional to adsorption capacity of the 10 adsorbent and n corresponds to its affinity (Avom et al., 2001; Horsfall and Spiff, 2004). It is 11

24Food

  Page 10 of 33 http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.uk concentration. Each result obtained was not significantly different to control samples 1 without DON. 2 Adsorbent effects on DON transfer in Caco-2 cells.

  3

5Page2

  this study that it is difficult to extrapolate adsorption results 20 obtained with one mycotoxin to another one, notably because of adsorbent specificity 21 (Phillips 1999). Our work, combining different in vitro tests, has permitted us to increase our 22 understanding of DON adsorption and to corroborate or refute different hypotheses presented 23 in the literature. Screening concentrations were chosen according to presumed norms in can vary from 1 to 5 kg/t. We obtained a rate between DON and adsorbent of about 0.25 1 to 1.7 mg/g, corresponding to about 0.75 to 5.1 µM DON with our dilutions. Screening 2 allowed us to identify 6 potential DON adsorbents, characteristics of which remained to be 3 specified. At studied doses, DON concentration did not show significant influence on 4 adsorption.Our results supported the hypothesis that DON adsorption by activated charcoal would 6 be of about 90% for DON concentrations lower than 2 mg/g and where the pH was between 3 7 and 8 (Avantaggiato et al. 2004; Galvano et al. 1998; Galvano et al. 2001). Variations of 8 adsorbed DON were maybe principally due to the charcoal used. However, Avantaggio et al. 9 (2004) described a 55% DON diminution for a concentration of 10 mg/g, suggesting that 10 DON linkage to charcoal might be a saturable mechanism for very high toxin concentrations. 11 Activated charcoal combined a high adsorption capacity and an important efficacy whatever 12 the pH was, according to the Freundlich adsorption isotherm. Charcoal did not seem to 13 significantly bind DON when not activated. 14 We demonstrated, as did Avantaggiato et al. (2004) in a previous study, the significant 15 effect of cholestyramin on DON concentration in buffer at pH 7. Using adsorption isotherms, 16 the resin showed average efficacy and capacity. However, the previous work showed an 17 adsorption of 10 ± 1 % and 4 ± 7 % for concentrations of 2 and 10 mg/g, at pH 8, whereas 18 our results gave a rate of about 60 % at pH 7 for concentrations corresponding to 0.3 to 3.6 19 mg/g. DON adsorption by cholestyramin could then be a saturable phenomenon. It seemed 20 that pH had a great influence on cholestyramin efficacy too as the amount of DON adsorbed 21 remained dramatically low at pH 3. It is believed that mannans can adsorb many 22 mycotoxins, notably fusariotoxins, and to a better extent than the other categories of 23 adsorbents (Devegowda and Castaldo, 2000, cited by Yiannikouris and Jouany, 2002). Our 24 study confirmed that Saccharomyces cerevisiae mannans could bind DON and specified that 25 decreased as DON concentration increased. Adsorption isotherms 1 suggested they have an average efficacy and capacity at both pH.Many recent studies are based on cell wall components, and β-glycans in particular.

Figure 1 . 6 Figure 2 .

 162 Figure 1. Adsorbed deoxynivalenol part in pH7 buffer for 2 toxin concentrations (1 µM (a.)

7Figure 3 .

 3 Figure 3. Adsorbent effects on cellular viability of proliferative Caco-2 cells exposed to

7Figure 4 .

 4 Figure 4. DON transfer in basolateral compartment after an apical exposure (10 µM) with or

Figure 5 :

 5 Figure 5: Inhibition percentages of deoxynivalenol (10 µM) transfer in Caco-2 cell

  France). TEER measurement was performed at the end of both the pre-incubation period and

	1	exposure. For every transport study, monolayer integrity was checked by measuring the
		transepithelial electrical resistance (TEER), using a Millicell ERS (Millipore, Molsheim,
	3
	4	the transport experiments. Monolayers displaying TEER values inferior to 300 Ω.cm 2 or in
	5	which TEER values fell more than 15 % were removed.
	6 7 8	F DON analysis o r For the analysis of BL medium in transport experiments, each sample was heated 10
	9	P min at 90°C and then centrifuged at 1600 g at room temperature. Every analysed sample was
	10	e evaporated to dryness under a stream of nitrogen at 60°C (Videmann et al., 2007). The dry
	11	e r residue was derivatized in 25 µl Trisil-TBT with 75 µl hexane, shaken and heated at 60°C for
	12 13	30 min. It was then extracted with twice 1 ml hexane and phosphate buffer, pH 7 (22% R e NaH 2 PO 4 , H 2 O 0.2M, pH 4.8, 32% Na 2 HPO 4 , H 2 O 0.2M pH 9.4). Hexane phases were
		v i e
		w
		O n l
		y
	25
	26	exposing cells or incorporating adsorbent material in apical transport medium1 h after DON

2

Table 1 .

 1 These results could not be extrapolated to an estimation of in vivo adsorption but it is These experiments could also allow us to better interpret the mechanisms involved. Results it appeared difficult to simply evaluate the efficacy of an adsorbent using only one mycotoxin 1 because neither in vitro nor in vivo tests described in literature prove satisfactory. We 2 proposed here experiments that allowed for a better evaluation of the efficacy of an adsorbent, It appeared difficult to classify DON adsorbents because of their heterogeneity. But it Hussein H, Ibrahim SF, Kandeel K, Moawad H. 2004. Biosorption of heavy metals from Miraglia M, Brera M, Colatosti M. 1996. Application of biomarkers of risk to human health 2 from exposure to mycotoxins. Microchem J. 54():472-477. Placinta CM, D'Mello JPF, MacDonald AMC. 1999. A review of worldwide contamination of Sudakin DL. 2003. Trichothecenes in the environment: relevance to human health. Toxicol Versantvoort CHM, Oomen AG, Van de Kamp E, Rompelberg CJM, Sips AJAM. 2005. Materials tested in aqueous buffer tests.

	20
	21	linear model appeared the least adapted. Moreover, it was sometimes difficult to discern
	22	which of the Freundlich or the Langmuir model was the most suitable, maybe because used
		concentrations did not allow us to reach the final steady-state if fixation sites were limited.

9

animal feed makes it of no interest. However, as it is composed of chlorophyll and Cu, it 10 could be suggested that chlorophyll has an adsorption action on DON. In parallel, we 11 expected long-bladed leguminous plants to be more efficient than short-bladed ones as 12

Strangoom and Smith (1984)

had observed that the fibrous fraction of alfalfa could neutralise 13 negative ZEA effects on growth by rats, but we observed that short-bladed material showed a 14 better adsorption rate than long-bladed material. Adsorption mechanisms in leguminous plant 15 might involve chlorophyllin more than fibre. These results corroborated the hypothesis 16 concerning the potential of chlorophyllin in DON adsorption. 17 The use of the three equilibrium sorption models permitted a better understanding of 18 the mechanisms involved in DON adsorption in relation to the considered material and gave 19 results in line with DON adsorption rate in buffer. Whatever the material and the pH, the 5 8 to specify what happens at the intestinal epithelium level. Caco-2 cell model first allowed 9 assessing consequence of adsorbent use on proliferative intestinal cell viability. It indicated 10 that only cholestyramin decreased cell viability and then might provoke lesions at the level of 11 intestinal integrity. 12 DON showed a growing cytotoxicity on proliferative intestinal cells as its 13 concentrations increased. Except for cholestyramin, the presence of adsorbent material 14 permitted to highly decrease, indeed to cancel, DON cytotoxicity on proliferative intestinal cells. Then, we focused on adsorbent effects on intestinal transepithelial transfer of DON. 16 DON transfer decrease remained quite stable except for cholestyramin and mannans which 17 showed a more and more increasing transfer, suggesting adsorption mechanisms depending 23 plant appeared more effective to diminish DON transfer than its adsorption tests could 24 predict. These results were in concordance with the impossibility to increase the scope of 25 buffer tests to in vivo reality. Intestinal cell absorption decreased apical DON concentration 7 8 demonstrated that pre-incubation allowed a more efficient adsorption for leguminous plant, 9 algae and fungus β-glycans. DON adsorption seemed thus to need an interval of time for these 10 adsorbents. Pre-incubation globally led to a rapid and better diminution of DON absorption in 11 vitro. This hypothesis should be tested in vivo to assess if the addition of adsorbents to feed 12 could enhance diminution of DON bioavailability. In mice, it was demonstrated that smectite 13 (2g/kg) did not succeed in preventing T-2 toxin (1mg/kg) affecting intestinal transit, but a 24h 14 pre-incubation period inhibited all the toxin effects (Fioramont et al., 1987). Otherwise, the 15

addition of an adsorbent after DON exposure could limit DON absorption, but only after at 16 least a 3h delay. We could extrapolate that in case of ingestion of contaminated feed, 17 adsorbent administration could reduce DON absorption at the intestinal level. The acceptable 18 delay could not be determined here, but as it seems obvious that adsorbent efficacy weakens 19 with time, as then the delay has to be as short as possible. 23 results obtained with one mycotoxin to another one (Phillips, 1999). Here, interesting DON 24 adsorbents were not always those already referenced for other mycotoxins, even for 25 fusariotoxins. It was then of great interest to carry out specific tests with DON. Furthermore, 3 even if results remain to be tested in vivo.

4 8 more interesting to explore this as multi-contamination is often observed in cereals and 9 adsorbent material has to prove efficient at dealing with several mycotoxins. 23 Azcona-Olivera J, Ouyang Y, Murtha J, Chu F.S., Pestka JJ. 1995. Induction of cytokine 24 Page 16 of 33 http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.uk 9 Danicke S, Valenta H, Doll S. 2004. On the toxicokinetics and the metabolism of 10 deoxynivalenol (DON) in the pig. Arch Anim Nutr. 58(2):169-180. 11 Edwards SG. 2004. Influence of agricultural practices on Fusarium infection of cereals and 12 subsequent contamination of grain by trichothecene mycotoxins: Toxicol Letters. 13 153(1):29-35. 14 Eriksen GS, Pettersson H, Lindberg, JE. 2003. Absorption, metabolism and excretion of 3-15 acetyl DON in pigs. Arch Anim Nutr 57(5):335-345. 16 Eriksen GS, Pettersson H. 2004. Toxicological evaluation of trichothecenes in animal feed: 17 Anim Feed Sci Tech. 114(4):205-239. 23 Galvano F, Piva A, Ritieni A, Galvano M. 2001. Dietary strategies to counteract the effects of 24 mycotoxins: a review. J Food Prot. 64(1):120-131.

2 8 Ledoux DR, Broomhead JN, Bermudez AJ, Rottinghaus GE. 2003. Individual and combined 9 effects of the Fusarium mycotoxins fumonisin B 1 and moniliformin in broiler chicks. 10 Avian Dis. 47(4):1368-1375. 11 MacDonald S, Prickett TJ, Wildey KB, Chan D. 2004. Survey of ochratoxin A and 12 deoxynivalenol in stored grains from the 1999 harvest in the UK. Food Addit Contam. 23 Development of a urinary biomarker of human exposure to deoxynivalenol. Food

1 7 8 cereal grains and animal feed with Fusarium mycotoxins. Anim Feed Sci Technol.

9 78():21-37. 10 Prelusky DB, Veira DM, Trenholm HL, Hartin KE. 1986. Excretion profiles of the mycotoxin 11 deoxynivalenol, following oral and intravenous administration to sheep. Fund Applied 6 8 http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.uk

Table 4 .

 4 Linear, Freundlich and Langmuir constants obtained from linear regression of DON

	adsorption isotherms at pH 7. (n = 18)
	Note: LPSB=leguminous plant short bladed.

Table 5 .

 5 Linear, Freundlich and Langmuir constants obtained from linear regression of DON

adsorption isotherms at pH 3. (n = 18) Note: LPSB=leguminous plant short bladed.

http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.ukFood Additives and Contaminants

Acknowledgments 12

We thank NEAR Company for its participation in this work. 13