Mikhail Lifshits
email: mikhail@lifshits.org

Marguerite Zani
email: marguerite.zani@univ-orleans.fr

Approximation of additive random fields based on standard information: average case and probabilistic settings

Keywords: Approximation complexity, Additive random fields, Gaussian processes, Standard information, Tensor product random fields

HAL is

1 Introduction: general information against standard information Let f be a function which we consider as an element of some Banach space (B, ||•||) of functions. Assume that the whole function f is unknown but we are able to measure the values of some functionals F 1 (f ), ..., F n (f ), such as values of f at certain points, its integrals, etc. Let Ψ : R n → B be some mapping. Then we may call Af := Ψ(F 1 (f ), ..., F n (f ))

an approximation algorithm and define its error at f by ||f -Af ||. A typical problem setting assumes minimization of the approximation error for given n by optimizing the choice of the set (F j (•)) 1≤j≤n within available class of functionals, as well as the choice of the mapping Ψ.

As for the target function f , we essentially have two options for problem setting. Either we can let it vary over some set, typically some compact subset of B, for example, the unit ball of some other Banach space compactly embedded into B, by taking the worst possible approximation error for algorithm evaluation. Or we may consider f as a random function having certain distribution in B and using the expectation of the error for algorithm evaluation. The two possibilities are often referred to as the "worst case setting", resp. "average case setting". We will rather stick to the latter one and consider random functions (or random fields, if their arguments are multivariate) as the objects of approximation.

Let us now specify the problem furthermore. We will use a Hilbert norm for error evaluation, which essentially means B = L 2 ([0, 1] d ), where d is understood as appropriate parametric dimension of a random field we consider. It is well known that for Hilbert norms the minimal average error is attained on a linear algorithm, which means

Af = n j=1 F j (f )φ j ,
where every F j (•) is a linear functional on B and φ j ∈ B are specific fixed elements. It is often said that such algorithms are based on linear information.

Since this article deals only with Hilbert norms, we will only work with linear algorithms.

In many applications it is too costly to calculate values of arbitrary linear functionals needed for such approximation. It is therefore preferable to restrict the choice of F j (•) by letting F j (f ) := f (t j ), i.e. by taking values of f at some points. One says that such algorithms are based on standard information.

It is clear that algorithms based on linear information form a larger class than those based on standard information, thus, after optimization, the average error would be smaller in linear case. The main question we are concerned with in this paper is "How much do we loose by considering standard information algorithms against those using general linear information?"

This problem already received much interest, see [START_REF] Hickernell | Tractability of linear multivariate problems in the average-case setting[END_REF][START_REF] Hinrichs | Linear information versus function evaluations for L 2 -approximation[END_REF][START_REF] Kuo | On the power of standard information for multivariate approximation in the worst case setting[END_REF][START_REF] Wasilkowski | The power of standard information for multivariate approximation in the randomized setting[END_REF] and especially the survey [START_REF] Novak | On the power of function values for the approximation problem in various settings[END_REF]. Most of the research was concentrated on the worst case setting. It was shown that in many cases the polynomial term of error decay (when considered as a function of the number n of functionals used) is the same for algorithms based on linear and standard information. However, there exist cases where the behavior of the error as a function of n for linear and standard information is drastically different. It means that the problem we consider is by far non-trivial.

In this paper we restrict investigation to one specific but very important class of averaging function distributions, or, equivalently, random fields under consideration. Namely we consider tensor product random fields and additive random fields. In short, tensor product random field on [0, 1] d is a zero mean random field with covariance function

K d (s, t) := d l=1 K(s l , t l ) where s = (s 1 , • • • , s d ), t = (t 1 , • • • , t d ) ∈ [0, 1] d , and K(•, •) is a covariance function on [0, 1] 2 .
Brownian sheet, Brownian pillow, and other famous random fields belong to this class.

Next, a d-parameter additive random field of order b, defined on [0, 1] d , appears as follows. We pick any set of b coordinates from d available ones, consider a tensor product random field introduced above but depending only of the picked coordinates and then sum up uncorrelated versions of those tensor products over all possible sets of coordinates of size b. As a simplest example, if b = 1, we just have a random field

X(t) = d l=1 X l (t l ),
where X l (s), 0 ≤ s ≤ 1, are uncorrelated copies of a one-parametric field with covariance K(•, •). See more definition details in Sections 3 and 4, respectively.

For both types of the fields, the error of linear algorithms is studied in great detail, cf. [START_REF] Lifshits | Curse of dimensionality in approximation of Gaussian random fields[END_REF][START_REF] Lifshits | Average case tractability of non-homogeneous tensor product problems[END_REF][START_REF] Lifshits | Approximation complexity of additive random fields[END_REF][START_REF] Wasilkowski | Weighted tensor-product algorithms for linear multivariate problems[END_REF], as well as closely related small deviation behavior [START_REF] Karol | Small ball probabilities for Gaussian random fields and tensor products of compact operators[END_REF]. However, the power of standard information was not addressed so far, which we do here. Our main result is that in most interesting cases there is no more than a logarithmic loss in approximation error when information is being restricted to the standard one.

Notice that we do not merely consider a fixed random field but also explore the case when one or even two parameters, dimension d and additivity order b go to infinity, in which case a concept of relative error is relevant.

There is still much room for the research in the direction we traced here. We do not really pretend that approximation rates for standard information case obtained in this work are optimal, although they are obviously quite close to those ones. Neither we provide concrete algorithms related to obtained rates. In fact, our results are rather existence theorems because they are obtained by randomization techniques. These interesting topics still wait for subsequent, much deeper investigation.

The paper is organized as follows. In Section 2 we recall a quite general approximation procedure providing upper bounds for the quality of algorithms based on standard information assuming that similar bounds are known for algorithms based on general linear functionals. This procedure is systematically applied in the sequel to different settings and to different random fields of interest.

In Section 3 we recall the notion of a tensor product random field and the known results about the quality of their approximation based on general linear functionals. Furthermore, we obtain new results for approximation of those fields based on standard information. Consideration concerns both fixed and increasing parametric dimension of random field.

In Section 4 we perform the same program for more complicated additive random fields and come out with the new estimates for approximation of those fields based on standard information.

Finally, in Section 5 we come back to tensor product random fields in fixed dimension and evaluate the power of standard information in the probabilistic setting, i.e. searching for the algorithms that assure given approximation quality with prescribed probability.

2 Approximation procedure based on standard information

The pointwise approximation algorithm

Let (D, ν) be a measure space. We first consider an approximation procedure for a deterministic function g ∈ L 2 (D, ν) based on the standard information, i.e. on the values of g at n points τ 1 , . . . , τ n .

The randomized algorithm below is a slight refinement of those presented in the series of works [START_REF] Hickernell | Tractability of linear multivariate problems in the average-case setting[END_REF], [START_REF] Hickernell | Integration and approximation in arbitrary dimension[END_REF], [START_REF] Kuo | On the power of standard information for multivariate approximation in the worst case setting[END_REF], and [START_REF] Wasilkowski | The power of standard information for multivariate approximation in the randomized setting[END_REF].

Fix two integers n, m. Let (η j ) j≥1 be an orthonormal system in L 2 (D, ν). Then

u m (t) := 1 m m j=1 η 2 j (t) is a density on D. Given n points τ = (τ 1 , • • • , τ n ) ∈ D n we define a linear approx- imation algorithm A τ : L 2 (D, ν) → L 2 (D, ν) as follows. Let g ∈ L 2 (D, ν).
Then g admits the Fourier expansion

g = m j=1 g j η j + g ⊥ m ,
where

g j = g, η j := D gη j dν ,
and g ⊥ m is orthogonal to the space spanned by {η j , j ≤ m}. Let

A τ g := m j=1 ĝj η j , where ĝj := 1 n n l=1 g(τ l ) η j (τ l ) u m (τ l )
is a pointwise approximation to the integral g j . Consider τ as an i.i.d. sample in D with density u m . For any j, l we can compute

E τ g(τ l ) η j (τ l ) u m (τ l ) = D g(t) η j (t) u m (t) u m (t) ν(dt) = g j .
Hence, E τ (ĝ j ) = g j . We can evaluate the variance as well,

Var τ g(τ l ) η j (τ l ) u m (τ l ) ≤ E τ g(τ l ) η j (τ l ) u m (τ l ) 2 = D g(t) 2 η j (t) 2 u m (t) ν(dt) .
Hence,

Var τ (ĝ j ) = E τ [(ĝ j -g j ) 2 ] ≤ 1 n D g(t) 2 η j (t) 2 u m (t) ν(dt) .
By summing up,

E τ   m j=1 (ĝ j -g j ) 2   ≤ 1 n D g(t) 2 m j=1 η j (t) 2 u m (t) ν(dt) = m n D g 2 (t) ν(dt) = m n g 2 2 .
Recall that

g -A τ g 2 2 = m j=1 (ĝ j -g j ) 2 + g ⊥ m 2 2 ,
therefore,

E τ g -A τ g 2 2 ≤ m n g 2 2 + g ⊥ m 2 2 . (1)
Now we pass to approximation of random functions. Let us consider a random field Y (t, ω) on D and, assuming the sample τ independent of Y , apply inequality (1) to the sample paths g(t) := Y (t, ω). By Fubini theorem,

E τ E Y Y -A τ Y 2 2 = E Y E τ Y -A τ Y 2 2 ≤ m n E Y 2 2 + E Y ⊥ m 2 2 ,
where Y ⊥ m is the orthogonal part of Y with respect to the finite system {η j } m j=1 ,

Y (t, ω) = m j=1 (Y, η j ) η j (t) + Y ⊥ m (t, ω) .
By the mean value theorem, there exists a point set

τ := τ (Y ) := {τ l (Y ), l = 1 • • • m} such that E Y Y -A τ Y 2 2 ≤ m n E Y 2 2 + E Y ⊥ m 2 2 . (2)
We stress the fact that the set τ (Y ) does not depend on the sample paths of Y , i.e. it is non-random.

Iteration procedure

Now we define a randomized (thus nonconstructive) iteration procedure, i.e. a sequence of approximations A k , k ≥ 0, for a field Y . Notice that the parameters n, m and the orthonormal system (η j ) m j=1 are fixed throughout the construction. Let A 0 := 0 and iteratively

A k Y := A k-1 Y + A τ (Y -A k-1 Y ) (Y -A k-1 Y ) , Let denote ∆ k := Y -A k Y . Then by construction, ∆ k = Y -A k-1 Y -A τ (Y -A k-1 Y ) (Y -A k-1 Y ) = ∆ k-1 -A τ (∆ k-1 ) (∆ k-1 ) .

By applying inequality (2) to the field ∆

k-1 instead of Y , we get E ∆ k 2 2 ≤ m n E ∆ k-1 2 2 + E ∆ ⊥ k-1 2 2 .
Notice that any algorithm used keeps invariant the orthogonal part, hence

E ∆ ⊥ k-1 2 2 = E Y ⊥ m 2 2 .
Then we obtain an iterative estimate

err k ≤ m n err k-1 + R, (3)
where

err k := E ∆ k 2 2 and R := E Y ⊥ m 2
2 does not depend on k. The iterative sequence like this quickly approaches a limit assuming that m n < 1. Indeed, let x := R/(1 -m n ) and write (3) as

err k ≤ m n err k-1 + (1 - m n ) x
which is equivalent to

err k -x ≤ m n k (err 0 -x), whence err k ≤ x + m n k err 0 .
By the previous definitions, the latter inequality reads as

E ∆ k 2 2 ≤ E Y ⊥ m 2 2 1-m n + m n k E Y 2 2 . (4)
Notice that the algorithm A k uses kn point evaluations (n new points for each step). Now we adjust the algorithm by connecting the variables m and n.

For any n ≥ 2 take m := ⌊n/2⌋ and apply algorithm A n,k := A k . We get by ( 4)

E Y -A n,k Y 2 2 ≤ 2E Y ⊥ ⌊n/2⌋ 2 2 + 2 -k E Y 2 2 . (5)
Let us now make an assumption about the average approximation rate of Y by finite parts of its Fourier expansion. We assume the following polynomial approximation rate: Assumption 2.1 There exist p > 0, C 1 > 0, and a real γ such that for any m ≥ 2,

E Y - m j=1 (Y, η j )η j 2 2 = E Y ⊥ m 2 2 ≤ C 1 m -2p (ln m) γ E Y 2 2 . (6)
For n ≥ 4, by letting k := ⌈Z log 2 n⌉ with large Z > 2p, we get the following

E Y -A n,k Y 2 2 ≤ 2C 1 ⌊n/2⌋ -2p (ln⌈n/2⌉) γ E Y 2 2 + n -Z E Y 2 2 ≤ Cn -2p (ln n) γ E Y 2 2 , (7)
while ⌈Z log 2 n⌉n point evaluations of Y are used and C = C(C 1 , p, b, Z).

We may summarize those findings in the following result.

E Y -A n Y 2 2 ≤ Cn -2p (ln n) γ E Y 2 2 . ( 8
)
Remark. Notice that the approximation rate while using point measurements (8) is the same, up to a constant, as in the case of using arbitrary functionals [START_REF] Karol | Small ball probabilities for Gaussian random fields and tensor products of compact operators[END_REF]. The minor loss consists in logarithmic increase of the number of used measurements: ⌈Z log 2 n⌉n instead of n. Unfortunately, our result does not provide construction of the measurement points because of the use of randomization technique.

3 Tensor product random fields

Simple tensor product

Let us apply this previous scheme to tensor product random fields. First, recall the latter notion. Let K(•, •) be a covariance kernel on [0, 1]. By the well known Mercer's theorem it can be represented as

K(s, t) = ∞ i=1 λ(i) 2 φ i (s)φ i (t)
where (φ i ) i>0 is an orthonormal system of functions in L 2 ([0, 1]), and (λ(i) 2 ) i>0 is the sequence of eigenvalues (corresponding to the integral operator with kernel K) satisfying condition

∞ i=1 λ(i) 2 < ∞.
We consider a random field X(t), t ∈ [0, 1] d , with zero mean and covariance function

K d (s, t) := d l=1 K(s l , t l ) (9)
where

s = (s 1 , • • • , s d ), t = (t 1 , • • • , t d ) ∈ [0, 1] d . It is natural to call X the d-th
tensor degree of a one-parametric random field with covariance K.

It is well known that X admits Karhunen-Loève expansion

X(t) = k∈N d d l=1 λ(k l ) d l=1 φ k l (t l ) ξ k := k∈N d λ k φ k ξ k , (10)
where {ξ k , k ∈ N d } are non-correlated random variables with zero mean and unit variance; and

λ k := d l=1 λ(k l ), φ k (t) := d l=1 φ k l (t l ) , k ∈ N d .
The sample paths of X are considered as elements of the space L 2 (D, ν) where D = [0, 1] d and ν is the d-dimensional Lebesgue measure.

In the paper [START_REF] Lifshits | Curse of dimensionality in approximation of Gaussian random fields[END_REF], Lifshits and Tulyakova studied approximation rate for such random fields, considering fixed and increasing dimension d for the case when the measurement of arbitrary linear functionals is available. In the present work, we aim to explore the power of approximation algorithms based upon pointwise evaluations of X. In terms of information based complexity theory [START_REF] Novak | Tractability of Multivariate Problems, Volumes I-III[END_REF][START_REF] Traub | Information-based Complexity[END_REF][START_REF] Woźniakowski | Tractability and strong tractability of linear multivariate problems[END_REF], we consider standard information.

Fixed dimension

Here and elsewhere in the sequel the equivalence notation a j ∼ b j means that lim j→∞ aj bj = 1. For fixed d we assume

Assumption 3.1 λ(i) ∼ µ i -r (ln i) q , i → ∞
for some µ > 0, r > 1/2 and q = r.

There are many important examples of such behavior, including Wiener process, fractional Brownian motion etc, see e.g. [START_REF] Lifshits | Lectures on Gaussian Processes[END_REF].

Let ( λ2 j , j ∈ N) be the decreasing rearrangement of the array (λ 2 k , k ∈ N d ). In fixed dimension, for a sequence of eigenvalues satisfying Assumption 3.1, we have the following elementary result for which we refer to [START_REF] Lifshits | Curse of dimensionality in approximation of Gaussian random fields[END_REF]:

Lemma 3.2 Let α := q/r. Then λ 2 j ∼ B 2 d j -2r (ln j) 2rβ , j → ∞, (11)
where

• for α > -1 :

B d = µ d Γ(α+1) d Γ(d(α+1)) r , β = (d -1) + dα,
• for α < -1 :

   B d = µd r i≥1 λ(i) 1/r (d-1)r , β = α.
Equivalent results can be found e.g. in Csáki [START_REF] Csáki | On small values of the square integral of a multiparameter Wiener process[END_REF], Li [START_REF] Li | Comparison results for the lower tail of Gaussian seminorms[END_REF], Papageorgiou and Wasilkowski [START_REF] Papageorgiou | On the average complexity of multivariate problems[END_REF] (for q = 0) and in Karol', Nikitin, and Nazarov [START_REF] Karol | Small ball probabilities for Gaussian random fields and tensor products of compact operators[END_REF].

The asymptotics for α = -1 is also known; it has a form λ 2 j ∼ B 2 d j -2r (ln j) -2r (ln ln j) 2r(d-1) , j → ∞, cf. [START_REF] Karol | Small ball probabilities for Gaussian random fields and tensor products of compact operators[END_REF]. We skip this case here and in the sequel just for the sake of brevity. By summing up and using (8), we have the following polynomial approximation error:

E X -X m 2 2 = j>m λ 2 j ≤ C 1 m 1-2r (ln m) 2rβ (12)
with some

C 1 = C 1 (X).
Let us consider the expansion [START_REF] Lifshits | Lectures on Gaussian Processes[END_REF] and apply the preceding procedure with the orthonormal system ( φj , j ∈ N) being the rearrangement of the system (φ k ) in [START_REF] Lifshits | Lectures on Gaussian Processes[END_REF] corresponding to the decreasing order of eigenvalues λ 2 k .

We obtain from (7) that for every n ≥ 4 there is a pointwise algorithm A n that uses ⌈Z log 2 n⌉n point evaluations and gives an error

E X -A n X 2 2 ≤ Cn 1-2r (ln n) 2rβ with some C = C(X) not depending on n.
A trivial variable change leads to the final result:

Proposition 3.3 Let X be a tensor product random field [START_REF] Lifshits | Lectures on Gaussian Processes[END_REF]. Under Assumption 3.1 there exists a constant C = C(X) and a sequence of linear approximation algorithms (A n ) n≥4 using respectively n point values such that for each n we have

E X -A n X 2 2 ≤ C n 1-2r (ln n) 2r(β+1)-1 . (13)

Increasing dimension

Now we let d go to infinity and consider d-parametric random fields as usually done in information based complexity theory [START_REF] Novak | Tractability of Multivariate Problems, Volumes I-III[END_REF][START_REF] Traub | Information-based Complexity[END_REF][START_REF] Woźniakowski | Tractability and strong tractability of linear multivariate problems[END_REF]. We still consider tensor product random fields [START_REF] Lifshits | Lectures on Gaussian Processes[END_REF] with covariance (9) generated by a fixed univariate covariance function K(•, •) but denote the field X d instead of X, in order to stress the dependence of parametric dimension.

In increasing dimension, one must consider relative error (also often called normalized error in the literature), since the total range of the random field

E X d 2 2 = ∞ i=1 λ(i) 2 d := Λ d (14)
may go to infinity as Λ > 1, resp. to zero as Λ < 1, whenever the dimension d increases. Therefore, for a given ǫ ∈ (0, 1) we look for a pointwise approximation algorithm A such that E X d -AX d 2 2 ≤ ǫ 2 Λ d , and ask how many point measurements of X d we need for achieving this.

Let the cardinality associated to the relative error md (ǫ) be defined by

md (ǫ) = inf    m ; j>m λ 2 j ≤ ǫ 2 Λ d    ,
From [START_REF] Lifshits | Curse of dimensionality in approximation of Gaussian random fields[END_REF] we know that the following result holds.

Theorem 3.4 ([12]

) Under assumption

M 2 := ∞ i=1 | ln λ(i)| 2 λ(i) 2 < ∞ , (15)
for any ǫ ∈ (0, 1) it is true that

lim d→∞ ln md (ǫ) -d ln Λ * √ d = 2q * ,
where

M := - ∞ i=1 ln λ(i) λ(i) 2 Λ , Λ * := Λe 2M , ( 16
)
and the quantile q * is given by the equation Φ q * σ = ǫ 2 , where σ 2 = M2 Λ -M 2 and

Φ(x) = 1 √ 2π ∞ x exp{-u 2 /2}du
is the tail of the standard normal law.

With pointwise estimation, from ( 5) and ( 14) we have with n = 2 md (ǫ)

E X d -A n,k X d 2 2 ≤ (2ǫ 2 + 2 -k )Λ d . Taking k = ⌈log 2 (ǫ -2
)⌉ points, we have the relative error 3ǫ 2 while using only 2⌈log 2 (ǫ -2 )⌉ md (ǫ) points.

Our conclusion is that the number of necessary measurements explodes almost in the same way, whenever we use general or pointwise functionals. Using notations from Theorem 3.4 we have again Proposition 3.5 Under assumption [START_REF] Novak | Tractability of Multivariate Problems, Volumes I-III[END_REF] for any ǫ ∈ (0, 1) there exists a sequence of positive integers m d (ǫ) and a sequence of linear approximation procedures We recall now the definition of a d-parameter additive random field of order b, cf.

A ǫ d based upon m d (ǫ) point measurements of X d such that for any d E X d -A ǫ d X d 2 2 ≤ 3ǫ 2 E X d
[1], [START_REF] Lifshits | Approximation complexity of additive random fields[END_REF]. Essentially it means that we pick any set of b coordinates from d available ones, consider a tensor product random field introduced in Subsection 3.1 above but depending only of the picked coordinates and then sum up uncorrelated versions of those tensor products over all possible sets of coordinates of size b.

By the reasons that will soon become clear, this time it is convenient for us to number the eigenfunctions and eigenvalues of the tensor product starting from zero. Thus let denote N 0 := {0, 1, 2, . . . } in addition to N := {1, 2, . . . }.

Let

λ(i) ≥ 0, i ∈ N 0 , satisfy ∞ i=0 λ(i) 2 < ∞ and let (φ i ) i∈N0 be an orthonormal basis in L 2 ([0, 1]).
Let denote by D and D b the following sets of indexes:

D = {1, • • • , d} , D b = {S ⊂ D , |S| = b} .
Here and elsewhere |S| stands for the number of elements in a finite set S. For each S ∈ D b we write S = {a 1 , • • • , a b } with elements ordered, say, as

a 1 < • • • < a d . Now an additive d-parametric random field X d,b (t), t ∈ [0, 1] d , of order b is defined by X d,b (t) := S∈D b k∈N S 0 a∈S λ(k a ) a∈S φ ka (t a ) ξ S k . (17)
Here (ξ S k , S ∈ D b , k ∈ N S 0 ) is a family of non-correlated random variables with mean zero and unit variance.

We assume furthermore

Assumption 4.1 ∀u ∈ [0, 1] , φ 0 (u) = 1 .
This technical assumption ensures (see [START_REF] Lifshits | Approximation complexity of additive random fields[END_REF], Lemma 2.2) that the family

φ S k (t) := a∈S φ ka (t a ), S ⊂ D, k ∈ N S is an orthonormal system in L 2 ([0, 1] d ).
It appears in [START_REF] Traub | Information-based Complexity[END_REF]. Many interesting and important processes satisfy Assumption 4.1, see [START_REF] Hickernell | Tractability of linear multivariate problems in the average-case setting[END_REF]. Under this assumption, ( 17) transforms into

X d,b (t) = S∈D b G⊂S λ(0) |S|-|G| k∈N G a∈G λ(k a ) φ G k (t)ξ S k0 , (18)
where k 0 ∈ (N 0 ) S is obtained from k ∈ N G by adding zeros.

Looking at the expansion ( 18), we see that the same eigenfunction φ G k (t) appears as many times as there are sets S ∈ D b containing G. Since the number of such sets is C b-|G| d-|G| , we may write

X d,b (t) = G⊂D |G|≤b C b-|G| d-|G| 1/2 λ(0) b-|G| k∈N G a∈G λ(k a ) φ G k (t) ξ G k = b h=0 G∈D h C b-h d-h 1/2 λ(0) b-h k∈N G a∈G λ(k a ) φ G k (t) ξ G k , (19)
where

( ξ G k , G ⊂ D, |G| ≤ b, k ∈ N G )
, is a family of non-correlated random variables with mean zero and unit variance.

Fixed dimensions

In this subsection, we assume that dimension d and order b ≤ d are fixed, and for an integer h satisfying 0 ≤ h ≤ b denote by

λ 2 k,h = h l=1 λ(k l ) 2 , k = (k 1 , • • • , k h ) ∈ N h .
Looking at [START_REF] Wasilkowski | The power of standard information for multivariate approximation in the randomized setting[END_REF], we see that eigenvalues of covariance operator have the form

C b-h d-h λ(0) 2(b-h) λ 2 k,h with multiplicity |D h | = C h d .
For every fixed h let denote by (λ 2 j,h , j ∈ N) the decreasing rearrangement of the array (λ2 k,h , k ∈ N h ). As in case of fixed dimension for simple tensor product (Subsection 3.1.1), let Assumption 3.1 on the asymptotics of λ(i) hold with some parameters r and q. Then by Lemma 3.2 the asymptotics of λ 2 j,h is given by [START_REF] Lifshits | Average case tractability of non-homogeneous tensor product problems[END_REF]. Subsequent analysis leads to different conclusions in two cases.

• If α := q/r > -1, then the exponent β = h -1 + hα in (11) depends on h. We observe the slowest decay of λ 2 j,h for h delivering maximal β, i.e. for h = b. When we merge all arrays and rearrange the total set of eigenvalues, the asymptotics will be the same as for the one for the dominating array. It follows that for appropriately chosen orthonormal system φ j in L 2 ([0, 1] d ) we have the Fourier expansion

X d,b = ∞ j=1 (X d,b , φ j ) φ j (20)
with the error

X ⊥ m := X - m j=1 (X d,b , φ j ) φ j = ∞ j=m+1 (X d,b , φ j ) φ j admitting the bound E X ⊥ m 2 2 ≤ Cm 1-2r (ln m) 2rβ
as in ( 12) with β = b -1 + bα.

Thus we arrive at the same conclusion as in the case of simple tensor field, i.e. Proposition 3.3. Namely, we have Proposition 4.2 Let X d,b be an additive tensor product random field of order b defined in [START_REF] Traub | Information-based Complexity[END_REF]. Under Assumption 3.1 (with q > -r) and Assumption 4.1 there exists a constant C = C(X d,b ) and a sequence of linear approximation algorithms (A n ) n≥4 using respectively n point values such that for each n we have

E X d,b -A n X d,b 2 2 ≤ C n 1-2r (ln n) 2b(r+q)-1 .
The proof is exactly the same as for Proposition 3.3. Just notice the calculation of logarithm's exponent: recall that h = b, α = q/r, and β = h -1 + hα yield for the exponent in ( 13)

2r(β + 1) -1 = 2rh(1 + α) -1 = 2b(r + q) -1.
• If α = q/r < -1, then β = α does not depend on h when Lemma 3.2 is applied. The rearranged eigenvalues λ

Recall how the asymptotically optimal finite rank approximation based on arbitrary functionals was constructed in [START_REF] Lifshits | Approximation complexity of additive random fields[END_REF]. Let us construct an optimal mterm approximation. Let denote by m h the number of largest eigenvalues in the array λ 2 j,h contributing to the approximation. We know from [START_REF] Lifshits | Approximation complexity of additive random fields[END_REF] that the quasi-optimal choice is

m h := m Q(h) 1/2r b l=1 Q(l) 1/2r
, where, taking into account the eigenvalue factor C b-h d-h λ(0)2(b-h) that was left out when moving from the true eigenvalues to λ 2 k,h , and the multiplicity

C h d , Q(h) := C b-h d-h λ(0) 2(b-h) B 2 h [C h d ] 2r , it is easy to see that the error is of order b h=1 C b-h d-h λ(0) 2(b-h) C h d j>m h /C h d B 2 h j -2r (ln j) 2q ∼ (2r -1) -1 b h=1 C b-h d-h λ(0) 2(b-h) C h d B 2 h m h /C h d 1-2r (ln m h ) 2q = (2r -1) -1 b h=1 Q(h)m 1-2r h (ln m h ) 2q ∼ (2r -1) -1 b h=1 Q(h) Q(h) (1-2r)/2r b l=1 Q(l) 1/2r 2r-1 m 1-2r (ln m) 2q = Q m 1-2r (ln m) 2q ,
where

Q := (2r -1) -1 b h=1 Q(h) 1 2r
2r

.

This means that an ordering of orthogonal basis is possible as in expansion [START_REF] Woźniakowski | Tractability and strong tractability of linear multivariate problems[END_REF] with the error of m-term approximation

E X ⊥ m 2
2 ∼ Q m 1-2r (ln m) 2q . Now we have the following. Proposition 4.3 Let X d,b be an additive tensor product random field of order b defined in [START_REF] Traub | Information-based Complexity[END_REF]. Under Assumption 3.1 (with q < -r) and Assumption 4.1 there exists a constant C = C(X d,b ) and a sequence of linear approximation algorithms (A n ) n≥4 using respectively n point values such that for each n we have

E X d,b -A n X d,b

Increasing dimension

As in Subsection 3.1.2, we let parameter dimension d go to infinity and consider d-parametric random fields as usually done in information based complexity theory [START_REF] Novak | Tractability of Multivariate Problems, Volumes I-III[END_REF]. In this context of growing d we consider additive random fields X d,b of order b described in [START_REF] Traub | Information-based Complexity[END_REF]. Recall that in increasing dimension, one must consider relative error, since the total range of the random field, measured by its mean square varies with dimension as

E( X d,b 2
L2([0,1] d ) = C b d Λ b ,
where Λ := ∞ i=0 λ(i) 2 . Recall that the spectrum of the process is described as follows. To any fixed h = 1, . . . , b associate an array of eigenvalues

λ 2 k,h := h l=1 λ(k l ) 2 , k = (k 1 , • • • , k h ) ∈ N h . ( 21
)
Then multiply all eigenvalues by C b-h d-h λ(0) 2(b-h) and take them with multiplicity C h d . Therefore, for given ǫ, we look for a minimal n such that an algorithm A n based on n point evaluations of

X d,b provides E X d,b -A n X d,b 2 L2([0,1] d ≤ C b d Λ d ǫ 2 .
Admitting that dimension d goes to infinity, we have a choice between keeping the additivity order b fixed or letting it go to infinity, too. Consider both cases.

Order b fixed and dimension d increasing. Let n avr d,b (ǫ) be the minimal m such that a representation

X d,b = m j=1 (X d,b , φ j ) φ j + X ⊥ , (22)
holds with some orthogonal system (φ j ) 1≤j≤m and

E||X ⊥ || 2 2 ≤ C b d Λ b ǫ 2 .
It is known from [START_REF] Lifshits | Approximation complexity of additive random fields[END_REF] that if Assumption 3.1 and Assumption 4.1 hold, if ǫ → 0, d → ∞, and b is fixed, that the main contribution to the approximation is given by the eigenvalue array (21) corresponding to h = b. Therefore,

n avr d,b (ǫ) ∼ d b b! Λ -b/(2r-1) n avr b (ǫ)
where the approximation cardinality n avr b (ǫ) is defined in [START_REF] Lifshits | Curse of dimensionality in approximation of Gaussian random fields[END_REF], for a simple tensor product random field in dimension b (additive field of order b = d) and behaves as follows

n avr b (ǫ) ∼ B b √ 2(r -1/2) rβ+1/2 | log ǫ| rβ ǫ 1/(r-1/2) , ǫ →
E X d,b -A n,k X d,b 2 2 ≤ 3C b d Λ b ǫ 2 , (23)
while using n k = 2 n avr d,b (ǫ) ⌈log 2 (1/ǫ)⌉ point measurements. Thus the polynomial dependence of the number of points on d remains the same. On the other hand, the number of points keeps its polynomial term in dependence of ǫ while getting an extra degree of logarithmic term. The relative error slightly grows up from ǫ to √ 3ǫ. One can, of course, arrange a trade off between the constants in the growth of relative error and the number of used points.

Order b and dimension d increasing proportionally. In this case we assume that both b and d tend to infinity while b/d → f ∈ (0, 1). It was observed in [START_REF] Lifshits | Approximation complexity of additive random fields[END_REF] that the main contribution to the field range E X d,b 2 2 is given by the eigenvalue arrays (21) corresponding to h such that h/b ∼ p where p = 1 -λ(0) 2 Λ . Using this observation, it was shown that for any fixed ǫ ∈ (0, 1) and assuming [START_REF] Novak | Tractability of Multivariate Problems, Volumes I-III[END_REF] along with Assumption 4.1, the representation ( 22) is possible with the same error

E||X ⊥ || 2 2 ≤ C b d Λ b ǫ 2
while the dimension n grows with a certain exponential rate. More precisely, for each small δ and all large d > d 0 (δ) we have n ≤ V (1+δ)d , where

V := (1 -f p) f p-1 f -f p (1 -p) (1-p)f Λ * f .
and Λ * depending on eigenvalues (λ(i)) was defined in [START_REF] Papageorgiou | On the average complexity of multivariate problems[END_REF].

Recall some special cases.

• If f = 1, then V = (1 -p) p-1 (1 -p) 1-p Λ * = Λ * .
This essentially corresponds to the case considered in Section 3.1.2.

• It is rather surprising that for f < 1 the explosion coefficient V depends on the "strange" quotient p.

• If p = 1, i.e. λ(0) = 0, then we have a nice formula

V = (1 -f ) f -1 f -f Λ * f .
• If f = 0, then V = 1, which means that there is no exponential explosion. This includes the case of b fixed while d growing to infinity.

By applying the pointwise algorithm A n,k with the same k = ⌈log 2 (1/ǫ)⌉ iterations we get again the bound (23). The number of points used n k ≤ 2V (1+δ)d ⌈log 2 (1/ǫ)⌉ is still exponential in variable d. On the exponential scale, the explosion coefficient V remains the same when we pass to pointwise algorithms.

Probabilistic setting

In this section, we consider a simple tensor product random field X(t, ω) X(t, ω) =

(exactly as in formula [START_REF] Lifshits | Lectures on Gaussian Processes[END_REF], with the notations explained there). But since we will deal now with probabilistic estimates, we assume additionally that random variables ξ k are jointly Gaussian. Since ξ k are supposed to be non-correlated, in Gaussian context this implies their independence. We also suppose that Assumption 3.1 holds.

For given ǫ > 0 and γ ∈ (0, 1), we are interested in a linear algorithm A based on point measurements of X, satisfying P( X -AX 2 > ǫ) ≤ γ (25) and using the minimal possible number of point values. . Then for some constant C = C(X) there exist a linear approximation algorithm A satisfying (25) and using at most C v 2 | ln v| 1-2r(β+1) -1/(2r-1)

(26) point values, where r comes from Assumption 3.1 and β is given in Lemma 3.2.

For proving Proposition 5.1, we will need the following simple fact from the theory of Gaussian vectors. Lemma 5.2 Let X be a zero mean Gaussian vector in a separable Banach space. Then for any γ ∈ (0, 1) we have P ||X|| ≥ E||X|| 2 1/2 1 + 2| ln γ| ≤ γ.

Proof. Let us write ||X|| in the dual form,

||X|| = sup ||f ||=1 (f, X),
where the supremum is taken over the unit sphere of the dual space. Notice that (f, X) is a centered Gaussian random variable and

E(f, X) 2 ≤ E(||f || ||X||) 2 = ||f || 2 E||X|| 2 .

It follows that

σ 2 := sup ||f ||=1 E(f, X) 2 ≤ E||X|| 2 .
We will also use the known fact that a median med(||X||) satisfies inequality med(||X||) ≤ E||X|| ≤ E||X|| 2 1/2 , see [START_REF] Lifshits | Gaussian Random Functions[END_REF]. Now by Gaussian isometric inequality, see [START_REF] Lifshits | Gaussian Random Functions[END_REF] Here Φ(•) is the distribution function of the standard normal law.

Proof of Proposition 5.1. According to [START_REF] Lifshits | Approximation complexity of additive random fields[END_REF], for every n we may find an algorithm A n using n point values such that E X -A n X 2 2 ≤ C 0 n 1-2r (ln n) 2r(β+1)-1 .

Now choose a minimal n so large that

C 0 n 1-2r (ln n) 2r(β+1)-1 ≤ v 2
which amounts to say that n exceeds expression (26) with appropriate constant C(X) independent of n.

1

 1 Additive random field of order b

  0, with B b and β given in Lemma 3.2 (let d = b in those notations) and depending on b and on parameters r, q from Assumption 3.1 but not depending on d. Notice especially that n avr d,b (ǫ) depends on d polynomially. Now we transform the approximation algorithm (22) that uses general linear functionals (φ j , •) into a point evaluation based algorithm A n,k from Section 2.2. We still use parameters n = 2n avr d,b (ǫ) and k = ⌈log 2 (1/ǫ)⌉ and obtain from (5)

φ

  k l (t l )ξ k (ω) := k∈N d λ k φ k ξ k ,

Proposition 5 . 1

 51 Let X be a tensor product random field (24) with eigenvalues satisfying Assumption 3.1. Let ǫ > 0 and γ ∈ (0, 1). Denote v := ǫ 1 + 2| ln γ| -1

2 1/2 1 +

 21 Let A := A n . Then E||X -AX 2 2| ln γ| ≤ v 1 + 2| ln γ| = ǫand by applying Lemma 5.2 to the Gaussian vector X -AX we getP( X -AX 2 > ǫ) ≤ P ||X -AX|| ≥ E||X -AX|| 2 1/2 1 + 2| ln γ| ≤ γ.

  Proposition 2.2 Let Y (t), t ∈ D, be a random field satisfying Assumption 2.1 with some parameters p, γ, C 1 . Let Z > 2p. Then there exists C = C(C 1 , p, γ, Z) such that for every integer n ≥ 4 there exist a linear approximation procedure A n based on ⌈Z log 2 n⌉ × n point evaluations of Y and satisfying

  again, P ||X|| ≥ E||X|| 2 1/2 1 + 2| ln γ| ≤ P ||X|| ≥ med(||X||) + σ 2| ln γ|

	≤ 1 -Φ	2| ln γ|
	≤ exp -	2| ln γ|

2

/2 = γ.

m,h have the same order of decay for all h, namely, λ 2 m,h ∼ B 2 h j -2r (ln j) 2q where the constants B h are defined in Lemma

3.2 above. Moreover, the same decay order corresponds to the rearranged total set of eigenvalues.

≤ C n 1-2r (ln n) 2(q+r)-1 .The proof is again the same as for Proposition

3.3, while the calculation of logarithm's exponent is as follows. The relations β = α = q/r yield for the exponent in[START_REF] Lifshits | Approximation complexity of additive random fields[END_REF] 2r(β + 1) -1 = 2rβ + 2r -1 = 2q + 2r -1.

Acknowledgements

We are grateful to the referees for their useful advice that improved the presentation of our results. The work of the first named author was supported by grants RFBR 13-01-00172 and SPbSU 6.38.672.2013.


Narimane Mati-Baouche

Hélène De Baynast

André Lebert

Shengnan Sun

Carlos Javier Sacristán López-Mingo

Philippe Leclaire

Philippe Michaud

Carlos Javier Sacristan Lopez-Mingo

Mechanical, thermal and acoustical characterizations of an insulating bio-based composite made from sunflower stalks particles and chitosan

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

Introduction

The implementation of agro-industrial residues in the development of composite materials for building trades is now a research of growing interest [START_REF] Cole | Energy and greenhouse gas emissions associated with the construction of alternative structural systems[END_REF][START_REF] Gustavsson | Variability in energy and carbon dioxide balances of wood and concrete building materials[END_REF][START_REF] Morel | Building houses with local materials: means to drastically reduce the environmental impact of construction[END_REF]. One of the major challenges of this industry for the next decade is to improve the energy performance of existing buildings taking into account the increasing social emphasis on issues of the environment, waste disposal, and the depletion of non-renewable resources. The motivation includes cost, mechanical, thermal and acoustical performance enhancements, weight reduction, and environment concerns [START_REF] Glé | Modelling of the acoustical properties of hemp particles[END_REF]. In this context, bio-based insulator materials made from by-products of agriculture are an interesting alternative to those obtained from fossil carbon. Thermal insulation is known to play a critical role in saving energy by reducing the rate of heat transfer [START_REF] Keynakli | A review of the economical and optimum thermal insulation thickness for building applications[END_REF]. A material is considered as a thermal insulator when its thermal conductivity (Ä) is lower than 0.1 W m -1 K -1 , some of them reaching 0.035 W m -1 K -1 [START_REF] Al-Homoud | Performance characteristics and practical applications of common building thermal insulation materials[END_REF]. They include fibrous and cellular inorganic materials, organic materials and metallic or metalized reflective membranes. Among them, those obtained from agricultural by-products have numerous advantages as they are eco-friendly and renewable. Moreover the high fiber content of some of them (wood, hemp, flax. . .) authorizes the reinforcement of composites [START_REF] Liu | Green composites from renewable ressources: preparation of epoxidized soybean oil and flax fiber composites[END_REF]. This property leads to the development of bio-based composites with natural fibers not only in the field of thermal insulators but also in other industrial areas such as paper or thermoplastic industries [START_REF] Ashori | Effects of nanoparticles on the mechanical properties of rice straw/polypropylene composites[END_REF][START_REF] Ashori | Bio-based composites from waste agricultural residues[END_REF][START_REF] Ashori | Thermoplastic hybrid composites using bagasse corn stalk and E-glass fibers; fabrication and characterization[END_REF][START_REF] Zahedi | A comparative study on some properties of wood plastic composites using canola stalk, paulownia and nanoclay[END_REF]. [START_REF] Onésippe | Sugar cane bagasse fibres reinforced cement composites: thermal considerations[END_REF] have used sugar cane bagasse fibers in cement composites. The bio-sourced material obtained had higher thermal conductivity and mechanical strength compared to cement alone. [START_REF] Korjenic | Development and performance evaluation of natural thermal insulation materials composed of renewable resources[END_REF] carried out a research project to develop a new insulating material from jute, flax and hemp. Results showed that the correct combination of natural materials is comparable with convectional materials for mechanical properties. Furthermore, [START_REF] Elfordy | Mechanical and thermal properties of lime and hemp concrete (hempcrete) manufactured by a projection process[END_REF], [START_REF] Nguyen | Effect of compactness on mechanical and thermal properties of hemp concrete[END_REF], Nozahic and Amziane (2012) and Nozahic et al. (2012) characterized the hemp concrete, a new insulating material. Its thermal conductivity and its compressive strength were evaluated ranging between 0.1 and 0.4 W m -1 K -1 and from 0.2 to 0.8 MPa, respectively.

Sunflower is widely cultivated all over the world. In 2010, the harvested area in Europe was 3.68 × 10 6 ha representing 16% of the total harvested area in the world [START_REF] Sun | Hygromechanical characterization of sunflower stems[END_REF]. The crushed sunflower stems have no real application in agriculture and it has been estimated that each hectare of sunflower can produce 3-7 tons of dry biomass including stems [START_REF] Marechal | Characterization of by-products of sunflower culture commercial applications for stalks and heads[END_REF]. The stems are usually burnt, used as natural fertilizer, for animal feed or for fuel production [START_REF] Chen | Progress in bio-fuel combustion in circulating fluidized bed boiler[END_REF][START_REF] Sun | Hygromechanical characterization of sunflower stems[END_REF]. In this context, non-food uses of a part of this biomass would have a significant economic impact on this culture [START_REF] Ashori | Bio-based composites from waste agricultural residues[END_REF][START_REF] Laufenberg | Transformation of vegetable waste into value added products: (A) the upgrading concept; (B) practical implementations[END_REF]. Several authors reported that building products/composites produced from various agro-waste materials, including sunflower stalks, have higher mechanical properties, lower thermal conductivity and density, are cheaper, durable, lightweight, available in abundance, much less abrasive and environmental friendly compared to conventional ones [START_REF] Ashori | Thermoplastic hybrid composites using bagasse corn stalk and E-glass fibers; fabrication and characterization[END_REF][START_REF] Díaz | Hydrothermal pretreatment and enzymatic hydrolysis of sunflower stalks[END_REF][START_REF] Madurwar | Application of agrowaste for sustainable construction materials: a review[END_REF][START_REF] Zahedi | A comparative study on some properties of wood plastic composites using canola stalk, paulownia and nanoclay[END_REF]. Bark of sunflower stems exhibit favorable mechanical properties whereas its pith has good insulation ones [START_REF] Sun | Hygromechanical characterization of sunflower stems[END_REF]. Hence, this by-product could find use in a total bio-sourced composite with thermal insulator properties using a natural organic binder contrary to classical strategies which employed cement [START_REF] Elfordy | Mechanical and thermal properties of lime and hemp concrete (hempcrete) manufactured by a projection process[END_REF][START_REF] Nguyen | Effect of compactness on mechanical and thermal properties of hemp concrete[END_REF]Nozahic andAmziane, 2012 andNozahic et al., 2012) or epoxy resin [START_REF] Binici | An environmentally friendly thermal insulation material from sunflower stalk textile waste and stubble fibres[END_REF]. Chitosan is a natural polymer obtained industrially by deacetylation of chitin from crustacean shells [START_REF] Dash | Chitosan -a versatile semisynthetic polymer in biomedical applications[END_REF][START_REF] No | Applications of chitosan for improvement of quality and shelf life of foods: a review[END_REF]. This polysaccharide, which is the most abundant polysaccharide after cellulose [START_REF] Barbosa | Polysaccharides as scaffolds for bone regeneration[END_REF], is a heteropolymer of ˇ-(1,4)-linked 2-acetamido-2-deoxy-d-glucopyranose and 2-amino-2-deoxy-d-glucopyranose units. It is the sole cationic polysaccharide described due to its positive charges (NH 3 + ) at acidic pH (pH < 6.5) [START_REF] Barbosa | Polysaccharides as scaffolds for bone regeneration[END_REF]. Over the past few decades, chitosan has received attention as a functional non-toxic, antimicrobial, biocompatible and biodegradable biopolymer useable in the area of biomaterials and/or biosourced materials [START_REF] Ashori | Effect of chitosan and cationic starch on the surface chemistry properties of bagasse paper[END_REF][START_REF] Dash | Chitosan -a versatile semisynthetic polymer in biomedical applications[END_REF][START_REF] Kumar | A review of chitin and chitosan applications[END_REF][START_REF] Muzzarelli | Chitosan, hyaluronan and chondroitin sulfate in tissue engineering for cartilage regeneration: a review[END_REF][START_REF] Nair | Biodegradable polymers as biomaterials[END_REF][START_REF] No | Applications of chitosan for improvement of quality and shelf life of foods: a review[END_REF][START_REF] Patel | Development of chitosan-based adhesive. Application to wood bonding[END_REF][START_REF] Shalaby | Chitosan-based systems[END_REF][START_REF] Umemura | Development of new natural polymerbased wood adhesives III: effects of glucose addition on properties of chitosan[END_REF]. In the light of all its features chitosan was selected as a natural binder for the conception of a thermal insulating material using sunflower stalks particles as reinforcement.

Material and methods

Raw materials

Shredded sunflower stalks

The sunflower (reference LG5474) stalks used in this study were harvested in 2009 (Perrier, France). Grinding of sunflower stalks was performed using a cutting mill SM 300 (Retsch) with a sieve of 20 mm mesh. The speed cut applied was 1000 rpm. The particles obtained were sieved at room temperature (20 • C) using Controlab sieve-tronic to obtain different particles sizes between 1 and 6.3 mm. Particles were stored at room temperature.

Binder

Chitosan from Shrimp Shell with a molecular weight of 98 kDa and a deacetylation degree of 90% was supplied by France-Chitin (France) with the reference number 342. The polysaccharide was solubilized at room temperature (20 • C) for 2 h under stirring at concentrations ranging between 2 and 9% (w/v) in acetic acid 1-2% (v/v) (Sigma Aldrich, 98.9%).

Bio-based composite

The process for obtaining bio-based composites from shredded stems of sunflower and chitosan is shown in Fig. 1. Chitosan solutions were mixed for 5 min with sunflower's stalk particles having different particle sizes (between 1.6 and 6.3 mm). Ratios chitosan/sunflower particles were between 0.04 and 0.15 g/g. Each type of mixture was prepared twice in two polyvinylchloride (PVC) molds: 180 mm × 50 mm × 40 mm for tensile tests and 180 mm × 50 mm × 120 mm for compressive measures. Each set was compacted during 1 min at 20 • C under pressures between 1 × 10 -3 and 32 × 10 -3 MPa using weights or compressive press machine (Zwick-Roell) equipped with a ±20 kN load cell for pressures between 156 × 10 -3 and 574 × 10 -3 MPa. After drying at 50 • C for 50 h in an oven, the resulting composites were firstly thermally characterized (Fig. 1) and then cut with a band saw to obtain slender shapes of 180 mm × 24 mm × 12 mm for tensile mechanical characterization (Fig. 2). Specimens obtained for compressive tests were also cut to obtain slender shape (Fig. 2). Composites failed by buckling when their critical load was reached. As a consequence, the shape of specimens was defined at 80 mm × 50 mm × 50 mm using Euler equation ( 1) to avoid the phenomenon of buckling during compressive tests:

P = 2 EI l 2 (1)
where "P" is the allowable load (N); "E" the Young modulus (kPa); "I" is the moment of inertia (mm 4 ) and "l" is the length of the sample (mm).

For the acoustical analysis, five composites made with sunflower particles of 5 mm, a ratio chitosan/sunflower stalks particles of 6% (w/w) and a compaction pressure of 32 kPa, were cut into circles with a diameter of 29 mm and a thickness of 13 mm.

Composites with labelled chitosan were built using 2 g of RITClabelled chitosan dissolved in a solution of 50 ml of 1% (v/v) acetic acid. This solution was mixed with 30 g of sunflower stalk particles. After that, the set was compacted with a stress pressure of 32 kPa in PVC mold of 180 mm × 50 mm × 20 mm before being dried in an oven for 50 h at 50 • C.

Biochemical characterization

An estimation of the parietal constituents of sunflower stalks on both pith and bark was performed using the [START_REF] Henneberg | Beiträge zur Begründung einer rationellen Fütterung der Wiederkäuer[END_REF] protocol to quantify cellulose and the procedure of [START_REF] Jarrige | Analyse des constituants des plantes fourragères. I. Fractionnement des constituants de la membrane par les hydrolyses acides[END_REF] to evaluate the amount of lignin. All assays were done in triplicate.

Thermal characterization

Thermal characterization of several specimens of sunflower stem particles/chitosan composites was carried out by measuring thermal conductivities (Ä) (W m -1 K -1 ) in different points on the samples by the hot wire method (NeoTIM FP2C) at 20 • C. Thermal conductivity is evaluated by tracking the thermal pulse propagation induced in the sample by a heating source consisting of a nickel alloy wire. The temperature is measured on the wire by means of two T type (Copper-Constantan) thermocouples [START_REF] Franco | An apparatus for routine measurement of thermal conductivity of materials for building application based on a transient hot-wire method[END_REF]. The nickel wire is used to slightly heat up the samples with the passage of an electrical current and to monitor the temperature increase by recording the variation of the voltage across the wire. The thermal conductivity is determined from the time-dependent temperature increase. A mean thermal conductivity was determined from five measurements on each sample for a temperature of 21 • C and a relative humidity close to 40%.

Mechanical characterization

Mechanical characterization was performed using a tensile testing machine (Instron 5543) equipped with a load cell of 5 kN. The cross-head speed was equal to 5 mm min -1 and the specimen clamping length was 140 mm. Compressive tests were carried out with a Zwick-Roell testing machine equipped with a ±20 kN load cell. The tests were displacement-controlled with a cross-head speed equal to 1.2 mm min -1 . The tested specimens were placed between two steel plates to have homogeneous displacement and pressure and the load was applied on the superior surface. The results of the mechanical tests were analyzed through stress-strain curves. These curves present typically two areas. One corresponds to the deformation (elongation), which is proportional to the stress and corresponds to an elastic reversible strain. The other one, corresponding to a plastic area, represents the part when the strain is non-linear and irreversible. The maximum stress ( max ) (MPa) was measured in both tensile and compressive modes. The strain at break ε b (%) (percentage of elongation at break) was evaluated only in the tensile mode.

Acoustical characterization

The measurements were carried out in an impedance tube and based on two-microphones transfer-function method. A sound wave is generated at one end of the tube by a loudspeaker. The wave can be considered as a plane wave if its frequency is below a cut off determined by the tube diameter. After a reflection at the surface of the material at the other end of the tube, the sound field created in the tube is a standing sound field and depends on the reflection coefficient at the surface of the material. The two microphones measure the sound field at two different positions. The loudspeaker can be fed with a white noise signal and the incident and reflected sound pressure recorded by the microphones are then processed to deduce the absorption coefficient of the sample in a frequency band. For a 29 mm diameter tube, the useful frequency band is between 500 Hz and 6400 Hz. Composite samples were placed at the end of the impedance tube and backed by a rigid surface.

Synthesis of rhodamine-labelled chitosan

The synthesis of rhodamine-labelled chitosan was done as described by [START_REF] Patel | Development of chitosan-based adhesive. Application to wood bonding[END_REF]. The final concentration of rhodamine isothiocyanate (RITC) in the reaction medium was controlled to give a ratio of labelled d-glucosamine residue ratio equal to 1:50. Two grams of dried chitosan powder were dissolved in 200 ml of acetic acid at 0.1 mol l -1 . Subsequently, 200 ml of dehydrated methanol were slowly added in this solution under continuous stirring. Rhodamine-isothiocynate (RITC, Sigma Aldrich) dissolved in methanol at 1.0 mg ml -1 was then slowly added to the solution of chitosan. The reaction was realized during 1 h in dark at room temperature (20 • C). RITC-labelled chitosan was then precipitated in a solution of sodium hydroxide at 0.1 mol l -1 . The precipitate was washed with deionized distilled water until complete absence of free RITC in the washing medium. The labelled chitosan was then freeze-dried. The yield was 85% as described by [START_REF] Qaqish | Synthesis of a fluorescent chitosan derivative and its application for the study of chitosan-mucin interactions[END_REF].

Microscopy

Composites of rhodamine-labelled chitosan/sunflower stalks particles were observed at magnification 6.3× under light microscopy in the aim to characterize the bonding area by investigating the adhesive penetration of chitosan into sunflower cells.

Observations in scanning electron microscopy (SEM, Jeol 820) were conducted on sections of sunflower stalks on both bark and pith. The samples were metalized with a physical vapor deposition (PVD) thin layer of gold and were observed with secondary electrons detector. The magnification range covered is between 200 and 15,000. The working distance and acceleration potential were equal to 15 mm and 10 kV, respectively.

Experimental design and statistical analysis

Preliminary testing of the design of composite chitosan/shredded sunflower stalks showed the feasibility of this mixture. To find the best values of the three process parameters affecting the thermal and mechanical properties, namely, the particle size, the ratio chitosan/shredded sunflower stalks and compaction stress, and to minimize the number of tests, a composite central design was chosen. This plan used consists of 24 trials (N) corresponding to N = 2 n + 2. N = 14 points plus 10 repetitions of the central point. Each factor is studied at five levels as shown in Table 1. The associated model is a quadratic polynomial with the linear and quadratic effects and all the interaction effects between the different operating variables effects. Three variables were modeled with this experimental design: thermal conductivity (k) (W m -1 K -1 ), Young's modulus E (MPa) and the maximum stress max (MPa). The results were analyzed by the statistical software R 2.15.2 (R Core Team, 2012). The optimal thermal insulation and mechanical strength were achieved by using a quadratic model with interactions.

Results and discussion

The elaboration of a new composite from agricultural byproducts has a great interest for the society. This study focuses on the development and the characterization of a thermal insulating agro-composite made from sunflower stalks particles glued with chitosan.

Table 2 shows the humidity, parietal content and global dimensions of the used sunflower stalk particles. The cellulose amounts were estimated at 48% for bark and 31.5% for pith, whereas the lignin contents were 14% and 2.5% for bark and pith, respectively. These results are in good agreement with those obtained by [START_REF] Marechal | Characterization of by-products of sunflower culture commercial applications for stalks and heads[END_REF] using chemical method leading to the conclusion that bark is responsible of the rigidity of sunflower stem.

Mechanical and thermal properties of the bio-based composites

Mechanical properties are of primary importance to quantify the performances of materials expected to undergo various types of stresses during their implementation. Young's modulus, also called the elastic modulus, corresponds to the stiffness of the specimen. More Young's modulus increases more rigid is the material. max is the maximum stress reached during tensile or compressive tests that the sample can sustain before being broken. These two parameters characterize the mechanical strength of the material. The thermal performances are evaluated when the conception of insulating material is aimed. More the thermal conductivity of a material is low (less than 0.1 W m -1 K -1 ) more the thermal insulation is significant. An experimental design centered on 24 trials was chosen to study the influence of the process parameters (particle size, compaction pressure and ratio binder/reinforcement) about values of Young's modulus (E ), maximum stress ( max ) and thermal conductivity (Ä) of the composites chitosan/shredded sunflower stalks. All the composites manufactured in this study had density between 150 and 200 kg m 3 . Table 3 summarizes results from this experimental design in terms of mechanical strength and insulating properties.

Knowing that a material is qualified as insulating when its thermal conductivity is less than 0.1 W m -1 K -1 [START_REF] Abdou | The variation of thermal conductivity of fibrous insulation materials under different levels of moisture content[END_REF] the composites obtained in this study can be considered as insulators because of their thermal conductivities between 0.056 and 0.058 W m -1 K -1 depending on the manufacturing conditions. These thermal performances are better than those of sunflower fiberboards (Ä = 0.088 W m -1 K -1 ) manufactured from sunflower whole plant using hydraulic press and other ecofriendly insulating materials available on the market, namely hemp concrete (Ä between 0.17 and 0.48 W m -1 K -1 ), coconut husk insulation boards (Ä = 0.081 W m -1 K -1 ) and date palm fibers insulation boards (Ä = 0.15 W m -1 K -1 ) [START_REF] Chikhi | Experimental investigation of new biocomposite with low cost for thermal insulation[END_REF][START_REF] Elfordy | Mechanical and thermal properties of lime and hemp concrete (hempcrete) manufactured by a projection process[END_REF][START_REF] Evon | New thermal insulation fiberboards from cake generated during biorefinery of sunflower whole plant in a twin-screw extruder[END_REF]. They are also slightly higher than the thermal conductivities of the current "non-bio-based" insulating products such as rock wool (Ä = 0.04 W m -1 K -1 ), mineral wool (Ä = 0.036 W m -1 K -1 ) and fiber glass (Ä = 0.035 W m -1 K -1 ) [START_REF] Abdou | The variation of thermal conductivity of fibrous insulation materials under different levels of moisture content[END_REF]. However, these insulators have very low mechanical resistance and require very specific processes for their maintenance in the building. Mechanical properties of the bio-based composites chitosan/sunflower stalks particles used in tensile mode attained up to 2.6 MPa for max , 250 MPa for E and 2% for the ε b whereas those used in compressive mode reached 2 MPa and for max , 25 MPa for E . These results revealed that this kind of bio-based composites present higher mechanical properties than the hemp concrete knowing as "the green insulator concrete". Indeed, the compressive strength (equivalent to the maximal stress in compressive mode) of this material is described lower than 2 MPa (Nozahic and Amziane, 2012). Furthermore, [START_REF] Elfordy | Mechanical and thermal properties of lime and hemp concrete (hempcrete) manufactured by a projection process[END_REF] have measured both Young's modulus and compressive strength of hemp concrete blocks with various densities (between 291 and 481 kg m -3 ). They have obtained Young's modulus ranging from 7 to 35 MPa and compressive strengths ranging between 0.18 and 0.8 MPa.

Analyses of results from Table 3 allow the following trends for each studied parameter:

The size of sunflower stalks particles greater than 3.1 mm permits the production of composites with optimum mechanical and thermal performances (E = 250 MPa, max = 2.6 MPa Ä = 0.056 W m -1 K -1 ).

Compaction pressures used to make composites (between 1 × 10 -3 and 574 × 10 -3 MPa) influence significantly their thermal and mechanical properties. Compaction is the ability of the sample to acquire cohesion and hence resistance for maximum sustained pressure. Results showed that the greater thermal conductivities were attained with compaction pressure of 10 -3 MPa for the samples made for compressive measurements (0.058 W m -1 K -1 ) and 574 × 10 -3 MPa for the samples made for tensile measurements (0.056 W m -1 K -1 ). This result is surprising because the thermal conductivity (Ä) of the most compacted material (574 × 10 -3 MPa), thus containing less air, is lower (0.056 W m -1 K -1 ). This phenomenon can be explained by the creation of radiative heat transfer on the highly compacted material as described by [START_REF] Pennec | A combined finite-discrete element method for calculating the effective thermal conductivity of bio-aggregates based materials[END_REF]. Higher mechanical performances (E and max ) appeared with significant compaction pressures for the two kinds of composites (tensile and compressive modes). From the mechanical point of view, this behavior can be linked to the work of [START_REF] Nguyen | Effect of compactness on mechanical and thermal properties of hemp concrete[END_REF] who have shown that compaction leads gains strength for hemp concrete composites.

The increase of chitosan/sunflower ratio is logically correlated with the increase of the mechanical strength. This result is not surprising because the chitosan solid state endowed a tensile strength of 70 MPa and Young modulus between 2 and 4 GPa, which reflects a rigid behavior [START_REF] Mati-Baouche | Physico-Chemical, Thermal and Mechanical Approaches for the Characterizations of Solubilized and Solid State Chitosans (under review)[END_REF]. The Young modulus of the bio-based composite was optimal (250 MPa) when the ratio (w/w) of chitosan/sunflower reached 15.3%, which means that the amount of binder has a direct impact on the strength of the final material. Logically, this increase in mechanical performances is correlated with a loss of thermal one (Ä = 0.066 W m -1 K -1 ) because of appearance of thermal bridges in the composite and reduction of the pore spaces of shredded sunflower stalks [START_REF] Pennec | A combined finite-discrete element method for calculating the effective thermal conductivity of bio-aggregates based materials[END_REF].

Acoustical properties of the composite chitosan/sunflower stalks particles

Nowadays, building materials are expected to perform several functions and to be sustainable. For example, building materials are supposed to satisfy structural, thermal and acoustical demands [START_REF] Glé | Acoustical properties of materials made of vegetable particles with several scales of porosity[END_REF]. Hence, acoustical characterization was performed on the composites presenting the best mechanical and thermal compromise. Samples made from 5 mm sunflower stalk's particles, with a 6.6% ratio chitosan/sunflower (w/w) and compacted with a pressure of 32 kPa during the manufacturing, were analyzed with the 29 mm diameter impedance tube. Fig. 3 represents the sound absorption coefficient (˛) of the composite, with thickness of 13 mm, in the frequency ranging from 0 to 4000 Hz. Note that this range of frequencies is used in the construction field to evaluate the acoustic insulating properties of materials. When toward 1, all the energy of the incident wave is absorbed and when ˛= 0, the sound is not dampened at all [START_REF] El Hajj | Development of thermal insulating and sound absorbing agro-sourced materials from auto linked flax-tows[END_REF]. The absorption coefficient of the bio-based composite tested is lower than 0.5 (˛≈ 0.2 at frequency 2500-4000 Hz). Knowing that a composite sugarcane waste fibers/polyester endowed a coefficient of absorption ˛≥ 0.5 [START_REF] Putra | Utilizing sugarcane wasted fibers as a sustainable acoustic absorber[END_REF] above 3500 Hz and that agro-sourced materials from auto link flax-tows had a ˛----0•8 at 3000 Hz [START_REF] El Hajj | Development of thermal insulating and sound absorbing agro-sourced materials from auto linked flax-tows[END_REF], the acoustical performance of the bio-based composite developed here can be considered as low. If the composite is too tight, the acoustic wave cannot penetrate in it as described for cellular concrete (Cerezo, 2005). The acoustic absorption in a porous material strongly depends upon several parameters among which the flow resistivity. For high values of flow resistivity, the sound attenuation can be low and explain the results of Fig. 3. Higher values of absorption can be expected for more porous materials [START_REF] Glé | Acoustical properties of materials made of vegetable particles with several scales of porosity[END_REF].

Penetration of chitosan in sunflower stalks's cells

The chitosan being not visible inside and around the sunflower particles and cells, RITC-labelled chitosan was synthesized as described in methods using the reactivity of isothiocyanate functional groups of RITC moiety with the primary amine group of d-glucosamine residues of chitosan. Fig. 4 provides evidence that rhodamine-labelled chitosan did not penetrate into the sunflower cell walls and therefore did not reach the cell lumens although a double porosity (50-200 m and 2-10 m) was observed by SEM (Fig. 5). Note that pith and bark have a honeycomb structure as described by El [START_REF] El Hajj | Innovant agromaterials formulated with flax shaves and proteinic binder: process and characterization[END_REF] on flax-shaves. So, rhodaminelabelled chitosan only interacted with the outer surface of the plant cell as reported by [START_REF] Patel | Development of chitosan-based adhesive. Application to wood bonding[END_REF] on wood cells and does not take the place of air preserving thermal insulating properties of the agrocomposite. This non-penetration of chitosan inside sunflower cells was explained by its viscosity and critical overlap concentration previously reported by [START_REF] Mati-Baouche | Physico-Chemical, Thermal and Mechanical Approaches for the Characterizations of Solubilized and Solid State Chitosans (under review)[END_REF].

Conclusion

The critical discussion about the preservation of natural resources and recycling has led to a renewed interest in natural materials with the focus on renewable raw materials. The objective of this work was to develop a method of manufacturing eco-insulating materials 100% natural based on sunflower stalks particles glued with binder based on chitosan solution, highlighting the factors that affect the mechanical and thermal properties of the bio-based composite. It was clearly shown that the thermal and mechanical performances of the composites were proportionately improved with the increasing of compaction pressure. The mechanical performances of the composites elaborated in this study are sufficient to ensure safe handling, transport and assembly. Composites with a thermal conductivity of 0.06 W m -1 K -1 and a maximum stress at break of about 2 MPa were obtained with a ratio chitosan/sunflower of 4.3%, a particle size of shredded sunflower stalks of 6.3 mm and a density of 150 kg m -3 . No penetration of the chitosan binder inside the sunflower cells was observed meaning the preservation of the natural properties of sunflower stalks's particles. In addition, acoustical study was performed on the best composite leading to the conclusion that its performance is low in the area of acoustic insulating.

Fig. 1 .

 1 Fig. 1. Process of obtaining agro-composite from sunflower stalks and chitosan. (A) sunflower stalks (LG5474), (B) chitosan solutions at 2-9% (w/v), (C) sunflower stalk particles (1-6 mm), (D) PVC mold, (E) composite sunflower/chitosan obtained for compressive analysis, (F) composites obtained for the tensile analysis.

Fig. 2 .

 2 Fig. 2. Composites chitosan/sunflower stalk aggregates before tensile (A) and compressive (B) analysis.

Fig. 3 .

 3 Fig. 3. Sound absorption (B) of composite (A) made from sunflower stalk's aggregates and chitosan with thickness of 13 mm.

Fig. 4 .

 4 Fig. 4. Light microscopic observation at a magnification 6.3× of the composite chitosan/sunflower stalks aggregates after labelling chitosan.

Fig. 5 .

 5 Fig. 5. SEM observations of pith and bark of sunflower stalk cells.

Table 1

 1 Area of variations of the operating parameters.

	Level	Particle size (mm)	Ratio (%)	Compaction pressure
				(10 -3 MPa)
	+1.68	6.30	15.3	574
	+1	5.00	13.3	373
	0	3.12	10.0	156
	-1	2.00	6.6	32
	-1.68	1.60	4.3	1

Table 2

 2 Characteristics of the studied sunflower's stalks.

		Humidity (%)	Cellulose (%)	Lignine (%)	Mean length (mm)	Mean width (mm)	Mean thickness (mm)
	Pith	13	31.5	2.5	1-6.3	1-6.3	0.5-2
	Bark	13	45	14	1-6.3	1-6.3	0.5-2

Table 3

 3 Results of the experimental design obtained on both mechanical and thermal properties of composites chitosan/sunflower stalks aggregates, as a function of the applied compaction pressure, the size of sunflower particles and the ratio chitosan/sunflower.

		Optimal properties	Compaction (10 -3 MPa)	Size of particles (mm)	Ratio (w/w) chitosan/sunflower (%)
	Tensile mode	E = 250 MPa	574	3.1	15.38
		max = 2 MPa	574	6.3	/ a
		Ä = 0.056 W m -1 K -1	574	6.3	4.38
	Compressive mode	E = 25 MPa	574	6.3	15.38
		max = 2.6 MPa	574	6.3	15.38
		Ä = 0.058 W m -1 K -1	1	6.3	4.38

a No significant effect was observed.

Acknowledgement

This work was supported by the French National Research Agency (ANR-DEMETHER-10-ECOT-004 grant) and the French clusters: Céréales Vallée and Viaméca.