Éphane St

Sabourau

SMALL VOLUME OF BALLS, LARGE VOLUME ENTROPY AND THE MARGULIS CONSTANT

Keywords: 2010 Mathematics Subject Classification. Primary 53C23; Secondary 53C20 Margulis constant, volume entropy, systolic inequality, filling radius, curvature-free estimates

published or not. The documents may come

Small volume of balls, large volume entropy and the

Margulis constant

Introduction

The volume entropy of a closed Riemannian manifold M , denoted by Ent(M ), is defined as the exponential growth rate of the volume of balls in the universal Riemannian cover M of M . More precisely,

Ent(M ) = lim R→∞ log[vol B(x, R)] R
where B(x, R) is the ball of radius R centered at any point x of M . Here, and throughout this article, all manifolds are supposed to be connected.

The following classical result of M. Gromov relates the volume entropynormalized by the volume -to the simplicial volume, a nonnegative topological invariant which does not vanish for closed manifolds admitting a negatively curved Riemannian metric.

Theorem 1.1 ( [START_REF] Gromov | Volume and bounded cohomology[END_REF]). Let M be a closed Riemannian n-manifold. Then

Ent(M ) n vol(M ) ≥ c n ||M ||
where ||M || is the simplicial volume of M and c n is an explicit positive constant depending only on n.

In short, if the volume is small then the volume entropy is large. The proof of this theorem relies on the notions of simplicial norm and bounded cohomology, and more specifically on the technique of diffusion of chains. A sharp estimate on the (scale invariant) product Ent(M ) n vol(M ) has been established by G. Besson, G. Courtois and S. Gallot for closed negatively curved locally symmetric Riemannian manifolds, cf. [BCG]. The proof of their result rests a different argument, namely the construction of a volumedescreasing map via the barycenter method.

In this article, we strengthen some aspects of M. Gromov's result, at least for closed manifolds admitting a negatively curved Riemannian metric, by relying on a yet different approach. More precisely, we prove the following.

Theorem 1.2. Let M be a closed n-manifold admitting a negatively curved Riemannian metric. Then, there exists δ > 0 such that for every Riemannian metric on M , if the volume of every ball B(R) of radius R in M satisfies volB(R) ≤ a n R n then Ent(M ) ≥ δ R where a n is an explicit positive constant depending only on n.

In short, if the volume of every ball of small radius is small, then the volume entropy is large. Therefore, this condition ensures a large volume entropy even if the volume of the whole manifold is large. Note that for R ≥ vol(M ) 1 n , we recover a lower bound on the product Ent(M ) n vol(M ) to be compared with Theorem 1.1. Actually, the conclusion of Theorem 1.2 holds for a larger class of manifolds, including manifolds which do not admit nonpositively curved Riemannian metrics, cf. Corollary 4.6. This class of manifolds is described in terms of the topology of the loop space of their classifying spaces, see Definition 3.1 for further details.

The proof of this theorem relies on the notion of filling radius and filling techniques introduced by M. Gromov in the context of systolic geometry, cf. [START_REF] Gromov | Volume and bounded cohomology[END_REF]. The filling radius of a closed Riemannian manifold M is a geometric invariant denoted by FillRad(M ), see Definition 4.4, related to the volume through the following fundamental bounds of M. Gromov and L. Guth.

Theorem 1.3 ([Gr83, Main Theorem 1.2.A], [Gu11, Theorem 1]). Let M be a closed Riemannian n-manifold. Then (1) vol(M ) ≥ c n FillRad(M ) n
where c n is an explicit positive constant depending only on n.

(2) For every R ≤ FillRad(M ), there exists a ball

B(R) of radius R in M with volB(R) ≥ a n R n
where a n is an explicit positive constant depending only on n.

These estimates make any lower bound on the filling radius highly desirable, as it would provide a universal lower bound on the volume of the manifold. In the context of curvature-free geometry, such lower bounds exist for the injectivity radius [START_REF] Gromov | Filling Riemannian manifolds[END_REF] (and more generally, for the local contractibility function [START_REF] Greene | Little topology, big volume[END_REF]), the length of the shortest noncontractible closed geodesic (i.e., the systole) on essential manifolds [START_REF] Gromov | Filling Riemannian manifolds[END_REF] and the length of the shortest closed geodesic on the two-sphere [START_REF] Sabourau | Filling radius and short closed geodesics of the 2-sphere[END_REF].

In this article, we establish another curvature-free lower bound in terms of the commutator systole. Here, the commutator systole of a closed Riemannian manifold M , denoted by sys 0 (M ), is defined as the length of the shortest loop representing a nontrivial element in the commutator subgroup

[π 1 (M ), π 1 (M )] of π 1 (M ). That is, sys 0 (M ) = inf{length(γ) | [γ] = 0 ∈ [π 1 (M ), π 1 (M )]}.
Clearly, the commutator systole is bounded from below by the systole, that is, sys 0 (M ) ≥ sys(M ).

We obtain the following filling radius estimate.

Theorem 1.4. Let M be a closed n-manifold admitting a negatively curved Riemannian metric. Then every Riemannian metric on M satisfies

FillRad(M ) ≥ 1 32 sys 0 (M ).
In particular, (1) vol(M ) ≥ c n sys 0 (M ) n (1.1) where c n is an explicit positive constant depending only on n.

(2) For every R ≤ sys 0 (M ), there exists a ball B(R) of radius R in M with volB(R) ≥ a n R n where a n is an explicit positive constant depending only on n.

As previously, this result holds for a larger class of manifolds formed of essential manifolds. However, it fails for some essential manifolds, even for closed nonpositively curved manifolds (e.g., the product of a closed hyperbolic surface with a small circle). Actually, this theorem is a consequence of a similar estimate, cf. Theorem 4.5, relating the filling radius to the Margulis constant defined in Definition 4.1.

Theorem 1.4 is related to a more general problem in systolic geometry connected to the notion of subgroup systole as follows. For every normal subgroup H ¡ π 1 (M ), we define the H-systole of M as sys

H (M ) = inf{length(γ) | [γ] = 0 ∈ H}.
By definition, sys 0 (M ) = sys H (M ) when H = [π 1 (M ), π 1 (M )]. We say that a closed n-manifold M satisfies a systolic inequality for H if there exists a positive constant c M depending only on the topology of M such that vol(M ) ≥ c M sys H (M ) n for every Riemannian metric on M . From [START_REF] Gromov | Filling Riemannian manifolds[END_REF] and [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF], a closed manifold M satisfies a systolic inequality for H = [π 1 (M ), π 1 (M )] if and only if M is essential (e.g., aspherical). A natural question is to determine which manifolds M satisfy a systolic inequality for a given proper normal subgroup H in π 1 (M ). For the notion of relative systole related to the quotients of the fundamental group (and not to its subgroups), the general theory of systolic inequality applies, cf. [START_REF] Gromov | Filling Riemannian manifolds[END_REF], [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF], and yields necessary and sufficient topological conditions for the existence of (relative) systolic inequalities.

Unfortunately, this does not work for normal subgroup systoles and a general theory for systolic inequalities for normal subgroups still needs to be developed. Indeed, except for the commutator subgroup, see the inequality (1.1) in Theorem 1.4, we do not know when such inequalities hold. (For surfaces, we actually have a stronger systolic inequality for the commutator subgroup, which is asymptotically optimal with the genus, cf. [START_REF] Sabourau | Asymptotic bounds for separating systoles on surfaces[END_REF].) Thus, Theorem 1.4 can be seen as a first step in the direction of a more general theory.

Review of algebraic and geometric entropies

In this short section, we introduce some definitions regarding algebraic and geometric entropies, and recall classical results relating the two notions without proof.

Ent |•| (Γ) = lim inf R→∞ log #{α ∈ Γ | |α| ≤ R} R (2.1)
In the special case where Γ = α, β is a two-generator group and d is the word distance with respect to the generating set {α, β}, we simply write Ent(α, β) and refer to it as the algebraic entropy of the pair α, β.

Let (M, g) be a closed Riemannian manifold. Fix x 0 ∈ M and a lift x0 of x 0 in the universal cover M . The group Γ = π 1 (M, x 0 ) acts on the universal Riemannian cover ( M , g) by isometries. The geometric norm on Γ induced by g is defined as

|α| g = d g(α • x0 , x0 )
for every α ∈ Γ. The geometric norm | • | g depends on the basepoint x0 . However, it is a classical result that the geometric entropy Ent |•|g (Γ) agrees with the volume entropy Ent(M, g) defined in the introduction. That is,

Ent |•|g (Γ) = Ent(M, g).
We denote by Ent |•|g (α, β) the entropy of the subgroup α, β of Γ with respect to the restriction of the geometric norm on Γ induced by g. We refer to it as the geometric entropy of the subgroup α, β .

In the sequel, we will only consider the algebraic and geometric entropies (for which the liminf in (2.1) is a limit by submultiplicativity of the counting function up to some constant shift). We recall the following classical estimates regarding the algebraic and geometric entropies of a pair of elements in the fundamental group of a closed Riemannian manifold (the proofs are straightforward and are left as an exercice).

Proposition 2.2. Let α, β ∈ Γ and φ : Γ → G be a group homomorphism.

(1)

Ent(φ(α), φ(β)) ≤ Ent(α, β) (2) Ent |•|g (α, β) ≤ Ent(M, g) (3) If |α| g ≤ L and |β| g ≤ L for some positive real L, then Ent(α, β) ≤ L × Ent |•|g (α, β).
In particular,

Ent(α, β) ≤ L × Ent(M, g).
(4) If α and β commute then

Ent(α, β) = 0.

Free loop space of Eilenberg-MacLane spaces

In this section, we describe the topology of the loop space of the Eilenberg-MacLane spaces corresponding to discrete groups satisfying some algebraic features. The topology of these loop spaces will play a crucial role in our arguments.

We first need to introduce some group-theoretical properties.

Definition 3.1. Let G be a discrete group. Consider the following algebraic property.

(P 0 ) The centralizer

C α = {β ∈ G | αβ = βα}
of every nontrivial element α of G is an infinite cyclic subgroup. Consider also the following algebraic property with δ > 0.

(P δ ) The algebraic entropy of every pair α, β of elements of G which do not lie in the same infinite cyclic subgroup is at least δ. That is,

Ent(α, β) ≥ δ.
From Proposition 2.2.(4), the property (P δ ) implies (P 0 ) and can be viewed as a quantitative version of (P 0 ).

Example 3.2. Torsion-free Gromov hyperbolic groups satisfy the property (P 0 ), cf. [START_REF] Gromov | Hyperbolic groups. Essays in Group Theory[END_REF]. In particular, this is the case for the fundamental group π 1 (M ) of every closed Riemannian n-manifold M of curvature κ ≤ -1.

Actually, the fundamental group π 1 (M ) also satisfies the stronger property (P δ ) for some δ uniformly bounded from above in terms of natural Riemannian invariants of M . More precisely, this holds true in either of the following cases:

(1) for every δ ≤ δ(n, λ), where λ is a (negative) lower bound on the curvature, cf. [BCG11]; (2) for every δ ≤ δ(n, i 0 ), where i 0 is a lower bound on the injectivity radius, cf. [BCG].

Definition 3.3. A nontrivial element α in a discrete group G is primitive if it cannot be written as the power of any element in G. That is, if α = ω m for some ω ∈ G and m ∈ Z then m = ±1.

The following lemma underlines some algebraic consequences of the property (P 0 ). Lemma 3.4. Let G be a discrete group satisfying the property (P 0 ). Let α and β be two nontrivial elements in G. Then

(1) The group G is torsion-free.

(2) The generators of the (infinite cyclic) centralizer C α are primitive.

(3) If and β commute then C α = C β . In particular, every nontrivial element of G lies in a unique maximal infinite cyclic subgroup, namely its centralizer.

Proof. The centralizers of the nontrivial elements of G are infinite cyclic subgroups and so are torsion-free. It follows that G itself is torsion-free since any nontrivial element of G lies in its own centralizer. Let α 0 be a generator of C α . Suppose that α 0 = ω m for some ω ∈ G and some integer m. Clearly, ω commutes with α, that is, ω ∈ C α = α 0 . Thus, ω = α k 0 for some integer k. Therefore, α 0 = α km 0 . Now, since G is torsion-free, we deduce that km = 1, and so m = ±1.

Let α 0 and β 0 be some generators of C α and C β . Since β ∈ C α = α 0 , we have β = α j 0 for some integer j. Thus, α 0 commutes with β, that is, α 0 ∈ C β . Therefore, the centralizer C α , which is generated by α 0 , is a subgroup of C β . Switching the roles of α and β, we derive C α = C β . Definition 3.5. Let G be a discrete group and BG = K(G, 1) be the Eilenberg-MacLane space of G. Denote by Λ(BG) = Map(S 1 , BG) the free loop space of BG and by Ω(BG) the space of loops based at the same fixed point * in BG. Both spaces are endowed with the compact-open topology.

The homotopy type of the free loop space of BG is closely related to the algebraic properties of G. We refer to [START_REF] Burghelea | The free loop space. I. Algebraic topology[END_REF] and references therein for a description of the homology group of the free loop space in terms of the Hochschild homology of the group ring Z[G]. In our case, we will need the following simpler description of its connected components which results from [H74].

Proposition 3.6. Let G be a discrete group. Then the connected components of Λ(BG) are in one-to-one correspondence with the conjugacy classes in G.

Moreover, if G satisfies the property (P 0 ) then (1) the connected component of Λ(BG) containing the constant loops is homotopy equivalent to BG;

(2) the connected components of Λ(BG) not containing the constant loops are homotopy equivalent to the circle S 1 = R/Z.

Proof. The first statement of the proposition is classical. We will focus on the topological description of the connected components of the free loop space following the argument of [H74]. Fix a loop γ * in Ω(BG) with basepoint * ∈ BG and consider the connected component of Λ(BG) containing γ * . All homotopy groups will have the point * or the loop γ * as their basepoint. The natural Serre fibration

Ω(BG) → Λ(BG) p -→ BG γ -→ γ(0)
induces the long exact sequence

• • • → π n+1 (BG) → π n (Ω(BG)) → π n (Λ(BG)) → π n (BG) → • • • → π 0 (BG).
Now, from the adjoint relation, cf. [Ha02, p. 395], we have π n (Ω(BG)) = π n+1 (BG) for every n ≥ 0 (hence the homotopy equivalence Ω(BG) G) and π n (BG) = 0 for n ≥ 2. Thus, for n ≥ 2, we obtain

π n (Λ(BG)) = 0
while for n = 1, we derive the inclusion

p # : π 1 (Λ(BG)) → π 1 (BG).
We claim that π 1 (Λ(BG)) agrees with the centralizer C [γ * ] of the homotopy class [γ * ] of γ * in π 1 (BG). Indeed, every element α in π 1 (BG) lies in the image of p # if and only if there exists a map To conclude, we need to examine two cases depending whether the homotopy class [γ * ] of γ * is trivial or not. In the first case, its centralizer C [γ * ] is the whole group G and the connected component Λ γ * is homotopy equivalent to K(G, 1) = BG. While, in the second case, its centralizer C [γ * ] is isomorphic to Z and the connected component Λ γ * is homotopy equivalent to the K(Z, 1)-space S 1 . Example 3.7. Let M be a closed negatively curved Riemannian manifold, see Example 3.2. The Birkhoff curve shortening flow gives rise to a deformation retract of the free loop space. This curve shortening flow retracts each noncontractible loop of M to the unique closed geodesic in its free homotopy class. It also retracts contractible loops of M to points.

S 1 × S 1 → BG such that • the loop c 1 : S 1 × {0} → BG
We conclude this section with the following important consequence of Proposition 3.6, key to the proof of the filling radius estimate of Theorem 1.4. Remark 3.8. Let G be a discrete group satisfying the property (P 0 ). Every noncontractible loop γ in BG is homotopic to the iterate γ k 0 of a loop γ 0 whose homotopy class [γ 0 ] generates the centralizer C [γ] and so is primitive from Lemma 3.4. The primitive loop γ 0 is unique up to orientation and homotopy. It can be chosen to depend only on the maximal infinite cyclic subgroup C [γ] generated by [γ], cf. Lemma 3.4.

From Proposition 3.6, there exists a deformation retract of the unparameterized free loop space Π(BG) = Λ(BG)/O(2), which homotopes contractible loops to points and noncontractible loops γ to the iterate γ k 0 of the primitive loop γ 0 modulo O(2). We will fix such a homotopy once and for all, and refer to it as the deformation retract of the (unparameterized free) loop space.

Note that this property fails for the torus T 2 , which obviously does not satisfies the property (P 0 ). Indeed, the free homotopy class of a noncontractible unparameterized loop γ of T 2 is homotopy equivalent to S 1 (the S 1 -family of closed geodesics homotopic to γ for any given flat metric, for instance). Therefore, there is no canonical representative -up to homotopy -to which the whole class of γ deformation retracts, instead, there is a whole family of natural representatives parameterized by S 1 .

Filling radius and Margulis constant

In this section, we establish a universal geometric inequality, cf. Theorem 4.5, between the filling radius, cf. Definition 4.4, and the Margulis constant, cf. Definition 4.1, for closed Riemannian manifolds satisfying some topological properties, cf. Proposition 3.1. As a consequence of this inequality, we derive a lower bound on the volume entropy in terms of the volume of small balls, cf. Corollary 4.6.

We first need to introduce the definition of the Margulis constant. Definition 4.1. Let M be a closed Riemannian manifold and φ : π 1 (M ) → G be a group homomorphism. For every x ∈ M and > 0, denote by

Γ φ,x = φ([γ]) ∈ G | x ∈ γ, φ([γ]
) = e and length(γ) ≤ the subgroup of G generated by the φ-image of the homotopy classes of the loops based at x of length at most . Observe that Γ φ,x is a subgroup of Γ φ,x for every ≤ . Hence, if Γ φ,x is cyclic, the same holds for Γ φ,x .

Define

φ (M ) = sup{ | Γ φ,
x is a cyclic subgroup of G for every x ∈ M }. The Margulis constant can be defined as

0 (M ) = id (M ).
It is minimal among the φ (M ). More precisely, we have the following. Proposition 4.2. Let M and φ be as above. Then 0 (M ) ≤ φ (M ).

Proof. Let

< 0 (M ). By definition of the Margulis constant, for every x ∈ M , the subgroup Γ id,x is cyclic. Since every homomorphic image of a cyclic group is cyclic, the subgroup Γ φ,x = φ(Γ id,x ) is cyclic too. Therefore, ≤ φ (M ) and the desired inequality follows.

The Margulis constant is related to the commutator systole through the straightforward bound.

Proposition 4.3. Let M be a closed Riemannian manifold whose fundamental group satisfies the property (P 0 ). Then

sys 0 (M ) ≤ 4 0 (M ).
Proof. By definition of the Margulis constant, there exist two loops γ 1 and γ 2 based at the same point of length at most 0 (M ) whose homotopy classes α 1 and α 2 do not lie in the same (infinite) cyclic subgroup. From (P 0 ), the homotopy classes α 1 and α 2 do not commute, that is, the commutator [α 1 , α 2 ] is nontrivial. By construction, this commutator can be represented by a loop of length at most 4 0 (M ), namely γ 1 γ 2 γ -1 1 γ -1 2 . Hence, the commutator systole of M satisfies the bound sys 0 (M ) ≤ 4 0 (M ).

Let us recall the notion of filling radius introduced by M. Gromov in [START_REF] Gromov | Filling Riemannian manifolds[END_REF] to established systolic inequalities on essential manifolds. Definition 4.4. Let (M, g) be a closed Riemannian n-manifold. Denote by d g the distance on M induced by the Riemannian metric g. The map

i : (M, d g ) → (L ∞ (M ), || • ||) defined by i(x)(•) = d g (x,
•) is an embedding from the metric space (M, d g ) into the Banach space L ∞ (M ) of bounded functions on M endowed with the sup-norm || • ||. This natural embedding, also called the Kuratowski embedding, is an isometry between metric spaces. We will consider M isometrically embedded into L ∞ (M ).

The filling radius of (M, g), denoted by FillRad(M, g), is the infimum of the positive reals ρ such that

(i ρ ) * ([M ]) = 0 ∈ H n (U ρ (M ); k) where i ρ : M → U ρ (M ) is the inclusion into the ρ-tubular neighborhood of M in L ∞ (M ), and [M ] ∈ H n (M ; k) is the fundamental class of M .
Here, the homology coefficients are in k = Z, if M is orientable, and in k = Z 2 , otherwise.

The following filling radius estimate, which holds, for instance, for every closed manifold admitting a negatively curved Riemannian metric, cf. Example 3.2, is key to all the volume and volume entropy bounds in this article.

Theorem 4.5. Let M be a closed n-manifold with n ≥ 2. Consider a group homomorphism φ : π 1 (M ) → G to a discrete group G satisfying the property (P 0 ). Suppose that the classifying map Φ :

M → K(G, 1) induced by φ satisfies Φ * [M ] = 0 ∈ H n (G; k). Then every Riemannian metric on M satisfies FillRad(M ) ≥ 1 8 φ (M ).

In particular,

(1) vol(M ) ≥ c n 0 (M ) n where c n is an explicit positive constant depending only on n.

(2) For every R ≤ 0 (M ), there exists a ball B(R) of radius R in M with volB(R) ≥ a n R n where a n is an explicit positive constant depending only on n.

Proof. By definition of the filling radius, the fundamental class [M ] of M vanishes in the ρ-neighborhood U ρ (M ) of M in L ∞ (M ), where ρ > FillRad(M ). Therefore, there exists a continuous map σ : P → U ρ (M ) defined on an (n + 1)-dimensional simplicial complex P such that the restriction σ :

∂P → M represents [M ] in H n (M ; k), that is, σ * [∂P ] = [M ] ∈ H n (M ; k). (4.1)
Suppose that ρ < 1 8 φ (M ). We would like to construct a continuous map F : P → K which agrees with Φ • σ on ∂P . Combined with (4.1), this would yield the relation Φ *

[M ] = F * [∂P ]. Hence a contradiction with Φ * [M ] = 0.
Actually, it might not be possible to construct such a map F . Still, we will show how to adapt the arguments to obtain a contradiction.

Denote by P k the k-skeleton of P . Subdividing P if necessary, we can assume that the diameter of the images by σ of the simplices of P is less than ε > 0, with ε < 1 4 φ (M ) -2ρ. We first define a map f :

P 0 ∪ ∂P → M
with f|∂P = σ by sending each vertex p i ∈ P to a nearest point σ(p i ) in M , as we wish. Since the inclusion i : M → U ρ (M ) is isometric, every pair p i , p j of adjacent vertices of P satisfies

d M ( f (p i ), f (p j )) ≤ d L ∞ ( f (p i ), σ(p i )) + d L ∞ (σ(p i ), σ(p j )) + d L ∞ (σ(p j ), f (p j )) ≤ r with r = 2ρ + ε < 1 4 φ (M ).
We extend the map f to P 1 by taking the edges of P \ ∂P to minimizing segments in M joining the images of their endpoints, as we wish.

By construction, the boundary of every 2-simplex of P is sent by f to a loop of length at most 3r. If the length was less than the systole of M , the loop would be contractible in M . We could define a map P 2 ∪ ∂P → M extending f to the 2-simplices of P . The composition of this map with Φ could then be extended to a map F : P → K by asphericality of K leading to the desired contradiction. This is the original argument developed by Gromov in the proof of the systolic inequality on essential manifolds, cf. [START_REF] Gromov | Filling Riemannian manifolds[END_REF]. However, in our case, the boundary of some 2-simplex of P might be mapped to a noncontractible loop of M . Indeed, we only know that it is sent to a loop of M of length at most 3r, not necessarily shorter than the systole of M .

Consider the composite map

f = Φ • f : P 1 ∪ ∂P → K
Let us introduce the following definition. Given a k-simplex ∆ of P , we will say that ∆ extends well to K if the image f (∂∆ 2 ) of the boundary ∂∆ 2 of every 2-simplex ∆ 2 of ∆ is contractible in K. As noticed above, for such a simplex, the map f : P 1 ∪ ∂P → K extends to ∆ by asphericality of K.

For other simplices, we need to make use of the far-reaching consequences of the property (P 0 ) summarized in Proposition 3.6 and Remark 3.8. Specifically, we proceed as follows. Let ∆ be a 2-simplex of P which does not extend well to K, that is, f (∂∆) is homotopically nontrivial in K. The deformation retract of the loop space of K given by Proposition 3.6 and Remark 3.8 gives rise to a homotopy from f (∂∆) to a closed curve depending only on the nontrivial homotopy class of f (∂∆). This closed curve is the iterate of a primitive loop γ ∆ , namely the generator (up to reparametrization) of the unique maximal infinite cyclic subgroup containing the homotopy class of f (∂∆). In particular, the image f (∂∆) is freely homotopic to γ k ∆ for some integer k.

At this point, we want to emphasize that our construction strongly relies on the deformation retract of the loop space of K given by Proposition 3.6 and Remark 3.8. In particular, all the loops in a homotopically nontrivial class -here, the homotopy class of f (∂∆) -naturally deform to the same loop up to homotopy. This will allow us to extend the map f to the simplex ∆ in a way compatible with an extension to the 3-skeleton P 3 , and ultimately to the whole complex P .

Define the cellular complex

X = K ∪ ∆∈C D ∆
by gluing a 2-cell D ∆ along γ ∆ for every 2-simplex ∆ of P such that f (∂∆) is noncontractible in K. By construction, every loop f (∂∆) is contractible in X. More precisely, every loop f (∂∆) is contractible in

K ∆ ∪ D ∆
where K ∆ is the cover of K with fundamental group the subgroup of G generated by the homotopy class of γ ∆ . By definition of γ ∆ , this subgroup is isometric to Z. Thus, the space K ∆ has the same homotopy type as S 1 = K(Z; 1). It follows that the space K ∆ ∪ D ∆ is homotopy equivalent to D ∆ and so to a point. Therefore, the map

f : P 1 ∪ ∂P → K → X extends to f : P 2 ∪ ∂P → X.
Now, contrary to K or K ∆ ∪ D ∆ , the whole space X might not be aspherical (it is not difficult to construct such examples). Therefore, we need to rely on other arguments to show that f : P 2 ∪ ∂P → X extends to P .

Let ∆ 1 and ∆ 2 be two 2-simplices of P lying in the same (n + 1)-simplex. These two simplices can be connected by an edge e = [p 1 , p 2 ] of P (which may be empty if they have a vertex in common) whose endpoints p i are vertices of ∆ i . The edge e decomposes into two segments e 1 = [p 1 , p] and e 2 = [p 2 , p] with a common endpoint p such that the arcs f (e 1 ) and f (e 2 ) have the same length in M . By construction, the two loops f (e

i ) ∪ f (∂∆ i ) ∪ f (e i )
of M based at f (p) are of length at most 4r. Since 4r < φ (M ), the image by Φ of these two loops generate a cyclic subgroup in homotopy. In particular, if none of the loops f (∂∆ i ) is contractible in K, then γ ∆ 1 = γ ∆ 2 up to reparametrization. This observation is key for our argument.

Let ∆ 3 be a 3-simplex of P which does not extend well to K. As observed above, the loops γ ∆ 2 , and so the spaces K ∆ 2 ∪ D ∆ 2 , are the same for every 2-simplex ∆ 2 of ∆ 3 such that f (∂∆ 2 ) is noncontractible in K. We will denote them by γ ∆ 3 and K ∆ 3 ∪ D ∆ 3 , since they only depend on ∆ 3 and not on the 2-simplices ∆ 2 . By construction, the map f takes the boundary ∂∆ 3 of ∆ 3 to K ∆ 3 ∪ D ∆ 3 before projecting it to X. As the space K ∆ 3 ∪ D ∆ 3 is contractible, the map ∂∆ 3 → K ∆ 3 ∪ D ∆ 3 extends to ∆ 3 and gives rise to a map

f : ∆ 3 → K ∆ 3 ∪ D ∆ 3 → K ∪ D ∆ 3 → X
where the second map is given by the covering map K ∆ 3 → K. Note that the loop γ ∆ 3 along which the 2-cell D ∆ 3 is attached remains fixed under this covering map. This argument carries over by induction on the k-skeleton P k of P as follows. By induction, the map f : P k-1 ∪ ∂P → X extends to a map f : P k ∪ ∂P → X, where the restriction of f to every k-simplex ∆ k which does not extend well to K decomposes as

f : ∆ k → K ∆ k ∪ D ∆ k → K ∪ D ∆ k → X (4.2)
and the second map is given by the covering map K ∆ k → K. The key point to extend the map f : ∂∆ k+1 → X to ∆ k+1 is to observe that the contractible spaces K ∆ k ∪ D ∆ k are the same for every k-simplex ∆ k of ∆ k+1 which does not extend well to K, as previously noticed. We will denote this space by K ∆ k+1 ∪ D ∆ k+1 . Thus, the map

∂∆ k+1 → K ∆ k+1 ∪ D ∆ k+1
given by the first map of (4.2) extends to ∆ k , which concludes the proof of the induction.

In conclusion, the map

∂P σ -→ M Φ -→ K → X
extends to f : P → X. For n ≥ 3, the composite map M Φ -→ K → X induces a nontrivial homomorphism between H n (M ; k) and H n (X; k). This leads to a contradiction since f * [∂P ] agrees with the nonzero homology class Φ * [M ] in H n (M ; k) = H n (X; k). For n = 2, we assume that all the maps are simplicial. Let B be an open 2-cell in the 2-skeleton K 2 of K lying in the support of the homologically nontrivial 2-cycle given by the image of M under Φ. Consider the map K 2 → X 2 → S 2 = B/∂B defined by collapsing all the points of X 2 lying outside B to a single point. The pre-composition of this map with Φ induces a monomorphism between H 2 (M ; k) and H 2 (S 2 ; k). Hence a contradiction as above.

It follows that the closed Riemannian manifold M does not bound in U ρ (M ) with ρ < 1 8 φ (M ). Hence the inequality FillRad(M ) ≥ 1 8 φ (M ). Corollary 4.6. Let M be a closed n-manifold as in Theorem 4.5, where the group G satisfying the property (P δ ) for some δ > 0 (see for instance Example 3.2). Then every Riemannian metric on M satisfies vol(M ) Ent(M ) n ≥ c n δ n where c n is an explicit positive constant depending only on n.

Moreover, there exists an explicit positive constant a n depending only on n such that if the volume of every ball B(R) of radius R ≤ vol(M )

1 n in M satisfies volB(R) ≤ a n R n then Ent(M ) ≥ δ R .
Proof. By definition of φ (M ), there exist two loops γ 1 and γ 2 based at the same point of M , of length at most φ (M ), such that the images by φ of the homotopy classes α 1 = [γ 1 ] and α 2 = [γ 2 ] do not lie in the same infinite cyclic subgroup of G. From the property (P δ ) and Proposition 2.2.(1)-(3), we have δ ≤ Ent(φ(α 1 ), φ(α 2 )) ≤ Ent(α 1 , α 2 ) ≤ φ (M ) Ent(M, g).

Now, the desired result follows from a combination of this inequality with Theorem 4.5 and Theorem 1.3.

  Definition 2.1. Let d be a left-invariant metric on a group Γ = S generated by a finite set S. Denote by e the neutral element of Γ. The metric d induces a norm | • | on Γ defined as |α| = d(e, α) for every α ∈ Γ. Define the entropy of (Γ, | • |) as

  agrees with γ * ; • the loop c 2 : {0} × S 1 → BG represents α. In this case, the homotopy classes of c 1 and c 2 commute. That is, α ∈ C [γ * ] . Conversely, given a loop c whose homotopy class α commutes with [γ * ], we can construct a map S 1 × S 1 → BG such that S 1 × {0} → BG and {0} × S 1 → BG agree with γ * and c by contracting the commutator loop γ * cγ -1 * c -1 to the basepoint * . Therefore, the connected component Λ γ * of Λ(BG) containing γ * is an aspherical space with fundamental group the centralizer C [γ * ] .


Thierry A Mara
email: mara@univ-reunion.fr

Benjamin Belfort

Vincent Fontaine

Anis Younes

Addressing factors fixing setting from given data: A comparison of different methods

Keywords: factors fixing setting, variance-based sensitivity measure, density-based sensitivity measure, derivative-based global sensitivity measure, simulated drainage experiment . . . . . . . .

This paper deals with global sensitivity analysis of computer model output.

Given a set of independent input sample and associated model output vector with possibly the vector of output derivatives with respect to the input variables, we show that it is possible to evaluate the following global sensitivity measures: (i) the Sobol' indices, (ii) the Borgonovo's density-based sensitivity measure, and (iii) the derivative-based global sensitivity measure of Sobol' and Kucherenko. We compare the efficiency of the different methods to address factors fixing setting, an important issue in global sensitivity analysis.

First, global sensitivity analysis of the Ishigami function is performed with the different methods. Then, they are applied to two different responses of a soil drainage model. The results show that the polynomial chaos expansion for estimating Sobol' indices is the most efficient approach.

Introduction

Computer models are widely used to mimic systems and processes of different natures (e.g. physical, environmental or economical) because running a computer model is less expensive than performing an experimental set-up.

Computer models can help understanding the process under study and/or improving its management. However, the different sources of uncertainty in the computer models should be taken into account in order to avoid erroneous inferences. For this purpose, the functional inputs should be treated as stochastic processes and the uncertain scalar inputs as random variables defined by a joint probability density function (see [START_REF] Blatman | Adaptive sparse polynomial chaos expansion based on least angle regression[END_REF][START_REF] Anstett-Collin | Sensitivity analysis of complex models: Coping with dynamic and static inputs[END_REF]. As a consequence, computer model predictions are random quantities too.

Both uncertainty and sensitivity analyses are helpful for measuring the impact of the different sources of uncertainty in the model response of interest. [START_REF] Saltelli | Sensitivity analysis practices: Strategies for model-based inference[END_REF] note that the response of interest "should not be concerned with the model output per se, but with the question that the model has been called to answer." While uncertainty analysis aims at assessing the uncertainty of model output, global sensitivity analysis (GSA) aims at pinpointing the different sources of uncertainty of the model response uncertainty. There are two situations to consider when GSA is to be undertaken: the case of independent inputs and the case of dependent inputs. The latter can be challenging (see for instance, [START_REF] Kucherenko | Estimation of global sensitivity indices for models with dependent variables[END_REF][START_REF] Mara | Variance-based sensitivity indices for models with dependent inputs[END_REF][START_REF] Mara | Non-parametric methods for global sensitivity analysis of model output with dependent inputs[END_REF], depending on the question addressed (see some possible issues in [START_REF] Saltelli | On the relative importance of input factors in mathematical models: Safety assessment for nuclear waste disposal[END_REF]. Generally, prior to a thorough data collection and statistical calibration of the computer model, independence of the inputs can be reasonably assumed. In the present work, independent input scalar variables are considered.

Let y = f (x) be the scalar response of a computer model function of d independent random variables x = (x 1 , . . . , x d ) ∼ p 1 (x 1 )ו • •×p d (x d ). We also consider the vector of local sensitivities j = (j 1 , . . . , j d ) (with j i = ∂y/∂x i ) possibly provided by the computer model itself. We denote by X, a N × d matrix, a given sample of x. The vector of responses associated to X is denoted y and J is the matrix of partial derivatives (also called the Jacobian matrix) obtained after evaluating the computer model with each of the N independent draws. The availability of the Jacobian matrix is only compulsory for the assessment of the so-called derivative-based global sensitivity measure (see below). It is not mandatory for the two other sensitivity measures studied in this work.

Recent reviews on sensitivity analysis have focused primarily on methods employing specific sampling designs (e.g., [START_REF] Iooss | Revue sur l'analyse de sensibilité globale de modèles numériques[END_REF][START_REF] Saltelli | Update 1 of: Sensitivity analysis for chemical models[END_REF][START_REF] Neumann | Comparison of sensitivity analysis methods for pollution degradation modelling: A case study from drinking water treatment[END_REF][START_REF] Norton | An introduction to sensitivity assessment of simulation models[END_REF]Borgonovo and Plischke, 2015, among others). In the present article, we address how to pinpoint the relevant input variables from a given set of data (X, y, J). Our aim is to demonstrate that, with a single sample of size N (say, thousands), several global sensitivity indices can be evaluated. Three global sensitivity measures are studied, namely: (i) the Sobol' indices [START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF][START_REF] Homma | Importance measures in global sensitivity analysis of nonlinear models[END_REF], (ii) the sensitivity measure of [START_REF] Borgonovo | Measuring uncertainty importance: Investigation and comparison of alternative approaches[END_REF], and (iii) the derivative-based global sensitivity measure [START_REF] Lamboni | Derivative-based global sensitivity measures: general links with Sobol' indices and numerical tests[END_REF]. These sensitivity indices are of different natures and may lead to different inferences about the model input importance.

The choice of the sensitivity index to compute and the model output to consider depends on the question that the model has been called to answer [START_REF] Saltelli | Sensitivity analysis in practice[END_REF]. In this work, we address the factors fixing setting (henceforth, FF setting) with the different sensitivity measures. [START_REF] Saltelli | Sensitivity analysis practices: Strategies for model-based inference[END_REF] define this setting as follows: "The objective of this setting, which could also be labelled as screening, is to identify the input factor or the subset of input factors that we can fix at any given value over their range of uncertainty without reducing significantly the output variance." One could even require " . . . without changing significantly the predicted output probability density function."

There are many sensitivity measures proposed in the literature to address the FF setting. Compared to those studied in the present work, some of them are of higher-order (e.g., [START_REF] Owen | Higher order Sobol' indices[END_REF][START_REF] Gamboa | Sensitivity analysis based on cramer von mises distance[END_REF]. In our study, we focus on the three aforementioned sensitivity indices because they are of different natures (i.e. variance-based, density-based and derivativebased sensitivity measures respectively) and can be assessed from a single sample. [START_REF] Plischke | Global sensitivity measures from given data[END_REF] define other possible moment-independent importance measures which can be estimated from given data. Their estimation is based on a method similar to the one described in this paper for assessing Borgonovo's importance measure (see Section 2.2). Therefore, we only consider the Borgonovo's importance measure in our study.

In the present work, we do not consider GSA methods based on surrogate models that are run to assess the sensitivity indices. The reasons are i) there is a plethora of different emulators, and it is not the point here to discuss their merits and flaws and ii) some emulators can require specific experimental designs for the sake of efficiency (e.g. the Gaussian emulator [START_REF] Oakley | Probabilistic sensitivity analysis of complex models: a Bayesian approach[END_REF], while it is assumed here that the sample is arbitrary (though independent). It can be notified that the polynomial chaos expansion method described in Section 2.1 is a surrogate-based approach. However, we argue that this approach employs ANOVA-like decomposition and as such is classified amongst the spectral methods (such as the Fourier amplitude sensitivity test, [START_REF] Cukier | Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I. theory[END_REF][START_REF] Schaibly | Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients[END_REF][START_REF] Saltelli | A quantitative model independent method for global sensitivity analysis of model output[END_REF]. Indeed, as detailed later (see Section 2.1), the variance-based sensitivity indices are directly obtained from the regression coefficients without needing to run the surrogate model.

The paper is organized as follows: In Section 2 we discuss the global SA methods based on the Sobol' indices (Section 2.1), the density-based importance measure of Borgonovo (Section 2.2), and the derivative-based global sensitivity measure (Section 2.3). First, their performance for addressing factors fixing setting of a well-known non-linear function is studied in Section 3.

Then, the different methods are applied to perform GSA of a soil drainage experiment in Section 4. Finally, a summary and conclusions are given in Section 5.

Global sensitivity analysis

Variance-based sensitivity measures

Variance-based sensitivity indices (also called Sobol' indices) are the most popular importance measures for GSA. By computing all Sobol' indices (there are 2 d -1 indices at most), one can guess the structure of the input/output relationship. In practice, two variance-based sensitivity indices are particularly interesting: the first-order sensitivity index S i (also called correlation ratio by [START_REF] Pearson | On the general theory of skew correlation and non-linear regression[END_REF][START_REF] Mckay | Variance-based methods for assessing uncertainty importance[END_REF] and the total sensitivity index ST i (first defined in [START_REF] Homma | Importance measures in global sensitivity analysis of nonlinear models[END_REF]. They are defined as follows: S i measures the amount of variance of y due to the uncertainty of x i alone, while ST i ≥ S i measures the amount of all contributions of x i to the variance of y, including its cooperative non-linear contributions with the other variables gathered in x ∼i . The higher ST i is, the more x i is an important input for the variance of y. The input/output relationship is additive when S i = ST i , ∀i = 1, . . . , d and, in this case, d i=1 S i = 1. Sobol' indices were introduced in the framework of ANOVA-HDMR (acronym for analysis of variance -high dimensional model representation, see [START_REF] Rabitz | Efficient input-output model representations[END_REF]. The latter relies on the existence of a unique ANOVA representation of the input/output relationship when model inputs are independent (otherwise, such a representation is not unique, see for instance [START_REF] Mara | Non-parametric methods for global sensitivity analysis of model output with dependent inputs[END_REF]. Thus, the 2 d -1 Sobol' indices reflect the structure of the model response. This is one of the appealing features of Sobol' indices. Indeed, (1 -d i=1 S i ) represents the amount of variance due to interactions between the input variables.

S i = V [E [y|x i ]] V [y] ∈ [0, 1] (1a)
ST i = E [V [y|x ∼i ]] V [y] ∈ [0, 1] (1b
There are several computationally affordable methods proposed in the literature to compute the set of first-order sensitivity indices [START_REF] Oakley | Probabilistic sensitivity analysis of complex models: a Bayesian approach[END_REF][START_REF] Tarantola | Random balance designs for the estimation of first-order global sensitivity indices[END_REF][START_REF] Ratto | State dependent parameter metamodelling and sensitivity analysis[END_REF][START_REF] Lewandowski | Sample-based estimation of correlation ratio with polynomial approximation[END_REF][START_REF] Mara | Comparison of some efficient methods to evaluate the main effect of computer model factors[END_REF]Plischke, 2010, among others). However, it is much more expensive to compute the entire set of total sensitivity indices. From given data, this task can be achieved with sparse polynomial chaos expansions (Blatman andSudret, 2010, 2011;[START_REF] Fajraoui | Reactive transport parameter estimation and global sensitivity analysis using sparse polynomial chaos expansion[END_REF][START_REF] Shao | Bayesian sparse polynomial chaos expansion for global sensitivity analysis[END_REF].

ANOVA-HDMR presumes that f (x) is square-integrable. This means that f ∈ H, where H is a Hilbert space equipped with the following inner-product:

< f, g > H = E [f (x)g(x)]
(2)

with g ∈ H. Polynomial chaos expansion (PCE) is a spectral method that casts the continuous scalar model response onto orthonormal polynomials as follows:

f (x) = α⊂N d a α Ψ α (x) (3)
where α = α 1 α 2 . . . α d ∈ N d is a d-dimensional index and the a α s are the PCE coefficients to be determined. The Ψ α s are the multidimensional polynomial basis elements of degree |α| = d i=1 α i . They form a complete orthonormal basis in H. For some specific random variables, they are derived from the Askey scheme (see [START_REF] Askey | Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials[END_REF][START_REF] Xiu | The Wiener-Askey polynomial chaos for stochastic differential equations[END_REF]. In the case of a vector of arbitrarily distributed variables x, one can either resort to isoprobabilistic transformations (see [START_REF] Sudret | Global sensitivity analysis using polynomial chaos expansions[END_REF] or build-up the orthonormal polynomials with the data-driven method of [START_REF] Oladyshkin | Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion[END_REF] or alternatively the stable procedure of Stieltjes (see [START_REF] Gautschi | On generating orthogonal polynomials[END_REF][START_REF] Gautschi | Is the recurrence relation for orthogonal polynomials always stable?[END_REF]. Because the numerical exercise in Section 3 and the application in Section 4 deal with parameters uniformly distributed, throughout this paper Legendre polynomials are employed in expansion (3).

The multidimensional polynomial Ψ α is simply the tensor product of univariate polynomials,

Ψ α (x) = d i=1 ψ α i (x i ) with ψ 0 = 1.
Due to the orthonormality of the polynomial basis elements, the total variance of f (x) reduces to the sum of squares of the PCE coefficients,

V [y] = +∞ |α|>0 a 2 α (4)
and the conditional expectation of y onto x i , also called x i 's marginal effect, writes

E [y|x i ] = a 0...0 + +∞ α i >0 a 0...α i ...0 ψ α i (x i ).
(5)

For practical reasons, the previous expansions are truncated up to a prescribed degree p. The resulting PCE coefficients are then evaluated by using a regression method yielding

â = argmin a    N k=1   y k - |α|≤p a α Ψ α (X k1 , . . . , X kd )   2    ( 6
)
where â is the vector of estimated coefficients and (X k1 , . . . , X kd ) is the k-th draw of the input vector.

The first-order and total sensitivity indices of x i are then inferred as follows:

ŜP CE i = p α i >0 â2 0...α i ...0 p |α|>0 â2 α (7a) ŜT P CE i = p α:α i >0 â2 α p |α|>0 â2 α . (7b)
Thus, evaluating the variance-based sensitivity indices with PCE amounts to estimating the polynomial coefficients. For a PCE of degree p, there are (p + d)!/p!d! coefficients to compute. However, most of these coefficients are not significant regarding the total variance. Hence, it is advisable to discard the irrelevant terms in the estimated truncated PCE.

This leads to the so-called sparse PCE [START_REF] Blatman | Efficient computation of global sensitivity indices using sparse polynomial chaos expansions[END_REF]. There are several algorithms proposed in the literature to build-up a sparse PCE (e.g. [START_REF] Blatman | Adaptive sparse polynomial chaos expansion based on least angle regression[END_REF][START_REF] Fajraoui | Reactive transport parameter estimation and global sensitivity analysis using sparse polynomial chaos expansion[END_REF]. The one proposed in [START_REF] Shao | Bayesian sparse polynomial chaos expansion for global sensitivity analysis[END_REF] [START_REF] Marelli | UQLab: A framework for uncertainty quantification in Matlab[END_REF].

Density-based sensitivity measure

The development of density-based sensitivity measures was motivated by the fact that variance-based sensitivity measures only reveal the relevant inputs for the output variance, which is only the second-order moment of the probability density function of the output. Hence, some authors have advocated the use of moment-independent importance measure [START_REF] Chun | An uncertainty importance measure using distance metric for the change in a cumulative distribution function[END_REF][START_REF] Borgonovo | Measuring uncertainty importance: Investigation and comparison of alternative approaches[END_REF][START_REF] Borgonovo | A new uncertainty importance measure[END_REF]. Several moment-independent sensitivity measures have been proposed in recent literature (some of them are described in [START_REF] Plischke | Global sensitivity measures from given data[END_REF][START_REF] Borgonovo | A common rationale for global sensitivity measures and their estimation[END_REF]. In the present work, we consider the density-based sensitivity measure introduced by [START_REF] Borgonovo | Measuring uncertainty importance: Investigation and comparison of alternative approaches[END_REF].

The sensitivity index of Borgonovo measures the importance of the input variable over the entire pdf and is defined as follows:

δ i = 1 2 E [s(x i )] ∈ [0, 1] (8) with, s(x i ) = R |p y (y) -p y|x i (y|x i )|dy (9)
where p y (y) is the probability density function (pdf) of y and p y|x i (y|x i ) is the conditional pdf of y onto x i . It is worth noting that s(x i ) measures the area between p y and p y|x i (see Figure 1) so that the latter is null if p y|x i = p y .

Eq. ( 8) simply computes the half expectation of s(x i ) with respect to x i . If δ i = 0 then fixing x i to any value in its uncertainty range does not change the predicted pdf. Borgonovo importance measure (BIM) is a strong measure of statistical dependence between two random variables [START_REF] Strong | An efficient method for computing partial expected value of perfect information[END_REF]. As opposed to Sobol' indices, BIM does not require the existence of an HDMR.

y p y|x i p y s(x i )
Thus, because δ i only measures the pairwise dependency between (x i , y), its definition is somehow simpler than ST i . Meanwhile, as shown in the following numerical exercises, accurate estimate of BIM may require a large simple size.

But, as also shown, rough estimate of the BIMs at modest sample size (say one thousand) may be sufficient for detecting irrelevant inputs. This is a desired feature for screening purposes of high-dimensional computer models.

Up to 2013, the use of BIM for GSA of computer models was hampered by its computational cost. Indeed, it can be noticed that, for each input, a double loop is necessary to assess its density-based sensitivity measue (Eqs.

(8-9)). In the first loop, the current variable x i is set to prescribed values, while in the second loop several draws of the remaining inputs x ∼i are generated. For each prescribed value, an output vector is obtained from which the conditional pdf (onto x i ) can be inferred. However, by noticing that setting

x i to different values is almost the same as varying x i within a narrow range, [START_REF] Plischke | Global sensitivity measures from given data[END_REF] proposed a cheaper computational method to estimate the BIM of each input variable with only one single sample (i.e. one single loop). Therefore, the proposed approach alleviates the so-called curse of dimensionality. The principle of the approach is explained hereafter.

In the approach proposed by Plischke et al. ( 2013), the draws of x i denoted {X ki , k = 1, . . . , N} are partitioned into M subsamples of equal size

(X 1 i , . . . , X M i ).
It must be ensured that there are enough draws in each partition for an accurate estimation of the conditional pdf (say a hundred).

In the case of equiprobable partitions, the number of draws per partition is P = N/M. Such a partitioning can be easily understood with a pairwise scatterplot (see the top plot of Figure 2). Then, a rough estimation of s(x i ) (see Eq. ( 9)) for

x i = x1 i = P k=1 X 1 ki /P is, ŝ(x 1 i ) = Ωy |p y (y) -py|x i (y|x i ∈ X 1 i )|dy (10)
where Ω y denotes the range of variation of y.

Numerically, Eq. ( 10) is computed by comparing the estimated pdf of y (i.e. py ) with the one of y 1 i (i.e. py|x i for x i = x1 i ). This is illustrated in the bottom plot of Figure 2 for each partition. The smoothing kernel-density approach [START_REF] Parzen | On estimation of a probability density function and mode[END_REF] is used by the authors to assess the pdfs from the sample and subsamples. In order to reduce bias in the estimate of the BIM, it must be ensured that py (y) and py|x i (y|x i = x1 i ) are significantly different. For this purpose, the Kolmogrov-Smirnov's test is employed [START_REF] Press | Numerical Recipes: The Art of Scientific Computing[END_REF].

Finally, the calculation of δ i is straightforward because the partitioning is equiprobable, and the result is

δi = 1 2M M j=1 ŝ(x j i ). (11)
The choice of M is one of the degrees of freedom of the proposed approach.

Given that each subsample must have a sufficient number of draws (say hundreds) to accurately infer the conditional pdf, the choice of M depends on the sample size N. In Section 3, we numerically show that, for small sample sizes, BIM estimates are not accurate but can allow for detecting irrelevant input variables. In the present work, the BIMs are estimated with the program developed by E. Plischke and available on the website of the Joint Research Center (https://ec.europa.eu/jrc/en/samo/simlab).

Derivative-based global sensitivity measure

Until the end of the last century, local sensitivity indices were intensively used to perform sensitivity analysis of a given model response y = f (x).

These indices rely on partial derivatives of the model response evaluated at some point x * . This point generally reflects the analyst's best knowledge about the input values. The success of local SA is partially explained by the usefulness of partial derivatives in model calibration and the existence of efficient computational methods to assess the sensitivity matrix (e.g. automatic differentiation, adjoint method [START_REF] Cacuci | Sensitivity theory for nonlinear systems. I. nonlinear functional analysis approach[END_REF][START_REF] Griewank | On automatic differentiation[END_REF][START_REF] Navon | Practical and theoretical aspect of adjoint parameter estimation and identifiability in meteorology and oceanography[END_REF].

The local sensitivity index of x i at x * is defined as follows:

LS i = x * i ∂y ∂x i x * . ( 12
)
The higher is LS i , the more x i is locally relevant.

Local sensitivity analysis suffers at least from two drawbacks: (i) the result is local and will differ from one point to another for non-linear functions and (ii) it does not capture the possible interactions amongst the inputs.

To circumvent these drawbacks, a derivative-based global sensitivity measure (DGSM) has been defined, thanks to the recent work of Lamboni et al.

(2013) and the earlier works of Sobol' andKucherenko (2009, 2010). DGSM is inferred by evaluating the partial derivatives at several points in the input space. Indeed, in [START_REF] Lamboni | Derivative-based global sensitivity measures: general links with Sobol' indices and numerical tests[END_REF] the authors define the following statistic:

ν i = R d ∂y ∂x i 2 d j=1 p j (x j )dx j . (13)
The authors then show that,

ST i ≤ 4C 2 i ν i V [y] (14)
where C i is called the Cheeger constant defined as follows:

C i = sup x i ∈R min (F i (x i ), 1 -F i (x i )) p i (x i ) (15)
with F i (x i ) the cumulative distribution function of x i (i.e. p i (x i ) = dF i dx i ). The Cheeger constant has been defined by Sobol' andKucherenko (2009, 2010) for the uniform distribution

p i (x i ) = U(a i , b i ) as C i = (b i -a i ) 2π and for the normal distribution p i (x i ) = N (µ i , σ 2 i ) as C i = σ i 2 .
This constant can be computed for any distribution that belongs to the class of Boltzmann probability measure (see [START_REF] Lamboni | Derivative-based global sensitivity measures: general links with Sobol' indices and numerical tests[END_REF], for more details).

From Eq. ( 14), it can be inferred that if the DGSM

Υ i = 4C 2 i ν i V[y]
is small, x i is an irrelevant input. Finally, we note that from a given (quasi) Monte Carlo sample of partial derivatives matrix J, ν i is estimated as follows:

νi = 1 N N k=1 J 2 ki . ( 16
)
It is then straightforward to get a (quasi) Monte Carlo estimate of Υ i .

GSA setting

Comparing different importance measures can be misleading because "importance" is not a mathematical concept [START_REF] Saltelli | Sensitivity analysis in practice[END_REF]. [START_REF] Saltelli | Sensitivity analysis practices: Strategies for model-based inference[END_REF] advice to define "importance" at the stage of framing the analysis.

We note that DGSM is particularly suited for screening purposes, which consists of investigating unessential inputs. Hence, in the present work, we study the ability of the different methods to pinpoint the irrelevant inputs of a given model response. This is known as the factors fixing setting [START_REF] Saltelli | On the relative importance of input factors in mathematical models: Safety assessment for nuclear waste disposal[END_REF].

We particularly investigate whether the different methods are subject to Type I error (i.e. failing at identifying irrelevant inputs) or Type II error (i.e.

identifying influential inputs as irrelevant ones). It is expected that the risk of error is high for small sample sizes. We can also expect that performing FF setting with DGSM might be subject to Type I error because it is an upper bound for the total sensitivity index (see Eq. ( 14)). Thereby, it is only informative when its value is very small (for irrelevant inputs).

Numerical test: the Ishigami function

GSA is performed on the Ishigami function defined as,

f (x 1 , x 2 , x 3 , x 4 ) = sin(x 1 ) + 7 sin 2 (x 2 ) + 0.1x 4 3 sin(x 1 ) + ǫx 4 (17)
with the input variables uniformly distributed as follows x i ∼ U(-π, π) and ǫ = 10 -6 . This function is non-monotonic and includes a strong interaction between x 1 and x 3 . Regarding FF setting, the first three input variables are important while x 4 is a non important parameter introduced in the analysis to make sure that the different methods are able to identify it as an irrelevant input.

The analysis is performed with quasi-Monte Carlo (QMC) sequences of different sizes, N = {64, 128, 256, 512, 1 024, 2 048, 4 096, 8 192, 16 384}. For this purpose, we used the MATLAB program LPtau51.m developed by [START_REF] Sobol | Quasi-random sequence generator (routine LPTAU51)[END_REF] and downloadable on the website of the Joint Research Center (see the link before § 2.3). BIM cannot be computed for small sample sizes because one must ensure that enough draws are contained in each subsample. We assign the following number of partitions to each sample M = {0, 0, 2, 4, 8, 16, 16, 16, 16, 32}, with M = 0 meaning that BIM cannot be assessed. To estimate the DGSMs, the partial derivatives evaluated at the different QMC draws are also computed, namely,

j 1 (x) = (1 + 0.1x 4 3 ) cos(x 1 ) (18)
j 2 (x) = 14 cos(x 2 ) sin(x 2 ) (19)

j 3 (x) = 0.4x 3 3 sin(x 1 ) (20) j 4 (x) = ǫ (21)
The estimated sensitivity measures are plotted in Figure 3 versus log 2 (N) = 6, . . . , 14. The results are compared with the analytical values. The latter are exactly calculated for the Sobol' indices. As far as the two other sensitivity measures are concerned, they are estimated from a QMC sample of size N = 2 19 . For the DGSM, a total of M = 64 partitions are selected which is much higher than what is recommended in practice (see [START_REF] Plischke | Global sensitivity measures from given data[END_REF]. This choice is done to get accurate estimates. The first-order sensitivity indices indicate that x 2 has the higher first-order effect while the one of x 3 is zero. This means that it is expected that by fixing x 2 one has more chance to reduce the output variance than fixing

x 3 [START_REF] Saltelli | On the relative importance of input factors in mathematical models: Safety assessment for nuclear waste disposal[END_REF]. The total sensitivity indices indicate that (x 1 , x 2 , x 3 ) are important and that x 4 is irrelevant.

It is worth mentioning that for N = 16 384, a 16-th degree polynomial has been identified with 26 terms in the expansion. A very sparse PCE is obtained due to i) the specific algorithm used (see [START_REF] Shao | Bayesian sparse polynomial chaos expansion for global sensitivity analysis[END_REF] and ii) the features of the Ishigami function. Indeed, the Ishigami function is odd with respect to x 1 , even with respect to x 2 and the interaction term between

x 1 and x 3 is the product of an even polynomial function of x 3 and an odd function of x 1 . The identified sparse PCE inherits from these features and contains only odd degree monomials of x 1 , even degree monomials of x 2 and products of odd degree monomials of x 1 with even degree monomials of x 3 .

Figure 3c depicts the BIM estimates. We can note that the estimations have not completely converged even after N ≥ 16 384, especially for δ 3 and δ 2 . This difference of convergence from one input to another seems to indicate that the partitioning should be adapted to the input variable for an overall better convergence of the estimator. At low sample sizes (i.e. N ≤ 512) the approach is not able to successfully identify the three first inputs as important (Type II error). Although the convergence is slow, when the sample size is sufficiently large (here from N = 1 024) the estimator of [START_REF] Plischke | Global sensitivity measures from given data[END_REF] is able to detect the relevant inputs.

Regarding the DGSMs, Figure 3d shows that the estimates are quite stable from N ≥ 256. We note that ( Υ1 , Υ2 , Υ3 ) are greater than one which indicates that (x 1 , x 2 , x 3 ) cannot be deemed as irrelevant. Note that the fact Υi can be greater than one contrasts with the two other sensitivity measures scaled within [0, 1]. Anyway, with the DGSM only x 4 is identified as non important ( Υ4 = 0).

For this exercise all the estimated sensitivity measures are able to detect the irrelevant input x 4 . PCE-based total sensitivity indices and DGSMs require less computational effort (i.e. draws) to effectively identify the irrelevant inputs. Note that because the sensitivity measures are based on different definitions, they can yield different rankings of importance. Indeed, accord-ing to Figure 3, the input vector ranked by order of importance is (x 1 , x 2 , x 3 )

with the total sensitivity index, (x 2 , x 1 , x 3 ) with BIM and (x 2 , x 3 , x 1 ) with DGSM. To simulate the unsaturated flow, the Mualem-van Genuchten (MvG) retention curve is chosen to relate the hydraulic conductivity of the unsaturated soil to its water content [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Van Genuchten | A closed form equation for predicting the hydraulic properties of unsaturated soils[END_REF]. This retention curve is defined as follows:

K = k s • S λ e 1 -1 -S 1/m e m 2 (22)
where,

S e = ω -ω r ω s -ω r =      1 (1 + (α|h|) n ) m h < 0 1 h ≥ 0 (23)
with m = 1 -1/n. The effective saturation S e links the pressure head h [cm]

to the water content ω [cm 3 .cm -3 ] at any location in the soil.

The soil hydraulic parameters are: the saturated hydraulic conductivity k s [m.s -1 ], the saturated water content ω s [cm 3 .cm -3 ], the residual water content ω r [cm 3 .cm -3 ] as well as the MvG parameters α [m -1 ], n [-] and λ

[-]. Hence, the random input vector is x = (k s , ω r , ω s , α, n, λ).

The state variables (i.e. h(t) and ω(t)) are solution of the Richard's equation, that is,

∂ω ∂t = ∂ ∂z K(h) ∂h ∂z -1 , (24)
where t [min] is time and z [cm] is the vertical coordinate (positive downward). This partial differential equation is solved with a standard Galerkin finite element method in conjunction with the Newton linearization method associated with the primary variable switching method [START_REF] Diersch | On the primary variable switching technique for simulating unsaturated-saturated flows[END_REF][START_REF] Hayek | Adaptive multi-scale parameterization for one-dimensional flow in unsaturated porous media[END_REF].

For this numerical exercise we consider a cylindrical cell, 8.5 cm in diameter and 6 cm in height, filled with a soil. The parameters of the soil are assumed to be uniformly distributed over the ranges reported in Table 1.

The soil, initially saturated with water, is drained by imposing a negative pressure head of -700 cm in the bottom of the cell. The responses of interest are the pressure head h at the center of the column, as well as the average soil water content ω avg in the column after a simulation period of 250 min. In addition to these responses, the numerical model provides the partial derivatives of these outputs with respect to the hydraulic parameters.

Hence, there are four scalar responses computed by the numerical model.

Preliminary studies showed that at least one thousand draws were necessary to obtain exploitable results with BIM. This finding is in agreement with the results of the Ishigami function (see the previous Section). Therefore, the computer model responses are evaluated for N = 1 024 different QMC draws.

GSA of the pressure head

First, let us analyse the results for the pressure head. ). This is one of the appealing features of variance-based sensitivity indices. Indeed, 6 i=1 Ŝi = 0.86 < 1 means that approximately 14% of the response variance is due to interactions amongst the input variables. According to the total sensitivity indices, the most important parameter is k s , followed by n and α. The same ranking is observed when using first-order or total sensitivity indices. The contribution of the parameters ω r , ω s and λ to the total variance is insignificant (less than 2%).

The marginal effects of the hydraulic parameters onto the pressure head are depicted in Figure 4. They highlight the non-linear effects of k s and n and the linear effect of α. This figure also indicates that k s and n have a monotonic decreasing effect onto the pressure head while α has an increasing effect. This means that increasing the value of k s alone (resp. α) tends to decrease (resp. increase) the predicted value of the pressure head.

The density-based sensitivity measures are evaluated using a small number of partitions (M = 4), which provides subsamples of size P = 256. The results in Table 1 show that the density-based sensitivity measures are able to detect the relevant inputs despite the small number of partitions. However, the ranking is different from that obtained with the Sobol' indices since the BIMs indicate that α is the most important parameter while n is the least important one. It is worth noting that 6 i=1 δi = 0.7 < 1, but in contrast with the first-order sensitivity indices, it does not mean that 30% of the output pdf is left unexplained by the BIMs. This is one drawback of Borgonovo's importance measure compared to the variance-based sensitivity measures, it does not provide any information about the model structure.

Both the indices ST i (Eq. ( 1b)) and δ i (Eq. ( 8)) have the advantage of measuring the importance of the input variable regardless the nature of the model (linear or non-linear, with continuous values or not, etc.). However, with the variance-based measures, it is assumed that the response uncertainty is represented by its variance, while the density-based measure is momentindependent. Hence, the two sensitivity measures can provide different rankings regarding the order of importance of the input variables. In our opinion, ST i measures the importance of x i in the model structure while δ i measures its importance for p y . Regarding the objective of the study, that is, detecting the irrelevant inputs (FF setting), the two methods provide the same results: ω r , ω s and λ are not important parameters for predicting the pressure head.

The derivative-based global sensitivity indices are reported on the last row of Table 1. Obviously, these indices can be greater than one and in this case are not very informative. For instance, the DGSM of α ( Υα = 0.36) should not be compared to that of n ( Υn = 42.8) because DGSM is useful for screening the inputs and not for their ranking by order of importance.

As expected, the DGSM is systematically greater than the respective total sensitivity index. This confirms that the DGSM is an upper bound for the total sensitivity index, that is, ST i ≤ Υ i . For the studied test case, it can be inferred that ω r , ω s and λ are irrelevant parameters for the predictive uncertainty of the pressure head. A dummy parameter is included to assess the significance of the estimated statistics (last column).

Parameter k s ω r ω s α n λ dummy [cm/min] [cm 3 /cm 3 ] [cm 3 /cm 3 ] [cm -1 ] [-] [-] [-] Range [0.

GSA of the average water content

The sensitivity indices for the average water content are gathered in Ta- Contrary to the results for the pressure head, the variance and density based sensitivity rankings are in good agreement for the average water content. The ranking of importance is the same with the first-order sensitivity indices, the total sensitivity indices and the density-based sensitivity indices.

The ranking of importance regarding the variance of the response indicates that the most important parameter is n, followed by ω r , then α and ω s .

Finally, the results of the DGSM must be interpreted with caution. As stated earlier, the DGSM cannot be used to rank the input variables by order of importance. The DGSM results in Table 2 show that the parameter k s is not considered to be irrelevant as found by all the other indicators. This result can be explained by analysing the scatterplot of the partial derivatives of the average water content with respect to k s in Figure 6. This scatterplot depicts high local sensitivities in a very small region near the smallest value of k s . However, this result does not mean that k s is globally a relevant parameter because the DGSM is an upper bound for the total sensitivity index which indicates that k s is irrelevant for the average water content (explaining only 1% of the variance). This result confirms that FF setting performed with the DGSM can be subject to Type I error.

Parameter k s ω r ω s α n λ dummy [cm/min] [cm 3 /cm 3 ] [cm 3 /cm 3 ] [cm -1 ] [-] [-] [-] Range [0.

Summary and conclusion

In this paper, we discussed and tested three global sensitivity analysis methods that can be performed using only a single independent sample of modest size (∼ a thousand). The GSA methods allow to compute the follow- Variance-based sensitivity indices (also called Sobol' indices) are model free. They measure the amount of the model response variance due to each input variable alone and in interaction with the other variables. In the present work, they were computed by using the Bayesian sparse polynomial chaos expansion method developed by [START_REF] Shao | Bayesian sparse polynomial chaos expansion for global sensitivity analysis[END_REF]. The results showed that this method provided reliable results at low computational cost. Moreover, PCE supplied quantitative insights into the input/output relationship.

Contrary to the Sobol' indices, the density-based sensitivity measure of Borgonovo is moment-independent. The BIMs were computed with the estimator proposed in [START_REF] Plischke | Global sensitivity measures from given data[END_REF]. This estimator showed some prob- bound for the total sensitivity index. Thus, it cannot help in ranking the input variables by order of importance. However, the method is interesting for screening purposes that involve the investigation of unessential inputs (i.e. FF setting). However, our results proved that DGSM could be subject to Type I error, that is, identifying irrelevant inputs as possible relevant ones.

In conclusion, GSA of computer models can be performed using different methods and free programs are often available for their computation when the inputs are independent. Several different global sensitivity measures can be computed from given data. In practice, the choice of the sensitivity measure to compute depends on the objective of the analysis. For the present work, it was found that sparse PCE was the most efficient method for addressing factors fixing setting.

  ) where x ∼i = x\x i . E [•] and V [•] represent the expectation and variance operators, respectively. E [•|•] and V [•|•] represent the conditional expectation and conditional variance operators, respectively.

Figure 1 :

 1 Figure 1: In blue, the unconditional predicted pdf (p y ) is represented. The red curve is the conditional predicted pdf (p y|xi ). The shaded area s(x i ) measures the shift between the two densities.

Figure 2 :

 2 Figure 2: On the top, the scatterplot of the model response y versus the input variablex i . The draws are partitioned into M = 6 subsamples of equal size (here x i is uniformly distributed). On the bottom, the pdf of the total draws of y (blue continuous curve) is compared to the conditional pdf of the draws localized in the partitions (red broken line).

Figure 3a -

 3a Figure 3a-b show that the variance-based sensitivity estimates are virtually constant. Actually, the PCE-based estimator have converged toward the analytical values proving that even for N = 64 the sparse PCEs are good approximations of the original function. Such a fast convergence is due to the fact that the Ishigami function is very smooth (infinitely derivable w.r.t. all the input variables). Note that, because the input variables are uniformly distributed, the sparse PCE was built with Legendre polynomials.

Figure 3 :

 3 Figure 3: GSA of the Ishigami function -estimated sensitivity indices versus sample size. The dashed lines are the reference values.

ble 2 .Figure 4 :

 24 Figure 4: Marginal effects of the hydraulic parameters for the predicted pressure head. The red curve is the estimated conditional expectation (i.e. E [y|x i ]) provided by the sparse PCE.

Figure 5 :

 5 Figure 5: Marginal effects of the hydraulic parameters for the average water content. The red curve is the estimated conditional expectation (i.e. E [y|x i ]) provided by the sparse PCE.

Figure 6 :

 6 Figure 6: Partial derivatives of the average water content w.r.t. the hydraulic parameters.

  is used in the present work. The approach developed in[START_REF] Shao | Bayesian sparse polynomial chaos expansion for global sensitivity analysis[END_REF] generally provides very sparse truncated PCEs because of the specific algorithm employed. The latter combines a model selection criterion with informative prior to infer the optimal sparse PCE in a Bayesian framework. The prior is chosen in order to favour low degrees and low interaction terms. The MATLAB programs are available upon request from the

first author of the present paper. Another alternative is to use UQLab, the MATLAB toolbox developed by

  The different sensitivity measure estimates are compiled in Table1. To assess the significance level of the sensitivity estimates, a dummy parameter has been included in the analysis. Its sensitivity indices were computed 1 000 times, each time by assigning different random draws to the dummy parameter. The ranges of variation (minimal value-maximal value) of its estimated sensitivity indices are reported on the last column of Table1. We note that, all methods are in relative good agreement and indicate that (k s , α, n) are the most important parameters.A sparse PCE containing only 54 terms in an 11th-degree polynomial expansion has been built for the computation of the Sobol' indices. Comparison of first-order sensitivity indices with total sensitivity indices in Table1indicate that there are non-negligible interactions amongst the relevant parame-

	P CE i	> ŜP CE i

ters (k s , α, n) (for these variables ŜT

Table 1 :

 1 GSA of the drainage model -estimated sensitivity indices for the pressure head.

			01,0.5] [0.01,0.20] [0.40,0.45] [0.005,0.02] [1.0,1.4] [-0.5,1.0]	[-]
	ŜP CE i	0.39	0.01	0.00	0.16	0.29	0.01	[0.0,0.0]
	ŜT	P CE i	0.50	0.01	0.00	0.28	0.34	0.02	[0.0,0.0]
		δi	0.23	0.06	0.00	0.26	0.10	0.05	[0.0,0.08]
	Υi	6.94	0.02	0.00	0.36	42.8	0.02	[0.0,0.0]

Table 2 :

 2 GSA of the drainage model -estimated sensitivity indices for the average water content.

			01,0.5] [0.01,0.20] [0.40,0.45] [0.005,0.02] [1.0,1.4] [-0.5,1.0]	[-]
	ŜP CE i	0.01	0.09	0.03	0.04	0.80	0.0	[0.0,0.0]
	ŜT	P CE i	0.01	0.11	0.03	0.05	0.84	0.0	[0.0,0.0]
		δi	0.0	0.09	0.04	0.06	0.44	0.0	[0.0,0.04]
	Υi	0.32	0.14	0.04	0.09	1.10	0.0	[0.0,0.0]

Acknowledgment: This work has been funded by the French National Research Agency through the research project RESAIN (n • ANR-12-BS06-0010-02). The authors would like to thank the reviewers for their insightful comments that helped improving the manuscript.