Anthony Nouy
email: anthony.nouy@univ-nantes.fr

Franck Schoefs

Nicolas Moës

X-SFEM, a computational technique based on X-FEM to deal with random shapes

Keywords: computational stochastic mechanics, X-FEM, level set, stochastic finite element, random geometry. MOTS-CLÉS : calcul stochastique, X-FEM, level set, éléments finis stochastiques, géométrie aléatoire

We propose a new method to deal with random geometries. It is an extension to the stochastic context of the eXtended Finite Element Method. This method lies on two majors points: the implicit description of geometry by the level set technique and the use of the partition of unity method for the enrichment of the finite element approximation space. This new technique leads by a direct calculus on a fixed finite element mesh to a solution which is explicit in terms of the basic random variables describing the geometry. We present here the basis of this approach and several examples to illustrate its performances.

RÉSUMÉ. Nous proposons une nouvelle méthode éléments finis stochastiques permettant de prendre en compte le caractère aléatoire de la géométrie dans le calcul de structures. Cette approche est basée sur la technique X-FEM, étendue au cadre stochastique. Elle repose sur deux points importants : la description fonctionnelle implicite de la géométrie par la technique des level sets et l'utilisation de la partition de l'unité pour l'enrichissement de l'espace d'approximation. Cette approche, qu'on pourrait baptiser naturellement eXtended Stochastic Finite Element Method (X-SFEM), permet d'obtenir par un calcul « direct » une solution explicite en fonction des variables aléatoires décrivant le caractère aléatoire de la géométrie. On présente ici les fondements de cette méthode et des exemples illustrent ses performances.

Introduction

A great challenge in the last few years has been to develop robust and efficient computational strategies to take into account uncertainties in structural analysis. Stochastic finite element methods [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF][START_REF] Ghanem | Ingredients for a general purpose stochastic finite elements implementation[END_REF][START_REF] Deb | Solution of stochastic partial differential equations using Galerkin finite element techniques[END_REF][START_REF] Matthies | Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations[END_REF] have been recently proposed and offers a significant tool to deal with random material properties and boundary conditions. They lead to a very fine solution at the stochastic level which is explicit in terms of the random variables describing the uncertainties. Their efficiency, constantly improved by the use of ad hoc resolution techniques [START_REF] Pellissetti | Iterative solution of systems of linear equations arising in the context of stochastic finite elements[END_REF][START_REF] Keese | Hierarchical parallelisation for the solution of stochastic finite element equations[END_REF][START_REF] Nouy | Technique de décomposition spectrale optimale pour la résolution d'équations aux dérivées partielles stochastiques[END_REF], allows one to use them in various fields of structural analysis such as design, reliability analysis. . . Dealing with uncertainties on applied loads or on material properties is a question relatively mastered within these methods. However, there is still no available efficient strategy to deal with uncertainties in the geometry although it could have a great interest in various applications: modeling of uncertainties in fabrication processes (size or position of borings, shape of a weld), of loss of matter due to corrosion, of random multi-phased materials. A traditional Monte-Carlo simulation could be performed to obtain statistics of the response. However, each event is associated with a new geometry and the use of classical finite element methods implies numerous remeshings and leads to prohibitive computational costs. To reduce the cost of remeshings, an alternative technique could consist in introducing variabilities on the position of the nodes of the finite element model. Nevertheless, this technique is limited to small variabilities of the geometry to avoid intolerable distorsion of the mesh and numerical instabilities. For large variabilities, this kind of techniques also need many remeshings.

Here, we propose a new stochastic finite element method which enables to deal with random geometry without remeshings. It is based on the extended finite element method [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF][START_REF] Sukumar | Modeling holes and inclusions by level sets in the extended finite-element method[END_REF], extended to the stochastic context. This method relies on two majors points: implicit description of the geometry by the level set technique [START_REF] Sethian | Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry[END_REF] and use of the partition of unity method [START_REF] Melenk | The partition of unity method: basic theory and applications[END_REF] for the enrichment of the finite element approximation space. The approach proposed here is a new intrusive technique which we call eXtended Stochastic Finite Element Method (X-SFEM). It leads by a direct calculus on a fixed finite element mesh to a solution which is explicit in terms of the basic random variables describing the geometry. A particular care is taken on the choice of approximation spaces. Notably, we show how the use of finite element approximation at the stochastic level can consequently improve the performances of this approach. In this article, we will focus on uncertainties on the shape of the domain. Enrichment by the partition of unity to model internal surfaces as cracks or material interfaces will be introduced in a subsequent paper. The method proposed here also offers a significant tool for parametric analysis which could be performed during the design stage of structures.

The plan of the paper is as follows: in Section 2, the problem involving random geometry is introduced mathematically. In Section 3, a probabilistic modeling of the geometry using the level set technique is presented. Approximation spaces are in-troduced in Section 4. The discretized problem, its construction and resolution are also presented in Section 4. Two examples will illustrate the capabilities of our approach in Section 5. We will emphasize in particular on the discretization choice at the stochastic level.

Formulation of a stochastic problem with random geometry

Formulation of the continuous problem

We will focus for simplicity on the case of a linear elliptic stochastic partial differential equation defined on a random domain. We denote by (Θ, B, P ) the probability space, where Θ is the set of outcomes, B the σ-algebra of events and P the probability measure. The random geometry is defined by the mapping θ ∈ Θ → Ω(θ) ⊂ R d . The stochastic problem is: find u(θ, x) which satisfies P -almost everywhere on Θ,

-∇ • (a(θ, x)∇u(θ, x)) = f (θ, x) on Ω(θ) [1] u = 0 on Γ 1 (θ) n • a∇u = g(θ, x) on Γ 2 (θ)
where a is the diffusion coefficient, possibly random. Γ 1 is the part of ∂Ω(θ) where Dirichlet conditions are imposed and Γ 2 (θ) is the complementary part to Γ 1 . In this article, we consider that Γ 1 is deterministic.

The variational formulation of the problem can be written: find

u ∈ V 0 = {u ∈ V ; u |Θ×Γ1 = 0} such that A(u, v) = L(v) ∀v ∈ V 0 [2] with A(u, v) = Θ Ω(θ) a∇u • ∇v dxdP (θ) L(v) = Θ Ω(θ) f v dxdP (θ) + Θ Γ2(θ)
gv dsdP (θ)

Choice of ad hoc function spaces

The question of the choice of an ad hoc function space V 0 in order to prove existence and uniqueness of the solution of problem [2] has not been treated in the litterature to the knowledge of the authors. Although presented in the deterministic context, we can find in [START_REF] Babuska | Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions[END_REF][START_REF] Babuska | Effects of uncertainties in the domain on the solution of Dirichlet boundary value problems[END_REF] some results on the effects of uncertainties in the domain on the solution of stochastic boundary value problems. Here, we introduce a mathematical framework which is adapted to the method proposed in this article.

Let us consider a domain B such that θ Ω(θ) ⊂ B. We assume that a is bounded and uniformly coercive. Under classical regularity assumptions on a and the right hand side of [2], the solution u, for a fixed θ, belongs to the space H 1 (Ω(θ)). By prolongation, this space can be considered as a subspace of H 1 (B). Then, we will work on the following function space:

V = L 2 (Θ, dP ; H 1 (B)) = {v : Θ × B → R | Θ v(θ, .) 2 H 1 (B) dP (θ) < ∞} [3]
The bilinear and linear forms in problem [2] are then rewritten as follows:

A(u, v) = Θ B a∇u • ∇vI Ω dxdP (θ) [4] L(v) = Θ B f vI Ω dxdP (θ) + Θ Γ2(θ)
gv dsdP (θ) [5] where I Ω (θ, •) : B → {0, 1} denotes the characteristic function of Ω(θ).

The bilinear form A is bounded but only semi-coercive on V. The kernel in

V 0 of v → A(v, v) is a set of functions whose support is a subset of what we can call the "non-physical" domain N = {(θ, x) ∈ Θ × B; x / ∈ Ω(θ)}. Problem [2]
have an infinite number of solutions which differ from one another by functions in this kernel. Let us briefly prove this statement. Let u 1 and u 2 be two solutions of [2] and w = u 1u 2 ∈ V 0 . We have A(w, w) = 0. Then, a∇w • ∇wI Ω = 0 almost surely on Θ × B. Let us denote by N c = {(θ, x) ∈ Θ × B; x ∈ Ω(θ)} the "physical" domain, which is the complementary set of N . We have I Ω (θ, x) = 1 on N c and 0 on N (i.e. I Ω is the characteristic function of N c ). Due to positivity properties of a, we have ∇w = 0 almost surely on N c and w |Θ×Γ1 = 0. We then conclude that w = 0 almost surely on N c . The "physical" part of the solution, i.e. its restriction to N c , is then unique. And this is the part of the solution we are interested in ! Prolongation of the solution in H 1 (B) in the stochastic context will not have any physical meaning. For example, if we consider a point x ∈ B such that P (x ∈ Ω(θ)) < 1, the quantity u(θ, x) will make sense only for outcomes of domain which include x. A good postprocessing will have to avoid such quantities of interest. We will rather be interested in quantities defined on a well defined zone (such as mean displacement on a part of the boundary. . . ) or on implicit quantities such as the maximum stress.

Representing random geometry by the level set technique

Definition and approximation of level sets

In the deterministic context, the level set technique [START_REF] Sethian | Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry[END_REF] consists in representing a surface by the zero level set of a function. We commonly use for this function the signed function to the surface. This function, denoted by φ(x), is gen-erally interpolated at nodes of a fixed finite element mesh which does not conform to the physical surface:

φ(x) = i N i (x)φ i = N(x)φ [6]
where N i are the classical finite element shape functions and where

N = [N 1 , . . . , N N ],
with N the number of nodes. In the probabilistic context, this function simply becomes a stochastic field φ(θ, x), where θ denotes the outcome. For a given outcome θ, the zero level set of φ(θ, x) represent the corresponding outcome of the random surface. In the same way, this field will be interpolated at nodes of a fixed finite element mesh and will then be described by a vector of random variables φ(θ).

The question is: how to build this stochastic field ? In all cases, for the resolution of the stochastic problem, we will try to reduce the source of randomness to a small number of independent random variables ξ(θ). The level set function will then be defined by

φ(ξ(θ), x) = N(x)φ(ξ(θ)) [7]
First, we can chose an a priori representation of the geometry parameterized in terms of a set of random variables, whose probabilistic laws can be identified from experimental data. Let us take for example the case of a circular hole with random center c(θ), whose position depends on two random variables in 2D, and with random radius R(θ). In this case, the level set can be explicitly written as follows:

φ(θ, x) = R(θ) -x -c(θ)
. Explicit expressions of level sets exist for several classical geometrical forms (ellipses, polygones. . . ) [START_REF] Sukumar | Modeling holes and inclusions by level sets in the extended finite-element method[END_REF].

If the geometry can be written as a function of a stochastic field, a discretization step of this field and a systematic construction of the corresponding level set will be necessary. These points will be detailed in a subsequent paper.

Representation of an external random geometry

When the domain Ω is random, we introduce a level set function φ(θ, x) whose iso-zero surface represents the boundary ∂Ω(θ). We suppose that φ(θ, x) is negative in Ω(θ) and positive in the complementary part. Denoting by H the Heaviside function, the characteristic function of the domain can simply be written:

I Ω (θ, x) = H(-φ(θ, x)) [8]

Discretization of the problem

In this part, we introduce the discretization of problem [2] at the spatial and stochastic levels. In this article, we only consider random shapes. It is a particular case of X-SFEM for which no enrichment is needed (see [START_REF] Daux | Arbitrary branched and intersecting cracks with the eXtended Finite Element Method[END_REF] for the deterministic case).

Preliminaries and notations

We assume that the probabilistic modeling stage has allowed us to represent the source of randomness with a finite number m of independent random variables ξ(θ) = (ξ 1 (θ), . . . , ξ m (θ)). A random variable v(θ) can then be expressed in terms of ξ: v(θ) ≡ v(ξ(θ)). We then work in the image probability space Θ = ξ(Θ) ⊂ R m endowed with probability measure P ξ defined for Borel sets b ∈ B R m by P ξ (b) = P (ξ -1 (b)). We denote by E(.) the mathematical expectation, defined by

E(v) = Θ v(θ)dP (θ) = Θ v(x)dP ξ (x) [9]

Choice of approximation spaces

We first notice that V, defined in [3], is the tensor product of the spaces S = L 2 (Θ) and U = H 1 (B). We then choose a tensor product approximation. The domain B is discretized by a mesh T h yielding a finite element approximation space U h ⊂ U . The set of shape functions is denoted by {N i } i∈I , where I denotes the set of nodes. At the stochastic level, we introduce {H α (ξ)} α∈IP , a basis of P functions of S P ⊂ S. The approximate solution, denoted by u h,P , will then be searched in the space

V h,P = S P ⊗ U h = { α∈I P i∈I N i (x)H α (ξ)u i,α , u i,α ∈ R} = { α∈IP N(x)u α H α (ξ) , u α ∈ R N } [10]
where N = [N 1 , . . . , N n ], n being the number of nodes in the mesh.

Several choices have been proposed for building approximation basis of L 2 (Θ). Spectral approaches as polynomial chaos [START_REF] Wiener | The homogeneous chaos[END_REF][START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF] or generalized polynomial chaos [START_REF] Xiu | The Wiener-Askey Polynomial Chaos for stochastic differential equations[END_REF] use orthogonal polynomial basis and exhibit exponential convergence rates [START_REF] Xiu | The Wiener-Askey Polynomial Chaos for stochastic differential equations[END_REF] in the case of regular solutions. For the case of less regular solutions, other approximation techniques were introduced (Wiener-Haar chaos (Le [START_REF] Maître | Uncertainty propagation using Wiener-Haar expansions[END_REF], hp finite elements [START_REF] Deb | Solution of stochastic partial differential equations using Galerkin finite element techniques[END_REF]). In practice, the basis functions will be taken orthonormal:

E(H α H β ) = 1 if α = β and 0 otherwise.

Polynomial chaos

We note that S is the product space S 1 ⊗ . . . ⊗ S m , where

S i = L 2 (Θ i , B i , dP ξi ).
The polynomial chaos discretization consists in choosing for S P a space of multidimensional polynomials H α = h 1 α1 (x 1 ) . . . h n αn (x n ), where h i αi ∈ S i and where α = (α 1 , . . . , α m ) denotes a multi-index of N m . Polynomials are chosen orthogonal with respect to the scalar product of S = L 2 (Θ, dP ξ ), defined by (u, v) = E(uv). The method initially proposed by [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF] consists in working with a set ξ of independent Gaussian variables. In this case, h i j denotes Hermite polynomials which are the orthogonal polynomials with respect to Gaussian measure. The same basis functions are used for all stochastic dimensions. Classically, we choose for S P the polynomial chaos of order p in m variables, defined with multi-indices:

I P = {α ∈ N m ; |α| = m i=1 α i p}.
The number of basis functions of

S P is P = (m+p)! p! m!
The generalized polynomial chaos method [START_REF] Xiu | The Wiener-Askey Polynomial Chaos for stochastic differential equations[END_REF] uses orthogonal polynomials with respect to measures dP ξi for h i j . In the case of uniform, beta or gamma random variables, we use respectively Legendre, Jacobi and Laguerre polynomial. For arbitrary measure and possibly dependent variables, it is also possible to introduce generalized chaos decompositions [START_REF] Soize | Physical systems with random uncertainties: chaos representations with arbitrary probability measure[END_REF].

Finite elements at the stochastic level

Finite element approximation at the stochastic level [START_REF] Deb | Solution of stochastic partial differential equations using Galerkin finite element techniques[END_REF][START_REF] Babuska | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation[END_REF] consists in choosing for S P a space of piecewise polynomial functions (discontinuous) on a mesh of Θ. This type of approximation allows one to use tools which are classical in the finite element context such as a posteriori error estimation, adaptative refinement. . .

Here, we suppose that Θ = Θ 1 × . . . × Θ m is a bounded open set of R m . On each stochastic dimension, we introduce a mesh T i k of n i elements of Θ i . We denote by T k = T 1 k × . . . × T m k the corresponding rectangular mesh of R m . The restriction of a function H α (x) of S P to an element τ of T k satisfies

H α|τ ∈ span m i=1
x qi i , q i ∈ N and q i p i [11] where p i is the approximation degree on Θ i . Here, we choose for I P the set:

I P = {α = (α 1 , . . . α m ) ∈ (N 2 ) m , α i = (j i , q i ) , 1 j i n i , 0 q i p i }.
The number of basis functions of S P is here P = m i=1 n i (p i + 1). We recall that the basis functions are orthonormalized.

Definition of the approximate solution

We introduce V h,P,0 = V h,P ∩ V 0 . The Galerkin approximation u h,P ∈ V h,P,0 of [2] is defined by

A(u h,P , v h,P ) = L(v h,P ) , ∀v h,P ∈ V h,P,0 [12]
To avoid singularity in the discretized problem [12], it is necessary to suppress functions of V h,P lying in the kernel of v → A(v, v). In particular, that leads to suppress degrees of freedom associated with the set of nodes I out whose support is P ξ -almost surely outside the physical domain. Denoting by ω i the support of shape function N i , i.e. the union of elements having node i as one of their vertices, the set I out can be written:

I out = {i ∈ I ; P ξ (ω i ∩ Ω(ξ) = ∅) = 1} = {i ∈ I ; ω i ⊂ Ω 0 } [13]
where Ω o is the biggest open set almost surely outside Ω, i.e. such that Suppressing these degrees of freedom is only a necessary condition for the nonsingularity of the discretized problem [12]. In general, we will then have to solve a semi-definite system of equations for which we will search a particular solution (see Section 2.2).

P ξ (Ω o ∩ Ω(ξ) = ∅) = 1.

The discretized problem and its resolution

Discretized problem [12] can also be written: find u ∈ V h,P,0 such that ∀v ∈ V h,P,0

A( α∈IP N(x)v α H α (ξ), β∈IP N(x)u β H β (ξ)) = L( α∈IP N(x)v α H α (ξ)) =⇒ α,β∈IP v T α E(KH α H β )u β = α∈IP v T α E(f H α ) [14]
where the random matrix K and the random vector f are defined by:

K(ξ) = B a(ξ, x)∇N T ∇NH(-φ(ξ, x)) dx [15] f (ξ) = B N T f H(-φ(ξ, x)) dx + Γ2(ξ) N T g ds [16] Let K αβ = E(KH α H β ) and f α = E(f H α )
. System [14] can then be rewritten:

      K α1α1 K α1α2 . . . K α1αP K α2α1 . . . . . . . . . . . . . . . K αP α1 . . . . . . K αP αP            u α1 u α2 . . . u αP      =      f α1 f α2 . . . f αP      [17]
In practice, system [17] can not be solved by a direct resolution technique. Indeed, the memory storage and computational costs of assembling and solving this huge system become prohibitive for common engineering problems. To avoid assembling and to take part of the sparsity of this system, we classically use Krylov-type iterative resolution techniques [START_REF] Ghanem | Numerical solution of spectral stochastic finite element systems[END_REF][START_REF] Pellissetti | Iterative solution of systems of linear equations arising in the context of stochastic finite elements[END_REF][START_REF] Keese | Hierarchical parallelisation for the solution of stochastic finite element equations[END_REF]).

An alternative approach has been proposed in [START_REF] Nouy | Technique de décomposition spectrale optimale pour la résolution d'équations aux dérivées partielles stochastiques[END_REF], which consists in searching an optimal spectral decomposition of the solution. An ad hoc construction of this decomposition allows one to transform the problem into the resolution of a few uncoupled deterministic spatial finite element problems and stochastic equations. This approach reduces drastically computational and storage costs.

Computing matrices and vectors of the discretized problem

For solving [17], matrices K αβ = E(KH α H β ) have to be computed. Computing these matrices independently would lead to prohibitive costs. In practice, we first decompose K on an ortho-normal basis of S P ′ ⊂ S:

K(ξ) = α∈I P ′ K α H α (ξ) with K α = E(KH α ) [18]
This decomposition being performed, we simply have:

E(KH α H β ) = γ∈I P ′ K γ E(H γ H α H β ) [19]
REMARK. -Here, S P ′ will be chosen different from S P , the approximation space of the solution. Indeed, in the case where S P is a space of orthogonal (piecewise) polynomials of degree less than p (for polynomial chaos or finite element approximation at the stochastic level), the value of K αβ will be exact if S P ′ contains orthogonal polynomials of degree less than 2p. With this choice, the approximation made on K will have no influence on the solution of the discretized problem [14] (see proof in [START_REF] Matthies | Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations[END_REF]).

In the decomposition [18], K α terms are calculated by Gauss-quadrature:

K α = E(KH α ) = ng k=1 K(ξ k )H α (ξ k )ω k [20]
where n k is the number of gauss points, ω k the weights and ξ k the integration points associated to the measure dP ξ . The calculation of all the matrices K α then only requires the assembling of n g matrices

K(ξ k ) = B a(ξ k , x)∇N(x) T ∇N(x)H(-φ(ξ k , x)) dx [21] φ(ξ k , x
) is a deterministic level set corresponding to an outcome of the geometry and, therefore, the calculation of matrices K(ξ k ) is a classical calculation within the context of deterministic X-FEM method. In practice, to perform spatial integration, we use a classical Gauss-quadrature on elements which are not cut by a level set. Elements which are cut by a level set are first split into sub-elements on which we perform a classical Gauss-quadrature [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF][START_REF] Dolbow | Discontinuous enrichment in finite elements with a partition of unity method[END_REF].

Vectors f α = E(f H α ) of the right hand side of [17] are calculated by a similar numerical integration. Their calculation is still classical within the X-FEM method.

Back to the choice of stochastic discretization

Parametrized problem equivalent to problem [14] can be written: K(ξ)u(ξ) = f (ξ), ξ ∈ Θ. In fact, matrices and vectors of this problem are only piecewise regular on Θ. This will be illustrated in example 1. Therefore, the use of a spectral discretization at the stochastic level such as polynomial chaos is not optimal. A better choice consists in using at the stochastic level a finite element discretization on a mesh T k of Θ such that matrices and vectors are regular on the elements of T k . We use for building this mesh an automatic procedure which will be presented in a forecoming paper. In the case where X-FEM method gives an exact solution to the associated deterministic problem, this choice of approximation can allow one to obtain the exact solution of the stochastic problem [2]. When using classical polynomial chaos approximation, this meshing procedure of Θ is also needed to well integrate matrices and vectors of the discretized problem [17].

Examples

Example 1: traction of a beam with random length

We consider an homogeneous beam in traction laying in the random domain Ω = ]0, ξ[, where ξ is a uniform random variable: ξ = U (0.25, 0.75). We take B =]0, 1[ as a prolongation of the random domain. The traction modulus is taken equal to 1. The beam is embedded at x = 0 and is submitted to a uniform and unitary body load. The problem can be written:

d 2 u dx 2 (ξ, x) + 1 = 0 f or x ∈]0, ξ[ , u(ξ, 0) = 0 , du dx (ξ, ξ) = 0
An exact solution u to this problem exists: u(ξ, x) = x(ξ -x 2 ) on ]0, ξ[. Domain B is meshed with n elements of length h = 1 n . The used level set set is φ(ξ, x) = xξ. At the spatial level, we use P 1 finite elements. Figure 2 shows a 10 elements mesh of B, possible outcomes of domain Ω(ξ) and shows the node set I out defined in [13] for which we must suppress the associated degrees of freedom. We first consider a generalized polynomial chaos approximation of degree p at the stochastic level (Section 4.2.1). Basis functions of S P are Legendre polynomials. To compare the solution u h,P of problem [12] to the exact solution u, we introduce two error indicators:

e h,P = u h,P -u L 2 (Θ;L 2 (Ω(ξ)) = E Ω(ξ) (u h,P (ξ, x) -u(ξ, x)) 2 dx 1/2 ẽh,P = u h,P (ξ, ξ) -u(ξ, ξ) L 2 (Θ) = E (u h,P (ξ, ξ) -u(ξ, ξ)) 2 1/2
The first indicator is a true error indicator. Indeed, when e h,P → 0, the physical part of the approximate solution (see Section 2.2 for this definition) converges toward the physical part of the exact solution. The second indicator represent the error which is made on a quantity of interest, namely the displacement at x = ξ (the right extremity of the random domain). Table 1 shows the evolution of these errors with respect to the degree (p) and mesh size (h).

We can observe a good h convergence of the approximation for both indicators, independent of the degree p (convergence in O(h 2 ) for e h,P and O(h) for ẽh,P ). Since the exact solution u(ξ, x) and the exact quantity of interest u(ξ, ξ) are respectively polynomials of degree 1 and 2 in ξ, it is natural to converge with h towards the exact solution, whatever the degree p 2. We obtain a good approximation, especially for the quantity of interest, even with coarse meshes and a low degree of approximation at the stochastic level. We know that within the deterministic context, X-FEM technique leads to an approximate solution u h (x) which gives an exact quantity of interest u h (ξ), whatever the mesh size h. We recover this property within the stochastic context for the coarser mesh (h = 1), for which we obtain an "exact" numerical solution.

REMARK. -For h = 1, the fact that the error increases with the degree p comes from round-off errors which become significant when we deal with higher order polynomials. For our purpose, we can then consider that these values of errors are approximately zero in a numerical sense.

Table 1. Example 1: generalized chaos approximation. Error indicators e P C h,P and ẽP C h,P with respect to the degree p of the chaos and the mesh size h e P C h,P p = 2 p = 4 p = 6 p = 8 h = 1.0 2.22 10 -2 2.22 10 -2 2.22 10 -2 2.22 10 -2 h = 0.5 1.32 10 -2 1.32 10 -2 1.32 10 -2 1.32 10 -2 h = 0.2 2.42 10 -3 2.38 10 -3 2.38 10 -3 2.38 10 -3 h = 0.1 6.72 10 -4 6.43 10 -4 6.39 10 -4 6.40 10 -4 ẽP C h,P p = 2 p = 4 p = 6 p = 8 h = 1.0 9.79 10 -15 2.55 10 -14 7.56 10 -13 1.61 10 -12 h = 0.5 2.34 10 -3 1.35 10 -3 9.66 10 -4 7.56 10 -4 h = 0.2 1.22 10 -3 5.76 10 -4 3.82 10 -4 2.87 10 -4 h = 0.1 5.76 10 -4 2.63 10 -4 1.91 10 -4 1.45 10 -4 However, we could have expected to obtain the exact quantity of interest for every mesh sizes. The fact that we do not obtain this property for more than 1 element is due to a non optimal choice of the approximation at the stochastic level. To better understand this, we consider the corresponding parametrized problem K(ξ)u(ξ) = f (ξ) with ξ ∈]0.25, 0.75[ for a mesh with 2 elements. Figure 3 shows the evolution of components of K, f and u with respect to ξ. We note that these components are continuous piecewise polynomials, with a discontinuity point in their derivative located at ξ = 0.5. This point corresponds to the outcome for which the extremity of the beam is located at the middle node. For example, let us prove this for K 11 (ξ). The support of shape function N 1 is the first element. On this element, ∂N1 ∂x = -2. Then, For n elements, we can also easily show that discontinuity points in the derivatives appear for values of ξ for which the extremity of the domain is located on a node. The use of spectral approximation is unadapted to represent non regular solutions and leads in this case to a bad p convergence of the approximation at the stochastic level (for a fixed mesh size h).

K 11 = 4
Following these remarks, we propose to use at the stochastic level finite elements of degree p (see Section 4.2.2) with a mesh Θ which coincides with the non-regular points. Of course, this mesh depends on the spatial mesh of B. Table 2 shows the evolution of error indicators with respect to degree p and mesh size h. We observe that with an adaptive choice of the stochastic basis, we obtain the "exact" value (in the numerical sense) for the quantity of interest for all mesh sizes. The fact that the error increases while h decreases or p increases is due to round-off errors (see remark above). However, for fixed degree of approximation p at the stochastic level, the global convergence is still in O(h 2 ): the values taken by indicators e F E h,P (Table 2) and e P C h,P

(Table 1) are quite the same. In fact, non-regularities at the stochastic level which was explained above become smoother while the mesh size h decrease. We can then expect that for small mesh size, polynomial chaos approximation gives satisfactory results.

Table 2. Example 1: finite element approximation of degree p at the stochastic level. The mesh of Θ is adapted to the one of B. Error indicators e F E h,P and ẽF E h,P with respect to p and h e F E h,P p = 2 p = 4 p = 6 p = 8 h = 1.0 2.22 10 -2 2.22 10 -2 2.22 10 -2 2.22 10 -2 h = 0.5 1.32 10 -2 1.32 10 -2 1.32 10 -2 1.32 10 -2 h = 0.2 2.39 10 -3 2.39 10 -3 2.39 10 -3 2.39 10 -3 h = 0.1 6.44 10 -4 6.44 10 -4 6.44 10 -4 6.44 10 -4 ẽF E h,P p = 2 p = 4 p = 6 p = 8 h = 1.0 9.79 10 -15 2.55 10 -14 7.56 10 -13 1.61 10 -12 h = 0.5 8.36 10 -15 2.79 10 -14 9.25 10 -13 1.64 10 -12 h = 0.2 7.82 10 -15 3.57 10 -14 9.94 10 -13 7.52 10 -12 h = 0.1 7.88 10 -15 3.72 10 -14 1.03 10 -12 5.62 10 -12

Example 2: plate with a random hole in tension

We consider the problem of a plate with a traction-free circular hole submitted to uniaxial tension. The plate lies in a square domain ]0, 3[×]0, 3[. The center of the hole is located at (1.5, 1.5) and has for radius a uniform random variable R(θ) = U (0.25, 0.35). The Young modulus is taken equal to 1 and the poisson coefficient to zero. Classical FEM approaches use sampling techniques to solve this stochastic problem. For each sample, corresponding to an outcome of the geometry, a new mesh conforming to the geometry is created and an associated finite element computation is performed. With the X-SFEM method proposed here, we build a unique mesh and we solve a unique system of Equation [12]. We use the mesh represented on Figure 4. This mesh is obtained by meshing a plate with a hole of radius R = 0.3 (mean radius) and by meshing also the interior of this hole. It is convenient for comparing our approach with a classical FEM technique. A totally unstructured mesh could also have been used. Nevertheless, creating a mesh based on outcomes of the geometry allows to better control the mesh size around zones of interest and also leads to a higher regularity of the approximate solution at the stochastic level (see Section 4.6). On Figure 4, elements are split into three groups: elements surely in the domain, element possibly cut by the boundary of the hole and elements surely out the domain. We don't associate degrees of freedom with nodes belonging to elements of the third group and not belonging to elements of the first two groups. At the stochastic level, we use a finite element approximation with piecewise polynomials of degree p = 2 (see Section 4.2.2): the mesh of Θ =]0.25, 0.35[ that we use is T k =]0.2500, 0.2903, 0.3321, 0.3500[. This mesh is obtained by an automatic procedure which will be detailed in a following paper. It is useful for numerical integration [20] and it allows one to obtain a good regularity per element of the approximate solution.

Figures 5 and6 show a compararison of FEM and X-SFEM solutions for outcomes of the geometry. Corresponding outcomes of the iso-zero of the level set representing the hole are shown on the right figures. On Figure 7, we compare the response surface of the horizontal elongation ∆u(R) = u(R, x b )u(R, x a ), where x a = (0, 1.5) and x b = (3, 1.5). We note that X-SFEM leads to a very high precision solution within the whole range of variation of R. We must also note that with X-SFEM, problem [12] being solved, the solution obtained has an explicit form in terms of R(θ). Postprocessing can then be performed at a very low cost. An example of post-processing is illustrated on Figure 8 where the probability density function of ∆u is plotted.

Conclusion

We proposed a stochastic finite element method based on X-FEM to solve stochastic partial differential equations defined on a random domain without remeshings. The random geometry is described in an implicit manner as the iso-zero of random level sets. A Galerkin approximation technique is then introduced which leads to a high precision approximate solution. At the stochastic level, a suitable finite element discretization is used which takes into account the lack of regularity of the approximate solution. It allows in some particular cases to obtain the exact solution of the stochastic problem and could allow in the general case to reach higher p convergence rates than with classical spectral approximation such as polynomial chaos. Of course, the strategy proposed here is well adapted for parametric studies, which can be viewed as a particular case of a probabilistic study.

Figure 1 .

 1 Figure 1. Circular hole with random radius (bounded): definition of the node set I out . Ω o is the biggest open set such that P ξ (Ω o ∩ Ω(ξ) = ∅) = 1

Figure 2 .

 2 Figure 2. Example 1: 10 elements mesh of B =]0, 1[. Outcomes of physical domain Ω(ξ) and node set I out whose support is almost surely outside Ω(ξ)

Figure 3 .

 3 Figure 3. Components of K(ξ), f (ξ) and u(ξ), ξ ∈]0.25, 0.75[, for a mesh with 2 elements of length 1 2 . Indices correspond to node numbers, numbered from left to right

  ξ > 0.5 4ξ f or ξ < 0.5

  Figure 4. X-SFEM mesh and 3 element groups: surely in the domain (e i ), possibly cut by the boundary of the hole (e c ) and surely out the domain (e o )

Figure 5 .Figure 6 .

 56 Figure 5. FEM (left) and X-SFEM (right) solutions: stresses σ xx for R = 0.275

Figure 7 .Figure 8 .

 78 Figure 7. Comparison between FEM and X-SFEM: response surface ∆u(R)

Acknowledgements

This work is supported by the French National Research Agency (ANR).


Rambling on the value of truth Pascal Engel

In Against Boredom , Essays dedicated to Nils-Eric Sahlin for his 6Oth birthday, Lund Universitet 2015

A cat is objectively valuable Ayn Rand

Although it has become a bit old-fashioned to use this kind of language, it is natural to think that Logic, Aesthetics and Ethics are "normative sciences", and to consider that they deal respectively with the values of Truth, Beauty and Goodness. Ramsey, however, for one, was not convinced that the correspondence is exact in the case of logic:

For whereas the chief question in Ethics is undoubtedly "What is good?", and in Aesthetics "What is beautiful?", the question "What is true?" is one which all the sciences answer, each in its own domain, and in no way the particular concern of Logic. What Logic studies is not so much the truth of the opinions, as the reasonableness of arguments or inferences. (Ramsey 1991, 3) Ramsey then hints, in the introduction to his unpublished manuscript Truth, that questions of value are to be answered through a psychological investigations about the kind of attitudes which are the source of these values, and in the case of logic about the nature of our opinion and judgments as psychological states and about their rationality in inferences. He was, in other words, a non-cognitivist, and, given his famous view that "there is no separate question of truth, but only a question about the nature of judgment" [START_REF] Ramsey | Philosophical Papers[END_REF]), a non-factualist both about truth and about the value of truth.

I have learnt most of what I know about Ramsey from Nils-Eric Sahlin. Although for long I have been sympathetic to Ramsey's view on truth and for his non-cognitivist stance on values, including epistemic, I have now come to doubt that they are correct. I try here to give some of the reasons why I prefer a cognitivist conception of the value of truth.

Truth as prima facie valuable

If we want to ask in what sense truth is valuable, we should attend some familiar distinctions about values (see e.g. [START_REF] Mulligan | Values[END_REF]). We ascribe to certain objects certain value properties (good, bad, beautiful). But to what kind of entities? What are the bearers of value? Objects? State of affairs? The more the values have content, the "thicker" they are, by opposition to "thin" values. Something can be a value or a disvalue in itself, or in relation to something else. We can conceive of truth as valuable in itself, as a final value, or in relation to another value, as an instrumental value. A value can be intrinsic, when the value is to be found in an object or property in itself, or extrinsic when the value is relative to another object. Something has an intrinsic value when it is valuable for its own sake, and an instrumental value when it is valuable for the sake of something else. There are also various kinds of values: practical, moral, epistemic, aesthetical, social, possibly others. What kind of properties are value properties? Do they form an exclusive kind or are they reducible to another more fundamental kind? In other words do values form a special domain, the domain of the axiological? Or do they have strong connexions, and possibly are they reducible to other normative properties such as the deontological ones or in the sense of being things for which we have reasons? When we ascribe value properties, do these properties denote a certain kind of entity -values -or are these properties a projection of our psychological attitudes -of our valuings? Ontologically speaking one can be a realist or an anti-realist about value. Finally one can take value properties to be reducible to natural properties, or to supervene upon these, or not. None of these various issues are independent from each other. I cannot hope to deal with all these distinctions, but we can try to apply these to the familiar idea that truth is valuable, hence at least a value property.

Although the fact that truth is valuable is a property of our ordinary concept of truth, it is not easy to specify what this property is, in what sense it is a value property and what its bearers are. It is often said that truth is a value. But of what is it a value? Truth by itself, as a property of our beliefs or assertions, has no value and is neither good nor bad, neither beautiful nor hideous. That grass is green or that manganese has atomic number 25 are truths is a fact about these sentences or propositions, and there is nothing valuable in that they are true or describe what is the case. Facts or truths as such do not have any value. If these propositions can be valuable or can have a value, it is as potential objects of our beliefs or of our assertions. Truth is a value property of our beliefs and assertions, which are its primary bearers. Moore said in Principia Ethica: "I cannot at any given moment distinguish what is true from what I believe" (Moore 1903, § 80). Commenting this passage, Marian [START_REF] David | How to take Truth as a Goal[END_REF] proposes the following test. I present you with a list of propositions and ask you: "Mark the ones that are true!" You comply. Imagine now that, concerning the very same list of propositions, I had asked you: "Mark the ones that you believe!" You would have marked the very same propositions. It would seem that if the possession of truth is valuable, the views which associate intrinsically the nature of truth to its possession by a believer will say that truth as a property is valuable. Thus verificationist views, which say that truth is warranted assertibility, coherentist views, which take truth to be coherent belief, or pragmatist theories, which take truth to be a property of successful beliefs, will readily associate the value of truth to some epistemic property. But we should be cautious here too: that truth is valuable relative what we say or what we believe does not entail that truth is an epistemic property. There is no reason to presuppose a form of anti-realism or epistemicism about truth when we attribute to true beliefs a value. The intuitive association noted by Moore between belief and truth does not even begin to indicate that being true entails or is equivalent to being believed. On the contrary it would seem that in order to be able to ascribe a value to truth, truth has to be a property which is independent of our believing anything about it. In particular the most radical of all epistemic theories, relativism, entails that truth cannot be a value. For in order to accept the idea of the value of truth, or of its disvalue, false belief must be possible. But relativism, or at least the crudest version of this doctrine, does not make room for false belief: according to it all our beliefs are equally true, just in virtue of being our beliefs. So all of our beliefs, if simple relativism holds, ought to be valuable. But if all beliefs are equally valuable, how can truth be a value? It cannot accommodate the idea of a value of truth in any objective sense.

If truth is a value property of our beliefs, it is presumably a "thin" and not a "thick" property, as many philosophers since Aristotle have claimed. Even Aristotle's famous "definition" of truth in Metaphysics 1011 b 26 -"To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true" -which is often interpreted as a first statement of the correspondence theory of truth, does not say very much. It is actually a platitude, which features among those which are said to be associated to our ordinary concept of truth [START_REF] Wright | Truth and Objectivity[END_REF]: transparency ("'P' is true" says the same thing as "P"), embedding ("that P is true" can be embedded in other contexts), correspondence ("'P' is true if P corresponds to the facts, to reality, to how things are), objectivity (truth contrasts with justification, is stable and absolute). To these platitudes one can add that truth is, as William James puts it, "the good in the way of belief": it is good, or better to have true, rather than false beliefs. It seems that without all these features, including the last one, our concept of belief would not be the one it is.

From the fact that truth is, on the face of it, a thin concept, does it follow that it is a thin value property of our beliefs? Not necessarily. Actually if being valuable is one of the "platitudes" which are attached to the concept of truth, it is not clear that this concept is so "lightweight" (Engel to appear). Certainly we do not seem to say very much when we say that truth is valuable because it is the goal of inquiry or what we "aim at" when we believe, and that error is what we try to avoid. We can express this positive and this negative goal respectively as:

(TG) (i) To believe P if P is true (TG) (ii) Not to believe P if P is false, and not to believe not P if P is true But, as a large literature shows (see e.g. [START_REF] Bykvist | Does Thought Imply Ought? Analysis[END_REF][START_REF] David | How to take Truth as a Goal[END_REF], Chan 2013), these goals are not easy to interpret. First, although it seems obvious that truth is what we try to get when we believe, it is not obvious that our aim should be to believe everything that is true. There are so many truths which are trivial or uninteresting, or dangerous to believe, that we ought to at least qualify (i) by saying that truth has only prima facie a value for our beliefs. Second, if it is a value is it a final value, or one which is only instrumental to something else? This question cannot be separated from the following: what kind of value is truth? This seems to depend upon the kind of goals one has when one tries to reach truth. If one is a scientist -if one attends primarily to epistemic value -presumably truth is a kind of final, or intrinsic value. But for many practical purposes -if one attend to practical values -truth seems to be of merely instrumental value. Things, however, are more complex, for in a number of circumstances, there are conflicts between theoretical and practical values. Can the former trump the latter and vice versa? Fourth, should we interpret (i) and (ii) as specifications of a value property of truth, or as specifications of other kinds of normative properties? Some [START_REF] Wedgwood | The Aim of Belief[END_REF][START_REF] Boghossian | The Normativity of Content[END_REF][START_REF] Engel | Truth and the Aim of Belief[END_REF]) take truth to be a norm of belief, or a standard of correctness of our believing in some constitutive sense. Is the truth of a belief something that we value, or something that we ought to attend to or to conform to? Is it something which we have most reason to attend or to conform to? It's one thing for truth to be what it is correct to believe and another thing for it to be what we aim at. The very nature of the normative concepts that we use here makes a lot of difference if we want to specify the nature of the relation of truth to belief. Is truth a value at all?

The eudaimonic value of truth

If truth is value (qua value of true belief) it can be either an intrinsic or an extrinsic value, and it can either be a final or an instrumental value. Most ancient philosophers -Aristotle first among them -claim that truth is a value not in itself but because it leads us to knowledge, which has a value not only because all men naturally seek it (Met. A, 980a22), but also because it leads to happiness or well-being as the supreme good. On this view, truth has eudaimonic value because it leads to knowledge and because knowledge is constitutive of well-being and happiness [START_REF] Hazlett | A Luxury of the Understanding, on the value of true belief[END_REF]. Thus truth would have only an instrumental value because knowledge is the primary value, to which truth is attached, in the sense that knowledge has more value than true belief. This view is reinforced by a famous argument in recent epistemology, the so-called "Swamping Argument" (Zagzebski 1996[START_REF] Kvanvig | The value of knowledge and the pursuit of Understanding[END_REF]. When one wants to go to Larissa, and with respect to that specific goal, having a true belief about the road to Larissa seems to be just as good and valuable as knowing the road to Larissa. Knowledge is thus swamped by true belief with respect to its value (here utility). Knowledge, however, as Plato noted (Meno 147b), is firmer and stronger than true belief, and for this reason, better and more valuable than true belief. If we accept this claim (although we shall see below a reason to qualify the idea that knowledge is always more valuable), true belief has a value, but this value is less than that of knowledge, hence not final. But whether it is truth or knowledge which carries the load of value, their value is the value of utility either in the narrow sense of serving our interests or in the wide sense of promoting well-being. The question is: to what extent has truth such an eudaimonic value?

To borrow Allan Hazlett's (2013) useful distinctions, knowledge and truth can have eudaimonic value: a) normally (in most cases), b) generally (in all cases), c) typically (in typical cases). This value can be either instrumental to well-being (when well-being is not constituted by knowledge) or constitutive of well-being (when it is of the essence of wellbeing to be constituted by knowledge), which can be either desire independent or desire independent. Hazlett formulates the eudaimonic ideal of true belief:

For any subject S and p normally believing what is true about p is better than believing what is false about p

To say that true belief is normally better than false belief is meant to avoid the easy objection that there can be cases where a true belief, or indeed a piece of knowledge, can be in some sense disvaluable. Cases abound, from the weak tennis player who would be better off not believing that she is going to lose her match rather than keeping the heartening belief that she is able to win, to the garden variety cases of rational self-deception (the spouse who prefers to ignore the lipstick on her husband's collar). Nobody would deny that "sometimes the value of truth is outweighted by other considerations" [START_REF] Horwich | The Value of Truth[END_REF]. In that respect, true beliefs may be only pro tanto valuable. "Valuable" here means: with respect to its contribution to the well-being of the agents who have them.

The problem, however, is that true belief or knowledge are not only sometimes disvaluable and false belief or ignorance sometimes valuable, but that they could also normally -more often than not -be so. Hazlett (2013: ch.2) argues, mobilizing a lot of evidence from cognitive and social psychology, that self-knowledge is not only sometimes, but actually very often, a bad thing, and ignorance of one's exact credentials can be a good thing. When people indulge in systematic self-esteem, and self-enhancement biases, when they nourish false hopes, are irrealistically optimist or entertain illusions of control over their plans or their lives, they not only sometimes but most often end up better off, happier and less depressed. This involve various form of self-deception or of wishful thinking, but this is all to the good for the individual. "Don't worry, be happy". Hazlett further argues that partiality and charity biases, by which we trust our friends and lovers sometimes against evidence or display systematic confidence in what they say, not only enhance well-being, but are also positive virtues constitutive of it. Emerson praised the value of "self-reliance". People care for other things than true belief (non-alethic goods) and there are cases where false belief is associated with non-alethic goods. Hazlett concludes that "there is no clearly identifiable pattern of cases where true belief is better than false belief". In any case, true belief seems, with respect to false belief, to enjoy no privilege and to have a quite neutral status with respect to their respective contribution to our well-being.

One might, however, wonder whether such biases are really constitutive of well-being. In the first place, it is hard to believe that well-being could normally depend upon lying to oneself or upon self-deception. In the second place, true belief may be useful to life in general, simply because it is necessary for action. If we take up a classical line of thinking that has been formulated most clearly by Frank Ramsey, true belief is required for successful action, and we act on the basis of our beliefs about how we could realize our desires. In this sense true belief always has instrumental value, just in virtue of the nature of action. Paul Horwich develops this line of thought in order to argue that true beliefs are always valuable because they lead to action: Directly action-guiding beliefs of the form, 'If I perform A, then X will occur'. It will clearly benefit me if I have many such beliefs and if they are all true. Because when I want a given thing and believe that a certain action will result in my getting it, then, very often, I will perform that action. And in that case, if my belief is true, this desire will be satisfied; whereas if it isn't true no such result is ensured. So true beliefs of the directly action-guiding form will indeed tend to benefit me. And the more such true beliefs I have the broader the spectrum of desires that will be easy for me to satisfy in this way. Moreover, these special beliefs are the results of inferences that tend to preserve truth; so it will benefit me for the premises of those inferences to be true. And there is no proposition that might not someday serve as such a premise. Therefore it will indeed be good for me-at least, that's what it's reasonable for me to suppose-if I believe every true proposition and if every proposition I believe is true. (Horwich 2006, 350) This general instrumental value of true belief is independent from the occasional disvalue of some true beliefs and from the sometimes valuable nature of false beliefs and biases, and is not threatened by these exceptions.

Neither is it threatened by the familiar examples of trivial and useless true beliefs which are often adduced against the positive TG (i) version of the goal of having true beliefs. Indeed counting the number of blades of grass in the garden or of grains of sands on the beach, trying to know how many people have a name beginning with the letter "D" in Wichita, Texas, or asking oneself whether Joe di Maggio had a 56-game hitting streak, are idle attempted believings or knowings. Other alleged counterexamples include beliefs about things which are so esoteric that no one would care to acquire them. Now as soon as we try to specify criteria for what kind of knowledge or belief is significant or potentially significant, we run into trouble. Some very idle or trivial beliefs might turn out to be significant in one circumstance or other, whether or not we can figure out how they can be such, and valuable for one reason or another. As soon as one attends to the particular cases, there is always room for either granting these beliefs value or disvalue. But along to what axis or criterion of evaluation? It is obvious that typically any true belief, as idle, trivial or useless it can be, is valuable, as the Ramsey-Horwich kind of reasoning establishes. The Ramsey-Horwich line takes the value of truth to be not a property which attaches to truth in general, but only a property which attaches to each particular truth which is a candidate for being believed. For each "action-guiding" proposition, there will a specific value in believing it, in so far as it leads to successful action. The value of truth in general is only the generalization on the list of such action-guiding propositions. But the value in question is utilitarian or success-inaction value, and one might ask whether true belief cannot be valuable in general, independently of whether it leads to successful action. For isn't it the case that any truth, however trivial or insignificant, is of epistemic value, in so far as it is a truth? (Lynch 2004: 152). Aren't truth and knowledge common goods just as water and fresh air are supposed to be common goods for mankind [START_REF] Zagzebski | The Search for the Source of Epistemic Good[END_REF])? Here we should remind ourselves that there are different kinds of value, and in particular not only practical values, but also epistemic ones. From the fact that true belief may be disvaluable or less valuable practically, it does not follow that it is disvaluable, period. In particular there is a dimension of evaluation along which true belief is prima facie valuable, which is epistemic evaluation. In so far as truth is the epistemic goal of inquiry, any truth whatsoever is epistemically valuable, including the most trivial or insignificant ones. Indeed this remark does not suppress the problem of distinguishing significant from insignificant true beliefs, but the fact that all truths are epistemically good does not mean that they are all equally epistemically good (Treanor 2013, Pritchard 2014, 121) We can indeed sort out those which are deep and which augment our knowledge of the world from those which are idle or shallow. But that does not prevent all truths to be, in variable degree, epistemically good.

At this point we should pause a bit to think again about the "swamping argument" alluded to above. It purports to show that knowledge is no better than true belief with respect to practical purposes. But from the fact that the practical value of true belief can swamp the practical value of knowledge, nothing follows about the epistemic value of true belief with respect to the epistemic value of knowledge. To use again Pritchard's terms, one should distinguish the value (or the disvalue of the epistemic) from epistemic value. And the latter is to be evaluated in terms of truth, evidence and knowledge. Now this distinction between epistemic and practical value seems to beg the question against those who asks: "Is true belief really valuable as such?" For what they ask, when they point out the value of self-confidence, of trust and of various biases, they are not evaluating our beliefs from the epistemological or cognitive point of view, but also from the practical one, and their point is that in spite of its bad epistemic credentials, belief without evidence or false belief can turn out to be beneficial for the individual and thus contribute to his overall well-being. Pragmatists of all sorts (e.g. [START_REF] Foley | Working without a Net[END_REF]) are fond of telling us that there is a dimension of comparability of the epistemic and of the practical, which makes the question "What should I believe?" both epistemological and practical, or perhaps neither. The ill person who knows that her belief that he will recover enhances her chances of recovering is asking a question which belongs to the two dimensions. James' alpinist who asks himself what his chances are to survive if he leaps across a dangerous mountain chasm, people who compare the advantage of believing at will over those of simply following the evidence clearly reason along the two dimensions. When we talk about the eudaimonic value of true belief we certainly evaluate it from the practical point of view, and we are obviously comparing epistemic value and practical value. But does it follow that when we engage in this sort of comparison we cease to evaluate our beliefs from the cognitive point of view? A wishful belief or a self-deceptive belief, a self-confident belief and an attitude of trust do not cease to be false, evidentially fragile or cognitively unreasonable when they play a positive role in our lives. Beliefs, like restaurants, can be evaluated from the standpoint of all kinds of values and normative standards. One may choose a restaurant for its food, but also for its atmosphere or for its proximity. Similarly for beliefs. They can be well-founded or not, beneficial or not to the believers, aesthetically satisfactory (dandies like to believe what is gracious or sublime) or simply preferred because they are popular (those who follow fashion or snobs like to believe what the rulers of fashion or of opinion dictate). But does it mean that there is no primary dimension of assessment of belief? No. Wishful thinking, as useful as it can be for ostriches or for men, is always prima facie wrong. False, fragile or biased beliefs too. The same is true for restaurants, which have to be evaluated for their food first: one can like a restaurant for its atmosphere and choose it for that reason, but if the food is bad there is something definitely wrong. In that respect belief cannot fall short of being evaluated epistemically. As Bernard [START_REF] Williams | Truth and truthfulness[END_REF] reminds us: falsity is a fatal defect for a belief.

Another way of expressing the same idea is to say that epistemic evaluation is exclusive for belief. And here, for reasons which I am going to give in the next paragraph, it is more appropriate to talk in terms of reasons rather that in terms of value. In a number of circumstances, we evaluate our beliefs on the basis of other criteria than epistemic: we have plenty of reasons to want to believe certain things. But our reasons for wanting to believe are not the same as our reasons to believe. The former are much more diverse than the latter. When believing is -in the cases when we have the power to acquire a belief -the object of a deliberation leading to an action, the reasons which we can have to want to have this belief are much more diverse than those that we have for believing , period. One can want to believe something because one finds it pleasant, comfortable, beautiful, useful, and in some sense good or valuable. But however good it can be to want to believe somethingand this goodness can be appreciated along many dimensions -there is only one kind of reason for believing proper: epistemic reasons, that is truth and evidence, which are "the right kind of reason" (Millar 2003[START_REF] Hieronymi | The wrong kind of reason[END_REF][START_REF] Parfit | On what Matters[END_REF]: appendix A, Engel 2013b).

The exclusivity of epistemic reasons does not entail that belief can be evaluated along the other dimensions. Truth can be good or bad in quite a number of respects. It can be bad for personal life, but also good for social and political life and for democracy in general [START_REF] Lynch | True to Life[END_REF]. Nietzschean thinkers can well tell us that truth as a general goal and value is a mythology in the service of power, and that worshiping truth can be in many ways dangerous. Pragmatists can well tell us that truth is useless or disvaluable in many ways, and that other social goals and values, such as solidarity. But one might wonder whether the potential disvalue of truth can be appreciated without attending to the central role of truth in any evaluation -good or bad -about the value of our beliefs -in other words how we could say anything about the value of truth without taking into account its epistemic value first. One might wonder also whether anything of value in personal or in social life could be achieved without it [START_REF] Williams | Truth and truthfulness[END_REF]. Truth is at least valuable by default: life would be much harder without it. This does not mean that it automatically adds value to life when it comes in.

The essentialist view: teleology

The prima facie epistemic nature of the evaluation of beliefs suggests that the goal of believing truths and only truth is not only a goal or an aim which could be a source of value, but that it is essentially so, and that the relation between belief and truth is in some sense constitutive. But in what sense?

According to what we can call essentialism about belief, the nature of epistemic evaluation derives from the nature of belief [START_REF] Hazlett | A Luxury of the Understanding, on the value of true belief[END_REF]: ch 5, see also Fassio 2012). There are, however, different forms of belief-essentialism, which can vary along two dimensions, which are respectively the kind of state that belief is and the kind of evaluation which is appropriate. One can, on the one hand, claim that belief has an essential nature because of (i) its metaphysical nature as a mental state, or because (ii) our concept of belief has certain constitutive a priori features. One can, on the other hand, evaluate beliefs by using different sorts of normative notions: (iii) one can assess beliefs as good or bad, as valuable or not, along an axiological dimension, or (iv) on can assess beliefs as being correct or incorrect, along a normative or deontic dimension. In general essentialist approaches of the metaphysical kind (i) are associated with the axiological dimension, because belief is supposed to aim, by nature, towards a certain kind of goal, hence to have a certain kind of teleology. Essentialist approaches of the conceptual kind (iv) are most often associated with the normative concepts of what one ought to believe, or of what one has most reason to believe.

Teleological views have in common the idea that it is the essence of belief to have a certain kind of aim, or goal, or direction or objective. The notion of an aim or of a goal suggests that aiming at true beliefs is the conscious or intentional objective of the believer. Some views are intentionalist in this sense. Thus [START_REF] Velleman | On the Aim of Belief, in his The possibility of Practical Reason[END_REF] holds that the distinguishing essence of belief with respect to other cognitive attitudes (such as guessing, imagining or supposing) is to be the attitude which is such that the believer aims at accepting its content as true if only if it is true, and Sosa (2011) compares believing with the intentional activity of an archer who tries to hit a target -here truth -and succeeds or not to reach this target. On such views, beliefs are, to a certain extent, active states of mind, and to a certain extent, kinds of actions or at least display a certain amount of epistemic agency. But it need not be so. Hume famously held that beliefs and desires have "distinct existences": belief or "reason" is concerned with only what is true or false, whereas desires are concerned with what we aim at. Anscombe (1958) and Searle (1989) have elaborated this distinction as that between two "directions of fit" -mind to world and world to mind -which respectively belief and desires have as mental states independently of whether we take them to have that direction and intend to exploit it. On this Humean view, beliefs are essentially directed at truth whether we desire or intend it or not. This is perfectly compatible with Hume's argument that belief is not a matter of the will and is an involuntary mental state. There are two variations upon the Humean view. One is the functionalist approached, pioneered by Ramsey, according to which belief is the kind of mental state which is such that it can be, together with desires, the cause of our actions. A functionalist theory of belief says that it is of the essence of belief to be the kind of state which receives input information from the environment and which, on the basis of desires, leads to behavioral outputs. The other is the Darwinian approach, according to which not only belief is that very kind of functional state, but also such that natural selection has selected it as the kind of state that it is [START_REF] Millikan | Truth Rules, Hoverflies, and the Kripke-Wittgenstein Paradox[END_REF][START_REF] Dretske | Norms, History, and the Mental, in Perception, Knowledge and Belief[END_REF][START_REF] Papineau | Normativity and Judgment[END_REF]). On the teleological view, the only normativity which is involved in epistemic evaluation is the one which is attached to the value that the agent -which can be nature itself -sets on having true belief.

One can raise at least four objections against teleological view, both in its intentionalist and in its nonintentionalist versions.

The first objection concerns the intentionalist version: believing is not, at least primarily, an intentional activity. Even if belief is the product of intentions in some specific kinds of believing -those which involve mental action, judgment and acceptance towards a certain content, this can hardly be true in general. One must leave room for unintentional kinds of believing -the paradigm example being beliefs based on perception and cases of unconscious belief formation -unless one espouses the implausible view that believing always involves some intentional activity -ranging from trying to reach its truth goal to doxastic control and commitment (Engel 2005).

The second problem affects both the intentionalist and the non-intentionalist version. It has to do with the fact that whatever the goal of belief can be -whether it aims at truth, or at knowledge, or at securing our well-being -this goal can in principle be balanced against other goals and changed. If one conceives of the aim of belief in intentional terms and as a goal, one has to accept the idea that the goal can be compared with others, and that it could change depending upon the aims of the believer. The problem is that belief is hardly a goal directed activity in this sense, and quite unlike an action. When one aims at something either in the sense of intending to do it or in the sense of having a long term plan, one typically can balance this objective or this goal against at least another one. But belief is quite unlike that. Believing is not like guessing, when one hesitates between various options, since there is actually no other choice than holding true or holding false, hence adopting an epistemic stance anyway. Even suspension of judgment, which comes close to having the choice between taking one option or another, cannot occur between choosing between an epistemic aim and a practical aim. When one believes there is no way to balance the truth goal against a practical goal. Truth is the only goal here is, and in this sense it cannot be a goal: epistemically there is no other choice, and as we saw above, when there is an apparent choice between an epistemic a non epistemic goal, the epistemic one is always the one which imposes itself by default.

The third problem about the teleological view is specific to its Darwinian or biological version. If belief is the kind of mental state that it is, with its specific direction of fit, and if i aims at truth in virtue of its having been selected by natural selection, how can it be the essence, in the metaphysical sense, of belief to have these characteristics? It has to be a merely contingent feature of our psychology. Can't we conceive of a distinct state -let us call it schmelief -which would be such that in most cases it would be directed at truth, but which in other cases it could be directed at falsity, and which in any case would not invariably be directed at truth (Papineau 2013)? On the Darwinian view there is no obstacle to such a supposition, and for that reason it cannot be part of the essence of belief that it aims at truth. Thus there is a tension in the Humean view and in the Darwinian view. On the one hand they say that aiming at truth is a general fact about belief, which is its essence. On the other hand these views tell us that this fact holds naturalistically, hence is contingent. Hence there is no essential aiming at truth in belief, no metaphysical nature of belief.

The fourth problem has to do with the normative force of the epistemic evaluation for teleological theories. On the Humean view there is no other normativity than the direction of fit of belief. Beliefs are the kind of attitudes that have the mind to world direction of fit. But no normative advice, even less a normative prescription or guidance can be involved in this bare fact: form the fact that my beliefs are supposed to be true or false in virtue of being the kind of natural mental state that they are, it follows nothing about what I ought to believe or not, which is normally what one can expect from a normative guidance [START_REF] Dretske | Norms, History, and the Mental, in Perception, Knowledge and Belief[END_REF]. The intentionalist version does not fare better. If the normative force of truth in believing is that of an intention to reach, through believing, the goal of reaching a truth, then this force is no stronger than that of a hypothetical imperative of the kind: if one wants to have true beliefs one ought to acquire the belief that p. But this imperative, being conditional on the desire or intention of the believer, is much too weak to capture the normative force of the evaluation, which is that of a categorical and unconditional imperative:

(TO) one ought to believe that p if and only if p which does not depend on the condition that the believer wants or intends to believe the truth. The normative force cannot be simply instrumental, as all the Humean views presuppose [START_REF] Kelly | Epistemic rationality and instrumental rationality: a critique[END_REF]. It depends not on a prior mental state but simply on how things are for the believer. Even on the teleological reading, the aim of belief should not depend on contingent desires or intentions. It should be a fixed aim. But how can it be, given that aims can, by nature, change?

The normative account

These objections lead us to favor an account according to which it is an a priori and constitutive property of our concept of belief that it is subject to a norm of correctness, which is truth. The constitutive correctness norm of belief, on such a view is the following (TN) Necessarily S's belief that p is correct if and only if p, and incorrect otherwise (TN) is supposed to be necessary, hence to treat the norm of belief as an essential feature of belief. In this sense, belief has a normative essence (Wedgwood 2007). (TO) is but one way of interpreting (TN). But (TN) can also be conceived as a conceptual a priori truth about belief [START_REF] Boghossian | The Normativity of Content[END_REF].

What speaks in favor of the correctness account is that, unlike the teleological one it captures the normative force of the relation between belief and truth. (TN) is not contingent upon the desires or the intentions of the believer. It accounts better (in the sense of inference to the best explanation) for some of the most pervasive features of belief. First, the fact that belief is involuntary and not under direct control of the will: if there is a norm to believe truths and only truths, any willful believing has to violate that norm. This is not to say that it cannot occur, but that if it does it has to occur against this norm. Second, the fact that "Moorean" beliefs or assertion of the form "P but I believe that not P" are paradoxical: if there were not a direct relation between one's asserting or believing that P and one's believing that P is true such Moorean assertions would not be strange. Third, the correctness norm for belief accounts for the "transparency" of belief: when one deliberates about whether to believe that P, the question is settled as soon as one realizes that P is the case (Shah 2003). Remember Moore's remark quoted above: "I cannot at any given moment distinguish what is true from what I believe". If there were not this direct connection between believing that P and believing that P is true, the remark would make no sense. Fourth, the normative account gives the best explanation for the centrality of belief among other belief-like attitudes and quasi-doxastic states, such as suppositions, acceptances, guesses, imaginings, partial beliefs, tacit beliefs, subdoxastic states, creedal feelings, feelings of knowing, pathological beliefs, phobias, "aliefs", delusions, biases. Some of these attitudes and states (like guessing, imagining or supposing) resemble belief in having propositional contents and being truth-evaluable. Others are "strange bedfellows" for belief, since it is not clear that they have a propositional content or are truth evaluable. The best criterion to distinguish these from beliefs is to see whether they are subject to the correctness norm and the transparency test. There are, however, reasons to think that they do not pass this test. This conceptualist account is Kantian in spirit because it involves the element of reflection: the standard or norm of right belief applies to us as soon as we reflect upon the nature of correct belief, which exists regardless whether one wants or not to conform to those standards (Hazlett 2013, 206). This reflective element is most clear in the transparency feature mentioned in the previous paragraph. One can further argue that the norms of belief, as well as those of action, form part of the normative order in the objective sense, as parts of the domain of reason [START_REF] Skorupski | The Domain of Reasons[END_REF]. In this sense the Kantian conception is essentialist. But it need not be essentialist in an objective metaphysical sense, in which the domain of reasons and of norms would be a further domain of facts, alongside natural facts. The Kantian view is rather constructivist: it does not say that there is a normative domain of facts, among which would featurenormative facts, and among which normative intentional facts. On this latter view one can conceive, in a more Platonist and cognitivist vein, of the normative nature of belief as an essential property of belief, and more widely of intentional states in general and take them to be (Wedgwood 2007[START_REF] Parfit | On what Matters[END_REF].

There are a number of objections against the normative account of belief, which I cannot examine here. A number have to do with the specific form which the biconditional (TN) is supposed to take in order to be able to guide properly belief formation [START_REF] Bykvist | Does Thought Imply Ought? Analysis[END_REF][START_REF] Gibbons | The Norm of Belief[END_REF], Chan 2013). The main objection is that it is not clear that the norm (TN) is normative at all. On the one hand, it is supposed to prescribe what one ought to believe it is much too strong: if (TN) were the correctness norm for belief, it would necessarily motivate us to believe that P if and only if P is true. But it need not do so. The norm does not inescapably motivate us to believe. As a prescriptive norm, it is implausible (Steglish-Pedersen 2006, Glüer and[START_REF] Glüer | Against Content Normativity[END_REF]. On the other hand, if it is supposed to be prescriptive about what one ought to believe, it is much too weak and it has no normative force. From the fact that it is correct to believe that P if and only if P, nothing at all seems to follow for how one has to go about with respect to believing that P. That these objections contradict each other shows that there must be something wrong with the premiss from which they start. They both presuppose that the correctness norm is normative in the sense that it ought to be prescriptive of our believing and guide our belief formation. But the presupposition is wrong. (TN) does not say, and doesn't have to say, what kinds of beliefs we have to adopt or how we have to adopt them (e.g whether we have to maintain them, to revise them or in which conditions we have to withdraw them). It not prescriptive in the sense of what J.J. Thomson (2008) calls "directives", and it prescribes no action, be it epistemic or not. It just says what we ought to believe in the most idealised sense (Engel 2013(Engel , 2013 a) a) Another strong objection against the normative account, especially in the cognitive essentialist sense (but also in the Kantian conceptualist form) is that it does not account for the supervenience of the normative properties or concepts upon the natural ones. The dilemma here is familiar: either the normative properties do not supervene on the natural ones and are left dangling without any natural basis. I cannot here deal with this objection. But one must remark here that any attempt to reconcile the normative essence of belief with natural facts will have at some point to assume that the norms of correct belief, and the objective reasons that there are to believe, have to depend in some sense from our psychological states, and most upon our desires. Only these can belong to the natural basis of our reasons, and only these can properly motivate us to accept the epistemic norms and to conform to them. If one takes this line (which is the one taken by most anti-realists and non-cognitivists about epistemic norms and values, especially expressivists) then one will have to reject two of the claims which I have taken to be central to epistemic norms: their categorical, non instrumental character on the one hand, and their exclusivity, the fact that epistemic reasons are by essence the "right kind" of reasons. One way or another we shall have to take exception to the supervenience of the normative on the natural.

Farewell to Plumpton

If the foregoing rambling thoughts are correct, there is no specific problem of the value of truth, because truth is not, primarily and constitutively, a value. One can ask whether it has value, including final and intrinsic value, but any appreciation of the value of truth will have to start from an appreciation of epistemic value in general. Epistemic value is best thought of not in terms of value, but in terms of norms. The normative stance has priority over the evaluative stance. Belief is subject to epistemic evaluation first. This does not prevent us from asking whether true belief can have a practical value or contribute to well-being.

I have not given any argument here in favor of non cognitivist and realist conceptions of epistemic normativity, as against anti-realist, non cognitivist and expressivist views. But if the considerations proposed in §3 against the teleological conception of epistemic normativity are correct, they favour a realist account. It remains to be seen whether it should take the form of a buck-passing account, of a Kantian constructivism, or of some form of Platonism about norms and reasons. In all this we shall very probably have to say goodbye to Ramsey's pragmatism. i

  . One can here compare the epistemic norm of truth with what Parfit (2011, 417 sq.) says about normative truths in ethics. They are not supposed to tell us what to do or to motivate us for certain actions. They are supposed to tell us what we ought to do, and what kinds of reasons we have. As Parfit says, if there were no such truths about our reasons, we could not begin to ask ourselves what kinds of decisions to take or how to live. Similarly, the correctness norm for belief tells us what we ought to believe, and what kinds of reasons we have. Such reasons do not depend upon our desires or upon our attitudes. They are objective.