Hamza Ennaji
email: hamza.ennaji@unilim.fr

Noureddine Igbida
email: noureddine.igbida@unilim.fr.

NOUREDDINE IGBIDA † Van Thanh Nguyen
email: nguyenvanthanh@qnu.edu.vn.

Thanh Van

Nguyen

Lambertian Continuous

Shape

CONTINUOUS LAMBERTIAN SHAPE FROM SHADING: A PRIMAL-DUAL ALGORITHM

published or not. The documents may come

Introduction

Shape from Shading (SfS) consists in reconstructing the 3D shape of an object from its given 2D image brightness. The shape of a surface u(x 1 , x 2 ) is related to the image brightness I(x 1 , x 2 ) by the Horn image irradiance equation:

R(n(x 1 , x 2 )) = I(x 1 , x 2 ), (1.1
) where I(x 1 , x 2 ) is the brightness greylevel measured in the image at point (x 1 , x 2 ); R(n(x 1 , x 2 )) is the reflectance map and n(x 1 , x 2 ) is the unit normal at point (x 1 , x 2 , u(x 1 , x 2 )) given by n(x 1 , x 2 ) = 1 1 + |∇u(x 1 , x 2 )| 2 (-∂ x 1 u(x 1 , x 2 ), -∂ x 2 u(x 1 , x 2 ), 1).

In (1.1), the irradiance function I(x 1 , x 2 ) is known since it is measured at each pixel of the brightness image, for example, in terms of greylevel in the interval [0, 1]. The implicit unknown is the surface u(x 1 , x 2 ), which has to be reconstructed.

In the case of Lambertian and the surface illuminated by a simple distant light source of direction = (w, r) = (w 1 , w 2 , r) ∈ R 3 , one has R(n(x 1 , x 2 )) = n(x 1 , x 2 ) • (w 1 , w 2 , r) and, by (1.1),

I 1 + |∇u| 2 + ∇u • w -r = 0.
This equation falls into the scope of Hamilton-Jacobi equations

H(x, ∇u) = 0 in Ω, (1.2)
where the Hamiltionian H is defined by H(x, p) = I 1 + |p| 2 + p • w -r. In particular, if the object is vertically enlightened, i.e., = (0, 0, 1), one obtains the standard Eikonal equation

|∇u(x 1 , x 2 )| = 1 I 2 (x 1 , x 2 ) -1. (1.3)
As pointed out in [START_REF] Durou | Numerical methods for shape-from-shading: A new survey with benchmarks[END_REF] (see also [START_REF] Zhang | Shape from shading: a survey[END_REF]), there are three major families of numerical methods allowing the resolution of the SfS problem. Namely, PDE methods (cf [START_REF] Bardi | Discrete approximation of the minimal time function for systems with regular optimal trajectories. Analysis and optimization of systems[END_REF][START_REF] Falcone | Level sets of viscosity solutions: Some applications to fronts and rendez-vous problems[END_REF][START_REF] Lions | Shape-from-shading, viscosity solutions and edges[END_REF][START_REF] Prados | A viscosity solution method for shape-from-shading without image boundary data[END_REF][START_REF] Rouy | A viscosity solutions approach to shape-from-shading[END_REF]), optimization methods (cf [START_REF] Crouzil | A multiresolution approach for shape from shading coupling deterministic and stochastic optimization[END_REF][START_REF] Horn | The variational approach to shape from shading[END_REF]) and approximating the image irradiance equation (cf [START_REF] Hayakawa | A computational model for shape estimation by integration of shading and edge information[END_REF][START_REF] Pentland | Local shading analysis[END_REF][START_REF] Ping-Sing | Shape from shading using linear approximation[END_REF]).

We are here interested in the study of the PDE formulation in terms of Hamilton-Jacobi equations (1.2). The theory of viscosity solutions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF][START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF][START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF] provides a suitable framework to study equations of the form (1.2). Applications of the viscosity theory to the SfS problem go back to the works of Lions, Rouy and Tourin [START_REF] Lions | Shape-from-shading, viscosity solutions and edges[END_REF][START_REF] Rouy | A viscosity solutions approach to shape-from-shading[END_REF] and recently in the work of Prados, Camilli and Faugeras [START_REF] Prados | A viscosity solution method for shape-from-shading without image boundary data[END_REF]. Several difficulties arise while dealing with the SfS problem, namely compatibility of boundary conditions and the degeneracy of the Hamiltonian. It is well known that for (1.2) coupled with the boundary condition u = g on ∂Ω, to admit a solution one needs to check that g(x)-g(y) ≤ d σ (y, x) for all x, y ∈ ∂Ω, where d σ is the intrinsic distance associated to the Hamiltonian, which will be defined later. In addition, imposing only boundary conditions is not sufficient to ensure the uniqueness of solution to the Hamiltons-Jacobi equations (1.2). It turns out that the set of degeneracy of the distance d σ , called the Aubry set, plays the role of a uniqueness set for (1.2) (see e.g. [START_REF] Fathi | PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians[END_REF]). In the case of Eikonal equation (1.3), the Aubry set A can be taken as the zero set [k = 0] of k = I -1/2 -1. In other words, it corresponds to the points with maximal intensity I, i.e., I(x 1 , x 2 ) = 1 so that the right hand side in (1.3) vanishes. Most of the authors (cf [START_REF] Camilli | Numerical approximation of the maximal solutions for a class of degenerate Hamilton-Jacobi equations[END_REF][START_REF] Rouy | A viscosity solutions approach to shape-from-shading[END_REF][START_REF] Prados | A viscosity solution method for shape-from-shading without image boundary data[END_REF] for example) choose to regularize the equation to avoid these points. We will not encounter this difficulty in our approach since we do not need to deal with the inverse of possibly vanishing functions. We only need to perform projections onto euclidean balls whose radii may be equal to zero.

In this paper, following our approach in [START_REF] Ennaji | Augmented Lagrangian method for Hamilton-Jacobi equations[END_REF], we characterize the maximal viscosity subsolution of (1.2) in terms of a concave maximization problem. We then associate a dual problem and exploit the saddle-point structure to approximate the solution of (1.2) using the Chambolle-Pock (CP) algorithm. Our approach lies between the PDE and optimization methods since we start by characterizing the maximal viscosity subsolution of the HJ equation thanks to the intrinsic metric of the Hamiltonian and we end up with an optimization problem under gradient constraint. Moreover, the convergence of discretization is also studied in detail.

The paper is organized as follows. In Section 2, we start by recalling briefly some notions on HJ equations, and we present the maximization problem following [START_REF] Ennaji | Augmented Lagrangian method for Hamilton-Jacobi equations[END_REF] as well as related duality results in continuous setting. Section 3 is devoted to the discretization issue and the proof of convergence. We show how to apply the CP algorithm and present some numerical results to illustrate our approach in Section 4.

Maximization problem and duality in continuous setting

In this section we recall the metric character of Hamilton-Jacobi (HJ) equations, and related duality results which will be useful to the proof of convergence of discretization in Section 3.

(H1) coercivity: Z(x) is compact; (H2) convexity: Z(x) is convex; (H3) H(x, 0) ≤ 0, i.e. 0 ∈ Z(x).
We consider the following HJ equation

H(x, ∇u) = 0, x ∈ Ω.
(2.4)

A continuous function u : Ω → R is said to be a viscosity subsolution (respectively supersolution) of (2.4) if H(x, ∇φ(x)) ≤ 0 (respectively H(x, ∇φ(x)) ≥ 0) for any x ∈ Ω and any C 1 function φ such that u -φ has a strict local maximizer (respectively minimizer) at x. Finally, u is a viscosity solution of (2.4) if it is both a subsolution and a supersolution. We denote by S - H (Ω) (respectively S + H (Ω)) the family of viscosity subsolutions (respectively supersolutions) of (2.4). For x ∈ Ω, we define the support function of the 0-sublevel set Z(x) by

σ(x, q) := sup q • Z(x) = sup{q • p | p ∈ Z(x)} for q ∈ R N .
The assumption (H1)-(H3) ensures that σ is a possibly degenerate Finsler metric, i.e., σ is a continuous nonnegative function in Ω × R N , convex and positively homogeneous with the second variable q. Due to the assumption (H3), σ(x, q) can possibly be equal to 0 for q = 0, which leads to the degeneracy and its dual σ * , as defined below, may take the value +∞. Here, the dual σ * (also called polar) is defined by

σ * (x, p) := sup q {p • q | σ(x, q) ≤ 1}.
Let us define the intrinsic distance by

d σ (x, y) := inf ζ∈Lip([0,1];Ω) ζ(0)=x,ζ(1)=y 1 0 σ(ζ(t), ζ(t))dt,
which is a quasi-distance, i.e. satisfying d σ (x, x) = 0 and the triangular inequality, but not necessarily symmetric. We summarize some basic characterizations of subsolutions in terms of the intrinsic distance d σ .

Proposition 2.1. ( [START_REF] Ennaji | Augmented Lagrangian method for Hamilton-Jacobi equations[END_REF][START_REF] Fathi | PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians[END_REF][START_REF] Igbida | Augmented Lagrangian method for optimal partial transportation[END_REF])

1) Compatibility condition: v ∈ S - H (Ω) if and only if v(x)-v(y) ≤ d σ (y, x) for any x, y ∈ Ω. 2) We have u(x) -u(y) ≤ d σ (y, x) ⇐⇒ σ * (x, ∇u) ≤ 1 a.e in Ω.

2.2.

A maximization problem and duality. Given a closed subset D ⊂ Ω (typically D = ∂Ω or D = {x} for some x ∈ Ω), we consider the following HJ equation

H(x, ∇u) = 0 in Ω \ D u = g on D (2.5)
where g : D → R is a continuous function satisfying the compatibility condition g(x) -g(y) ≤ d σ (y, x) for any x, y ∈ D.

Thanks to Proposition 2.1, the following result allows us to approach the SfS problem via a maximization problem.

Theorem 2.2. [START_REF] Ennaji | Augmented Lagrangian method for Hamilton-Jacobi equations[END_REF] The unique maximal viscosity subsolution of the equation (2.5) can be recovered via the following maximization problem

(M) := max u∈W 1,∞ (Ω) Ω u(x)dx, σ * (x, ∇u(x)) ≤ 1 and u = g on D .
This problem can be linked to the following dual problem. For simplicity, we will state it for the case D = ∂Ω (which is essentially the case for other numerical computations).

Theorem 2.3. We have

max u∈W 1,∞ (Ω) Ω u(x)dx, σ * (x, ∇u(x)) ≤ 1 and u = g on ∂Ω = inf φ∈L 2 (Ω) N Ω σ(x, φ)dx -g, φ • n H 1/2 ,H -1/2 : -div(φ) = 1 in D (Ω) := (OF).
Proof. To prove the duality between (M) and (OF) in Theorem 2.3, we use a perturbation technique as follows. Define on L 2 (Ω) N the following functional

E(p) := -sup Ω udx : u ∈ Lip(Ω), σ * (x, ∇u(x) -p(x)) ≤ 1, u = g on ∂Ω .
Then, one can check that E is convex and lower semicontinuous. To compute E * we start by observing that since u = g on ∂Ω, we can assume thanks to trace lifting Theorem that g = γ 0 (w) for some w in H 1 (Ω), and

u = ξ + w with ξ ∈ H 1 0 (Ω) ∩ W 1,∞ (Ω). We then have for any φ ∈ L 2 (Ω) N : E * (φ) = sup p∈L 2 (Ω) N Ω φ • p dx -E(p) = sup p∈L 2 (Ω) N ,ξ∈H 1 0 (Ω) Ω φ • p dx + Ω ξdρ + Ω wdx : σ * (x, ∇(ξ + w) -p) ≤ 1 .
Set q = ∇(ξ + w) -p we get p = ∇(ξ + w) -q, we then have

E * (φ) = sup q∈L 2 (Ω) N ,ξ∈H 1 0 (Ω) Ω φ • (∇(ξ + w) -q)dx + Ω ξdx + Ω wdx : σ * (x, q(x)) ≤ 1 = sup ξ∈H 1 0 (Ω) q∈L 2 (Ω) N , σ * (x,q(x))≤1 Ω φ • ∇ξdx + Ω ξdx + Ω φ • ∇wdx + Ω wdx - Ω φ • qdx .
The last quantity is finite if we impose that

Ω φ • ∇ξdx + Ω ξdx = 0 for all ξ ∈ H 1 0 (Ω), which means that -div(-φ) = 1 and consequently φ ∈ H div (Ω). Thus the normal trace of φ is well- defined and φ.n ∈ H -1/2 (∂Ω). Taking ν = -φ.n, then -div(-φ) = 1 -ν in D (R N
) and therefore, for such a φ, integrating by parts we get

E * (-φ) = sup q∈L 2 (Ω) N Ω φ • q dx : σ * (x, q(x)) ≤ 1 + w, ν H 1/2 ,H -1/2 = Ω σ(x, φ)dx -g, φ.n H 1/2 ,H -1/2 .
Finally,

max(M) = -E(0) = -E * * (0) = -sup -E * (-φ) = inf E * (-φ) = inf(OF),
as desired.

Remark 2.4. Notice here that the proof remains to be true also for the general maximization problem

max u∈W 1,∞ (Ω) Ω u(x) ρ(x)dx, σ * (x, ∇u(x)) ≤ 1 and u = g on ∂Ω ,
where ρ ∈ L 2 (Ω). See that dealing with this duality, the trace of the dual variable σ on the boundary plays an important role in the dual problem. For the context of general Radon measure ρ, one can see the paper [START_REF] Ennaji | Augmented Lagrangian method for Hamilton-Jacobi equations[END_REF]. In this case one needs to deal with technical functional space DM p (Ω) (space of vector fields φ ∈ L p (Ω) N whose divergence are bounded measures). On such a space, one can give a sense to φ • n, the normal trace of φ on ∂Ω (cf [START_REF] Ennaji | Augmented Lagrangian method for Hamilton-Jacobi equations[END_REF]). For a different duality approach with free Radon measure boundary trace one can see the paper [START_REF] Ennaji | Beckmann-type problem for degenerate Hamilton-Jacobi equations[END_REF].

Remark 2.5. The so-called Aubry set, denoted by A, is defined as the set where the quasidistance d σ degenerates. Prescribing a boundary value on ∂Ω doesn't guarantee the uniqueness of viscosity solutions to (2.4) unless if A = ∅. The Aubry set A appears then to be a uniqueness set for (2.4) (see [START_REF] Fathi | PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians[END_REF] for details). As a typical example, for the case of the vertical light = (0, 0, 1) and u = 0 on ∂Ω, the SfS problem amounts to solve the following Eikonal equation

|∇u| = k in Ω u = 0 on ∂Ω (2.6)
where k(x) = I -1/2 (x) -1. In this case, the duality in Theorem 2.3 reads as

max u∈W 1,∞ (Ω) Ω udx : |∇u| ≤ k, u = 0 on ∂Ω = inf φ∈L 2 (Ω) N Ω k(x)|φ|dx : -div(φ) = 1 in D (Ω) .
The Aubry set A can then be taken as the zero set [k = 0] of k = I -1/2 -1, which corresponds to the points with maximal intensity I, i.e., I(x) = 1 so that k(x) vanishes. As we will see in the next section, dealing with nonempty Aubry set does not represent an obstacle in our approach. Contrary to the works (e.g. [START_REF] Camilli | Numerical approximation of the maximal solutions for a class of degenerate Hamilton-Jacobi equations[END_REF][START_REF] Prados | A viscosity solution method for shape-from-shading without image boundary data[END_REF][START_REF] Rouy | A viscosity solutions approach to shape-from-shading[END_REF]) where the authors approximate the degenerate HJ equation via non-degenerate one (typically, by considering (2.6) with k = max(k, ) for > 0), the only step where we deal with degeneracy points is the projection onto a ball of radius k, which may be equal to zero.

Remark 2.6 (Boundary conditions). It is well known that a natural choice for boundary conditions is the Dirichlet boundary condition. As pointed out in [START_REF] Durou | Numerical methods for shape-from-shading: A new survey with benchmarks[END_REF], the images we will consider in this paper contain an occluding boundary (see Fig 1) which will be taken as the boundary ∂Ω.

Particularly, assuming that the object is placed on a flat table suggests taking u = 0 on ∂Ω or more generally, if the height g of the surface on which is placed is known one can take u = g on ∂Ω.

Ω ∂Ω Figure 1. An object and its occluding boundary

Discretization

In this section we focus on the the discretized (finite-dimensional) problems associated with the problems (M) and (OF).

Discretization of the domain and operators.

Let Ω ⊂ R d be an image domain, which can be taken as Ω = [0, 1] 2 . Following [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF], we discretize the domain Ω using a regular grid m × n: {(ih, jh) : 1 ≤ i ≤ m, 1 ≤ j ≤ n} for a fixed h > 0. We denote by D d = {(i, j) : (ih, jh) ∈ D} the indexes whose spatial positions belong to D and by u i,j the values of u at (ih, jh). The space X = R m×n is equipped with a scalar product and an associated norm as follows:

u, v = h 2 m i=1 n j=1 u i,j v i,j and u = u, u .
For 1 ≤ i ≤ m and 1 ≤ j ≤ n, we define the components of the discrete gradient operator via finite differences:

(∇ h u) 1 i,j =    u i+1,j -u i,j h if i < m 0 if i = m , (∇ h u) 2 i,j =    u i,j+1 -u i,j h if j < n 0 if j = n.
Then the discrete gradient

∇ h : X -→ Y = R m×n×2 given by (∇ h u) i,j = (∇ h u) 1 i,j , (∇ h u) 2 i,j
. Similar to the continuous setting, we define a discrete divergence operator div h : Y → X, which is the minus of the adjoint of ∇ h , given by div

h = -∇ * h . That is, -div h φ, u X = φ, ∇ h u Y for any φ = (φ 1 , φ 2 ) ∈ Y and u ∈ X. It follows that div is explicitly given by (div h φ) i,j =              φ 1 i,j h if i = 1 φ 1 i,j -φ 1 i-1,j h if 1 < i < m -φ 1 m-1,j h if i = m +              φ 2 i,j h if j = 1 φ 2 i,j -φ 2 i,j-1 h if 1 < j < n -φ 2 i,n-1 h if j = n.
Proposition 3.7. ( [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF][START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF]) Under the above-mentioned definitions and notations, one has that

• The adjoint operator of ∇ h is ∇ * h = -div h . • Its norm satisfies: ∇ h 2 = div h 2 ≤ 8/h 2 .

Discretization of the optimization problem.

Based on the discrete gradient and divergence operators, we propose a discrete version of (M) as follows

(M) d : min u∈X u i,j =g i,j ∀(i,j)∈D d -h 2 m i=1 n j=1 u i,j + I B σ * (∇ h u)
where B σ * := {v ∈ Y : σ * (ih, jh, v i,j ) ≤ 1, ∀(i, j)} the unit ball w.r.t. σ * , and I B σ * is the indicator function in the sense of convex analysis, that is,

I B σ * (v) = 0 if v ∈ B σ * +∞ otherwise.
In other words, the discrete version (M) d can be written as

min u∈X F h (u) + G h (∇ h u),
where

F h (u) =      -h 2 m i=1 n j=1 u i,j if u i,j = g i,j ∀(i, j) ∈ D d +∞ otherwise
, and G h = I B σ * .

Let u * ∈ X * , we then have

F * h (u * ) = sup u∈X u, u * X -F h (u) = sup u∈X u i,j =g i,j ∀(i,j)∈D d h 2 m i=1 n j=1 u i,j u * i,j + h 2 m i=1 n j=1 u i,j = sup u∈X u i,j =g i,j ∀(i,j)∈D d h 2 m i=1 n j=1 u i,j (u * i,j + 1) =      h 2 (i,j)∈D d g i,j u * i,j + 1 if -u * i,j = 1 for (i, j) / ∈ D d +∞ otherwise.
It follows that

F * h (div h φ) =      h 2 (i,j)∈D d g i,j ((div h φ) i,j + 1) if (-div h φ) i,j = 1 for (i, j) / ∈ D d +∞ otherwise.
On the other hand, we have for q = (q

1 , q 2 ) ∈ Y * G * h (q) = sup p=(p 1 ,p 2 )∈Y p, q Y -G h (p) = sup p∈B σ * h 2 m i=1 n j=1 p 1 i,j q 1 i,j + p 2 i,j q 2 i,j = h 2 m i=1 n j=1
σ(ih, jh, q i,j ).

Consequently, the corresponding discrete dual problem is given by

(OF) d : max φ∈Y -F * h (div h φ) -G * h (φ) = - min φ∈Y (-div h φ) i,j =1 for (i,j) / ∈D d h 2    m i=1 n j=1 σ(ih, jh, φ i,j ) + (i,j)∈D d g i,j ((div h φ) i,j + 1)    .
(3.7) In particular, for the case of Eikonal equations |∇u(x)| = k(x), the primal-dual relations can be explicitly written as

min u∈X u i,j =g i,j ∀(i,j)∈D d -h 2 m i=1 n j=1 u i,j + I B(0,k i,j ) (∇ h u i,j ) = - min φ∈Y (-div h φ) i,j =1 for (i,j) / ∈D d h 2    m i=1 n j=1 k i,j φ i,j + (i,j)∈D d g i,j ((div h φ) i,j + 1)    ,
where I B(0,k i,j ) is the indicator function of the Euclidean ball with center 0 and radius k i,j , the latter being the value of k at (ih, jh).

To end this subsection, let us recall that a pair (u, φ) ∈ X × Y solves the primal and dual problems (M) d and (OF) d if and only if

div h (φ) ∈ ∂F h (u) and φ ∈ ∂G h (∇ h u),
or equivalently, they satisfy the following system

           -(div(φ)) i,j = 1 for (i, j) ∈ D d φ i,j • ∇ h u i,j = σ(ih, jh, φ i,j
) for all (i, j) u i,j = g i,j for every (i, j) ∈ D d .

(3.8)

3.3. The convergence of discretization. In this subsection, we will show a result on the convergence of discretization, i.e, where the solutions of the discrete optimization and its discrete dual problem converge to the ones of the corresponding problems in continuous setting. For technical reason (see Remarks 3.10, 3.11) we focus on the non-degenerate case; i.e. H(x, 0) < 0 for any x ∈ Ω, as well as to the case where g ≡ 0.

First, let us describe how to interpolate elements of X and Y . We know the values of u h ∈ X at the vertices (i, j), (i, j + 1), (i + 1, j + 1), (i + 1, j) of a small square (see Fig 2). We interpolate u h ∈ X by piecewise affine functions on the sub-triangles, i.e., taking ũh ∈ L 2 (Ω) as an affine function on the sub-triangles and coincides with u h on all the vertices. Then ũh is Lispchitz function and its gradient is, by the definition of ũh , given by

∇ũ h (x, y) = u h (i + 1, j) -u h (i, j) h , u h (i, j + 1) -u h (i, j) h (3.9)
on the sub-triangle of the vertices (i, j), (i, j + 1), (i + 1, j); and

∇ũ h (x, y) = u h (i + 1, j + 1) -u h (i, j + 1) h , u h (i + 1, j + 1) -u h (i + 1, j) h (3.10)
on the sub-triangle of the vertices (i, j + 1), (i + 1, j + 1), (i + 1, j).

Let φh ∈ L 2 (Ω) 2 be an interpolation of φ h ∈ Y such that Ω σ(x, φh )dx = h 2 m i=1 n j=1
σ(ih, jh, (φ h ) i,j ).

(i,j) (i+1,j) (i+1,j+1) (i,j+1) Figure 2. Sub-triangles Proposition 3.8 (Convergence of discretization). Assume that the Finsler metric σ associated with the Hamiltionian H is non-degenerate (i.e. H(x, 0) < 0, ∀x ∈ Ω) and that g = 0. Let u h ∈ X and φ h = (φ 1 h , φ 2 h ) ∈ Y be a pair of primal-dual solutions to the discrete optimization problem (M) d and its dual problem (3.7). Then ũh ⇒ u and φh φ weakly* in M b (Ω), as the step size h → 0. Moreover, u and φ are optimal solutions to (M) and its dual problem, respectively, in the following sense

(OF) = min ψ∈M b (Ω) N Ω σ x, ψ |ψ| d|ψ| : -div(ψ) = 1 in D (Ω) = Ω σ x, φ |φ| d|φ| = Ω udx = (M),
where φ |φ| the density of φ with respect to |φ|, the total variation of φ.

Proof. Since u h is feasible for the discrete optimization problem (M) d , its discrete gradient

∇ h u h ∈ B σ * is bounded for all small h > 0.
In other words, the sequences u h (i + 1, j) -u h (i, j) h and u h (i, j + 1) -u h (i, j) h are bounded for h > 0 and i = 1, ..., m, j = 1, ..., n. Following (3.9) and (3.10), the sequence {ũ h } is equi-Lispchitz. Combining with the fact that u h = 0 on D d , by Ascoli-Arzela's Theorem, up to a subsequence, ũh converges uniformly to some Lipschitz function u on Ω as the step size h → 0. By the optimality of u h and φ h , we have

F h (u h ) + G h (∇ h u h ) = -F * h (div h φ h ) -G * h (φ h ). More concretely, h 2 m i=1 n j=1 u i,j = h 2 m i=1 n j=1 σ(ih, jh, (φ h ) i,j ) or equivalently Ω ũh dx = Ω σ(x, φh )dx.
Since σ is non-degenerate and ũh is bounded, φh is also bounded in L 1 (Ω). Hence, φh φ weakly* in M b (Ω) N . Using the lower-semicontinuity of the integrand (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Theorem 2.38]), we deduce that

Ω σ x, φ |φ| d|φ| ≤ lim h→0 Ω σ(x, φh )dx = lim h→0 Ω ũh dx = Ω udx.
This implies that

(OF) = min ψ∈M b (Ω) N Ω σ x, ψ |ψ| d|ψ| : -div(ψ) = 1 in D (Ω) ≤ Ω σ x, φ |φ| d|φ| ≤ Ω udx ≤ (M).
By the duality result in the continuous setting given in Section 2 (Theorem 2.3), we deduce the optimality of u and φ.

Remark 3.9. In general we do not know if (OF) has a solution in L 2 (Ω) N , even if we do believe that this is not true in general. We are convinced that the weak* convergence of φh in M b (Ω) N , as the step size h → 0, is optimal. For the special case where σ is given by the Euclidean norm, (OF) admits a solution in L 2 (Ω) N (see e.g. [START_REF] Dweik | Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques[END_REF]).

Remark 3.10. See that in the case where σ is a degenerate Finsler metric, even if the duality between (OF) and (M) holds to be true and the dual problem (OF) still have a solution in M b (Ω) N , we loose the compactness of φh . However, we still have the uniform convergence of ũh to the optimal solution of the maximization problem (M).

Remark 3.11. The convergence of φh , even in the non degenerate Finsler metric case, is more subtle. This connected to the weak* convergence of φh in M b (Ω) N , and the normal trace of Radon measure vectors valued measure whose divergence is a bounded measures. In other words, it is not clear how to handle the convergence of the boundary term

(i,j)∈D d g i,j ((div h φ) i,j + 1)
to the corresponding continuous one of the type g, φ • n .

Numerical resolution

In this section we focus on the case where the light direction is vertical, i.e., = (0, 0, 1).

4.1.

Saddle-point structure. As we pointed out in Section 3, the discrete version (M) d of (M) can be rewritten in the form

inf u∈X F h (u) + G h (∇ h u) (4.11)
or in an inf-sup form as inf

u∈X sup φ∈Y F h (u) + φ, ∇ h u -G * h (φ)
where

F h (u) =      -h 2 m i=1 n j=1 u i,j if u i,j = g i,j ∀(i, j) ∈ D d +∞ otherwise
, and

G h = I B σ * .
Both the functions F h and G h are lower-semicontinuous, convex and they are "proximable", i.e. we can compute their proximal operators:

prox τ F h (u) = argmin v∈X 1 2 u -v 2 + τ F h (v) prox ηG h (ψ) = argmin φ∈Y 1 2 ψ -φ 2 + ηG h (φ)
where τ, η > 0. Then the Chambolle-Pock algorithm [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] can be applied to (4.11):

Algorithm 1 Chambolle-Pock iterations 1st step. Initialization: choose η, τ > 0, θ ∈ [0, 1], u 0 and take φ 0 = ∇ h u 0 , ū0 = u 0 . 2nd step. For k ≤ Iter max do

φ k+1 = prox ηG * h (φ k + η∇ h (ū k )); u k+1 = prox τ F h (u k -τ ∇ * h (φ k+1 )); ūk+1 = u k+1 + θ(u k+1 -u k ).
It was shown in [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] that when θ = 1 and ητ ∇ h 2 < 1, the sequence {u k } converges to an optimal solution of (4.11). Contrary to the augmented Lagrangian approach in [START_REF] Ennaji | Augmented Lagrangian method for Hamilton-Jacobi equations[END_REF] (see also [START_REF] Benamou | Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations[END_REF][START_REF] Igbida | Augmented Lagrangian method for optimal partial transportation[END_REF][START_REF] Igbida | On the uniqueness and numerical approximations for a matching problem[END_REF]), the Chambolle-Pock algorithm does not require to solve a Laplace equation at each iteration, we only need to perform some algebraic operations, namely the multiplication by apply the gradient and the divergence in each iteration. The Chambolle-Pock algorithm is easy to implement on Matlab which allows working on images easily contrary to the augmented Lagrangian approach which was implemented using FreeFem++ to solve linear PDEs.

In order to compute prox ηG * h we make use of the celebrated Moreau identity

φ = prox ηG * h (φ) + ηprox η -1G h (φ/η), ∀φ ∈ Y.
Moreover, prox η -1 G h is nothing but the projection onto B(0, k i,j ). Indeed

prox η -1 G h (ψ) = arg min q∈Y 1 2 |q -ψ| 2 + 1 η G h (q)
= arg min q i,j ∈B(0,k i,j )

1 2 |q -ψ| 2 = Proj B(0,k i,j ) (ψ i,j ).
Consequently,

prox ηG * h (ψ) i,j
= ψ i,j -ηProj B(0,k i,j ) (ψ i,j /η).

Let us now compute the proximal operator of F h . We have

prox τ F h (u) = argmin v∈X 1 2 v -u 2 + τ F h (v) = argmin v=g on D d 1 2 v -u 2 -τ h 2 m i=1 n j=1 v i,j .
Writing the first-order optimality condition we get

(prox γF h (u)) i,j -u i,j -τ h 2 = 0 ⇔ (prox γF h (u)) i,j = u i,j + τ h 2 , ∀i = 1, ..., m, j = 1, ..., n.
So in practice, we update u n+1 via the previous formula and we then set its values to g on the Dirichlet domain.

For applications in image, as usual, one can always assume that h = 1 since it only scales the domain. The details of the 2nd step in Algorithm 1 are then given by

• compute φ k+1 : φk+1 = φ k + η∇ h ūk φ k+1 i,j = φk+1 i,j -ηProj B(0,k i,j ) ( φk+1 i,j /η), 1 ≤ i ≤ m, 1 ≤ j ≤ n; • compute u k+1 : v k+1 = u k + τ div h (φ k+1 ) u k+1 i,j = v k+1 i,j + τ, 1 ≤ i ≤ m, 1 ≤ j ≤ n.
Remark 4.12. Another way to formulate the problem (M) (in the continuous setting) is to take

F(u) = - Ω udx, and G(q, v) = 0 if |q| ≤ k and v = g on ∂Ω ∞ otherwise, for all u ∈ W 1,∞ (Ω), and (q, v) ∈ L ∞ (Ω) d × L 2 (∂Ω).
In this case, the problem (M) can be rewritten as

inf u F(u) + G(K(u))
where K = (∇, γ 0 ), and γ 0 is the trace operator on the boundary. This being said, at the second step of the Algorithm 1 we need to compute γ * 0 which turns requiring to solve a PDE. Indeed, we define

γ 0 : H 1 (Ω) → L 2 (∂Ω) through γ 0 (u) = u |∂Ω for every u ∈ H 1 (Ω). By definition, for any (u, v) ∈ H 1 (Ω) × L 2 (∂Ω) γ 0 u, v L 2 (∂Ω) = u, γ * 0 v H 1 (Ω) . This means that ∂Ω uvdS = Ω u(γ * 0 v)dx + Ω ∇u∇(γ * 0 v)dx
for any u ∈ H 1 (Ω). In other words γ * 0 v solves the following PDE -∆z + z = 0 in Ω and ∂ n z = v on ∂Ω.

Thus we opt for the first formulation in order to avoid additional costs to the computations.

Error criterion.

As usual, we can check the optimality conditions (3.8) associated to (M) d and (OF) d . Namely we check the following conditions:

• Divergence error: -div h (φ) -1 2 . • Dual error: σ(x, φ) -∇ h u • φ 1 . • Lip error: sup i,j σ * (ih, jh, ∇ h u i,j ).
We expect Divergence error and Dual error to be small. Note that for vertical light direction, the support function σ is easy to compute. More particularly, one has for every p ∈ R d , σ(x, p) = k(x)|p| where |p| is the euclidean norm of p. Thus, for the Lip error, we can check the value sup i,j ( ∇ h u i,j -k i,j ) and expect it to be close to zero.

Numerical examples.

We test for some commonly used images: Mozart and vase images taken from [START_REF] Zhang | Shape from shading: a survey[END_REF] and Basilica and vaso images taken from [START_REF] Festa | Analysis and Approximation of Hamilton Jacobi equations on irregular data[END_REF]. In these cases, the shapes are reconstructed by solving the Eikonal equation |∇u(x, y)| = k(x, y) in 2D with g ≡ 0, i.e, with homogeneous Dirichlet boundary condition u = 0 on ∂Ω. The algorithm was implemented in Matlab and executed on a 2,3 GHz CPU running macOs Catalina system.

2. 1 .

 1 Metric character of HJ equations. Given a regular connected open domain Ω ⊂ R d , and a continuous Hamiltonian H : Ω×R d → R satisfying, for x ∈ Ω, Z(x) := {p ∈ R d : H(x, p) ≤ 0}:

Figure 3 .

 3 Figure 3. Left to right: Initial image, the reconstructed shape

Figure 4 .

 4 Figure 4. Error criterion for 5000 iterations and τ = 0.01 and η = 8/τ

Figure 5 .Figure 6 .

 56 Figure 5. Left to right: Initial image, the reconstructed shape

Figure 7 .Figure 8 .

 78 Figure 7. Left to right: Initial image, the reconstructed shape

Figure 9 .Figure 10 .

 910 Figure 9. Left to right: Initial image, the reconstructed shape

Let us mention that for most shapes, only a hundred of iterations is enough to reconstruct a reasonable solution. We took 5000 iterations in order to check that the error criteria are getting smaller.


Rihab Boukil

Nazym Tuleushova

Didier Cot

Bertrand Rebière

Valerie Bonniol

Julien Cambedouzou

Sophie Tingry

David Cornu

Yaovi Holade
email: yaovi.holade@enscm.fr

Bertrand Rebiere

Enhanced electrocatalytic activity and selectivity of glycerol oxidation triggered by nanoalloyed silver-gold nanocages directly grown onto gas diffusion electrodes

Enhanced electrocatalytic activity and selectivity of glycerol oxidation triggered by nanoalloyed silver-gold nanocages directly grown on gas diffusion electrodes

Au and Pt-based catalysts for efficient alcohols electroreforming in low energy consumption and cogeneration H2, CO2 or N2 electrolyzers. The present method that combines the direct growth, galvanic replacement and nanostructuration provides a smart route to carving out new designs in catalysts nanofabrication in order to target high electrocatalytic performance.

Main text

Electrolysis has been historically a powerful tool to produce a number of inorganic or organic materials. It has experienced a renaissance over the last decade with research efforts seeking low electrical energy consumption platforms for hydrogen evolution reaction (HER) and reaction of CO2 reduction (CO2RR) or N2 reduction (N2RR) in order to electro-synthesize compounds. The counter reaction, i.e., oxidation at the anode is typically the oxygen evolution reaction (OER) that not only occurs at high electrode potential of E ≥ 1.4 V vs reversible hydrogen electrode (RHE), but also uses precious/expensive catalysts (Ru, Ir, Co). [1][2][3] Consequently, a significant portion of the energy required to electrochemically reduce H + /H2O, CO2 or N2 to fuels and chemicals is typically consumed by the accompanying OER. [START_REF] Tatin | Proc. Natl. Acad. Sci[END_REF][START_REF] Bardow | [END_REF][6][7] Given their low oxidation potential (E = 0-1 V vs RHE) compared to OER, the use of organic molecules instead of water molecules greatly lowers the input energy. [START_REF] Bardow | [END_REF]6,[8][9][10] Verma and coworkers reported alternative oxidative reactions using biodiesel-waste in order to achieve not only a low cell voltage, but also a high current density. [START_REF] Bardow | [END_REF]6 More interesting, a tight control over the selectivity is extremely interesting in co-generation systems. Indeed, glycerol, a by-product from several industrial pathways such as biodiesel (100 kg of biodiesel release 10 kg of crude glycerol as side-product 11 ) and fatty acids exceeds by far its consumption. So, a selective oxidation to specific products such as glycolic or formic acid can effectively cut down the cost of electrolysis. 9,12 However, the simultaneous achievement of high activity and selectivity is a challenge. The mostly reported catalysts rely on the chemical or physical reduction of metallic precursors in the presence of a capping agent or surfactant, whereby catalysts are produced in the form of powder or liquid and a relatively high loading of metals of at least 0.1 mg cm -2 is used during the catalysis. 6,8 The used surfactants also limit the accessibility to the entire active sites. In addition, a binder, such as Nafion, is required to immobilize the NPs on a substrate (i.e. glassy carbon) prior to use, which adds a layer of complexity to the fabrication process as well as the loss of efficiency during operation. 13 The combination of seed mediated growth and galvanic replacement methods enables to develop porous and alloyed structures comprising binary/ternary systems. [14][15][16][17][18][19][20][21][22][23] For Au-based materials, Ag routinely acts as the template during the galvanic displacement since the standard redox potential of Ag + /Ag (0.80 V vs. standard hydrogen electrode (SHE)) is lower than that of AuCl4 -/Au (1.00 V vs. SHE). While it was observed that residual Ag could locally stabilizes the surface atomic steps and increase the electrocatalysis, 15 it is still unknown how to rationally incorporate Au during galvanic displacement directly on a 3D gas diffusion electrode (GDE) to subsequently optimize the activity, selectivity and durability of the nanoalloyed and nanoporous catalysts. In addition, the monitoring of the phenomenon of galvanic replacement to directly visualize the change in the galvanic potential and optimize the displacement duration has not yet reported. To address all these challenges, we report in this study a versatile surfactant-and binder-free method to directly grow Ag NPs onto a 3D GDE and further tightly perform a partial replacement of Ag by Au in order to fabricate free-standing electrocatalytic electrodes readyto-use. We found that the incorporation of chloride into the synthesis mixture is critical for tailoring highly porous AgAu decorated carbon paper electrode (GDE@AgAu), thus contributing to its intrinsic high electrocatalytic activity, selectivity and durability towards glycerol oxidation into glycolate (32%) and formate (66%).

To facilitate the implementation in the glycerol electrocatalysis for those AgAu nanoporous and nanoalloyed materials resulting from direct galvanic displacement onto a large surface electrode (GDE, at least 3 cm × 3 cm), we have designed the experiments shown in Scheme 1a.

The experimental procedures are reported in the Electronic Supplementary Information (ESI †).

Herein, the pristine Ag NPs take the advantage of the metal-support interaction and do not have to be involved in a catalytic ink preparation process before immobilization on a substrate during various post-synthesis steps. This is an alternative to conventional galvanic displacement protocols, 19,20,24 whereby the least noble NPs, here Ag, are initially prepared in solution before introduction in a replacement mixture made of the cation of the most noble metal, here Au; the most used being AuCl4 -. Since the introduction of 1 atom of Au requires the removal of 3 atoms of Ag (3Ag + AuCl4 -→ 3Ag + + Au + 4Cl -), the process is expected to yield nanoporous structures. This illustration shows that the galvanic displacement of Ag atoms directly grown at the surface of a GDE's microfibers (template electrode, GDE@Ag) by those of Au leads to a nanoporous and nanoalloyed AgAu strongly attached to the support, and yields a free-standing catalyst (target electrode, GDE@AgAu). As shown in Scheme 1b, the further use of such GDE@AgAu electrode for glycerol electrooxidation reaction (GOE) can replace OER during H2 production by the electrocatalytic protons or water reduction reaction (HER) in aqueous media. More broadly, the illustration shows that any electrochemical reaction is always accompanied by a counter reaction; this dual electrosynthesis is termed "paired electrosynthesis". Hence, in aqueous media, an electro-reduction reaction such as HER, N2RR or CO2, etc. is always coupled with an electro-oxidation; OER or an organic molecule such as glycerol (refs. [5,6]). The overall required energy being the difference between the electrode potentials of the two processes, it becomes obvious that GOR enables lowering significantly the total consumed electrical energy. Fig. S1a (ESI †) of the control linear sweep voltammetry (LSV) demonstrates that the introduction of the ammonia solution (NH4OH) to form a [Ag(NH3)2] + complex (R = n(NH4OH)/n(Ag + ), referred to as Rx, x = 0, 4) is accompanied by the decrease of the reduction potential. In fact, the complexation of Ag + by NH3 leads to a strong decrease of the standard redox potential of Ag(I) from 0.80 to 0.37 V vs SHE. This enables lowering the reduction kinetics in order to have a tight control on the seeds growth into homogeneous particles. The electrochemical program to grow directly Ag NPs onto GDE support of 3 cm × 3 cm (one side) is composed of an OFF step (Iappl = 0 mA, tOFF = 5 s), an ON step (Iappl = -4.5 mA, i.e., jappl = -0.5 mA cm -2 , tON = 5 s) and Ncycles = 126 (0.15 mM AgNO3) or 252 (0.30 mM AgNO3) as the total cycles to allow the seeds formation and electrode relaxation (Fig. S1b, ESI †). Fig. 1a shows the cyclic voltammetry (CV) profiles of the as-synthesized CP-Ag electrodes at different ratios of R0 and R4. The oxidized silver during the forward scan is reduced during the backward scan at ca. 1.1 V vs RHE (the calibration curve vs the mercury-mercury oxide electrode (MOE) is reported in Fig. S2,ESI †). The reduction peak area of Ag2O with a monolayer charge of 420 μC cm -2 was considered to determine the electrochemically active surface area (ECSA inset of Fig. 1a). The ECSA of CP-Ag-R4 is 4.4× higher in comparison to CP-Ag-R0. This indicates that the introduction of a complexation step has a beneficial effect, which could be attributed to the reduction of the NPs size as well as a better distribution of NPs over the electrode surface. 25,26 A further increase of the AgNO3 concentration is accompanied by an augmentation of the ECSA. We further examined numerous scanning electron microscopy (SEM) and energydispersive X-ray (EDX) images to determine the coverage of GDE's fibers by Ag NPs (Figs. 1b-f4). The SEM of the pristine GDE shows fibers (filled tubes) with a diameter of 7-9 µm (Fig. ESI †). This suggests that NPs are preferentially deposited at the surface of the first three layers of the GDE on a total of approximately 25, which is similar to previous observations with metallic nanostructures obtained by chemical reduction onto GDE. 13 In that case, it has been argued that the metal salt within the GDE interior diffuses towards the surface. The present observation can more rationally be explained by the applied electrical field between the working and counter electrodes in which the most exposed fibers allow the easy grow of particles. This means that the first step of the reduction of metal salts takes place at the external fibers' surface and then, the formation of the first seeds' layer of particles acts as an active site where a further reduction occurs. As shown in Figs. 1f1-f4 (and Figs. S3, S4-8, Table S1, ESI †), the maps of Ag and C confirm the homogeneous distribution.

The previous results demonstrate that we were able to manipulate the synthesis towards different localizations in the 3D structure of the GDE. So, having demonstrated the ability to use NH3 as a non-organic capping or surfactant agent to homogeneously grow Ag NPs directly onto a GDE support, we sought to study more carefully the conditions that enable a partial galvanic displacement of Ag atoms by those of Au for a better glycerol electrocatalysis. Since the GDE@Ag electrode has been obtained in KNO3 electrolyte, we firstly implemented the galvanic replacement in KNO3 solution (referred to as GR(KNO3)) and secondly in pure water (referred to as GR(H2O)) to evaluate the impact on the replacement process. The objective was to mimic the GDE@Ag synthesis and determine whether Au(III) species could be injected directly into the Ag synthesis reactor once Ag NPs were formed or not. To evaluate the influence of the reaction duration on the growth and subsequently the morphology or electrocatalytic ability of particles, we quenched the reaction at different times. Figs. 2a-b show the CV profiles of the as-synthesized GDE@AgAu electrodes in the absence of glycerol. Those of the electrocatalytic performance in the presence of glycerol are depicted in Fig. 2c and Fig. S9 (ESI †). The CVs show that the material with the largest area of the Ag2O reduction peak GR(KNO3)-30min is not necessary the one which gives the highest peak current density during the electrocatalytic glycerol oxidation. Specifically, for the materials obtained by galvanic displacement in KNO3, the best kinetics and lowest charge transfer resistance (Fig. S9, ESI †) at potential E ≤ 1 vs RHE is GR(KNO3)-30min. For higher potentials, GR(KNO3)-10min becomes the best with a peak current density of jp = 140 mA cm -2 in comparison to jp = 70 mA cm -2 for GR(KNO3)-30min. For the materials synthesized in water, GR(H2O)-5min is the most active catalyst with jp = 171 mA cm -2 . A longer duration results in a decrease of the performance, presumably because of the oxidative etching of the formed nanostructures and the excess formation of the insoluble AgCl. 27,28 The last hypothesis is supported by SEM and EDX observations (Figs. 2d-g, Fig. S10-14, ESI †) that show more AgCl nanocubes. Quantitative EDX in Fig. 2h and Tables S2-5 (ESI †) support the significant presence of chloride in the materials. The gold content is below 20 at%. However, the galvanic replacement was targeting to fabricate a bimetallic AgAu material in which the gold content is about 80 at%. Those results indicate that the maximum concentrations of chloride and silver ions in the final reaction solution lead to [Ag + ]×[Cl -] that falls below the solubility product of AgCl, Ks = 1.8×10 -10 at 20 °C. 23 Since the AgCl precipitate can be easily dissolved with a saturated NaCl solution (further called "brine solution") through the coordination reaction AgCl(s) + 3Cl -(aq) → AgCl4 -(aq), 23,29 we next tested the catalytic performance of three types of materials derived from GR(H2O)-5min: the sample was thoroughly washed with NaCl after drying (referred to as GR(H2O)-driedwashed(NaCl)), the sample was immediately washed with NaCl (referred to as GR(H2O)washed(NaCl)), or NaCl was directly dissolved in the galvanic displacement water (referred to as GR(NaCl)). Fig. 3a shows the CVs normalized by the geometric surface area and those with the metal content are reported in Fig. 3b. A comparison is made with the commercial Vulcan-Au (20 wt% Au/C) and the homemade CP-Au (Fig. S15, ESI †). Among the tested conditions for the galvanic displacement, jp = 150, 173 and 290 mA cm -2 for GR(H2O)-driedwashed(NaCl), GR(H2O)-washed(NaCl), and GR(NaCl), respectively. This suggests that the optimal surface composition and morphology of the AgAu particles after the galvanic displacement is reached when the latter is done directly in the presence of an excess amount of chloride. Controls experiments have been carried out with duration much shorter (2 min) and longer (10 min); and lower catalytic activities were observed. For CP-Au and Vulcan-Au, jp = 84, and 44 mA cm -2 , respectively. Those peak current densities are 3.5× and 6.5× times lower in comparison to GR(NaCl). The bulk gold content determined by inductively coupled plasma optical emission spectrometry (ICP-OES) is 0.17 wt% for GR(NaCl) (i.e. 16 µgAu cm -2 ) and 0.75 wt.% for CP-Au (i.e. 68 µgAu cm -2 ). So, in terms of the mass-activity, jp = 18.2, 1.2, and 2.8 A mgAu -1 for GR(NaCl), CP-Au, and Vulcan-Au, respectively. This represents about one order of magnitude increase of the catalytic activity for the material obtained by the galvanic displacement. Table S6 (ESI †) gathers the comparative performance of relevant glycerol electrooxidation reaction on metallic catalysts in alkaline media from literature, wherein jp is 20 to 70 mA cm -2 (geometric surface area), 0.2 to 4 A mgAu -1 (gold content) and the metal loading is relatively high, about 30-500 µgAu cm -2 . 8,14,[30][31][32][33][34][35] Furthermore, a low onset potential (Eonset) of ca. 0.3 V vs RHE was achieved with catalyst from GR(NaCl). It can thus be concluded that the electrocatalytic kinetics of the present GDE@AgAu free-standing electrocatalyst is higher than relevant values for the conventional chemically synthesized Au NPs by an order of magnitude and outperformed all data reported in the literature for metal and metal alloy-based electrocatalysts. It should be noted that ECSA cannot be precisely determined for a bimetallic AgAu electrode because there is a common potential window where the oxides of both involved metals are reduced and there is no reference value for the monolayer charge. Even though the ECSA of AgAu cannot be fairly evaluated for a comparison with the monometallic GDE@Au and Vulcan-Au materials, Fig. S16 (ESI †) report an estimation of the performance by normalizing the current by the ECSA or both the ECSA and the Au weight. The results still show a significant enhancement of the performance for the bimetallic electrode. For Au/C and GDE@AgAu, jp = 27 mA cm -2 Au (= 3.3 mA cm -2 Au µg -1 Au) and 92 mA cm -2 Au (= 22.9 mA cm -2 Au µg -1 Au), respectively. In addition, ECSA is about 2, 11 and 20 m 2 g -1 Au for GDE@Au, commercial Vulcan-Au and GDE@AgAu, respectively. The enhanced electrocatalytic characteristics of GDE@AgAu are explained by the absence of any organic contaminants or protective ligands on their surface and by the possible porous and alloyed structure (will be fixed in the next sections). Indeed, Li et al. 15 observed that residual Ag could stabilize the surface active sites during not only the dealloying of Au-Ag-Cu ternary alloy nanoparticles, but also the electrocatalytic oxidation of alcohols. Authors have observed that while the binding energy of the Au 4f peak remains essentially unchanged in regards to that of bulk Au, both Cu 2p and Ag 3d peaks significantly down-shift by 0.4 and 0.5 eV, respectively compared to their bulk materials. Thus, the present high efficiency could indicate that a fine balance has been achieved between the introduced Au and remaining Ag within the 3D materials to maximize the catalytic performance.

To gain a deeper understanding on the origin of the exceptional electrocatalytic performance of the GR(NaCl) material, we next utilized the electrochemical impedance spectroscopy (EIS, Fig. S17 (ESI †)). The obtained results are shown in Fig. 3c in terms of the complex-plane Nyquist impedance plots and Fig. 3d by the Tafel plots. The representative equivalent electrical circuit that best describes the electrocatalytic interface is R+QCPE//Rct and shown in Fig. 3c (Tables S7-9 (ESI †) resume the fitted data) wherein R is the ohmic resistance or the uncompensated resistance, Rct is the charge transfer resistance and QCPE is the constant phase element to model "imperfect" capacitance of the double-layer. [START_REF] Lasia | Electrochemical Impedance Spectroscopy and its Applications[END_REF][START_REF] Orazem | Electrochemical Impedance Spectroscopy[END_REF] The lowest values of Rct (inversely proportional to the standard rate constant k°, thus the exchange current density j0) and Tafel slope (semi-quantitative indicator of charge-transfer kinetics and/or mechanisms) for CP-AgAu electrode confirm its excellent ability for fast electron transfer. Furthermore, the CVs collected at different scan rates (v) in the presence of glycerol and the plot of log(jp) vs log(v) suggest that the overall reaction is limited by the adsorption, diffusion and electron transfer (Fig. S16,(ESI †)). In fact, glycerol electrooxidation is a multi-proton and multi-electron transfer process, at least two electrons and two protons are transferred to electro-oxidize the primary or secondary alcohol function. 8,9,12,35 Hence, the previous outcomes demonstrate only our ability to regulate the kinetics of those processes. Suitability of GDE@AgAu electrodes for wider applications in the heterogeneous electrocatalysis may require lower number of electrons. So, we next theorized a same trend. To validate such hypothesis, we then electrochemically scrutinized the quasi-reversible redox probe Fe(CN)6 3-/Fe(CN)6 4-. Therefore, in order to provide a better qualitative and quantitative analysis of the electrochemical parameters, we used two different methods, namely the peak-to-peak separation ΔEp between the anodic and cathodic peaks on a CV (Fig. 3e) and the EIS at the open circuit potential (OCP) (Fig. 3f, Figs. S10-11 (ESI †). The above equivalent electrical circuit is modified to add in serial with Rct, a Warburg element (W) to model the mass transport, i.e., diffusion. [START_REF] Lasia | Electrochemical Impedance Spectroscopy and its Applications[END_REF][START_REF] Orazem | Electrochemical Impedance Spectroscopy[END_REF] k°(10 -3 cm s -1 ) = 0.3, 0.7, 2.0 for CP, CP-Ag and CP-AgAu, respectively. Those data correspond to a significant enhancement of the electrochemical kinetics of CP by factor of 2.3 upon the growth of Ag and a factor of 6.7 following the galvanic displacement. The trend can be translated to as a decrease of Rct, which is confirmed by EIS where Rct = 1844, 627, 170 and 38  cm 2 for CP, CP-Ag, CP-AgAu and Pt-bulk, respectively. It features the ability to regulate the kinetics of electrodes as previously seen.

S18-19 and Tables

Having demonstrated the ability to use brine solution to improve the electrocatalytic efficiency of materials resulting from the galvanic displacement, we sought to study the composition and morphology more carefully. To these ends, we combined physical techniques of SEM, EDX, X-ray photoelectron spectroscopy (XPS) and small angle X-ray scattering (SAXS). For the sample GR(H2O)-washed(NaCl), SEM images in Figs. 4a-b (and Fig. S20 (ESI †)) show the formation of porous donut-like nanostructures, which could be explained by the dissolution of AgCl so that the material reconstruction is not possible. For the GR(NaCl) material, SEM images in Figs. 4c-f (and Fig. S21,ESI †) highlight the formation of a nanocage structure, which is similar to AgAu materials routinely obtained by the galvanic displacement, but in solution. 15,19,20,23,24 The EDX maps of Figs. 4g-i underscore important current trends that the obtained metals are not oxidized. For the GR-5min, the doublet of gold metal are situated at 84.0 eV for Au 4f7/2 and 87.7 eV for Au 4f5/2 while those of Ag 3d5/2 and Ag 3d3/2 are situated at 367.9 and 373.9 eV, respectively. For the GR-0min, silver metal is situated at 368.2 eV for Ag 3d5/2 and 374.2 eV for Ag 3d3/2. This indicates a negative shift of about 0.3 eV in bimetallic material. Similar observations have been reported by Li et al. 15 where the binding energy of the Au 4f peak remains unchanged while Ag 3d peak down-shifts by 0.5 eV. So, it appears that in addition to the highly nanoporous structure, both Ag and Au elements are forming an alloyed phase in which there is a strong electronic interaction between silver and gold. Quantitative data from EDX (Figs. S22-23, Tables S12-13 (ESI †)) are displayed in Fig. 4l and compared with the results from XPS in the case of GR(NaCl). When the galvanic replacement is performed in H2O and then immediately washed by NaCl, the remaining chloride represents 5 at% (considering Ag, Au and Cl species). By carrying out directly in NaCl, all the chloride is removed and the final atomic ratio is Ag/Au = 10:90 by XPS and 13:87 by EDX. Hence, the results testify that we successfully realized the partial galvanic replacement of Ag atoms by those of Au directly at the surface of free-standing GDE. SAXS analyses were conducted on GDE, GDE@Ag and GDE@AgAu (Figs. 4m-n). Fig. 4n shows that the graphite (002) peak at 18.8 nm -1 overlap well in all samples, meaning that the graphitic carbon amount remains constant. The (111) Bragg peak from Ag is visible at ~27 nm - 1 for GDE@Ag, but absent in GDE@AgAu. This highlights the alteration of the Ag crystalline structure. SAXS profiles show a Porod regime in the 0.5-3 nm -1 range, revealing sharp interfaces at nanoscale. Interestingly, GDE@AgAu shows high intensity for scattering vectors lower than 0.4 nm -1 . This confirms the presence of new nanoscale objects. Conclusively, the combination of nanoscale defects in the channels, pore walls and bifunctional/ligand effects 15,31,32,35,[START_REF] Nguyen | [END_REF]39 explained the substantial improvement of the catalysis of the GDE@AgAu material.

To gain deeper insights about the galvanic displacement in brine solution and why the reaction duration of 5 min gave the best electrocatalytic activity, we designed an experiment to determine the electrode potential of CP-Ag following the addition of AuCl4 -. Fig. 5a shows the OCP profiles for two different CP-Ag free-standing electrodes. Initially, OCP is about -0.13 V vs Ag/AgCl. Upon the one-pot injection of Au(III), there is a first sharp increase towards +0.03 V vs Ag/AgCl, followed by a stabilization before a second important increase to yield a steadystate OCP = +0.69 V vs Ag/AgCl. Those three characteristic domains could be assigned to the following processes. The first is attributed to an undefined potential of an Ag electrode immerged in a not degassed NaCl solution (OCP = -0.13 V vs Ag/AgCl). The second is a fabricated "pseudo Ag/AgCl electrode" after the first galvanic displacement so that the recorded OCP is just the potential difference between two Ag/AgCl reference electrodes (that is why OCP is quiet zero). Given the composition of Ag10Au90 after the galvanic replacement, the third is the potential of an "Au-like" electrode immerged in a not degassed NaCl solution containing AuCl4 -since E°(AuCl4 -/Au) = 1.0 V vs SHE, about 0.8 V vs Ag/AgCl. The exact origin of the spikes is unknown, but they could be the consequence of a mass substitution of Ag atoms as well as the nanostructuration of particles in the presence of O2 and Cl -; those species form a strong oxidative etching agent. 21 From Fig. 5a, it becomes clear that 5 min is the optimal duration for the galvanic displacement in a brine solution, which supports the previous results.

We further run the long-term bulk glycerol electrolysis to simultaneously evaluate the reaction selectivity as well as the accelerated ageing on the catalyst's morphology and composition. Fig. 5b shows the evolution of the oxidation current (I) and quantity of electricity (Q) during a 3 h electrolysis. Given the 3D nanostructure of the GDE@AgAu material, we utilized a programmed potential electrolysis (PPE) to facilitate the mass transport by repeating potentiostatic discharging cycles of 10 s ON (E = Eappl = 1.246 V vs RHE, no iR-drop correction) and 10 s OFF (E = OCP). The results show a good stability. Fig. 5c displays the chromatograms obtained from HPLC analysis (Table S14, ESI †) while Fig. 5d shows the selectivity trend for identified products after electrolysis; Fig. S25 (ESI †) for the standards.

While many products used to be obtained at metallic electrocatalysts, 35,[40][41][42] only two main products have been obtained in this study, for instance formic and glycolic acids at the selectivity of 66 and 32%, respectively (oxalic acid is 1.8%, glyceric acid is 0.7 % and tartronic acid 0.1%). The determined exchanged number of electrons that are transferred is about 6, which is in agreement with the production of the carboxylate forms of those products in alkaline media so that the faradaic yield is about 86%. Based on chromatographic analyses and in order to account the glycerol electrooxidation, Scheme S1 was proposed for glycerol electrooxidation on the as-synthesized silver-gold nanoporous and nanoalloyed nanocatalysts in alkaline media.

Given the high selectivity towards formate, we postulated that it was produced by two parallel pathways, one from glycerate and another from glycolate. Finally, Figs. 5e-j show the postmortem SEM, HRSEM and EDX after the electrolysis. Interestingly, HRSEM analyses do not show any morphology modification; hence the material keeps its nanoporous structure. EDX spectra and quantitative data are reported in Fig. S26 and Table S15 (ESI †). The atomic ratio Ag/Au is about 18:82, a slight change when compared to raw material (13:87). It can thus be summed up that the electrode exhibits good electrochemical stability.

Conclusions

In summary, a novel class of electrode material, composed of a gas diffusion electrode (GDE) decorated by metallic silver-gold nanoporous and nanoalloyed particles, was fabricated to selectively electro-oxidize glycerol in alkaline media. The newly developed convenient approach without introducing any template or organic surfactant enables simplifying the various preparation and post-synthesis steps. The nanoparticles were directly attached to the GDE so that there was no need to prepare a catalytic ink, which enabled bypassing the use of binders, avoiding active sites obstruction, minimizing nanoparticles detachment from the electrode, and preventing particles agglomeration. Owing to its nanoporous and nanoalloyed features, the as-synthesized free-standing GDE@AgAu electrode demonstrated a drastic enhancement of the electrocatalytic performance with an onset potential below 0.5 V vs RHE and a record current density of 18 A mgAu -1 (290 mA cm -2 ). This is at least one order of magnitude larger than the tested commercial Au/C catalyst under our conditions and other reported mono-and multi-metallic electrocatalysts. The material also exhibited an unprecedented selectivity in glycerol electrooxidation, whereby only two products of high interest, formate (66%) and glycolate (32%) have been obtained. This study contributes toward advances in the design of efficient and low energy input anode to be coupled with emerging hydrogen evolution (HER), CO2 reduction reaction (CO2RR) or N2 reduction reaction (N2RR) in order to electro-synthesize high value-added fuels and chemicals.

  1b and Fig. S3, ESI †). Since the thickness is 190 µm, it can be deduced that the entire 3D network is composed of ca. 27-21 layers of microfibers. We first observed that in the absence of NH3 ligand (Figs. 1c1-c2), the Ag NPs are randomly distributed on the surface of the microfibers with much agglomeration. Upon the introduction of NH3, homogenous Ag NPs are obtained; about 10-50 nm for particles situated at the surface of the first layer of microfibers (Figs. 1d1-d2 and Fig. S4, ESI †). Figs. 1e1-e2 and Fig. S5, ESI † show the images for a higher concentration of AgNO3 where the surface coverage by Ag NPs is much important. The size of NPs is much smaller for those situated inside the 3D network of microfibers (Figs. S4 and S6,

  show the homogeneous distribution of Ag and Au atoms, supported by EDX spectra (Figs. S22-23, ESI †). The high-resolution XPS spectra of Au 4f and Ag 3d regions are depicted in Figs. 4j-k (and Fig. S24, ESI †), which

Scheme 1 .

 1 Scheme 1. Sketch of electrode fabrication for the bioeconomy production of value -added fuels or chemicals. (a) Schematic illustration of the process of galvanic displacement (GR) of Ag atoms by those of Au directly at the surface of a gas diffusion electrode (GDE), leading to nanoporous structures. (b) Typical half-cell polarization curves illustrating the working principle of different scenarios of cogenerative electrosynthesis that couple a process at negative electrode (cathode, (-)) to one at positive electrode (anode, (+)).

Fig. 1 .

 1 Fig. 1. Characterization of template electrodes. (a) iR-free CVs recorded at 50 mV s -1 in 1 M KOH and 25 °C (in inset the profiles of the corresponding ECSA) for the as-synthesized materials at AgNO3 concentration of 0.15 and 0.30 mM for different ratios R(= n(NH4OH)/n(AgNO3)) = 0 and 4. SEM micrographs of Ag grown onto a GDE for different ratios: (c1, c2) R = 0 and 0.15 mM AgNO3, (d1, d2) R = 4 and 0.15 mM AgNO3; (e1, e2) R = 4 and 0.30 mM AgNO3. (f1-f4) SEM and EDX elemental maps (scale bar = 5 µm) for GDE@Ag obtained with R = 4 and 0.30 mM AgNO3: (f2) C, (f3) Ag, (f4) C+Ag.

Fig. 2 .

 2 Fig. 2. Characterization and performance of GDE@AgAu materials obtained by galvanic displacement (GR) performed in KNO3 or H2O. (a) iR-free CVs (50 mV s -1 , 1 M KOH, 25 °C) for GR performed in KNO3. iR-free CVs (50 mV s -1 , 1 M KOH, 25 °C) in the absence (b) and presence (c) of 0.1 M glycerol for GR performed in KNO3 or H2O. SEM images for GR performed in KNO3: (d) 4 h and (e) 30 min and its corresponding EDX elemental maps (Ag, Au, and Cl from top to down). SEM images for GR performed in H2O: (f) 10 min and (g) 5 min. (h) Atomic composition from EDX maps: error bars represent 1 SD (n ≥ 3).

Fig. 3 .

 3 Fig. 3. Comparative performance of the commercial Vulcan-Au and the as-synthesized CP-Ag, CP-Au, CP-AgAu materials obtained by galvanic displacement (GR) in either H2O before washing by a brine

Fig. 4 .

 4 Fig.4. Morphological and compositional characterization of the as-synthesized GDE@AgAu materials by the galvanic replacement (GR). (a, b) SEM/HRSEM images of the as-synthesized GDE@AgAu material by GR performed in H2O (5 min) and subsequently washed with a brine solution, referred to as GR(H2O)-washed(NaCl). (c-f) SEM/HRSEM images of the as-synthesized GDE@AgAu material by GR performed in a brine solution, referred to as GR(NaCl). (g-i) EDX mapping (scale bar = 5 µm) of GR(NaCl) material: (g) Au+Ag, (h) Ag, and (i) Au. (j, k) High-resolution XPS spectra of GR(NaCl) material: (j) Au 4f and (k) Ag 3d. (l) Atomic composition from EDX and XPS: error bars represent 1 SD (n ≥ 3). (m, n) SAXS patterns of the pristine GDE (black), GDE@Ag (blue), and GDE@AgAu (red). The black dashed line represents a q-4 Porod power law: Panel (n) is a magnification of the wide angles.

Fig. 5 .

 5 Fig. 5. Study of the galvanic replacement (GR) in the brine solution and the efficiency of the obtained GDE@AgAu material. (a) OCP profiles recorded during GR (3.5 M NaCl, room temperature, gentle stirring) at two different CP-Ag free-standing electrodes. (b) Evolution of the oxidation current (I, left y-axis) and quantity of electricity (Q, right y-axis) during a 3 h long-term electrolysis (1 M KOH, 0.1 M glycerol, 25 °C) performed by repeating potentiostatic discharging cycles of 10 s ON (E = Eappl = 1.246 V vs RHE, no iR-drop correction) and 10 s OFF (E = OCP). The corresponding: (c) chromatograms obtained from HPLC analysis (Eluent: 3 mM H2SO4; injection: 25 μL; column: BP-OA Benson 2000-0, 30 cm, room temperature of 22 ± 2°C) and (d) Selectivity trend. Post-mortem SEM and EDX (for mapping, scale bar = 5 µm) of the GDE@AgAu electrode after bulk electrolysis for 3 h.

Acknowledgements

This work has been partially funded by the French National Agency (ANR) through the LabEx CheMISyst (ANR-10-LABX-05-01) and the European Institute of Membranes of Montpellier through the Exploratory Project COGENFC (PAT-Axis-Energy-2018). The authors thank J.

Causse for assistance in SAXS operation and Valerie Flaud for XPS measurements.

Author Contributions

Conflicts of interest

The authors declare no competing financial interest.