H S Aghamiry
email: aghamiry@geoazur.unice.fr

A Gholami
email: agholami@ut.ac.ir

S Operto
email: operto@geoazur.unice.fr

Full Waveform Inversion by Proximal Newton Method using Adaptive Regularization

Keywords: Inverse theory, Waveform inversion, Controlled source seismology, Numerical modelling

Regularization is necessary for solving nonlinear ill-posed inverse problems arising in different fields of geosciences. The base of a suitable regularization is the prior expressed by the regularizer, which can be non-adaptive or adaptive (data-driven), smooth or non smooth, variational-based or not. Nevertheless, tailoring a suitable and easy-to-implement prior for describing geophysical models is a nontrivial task. In this paper, we propose two generic optimization algorithms to implement arbitrary regularization in nonlinear inverse problems such as full-waveform inversion (FWI), where the regularization task is recast as a denoising problem. We assess these optimization algorithms with the plug-and-play block-matching BM3D regularization algorithm, which determines empirical priors adaptively without any optimization formulation. The nonlinear inverse problem is solved with a proximal Newton method, which generalizes the traditional Newton step in such a way to involve the gradients/subgradients of a (possibly non-differentiable) regularization function through operator splitting and proximal mappings. Furthermore, it requires to account for the Hessian matrix in the regularized leastsquares optimization problem. We propose two different splitting algorithms for this task. In the first, we compute the Newton search direction with an iterative method based upon the firstorder generalized iterative shrinkage-thresholding algorithm (ISTA), and hence Newton-ISTA (NISTA). The iterations require only Hessian-vector products to compute the gradient step of the quadratic approximation of the nonlinear objective function. The second relies on the alternating direction method of multipliers (ADMM), and hence Newton-ADMM (NADMM), where the least-squares optimization subproblem and the regularization subproblem in the composite objective function are decoupled through auxiliary variable and solved in an alternating mode. The least-squares subproblem can be solved with exact, inexact, or quasi-Newton methods. We compare NISTA and NADMM numerically by solving FWI with BM3D regularization. The tests show promising results obtained by both algorithms. However, NADMM shows a faster convergence rate than Newton-ISTA when using L-BFGS to solve the Newton system.

INTRODUCTION

Non-linear inverse problems frequently arise in different fields of geosciences [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]. Large-scale problems are typically solved with iterative local optimization (gradient-based) techniques such as Newton's method. Furthermore, such problems are inherently ill-posed and thus require regularization techniques to be implemented such that assumptions and priors about the unknown models are encoded in the optimization. At the heart of a suitable regularization is a priori information expressed by the regularizer or regularization function [START_REF] Gholami | Regularization of linear and nonlinear geophysical ill-posed problems with joint sparsity constraints[END_REF]. A proper reg-ularizer can be added to the objective function as penalty terms or constrains [START_REF] Peters | Constraints versus penalties for edgepreserving full-waveform inversion[END_REF][START_REF] Peters | Projection methods and applications for seismic nonlinear inverse problems with multiple constraints[END_REF], potentially renders the solution unique, increases its stability, and prevents data overfitting. It should be able to (mathematically) describe the solution while being easy to implement with iterative linearization methods. These specifications make tailoring a suitable regularizer nontrivial. A prior can be adaptive or non-adaptive, where by adaptive is meant the adaptation of the regularization function to the problem of interest. Traditional priors used to solve inverse problems such as smoothness, sparseness, blockiness are non-adaptive [START_REF] Tikhonov | Solution of ill-posed problems[END_REF][START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]. They are defined ac-cording to the preliminary assumptions about the targeted model, which are independent of the input of the problem to be solved. In contrast, adaptive priors are solely derived from the input and tailored to the output model accordingly. Complex models require complex priors, which can be hard to derive. Different priors lead to different forms of regularization, ranging from smooth and convex single-parameter regularizers [START_REF] Tikhonov | Solution of ill-posed problems[END_REF] to nonsmooth and non-convex multi-parameter ones (Gholami & Hosseini 2011;[START_REF] Selesnick | Sparse signal approximation via nonseparable regularization[END_REF].

Denoising as the simplest inverse problem (Section 2.1) has contributed to enormous progress in developing sophisticated adaptive and non-adaptive priors for complicated signal recovery from noisy signals [START_REF] Milanfar | A tour of modern image filtering: New insights and methods, both practical and theoretical[END_REF]. Some recently proposed excellent denoising methods include nonlocal means filters [START_REF] Milanfar | A tour of modern image filtering: New insights and methods, both practical and theoretical[END_REF][START_REF] Goyal | Image denoising review: From classical to state-of-the-art approaches[END_REF]) and block matching and 3D filtering (BM3D) (Dabov et al. 2007) and its variants [START_REF] Goyal | Image denoising review: From classical to state-of-the-art approaches[END_REF]). These patch-based methods use both local and nonlocal redundancy of information in the input signal to preserve structures in the solution by yielding locally adaptive filters via similarity kernels. Specifying the kernel function in these methods is essentially equivalent to estimating a particular type of empirical prior from the input signal [START_REF] Milanfar | A tour of modern image filtering: New insights and methods, both practical and theoretical[END_REF]. This somehow contrasts with the traditional non-adaptive regularization methods, for which the prior is fixed and independent from the input signal [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]. Such an adaptive regularization has been applied to linear inverse problems in, e.g., [START_REF] Danielyan | Bm3d frames and based denoising and inversion[END_REF] and [START_REF] Venkatakrishnan | Plugand-play priors for model based reconstruction[END_REF]. We refer the reader to Appendix A for a more detailed review of the BM3D method that will be used in this study.

The focus of this paper is to present a generic optimization framework to implement arbitrary regularizer in nonlinear inverse problems such as Full Waveform Inversion (FWI) [START_REF] Tarantola | Inversion of seismic reflection data in the acoustic approximation[END_REF][START_REF] Pratt | Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[END_REF][START_REF] Virieux | An overview of full waveform inversion in exploration geophysics[END_REF]. This framework extends proximal Newton-type algorithms to deal with adaptive plug-and-play regularizer, which may not have an explicit optimization (or variational) formulation as BM3D (Dabov et al. 2007). Similar to the classical Newton-type methods, proximal Newton methods solve a nonlinear inverse problem iteratively as m k+1 = m k + α k ∆m k , where m k is the model parameters at iteration k, ∆m k is the search direction and α k is the step length. When a composite objective function includes a general (and possibly non-differentiable) regularization term, ∆m k must further involve the gradients/subgradients of the regularization function [START_REF] Lee | Proximal Newton-type methods for minimizing composite functions[END_REF]. Proximal Newton methods achieve this task by breaking down the original complex problem into simpler subproblems through operator splitting and proximal mappings. We propose two distinct algorithms to solve the regularized problem with proximal Newton methods. In the first, called NISTA, the Newton search direction ∆m k is computed at iteration k by minimizing a composite objective function given by the sum of the locally quadratic approximation of the nonlinear misfit function involving the Hessian and the regularization function. The minimum of this surrogate objective function is found iteratively in an inner loop inside iteration k with a proximal gradient method (Section 2.2) based upon the shrinkage-thresholding algorithm (ISTA) (Daubechies et al. 2004;[START_REF] Attouch | The rate of convergence of nesterov's accelerated forward-backward method is actually faster than 1/kˆ2[END_REF]. This method requires one Hessianvector product per inner iteration to build the gradient of the surrogate linearized misfit function.

The second algorithm, called NADMM, relies on the alternating direction method of multipliers (ADMM) [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction of multipliers[END_REF]Aghamiry et al. 2019b). ADMM decouples the linearized leastsquares objective function and the regularization term via an auxiliary variable and solves the two subproblems in alternating mode with the primal-dual method of multipliers. The first subproblem requires to solve at each iteration k a linear system involving the Hessian, just like classical Newton-type methods. This system can be solved exactly or approximately with inexact or quasi-Newton algorithms [START_REF] Nocedal | Numerical Optimization[END_REF][START_REF] Métivier | Full waveform inversion and the truncated Newton method[END_REF]).

An important property of the proposed algorithms is that they only need the output of the regularizer without requiring any information about its functional form and statistical properties. This black-box implementation brings flexibility to the algorithms for implementing arbitrary adaptive and non-adaptive regularizations in the nonlinear inverse problem. Therefore, the main properties of the proposed regularization method can be summarized as follow: [START_REF] Attouch | The rate of convergence of nesterov's accelerated forward-backward method is actually faster than 1/kˆ2[END_REF] It can be easily implemented for existing nonlinear optimization algorithms.

[2] The regularizer is treated as black-box denoiser, and thus, regularization which has not an explicit optimization formulation such as plug-and-play denoiser can be used. [3] Irrespective of the differentiability of the regularizer, it can be implemented with iterative gradient-based solvers. [4] The computational overhead generated by the regularization is the computation of the proximal/denoising operator at each iteration, and hence is negligible in most cases.

We implement the proposed adaptive regularization to solve full-waveform inversion (FWI), an ill-posed PDE-constrained nonlinear optimization problem, in which the subsurface parameters and the wavefields are defined as the minimizers of the Euclidean distance between observed and calculated data [START_REF] Tarantola | Inversion of seismic reflection data in the acoustic approximation[END_REF][START_REF] Pratt | Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[END_REF]). Among different methods to solve this constrained optimization problem, we consider a variable projection formulation leading to the classical FWI [START_REF] Pratt | Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[END_REF]) and a PDE-relaxation formulation implemented with ADMM (Aghamiry et al. 2019c). The ADMM formulation, which updates the parameters and the wavefields in alternating mode, is referred to iteratively refined wavefield reconstruction inversion (IR-WRI) (Aghamiry et al. 2019c,b). Numerical tests performed show outstanding performance of the adaptive regularization in building complicated velocity models by the above waveform inversion methods.

PRELIMINARIES

We start with a brief review of the concepts and formulas of the linear inverse theory used in this paper. This section is intended for background, and the readers who are familiar with linear inverse theory can skim over this section. In linear inverse problems, the desired model, denoted by the column vector m, needs to be estimated from measurements d that are related to m via a linear operator/matrix A, i.e. d = Am + e for some random noise e. For a Gaussian distributed random noise, the estimation problem usually appears as determination of the minimizer of a suitably defined objective function

arg min m 1 2 d -Am 2 2 + λR(m), (2.1)
where R is a regularizer or regularization function, which somehow prevents data overfitting and λ determines regularization weight. Different forms of R have been proposed, ranging from smooth and convex single-parameter functions [START_REF] Tikhonov | Solution of ill-posed problems[END_REF] to non-smooth and non-convex multi-parameter ones (Gholami & Hosseini 2011;[START_REF] Selesnick | Sparse signal approximation via nonseparable regularization[END_REF]. In its simplest form, R(m) = mm prior 2 2 is a damping term that encourages m not to be very far from the prior model m prior (Tarantola 2005).

Denoising and Proximal Operator

In denoising problem, A = I (the identity matrix) and the estimate is simply defined as

prox λR (d) = arg min m 1 2 d -m 2 2 + λR(m). (2.2)
This is called the proximal operator of R [START_REF] Combettes | Video denoising by sparse 3D transform-domain collaborative filtering[END_REF][START_REF] Parikh | Proximal algorithms[END_REF]. Despite its simple definition, proximal operators are powerful tools in optimization because 1) the general optimization problem (2.1) can be solved by proximal algorithms, which merely require to evaluate the gradient of the misfit function, M(m) = 1 2 d -Am 2 2 , and the proximal operator (2.2). 2) Since a proximal operator involves the information about gradients/subgradients of R, proximal algorithms handle both differentiable and nondifferentiable forms of R. This contrasts with traditional algorithms, such as the Newton's algorithm, which requires the objective to be differentiable. Furthermore, the interpretation of the proximal operator as a denoising problem [START_REF] Kamilov | A plug-and-play priors approach for solving nonlinear imaging inverse problems[END_REF] allows us to solve optimization problem (2.1) with advanced regularizations embedded in sophisticated denoising algorithms such as non-local means (NLM), Block-matching and 3D filtering (BM3D) (Appendix A) or deep learning denoisers [START_REF] Meinhardt | Learning proximal operators: Using denoising networks for regularizing inverse imaging problems[END_REF]).

The Proximal Gradient Method

The proximal-gradient method is an important tool for solving non-linear problems we describe in subsequent sections. In order to see how the proximal operator (2.2) helps to solve (2.1), we use the majorization-minimization (MM) approach [START_REF] Lange | MM optimization algorithms[END_REF], which has a simple convergence proof. The governing idea of MM is to find the minimum of non-convex/convex function M(m) + λR(m) via the iterative minimization of a simpler convex surrogate function It is interesting to note that, for M(m) = 1 2 d -Am 2 2 and c ∈ (0, 1/|||A||| 2 ) with |||A||| the largest singular value of A, we have that

M k (m) + λR(m) that majorizes M(m) + λR(m) at step k (i.e. M k (m) + λR(m) ≥ M(m) + λR(m)).
M(m) + λR(m) ≤ M k (m) + λR(m), (2.3)
with equality at m = m k , where

M k (m) = M(m k ) + (m -m k ) T ∇M(m k ) + 1 2c m -m k 2 2 , (2.4)
in which m k is a reference model (previous iterate) and ∇M(m k ) is the gradient vector. This approximation allows us to minimize (2.1) by iteratively minimizing a simpler problem

m k+1 = arg min m M k (m) + λR(m).
(2.5)

Simple algebra shows that equation (2.5) is equivalent to

m k+1 = prox cλR (m k -c∇M k (m k )).
(2.6) This is nothing but the famous iterative shrinkage-thresholding algorithm (ISTA) (Daubechies et al. 2004) (also known as forwardbackward splitting algorithm and proximal gradient method). FISTA [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF]) is an accelerated version that uses a particular linear combination of the two last iterates to perform the update. A simple acceleration is obtained by using the extrapolation method of Nesterov [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF], leading to the generalized form of ISTA [START_REF] Attouch | The rate of convergence of nesterov's accelerated forward-backward method is actually faster than 1/kˆ2[END_REF])

m k+1 = prox cλR (p k -c∇M k (p k )) p k+1 = m k+1 + k-1 k+2 (m k+1 -m k ).
(2.7)

METHOD

A nonlinear inverse problem such as FWI with a general form of regularization can be written as

min m M(m) + λR(m), (3.1)
where m gathers the model parameters. In optimization problem (3.1), M(m) is the data misfit function. Its minimization ensures that the simulated data F (m) are close to the measurements d, where F is a nonlinear differentiable function. R(m) is the possibly non-differentiable regularization, which encodes the prior knowledge about the model parameters and prevents data overfitting. λ is the trade-off parameter that balances between the data misfit and regularization terms.

A Newton-type method majorizes the misfit term with a local quadratic function of form

M k (m) = M(m k ) + (m -m k ) T ∇M(m k ) + 1 2 (m -m k ) T H k (m -m k ), (3.2)
where m k is the iterate at iteration k, ∇M(m k ) is the gradient vector, and H k is the Hessian matrix ∇ 2 M(m k ) or an approximation of it.

Using the approximation in equation (3.2), proximal Newtontype methods solve problem (3.1) iteratively as

m k+1 = m k + α k ∆m k , (3.3)
where α k is the step length, which can be determined by a line search method, and

∆m k = arg min ∆m M k (m k + ∆m) + λR(m k + ∆m) (3.4)
is a search direction [START_REF] Lee | Proximal Newton-type methods for minimizing composite functions[END_REF]. Computation of the search direction ∆m k is the most computationally expensive part of this algorithm because it requires the minimization of a composite function given by the sum of a quadratic term involving the Hessian matrix, equation (3.2), and the regularization term R. For λ = 0, the algorithm reduces to a classical Newton method, where an approximation of the Hessian can be employed, leading to quasi-Newton methods or gradient method if H k reduces to a scaled version of the identity matrix. For λ = 0, however, determination of the search direction in problem (3.4) is more challenging. In the following, we propose two methods for this task.

Newton-ISTA (NISTA)

NISTA relies on the first-order iterative shrinkage-thresholding algorithm (ISTA). We refer the reader to Daubechies et al. ( 2004) and Section 2.2 for a more detailed review of this method to estimate iteratively the Newton search direction of problem (3.4). This requires to implement the following inner loop within the outer loop over k

       ∆m + 1 2 k = ∆p -c k [H k ∆p + ∇M(m k )], ∆m +1 k = prox c k λR (m k + ∆m + 1 2 k ) -m k , ∆p +1 = ∆m +1 k + -1 +2 (∆m +1 k -∆m k ), (3.5)
where l is the inner iteration count, ∆p 0 = 0, and

c k ∈ (0, 1/|||H k ||| 2 ).
The term in bracket in the first line of equation (3.5) is the gradient of the surrogate function M k (m), equation (3.2). Also, the operator in the second line of equation (3.5), prox(•), is the proximal operator of the regularization term R(m), which can be viewed as a denoiser (Section 2.1). For many choices of the regularizer R, there can be a closed-form expression for the denoiser in the second subproblem of (3.5). The main property of this formulation is that it can be generalized to exploit multiple (even data-driven) priors by using different denoisers (e.g., BM3D (Dabov et al. 2007), the weighted nuclear norm [START_REF] Kamilov | A plug-and-play priors approach for solving nonlinear imaging inverse problems[END_REF])) instead of the prox operator. It is important that the denoiser function is treated as a black box, i.e., we only need access to the output of the denoiser for a given input, irrespective of its functional form.

The NISTA is summarized in Algorithm 1. The algorithm is started with ∆p0 = 0. However, to reduce the number of iterations, we can perform a warm start of the inner loop by using the results of the previous iteration. Any approximation of the Hessian (diagonal, BFGS approximation) can be used to perform the Hessian-vector product in line 5 of Algorithm 1. Alternatively, a second-order adjoint-state method can be used to perform this product [START_REF] Métivier | Full Waveform Inversion and the truncated Newton method[END_REF]. ∆m

+ 1 2 k = ∆p -c k (H k ∆p + ∇M(m k )) 6: ∆m +1 k = prox c k λR (m k + ∆m + 1 2 k ) -m k 7: ∆p +1 = ∆m +1 k + -1 +2 (∆m +1 k -∆m k ) 8:
end for 9:

Select step length α k with a line search algorithm.

10:

Update: m k+1 = m k + α k ∆m N k . 11: until stopping conditions are satisfied.

Newton-ADMM (NADMM)

NADMM is obtained by solving optimization problem (3.4) via the alternating direction method of multipliers (ADMM) [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction of multipliers[END_REF]. By introducing the auxiliary variable p = m k +∆m, we recast the minimization problem in (3.4) as the following constrained problem:

min ∆m,p M k (m k + ∆m) + λR(p) subject to m k + ∆m = p. (3.6)
Solving problem (3.6) with an augmented Lagrangian method leads to the following saddle point problem

min ∆m,p max q M k (m k + ∆m) + λR(p) + q, m k + ∆m -p + 1 2c k m k + ∆m -p 2 2 , (3.7)
where q is the Lagrange multiplier and 1/c k serves as a penalty parameter. Applying the scaled form of ADMM (Boyd et al. 2010, section 3.1.1) to problem (3.7), when combined with equation (3.3), gives the iteration

               ∆m k = arg min∆m M k (m k + ∆m) + 1 2c k m k + ∆m -p k -q k 2 2 m k+1 = m k + α k ∆m k p k+1 = prox c k λR (m k+1 -q k ) q k+1 = q k + p k+1 -m k+1 , (3.8)
where the primal and dual variables are updated in alternating mode. A key property of the ADMM algorithm, equation (3.8), is that we don't need to solve it exactly at each iteration. The errors which are made by inexact solving of equation (3.8) are compensated by the Lagrange multipliers. This statement was corroborated by numerical experiments that showed that one (inner) iteration of equation (3.8) generates solutions that are accurate enough to guarantee the fast convergence of the ADMM algorithm [START_REF] Goldstein | The split Bregman method for L1regularized problems[END_REF][START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction of multipliers[END_REF]Aghamiry et al. 2019bAghamiry et al. , 2020a)).

With a change of variable m prior k = p k + q k , the first subproblem in (3.8) requires us to solve

min ∆m M k (m k + ∆m) + 1 2c k m k + ∆m -m prior k 2 2 , (3.9)
which has a closed-form minimizer given by

∆m k = (c k H k + I) -1 (-c k ∇M(m k ) + ∆m prior k ), (3.10)
where ∆m prior k = m prior k -m k . This is a generalized gradient step because it implicitly includes the information carried out by the gradient/subgradient of the possibly non-differentiable regularizer. It is seen that the prior information introduced by R(m) in the original problem (3.1), regardless of its mathematical form or its differentiability, is replaced by the a priori information that the (unknown) model at each iteration is a sample of a known Gaussian probability density whose mean is m prior k and whose covariance matrix is a scaled identity matrix. The regularization appeared as a damping term that encourages the model not to be very far from the dynamic prior/reference model m prior k , unlike traditional Bayesian approach [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF] where the a priori model is static. The (c k H k + I) -1 in Newton system (3.10), line 4 of Algorithm 2, can be approximated with any quasi-Newton or inexact Newton methods such as L-BFGS [START_REF] Nocedal | Numerical Optimization[END_REF] or the truncated Newton method [START_REF] Métivier | Full waveform inversion and the truncated Newton method[END_REF]). The proposed NADMM method is summarized in Algorithm 2.

Algorithm 2 Adaptive regularization by NADMM.

Require: starting point m0

1: set p0 = q0 = 0 2: repeat 3:

Compute the Hessian H k or an approximation to it.

4:

Compute the step direction:

∆m k = (c k H k + I) -1 (-c k ∇M(m k ) + p k + q k -m k ).
5:

Select step length α k with a line search algorithm.

6:

Update:

m k+1 = m k + α k ∆m k . 7:
Update: p k+1 = prox c k λR (m k+1 -q k ).

8:

Update: q k+1 = q k + p k+1 -m k+1 9: until stopping conditions are satisfied.

Application to Full Waveform Inversion

In the Numerical example section, we assess the algorithms 1 and 2 against seismic full-waveform inversion (FWI) [START_REF] Virieux | An overview of full waveform inversion in exploration geophysics[END_REF] methods with a series of benchmark velocity models. FWI is a nonlinear multivariate PDE-constrained optimization problem, which can be formulated in the frequency domain as

min m,u Pu -d 2 2 , subject to A(m)u = b, (3.11)
where m ∈ R N ×1 denotes the discrete subsurface model parameters, b ∈ C N ×1 the source term, u ∈ C N ×1 the modelled wavefield, d ∈ C M ×1 the recorded seismic data and P ∈ R M ×N a linear observation operator that samples u at the receiver positions. This objective function is for one frequency and one source. Extension to multiple frequencies and sources is simply implemented by summation over sources and frequencies in the objective function and by adding multiple right-hand sides in the constraint, equation (3.11) (Aghamiry et al. 2019a). The matrix A(m) ∈ C N ×N , whose coefficients depend on m, represents the discretized PDE [START_REF] Pratt | Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[END_REF][START_REF] Plessix | A Helmholtz iterative solver for 3D seismic-imaging problems[END_REF]). Here, we briefly review the two different formulations of FWI that will be used. The first classical one relies on variable projection to recast the nonlinear constrained problem as an unconstrained problem with a reduced search space [START_REF] Plessix | A review of the adjoint-state method for computing the gradient of a functional with geophysical applications[END_REF]. The second extends the linear regime of the waveform inversion by relaxation of the PDE constraint with ADMM (Aghamiry et al. 2019c) in the framework of the wavefield reconstruction inversion (WRI) method (van Leeuwen & Herrmann 2013, 2016) (see also Huang et al. (2018a) and [START_REF] Huang | Volume source-based extended waveform inversion[END_REF] for some variants). This recasts the original nonlinear constrained problem as a biconvex problem according to the bilinearity of the wave equation in wavefield and model parameters.

Reduced-space FWI

Classical FWI [START_REF] Pratt | Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[END_REF][START_REF] Plessix | A review of the adjoint-state method for computing the gradient of a functional with geophysical applications[END_REF]) strictly enforces the PDE constraint, u = A(m) -1 b, in the misfit function at each iteration by projection of the full multivariate search space onto the parameter search space for sake of computational efficiency. This leads to the following monovariate misfit function

M(m) = 1 2 d -F (m) 2 2 , (3.12)
where F (m) = PA -1 (m)b is the calculated data. The gradient and the Hessian of equation (3.12), which are required for Proximal-Newton methods, are given by [START_REF] Pratt | Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[END_REF])

∇M(m) = -J T ∆d, (3.13)
and

∇ 2 M(m) = J T J + ∂J T ∂m T , [∆d| • • • |∆d], (3.14)
where ∆d = d -F (m) and J is the sensitivity or the Fréchet derivative matrix, defined as

Jij(m) = ∂[F (m)]i ∂mj . (3.15)
In practice, the gradient is calculated with the adjoint-state method [START_REF] Plessix | A review of the adjoint-state method for computing the gradient of a functional with geophysical applications[END_REF]. The action of the Hessian can be taken into account approximately with preconditioned quasi-Newton method (l-BFGS) or truncated Newton methods (see [START_REF] Métivier | The seiscope optimization toolbox: A large-scale nonlinear optimization library based on reverse communication[END_REF] for an overview).

ADMM-based Wavefield Reconstruction Inversion (IR-WRI)

In classical FWI, the wave-equation A(m)u = b is solved exactly at each iteration to generate the reduced form of the objective function (3.12). In the wavefield reconstruction inversion (WRI) method (van Leeuwen & Herrmann 2013, 2016), the wave-equation is processed as a weak constraint (namely, it is satisfied approximately) with a penalty method such that the simulated wavefields match the observations to a great extent. Then, the parameters are updated from the wavefields by least-squares minimization of the wave equation errors [START_REF] Van Leeuwen | Mitigating local minima in fullwaveform inversion by expanding the search space[END_REF]Aghamiry et al. 2019c). Updating the wavefields and the subsurface parameters in alternating mode at iteration k leads to the following objective function for m

M(m) = 1 2 b -A(m)u k 2 2 , (3.16)
where the so-called data-assimilated wavefield u k (Aghamiry et al. 2020b) is the least-squares solution of the overdetermined system gathering the wave equation and the observation equation

A(m k-1 ) µP u k = b µd , (3.17)
where µ > 0 is the penalty parameter. Note that equations (3.16) and (3.17) are provided assuming a single source experiment. For multiple sources, the objective function (3.16) is simply obtained by summation over sources, while one augmented system (3.17) per source needs to be solved (Aghamiry et al. 2019a). For (3.16), the gradient and Hessian are given by

∇M(m) = -L T (b -A(m)u k ), (3.18)
where (3.19) and

∇ 2 M(m) = L T L,
Lij = ∂[A(m)u k ]i ∂mj .
(3.20)

The columns of the matrix L contain the so-called virtual sources [START_REF] Pratt | Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[END_REF], that makes this matrix diagonal when optimization parameters are squared slownesses.

NUMERICAL EXAMPLES

Wavefield Inversion of Inclusions Models

We first assess the NISTA and NADMM algorithms in the FWI/WRI framework using BM3D as denoiser. To highlight the adaptivity power of BM3D, we design four different velocity models, where inclusions of different shape are added to a 2 km × 2 km homogeneous background velocity model (v = 2 km/s) (first column of Fig. 2). Also, to show the flexibility of the proposed adaptive regularization in managing different priors simultaneously, we put all the four inclusions together in the background model and reconstruct them jointly (Fig. 3). For all the tests of this section, data are generated by five sources at the surface (with 400 m spacing) and 50 m equally spaced receivers placed on all the boundaries except the surface. The forward modelling is performed with a 9-point stencil finite-difference method implemented with antilumped mass and PML absorbing boundary conditions to solve the Helmholtz equation, where the stencil coefficients are optimized to the frequency [START_REF] Chen | An optimal 9-point finite difference scheme for the Helmholtz equation with PML[END_REF]) (this scenario is considered for all wave-propagation examples in this paper). The source signature is a Ricker wavelet with a 10 Hz dominant frequency. We start the inversion from the homogeneous background model (v = 2 km/s) and invert simultaneously four frequency components (5, 7, 10, and 12.5 Hz) of noiseless and noisy data with FWI and IR-WRI with and without the BM3D regularization. We first apply FWI and IR-WRI via NADMM without and with BM3D regularization. It is worth to mention that, when a denoiser is used for regularization, the regularization weight λ is embedded in the standard deviation of noise (σ) of the denoiser, σ = √ c k λ (see equation 2.6). We set σ of BM3D equal to 25 and 45 for noiseless and noisy data, respectively (Appendix A). We perform FWI with the L-BFGS quasi-Newton method with line search to perform NADMM. We perform the inversion with noiseless data and set the maximum number of iteration to 70 as stopping criterion for IR-WRI in both cases (without and with BM3D). For FWI, the stopping criteria is set to the model error (l2-norm of the difference between true and estimated model) achieved by IR-WRI for a fair comparison between the two waveform inversion methods. Fig. 2 shows the results obtained by FWI and IR-WRI for all four models without and with regularization. It is clearly seen that, for both methods, regularization improved the results and successfully recovered the shape of the different anomalies, thanks to the adaptive nature of the BM3D. It is worth noting that IR-WRI performed better than FWI, although the focus of this paper is not to compare the two methods as this topic has been already addressed in Aghamiry et al. (2019c). Instead, we aim to show how adaptive regularization can improve the results of both methods when it is implemented with proximal Newton algorithms. We continue by using a model that includes all four inclusions (Fig. 3a). Fig. 3 shows the velocity models estimated by FWI (Figs. 3b-c) and IR-WRI (Figs. 3d-e) with and without regularization. The different inclusions are jointly reconstructed with the same accuracy as in the case where they are reconstructed independently (Fig. 2). This highlights the local-adaptivity property of the BM3D regularization.

Sensitivity to the regularization parameter

The only free parameter of BM3D is σ (Appendix A) which is selected by trial and error in this paper. Here we want to asses the sensitivity of the final results against σ. We apply NADMMbased FWI and IR-WRI with BM3D as regularizer on noiseless data using different values of σ. For each σ, the stopping criteria is chosen as in the previous test ( 70

Robustness against noise

We continue by assessing the robustness of the proposed method against random noise. We apply NADMM-based FWI and IR-WRI without and with BM3D when the data are contaminated with different level of random noises. The relative root mean square error (RMSE) curves versus signal to noise ratio (SNR) is depicted in Fig. 5, where RMSE and SNR are defined as

RMSE = 100 m -m * 2 m * 2 , (4.1)
in which m and m * are the estimated and true models, respectively, and SNR = 20 log Signal RM S amplitude N oise RM S amplitude . (4.2) Fig. 5 shows the average value (over 20 runs) for each SNR. Furthermore, we use Pu k -d 2 = 1.01ε as the stopping criterion of iteration, where ε is the 2 norm of the noise. The velocity models estimated by FWI and IR-WRI without/with BM3D regularization for SNR=5db are shown in Fig. 6. In order to show how the data are fitted, the difference between the estimated and noiseless (10 Hz) data for the tests of Fig. 6 are shown in Fig. 7.

A comparison between NISTA and NADMM

Here, we use the BM3D regularized FWI with noiseless and noisy data to compare NISTA and NADMM. Both NISTA and NADMM algorithms are implemented with L-BFGS to account for the action of the Hessian (Line 5 and 4 of Algorithms 1 and 2, respectively). Fig. 8 shows the velocity models reconstructed by both algorithms and Fig. 9 shows the corresponding convergence history (the value of the objective function as a function of the iteration count). Although we perform approximately 100 inner iterations of proximal gradient to estimate the search direction of NISTA, NADMM still performs better. Furthermore, since we implement both algorithms with L-BFGS, the results show that in practice NADMM should be preferred to NISTA.

Performance on benchmark models

We continue by assessing the performance of the NADMM-based adaptive IR-WRI algorithm using well-documented 2D benchmark subsurface velocity models in exploration seismic, e.g. the Marmousi II [START_REF] Martin | Marmousi2: An elastic upgrade for Marmousi[END_REF], SEG/EAGE overthrust [START_REF] Aminzadeh | 3-D Salt and Overthrust models[END_REF], SEG/EAGE salt [START_REF] Aminzadeh | 3-D Salt and Overthrust models[END_REF]), synthetic Valhall (Prieux et al. 2011[START_REF] Billette | The 2004 BP velocity benchmark[END_REF]BP salt (Billette & Brandsberg-Dahl 2004) benchmark velocity models. The selected target from these benchmark models are shown in the first column of Fig. 10, respectively. The fixed-spread acquisition with a few equally spaced sources at the sea bottom and a line of equally spaced receivers at the 25 m depth is used for all of the tests (see Table 1 for more technical details). Also, the models are discretized with 25 m spacing in horizontal and vertical directions. We compute the wavefields using Perfectly-Matched Layer (PML) absorbing boundary conditions along the bottom, right, and left sides of the model using 10 grid points in the PMLs and a free-surface boundary condition at the surface and a 10 Hz Ricker wavelet is used as the source signature. We design a multiscale inversion with a classical continuation frequency strategy in the selected frequency band by proceeding over small batches of two frequencies with a frequency interval of 0.5 Hz. We also perform three paths through the batches, where the starting and finishing frequencies of the paths and other technical details about the models are reported in Table 1. Also, we use 60, 40, and 25, respectively, for σ of BM3D in the first, second and third path. The initial velocity models are crude models, as shown in the second column of Fig. 10. Accordingly, we tackle these benchmarks with IR-WRI only since FWI would remain stuck in a local minimum due to cycle skipping.

We set the number of IR-WRI iterations per frequency batch equal to 10 or 2-norm of source residuals equal to 1e-3 as the stopping criteria. The estimated models without and with BM3D regularization with NADMM (Algorithm 2) are shown in the third and fourth columns of Fig. 10, respectively. A direct comparison along vertical logs (as depicted with dashed lines in first column of Fig. 10) between the true velocity, the initial and the final velocity models without/with BM3D regularization are shown in Figs. 11a-d, for Marmousi II, SEG/EAGE salt, Synthetic Valhall and 2004 BP salt models, respectively. The results show that, although the different benchmark models are characterized by different kinds of structures, adaptive regularization combined with IR-WRI manages to reconstruct accurately each of them with a significant jump of quality compared to the case where BM3D is not used.

CONCLUSIONS

In this paper, we proposed a flexible optimization framework based upon proximal Newton methods to implement arbitrary regularizations (smooth, nonsmooth, adaptive) in nonlinear inverse problem such as FWI and its WRI variant. The regularization problem is recast as a denoising problem through proximal mapping such that the optimization algorithm only needs the output of the denoiser. This means that any denoiser can be easily implemented as a black box in the optimization algorithm. This open the door to leadingedge plug-and-play denoisers such as the block-matching BM3D method, which has no explicit optimization formulation (it can not be cast as a variational problem). Two different algorithms (NISTA and NADMM) based upon FISTA and ADMM are proposed to implement the proximal Newton method. Both of them requires to perform Hessian-vector products, which can be implemented with quasi-Newton or second-order adjoint-state method. Both algorithms are assessed against a series of well-documented subsurface models. In all cases, the results show that NADMM should be preferred at the expense to NISTA. Although the adaptive BM3D regularizer was used in this study, more conventional Tikhonov, Total Variation or a combination of the two can be easily implemented in NISTA-based and NADMM-based FWI and IR-WRI. To allow people to evaluate the NISTA and NADMM algorithm, we have made some codes available at https://gitlab.oca.eu/wind.

APPENDIX A: BLOCK-MATCHING AND 3D FILTERING (BM3D)

Most of the denoising techniques are based on shrinking small coefficients in a transformed domain where the coefficient distribution is sparse. BM3D (Dabov et al. 2007) is a novel image denoising technique, which is based on an enhanced sparse representation in a transformed domain where the self-similarities in the image are exploited. This method contains four steps [START_REF] Lebrun | An analysis and implementation of the bm3d image denoising method[END_REF]):

• Select an image patch with size k × k and find all the similar patches in the original image and group them in a 3D cube.

• Apply a 3D linear transform on the cube.

• Filter the transform coefficients by thresholding or Wiener filtering.

• Apply inverse 3D transform and return the filtered patches to their correct positions.

Redundancy between patches enables BM3D to reconstruct smooth and flat parts properly. Furthermore, it is capable to reconstruct fine details and sharp edges. Despite good performance of this technique, it has some drawback like artifacts in denoised images when the signal to noise ratio is poor. Also, the computational complexity is much higher than that of simple denoising methods. The most computational burden of this method is generated by the first step, which is grouping and comparing similar patches. For a N × N image with a k × k patch, it is in order of O(N 4 k 2 ). The burden of the remaining steps is related to the used sparsifying transform. In this paper, we use 3D discrete cosine transform (DCT) and the computational complexity can be expressed as O(k 2 R log(k 2 R)) where R is the number of similar blocks in each cube. Moreover, BM3D and all of the patch-based denoising methods have a large number of parameters that are difficult to adjust properly. In this paper, we used the official package of BM3D, just set the standard deviation of noise (σ) and used the default values for the rest of the parameters according to table I of Dabov et al. (2007). The σ is selected by trial and error in this paper.

  Fig. 1 shows a schematic of the MM process. The non-convex function M(m) + λR(m) is shown in blue while a few surrogate functions M k (m) + λR(m) for points m k , k ∈ {0, 1, 2}, are shown in orange, pink and red, respectively. This figure shows how the iterative MM algorithm approaches a local minimum of M(m) + λR(m) through the minimum of easy to minimize surrogate functions M k (m) + λR(m).

Figure 1 .

 1 Figure 1. The function M(m) is shown in green where it has a global minimum at (4,0). Also M(m) + λR(m) is shown in blue with a global minimum at (6,0) while a few surrogate functions M k (m) + λR(m), k ∈ {0, 1, 2} are shown in orange, pink and red, respectively. The MM algorithm seeks to find a local minimizer of M(m) + λR(m), blue curve, by iteratively minimizing easy-to-minimize functions M k (m) + λR(m). Minimization of M k (m) + λR(m) gives a new optimal point,m 1 , m 2 and m 3 , and it reaches to the minimum of M(m) + λR(m) when k tends to infinity.

Algorithm 1

 1 Adaptive regularization by NISTA. Require: starting point m k 1: set ∆p 0 = 0

  iterations for IR-WRI and model error achieved by IR-WRI for FWI). The velocity models estimated by FWI and IR-WRI with BM3D regularization for σ = [160, 40, 5, 1] are shown in Fig. 4. Note that the results with σ = 20 are shown in Figs. 3c and 3e for FWI and IR-WRI, respectively. It is seen that both of FWI and IR-WRI can reconstruct an acceptable model for a wide range of values of σ, i.e. σ = [5 -40].

Figure 2 .

 2 Figure2. The performances of FWI and IR-WRI with adaptive NADMM for reconstruction of different velocity structures. The data were generated by five sources at the surface (with 400 m spacing) and 50 m equally spaced receivers positioned on all the boundaries except the surface. In all figures, the colour bar varies between 2 km/s and 2.5 km/s with low velocities in black and high velocities in white.

Figure 3 .

 3 Figure 3. Inclusion test. (a) True velocity model. (b-e) Velocity models estimated by (b) FWI without regularization, (c) FWI with regularization, (d) IR-WRI without regularization, (e) IR-WRI with regularization.

Figure 4 .

 4 Figure 4. Inclusion test with different σ for BM3D. (a-d) Estimated velocity model using FWI with BM3D regularization with σ = 160 (a), 40 (b), 5 (c), and 1 (d). (e-h) Same as (a-d) but for IR-WRI.

Figure 5 .Figure 6 .

 56 Figure 5. RMSE for FWI and IR-WRI without and with BM3D using NADMM when data are contaminated with different level of noises.

Figure 7 .

 7 Figure 7. Real part of the 10 Hz data. (a) The noisy data with SNR=5db are shown in blue, while the noiseless data are shown in red. (b) The difference between predicted data (Pu k ) and noiseless data (d 0 ) at the final iteration of Fig. 6. The residual curves are scaled by factor 2.

Figure 8 .

 8 Figure 8. Inclusion test with BM3D regularized FWI using NISTA (a and c) and NADMM (b and d). (a-b) Noiseless data, (c-d) Noisy data with SNR=5db.

Figure 10 .

 10 Figure 10. IR-WRI without/with BM3D regularization on benchmark models. (a) Marmousi II, (b) SEG/EAGE salt, (c) Synthetic Valhalla and (d) 2004 BP salt models. The columns of this figure are as follows: True velocity model, initial velocity model, IR-WRI and BM3D regularized IR-WRI with NADMM. The vertical dashed lines indicate the location of vertical logs of Fig. 11

Figure 11 .

 11 Figure 11. Direct comparisons between velocity models estimated by IR-WRI without/with BM3D regularization . (a) Marmousi II, (b) SEG/EAGE salt, (c) Synthetic Valhall and (d) 2004 BP salt models. In each panel, the true model is solid black, initial model is dashed black, the estimated model without regularization is blue and estimated model with regularization is orange. The horizontal location of each log is written in each panel and depicted with vertical lines in the first column of Fig. 10.

Table 1 .

 1 Technical details of the benchmark models

		Size (km × km)	Source interval(m)	Receiver interval(m)	Inverted frequency band (Hz) [starting-finishing]	Outer iterations frequencies (Hz) [starting-finishing]
	Marmousi II	4.25 × 11.5	250	50	[3-10]	[3-8], [4-9], [5-10]
	SEG/EAGE salt model	2.1 × 7.8	100	25	[3-7]

ACKNOWLEDGEMENTS

We would like to thank editors Frederik Simons, Fern Storey, and two anonymous reviewers for their comments which help improving the manuscript. This study was partially funded by the WIND consortium (https://www.geoazur.fr/WIND), sponsored by Chevron, Shell, and Total. This study was granted access to the HPC resources of SIGAMM infrastructure (http://crimson.oca.eu), hosted by Observatoire de la Côte d'Azur and which is supported by the Provence-Alpes Côte d'Azur region, and the HPC resources of CINES/IDRIS/TGCC under the allocation A0050410596 made by GENCI. Hossein S. Aghamiry would like to thank the IDEX UCA JEDI with the WIMAG project for their support. Ali Gholami would like to acknowledge the financial support of university of Tehran for this research under grant number 27711-1-06.


Aurélien Habrioux

Claudia Morais

Teko W Napporn

Boniface Kokoh

Recent trends in hydrogen and oxygen electrocatalysis for anion exchange membrane technologies

Keywords: Hydrogen evolution reaction, hydrogen oxidation reaction, oxygen reduction reaction, oxygen evolution reaction, electrocatalysis, alkaline electrolytes

This review aims at presenting recent findings in the understanding of oxygen and hydrogen electrocatalysis in alkaline electrolytes that are key processes for the emergence of sustainable energy storage and conversion devices such as anion exchanged membrane fuel cells and electrolyzers. In these systems, the exchange of electrons through electrochemical reactions provides a unique pathway to reversibly convert the electricity vector into chemical one: hydrogen. A concise and critical review of advances made during the last past years in the design of catalysts is provided. Challenges and opportunities for the development of the next catalyst generation are also addressed.

Introduction

Storing and then using renewable resources are sustainable processes to reduce fossil fuel consumption. The production of clean hydrogen in a water electrolyzer and its further combustion in a fuel cell to deliver heat, electricity and only water as by-product are thereby sustainable perspectives [START_REF] Zou | Noble metal-free hydrogen evolution catalysts for water splitting[END_REF] for the implementation of a future energy grid based on a hydrogen economy. Hydrogen as well as oxygen electrocatalysis will govern electrical performances of the above-mentioned applications. The recent development of anion exchange membranes [START_REF] Varcoe | Prospects for Alkaline Anion-Exchange Membranes in Low Temperature Fuel Cells[END_REF] with useful conductivities in the range of 20-40 mS cm -1 [START_REF] Ramaswamy | Hydrogen oxidation reaction in alkaline media: Relationship between electrocatalysis and electrochemical double-layer structure[END_REF] paved the way for the design of solid anion exchange membrane technologies, recently reinforcing the interest to design catalysts evolving in alkaline environments. These media allow using catalysts free from PGM and even free from critical raw materials [START_REF] Varcoe | Anion-exchange membranes in electrochemical energy systems[END_REF]. In this review, recent advances in the design of catalysts and in the understanding of their activity towards key hydrogen evolution (HER), hydrogen oxidation (HOR), oxygen evolution (OER) and oxygen reduction (ORR) reactions in alkaline medium are discussed. In this review, the emphasis will be mainly on recent developments made in the design of non-PGM based catalysts.

Hydrogen electrocatalysis in alkaline medium

Understanding factors affecting hydrogen electrocatalysis in alkaline medium

Hydrogen electrocatalysis has been extensively studied to lay the basis of modern electrocatalysis [START_REF] Tafel | Über die Polarisation bei kathodischer Wasserstoffentwicklung[END_REF]. Reaction mechanisms generally imply three different elementary steps (Heyrovsky-Volmer or Tafel-Volmer mechanisms) that can be described as follows in alkaline electrolytes (* refers to a catalytic site): To be more informative, HER/HOR mechanisms have been illustrated in Figure 1.

Figure 1:

Illustration of HER and HOR mechanisms in alkaline medium HOR/HER have been largely investigated in acidic electrolytes [START_REF] Parsons | The rate of electrolytic hydrogen evolution and the heat of adsorption of hydrogen[END_REF] and reactivity of PGMbased catalysts was shown to be lower by at least two orders of magnitude in alkaline media [START_REF] Durst | New insights into the electrochemical hydrogen oxidation and evolution reaction mechanism[END_REF][START_REF] Rheinländer | Kinetics of the Hydrogen Oxidation/Evolution Reaction on Polycrystalline Platinum in Alkaline Electrolyte Reaction Order with Respect to Hydrogen Pressure[END_REF] making it difficult to explain reactivity with a simple descriptor. It was first proposed that metal-hydrogen bond strength increased with pH value in agreement with the position of Hupd peaks [START_REF] Sheng | Correlating hydrogen oxidation and evolution activity on platinum at different pH with measured hydrogen binding energy[END_REF]. Nevertheless, this quite simple correlation has to be reviewed since Hupd peaks position depends both on Pt surface oxophilicity [START_REF] Mccrum | First Principles Simulations of Cyclic Voltammograms on Stepped Pt(553) and Pt(533) Electrode Surfaces[END_REF] and the adsorption of cations from the alkaline electrolyte [START_REF] Chen | Co-adsorption of Cations as the Cause of the Apparent pH Dependence of Hydrogen Adsorption on a Stepped Platinum Single-Crystal Electrode[END_REF]. Recently, Strmcnik et al. [START_REF] Strmcnik | Design principles for hydrogen evolution reaction catalyst materials[END_REF][START_REF] Strmcnik | Improving the hydrogen oxidation reaction rate by promotion of hydroxyl adsorption[END_REF], showed that three main factors affect HOR and HER kinetics. In addition to H adsorption energy, the nature of the proton source (oxonium ion in acidic medium or water in alkaline medium) as well as the presence of surface species (hydroxides, electrolyte ions) should be considered to explain HOR/HER activity of a material. The two latter phenomena are responsible for modifying both adsorption energy of hydrogen and water molecules, thus particularly affecting kinetics of the Volmer step [START_REF] Ramaswamy | Hydrogen oxidation reaction in alkaline media: Relationship between electrocatalysis and electrochemical double-layer structure[END_REF]. Furthermore, very recently, the role played by the formation of OHad-(H2O)x-AM + (with AM + = alkali metal cation) adducts in alkaline electrolytes has been brought to light and has been shown to be responsible for destabilizing OHad, thus limiting the rate of HOR whereas promoting HER kinetics [START_REF] Liu | Unifying the Hydrogen Evolution and Oxidation Reactions Kinetics in Base by Identifying the Catalytic Roles of Hydroxyl-Water-Cation Adducts[END_REF]. From the previous discussion, it appears that tuning of OH adsorption energy at a catalyst surface is a promising pathway to improve its catalytic efficiency towards hydrogen electrocatalysis.

Catalysts for HOR/HER in alkaline medium

PGM-based catalysts (based on the use of Pd, Pt, Ir, Ru) are the most efficient ones towards HOR in alkaline medium. Numerous strategies (substrate engineering, alloying effect, particle size effect) have been employed to improve their activity and recently reviewed [START_REF] Campos-Roldán | The Hydrogen Oxidation Reaction in Alkaline Medium: An Overview[END_REF]. During the last years, research efforts were focused on the alloying of noble metals (Pt or Pd) with oxophilic atoms such as Ru and Ni [START_REF] Strmcnik | Improving the hydrogen oxidation reaction rate by promotion of hydroxyl adsorption[END_REF][START_REF] Alesker | Palladium/nickel bifunctional electrocatalyst for hydrogen oxidation reaction in alkaline membrane fuel cell[END_REF][START_REF] Scofield | Role of Chemical Composition in the Enhanced Catalytic Activity of Pt-Based Alloyed Ultrathin Nanowires for the Hydrogen Oxidation Reaction under Alkaline Conditions[END_REF]. As an example, resulting Pt/Ru catalyst exhibited a very high HOR activity (see Table 1) in reason of the promotion of OH adsorption. The same kind of effects was observed by depositing Pd nanocrystals onto oxophilic ceria [START_REF] Miller | Highly active nanostructured palladiumceria electrocatalysts for the hydrogen oxidation reaction in alkaline medium[END_REF]. Non-noble metal-based catalysts presenting a significant HOR activity are exclusively based on the use of Ni. Performances of such catalysts are presented in Table 1. Alloying Ni with Co and Mo allowed obtaining a very active CoNiMo catalyst [START_REF] Sheng | Nonprecious metal electrocatalysts with high activity for hydrogen oxidation reaction in alkaline electrolytes[END_REF] presenting a hydrogen binding energy close to that of Pt. The synthesis and the use of Ndoped carbon nanotubes supported Ni nanoparticles as catalyst allowed achieving a very high HOR kinetics [START_REF] Zhuang | Nickel supported on nitrogen-doped carbon nanotubes as hydrogen oxidation reaction[END_REF], in reason of the electronic modulation of Ni resulting from the interaction between Ni and N-edge atoms grafted onto the carbon substrate. The substrate engineering appears then as a reliable strategy to modulate hydrogen bonding energy at a catalytic surface.

More recently, a Ni3N/C catalyst was synthesized and exhibited a huge activity towards HOR in reason of a downshift of the Ni d-band weakening binding energy of hydrogen as well as oxygenated species [21]. As reviewed in 2015 [START_REF] Zou | Noble metal-free hydrogen evolution catalysts for water splitting[END_REF], HER catalysts can be classified into three main subclasses comprising PGM-based catalysts [22], transition-metal carbides [START_REF] Huang | Mo2C Nanoparticles Dispersed on Hierarchical Carbon Microflowers for Efficient Electrocatalytic Hydrogen Evolution[END_REF], oxides [START_REF] Hao | A bimetallic oxide Fe1.89Mo4.11O7 electrocatalyst with highly efficient hydrogen evolution reaction activity in alkaline and acidic media[END_REF], sulfides [START_REF] Kuang | Situ Fabrication of Ni-Mo Bimetal Sulfide Hybrid as an Efficient Electrocatalyst for Hydrogen Evolution over a Wide pH Range[END_REF], selenides [START_REF] Yang | High-index faceted binary-metal selenide nanosheet arrays as efficient 3D electrodes for alkaline hydrogen evolution[END_REF], phosphides [START_REF] Zhang | A Mn-doped Ni2P nanosheet array: an efficient and durable hydrogen evolution reaction electrocatalyst in alkaline media[END_REF], nitrides [START_REF] Lai | Strongly Coupled Nickel-Cobalt Nitrides/Carbon Hybrid Nanocages with Pt-Like Activity for Hydrogen Evolution Catalysis[END_REF], borides [START_REF] Sheng | Network-like porous Co-Ni-B grown on carbon cloth as efficient and stable catalytic electrodes for hydrogen evolution[END_REF] and non-metal catalysts based on the use of C, N, S elements [START_REF] Peng | Homologous metal-free electrocatalysts grown on three-dimensional carbon networks for overall water splitting in acidic and alkaline media[END_REF]. of the lowering of adsorption free energy of H at heterointerfaces [START_REF] Li | Metal-Organic Precursor-Derived Mesoporous Carbon Spheres with Homogeneously Distributed Molybdenum Carbide/Nitride Nanoparticles for Efficient Hydrogen Evolution in Alkaline Media[END_REF]. These heterostructured catalysts often combine at least two different efficient catalysts towards HER [START_REF] Kuang | Situ Fabrication of Ni-Mo Bimetal Sulfide Hybrid as an Efficient Electrocatalyst for Hydrogen Evolution over a Wide pH Range[END_REF]. This concept was then used to perform the growth of heterostructured catalysts composed of sulfides [START_REF] Zhang | Interface Engineering of MoS2/Ni3S2 Heterostructures for Highly Enhanced Electrochemical Overall-Water-Splitting Activity[END_REF] or phosphides [START_REF] Yu | Highperformance bifunctional porous non-noble metal phosphide catalyst for overall water splitting[END_REF] directly onto Ni foam. The established interfaces allow facilitating both chemisorption of hydrogen and oxygen intermediates, thus improving the overall water splitting activity [START_REF] Zhang | Interface Engineering of MoS2/Ni3S2 Heterostructures for Highly Enhanced Electrochemical Overall-Water-Splitting Activity[END_REF]. Another recent trend consists in atomically engineering active sites by doping a poorly active HER catalyst. This allows tuning electron density of active sites, thus modifying H adsorption energy and favoring the cleavage of water molecule [START_REF] Huang | Atomically engineering activation sites onto metallic 1T-MoS2 catalysts for enhanced electrochemical hydrogen evolution[END_REF][START_REF] Wu | Electron density modulation of NiCo2S4 nanowires by nitrogen incorporation for highly efficient hydrogen evolution catalysis[END_REF][START_REF] Ling | Well-Dispersed Nickel-and Zinc-Tailored Electronic Structure of a Transition Metal Oxide for Highly Active Alkaline Hydrogen Evolution Reaction[END_REF]. HER/HOR activities of previously described materials are reported in Table 1.

Table 1: Non-exhaustive list presenting representative recently developed catalysts for HER/HOR in alkaline medium. j0 represents the exchange current density. It has been normalized using either platinum active surface area (in that case the unit is denoted as mg cm -2 Pt) or using geometric surface area. The mass normalized j0 value is also given for HOR catalysts. η represents the overpotential. For HER, overpotential values at -10 mA cm -2 (η-10) as well as at -100 mA cm -2 (η-100) are given. b is the Tafel slope.

Oxygen electrocatalysis in alkaline medium

Origin of oxygen electrode overpotentials

OER/ORR kinetic limitations can be described assuming a reaction mechanism composed of four proton coupled electron transfer steps and implying three reaction intermediates (S=O, S-OH and S-OOH, where S denotes a catalytic site). This reaction mechanism can be described using the following equations ( 4)-( 7):

S + H2O = S-OH + H + + e - (4)
S-OH = S=O + H + + e - ( 5)

S=O + H2O = S-OOH + H + + e - (6)
S-OOH = S + O2+ H + + e - (7)
From these reactions, the different reaction energies can be expressed as follows taking water as the origin of the energy scale:

∆G 1 = ∆G OH -∆G H 2 O = ∆G OH (8)
∆G 2 = ∆G O -∆G OH (9)
∆G 3 = ∆G OOH -∆G O (10)
∆G 4 = ∆G O 2 -∆G OOH = 4.92 -∆G OOH (11)
Considering that total energy required to transform H2O into O2 is 4.92 eV (41.23 eV) as well as that S-OH and S=O are respectively singly and doubly bonded reaction intermediates, equations ( 8)-( 11) can be written, as follows [START_REF] Busch | Beyond the top of the volcano? -A unified approach to electrocatalytic oxygen reduction and oxygen evolution[END_REF]:

∆G 1 = ∆G 2 (12)
∑ ΔG i = i=1 to 4

4.92 eV ( 13)

∆G 4 = 4.92 -∆G 3 -2∆G 1 (14)
Thus ΔG i can be described using two independent parameters, that is to know ∆G 1 and ∆G 3 .The overpotential required for OER/ORR is obviously governed by the step with highest energy barrier.

Thus, ORR/OER overpotentials can be described with the following equations [START_REF] Busch | Beyond the top of the volcano? -A unified approach to electrocatalytic oxygen reduction and oxygen evolution[END_REF][START_REF] Govindarajan | Does the breaking of adsorption-energy scaling relations guarantee enhanced electrocatalysis?[END_REF]:

𝜂 𝑂𝐸𝑅 = 𝑀𝑎𝑥(ΔG i ) 𝑖 = 1 𝑡𝑜 4 𝑒 -1.23 (15)
𝜂 𝑂𝑅𝑅 = 𝑀𝑖𝑛(ΔG i ) 𝑖 = 1 𝑡𝑜 4 𝑒 -1.23 ( 16
)
where e is the elementary charge.

For an ideal catalyst, it is observed that (ΔG i ) 𝑖=1 𝑡𝑜 4 = [START_REF] Busch | Beyond the top of the volcano? -A unified approach to electrocatalytic oxygen reduction and oxygen evolution[END_REF]. Nevertheless, as oxygenated intermediates are chemically bonded to the catalyst surface through O atom, binding energies of S-OH and S-OOH are interdependent [START_REF] Koper | Thermodynamic theory of multi-electron transfer reactions: Implications for electrocatalysis[END_REF][START_REF] Rossmeisl | Electrolysis of water on oxide surfaces[END_REF] and linearly scaled, rendering impossible their separate optimization. Furthermore, the energy difference ∆G OOH -∆G OH is nearly constant whatever the considered materials and is equal to 3.2 eV [START_REF] Govindarajan | Does the breaking of adsorption-energy scaling relations guarantee enhanced electrocatalysis?[END_REF], far from the ideal value of 2.46 eV. For the real catalyst, at 1.23 V vs. RHE, all steps are not downhill processes. Applying an electrode potential away from 1.23 V vs. RHE is thus required.

Comparison between different catalysts families using this theoretical framework is nevertheless only possible if catalytic surfaces follow the same reaction mechanism. That is not the case for all catalysts families and OER/ORR mechanisms are still under debate. Some recent studies pointed out that breaking scaling relations are a required but not sufficient condition to optimize the activity of a catalyst and the potential determining step has to be considered through the examination and comparison of the electrochemical-step symmetry index value with overpotential [START_REF] Govindarajan | Does the breaking of adsorption-energy scaling relations guarantee enhanced electrocatalysis?[END_REF][START_REF] Govindarajan | Outlining the Scaling-Based and Scaling-Free Optimization of Electrocatalysts[END_REF]. It is widely accepted that these two adsorption modes lead to a fourelectron pathway and a two-electron pathway, respectively [START_REF] Jörissen | Bifunctional oxygen/air electrodes[END_REF].

On platinum surfaces the first electron transfer step to adsorbed oxygenated species (Osa resulting from the dissociation of O2,a or O2a) is widely considered as the rate determining step [START_REF] Ge | Oxygen Reduction in Alkaline Media: From Mechanisms to Recent Advances of Catalysts[END_REF] and the reaction pathways (Figure 2a) can be described using the following electrochemical equations as already reported [START_REF] Anastasijević | Determination of the kinetic parameters of the oxygen reduction reaction using the rotating ring-disk electrode: Part I[END_REF][START_REF] Ge | Oxygen Reduction in Alkaline Media: From Mechanisms to Recent Advances of Catalysts[END_REF]:

𝑂 2,𝑎 + 𝑒 -𝑘 1 → 𝑂 2,𝑎 - (17)
𝑂 2,𝑎 -+ 𝐻 2 𝑂 + 𝑒 -𝑘 2 ,𝑘 -2 ↔ 𝐻𝑂 2,𝑎 -+ 𝐻𝑂 - (18)
𝑂 𝑠𝑎 + 𝑒 -𝑘 3 → 𝑂 𝑠𝑎 - ( 19
)
𝑂 2,𝑠𝑎 -+ 𝐻 2 𝑂 + 𝑒 -𝑘 4 → 𝐻𝑂 2,𝑠𝑎 -+ 𝐻𝑂 - (20)
𝐻𝑂 2,𝑠𝑎 -+ 𝑒 -𝑘 5 ,𝑘 -5 ↔ 𝑂 𝑠𝑎 -+ 𝐻𝑂 - (21)
𝑂 𝑠𝑎 -+ 𝐻 2 𝑂 + 𝑒 -𝑘 6 → 2𝐻𝑂 - (22)
All adsorbed species may be subjected to several chemical transformations, as described by the following equations:

2𝑂 2,𝑎 -+ 𝐻 2 𝑂 𝑘 10 → 𝑂 2,𝑎 + 𝐻𝑂 2,𝑎 -+ 𝐻𝑂 - (23)
2𝐻𝑂

Finally, equilibria exist between weakly adsorbed species (adsorption energy in the range of kT)

and strongly adsorbed species. The strength of adsorption greatly affects the rate of chemical and electrochemical steps. These equilibria can be described by the following equations:

𝑂 2,𝑎 - 𝑘 23 → 𝑂 2,𝑠𝑎 - (27)
𝑂 2,𝑎 𝑘 22 → 2𝑂 𝑠𝑎 (28)
As for platinum-based catalysts, ORR at metal free surfaces can occur through associative or dissociative mechanisms (Figure 2b) [START_REF] Li | Recent Progresses in Oxygen Reduction Reaction Electrocatalysts for Electrochemical Energy Applications[END_REF][START_REF] Ma | A review of oxygen reduction mechanisms for metal-free carbon-based electrocatalysts[END_REF]. It was nevertheless found that the energy barrier to form adsorbed peroxide from adsorbed O2 is lower than that corresponding to O2 dissociation, thus associative mechanism is energetically favored [START_REF] Yu | Oxygen reduction reaction mechanism on nitrogendoped graphene: A density functional theory study[END_REF]. Associative mechanism first involves the adsorption of molecular oxygen onto the catalyst surface and the overall reaction mechanism proceeds as follows (S represents the active site):

𝑂 2 + 𝑆 → 𝑆 -𝑂 2 (29)
𝑆 -𝑂 2 + 𝐻 2 𝑂 + 𝑒 -→ 𝑆 -𝑂𝑂𝐻 + 𝐻𝑂 -(30)

𝑆 -𝑂𝑂𝐻 + 𝑒 -→ 𝑆 -𝑂 + 𝐻𝑂 - (31)
𝑆 -𝑂 + 𝐻 2 𝑂 + 𝑒 -→ 𝑆 -𝑂𝐻 + 𝐻𝑂 -

𝑆 -𝑂𝐻 + 𝑒 -→ 𝐻𝑂 -+ 𝑆

Alternatively, depending on the relative stability of S-O and S-OOH intermediates, S-OOH may accept an electron before the releasing of peroxide anion into the electrolyte according to the following equation:

𝑆 -𝑂𝑂𝐻 + 𝑒 -→ 𝐻𝑂𝑂 -+ 𝑆 (34)
For the dissociative mechanism ORR begins with adsorption of O2 and its further dissociation into two S-O intermediates. This reaction pathway only leads to the formation of hydroxyl ions as reduction products. This reaction pathway can be described as follows:

𝑂 2 + 2𝑆 → 2𝑆 -𝑂 (35)
𝑆 -𝑂 + 𝐻 2 𝑂 + 𝑒 -→ 𝑆 -𝑂𝐻 + 𝐻𝑂 -

𝑆 -𝑂𝐻 + 𝑒 -→ 𝐻𝑂 -+ 𝑆

ORR at transition metal oxide surfaces has a quite different pathway. First, surface transition metal cations (M) of the oxide react with water molecules to form adsorbed OH -species. The charge modification occurring subsequently to protonation of surface oxygen ligand is counterbalanced by the reduction of the metallic cation (M). The overall reaction process can be described as follows:

𝑀 𝑚+ -𝑂 2-+ 𝐻 2 𝑂 + 𝑒 -→ 𝑀 (𝑚-1)+ -𝑂𝐻 -+ 𝐻𝑂 - (36)
𝑂 2 + 𝑒 -+ 𝑀 (𝑚-1)+ -𝑂𝐻 -→ 𝑀 𝑚+ -𝑂𝑂 2-+ 𝐻𝑂 -

𝑀 𝑚+ -𝑂𝑂 2-+ 𝐻 2 𝑂 + 𝑒 -→ 𝑀 (𝑚-1)+ -𝑂𝑂𝐻 -+ 𝐻𝑂 -

𝑀 (𝑚-1)+ -𝑂𝑂𝐻 -+ 𝑒 -→ 𝑀 𝑚+ -𝑂 2-+ 𝐻𝑂 -

In this reaction pathway, the displacement or the regeneration of surface hydroxide can be considered as the rate determining step [START_REF] Suntivich | Design principles for oxygen-reduction activity on perovskite oxide catalysts for fuel cells and metal-air batteries[END_REF]. The reduction process can also lead to the formation of the two-electron product that is to know peroxide anion according to the following equation:

𝑀 𝑚+ -𝑂𝑂 2-+ 𝐻 2 𝑂 + 𝑒 -→ 𝑀 (𝑚-1)+ -𝑂𝐻 -+ 𝐻𝑂 2 - ( 40
)
The uniqueness of double layer structure in alkaline medium allows modifying ORR kinetics independently on the chemical nature of the electrode. This is made possible in reason of H-bond formation between H atom of OH -species at the inner Helmholtz plane and water solvating O2 molecules located at the outer Helmholtz plane. This facilitates the outer electron transfer towards O2 molecule leading to the formation of superoxide anions [START_REF] He | Review-Recent Progress in Electrocatalysts for Oxygen Reduction Suitable for Alkaline Anion Exchange Membrane Fuel Cells[END_REF]. This latter phenomenon allows compensating the possible weak interaction between O2 and a given catalyst and consequently explains the high reduction potentials of O2 observed for a large number of catalysts in alkaline medium. The promotion of the outer sphere electron transfer may be also responsible for the release of large amounts of peroxide species. Nevertheless, at ORR potentials, the electrostatic interaction between negatively charged peroxide anion and positively charged electrode surface could allow the fast peroxide reduction to hydroxyl ion [START_REF] He | Review-Recent Progress in Electrocatalysts for Oxygen Reduction Suitable for Alkaline Anion Exchange Membrane Fuel Cells[END_REF].

Many research groups have proposed OER mechanisms in alkaline conditions. Some similarities and discrepancies appear in the reported studies. Most of the proposed mechanisms involve -OH as well as -O reaction intermediates and some differences are encountered in what concerns the reaction of formation of molecular O2. These different reaction mechanisms have been proposed in Figure 2d and can be described using the following equations, where S denotes a general catalytic site:

𝑆 + 𝐻𝑂 -→ 𝑆 -𝑂𝐻 + 𝑒 - (41)
𝑆 -𝑂𝐻 + 𝐻𝑂 -→ 𝑆 -𝑂 + 𝐻 2 𝑂 + 𝑒 -

2𝑆 -𝑂 → 2𝑆 + 𝑂 2 (43)

𝑆 -𝑂 + 𝐻𝑂 -→ 𝑆 -𝑂𝑂𝐻 + 𝑒 - (44)
𝑆 -𝑂𝑂𝐻 + 𝐻𝑂 -→ 𝑆 + 𝑂 2 + 𝐻 2 𝑂 + 𝑒 -

𝑆 -𝑂𝑂𝐻 + 𝐻𝑂 -→ 𝑆 -𝑂𝑂 + 𝐻 2 𝑂 + 𝑒 -

𝑆 -𝑂𝑂 → 𝑆 + 𝑂 2 [START_REF] Cai | Recent advances in layered double hydroxide electrocatalysts for the oxygen evolution reaction[END_REF] As already mentioned for ORR, HER and HOR, the structure and properties of the double layer strongly influence OER kinetics. In particular, specific interactions developed between cations of the electrolyte and reaction intermediates may affect the strength of hydrogen bonds with interfacial water. As a result, the energy required for the formation of oxygenated intermediates governing the rate determining step (i.e. the formation of O-O bond) is modified [START_REF] Yang | Chemical Recognition of Active Oxygen Species on the Surface of Oxygen Evolution Reaction Electrocatalysts[END_REF].

Catalysts for ORR/OER in alkaline medium

Numerous efficient non-noble transition metal-based hydroxides (especially Co-Fe, Ni-Fe and Ni-Co-based layered double hydroxides [START_REF] Cai | Recent advances in layered double hydroxide electrocatalysts for the oxygen evolution reaction[END_REF] and oxides (perovskites [START_REF] Wang | Nickel-Doped La0.8Sr0.2Mn1-xNixO3 Nanoparticles Containing Abundant Oxygen Vacancies as an Optimized Bifunctional Catalyst for Oxygen Cathode in Rechargeable Lithium-Air Batteries[END_REF],..) with performances overcoming those of Pt-based catalysts for ORR and those of Ir-or Ru-based oxides for OER have been reported during the past few years. The ABO3 (where A is a lanthanide or an alkaline metal and B is a transition metal) structure of perovskites is of large interest for ORR and OER since their lattice can accommodate a wide range of cations substitution both on A and B sites as well as oxygen deficiencies. Varying their composition can adjust their electronic properties and crystal structure and consequently being responsible for modifying their catalytic activity. In these structures B site is considered to be the active site for ORR/OER and their activity can be correlated with the eg orbital filling [START_REF] Xu | Recent Advances in Oxygen Electrocatalysts Based on Perovskite Oxides[END_REF]. As a result of these lattice modifications, effective materials have been designed for ORR and OER [START_REF] Wang | Nickel-Doped La0.8Sr0.2Mn1-xNixO3 Nanoparticles Containing Abundant Oxygen Vacancies as an Optimized Bifunctional Catalyst for Oxygen Cathode in Rechargeable Lithium-Air Batteries[END_REF] (Table 2). Layered double hydroxide is also a material family which has been largely investigated in the past few years, especially for the design of efficient catalyst towards OER. These materials are composed of hexa-coordinated transition metals.

Octahedrons form a two-dimensional layer by edge-sharing. The general chemical formula of layered double hydroxides is [𝑀 1-𝑥 2+ 𝑀 𝑥 3+ 𝑂𝐻 2 ] 𝑥+ (𝐴 𝑛-)𝑥 𝑛 . 𝑚𝐻 2 𝑂, where M 2+ cations can be substituted by M 3+ cations. The interlayer spacings are filled with anions to counterbalance the positive charge of two-dimensional layers. By exfoliating these two-dimensional materials, modifying their chemical composition of adjusting their morphology, very active OER catalysts have been designed. As an example, the high OER activity of NiFe layered double hydroxide hollow microspheres were recently reported [START_REF] Zhang | Hierarchical NiFe Layered Double Hydroxide Hollow Microspheres with Highly-Efficient Behavior toward Oxygen Evolution Reaction[END_REF]. An overpotential of only 239 mV was indeed required to reach a current density of 10 mA cm -2 (Table 2).

Additionally, since the amazing catalytic behavior of Co3O4/N-rGO composites towards OER/ORR was reported in 2011 [START_REF] Liang | Co3O4 nanocrystals on graphene as a synergistic catalyst for oxygen reduction reaction[END_REF], studies have highlighted the design of highly efficient heterostructured catalysts for ORR/OER mainly resulting from the coupling of heteroatom-doped carbon-based materials and 3d transition metal (Co, Ni, Fe, Mn) (hydr)oxides (perovskites [START_REF] Fabbri | Superior Bifunctional Electrocatalytic Activity of Ba0.5Sr0.5Co0.8Fe0.2O3-δ/Carbon Composite Electrodes: Insight into the Local Electronic Structure[END_REF], spinels [START_REF] Liang | Co3O4 nanocrystals on graphene as a synergistic catalyst for oxygen reduction reaction[END_REF], layered double hydroxides [START_REF] Cai | Recent advances in layered double hydroxide electrocatalysts for the oxygen evolution reaction[END_REF] as recently reviewed by Chen et al. [START_REF] Chen | Nanocarbon/oxide composite catalysts for bifunctional oxygen reduction and evolution in reversible alkaline fuel cells: A mini review[END_REF]. The activity and high stability of these heterostructures result from a higher dispersion degree of the oxide phase, synergistic effects as well as in some cases from the formation of a covalent heterointerface between phases notably favoring charge transfer during OER/ORR [START_REF] Liang | Covalent Hybrid of Spinel Manganese-Cobalt Oxide and Graphene as Advanced Oxygen Reduction Electrocatalysts[END_REF]. The presence of a highly active and stable oxide phase is responsible for mitigating carbon corrosion rate [START_REF] Chen | Nanocarbon/oxide composite catalysts for bifunctional oxygen reduction and evolution in reversible alkaline fuel cells: A mini review[END_REF] but it is not sure it is sufficiently stable to be used in real cells. To improve stability and possibly activity of such catalysts the use of oxygen deficient oxides like TiO2-x or WO3-x presenting conductivities in the range of several S cm -1 could be of major interest [START_REF] Ibrahim | A review of transition metal-based bifunctional oxygen electrocatalysts[END_REF]. In the same way, recently developed 2-D carbonitrides known as MXene could lead to very efficient heterostructures in reason of their very high electronic conductivity [START_REF] Benchakar | MXene Supported Cobalt Layered Double Hydroxide Nanocrystals: Facile Synthesis Route for a Synergistic Oxygen Evolution Reaction Electrocatalyst[END_REF].

Oxygen vacancies at oxide surfaces can be also responsible for modifying adsorption energies of intermediates especially by tuning density of states on metal cations [START_REF] Zhuang | Ultrathin Iron-Cobalt Oxide Nanosheets with Abundant Oxygen Vacancies for the Oxygen Evolution Reaction[END_REF]. Another way to tune electronic properties of active sites and the to enhance ORR and/or OER electroactivity of metal oxides consists in substituting oxygen for another or several other anions resulting in the formation of very active chalcogenides [START_REF] Dong | Two-step synthesis of binary Ni-Fe sulfides supported on nickel foam as highly efficient electrocatalysts for the oxygen evolution reaction[END_REF], nitrides [START_REF] Yuan | Zirconium nitride catalysts surpass platinum for oxygen reduction[END_REF], phosphides [START_REF] Huang | Unveiling the active sites of Ni-Fe phosphide/metaphosphate for efficient oxygen evolution under alkaline conditions[END_REF] and borides [START_REF] Nsanzimana | Ultrathin Amorphous Iron-Nickel Boride Nanosheets for Highly Efficient Electrocatalytic Oxygen Production[END_REF]. These materials can be obviously coupled to an electron conductive and/or active matrix to obtain heterostructured catalysts with improved efficiency [START_REF] Ibrahim | A review of transition metal-based bifunctional oxygen electrocatalysts[END_REF]. As mentioned in section 2.2, the growth of heterostructured catalysts composed of sulfides [START_REF] Zhang | Interface Engineering of MoS2/Ni3S2 Heterostructures for Highly Enhanced Electrochemical Overall-Water-Splitting Activity[END_REF] or phosphides [START_REF] Yu | Highperformance bifunctional porous non-noble metal phosphide catalyst for overall water splitting[END_REF] directly onto Ni foam lead to the design of very efficient catalyst for overall water splitting and thus towards OER. The established interfaces with Ni oxides at OER potentials allow modifying adsorption energies of oxygenated intermediates, thus improving OER kinetics [START_REF] Zhang | Interface Engineering of MoS2/Ni3S2 Heterostructures for Highly Enhanced Electrochemical Overall-Water-Splitting Activity[END_REF].

Another trend consists in developing carbon-based metal free materials in which defects and/or heteroatom are introduced to modulate electronic properties of carbon [START_REF] Liu | Carbon-based metal-free catalysts[END_REF]. As an example, in N doped carbon-based catalysts, electronic structure of carbon atoms adjacent to N atoms (which are believed to be active sites for ORR/OER) are subject to a strong electron density modification.

Pyridinic and graphitic nitrogen speciation are mainly responsible for the activity [START_REF] Guo | Active sites of nitrogen-doped carbon materials for oxygen reduction reaction clarified using model catalysts[END_REF]. Based on these results, C3N4-based materials containing high amounts of nitrogen were recently investigated [START_REF] Niu | Surface-Modified Porous Carbon Nitride Composites as Highly Efficient Electrocatalyst for Zn-Air Batteries[END_REF]. Other heteroatoms (S, P, B) [START_REF] Daems | Metal-free doped carbon materials as electrocatalysts for the oxygen reduction reaction[END_REF] are also efficient to improve catalytic properties of carbonaceous materials. Electronic properties of doped carbon are related to the number of electrons in the external shell of the heteroatom, its electronegativity, chemical hardness and size [START_REF] Daems | Metal-free doped carbon materials as electrocatalysts for the oxygen reduction reaction[END_REF].

Synthesizing multiple-doped materials is furthermore recognized as an efficient way to get an active catalyst. As an example, for nitrogen and phosphorous dual-doped carbon-based materials it was shown from DFT calculations that formation of P-N bonds and the presence of N dopants were respectively at the origin of the OER and ORR activity [START_REF] Chai | Active sites engineering leads to exceptional ORR and OER bifunctionality in P,N Co-doped graphene frameworks[END_REF]. These theoretical considerations led to the synthesis of a highly active P/N doped graphene framework exhibiting an ORR half wave potential of 0.85 V vs. RHE and a potential of 1.55 V vs. RHE to drive a current density of 10 mA cm -2 during OER. Another strategy consists in introducing structural defects in graphitic materials to tune carbon electronic structure. Especially, DFT calculations shown that a five-carbon ring adjacent to a seven-carbon ring leads to an optimal adsorption energy for oxygenated intermediates [START_REF] Tang | Topological Defects in Metal-Free Nanocarbon for Oxygen Electrocatalysis[END_REF]. Based on this work, numerous catalysts were designed for OER and ORR [START_REF] Hang | Situ Exfoliating and Generating Active Sites on Graphene Nanosheets Strongly Coupled with Carbon Fiber toward Self-Standing Bifunctional Cathode for Rechargeable Zn-Air Batteries[END_REF]. Nevertheless, stability of carbon-based catalysts at electrode potentials, required for the OER and under harsh conditions, remains questionable. OER/ORR activities of materials described in this section are reported in

Conclusions and perspectives

This short review provides a full understanding of HER, HOR, OER and ORR mechanisms in alkaline electrolyte and gives an overview of state-of-the-art catalysts designed for these reactions.

Even if very efficient materials have been reported during the past few years, the design of catalysts remains based on a trial and error approach. A general guideline to design efficient catalysts can be summarized in the following points: 1) engineering the porosity of the structure (microporosity, mesoporosity, macroporosity) to increase the number and density of exposed active sites as well as to improve diffusion of reactants and electrolyte 2) using a stable electron conductive substrate improving dispersion of catalysts and active sites activity by accelerating charge transfer and tuning electronic properties of active sites 3) engineering heterostructured catalysts with exposed heterointerfaces leading to the optimization of adsorption energies of reaction intermediates 4) modulating electronic properties of active sites by using a suitable heteroatom doping strategy or by inducing the formation of structural vacancies. Additionally, the better understanding of electrode/electrolyte interface in alkaline media and particularly the interaction between species (water, alkali cations,…) in the double layer and the inner-sphere HO -adsorption is required to improve our understanding of phenomena and then to extract better rational design principles. to carbon are responsible for the weakening of binding energies of hydrogenatedand oxygenated intermediates.

Table 1: Non-exhaustive list presenting representative recently developed catalysts for HER/HOR in alkaline medium. j0 represents the exchange current density. It has been normalized using either platinum active surface area (in that case the unit is denoted as mg cm -2 Pt) or using geometric surface area. The mass normalized j0 value is also given for HOR catalysts. η represents the overpotential. For HER, overpotential values at -10 mA cm -2 (η-10) as well as at -100 mA cm -2 (η-100) are given. b is the Tafel slope .

1 )

 1 Tafel step (30 mV dec -1 ): H2 + 2* = 2*-Had (Heyrovsky step (40 mV dec -1 ): H2 + HO -+* = *-Had + H2O + e - (2) Volmer step (120 mV dec -1 ): *-Had + HO -= H2O + e -+* (3)

Furthermore, recent progressFigure 2 :

 2 Figure 2: a) Proposed ORR mechanisms in alkaline medium at Pt-based surfaces. Reproduced from

  In 2011, Markovic et al. reported a heterostructured catalyst composed of Pt decorated with Ni(OH)2 clusters. This material exhibited an enhanced HER activity in comparison with the Pt platinum pristine surface. The presence of Ni(OH)2 favored water dissociation and then faster kinetics for the Volmer step [31]. Based on the same concept a Pt/Ni(HCO3)2 catalyst was very recently designed and exhibited a huge HER activity with an overpotential as low as 27 mV to achieve a current density of -10 mA cm -2 [22]. Recently, numerous nanostructured non-noble metal-based with heterojunctions were developed, some of them being more active than Pt/C catalysts in reason

Hydrogen oxidation reaction (HOR) Catalyst Morphology / structure Loading (mgmetal cm -2 ) Working electrode Electrolyte j0 (mA mg -1 metal) j0 (mA cm -2 metal) Activity at η = 50 mV Ref.

	Pt	Nanowires	0.02	Vulcan XC 72 R coated glassy carbon	0.1 M KOH	16 mA mg -1 Pt	0.229 mA cm -2 Pt	69 mA mg -1 Pt
	PtRu	Nanowires	0.02	Vulcan XC 72 R coated glassy carbon	0.1 M KOH	32 mA mg -1 Pt	0.493 mA cm -2 Pt	160 mA mg -1 Pt	[17]
	PtFe	Nanowires	0.02	Vulcan XC 72 R coated glassy carbon	0.1 M KOH	27 mA mg -1 Pt	0.459 mA cm -2 Pt	122 mA mg -1 Pt
	CoMoNi Pt	Electroplated material polycristalline disk	n/a n/a	Au Pt	0.1 M KOH 0.1 M KOH	n/a n/a	0.015 0.61	0.044 mA cm -2 metal 1.4 mA cm -2 metal	[19]
	Ni/N-CNT	Nanoparticles (20 nm)	0.25	Glassy carbon	0.1 M KOH	3.5	0.028	9.3 mA mg -1 metal	[20]
	Ni3N/C	Nanoparticles (4.6 nm)	0.16	Glassy carbon	0.1 M KOH	12	0.014	24.4 mA mg -1 metal	[21]

Hydrogen evolution reaction (HER) Catalyst Morphology / structure Loading (mg cm -2 ) Working electrode Electrolyte η -10 (mV) η -100 (mV) j0 (mA cm -2 ) b (mV dec -1 ) Ref.

	Mn-doped Ni2P NiO@1T-MoS2/CFP	Mn-doped Ni2P Nanosheets array on Ni Nanosheets	n/a 1.02	Ni foam Carbon fiber paper	1 M KOH 1 M KOH	ca. 50 -46	ca. -220 ca. -120	n/a 0.44	135 52
	Pt/C Pt/C	foam Commercial 20 wt. % Pt/C (Sigma-Aldrich) Commercial 20 wt. % Pt/C (Johnson Matthey)	7.0 1	Ni foam Carbon fiber paper	1 M KOH 1 M KOH	ca. 20 -23	ca. -120 ca. -130	n/a 0.69	98 32	[27] [35]
		Hybrid nanocages
	NiCoN/C	composed of nanosheets with a thickness of 2-3 nm	0.2	Glassy carbon	1 M KOH	-103	n/a	n/a	n/a	[28]
	Pt/C	Commercial Pt/C	0.05	Glassy carbon	1 M KOH	-91	n/a	n/a	n/a
	Co-Ni-B/carbon cloth	Co-Ni-B nanoparticles (5-10 nm)	0.34	Carbon cloth	1 M KOH	-80	ca. -250	n/a	88	[29]
	Pt/C	20 wt. % Pt nanoparticles Commercial Pt/C	0.34	Carbon cloth	1 M KOH	ca. -25	ca. -240	n/a	30
	Pt/Ni(HCO3)2 S-doped Pt C3N4/carbon naotubes/carbon	(2-3 nm) on Ni(HCO3) nanoplates Bare Pt nanoparticles Fibers decorated with S-doped C3N4	0.04 0.2 0.5	Glassy carbon Glassy carbon Carbon fiber	1 M KOH 1 M KOH 1 M KOH	-27 -44 -131	ca. -70 mV ca. -125 mV ca. -250	5.6 mA cm -2 Pt 2.8 mA cm -2 Pt 0.28	79	[22] [30]
	Pt/C fiber Mo2C/N-doped carbon microflowers Mo2C/Mo2N Pt/C Pt/C Fe1.89Mo4.11O7 MoS2/Ni3S2/nickel foam Pt/C	Commercial 10 wt.% Pt/C (Sigma-Aldrich) Mesoporous material N-doped microflowers consisting of nanoflakes 20 wt. % Pt/C (Alfa Aesar) decorated with β-Mo2C (particles) Commercial 20 wt. % Pt/C Microparticles Porous heterostructure grown onto Ni foam 20 wt. % Pt/C (Fuel Cell store)	0.02 0.38 0.28 0.38 0.28 2.4 9.7 0.8	Glassy carbon Glassy carbon Glassy carbon Glassy carbon Glassy carbon Bare carbon paste Ni foam Ni foam	1 M KOH 1 M KOH 1 M KOH 1 M KOH 1 M KOH 1 M KOH 1 M KOH 1 M KOH	ca. -50 -145 -100 -60 -50 -197 ca. -110 ca. -25	n/a -222 n/a -263 n/a n/a n/a n/a	n/a 0.05 n/a 5.87 mA cm -2 Pt n/a 0.069 n/a n/a	104 55 47 83 32	[32] [23] [33]
	Pt/C FeP/Ni2P/nickel foam	Porous heterostructures Commercial 20 wt. % Pt/C Hybrid porous nanowires composed of FeP and Ni2P nanoparticles grown onto Ni foam	n/a 8	n/a Ni foam	1 M KOH 1 M KOH	-103 -14	n/a ca. -130	n/a n/a	24	[24] [34]
	NiS2/MoS2 Pt	composed of MoS2 nanoplates and NiS2 Pt wire	0.2 n/a	Glassy carbon n/a	1 M KOH 1 M KOH	-204 -59	n/a ca. -130	n/a n/a	37
	N-NiCo2S4	particles with heterojunctions Nanowires	n/a	Ni foam	1 M KOH	-41	ca. -140	n/a	37	[25]
	Pt/C Pt/C FeNiSe/exfoliated graphene Ni, Zn-doped CoO Pt/C Pt/C	n/a Commercial 20 wt. % Pt/C Mesoporous FeNiSe nanosheets grown onto Nanorods exfoliated graphene Commercial catalyst Commercial 20 wt. % Pt/C (Fuel cell store)	0.2 2.3 3.3 0.486 3.3 0.4	Glassy carbon Ni foam Graphite foil Carbon fiber paper n/a Carbon fiber paper	1 M KOH 1 M KOH 1 M KOH 1 M KOH 1 M KOH 1 M KOH	-50 -28 -187 -53 ca.-75 ca. -60	n/a ca. -140 n/a ca. -160 n/a ca. -175	n/a n/a 0.57 n/a n/a n/a	29 47 33	[36] [26] [37]

Table 2 Table 2 :

 22 Representative recently developed catalysts for OER/ORR in alkaline medium.Performances of reported catalysts can be systematically compared with those of reference catalysts tested under the same operating conditions. E1/2 represents the half-wave potential measured from ORR polarization curves. n is the number of electrons exchanged per oxygen molecule. Ej10 and Ej50 corresponds to the electrode potential required to drive respective current densities of 10 and 50 mA cm -2 . b is the Tafel slope.

			Loading	Working		E1/2
	Catalyst	Morphology			Electrolyte		n	Ref.
			(mg cm -2 )	electrode		(V vs. RHE)
	La0.8Sr0.2Mn0.6Ni0.4O3	Nanoparticles (14 nm)	0.14	Glassy carbon	1 M KOH	0.66 (ca. -0.4 V/ SCE)	3.92
								[56]
	Pt/C	Commercial 20 wt.% Pt/C	0.028	Glassy carbon	1 M KOH	0.83 (ca. -0.24 V/ SCE)	4
	Ba0.5Sr0.5Co0.5Fe0.2O3-δ
		Microparticles	ca. 0.4	Glassy carbon	0.1 M KOH	ca. 0.65	n/a	[60]
	/acetylene black
	MnCo2O4/N-rGO	Particles (5 nm) deposited onto N-rGO	0.1	Glassy carbon	1 M KOH	0.87	3.90	[62]
	Pt/C	Commercial 20 wt.% Pt/C	0.1	Glassy carbon	1 M KOH	0.89	n/a
	ZrN	Nanoparticles (45 nm)	0.024	Glassy carbon	0.1 M KOH	0.8	3.9 -4.0
	Pt/C	Commercial 20 wt.% Pt/C (Johson Matthey)	0.024	Glassy carbon	0.1 M KOH	0.8	3.80	[67]
		Porous C3N4 matrix supported
	CoSx@P-C3N4/rGO	onto rGO and containing CoSx	0.4	Glassy carbon	0.1 M KOH	0.78	3.90
		nanoparticles						[72]
	Pt/C	Commercial 20 wt.% Pt/C	0.2	Glassy carbon	0.1 M KOH	0.81	3.90
	P,N-doped graphene	Graphene framework	0.1	Glassy carbon	0.1 M KOH	0.85	3.99
	Pt/C	Commercial 20 wt.% Pt/C (Alfa Aesar)	0.1	Glassy carbon	0.1 M KOH	0.82	n/a	[74]
	N,P-doped graphene/carbon fiber	P,N dual doped defective carbon fibers graphene interacting with	N/A	Glassy carbon	0.1 M KOH	0.79	3.8	[76]

as an Active Hydrogen Oxidation Reaction Catalyst in Alkaline Medium

  The stability of designed catalysts should be evaluated under conditions mimicking the working environment encountered in real cells. Finally, measurements of catalysts activity should be rationalized by following already reported guidelines in order to establish unbiased comparisons between different studies. ://dx.doi.org/10.1038/ncomms10141  [21] Ni W, Krammer A, Hsu C-S, Chen HM, Schüler A, Hu X: Ni3N . Angew. Chem. Int. Ed. 2019 58: 7445-7449. http://dx.doi.org/10.1002/anie.201902751  This article deals with the synthesis and characterization of Ni3N/C catalyst exhibiting the highest HOR activity ever reported for a non-noble metal-based material in alkaline medium. A downshift of the Ni d-band as well as and interfacial charge transfer from Ni3N

	catalyst	in	alkaline	electrolyte.	Nat.	Commun.	2016	7:	10141.

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Acknowledgements:

The authors are grateful to the European Union (ERDF) and "Region Nouvelle Aquitaine" for financial support.

Papers of outstanding interest, published within the past two years have been highlighted with .

Conflict of interest

Nothing declared