Martín I Idiart
email: mii23@cam.ac.uk.

Nonlinear sequential laminates reproducing hollow sphere assemblages

Keywords: Porous media, Homogenization, Nonlinear behavior Mots-clés : Milieux poreux, Homogénéisation, Comportement non linéaire

published or not. The documents may come

Nonlinear sequential laminates reproducing hollow sphere assemblages

Martín I.

Introduction

In this Note, a special class of nonlinear porous materials with isotropic 'sequentially laminated' microstructures is found to reproduce exactly the hydrostatic behavior of 'hollow sphere assemblages' (HSAs). The interest in this result 1 stems from the fact that, as will be argued below, it supports the conjecture that Gurson's approximate criterion [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media[END_REF] for plastic porous materials, and its viscoplastic extension [START_REF] Leblond | Exact results and approximate models for porous viscoplastic solids[END_REF], may actually yield rigorous upper bounds for the hydrostatic flow stress of porous materials containing an isotropic, but otherwise arbitrary, distribution of porosity [START_REF] Chaboche | Endommagement et changement d'échelle[END_REF][START_REF] Leblond | [END_REF][START_REF] Gȃrȃjeu | On the influence of local fluctuations in volume fraction of constituents on the effective properties of nonlinear composites. Application to porous materials[END_REF]. Given that Gurson's criterion is the most widely used estimate for plastic porous materials, and the fact that for sufficiently high triaxialities it can be much more restrictive than the best upper bounds for porous materials known to date [START_REF] Castañeda | The effective mechanical properties of nonlinear isotropic composites[END_REF][START_REF] Willis | On methods for bounding the overall properties of nonlinear composites[END_REF][START_REF] Suquet | On bounds for the overall potential of power law materials containing voids with an arbitrary shape[END_REF]-especially at small porosities-, the validity of this conjecture would have important implications, both, practical and theoretical.

The subsequent analysis is set in the theoretical framework of nonlinear composites. Composites refer to heterogeneous materials made of N different homogeneous constituents, or phases, which are randomly distributed in a volume Ω, at a length scale that is much smaller than the size of Ω and the scale of variation of the loading conditions. The constitutive behavior of each phase is taken to be characterized by convex strain potentials w (r) (r = 1,...,N) such that

σ = ∂ ε w (r) (ε) (1)
where ∂ ε denotes differentiation with respect to ε. Within the context of the deformation theory of plasticity, ε and σ represent the infinitesimal strain and stress, while in the context of viscoplasticity, they represent the Eulerian strain rate and Cauchy stress, respectively. Then, the effective behavior of the composite, which is defined as the relation between the volume averages of the stress σ = σ and the strain ε = ε , can be characterized by an effective strain potential W , such that (e.g., [START_REF] Castañeda | Nonlinear composites[END_REF])

σ = ∂ ε W(ε), W(ε) = min ε∈K(ε) N r=1 c (r) w (r) (ε) (r) (2)
Here, • and • (r) denote the volume averages over the composite (Ω) and over phase r (Ω (r) ), respectively, c (r) is the volume fraction of phase r, and K is the set of kinematically admissible strain fields with a prescribed volume average ε. In general, the function W is extremely difficult to compute, since it amounts to solving a set of nonlinear partial differential equations with randomly oscillating coefficients. However, for the special class of sequentially laminated composites considered in Section 2, expression (2) simplifies considerably, becoming amenable of analytical treatment.

Two-phase composites with sequentially laminated microstructures

A sequential laminate is an iterative construction obtained by layering laminated materials (which in turn have been obtained from lower-order lamination procedures) with other laminated materials, or directly with the homogeneous phases that make up the composite, in such a way as to produce hierarchical microstructures of increasing complexity (e.g., [START_REF] Milton | The Theory of Composites[END_REF]). The rank of the laminate refers to the number of layering operations required to reach the final sequential laminate. Of the many possible types of sequential laminates, we restrict attention to two-phase, sequential laminates formed by layering at every step a laminate with one of the original phases, identified with r = 1. Thus, a rank-1 laminate corresponds to a simple laminate with a given layering direction n (1) , with phases 1 and 2 in proportions 1f 1 and f 1 (see Fig. 1(a)). In turn, a rank-2 laminate is constructed by layering the rank-1 laminate with phase 1, in a different layering direction n (2) , in proportions f 2 and 1f 2 , respectively (see Fig. 1(b)). Rank-M laminates are obtained by iterating this procedure M times, layering the rank-(M -1) laminate with phase 1 in direction n (M) , in proportions f M and 1f M , respectively. In this procedure, it is assumed that the length scale of the embedded laminate is much smaller than the length scale of the embedding laminate (i.e., δ 1 ≪ δ 2 in Fig. 1(b)). This assumption allows one to regard the rank-(M -1) laminate in the rank-M laminate as a homogeneous phase, so that available expressions for the effective potential of simple laminates (e.g., [START_REF] Debotton | On the ductility of laminated materials[END_REF]) can be used at each step of the process to obtain an exact expression for the effective potential of the rank-M sequential laminate (e.g., [START_REF] Castañeda | Bounds and estimates for the properties of nonlinear heterogeneous systems[END_REF][START_REF] Debotton | Transversely isotropic sequentially laminated composites in finite elasticity[END_REF]).

It is important to note that the effective behavior of the sequential laminates considered here, unlike that of typical nonlinear composites, does not depend on all the details of the microstructure, but only on partial information of it in the form of the volume fractions f i and lamination directions n (i) . Furthermore, from their construction process described above, it follows that the microstructure of these sequential laminates can be regarded as random and particulate, with phase 2 playing the role of the inclusion (discontinuous) phase embedded in a (continuous) matrix made up of phase 1 (see Fig. 1(b)). A distinctive feature of this very special class of 'particulate' composites is that the local fields are uniform in the inclusion phase, but not in the matrix phase.

In general, sequentially laminated microstructures are anisotropic. However, deBotton [START_REF] Debotton | Transversely isotropic sequentially laminated composites in finite elasticity[END_REF] has recently proposed a lamination sequence, namely a set of f i and n (i) (i = 1,...,M), such that isotropic microstructures result in the limit of infinite rank (M →∞). In this sequence, the M unit vectors n (i) are chosen so that their endpoints are uniformly distributed over the unit sphere S, and the corresponding f i arealltakentobef i = 1c (1) /M, where the volume fraction of the matrix phase c (1) in the rank-M laminate is assumed infinitesimally small (i.e., c (1) ≪ 1). Thus, expanding the effective potential of such rank-M laminate (e.g., [START_REF] Debotton | Transversely isotropic sequentially laminated composites in finite elasticity[END_REF]) to first order in c (1) , and then letting M →∞in the resulting expression, the following result is obtained for the effective potential of the infinite-rank laminate [START_REF] Idiart | Macroscopic behavior and field statistics in viscoplastic composites[END_REF]:

W(ε) = min a(n) 1 -c (1) w (2) (ε) + c (1) 1 |S| S w (1) ε + a(n) ⊗ s n -n • ∂ ε w (2) (ε)a(n) dS(n) (3)
Here, |S| and dS(n) denote, respectively, the total and differential surface area over the unit sphere S, a(n) is the (three-dimensional) amplitude vector associated with the strain jump between laminae, and the symbol ⊗ s denotes the symmetric part of the outer product.

It is emphasized that expression (3) constitutes an exact result for nonlinear composites with the class of random and isotropic, 'particulate' microstructures described above. Even though this result, being valid only for composites with a dilute matrix concentration, may not seem very useful at first, in fact, it can be used in an iterative process to obtain exact results for composites with non-dilute matrix concentrations, as will be seen in the next section.

Finally, it is useful to note that the optimization problem in (3) simplifies when the matrix phase is incompressible. Indeed, the argument of w (1) in (3) must be traceless in this case, and therefore, the minimization must be carried out over functions a(n) of the form

a(n) =-tr(ε)n + a 1 (n)m 1 + a 2 (n)m 2 (4)
where m 1 and m 2 are two mutually orthogonal, unit vectors lying on the plane with normal n. Thus, incompressibility reduces by one the dimension of the optimization problem in (3).

Porous, power-law materials under hydrostatic loadings

In this section, the result provided above is used to generate an exact expression for the effective potential of a special class of sequential laminates made of a power-law matrix containing an arbitrary concentration of pores. Thus, the inclusion phase is taken to be vacuous, so that w (2) = 0, and the matrix phase is taken to be characterized by an isotropic, incompressible potential of the form w (1)

(ε) = σ 0 ε 0 1 + m ε e ε 0 1+m ( 5
)
where m is the strain-rate sensitivity, such that 0 m 1, σ 0 is a flow stress, ε 0 is a reference strain, and the von Mises equivalent strain is given in terms of the deviatoric part of the strain tensor by

ε e = √ (2/3)ε d • ε d .
Following deBotton [START_REF] Debotton | Transversely isotropic sequentially laminated composites in finite elasticity[END_REF], porous laminates with non-dilute matrix concentrations are constructed by means of an iterative process based on a differential scheme. The idea is to take an infinite-rank porous laminate with a dilute matrix concentration, as the inclusion phase of an infinite-rank laminate with a dilute matrix concentration, which in turn is taken as the inclusion phase of an infinite-rank laminate with a dilute matrix concentration, and so on. Thus, each time, an infinitesimal amount of matrix material is added to the porous material, in such a way that, after repeating the process an infinite number of times, a non-dilute concentration of the matrix material is reached. This process allows repeated use of the exact result (3), so that, at the end, an exact expression for the effective potential of the non-dilute porous laminate is obtained in the form of a differential equation. It is noted that this process is contrary to those followed in common differential schemes, where the starting point is a composite with a dilute concentration of the inclusion phase (see Ref. [START_REF] Milton | The Theory of Composites[END_REF] and references therein).

Thus, the first step in this process is to consider an infinite-rank porous laminate with a dilute concentration c [1] of the power-law matrix [START_REF] Gȃrȃjeu | On the influence of local fluctuations in volume fraction of constituents on the effective properties of nonlinear composites. Application to porous materials[END_REF], subjected to hydrostatic loadings. A superscript [k] is used here to denote quantities corresponding to the kth iteration. According to ( 3) and ( 4), the corresponding effective potential is given by W [1] (ε) = min a [1] 1 (n),a [1] 2 (n)

c [1] 1

|S| S σ 0 ε 0 1 + m (2ε m ) 2 + (a [1] 1 (n)) 2 + (a [1] 2 (n)) 2 3 (1+m)/2 dS(n) ( 6
)
where εm = (1/3) tr(ε) is the macroscopic mean strain. Clearly, the minimum in this expression is attained at a [1] 1 (n) = a [1] 2 (n) = 0, the integrand in ( 6) becomes independent of n, and so the effective potential W [1] can be written explicitly as

W [1] (ε) = σ [1] h ε 0 1 + m 2 εm ε 0 1+m , σ [1] h σ 0 = c [1] (7)
where σ [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media[END_REF] h denotes the hydrostatic effective flow stress of the porous laminate in the first iteration. Note that, because of the isotropy of W [1] , the derivative ∂ ε W [1] (ε) evaluated at hydrostatic loadings must be hydrostatic, the resulting mean stress being σm = (2/3) σ [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media[END_REF] h |ε m /ε 0 | m sgn(ε m ). Next, let the potential [START_REF] Willis | On methods for bounding the overall properties of nonlinear composites[END_REF] be that of the inclusion phase in an infinite-rank laminate with a dilute concentration c [2] of a power-law matrix phase. It then follows from the specialized version of (3), together with (4), that the effective potential W [2] of this laminate subjected to hydrostatic loadings is of the same form (7) 1 , but with an effective flow stress σ [START_REF] Leblond | Exact results and approximate models for porous viscoplastic solids[END_REF] h given by

σ [2] h σ 0 = σ [1] h σ 0 + c [2] 1 + m σ [1] h σ 0 (8)
where σ [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media[END_REF] h isgi venby(7) 2 . In deriving this expression, use has been made of the facts that the derivative ∂ ε W [1] (ε) evaluated at hydrostatic loadings is hydrostatic, and the minimum in (3) is, once again, attained at a [2] 1 (n) = a [2] 2 (n) = 0. Note that σ [START_REF] Leblond | Exact results and approximate models for porous viscoplastic solids[END_REF] h corresponds to an infinite-rank, porous laminate with a total porosity (1c [2] )(1c [1] ). By following exactly similar arguments, it can be shown that the effective potential at the k + 1 iteration is of the same form [START_REF] Willis | On methods for bounding the overall properties of nonlinear composites[END_REF] 1 , but with an effective flow stress given by

σ [k+1] h σ 0 = σ [k] h σ 0 + c [k+1] 1 + m σ [k] h σ 0 ( 9
)
where σ [k] h is the effective flow stress of the porous laminate in the previous iteration, with σ [0] h = 0, and c [k+1] denotes the (infinitesimal) concentration of matrix material added to the porous laminate at the k + 1 iteration. The total concentration c (1) k+1 of matrix material in the porous laminate at the k + 1 iteration is given by

c (1) k+1 = 1 - k+1 j =1 1 -c [j ] (10)
and the increment of total matrix concentration at each iteration is thus given by c

(1)

k+1 -c (1) k = k j =1 1 -c [j ] - k+1 j =1 1 -c [j ] = c [k+1] k j =1 1 -c [j ] = c [k+1] 1 -c (1) k (11)
Then, from relations ( 9) and [START_REF] Debotton | On the ductility of laminated materials[END_REF], it follows that

1 + m σ [k] h σ 0 -1 ( σ [k+1] h /σ (1) 0 ) -( σ [k] h /σ (1) 0 ) c (1) k+1 -c (1) k = 1 1 -c (1) k (12)
Since the c [k] are infinitesimally small, relation [START_REF] Castañeda | Bounds and estimates for the properties of nonlinear heterogeneous systems[END_REF] leads to the following differential equation for the effective flow stress σh of a porous laminate with a total, non-dilute matrix concentration c (1) :

1 + m σh σ 0 -1 d( σh /σ 0 ) dc (1) = 1 1 -c (1) , σh σ 0 c (1) =0 = 0 (13)
Finally, integrating this equation gives

σh σ 0 = 1 m f -m -1 (14)
where f = 1c (1) denotes the total porosity in the final infinite-rank, porous laminate. In the limiting case of an ideally plastic matrix (m → 0), this expression becomes

σh σ 0 =-ln f (15)
Note that the corresponding hydrostatic yield stress is |σ m |=(2/3) σh =-(2/3)σ 0 ln f . Expressions ( 14) and ( 15) agree exactly with those corresponding to HSAs (e.g., [START_REF] Leblond | Exact results and approximate models for porous viscoplastic solids[END_REF]). Finally, it is worth noting that the two-dimensional analogue of this result, i.e., transversely isotropic laminates reproducing the in-plane hydrostatic behavior of 'hollow cylinder assemblages', follows from exactly similar arguments.

Discussion

Thus, isotropic porous materials with HSAs and sequentially laminated microstructures exhibit identical hydrostatic behaviors. In turn, this behavior is reproduced exactly by the hydrostatic limit of Gurson's approximate criterion [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media[END_REF] and its viscoplastic extension [START_REF] Leblond | Exact results and approximate models for porous viscoplastic solids[END_REF]. Furthermore, since expression [START_REF] Idiart | Macroscopic behavior and field statistics in viscoplastic composites[END_REF] with m = 1 reduces to the linear Hashin-Shtrikman upper bound for the class of porous materials exhibiting overall isotropy (e.g., [START_REF] Milton | The Theory of Composites[END_REF]), it follows that HSAs and sequentially laminated microstructures are 'optimal' within that class. Even though this 'optimality' is only valid in the linear case, the fact that porous materials with these two very different types of microstructures exhibit identical hydrostatic behaviors for any degree of matrix nonlinearity suggests that these microstructures may still be 'optimal' in the nonlinear domain. In turn, this supports the conjecture that the hydrostatic limit of Gurson's criterion and its viscoplastic extension may constitute rigorous upper bounds for the hydrostatic flow stress of porous materials containing an isotropic, but otherwise arbitrary, distribution of porosity [START_REF] Chaboche | Endommagement et changement d'échelle[END_REF][START_REF] Leblond | [END_REF][START_REF] Gȃrȃjeu | On the influence of local fluctuations in volume fraction of constituents on the effective properties of nonlinear composites. Application to porous materials[END_REF]. In this connection, it is noted that recent numerical simulations [START_REF] Bilger | Effect of a nonuniform distribution of voids on the plastic response of voided materials: a computational and statistical analysis[END_REF] of porous, ideally plastic materials with random distributions of porosity different than HSAs, indeed exhibit a weaker hydrostatic stress than that given by Gurson's criterion.

A distinctive feature of porous materials with the above-mentioned HSAs and sequentially laminated microstructures is that the fields in the 'inclusion' phase (i.e., the pores) are uniform, what may hint at proving Gurson's bounding character. A proof of this conjecture, however, remains elusive.

Fig. 1 .

 1 Fig. 1. Two-phase laminates: (a) simple or rank 1 laminate, (b) rank 2 laminate (δ 2 ≫ δ 1 ).

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grants CMS-02-01454 and OISE-02-31867. Insightful discussions with Professor P. Ponte Castañeda and K. Danas are gratefully acknowledged.


Hedy Attouch

Zaki Chbani

Jalal Fadili

Hassan Riahi

FIRST-ORDER OPTIMIZATION ALGORITHMS VIA INERTIAL SYSTEMS WITH HESSIAN DRIVEN DAMPING

Keywords: Hessian driven damping, inertial optimization algorithms, Nesterov accelerated gradient method, Ravine method, time rescaling AMS subject classification. 37N40, 46N10, 49M30, 65B99, 65K05, 65K10, 90B50, 90C25

In a Hilbert space setting, for convex optimization, we analyze the convergence rate of a class of first-order algorithms involving inertial features. They can be interpreted as discrete time versions of inertial dynamics involving both viscous and Hessian-driven dampings. The geometrical damping driven by the Hessian intervenes in the dynamics in the form ∇ 2 f (x(t)) ẋ(t). By treating this term as the time derivative of ∇f (x(t)), this gives, in discretized form, first-order algorithms in time and space. In addition to the convergence properties attached to Nesterov-type accelerated gradient methods, the algorithms thus obtained are new and show a rapid convergence towards zero of the gradients. On the basis of a regularization technique using the Moreau envelope, we extend these methods to non-smooth convex functions with extended real values. The introduction of time scale factors makes it possible to further accelerate these algorithms. We also report numerical results on structured problems to support our theoretical findings.

Introduction

Unless specified, throughout the paper we make the following assumptions

    
H is a real Hilbert space; f : H → R is a convex function of class C 2 , S = argmin f = ∅; γ, β, b : [t 0 , +∞[→ R + are non-negative continuous functions, t 0 > 0.

As a guide in our study, we will rely on the asymptotic behavior, when t → +∞, of the trajectories of the inertial system with Hessian-driven damping ẍ(t) + γ(t) ẋ(t) + β(t)∇ 2 f (x(t)) ẋ(t) + b(t)∇f (x(t)) = 0. γ(t) and β(t) are damping parameters, and b(t) is a time scale parameter.

The time discretization of this system will provide a rich family of first-order methods for minimizing f . At first glance, the presence of the Hessian may seem to entail numerical difficulties. However, this is not the case as the Hessian intervenes in the above ODE in the form ∇ 2 f (x(t)) ẋ(t), which is nothing but the derivative wrt time ∇f (x(t)). This explains why the time discretization of this dynamic provides first-order algorithms. Thus, the Nesterov extrapolation scheme [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF][START_REF] Nesterov | Gradient methods for minimizing composite objective function[END_REF] is modified by the introduction of the difference of the gradients at consecutive iterates. This gives algorithms of the form

y k = x k + α k (x k -x k-1 ) -β k (∇f (x k ) -∇f (x k-1 )) x k+1 = T (y k ),
where T , to be specified later, is an operator involving the gradient or the proximal operator of f .

Coming back to the continuous dynamic, we will pay particular attention to the following two cases, specifically adapted to the properties of f :

• For a general convex function f , taking γ(t) = α t , gives (DIN-AVD) α,β,b ẍ(t) + α t ẋ(t) + β(t)∇ 2 f (x(t)) ẋ(t) + b(t)∇f (x(t)) = 0.

In the case β ≡ 0, α = 3, b(t) ≡ 1, it can be interpreted as a continuous version of the Nesterov accelerated gradient method [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]. According to this, in this case, we will obtain O t -2 convergence rates for the objective values. • For a µ-strongly convex function f , we will rely on the autonomous inertial system with Hessian driven damping (DIN) 2 √ µ,β ẍ(t) + 2 √ µ ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) = 0, and show exponential (linear) convergence rate for both objective values and gradients. For an appropriate setting of the parameters, the time discretization of these dynamics provides first-order algorithms with fast convergence properties. Notably, we will show a rapid convergence towards zero of the gradients.

1.1. A historical perspective. B. Polyak initiated the use of inertial dynamics to accelerate the gradient method in optimization. In [START_REF] Polyak | Some methods of speeding up the convergence of iteration methods[END_REF][START_REF] Polyak | Introduction to optimization[END_REF], based on the inertial system with a fixed viscous damping coefficient γ > 0 (HBF) ẍ(t) + γ ẋ(t) + ∇f (x(t)) = 0, he introduced the Heavy Ball with Friction method. For a strongly convex function f , (HBF) provides convergence at exponential rate of f (x(t)) to min H f . For general convex functions, the asymptotic convergence rate of (HBF) is O( 1 t ) (in the worst case). This is however not better than the steepest descent. A decisive step to improve (HBF) was taken by Alvarez-Attouch-Bolte-Redont [START_REF] Álvarez | A second-order gradient-like dissipative dynamical system with Hessiandriven damping. Application to optimization and mechanics[END_REF] by introducing the Hessian-driven damping term β∇ 2 f (x(t)) ẋ(t), that is (DIN) 0,β . The next important step was accomplished by Su-Boyd-Candès [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF] with the introduction of a vanishing viscous damping coefficient γ(t) = α t , that is (AVD) α (see Section 1.1.2). The system (DIN-AVD) α,β,1 (see Section 2) has emerged as a combination of (DIN) 0,β and (AVD) α . Let us review some basic facts concerning these systems.

1.1.1. The (DIN) γ,β dynamic. The inertial system (DIN) γ,β ẍ(t) + γ ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) = 0, was introduced in [START_REF] Álvarez | A second-order gradient-like dissipative dynamical system with Hessiandriven damping. Application to optimization and mechanics[END_REF]. In line with (HBF), it contains a fixed positive friction coefficient γ. The introduction of the Hessian-driven damping makes it possible to neutralize the transversal oscillations likely to occur with (HBF), as observed in [START_REF] Álvarez | A second-order gradient-like dissipative dynamical system with Hessiandriven damping. Application to optimization and mechanics[END_REF] in the case of the Rosenbrook function. The need to take a geometric damping adapted to f had already been observed by Alvarez [START_REF] Álvarez | On the minimizing property of a second-order dissipative system in Hilbert spaces[END_REF] who considered ẍ(t) + Γ ẋ(t) + ∇f (x(t)) = 0, where Γ : H → H is a linear positive anisotropic operator. But still this damping operator is fixed. For a general convex function, the Hessian-driven damping in (DIN) γ,β performs a similar operation in a closedloop adaptive way. The terminology (DIN) stands shortly for Dynamical Inertial Newton. It refers to the natural link between this dynamic and the continuous Newton method.

1.1.2. The (AVD) α dynamic. The inertial system

(AVD) α ẍ(t) + α t ẋ(t) + ∇f (x(t)) = 0,
was introduced in the context of convex optimization in [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]. For general convex functions it provides a continuous version of the accelerated gradient method of Nesterov. For α ≥ 3, each trajectory x(•) of (AVD) α satisfies the asymptotic rate of convergence of the values f (x(t)) -inf H f = O 1/t 2 . As a specific feature, the viscous damping coefficient α t vanishes (tends to zero) as time t goes to infinity, hence the terminology. The convergence properties of the dynamic (AVD) α have been the subject of many recent studies, see [START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF][START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF][START_REF] Attouch | Convergence rates of inertial forward-backward algorithms[END_REF][START_REF] Attouch | Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient[END_REF][START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF][START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF][START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF][START_REF] Aujol | Stability of over-relaxations for the Forward-Backward algorithm, application to FISTA[END_REF][START_REF] Aujol | Optimal rate of convergence of an ODE associated to the Fast Gradient Descent schemes for b > 0[END_REF][START_REF] May | Asymptotic for a second-order evolution equation with convex potential and vanishing damping term[END_REF][START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]. They helped to explain why α t is a wise choise of the damping coefficient.

In [START_REF] Cabot | On the long time behavior of second order differential equations with asymptotically small dissipation[END_REF], the authors showed that a vanishing damping coefficient γ(•) dissipates the energy, and hence makes the dynamic interesting for optimization, as long as +∞ t0 γ(t)dt = +∞. The damping coefficient can go to zero asymptotically but not too fast. The smallest which is admissible is of order 1 t . It enforces the inertial effect with respect to the friction effect.

The tuning of the parameter α in front of 1 t comes from the Lyapunov analysis and the optimality of the convergence rates obtained. The case α = 3, which corresponds to Nesterov's historical algorithm, is critical. In the case α = 3, the question of the convergence of the trajectories remains an open problem (except in one dimension where convergence holds [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF]). As a remarkable property, for α > 3, it has been shown by Attouch-Chbani-Peypouquet-Redont [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF] that each trajectory converges weakly to a minimizer. The corresponding algorithmic result has been obtained by Chambolle-Dossal [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage Thresholding Algorithm[END_REF]. For α > 3, it is shown in [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF] and [START_REF] May | Asymptotic for a second-order evolution equation with convex potential and vanishing damping term[END_REF] that the asymptotic convergence rate of the values is actually o(1/t 2 ). The subcritical case α ≤ 3 has been examined by Apidopoulos-Aujol-Dossal [START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF] and Attouch-Chbani-Riahi [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF], with the convergence rate of the objective values O t -2α 3 . These rates are optimal, that is, they can be reached, or approached arbitrarily close:

• α ≥ 3: the optimal rate O t -2 is achieved by taking f (x) = x r with r → +∞ (f become very flat around its minimum), see [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF].

• α < 3: the optimal rate O t -2α 3 is achieved by taking f (x) = x , see [START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF].

The inertial system with a general damping coefficient γ(•) was recently studied by Attouch-Cabot in [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF][START_REF] Attouch | Convergence rates of inertial forward-backward algorithms[END_REF], and Attouch-Cabot-Chbani-Riahi in [START_REF] Attouch | Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient[END_REF].

1.1.3. The (DIN-AVD) α,β dynamic. The inertial system (DIN-AVD) α,β ẍ(t) + α t ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) = 0,
was introduced in [START_REF] Attouch | A dynamical approach to an inertial forward-backward algorithm for convex minimization[END_REF]. It combines the two types of damping considered above. Its formulation looks at a first glance more complicated than (AVD) α . In [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF], Attouch-Peypouquet-Redont showed that (DIN-AVD) α,β is equivalent to the first-order system in time and space

   ẋ(t) + β∇f (x(t)) -1 β -α t x(t) + 1 β y(t) = 0; ẏ(t) -1 β -α t + αβ t 2 x(t) + 1 β y(t) = 0.
This provides a natural extension to f : H → R ∪ {+∞} proper lower semicontinuous and convex, just replacing the gradient by the subdifferential.

To get better insight, let us compare the two dynamics (AVD) α and (DIN-AVD) α,β on a simple quadratic minimization problem, in which case the trajectories can be computed in closed form as explained in Appendix A. [START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF].

Take H = R 2 and f (x 1 , x 2 ) = 1 2 (x 2 1 + 1000x 2 2
), which is ill-conditioned. We take parameters α = 3.1, β = 1, so as to obey the condition α > 3. Starting with initial conditions: (x 1 (1), x 2 (1)) = (1, 1), ( ẋ1 (1), ẋ2 (1)) = (0, 0), we have the trajectories displayed in Figure 1. This illustrates the typical situation of an ill-conditioned minimization problem, where the wild oscillations of (AVD) α are neutralized by the Hessian damping in (DIN-AVD) α,β (see Appendix A.3 for further details).

1.2. Main algorithmic results. Let us describe our main convergence rates for the gradient type algorithms. Corresponding results for the proximal algorithms are also obtained. General convex function. Let f : H → R be a convex function whose gradient is L-Lipschitz continuous. Based on the discretization of (DIN-AVD) α,β,1+ β t , we consider Strongly convex function. When f : H → R is µ-strongly convex for some µ > 0, our analysis relies on the autonomous dynamic (DIN) γ,β with γ = 2 √ µ. Based on its time discretization, we obtain linear convergence results for the values (hence the trajectory) and the gradients terms. Explicit discretization gives the inertial gradient algorithm

y k = x k + 1 -α k (x k -x k-1 ) -β √ s (∇f (x k ) -∇f (x k-1 )) -β √ s k ∇f (x k-1 ) x k+1 = y k -s∇f (y k ). Suppose that α ≥ 3, 0 < β < 2 √ s, sL ≤ 1. In Theorem 3.3, we show that i) f (x k ) -min H f = O 1 k 2 as k → +∞; ii) k k 2 ∇f (y k ) 2 < +∞ and k k 2 ∇f (x k ) 2 < +∞.
x k+1 = x k + 1 - √ µs 1 + √ µs (x k -x k-1 ) - β √ s 1 + √ µs (∇f (x k ) -∇f (x k-1 )) - s 1 + √ µs ∇f (x k ).
Assuming that ∇f is L-lipschitz continuous, L sufficiently small and β ≤ 1 √ µ , it is shown in Theorem 5.4

that, with q = 1 1 + 1 2 √ µs ( 0 < q < 1)

f (x k ) -min H f = O q k and x k -x = O q k/2 as k → +∞,
Moreover, the gradients converge exponentially fast to zero.

1.3. Contents. The paper is organized as follows. Sections 2 and 3 deal with the case of general convex functions, respectively in the continuous case and the algorithmic cases. We improve the Nesterov convergence rates by showing in addition fast convergence of the gradients. Sections 4 and 5 deal with the same questions in the case of strongly convex functions, in which case, linear convergence results are obtained. Section 6 is devoted to numerical illustrations. We conclude with some perspectives.

Inertial dynamics for general convex functions

Our analysis deals with the inertial system with Hessian-driven damping

(DIN-AVD) α,β,b ẍ(t) + α t ẋ(t) + β(t)∇ 2 f (x(t)) ẋ(t) + b(t)∇f (x(t)) = 0.
2.1. Convergence rates. By specializing the functions β and b, the convergence rates obtained in the following theorem make it possible to find most of the related results existing in the literature. The following quantities play a central role in our analysis:

(

(G 2 ) b(t) > β(t) + β(t) t ; (G 3 ) t ẇ(t) ≤ (α -3)w(t).
Then, w(t) is positive and

i) f (x(t)) -min H f = O 1 t 2 w(t) as t → +∞; ii) +∞ t0 t 2 β(t)w(t) ∇f (x(t)) 2 dt < +∞; iii) +∞ t0 t (α -3)w(t) -t ẇ(t) (f (x(t)) -min H f )dt < +∞. Proof. Given x ∈ argmin H f , define for t ≥ t 0 (2) E(t) := δ(t)(f (x(t)) -f (x )) + 1 2 v(t) 2 ,
where v(t) := (α -1)(x(t) -x ) + t ( ẋ(t) + β(t)∇f (x(t)) .

The function E(•) will serve as Lyapunov's function. Differentiating E gives

(3)

d dt E(t) = δ(t)(f (x(t)) -f (x )) + δ(t) ∇f (x(t)), ẋ(t) + v(t), v(t) .
Using equation (DIN-AVD) α,β,b , we have

v(t) = α ẋ(t) + β(t)∇f (x(t)) + t ẍ(t) + β(t)∇f (x(t)) + β(t)∇ 2 f (x(t)) ẋ(t) = α ẋ(t) + β(t)∇f (x(t)) + t -α t ẋ(t) + ( β(t) -b(t))∇f (x(t)) = t β(t) + β(t) t -b(t) ∇f (x(t)).
Hence,

v(t), v(t) = (α -1)t β(t) + β(t) t -b(t) ∇f (x(t)), x(t) -x +t 2 β(t) + β(t) t -b(t) ∇f (x(t)), ẋ(t) +t 2 β(t) β(t) + β(t) t -b(t) ∇f (x(t)) 2 .
Let us go back to [START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF]. According to the choice of δ(t), the terms ∇f (x(t)), ẋ(t) cancel, which gives

d dt E(t) = δ(t)(f (x(t)) -f (x )) + (α-1) t δ(t) ∇f (x(t)), x -x(t) -β(t)δ(t) ∇f (x(t)) 2 .
Condition (G 2 ) gives δ(t) > 0. Combining this equation with convexity of f ,

f (x ) -f (x(t)) ≥ ∇f (x(t)), x -x(t) ,
we obtain the inequality

(4) d dt E(t) + β(t)δ(t) ∇f (x(t)) 2 + (α -1) t δ(t) -δ(t) (f (x(t)) -f (x )) ≤ 0. Then note that (5) (α -1) t δ(t) -δ(t) = t (α -3)w(t) -t ẇ(t) .
Hence, condition (G

f (x(t)) -f (x ) ≤ E(t 0 ) t 2 b(t) -β(t) - β(t) t .
Then, by integrating (4) we obtain

+∞ t0 β(t)δ(t) ∇f (x(t)) 2 dt ≤ E(t 0 ) < +∞,
and

+∞ t0 t (α -3)w(t) -t ẇ(t) (f (x(t)) -f (x ))dt ≤ E(t 0 ) < +∞,
which gives ii) and iii), and completes the proof.

Particular cases.

Case 1. The (DIN-AVD) α,β system corresponds to β(t) ≡ β and b(t) ≡ 1. In this case, w(t) = 1 -β t . Conditions (G 2 ) and (G 3 ) are satisfied by taking α > 3 and t > α-2 α-3 β. Hence, as a consequence of Theorem 2.1, we obtain the following result of Attouch-Peypouquet-Redont [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF]:

Theorem 2.2 ([12]). Let x : [t 0 , +∞[→ H be a trajectory of the dynamical system (DIN-AVD) α,β . Suppose α > 3. Then f (x(t)) -min H f = O 1 t 2 and ∞ t0 t 2 ∇f (x(t)) 2 dt < +∞.
Case 2. The system(DIN-AVD) α,β,1+ β t , which corresponds to β(t) ≡ β and b(t) = 1 + β t , was considered in [START_REF] Shi | Understanding the acceleration phenomenon via high-resolution differential equations[END_REF]. Compared to (DIN-AVD) α,β it has the additional coefficient β t in front of the gradient term. This vanishing coefficient will facilitate the computational aspects while keeping the structure of the dynamic. Observe that in this case, w(t) ≡ 1. Conditions (G 2 ) and (G 3 ) boil down to α ≥ 3. Hence, as a consequence of Theorem 2.1, we obtain

Theorem 2.3. Let x : [t 0 , +∞[→ H be a solution trajectory of the dynamical system (DIN-AVD) α,β,1+ β t . Suppose α ≥ 3. Then f (x(t)) -min H f = O 1 t 2 and ∞ t0 t 2 ∇f (x(t)) 2 dt < +∞.
Case 3. The dynamical system (DIN-AVD) α,0,b , which corresponds to β(t) ≡ 0, was considered by Attouch-Chbani-Riahi in [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF]. It comes also naturally from the time scaling of (AVD) α . In this case, we have

w(t) = b(t). Condition (G 2 ) is equivalent to b(t) > 0. (G 3 ) becomes t ḃ(t) ≤ (α -3)b(t),
which is precisely the condition introduced in [7, Theorem 8.1]. Under this condition, we have the convergence rate

f (x(t)) -min H f = O 1 t 2 b(t) as t → +∞.
This makes clear the acceleration effect due to the time scaling. For b(t

) = t r , we have f (x(t)) -min H f = O 1 t 2+r , under the assumption α ≥ 3 + r.
(t) = ct b , β(t) = t β . We have w(t) = ct b -(β+1)t β-1 , w (t) = cbt b-1 -(β 2 -1)t β-2 .
The conditions (G 2 ), (G 3 ) can be written respectively as:

(7) ct b > (β + 1)t β-1 and c(b -α + 3)t b ≤ (β + 1)(β -α + 2)t β-1 .
When b = β -1, the conditions (7) are equivalent to β < c -1 and β ≤ α -2, which gives the convergence

rate f (x(t)) -min H f = O 1 t β+1 .
Let us apply these choices to the quadratic function f

: (x 1 , x 2 ) ∈ R 2 → (x 1 + x 2 )
2 . f is convex but not strongly so, and argmin f = {(x 1 , x 2 ) ∈ R 2 : x 2 = -x 1 }. The closed-form solution of the ODE with this choice of β(t) and b(t) is given in Appendix A.3. We choose the values α = 5, β = 3, b = β -1 = 2 and c = 5 in order to satisfy condition [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF]. The left panel of Figure 2 depicts the convergence profile of the function value, and its right panel the trajectories associated with the system (DIN-AVD) α,β,b for different scenarios of the parameters. Once again, the damping of oscillations due to the presence of the Hessian is observed.

Inertial algorithms for general convex functions

3.1. Proximal algorithms.

3.1.1. Smooth case. Implicit time discretization of (DIN-AVD) α,β,b , with step size h > 0, gives

x k+1 -2x k + x k-1 h 2 + α kh x k+1 -x k h + β k h (∇f (x k+1 ) -∇f (x k )) + b k ∇f (x k+1 ) = 0. Equivalently k(x k+1 -2x k + x k-1 ) + α(x k+1 -x k ) + β k hk(∇f (x k+1 ) -∇f (x k )) +b k h 2 k∇f (x k+1 ) = 0. ( 8
)
Set s = h 2 . We obtain the following algorithm with β k and b k varying with k:

(IPAHD): Inertial Proximal Algorithm with Hessian Damping.

Step k : Set

µ k := k k+α (β k √ s + sb k ).
(IPAHD)

y k = x k + 1 -α k+α (x k -x k-1 ) + β k √ s 1 -α k+α ∇f (x k ) x k+1 = prox µ k f (y k ). Theorem 3.1. Suppose that α ≥ 1. Set (9) δ k := h b k hk -β k+1 -k(β k+1 -β k ) (k + 1),
and suppose that the following growth conditions are satisfied:

(G 2 ) b k hk -β k+1 -k(β k+1 -β k ) > 0; (G 3 ) δ k+1 -δ k ≤ (α -1) δ k k + 1 .
Then, δ k is positive and, for any sequence (x k ) k∈N generated by (IPAHD)

i) f (x k ) -min H f = O 1 δ k = O 1 k(k + 1) b k h -1 k β k+1 -(β k+1 -β k ) ii) k δ k β k+1 ∇f (x k+1 ) 2 < +∞. Proof. Given x ∈ argmin f , set E k := δ k (f (x k ) -f (x )) + 1 2 v k 2 ,
where

v k := (α -1)(x k -x ) + k(x k -x k-1 + β k h∇f (x k )),
and (δ k ) k∈N is a positive sequence that will be adjusted. Set ∆E k := E k+1 -E k , i.e.,

∆E k = (δ k+1 -δ k )(f (x k+1 ) -f (x )) + δ k (f (x k+1 ) -f (x k )) + 1 2 ( v k+1 2 -v k 2 )
Let us evaluate the last term of the above expression with the help of the three-point identity

1 2 v k+1 2 - 1 2 v k 2 = v k+1 -v k , v k+1 -1 2 v k+1 -v k 2 .
Using successively the definition of v k and (8), we get

v k+1 -v k = (α -1)(x k+1 -x k ) + (k + 1)(x k+1 -x k + β k+1 h∇f (x k+1 )) -k(x k -x k-1 + β k h∇f (x k )) = α(x k+1 -x k ) + k(x k+1 -2x k + x k-1 ) + β k+1 h∇f (x k+1 ) +hk(β k+1 ∇f (x k+1 ) -β k ∇f (x k )) = [α(x k+1 -x k ) + k(x k+1 -2x k + x k-1 ) + khβ k (∇f (x k+1 ) -∇f (x k ))] +β k+1 h∇f (x k+1 ) + kh(β k+1 -β k )∇f (x k+1 ) = -b k h 2 k∇f (x k+1 ) + β k+1 h∇f (x k+1 ) + kh(β k+1 -β k )∇f (x k+1 ) = h β k+1 + k(β k+1 -β k ) -b k hk ∇f (x k+1 ). Set shortly C k = β k+1 + k(β k+1 -β k ) -b k hk. We have obtained 1 2 v k+1 2 - 1 2 v k 2 = - h 2 2 C 2 k ∇f (x k+1 ) 2 ∇f (x k+1 ), (α -1)(x k+1 -x ) + (k + 1)(x k+1 -x k + β k+1 h∇f (x k+1 )) = -h 2 1 2 C 2 k -C k β k+1 ∇f (x k+1 ) 2 -(α -1)hC k ∇f (x k+1 ), x -x k+1 -hC k (k + 1) ∇f (x k+1 ), x k -x k+1 .
Let us assume that, for k large enough

-C k = b k hk -β k+1 -k(β k+1 -β k ) ≥ 0.
Then, in the above expression, the coefficient of ∇f (x k+1 ) 2 is less or equal than zero, which gives

1 2 v k+1 2 - 1 2 v k 2 ≤ -(α -1)hC k ∇f (x k+1 ), x -x k+1 -hC k (k + 1) ∇f (x k+1 ), x k -x k+1 .
According to the (convex) subdifferential inequality and C k ≤ 0, we infer

1 2 v k+1 2 - 1 2 v k 2 ≤ -(α -1)hC k (f (x ) -f (x k+1 ) -hC k (k + 1)(f (x k ) -f (x k+1 )).
Take

δ k := -hC k (k + 1) = h b k hk -β k+1 -k(β k+1 -β k ) (k + 1) so that the terms f (x k ) -f (x k+1 ) cancel in E k+1 -E k .
We obtain

E k+1 -E k ≤ δ k+1 -δ k -(α -1)h(b k hk -β k+1 -k(β k+1 -β k )) (f (x k+1 ) -f (x )) Equivalently E k+1 -E k ≤ δ k+1 -δ k -(α -1) δ k k + 1 (f (x k+1 ) -f (x )).
By assumption (G 3 ), we have δ k+1 -δ k -(α -1) δ k k+1 ≤ 0. Therefore, the sequence (E k ) k∈N is non-increasing, which, by definition of E k , gives, for k ≥ 0

f (x k ) -min H f ≤ E 0 δ k .
By summing the inequalities

E k+1 -E k + h h 2 (β k+1 + k(β k+1 -β k ) -b k hk) 2 + δ k β k+1 ∇f (x k+1 ) 2 ≤ 0 we finally obtain k δ k β k+1 ∇f (x k+1 ) 2 < +∞.
3.1.2. Non-smooth case. Let f : H → R ∪ {+∞} be a proper lower semicontinuous and convex function.

We rely on the basic properties of the Moreau-Yosida regularization. Let f λ be the Moreau envelope of f of index λ > 0, which is defined by:

f λ (x) = min z∈H f (z) + 1 2λ z -x 2 , for any x ∈ H.
We recall that f λ is a convex function, whose gradient is λ -1 -Lipschitz continuous, such that argmin f λ = argmin f . The interested reader may refer to [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF][START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF] for a comprehensive treatment of the Moreau envelope in a Hilbert setting. Since the set of minimizers is preserved by taking the Moreau envelope, the idea is to replace f by f λ in the previous algorithm, and take advantage of the fact that f λ is continuously differentiable. The Hessian dynamic attached to f λ becomes

ẍ(t) + α t ẋ(t) + β∇ 2 f λ (x(t)) ẋ(t) + b(t)∇f λ (x(t)) = 0.
However, we do not really need to work on this system (which requires f λ to be C 2 ), but with the discretized form which only requires the function to be continuously differentiable, as is the case of f λ . Then, algorithm (IPAHD) now reads

y k = x k + 1 -α k+α (x k -x k-1 ) + β √ s 1 -α k+α ∇f λ (x k ) x k+1 = prox k k+α (β √ s+sb k )f λ (y k ).
By applying Theorem 3.1 we obtain that under the assumption (G 2 ) and (G 3 ),

f λ (x k ) -min H f = O 1 k 2 b k , k k 2 b 2 k ∇f λ (x k+1
) 2 < +∞. Thus, we just need to formulate these results in terms of f and its proximal mapping. This is straightforward thanks to the following formulae from proximal calculus [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF]:

• f λ (x) = f (prox λf (x)) + 1 2λ x -prox λf (x)) 2 . • ∇f λ (x) = 1 λ x -prox λf (x) . • prox θf λ (x) = λ λ+θ x + θ λ+θ prox (λ+θ)f (x)
. We obtain the following relaxed inertial proximal algorithm (NS stands for Non-Smooth):

(IPAHD-NS) :

Set µ k := λ(k+α) λ(k+α)+k(β √ s+sb k )    y k = x k + (1 -α k+α )(x k -x k-1 ) + β √ s λ 1 -α k+α x k -prox λf (x k ) x k+1 = µ k y k + (1 -µ k ) prox λ µ k f (y k ).
Theorem 3.2. Let f : H → R ∪ {+∞} be a convex, lower semicontinuous, proper function. Suppose that the following growth conditions are satisfied

(G 2 ) b k hk -β k+1 -k(β k+1 -β k ) > 0; (G 3 ) δ k+1 -δ k ≤ (α -1) δ k k + 1 ,
where the sequence (δ k ) has been defined in [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF]. Then, for any sequence (x k ) k∈N generated by (IPAHD-NS) , the following holds

f (prox λf (x k )) -min H f = O 1 k 2 b k , k δ k β k+1 x k+1 -prox λf (x k+1 ) 2 < +∞.
3.2. Gradient algorithms. Take f a convex function whose gradient is L-Lipschitz continuous. Our analysis is based on the dynamic (DIN-AVD) α,β,1+ β t considered in Theorem 2.3 with damping parameters α ≥ 3, β ≥ 0. Consider the time discretization of (DIN-AVD) α,β,1+

β t 1 s (x k+1 -2x k + x k-1 ) + α ks (x k -x k-1 ) + β √ s (∇f (x k ) -∇f (x k-1 )) + β k √ s ∇f (x k-1 ) + ∇f (y k ) = 0,
with y k inspired by Nesterov's accelerated scheme. We obtain the following scheme:

(IGAHD) : Inertial Gradient Algorithm with Hessian Damping.

Step

k: α k = 1 -α k . y k = x k + α k (x k -x k-1 ) -β √ s (∇f (x k ) -∇f (x k-1 )) -β √ s k ∇f (x k-1 ) x k+1 = y k -s∇f (y k ), Following [5], set t k+1 = k α-1 , whence t k = 1 + t k+1 α k .
Given x ∈ argmin f , our Lyapunov analysis is based on the sequence (E k ) k∈N

E k := t 2 k (f (x k ) -f (x )) + 1 2s v k 2 (10)
v k := (x k-1 -x ) + t k x k -x k-1 + β √ s∇f (x k-1 ) . ( 11
)
Theorem 3.3. Let f : H → R be a convex function whose gradient is L-Lipschitz continuous. Let (x k ) k∈N be a sequence generated by algorithm (IGAHD) , where α ≥ 3, 0 ≤ β < 2 √ s and sL ≤ 1. Then the sequence (E k ) k∈N defined by (10)-( 11) is non-increasing, and the following convergence rates are satisfied:

i) f (x k ) -min H f = O 1 k 2 as k → +∞; ii) Suppose that β > 0. Then k k 2 ∇f (y k ) 2 < +∞ and k k 2 ∇f (x k ) 2 < +∞.
Proof. We rely on the following reinforced version of the gradient descent lemma (Lemma A.1 in Appendix A.1). Since s ≤ 1 L , and ∇f is L-lipschitz continuous,

f (y -s∇f (y)) ≤ f (x) + ∇f (y), y -x - s 2 ∇f (y) 2 - s 2 ∇f (x) -∇f (y) 2
for all x, y ∈ H. Let us write it successively at y = y k and x = x k , then at y = y k , x = x . According to x k+1 = y k -s∇f (y k ) and ∇f (x ) = 0, we get

f (x k+1 ) ≤ f (x k ) + ∇f (y k ), y k -x k - s 2 ∇f (y k ) 2 - s 2 ∇f (x k ) -∇f (y k ) 2 (12) f (x k+1 ) ≤ f (x ) + ∇f (y k ), y k -x - s 2 ∇f (y k ) 2 - s 2 ∇f (y k ) 2 . ( 13
)
Multiplying ( 12) by t k+1 -1 ≥ 0, then adding (13), we derive that

t k+1 (f (x k+1 ) -f (x )) ≤ (t k+1 -1)(f (x k ) -f (x )) + ∇f (y k ), (t k+1 -1)(y k -x k ) + y k -x - s 2 t k+1 ∇f (y k ) 2 . - s 2 (t k+1 -1) ∇f (x k ) -∇f (y k ) 2 - s 2 ∇f (y k ) 2 . ( 14
)
Let us multiply [START_REF] Aujol | Stability of over-relaxations for the Forward-Backward algorithm, application to FISTA[END_REF] by t k+1 to make appear E k . We obtain

t 2 k+1 (f (x k+1 ) -f (x )) ≤ (t 2 k+1 -t k+1 -t 2 k )(f (x k ) -f (x )) + t 2 k (f (x k ) -f (x )) +t k+1 ∇f (y k ), (t k+1 -1)(y k -x k ) + y k -x - s 2 t 2 k+1 ∇f (y k ) 2 - s 2 (t 2 k+1 -t k+1 ) ∇f (x k ) -∇f (y k ) 2 - s 2 t k+1 ∇f (y k ) 2 . Since α ≥ 3 we have t 2 k+1 -t k+1 -t 2 k ≤ 0, which gives t 2 k+1 (f (x k+1 -f (x )) ≤ t 2 k (f (x k ) -f (x )) +t k+1 ∇f (y k ), (t k+1 -1)(y k -x k ) + y k -x - s 2 t 2 k+1 ∇f (y k ) 2 - s 2 (t 2 k+1 -t k+1 ) ∇f (x k ) -∇f (y k ) 2 - s 2 t k+1 ∇f (y k ) 2 .
According to the definition of E k , we infer

E k+1 -E k ≤ t k+1 ∇f (y k ), (t k+1 -1)(y k -x k ) + y k -x - s 2 t 2 k+1 ∇f (y k ) 2 - s 2 (t 2 k+1 -t k+1 ) ∇f (x k ) -∇f (y k ) 2 - s 2 t k+1 ∇f (y k ) 2 + 1 2s v k+1 2 - 1 2s v k 2 .
Let us compute this last expression with the help of the elementary inequality

1 2 v k+1 2 - 1 2 v k 2 = v k+1 -v k , v k+1 - 1 2 v k+1 -v k 2 .
By definition of v k , according to (IGAHD) and t k -1 = t k+1 α k , we have

v k+1 -v k = x k -x k-1 + t k+1 (x k+1 -x k + β √ s∇f (x k )) -t k (x k -x k-1 + β √ s∇f (x k-1 )) = t k+1 (x k+1 -x k ) -(t k -1)(x k -x k-1 ) + β √ s t k+1 ∇f (x k ) -t k ∇f (x k-1 ) = t k+1 x k+1 -(x k + α k (x k -x k-1 ) + β √ s t k+1 ∇f (x k ) -t k ∇f (x k-1 ) = t k+1 (x k+1 -y k ) -t k+1 β √ s(∇f (x k ) -∇f (x k-1 )) -t k+1 β √ s k ∇f (x k-1 ) +β √ s(t k+1 ∇f (x k ) -t k ∇f (x k-1 )) = t k+1 (x k+1 -y k ) + β √ s t k+1 (1 - 1 k ) -t k ∇f (x k-1 ) = t k+1 (x k+1 -y k ) = -st k+1 ∇f (y k ).
Hence

1 2s v k+1 2 - 1 2s v k 2 = - s 2 t 2 k+1 ∇f (y k ) 2 -t k+1 ∇f (y k ), x k -x + t k+1 x k+1 -x k + β √ s∇f (x k ) .
Collecting the above results, we obtain

E k+1 -E k ≤ t k+1 ∇f (y k ), (t k+1 -1)(y k -x k ) + y k -x -st 2 k+1 ∇f (y k ) 2 -t k+1 ∇f (y k ), x k -x + t k+1 x k+1 -x k + β √ s∇f (x k ) - s 2 (t 2 k+1 -t k+1 ) ∇f (x k ) -∇f (y k ) 2 - s 2 t k+1 ∇f (y k ) 2 .
Equivalently

E k+1 -E k ≤ t k+1 ∇f (y k ), A k -st 2 k+1 ∇f (y k ) 2 - s 2 (t 2 k+1 -t k+1 ) ∇f (x k ) -∇f (y k ) 2 - s 2 t k+1 ∇f (y k ) 2 , with A k = (t k+1 -1)(y k -x k ) + y k -x k -t k+1 x k+1 -x k + β √ s∇f (x k ) = t k+1 y k -t k+1 x k -t k+1 (x k+1 -x k ) -t k+1 β √ s∇f (x k ) = t k+1 (y k -x k+1 ) -t k+1 β √ s∇f (x k ) = st k+1 ∇f (y k ) -t k+1 β √ s∇f (x k ) Consequently E k+1 -E k ≤ t k+1 ∇f (y k ), st k+1 ∇f (y k ) -t k+1 β √ s∇f (x k ) -st 2 k+1 ∇f (y k ) 2 - s 2 (t 2 k+1 -t k+1 ) ∇f (x k ) -∇f (y k ) 2 - s 2 t k+1 ∇f (y k ) 2 = -t 2 k+1 β √ s ∇f (y k ), ∇f (x k ) - s 2 (t 2 k+1 -t k+1 ) ∇f (x k ) -∇f (y k ) 2 - s 2 t k+1 ∇f (y k ) 2 = -t k+1 B k ,
where

B k := t k+1 β √ s ∇f (y k ), ∇f (x k ) + s 2 (t k+1 -1) ∇f (x k ) -∇f (y k ) 2 + s 2 ∇f (y k ) 2 .
When β = 0 we have B k ≥ 0. Let us analyze the sign of B k in the case β > 0. Set Y = ∇f (y k ), X = ∇f (x k ). We have

B k = s 2 Y 2 + s 2 (t k+1 -1) Y -X 2 + t k+1 β √ s Y, X = s 2 t k+1 Y 2 + t k+1 (β √ s -s) + s Y, X + s 2 (t k+1 -1) X 2 ≥ s 2 t k+1 Y 2 -t k+1 (β √ s -s) + s Y X + s 2 (t k+1 -1) X 2 .
Elementary algebra gives that the above quadratic form is non-negative when

t k+1 (β √ s -s) + s 2 ≤ s 2 t k+1 (t k+1 -1).
Recall that t k is of order k. Hence, this inequality is satisfied for

k large enough if (β √ s -s) 2 < s 2 , which is equivalent to β < 2 √ s.
Under this condition E k+1 -E k ≤ 0, which gives conclusion i). Similar argument gives that for 0 < < 2 √ sβ -β 2 (such exists according to assumption 0 < β < 2 √ s)

E k+1 -E k + 1 2 t 2 k+1 ∇f (y k ) 2 ≤ 0.
After summation of these inequalities, we obtain conclusion ii).

Remark 3.4. From k k 2 ∇f (x k ) 2 < +∞ we immediately infer that for k ≥ 1 inf i=1,...,k ∇f (x i ) 2 k i=1 i 2 ≤ k i=1 i 2 ∇f (x i ) 2 ≤ i∈N i 2 ∇f (x i ) 2 < +∞.
A similar argument holds for y k . Hence

inf i=1,...,k ∇f (x i ) 2 = O 1 k 3 , inf i=1,...,k ∇f (y i ) 2 = O 1 k 3 .
Remark 3.5. In Theorem 3.3, the convergence property of the values is expressed according to the sequence (x k ) k∈N . It is natural to know if a similar result is true for the sequence (y k ) k∈N . This is an open question in the case of Nesterov's accelerated gradient method and the corresponding FISTA algorithm for structured minimization [START_REF] Nesterov | Gradient methods for minimizing composite objective function[END_REF][START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF]. In the case of the Hessian-driven damping algorithms, we give a partial answer to this question. By the classical descent lemma, and the monotonicity of ∇f we have

f (y k ) ≤ f (x k+1 ) + y k -x k+1 , ∇f (x k+1 ) + L 2 y k -x k+1 2 ≤ f (x k+1 ) + y k -x k+1 , ∇f (y k ) + L 2 y k -x k+1 2 According to x k+1 = y k -s∇f (y k ) we obtain f (y k ) -min H f ≤ f (x k+1 ) -min H f + s ∇f (y k ) 2 + s 2 L 2 ∇f (y k ) 2 .
From Theorem 3.3 we deduce that

f (y k ) -min H f ≤ O 1 k 2 + s + s 2 L 2 ∇f (y k ) 2 = O 1 k 2 + o 1 k 2 .
Remark 3.6. When f is a proper lower semicontinuous proper function, but not necessarily smooth, we follow the same reasoning as in Section 3.1.2. We consider minimizing the Moreau envelope f λ of f , whose gradient is 1/λ-Lipschitz continuous, and then apply (IGAHD) to f λ . We omit the details for the sake of brevity. This observation will be very useful to solve even structured composite problems as we will describe in Section 6.

Inertial dynamics for strongly convex functions

4.1. Smooth case. Recall the classical definition of strong convexity:

Definition 4.1. A function f : H → R is said to be µ-strongly convex for some µ > 0 if f -µ 2 • 2 is convex.
For strongly convex functions, a suitable choice of γ and β in (DIN) γ,β provides exponential decay of the value function (hence of the trajectory), and of the gradients. This corresponds to linear convergence in the algorithmic case. It can be seen as an extension of the Nesterov accelerated method for strongly convex functions that corresponds to the particular case β = 0. The result in the case β = 0 was considered in [START_REF] Siegel | Accelerated first-order methods: Differential equations and Lyapunov functions[END_REF]Theorem 2.2]. In the case β > 0, a related but different result can be found in [START_REF] Wilson | A Lyapunov analysis of momentum methods in optimization[END_REF]Theorem 1]. The gradient estimate is new.

Theorem 4.2. Suppose that f : H → R is µ-strongly convex for some µ > 0. Let x(•) : [t 0 , +∞[→ H be a solution trajectory of (15) ẍ(t) + 2 √ µ ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) = 0. Suppose that 0 ≤ β ≤ 1 2 √ µ .
Then, the following hold:

i) for all t ≥ t 0 µ 2 x(t) -x 2 ≤ f (x(t)) -min H f ≤ Ce - √ µ 2 (t-t0)
where

C := f (x(t 0 )) -min H f + µdist(x(t 0 ), S) 2 + ẋ(t 0 ) + β∇f (x(t 0 )) 2 . ii) There exists some constant C 1 > 0 such that, for all t ≥ t 0 e -√ µt t t0 e √ µs ∇f (x(s)) 2 ds ≤ C 1 e - √ µ 2 t . Moreover, ∞ t0 e √ µ 2 t ẋ(t) 2 dt < +∞. When β = 0, we have f (x(t)) -min H f = O e -√ µt as t → +∞.

Proof.

i) Let x be the unique minimizer of f . Define E : [t 0 , +∞[→ R + by

E(t) := f (x(t)) -min H f + 1 2 √ µ(x(t) -x ) + ẋ(t) + β∇f (x(t)) 2 . Set v(t) = √ µ(x(t) -x ) + ẋ(t) + β∇f (x(t)). Derivation of E(•) gives d dt E(t) := ∇f (x(t)), ẋ(t) + v(t), √ µ ẋ(t) + ẍ(t) + β∇ 2 f (x(t)) ẋ(t) .
Using [START_REF] Aujol | Optimal rate of convergence of an ODE associated to the Fast Gradient Descent schemes for b > 0[END_REF], we get

d dt E(t) = ∇f (x(t)), ẋ(t) + v(t), - √ µ ẋ(t) -∇f (x(t)) .
After developing and simplification, we obtain

d dt E(t) + √ µ ∇f (x(t)), x(t) -x + µ x(t) -x , ẋ(t) + √ µ ẋ(t) 2 +β √ µ ∇f (x(t)), ẋ(t) + β ∇f (x(t)) 2 = 0.
By strong convexity of f we have

∇f (x(t)), x(t) -x ≥ f (x(t)) -f (x ) + µ 2 x(t) -x 2 .
Thus, combining the last two relations we obtain

d dt E(t) + √ µA ≤ 0,
where (the variable t is omitted to lighten the notation)

A := f (x) -f (x ) + µ 2 x -x 2 + √ µ x -x , ẋ + ẋ 2 + β ∇f (x), ẋ + β √ µ ∇f (x) 2 Set Z(t) := 2β(f (x(t)) -f (x )) + √ µ x(t) -x 2 ]. We have d dt Z(t) + √ µZ(t) + β 2 ∇f (x(t)) 2 ≤ Ce - √ µ 2 t .
By integrating this differential inequality, elementary computation gives

e -√ µt t t0 e √ µs ∇f (x(s)) 2 ds ≤ Ce - √ µ 2 t .
Noticing that the integral of e √ µs over [t 0 , t] is of order e √ µt , the above estimate reflects the fact, as t → +∞, the gradient terms ∇f (x(t)) 2 tend to zero at exponential rate (in average, not pointwise).

Remark 4.3. Let us justify the choice of γ = 2 √ µ in Theorem 4.2. Indeed, considering

ẍ(t) + 2γ ẋ(t) + β∇ 2 f (x(t)) + ∇f (x(t)) = 0,
a similar proof to that described above can be performed on the basis of the Lyapunov function

E(t) := f (x(t)) -min H f + 1 2 γ(x(t) -x ) + ẋ(t) + β∇f (x(t)) 2 .
Under the conditions γ ≤ √ µ and β ≤ µ 2γ 3 we obtain the exponential convergence rate

f (x(t)) -min H f = O e -γ 2 t
as t → +∞.

Taking γ = √ µ gives the best convergence rate, and the result of Theorem 4.2.

4.2.

Non-smooth case. Following [START_REF] Álvarez | A second-order gradient-like dissipative dynamical system with Hessiandriven damping. Application to optimization and mechanics[END_REF], (DIN) γ,β is equivalent to the first-order system

   ẋ(t) + β∇f (x(t)) + γ -1 β x(t) + 1 β y(t) = 0; ẏ(t) + γ -1 β x(t) + 1 β y(t) = 0.
. This permits to extend (DIN) γ,β to the case of a proper lower semicontinuous convex function f : H → R ∪ {+∞}. Replacing the gradient of f by its subdifferential, we obtain its Non-Smooth version :

(DIN-NS) γ,β    ẋ(t) + β∂f (x(t)) + γ -1 β x(t) + 1 β y(t) 0; ẏ(t) + γ -1 β x(t) + 1 β y(t) = 0.
Most properties of (DIN) γ,β are still valid for this generalized version. To illustrate it, let us consider the following extension of Theorem 4.2.

Theorem 4.4. Suppose that f : H → R ∪ {+∞} is lower semicontinuous and µ-strongly convex for some µ > 0. Let x(•) be a trajectory of (DIN-NS)

2 √ µ,β . Suppose that 0 ≤ β ≤ 1 2 √ µ . Then µ 2 x(t) -x 2 ≤ f (x(t)) -min H f = O e - √ µ 2 t as t → +∞,
and

∞ t0 e √ µ 2 t ẋ(t) 2 dt < +∞.
Proof. Let us introduce E : [t 0 , +∞[→ R + defined by

E(t) := f (x(t)) -min H f + 1 2 √ µ(x(t) -x ) -2 √ µ - 1 β x(t) - 1 β y(t) 2 ,
that will serve as a Lyapunov function. Then, the proof follows the lines of Theorem 4.2, with the use of the derivation rule of Brezis [19, Lemme 3.3, p. 73].

5.

Inertial algorithms for strongly convex functions 5.1. Proximal algorithms.

5.1.1. Smooth case. Consider the inertial dynamic [START_REF] Aujol | Optimal rate of convergence of an ODE associated to the Fast Gradient Descent schemes for b > 0[END_REF]. Its implicit discretization similar to that performed before gives

1 h 2 (x k+1 -2x k + x k-1 ) + 2 √ µ h (x k+1 -x k ) + β h (∇f (x k+1 ) -∇f (x k )) + ∇f (x k+1 ) = 0,
where h is the positive step size. Set s = h 2 . We obtain the following inertial proximal algorithm with hessian damping (SC refers to Strongly Convex):

(IPAHD-SC)    y k = x k + 1 - 2 √ µs 1+2 √ µs (x k -x k-1 ) + β √ s 1 - 2 √ µs 1+2 √ µs ∇f (x k ) x k+1 = prox β √ s+s 1+2 √ µs f (y k ).
Theorem 5.1. Take f : H → R µ-strongly convex, µ > 0, and suppose that

0 ≤ β ≤ 1 2 √ µ and √ s ≤ β. Set q = 1 1+ 1 2
√ µs , which satisfies 0 < q < 1. Then, the sequence (x k ) k∈N generated by the algorithm (IPAHD-SC) obeys, for any k ≥ 1

µ 2 x k -x 2 ≤ f (x k ) -min H f ≤ E 1 q k-1 ,
where

E 1 = f (x 1 ) -f (x ) + 1 2 √ µ(x 1 -x ) + 1 √ s (x 1 -x 0 ) + β∇f (x 1 ) 2 .
Moreover, the gradients converge exponentially fast to zero: setting θ = 1 1+ √ µs which belongs to ]0, 1[, we have

θ k k-2 p=0 θ -j ∇f (x j ) 2 = O q k as k → +∞.
Proof. Take x ∈ argmin f , and consider the sequence (E k ) k∈N

E k := f (x k ) -f (x ) + 1 2 v k 2 ,
where

v k = √ µ(x k -x ) + 1 √ s (x k -x k-1 ) + β∇f (x k
). We will use the following equivalent formulation of the algorithm (IPAHD-SC)

(16) 1 √ s (x k+1 -2x k + x k-1 ) + 2 √ µ(x k+1 -x k ) + β(∇f (x k+1 ) -∇f (x k )) + √ s∇f (x k+1 ) = 0.
We have

E k+1 -E k = f (x k+1 ) -f (x k ) + 1 2 v k+1 2 - 1 2 v k 2 .
Using successively the definition of v k and ( 16), we get

v k+1 -v k = √ µ(x k+1 -x k ) + 1 √ s (x k+1 -2x k + x k-1 ) + β(∇f (x k+1 ) -∇f (x k )) = √ µ(x k+1 -x k ) -2 √ µ(x k+1 -x k ) - √ s∇f (x k+1 ) = = - √ µ(x k+1 -x k ) - √ s∇f (x k+1 ). Write shortly B k = √ µ(x k+1 -x k ) + √ s∇f (x k+1 ). We have 1 2 v k+1 2 - 1 2 v k 2 = v k+1 -v k , v k+1 - 1 2 v k+1 -v k 2 = -B k , √ µ(x k+1 -x ) + 1 √ s (x k+1 -x k ) + β∇f (x k+1 ) - 1 2 B k 2 = -µ x k+1 -x k , x k+1 -x - µ s x k+1 -x k 2 -β √ µ ∇f (x k+1 ), x k+1 -x k - √ µs ∇f (x k+1 ), x k+1 -x -∇f (x k+1 ), x k+1 -x k -β √ s ∇f (x k+1 ) 2 - 1 2 µ x k+1 -x k 2 - 1 2 s ∇f (x k+1 2 - √ µs ∇f (x k+1 ), x k+1 -x k By virtue of strong convexity of f f (x k ) ≥ f (x k+1 ) + ∇f (x k+1 ), x k -x k+1 + µ 2 x k+1 -x k 2 ; f (x ) ≥ f (x k+1 ) + ∇f (x k+1 ), x -x k+1 + µ 2 x k+1 -x 2 .
Combining the above results, and after dividing by √ s, we get

1 √ s (E k+1 -E k ) + √ µ[f (x k+1 ) -f (x ) + µ x k+1 -x 2 ] ≤ - µ √ s x k+1 -x k , x k+1 -x - √ µ s x k+1 -x k 2 -β µ s ∇f (x k+1 ), x k+1 -x k - µ 2 √ s x k+1 -x k 2 -β ∇f (x k+1 ) 2 - µ 2 √ s x k+1 -x k 2 - 1 2 √ s ∇f (x k+1 2 - √ µ ∇f (x k+1 ), x k+1 -x k ,
which gives, after developing and simplification

1 √ s (E k+1 -E k ) + √ µE k+1 -βµ ∇f (x k+1 ), x k+1 -x ≤ - √ µ 2s + µ √ s x k+1 -x k 2 -β - β 2 √ µ 2 + √ s 2 ∇f (x k+1 ) 2 - √ µ ∇f (x k+1 ), x k+1 -x k .
According to 0 ≤ β ≤ 1 2 √ µ , we have β -

β 2 √ µ 2 ≥ 3β 4
, which, with Cauchy-Schwarz inequality, gives

1 √ s (E k+1 -E k ) + √ µE k+1 + √ µ 2s + µ √ s x k+1 -x k 2 + 3β 4 ∇f (x k+1 ) 2 -βµ ∇f (x k+1 ) x k+1 -x - √ µ ∇f (x k+1 ) x k+1 -x k ≤ 0.
Let us use again the strong convexity of f to write

E k+1 ≥ 1 2 E k+1 + 1 2 (f (x k+1 ) -f (x )) ≥ 1 2 E k+1 + µ 4 x k+1 -x 2 .
Combining the two inequalities above, we get

1 √ s (E k+1 -E k ) + 1 2 √ µE k+1 + √ µ µ 4 x k+1 -x 2 + √ µ 2s + µ √ s x k+1 -x k 2 + 3β 4 ∇f (x k+1 ) 2 -βµ ∇f (x k+1 ) x k+1 -x - √ µ ∇f (x k+1 ) x k+1 -x k ≤ 0.
Let us rearrange the terms as follows

1 √ s (E k+1 -E k ) + 1 2 √ µE k+1 + √ µ µ 4 x k+1 -x 2 + β 2 ∇f (x k+1 ) 2 -βµ ∇f (x k+1 ) x k+1 -x Term 1 + √ µ 2s + µ √ s x k+1 -x k 2 + β 4 ∇f (x k+1 ) 2 - √ µ ∇f (x k+1 ) x k+1 -x k Term 2 ≤ 0
Let us examine the sign of the last two terms in the rhs of inequality above. Term 1 Set X = x k+1 -x , Y = ∇f (x k+1 ) . Elementary algebra gives that

√ µ µ 4 X 2 + β 2 Y 2 -βµXY ≥ 0, holds true under the condition 0 ≤ β ≤ 1 2 √ µ . Hence, under this condition √ µ µ 4 x k+1 -x 2 + β 2 ∇f (x k+1 ) 2 -βµ ∇f (x k+1 ) x k+1 -x ≥ 0. Term 2 Set X = x k+1 -x k , Y = ∇f (x k+1 ) . Elementary algebra gives √ µ 2s + µ √ s X 2 + β 4 Y 2 - √ µXY ≥ 0 holds true under the condition √ µ 2s + µ √ s ≥ µ β . Hence, under this condition √ µ 2s + µ √ s x k+1 -x k 2 + β 4 ∇f (x k+1 ) 2 - √ µ ∇f (x k+1 ) x k+1 -x k ≥ 0.
In turn, the condition

√ µ 2s + µ √ s ≥ µ β is equivalent to √ s ≤ β 2 1 + 1 + 2 β √ µ .
Clearly, this condition is satisfied if √ s ≤ β. Let us put the above results together. We have obtained that, under the conditions 0

≤ β ≤ 1 2 √ µ and √ s ≤ β, 1 √ s (E k+1 -E k ) + 1 2 √ µE k+1 ≤ 0. Set q = 1 1+ 1 2
√ µs , which satisfies 0 < q < 1. From this, we infer E k ≤ qE k-1 which gives ( 17)

E k ≤ E 1 q k-1 . Since E k ≥ f (x k ) -f (x ), we finally obtain f (x k ) -f (x ) ≤ E 1 q k-1 = O q k .
Let us now estimate the convergence rate of the gradients to zero. According to the exponential decay of (E k ) k∈N , as given in [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF], and by definition of E k , we have, for all k ≥ 1

√ µ(x k -x ) + 1 √ s (x k -x k-1 ) + β∇f (x k ) 2 ≤ 2E k ≤ 2E 1 q k-1 .
After developing, we get

µ x k -x 2 + 1 s x k -x k-1 2 + β 2 ∇f (x k ) 2 + 2β √ µ x k -x , ∇f (x k ) + 1 √ s x k -x k-1 , 2β∇f (x k ) + 2 √ µ(x k -x ) ≤ 2E 1 q k-1 .
By convexity of f , we have

x k -x , ∇f (x k ) ≥ f (x k ) -f (x ) and x k -x k-1 , ∇f (x k ) ≥ f (x k ) -f (x k-1 ) Moreover, x k -x k-1 , x k -x ≥ 1 2 x k -x 2 -1 2 x k-1 -x 2 .
Combining the above results, we obtain

√ µ 2β(f (x k ) -f (x )) + √ µ x k -x 2 + β 2 ∇f (x k ) 2 + 1 √ s 2β(f (x k ) -f (x )) + √ µ x k -x 2 - 1 √ s 2β(f (x k-1 ) -f (x )) + √ µ x k-1 -x 2 ≤ 2E 1 q k-1 . Set Z k := 2β(f (x k ) -f (x )) + √ µ x k -x 2 . We have, for all k ≥ 1 (18) 1 √ s (Z k -Z k-1 ) + √ µZ k + β 2 ∇f (x k ) 2 ≤ 2E 1 q k-1 . Set θ = 1 1+
√ µs which belongs to ]0, 1[. Equivalently

Z k + θβ 2 √ s ∇f (x k ) 2 ≤ θZ k-1 + 2E 1 θ √ sq k-1 .
Iterating this linear recursive inequality gives

(19) Z k + θβ 2 √ s k-2 p=0 θ p ∇f (x k-p ) 2 ≤ θ k-1 Z 1 + 2E 1 θ √ s k-2 p=0 θ p q k-p-1 .
Then notice that θ q =

1+ 1 2 √ µs 1+ √ µs < 1, which gives k-2 p=0 θ p q k-p-1 = q k-1 k-2 p=0 θ q p ≤ 2 1 + 1 √ µs q k-1 .
Collecting the above results, we obtain (20)

θβ 2 √ s k-2 p=0 θ p ∇f (x k-p ) 2 ≤ θ k-1 Z 1 + 4E 1 √ µ q k-1 .
Using again the inequality θ < q, and after reindexing, we finally obtain

θ k k-2 p=0 θ -j ∇f (x j ) 2 = O q k .
5.1.2. Non-smooth case. Let f : H → R ∪ {+∞} be a proper, lower semicontinuous and convex function. We argue as in Section 3.1.2 by replacing f with its Moreau envelope f λ . This operation also preserves strong convexity, thought with a different modulus as shown by the following result.

Proposition 5.2. Suppose that f : H → R ∪ {+∞} is a proper, lower semicontinuous convex function.

Then, for any λ > 0 and µ > 0 f is µ-strongly convex =⇒ f λ is strongly convex with modulus µ 1 + λµ .

Proof. If f is strongly convex with constant µ > 0, we have f = g + µ 2 • 2 for some convex function g. Elementary calculus (see e.g., [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF]Exercise 12.6]) gives, with θ = λ 1+λµ ,

f λ (x) = g θ 1 1 + λµ x + µ 2(1 + λµ) x 2 . Since x → g θ 1 1+λµ
x is convex, the above formula shows that f λ is strongly convex with constant µ 1+λµ .

According to the expressions ∇f λ (x) =

y k = x k + (1 -a)(x k -x k-1 ) + β √ s λ (1 -a) x k -prox λf (x k ) x k+1 = λ λ+θ y k + θ λ+θ prox (λ+θ)f (y k )
It is a relaxed inertial proximal algorithm whose coefficients are constant. As a result, its computational burden is equivalent to (actually twice) that of the classical proximal algorithm. A direct application of the conclusions of Theorem 5.1 to f λ gives the following statement.

Theorem 5.3. Suppose that f : H → R ∪ {+∞} is a proper, lower semicontinuous and convex function which is µ-strongly convex for some µ > 0. Take λ > 0. Suppose that

0 ≤ β ≤ 1 2 λ + 1 µ and √ s ≤ β. Set q = 1 1 + 1 2 µ 1+λµ s
, which satisfies 0 < q < 1. Then, for any sequence (x k ) k∈N generated by algorithm (IPAHD-NS-SC)

x k -x = O q k/2 and f (prox λf (x k )) -min H f = O q k as k → +∞, and x k -prox λf (x k ) 2 = O q k as k → +∞.
+ x k-1 ) + √ µ √ s (x k+1 -x k-1 ) + β 1 √ s (∇f (x k ) -∇f (x k-1 )) + ∇f (x k ) = 0. Equivalently, (21)
(x k+1 -2x k + x k-1 ) + √ µs(x k+1 -x k-1 ) + β √ s(∇f (x k ) -∇f (x k-1 )) + s∇f (x k ) = 0,
which gives the inertial gradient algorithm with Hessian damping (SC stands for Strongly Convex):

(IGAHD-SC)

x k+1 = x k + 1- √ µs 1+ √ µs (x k -x k-1 ) -β √ s 1+ √ µs (∇f (x k ) -∇f (x k-1 )) -s 1+ √ µs ∇f (x k ).
Let us analyze the linear convergence rate of (IGAHD-SC) .

Theorem 5.4. Let f : H → R be a function µ-strongly convex for some µ > 0, and whose gradient ∇f is L-Lipschitz continuous. Suppose that

β ≤ 1 √ µ and L ≤ min    √ µ 8β , √ µ 2s + µ √ s 2βµ + 1 √ s + √ µ 2    . Set q = 1 1 + 1 2 √ µs
, which satisfies 0 < q < 1. Then, for any sequence (x k ) k∈N generated by algorithm (IGAHD-SC) , we have

x k -x = O q k/2 and f (x k ) -min H f = O q k as k → +∞.
Moreover, the gradients converge exponentially fast to zero: setting θ =

1+

√ µs which belongs to ]0, 1[, we have

θ k k-2 p=0 θ -j ∇f (x j ) 2 = O q k as k → +∞.
Proof. The proof is based on Lyapunov analysis, and the decrease property at linear rate of the sequence (E k ) k∈N defined by

E k := f (x k ) -f (x ) + 1 2 v k 2 ,
where x is the unique minimizer of f , and

v k = √ µ(x k-1 -x ) + 1 √ s (x k -x k-1 ) + β∇f (x k-1 ).
We have

E k+1 -E k = f (x k+1 ) -f (x k ) + 1 2 v k+1 2 -1 2 v k 2 .
Using successively the definition of v k and (21), we obtain

v k+1 -v k = √ µ(x k -x k-1 ) + 1 √ s (x k+1 -2x k + x k-1 ) + β(∇f (x k ) -∇f (x k-1 )) = 1 √ s (x k+1 -2x k + x k-1 ) + √ µs(x k -x k-1 ) + β √ s(∇f (x k ) -∇f (x k-1 )) = 1 √ s -s∇f (x k ) - √ µs(x k+1 -x k-1 ) + √ µs(x k -x k-1 )) = - √ µ(x k+1 -x k ) - √ s∇f (x k ). Since 1 2 v k+1 2 -1 2 v k 2 = v k+1 -v k , v k+1 -1 2 v k+1 -v k 2 , we have 1 2 v k+1 2 - 1 2 v k 2 = - 1 2 √ µ(x k+1 -x k ) + √ s∇f (x k ) 2 - √ µ(x k+1 -x k ) + √ s∇f (x k ), √ µ(x k -x * ) + 1 √ s (x k+1 -x k ) + β∇f (x k ) = -µ x k+1 -x k , x k -x * - µ s x k+1 -x k 2 -β √ µ ∇f (x k ), x k+1 -x k - √ µs ∇f (x k ), x k -x * -∇f (x k ), x k+1 -x k -β √ s ∇f (x k ) 2 - 1 2 µ x k+1 -x k 2 - 1 2 s ∇f (x k 2 - √ µs ∇f (x k ), x k+1 -x k .
By strong convexity of f and L-lipschitz continuity of ∇f we have

f (x ) ≥ f (x k ) + ∇f (x k ), x -x k + µ 2 x k -x 2 f (x k ) ≥ f (x k+1 ) + ∇f (x k+1 ), x k -x k+1 + µ 2 x k+1 -x k 2 ≥ f (x k+1 ) + ∇f (x k ), x k -x k+1 + ( µ 2 -L) x k+1 -x k 2 .
Combining the results above, and after dividing by √ s, we get

1 √ s (E k+1 -E k ) + √ µ[f (x k+1 ) -f (x ) + µ 2 x k -x 2 ] + √ µ(f (x k ) -f (x k+1 )) ≤ - µ √ s x k+1 -x k , x k -x - √ µ s x k+1 -x k 2 -β µ s ∇f (x k ), x k+1 -x k + 1 √ s (L - µ 2 ) x k+1 -x k 2 - µ 2 √ s x k+1 -x k 2 -β + 1 2 √ s ∇f (x k 2 - √ µ ∇f (x k ), x k+1 -x k . Let us make appear E k 1 √ s (E k+1 -E k ) + √ µE k+1 ≤ √ µ ∇f (x k ), x k+1 -x k + √ µ L 2 x k+1 -x k 2 + √ µ 2 1 √ s (x k+1 -x k ) + β∇f (x k ) 2 + µ x k -x , 1 √ s (x k+1 -x k ) + β∇f (x k ) - µ √ s x k+1 -x k , x k -x - √ µ s x k+1 -x k 2 -β µ s ∇f (x k ), x k+1 -x k + 1 √ s (L - µ 2 ) x k+1 -x k 2 - µ 2 √ s x k+1 -x k 2 -β + 1 2 √ s ∇f (x k 2 - √ µ ∇f (x k ), x k+1 -x k .
After developing and simplification, we get

1 √ s (E k+1 -E k ) + √ µE k+1 ≤ - √ µ 2s + µ √ s -L 1 √ s + √ µ 2 x k+1 -x k 2 -β - β 2 √ µ 2 + √ s 2 ∇f (x k+1 ) 2 + βµ ∇f (x k ), x k -x .
Let us majorize this last term by using the Lipschitz continuity of ∇f ∇f

(x k ), x k -x = ∇f (x k ) -∇f (x ), x k -x ≤ L x k -x 2 ≤ 2L x k+1 -x 2 + 2L x k+1 -x k 2 . Therefore 1 √ s (E k+1 -E k ) + √ µE k+1 + √ µ 2s + µ √ s -L 2βµ + 1 √ s + √ µ 2 x k+1 -x k 2 + β - β 2 √ µ 2 + √ s 2 ∇f (x k+1 ) 2 -2βµL x k+1 -x 2 ≤ 0.
According to 0 ≤ β ≤ 1 √ µ , we have β -

β 2 √ µ 2 ≥ β 2 , which gives 1 √ s (E k+1 -E k ) + √ µE k+1 + √ µ 2s + µ √ s -L 2βµ + 1 √ s + √ µ 2 x k+1 -x k 2 + β 2 ∇f (x k+1 ) 2 -2βµL x k+1 -x 2 ≤ 0.
Let us use again the strong convexity of f to write

E k+1 ≥ 1 2 E k+1 + 1 2 (f (x k+1 ) -f (x )) ≥ 1 2 E k+1 + µ 4 x k+1 -x 2 .
Combining the two above relations we get Set q = 1 1+ 1 2 √ µs , which satisfies 0 < q < 1. By a similar argument as in Theorem 5.1 where λ i > 0 is an eigenvalue of A. In the following, we drop the subscript i. Using the relationship between the Whitaker functions and the Kummer's confluent hypergeometric functions M and U , see [START_REF] Bateman | Higher transcendental functions[END_REF], the solution to [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF] can be shown to take the form It is clear that this is a special case of [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF]. Since β and λ > 0, set

1 √ s (E k+1 -E k ) + 1 2 √ µE k+1 + √ µ µ 4 -2βµL x k+1 -x 2 + √ µ 2s + µ √ s -L 2βµ + 1 √ s + √ µ 2 x k+1 -x k 2 + β 2 ∇f (x k+1
E k ≤ E 1 q k-1 .
ξ = λ 1 + β , κ = - α + β -c 1 + β , σ = α + β 2(1 + β) - 1 2 .
It follows from the first case above that

x(t) = ξ σ+1/2 e -λτ 1+β c 1 M σ -κ + 1/2, α + β 1 + β , ξτ + c 2 U σ -κ + 1/2, α + β 1 + β , ξτ .
Asymptotic estimates can also be derived similarly to above. We omit the details for the sake of brevity.

Figure 1 .

 1 Figure 1. Evolution of the objective (left) and trajectories (right) for (AVD) α (α = 3.1) and (DIN-AVD) α,β (α = 3.1, β = 1) on an ill-conditioned quadratic problem in R 2 .

Figure 2 .

 2 Figure 2. Convergence of the objective values and trajectories associated with the system (DIN-AVD) α,β,b for different choices of β(t) and b(t).

Case 4 .

 4 Let us illustrate our results in the case b

) 2 ≤ 0 2 √

 02 Let us examine the sign of the above quantities: Under the condition L ≤ -E k ) + 1 µE k+1 + β 2 ∇f (x k+1 ) 2 ≤ 0.

2 -= argmin z∈R n 1 2s z -x 2 -A. 3 .

 223 A(x -z), Ax -y + g(z) z -x, A * (y -Ax) + g(z)= argmin z∈R n 1 2s z -(x -sA * (Ax -y)) 2 + g(z)= prox sg (x -sA * (Ax -y)) . Closed-form solutions of (DIN-AVD) α,β,b for quadratic functions. We here provide the closed form solutions to (DIN-AVD) α,β,b for the quadratic objective f : R n → Ax, x , where A is a symmetric positive definite matrix. The case of a semidefinite positive matrix A can be treated similarly by restricting the analysis to ker(A) . Projecting (DIN-AVD) α,β,b on the eigenspace of A, one has to solve n independent one-dimensional ODEs of the form ẍi (t) + α t + β(t)λ i ẋi (t) + λ i b(t)x i (t) = 0, i = 1, . . . , n.

•

  Case β(t) ≡ β, b(t) = b + γ/t, β ≥ 0, b > 0, γ ≥ 0: The ODE reads If β 2 λ 2 = 4bλ: set ξ = β 2 λ 2 -4bλ, κ = λ γ -αβ/2 ξ , σ = (α -1)/2.

x 2 t 1 β+1.

 21 (t) = ξ α/2 e -(βλ+ξ)t/2 [c 1 M (α/2 -κ, α, ξt) + c 2 U (α/2 -κ, α, ξt)] ,where c 1 and c 2 are constants given by the initial conditions.• If β 2 λ 2 = 4bλ: set ζ = 2 λ (γ -αβ/2). The solution to[START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF] takes the formx(t) = t -(α-1)/2 e -βλt/2 c 1 J (α-1)/2 (ζ √ t) + c 2 Y (α-1)/2 (ζ √ t) ,where J ν and Y ν are the Bessel functions of the first and second kind.When β > 0, one can clearly see the exponential decrease forced by the Hessian. From the asymptotic expansions of M , U , J ν and Y ν for large t, straightforward computations provide the behaviour of |x(t)| for large t as follows:• If β 2 λ 2 > 4bλ, we have |x(t)| = O t -α 2 +|κ| e -βλ-ξ = O e -2b β t-( α 2 -|κ|) log(t) .• If β 2 λ 2 < 4bλ, whence ξ ∈ iR + * and κ ∈ iR, we have|x(t)| = O t -α 2 e -βλ 2 t . • If β 2 λ 2 = 4bλ, we have |x(t)| = O t -2α-1 4 e -βλ 2 t . Case β(t) = t β , b(t) = ct β-1 , β ≥ 0, c > 0: The ODE reads now ẍ(t) + α t + t β λ ẋ(t) + cλt β-1 x(t) = 0.Let us make the change of variable t := τ Let y(τ ) := x τ 1 β+1. By the standard derivation chain rule, it is straightforward to show that y obeys the ODE ÿ

  Take α ≥ 1. Let x : [t 0 , +∞[→ H be a solution trajectory of (DIN-AVD) α,β,b . Suppose that the following growth conditions are satisfied:

	Theorem 2.1.
	1)	w(t) := b(t) -β(t) -	β(t) t	and δ(t) := t 2 w(t).

  1 λ x -prox λf (x) , and prox θf λ (x) = λ λ+θ x + θ λ+θ prox (λ+θ)f (x), (IPAHD-SC) becomes with θ = β

	√ √ µ s+s 1+2 1+λµ s and a =	2 1+2 √ µ 1+λµ s √ µ 1+λµ s :
	(IPAHD-NS-SC)

Let us formulate A with E(t).

After developing and simplifying, we obtain

Let us use again the strong convexity of f to write

By combining the two inequalities above, we obtain

Elementary algebraic computation gives that, under the condition 0

By integrating the differential inequality above we obtain

By definition of E(t), we infer

. Developing the above expression, we obtain

By convexity of f we have

Combining the above results, we obtain

According to the definition of E k ≥ f (x k ) -f (x ), we finally obtain

and the linear convergence of x k to x and that of the gradients to zero.

Numerical results

Here, we illustrate our results on the composite problem on

where A is a linear operator from R n to R m , m ≤ n, g : R n → R ∪ {+∞} is a proper lsc convex function which acts as a regularizer. Problem (RLS) is extremely popular in a variety of fields ranging from inverse problems in signal/image processing, to machine learning and statistics. Typical examples of g include the 1 norm (Lasso), the 1 -2 norm (group Lasso), the total variation, or the nuclear norm (the 1 norm of the singular values of x ∈ R N ×N identified with a vector in R n with n = N 2 ). To avoid trivialities, we assume that the set of minimizers of (RLS) is non-empty. Though (RLS) is a composite non-smooth problem, it fits perfectly well into our framework. Indeed, the key idea is to appropriately choose the metric. For a symmetric positive definite matrix S ∈ R n×n , denote the scalar product in the metric S as S•, • and the corresponding norm as • S . When S = I, then we simply use the shorthand notation for the Euclidean scalar product •, • and norm • . For a proper convex lsc function h, we denote h S and prox S h its Moreau envelope and proximal mapping in the metric S, i.e.

Similarly, when S = I, we drop S in the above notation. Let M = s -1 I -A * A. With the proviso that 0 < s A 2 < 1, M is a symmetric positive definite matrix. It can be easily shown (we provide a proof in Appendix A.2 for completeness; see also the discussion in [START_REF] Chambolle | An introduction to continuous optimization for imaging[END_REF]Section 4.6]), that the proximal mapping of f as defined in (RLS) in the metric M is [START_REF] Chambolle | An introduction to continuous optimization for imaging[END_REF] prox M f (x) = prox sg (x + sA * (y -Ax)), which is nothing but the forward-backward fixed-point operator for the objective in (RLS). Moreover, f M is a continuously differentiable convex function whose gradient (again in the metric M ) is given by the standard identity ∇f M (x) = x -prox M f (x), and ∇f M (x) -∇f M (z) M ≤ x -z M , i.e. ∇f M is Lipschitz continuous in the metric M . In addition, a standard argument shows that argmin(f ) = Fix(prox M f ) = argmin(f M ). We are then in position to solve (RLS) by simply applying (IGAHD) (see Section 3.2) to f M . We infer from Theorem 3.3 and properties of f M that

). (IGAHD) and FISTA (i.e. (IGAHD) with β = 0) were applied to f M with four instances of g: 1 norm, 1 -2 norm, the total variation, and the nuclear norm. The results are depicted in Figure 3. One can clearly see that the convergence profiles observed for both algorithms agree with the predicted rate. Moreover, (IGAHD) exhibits, as expected, less oscillations than FISTA, and eventually converges faster.

Conclusion, Perspectives

As a guideline to our study, the inertial dynamics with Hessian driven damping give rise to a new class of first-order algorithms for convex optimization. While retaining the fast convergence of the function values reminiscent of the Nesterov accelerated algorithm, they benefit from additional favorable properties among which the most important are:

• fast convergence of gradients towards zero;

• global convergence of the iterates to optimal solutions;

• extension to the non-smooth setting;

• acceleration via time scaling factors. This article contains the core of our study with a particular focus on the gradient and proximal methods. The results thus obtained pave the way to new research avenues. For instance:

• as initiated in Section 6, apply these results to structured composite optimization problems beyond (RLS) and develop corresponding splitting algorithms; • with the additional gradient estimates, we can expect the restart method to work better with the presence of the Hessian damping term; • deepen the link between our study and the Newton and Levenberg-Marquardt dynamics and algorithms (e.g., [START_REF] Attouch | A continuous dynamical Newton-Like approach to solving monotone inclusions[END_REF]), and with the Ravine method [START_REF] Gelfand | Printszip nelokalnogo poiska v sistemah avtomatich, Optimizatsii[END_REF].

• the inertial dynamic with Hessian driven damping goes well with tame analysis and Kurdyka-Lojasiewicz property [START_REF] Álvarez | A second-order gradient-like dissipative dynamical system with Hessiandriven damping. Application to optimization and mechanics[END_REF], suggesting that the corresponding algorithms be developed in a non-convex (or even non-smooth) setting. Then for all (x, y) ∈ H 2 , we have

Proof. Denote y + = y -s∇f (y). By the standard descent lemma applied to y + and y, and since sL ≤ 1 we have ( 24)

We now argue by duality between strong convexity and Lipschitz continuity of the gradient of a convex function. Indeed, using Fenchel identity, we have f (y) = ∇f (y), y -f * (∇f (y)).

L-Lipschitz continuity of the gradient of f is equivalent to 1/L-strong convexity of its conjugate f * . This together with the fact that (∇f ) -1 = ∂f * gives for all (x, y) ∈ H 2 ,

Inserting this inequality into the Fenchel identity above yields

Inserting the last bound into (24) completes the proof.

A.2. Proof of [START_REF] Chambolle | An introduction to continuous optimization for imaging[END_REF].

Proof. We have prox M f (x) = argmin z∈R n


Méline Wery
email: meline.wery@irisa.fr

Emmanuelle Becker

Franck Auge

Charles Bettembourg

Olivier Dameron

Anne Siegel

Identification of causal signature using omics data integration and network reasoning-based analysis

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

Introduction

Identifying a pathological signature for a complex disease remains a challenge.

The actual definition of a signature in the context of a disease is :

• a set of features able to separate two populations • based on statistical test Features can be further used as candidate therapeutic targets.

Limits of current approaches

1.1

  Limited to one omic layer 2. Require a clear stratification of population 3. Identified features = mix causes, consequences and noise 4. Limited to the most striking effects ⇒ regulation ? Objective : Pathological signature based on individual features ⇒ Individualize the identification of signature based on patient specific information (Polymorphism + Gene expression) ⇒ Integrate multi-omics data with prior knowledge on interaction network Conclusion & Perspectives Diagnosis signature from literature is insufficiant for complex disease Identify a signature by taking into account the dependencies of regulation Causal signature ⇒ Minimal set of MIS between patients ⇒ New stratification of patients ⇒ Enrich signature with clinical criteria ⇒ Propose candidate therapeutic targets One-by-one patient approach Analysis of each gene expression < Interval of expression for each gene in control population < [ Blavy, P. et al (2014) BMC Systems Biology 2 Samaga, R. et al (2010) Journal of Computational Biology 3 Videla, S. et al.. (2017). Bioinformatics eQTL Filter and discretization over gene expression Value in { 1 ; -1 } Caspo 3