Guillaume Leloup
email: guillaume.leloup@univ-rennes.fr

Roger Lewandowski
email: roger.lewandowski@univ-rennes1.fr

Cherif Amrouche
email: cherif.amrouche@univ-pau.fr

TKE model involving the distance to the wall. Part 1: the relaxed case

Keywords: Fluid mechanics, Turbulence models, degenerate operators, Navier-Stokes Equations, Turbulent Kinetic Energy. 2010 MSC: 76D05, 35Q30, 76F65, 76D03, 35Q30

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

Introduction

This paper is dedicated to the analysis of a stationary coupled PDE system (see (1.4) below) that arises in incompressible turbulence modeling. The modeling of turbulence in fluid mechanics involves decomposing the velocity and pressure fields into mean fields, denoted as u and p in this work. These mean fields are assumed to be stationary, representing long-time averages. The fluctuations are denoted as u and p . By averaging the Navier-Stokes equations, we obtain the Reynolds stress tensor R, representing the long-time average of the tensor product u ⊗ u [START_REF] Chacòn-Rebollo | Mathematical and Numerical Foundations of Turbulence Models and Applications[END_REF][START_REF] Mohammadi | Analysis of the k-epsilon turbulence model[END_REF][START_REF] Pope | Turbulent flows[END_REF]. It is crucial to model this Reynolds tensor, and it is widely accepted to be dissipative [START_REF] Berselli | On the reynolds time-averaged equations and the long-time behavior of leray-hopf weak solutions, with applications to ensemble averages[END_REF][START_REF] Foias | Encyclopedia of Mathematics and its Applications[END_REF]. According to the Boussinesq hypothesis, it is made proportional to the average strain tensor via a turbulent viscosity denoted as ν t . The question then arises as to whether the molecular diffusion term is negligible compared to the turbulent diffusion term. Various approaches exist for modeling ν t , either within the framework of statistical averages (RANS-type models) or by filtering the high frequencies of the flow (LES-type models). In this article, we focus on a RANS model, while the case of LES models is studied in [START_REF] Legeais | Solutions obtained by approximations of a degenerated smagorinsky's model[END_REF]. Generally speaking, in RANS models, turbulent viscosity is a function of turbulent kinetic energy, k (TKE), and turbulent dissipation . This can be simplified by relating TKE k to through the Prandtl mixing length, denoted by , by the dimensional law = k|k|1/2 , yielding the following law for the eddy viscosity:

(1.1)

ν t = C √ k,
where C is a dimensionless constant (and k ≥ 0). After a modeling process [START_REF] Chacòn-Rebollo | Mathematical and Numerical Foundations of Turbulence Models and Applications[END_REF][START_REF] Mohammadi | Analysis of the k-epsilon turbulence model[END_REF], one obtains the following equation for k in the steady state case, named TKE model:

(1.2) u • ∇k -∇ • (µ t (k)∇k) + k|k| 1/2 = ν t (k)|Du| 2 ,
where u is the fluid velocity, Du = 1 2 (∇u + (∇u) T ) is the strain tensor, µ t is an eddy diffusion similar to the eddy viscosity given by (1.1). Assuming that is a smooth bounded function such that (x) ≥ 0 > 0 in the fluid domain Ω, the coupling of the resulting steady-state Navier-Stokes equation with eddy diffusion to the TKE model (1.2) has been extensively studied [START_REF] Bernardi | Automatic insertion of a turbulence model in the finite element discretization of the Navier-Stokes equations[END_REF][START_REF] Chacòn-Rebollo | Mathematical and Numerical Foundations of Turbulence Models and Applications[END_REF][START_REF] Gallouët | On a turbulent system with unbounded eddy viscosities[END_REF][START_REF] Lederer | A RANS 3D model with unbounded eddy viscosities[END_REF][START_REF] Lewandowski | The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity[END_REF][START_REF] Lewandowski | On a one dimensional turbulent boundary layer model[END_REF]. In most cases, we assume that ν t is bounded below and above. This assumption can be implemented by replacing C √ k with a truncation function ν defined on the interval [1/M, M ], where ν is equal to C √ k, and on [0, 1/M ], where it is equal to C/ √ M , and on [M, ∞[, where it is equal to C √ M , for some M > 0. In this paper, we remain in this framework and focus on the problem posed by the Prandtl mixing length . In many physical applications, such as meteorology or oceanography [START_REF] Lewandowski | Analyse mathématique et océanographie[END_REF][START_REF] Monin | Basic laws of turbulent mixing in the surface layer of the atmosphere[END_REF], = (x) vanishes at the boundary, equivalent to the distance to the boundary Γ, denoted d(x, Γ), which poses a major difficulty. Notice that it may also be of order √ d, as in the case of shear flows [START_REF] Kean | On the Prandtl-Kolmogorov 1-equation model of turbulence[END_REF]. Throughout the paper, we consider the case (x) ≈ d(x, Γ) as x → Γ. A similar problem was already considered in the case of some LES models [START_REF] Berselli | On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations[END_REF][START_REF] Berselli | On the existence of weak solutions for an unsteady rotational Smagorinsky model[END_REF][START_REF] Rappaz | On non-linear Stokes problems with viscosity depending on the distance to the wall[END_REF][START_REF] Rappaz | On some weighted Stokes problems: applications on Smagorinsky models[END_REF], where the eddy viscosity is proportional to 2 (instead of as in the TKE model), leading to challenging open problems. In these papers, the authors suggest relaxing this law and taking an eddy viscosity proportional to α , for some 0 < α < 2. The same situation occurs in the case of RANS problems. Therefore, in this paper, we aim to relax the law given by (1.1) by taking

(1.3) ν t (k) = η ν(k),
for 0 < η < 1, and we study the coupling of the resulting steady-state Navier-Stokes equation to (1.2), within the scope of the viscous approximations posed by the question of whether or not we can neglect the molecular viscosity in comparison with the turbulent viscosity.

To be more specific, let Ω be a C 0,1 bounded open subset of R N , where N = 2, 3, 0 < η < 1, 0 < β < 1, and 0 < λ. As mentioned earlier, : Ω → R * + is a C ∞ function equivalent to d(x, Γ) near Γ (see the precise assumption in Section 2.1). We consider the following stationnay system, where p denotes the pressure 12 :

(1.4)

               λu + (u • ∇)u -div( η ν(k)∇u) + ∇p = f in Ω, div u = 0 in Ω, u • ∇k -div( β µ(k)∇k) + k|k| 1/2 = η ν(k)|∇u| 2 in Ω, k ≥ 0 in Ω, u = k = 0 on Γ.
The term λu is a stabilization term, which can be seen as a discrete evolution term coming from a finite difference and plays an important role in the proofs of the main results. We next introduce the approximated system obtained by viscous regularization, where ε > 0,

(1.5)

               λu + (u • ∇)u -div( η ν(k)∇u) -ε∆u + ∇p = f in Ω, div u = 0 in Ω, u • ∇k -div( β µ(k)∇k) -ε∆k + k|k| 1/2 + ε = η ν(k)|∇u| 2 in Ω, k ≥ 0 in Ω, u = k = 0 on Γ.
For this system, we know from [START_REF] Lewandowski | The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity[END_REF] the existence of a weak solution in an appropriate sense. We transcribe the question "Is the molecular diffusion term negligible compared to the turbulent diffusion term?" by "Do the solutions of the system (1.5) converge towards a solution -in some weak sense-of the system (1.4)" when ε goes to 0? As we will see, the situation is seriously complicated. We only get a positive answer at the cost of severe restrictions on the exponents η and β, i.e., when 0 < η < 1 2 and 0 < β < 1 3 in dimension 2, and 0 < η, β < 1 5 in dimension 3. A specific statement is given by Theorem 4.2 below, which is the main result of this paper. The paper is structured into three main sections and organized as follows. In the first section, we focus on the study of the steady Navier-Stokes equations with an eddy viscosity, as given in (1.3), for a specified k. To achieve this, we meticulously define the function space in which we seek solutions. We employ the natural weighted Sobolev space of divergencefree vector fields, deduced from the energy balance. Utilizing results from Kufner [START_REF] Kufner | Weighted Sobolev spaces[END_REF], we embed this space into W 1,p for 1 ≤ p < p η = 2 1+η , and subsequently into H 1-η/2 through an interpolation argument, following Amrouche-Moussaoui [START_REF] Amrouche | The dirichlet and neumann problems in lipschitz and in C 1,1 domains[END_REF]. Once the functional framework is established, we demonstrate the existence of a weak solution (see Definition 2.1 below), obtained as a limit of the viscous regularization (SOLA3 ) by introducing the term -ε∆u, for 0 < ε. However, for the coupling with the TKE equation, strong convergence of the approximations in our weighted space is necessary, and the obtained (SOLA) must satisfy the energy balance. This requirement holds only if 0 < η < 1/2 when N = 2 and 0 < η < 1/5 when N = 3. In the second section, our focus shifts to the TKE equation for a fixed vector field u, especially in deriving a priori estimates in regular Sobolev spaces. This task is not straightforward due to the presence of the right-hand side in L 1 and the degeneracy of the diffusive term at the boundary. To address this, we continue to use a weighted Sobolev space and adapt the ladder method initially introduced by Boccardo-Gallouët [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]. This adaptation stands as one of the principal contributions of this paper. The third section synthesizes the findings from the previous two sections. Here, we successfully take the limit in the viscous regularizations (1.5) as ε tends to 0 and establish the main theorem, namely Theorem 4.2. This also verifies the existence of a weak solution to system (1.4), namely a (SOLA). It is important to note that the equation for TKE only holds in the sense of distributions, and achieving better results would require further reductions in the size of the β parameter.

2 Velocity equation with a fixed TKE

General setting and space function

Let Ω be a C 0,1 bounded open subset of IR N (N = 2, 3), Γ its boundary, : Ω → IR + a C ∞ function equivalent to the distance to the boundary, which satisfies

lim d(x,Γ)→0 x∈Ω (x) d(x, Γ) = 1, (2.1) ∀ n > 0, n = inf d(x,Γ)≥ 1 n x∈Ω (x) > 0 (2.2)
The functions ν, µ : IR → IR are continous, bounded by above and by below, namely

(2.3) ∀ k ∈ IR, 0 < ν m ≤ ν(k), µ(k) ≤ ν M ,
We start by studying the velocity equation for a fixed TKE function k ∈ L 1 (Ω), and a source term f ∈ L 2 (Ω) N . The corresponding Navier-Stokes equations are as follows:

(2.4)

   λu + (u • ∇)u -div ( η ν(k)∇u) + ∇p = f in Ω, div u = 0 in Ω, u = 0 at Γ.
To obtain an a priori estimate, we formally take the dot product of the momentum equation with u and integrate by parts. A standard calculation, utilizing the no-slip boundary condition and incompressibility, yields ((u • ∇)u, u) = 0 and (∇p, u) = 0, resulting in the energy balance:

(2.5)

λ Ω |u| 2 + Ω η ν(k)|∇u| 2 = f , u .
Consequently, as f ∈ L 2 (Ω) N , we deduce from (2.3) the a priori estimate:

(2.6)

Ω |u| 2 + Ω η |∇u| 2 ≤ C = f 2 0,2;Ω 2λ min(λ/2, ν m ) .
This suggests to introduce the space fonction V η defined as the closure of V for the norm

(2.7) u 1,2; ,η = u 2 0,2,Ω + η/2 ∇u 2 0,2,Ω 1/2
, where (2.8)

V = {u ∈ D(Ω) N / div u = 0}.
In particular, V η is a Hilbert space with the scalar product

(2.9) (u, v) 1,2; ,η = Ω u • v + Ω η ∇u : ∇v.
The norm given by (2.7) corresponds to the H 1 -norm when η = 0. Consequently, the V η functions are not likely to be in H 1 . The following result from [START_REF] Amrouche | The dirichlet and neumann problems in lipschitz and in C 1,1 domains[END_REF] gives H s -regularity for 1/2 < s ≤ 1.

Theorem 2.1. Let η ∈]0; 1[, the continuous embedding

(2.10) V η → H 1-η 2 (Ω) N stands.
Consequently, the Sobolev embedding theorem yields the compact embedding

(2.11) V η → L r (Ω) N ,
for all

(2.12) r < r η := 2N N + η -2 =      4 η if N = 2, 6 1 + η if N = 3.
We present the proof of Theorem 2.1 in Appendix A for the sake of self-containment. However, this result does not guarantee that gradients of functions in V η belong to certain L q spaces, which, nevertheless, is necessary. To address this, we will rely on a result from [14, Proposition 6.5], as stated below.

Theorem 2.2. Assume that Ω satisfies the cone property, and let η ∈ ]0, 1[. For any p such that

(2.13) 1 ≤ p < 2 1 + η =: p η ,
the following continuous embedding holds

(2.14) V η → W 1,p 0 (Ω) N .
Remark 2.1. We deduce from Theorem 2.2 that vector fields in V η have a trace at Γ which is equal to 0, giving a sense to the no slip boundary condition for vector fields in V η .

Remark 2.2. The critical exponent p η is given by

(2.15) q < p η := 2N N (1 + η) -2 =      2 η if N = 2, 6 1 + 3η if N = 3.
Therefore, Theorem 2.1 establishes a superior Lebesgue regularity for vector fields in V η compared to Theorem 2.2. Specifically, we have V η → L q (Ω) N for a given q > 2, where N = 2, 3.

Remark 2.3. The following Poincaré inequality holds, for all p < p η and all u ∈ V η ,

(2.16) u 0,p,Ω ≤ C u 1,2; ,η , where C = C(p, Ω).
Throughout the rest of the paper, we assume that Ω satisfies the cone property, in order to verify (2.14).

Remark 2.4. Let η ∈]0; 1[. The space V η is a distributional space, and in particular we conjecture that

(2.17)

V η = u ∈ D (Ω) N , div u = 0, u ∈ L 2 (Ω) N , η/2 ∇u ∈ L 2 (Ω) N ×N , u| Γ = 0 .
Unfortunetely, we were not able to prove this result, which is open.

We also denote in what follows

(2.18) V 0 = {u ∈ H 1 0 (Ω) N ; div u = 0}.
This space is well-defined, and V is dense in it.

Weak solutions and existence result

We begin by presenting the weak formulation of Problem (2.4), excluding the pressure as it involves free divergence vector fields as test functions. However, it is essential to account for the transport term, which gets as div u = 0,

(u • ∇)u = div (u ⊗ u),
and which formally acts in the sense of the distributions as

(2.19) ∀ ϕ ∈ D (Ω)N, div (u ⊗ u, ϕ = - Ω u ⊗ u : ∇ϕ T .
We deduce from (2.12) that for all 0 < η < 1, r η > 2. Therefore,

∀ u ∈ V η , ∀ q ∈ ]2; r η [, u ⊗ u ∈ L q/2 (Ω) N ×N .
Hence, for any δ ∈ [2; r η [ we are led to take tests v such that

(2.20) v ∈ δ<q<r η V η ∩ W 1,(q/2) 0 (Ω) N =: H η,δ In particular, when v ∈ H η,δ , u ⊗ u : ∇v ∈ L 1 (Ω), as well as λu • v + η ν(k)∇u : ∇v ∈ L 1 (Ω), since ν(k) ∈ L ∞ (Ω), and f • v ∈ L 1 (Ω).
Unfortunately, in view of the construction of weak solutions to Problem (2.4), we must reduce the space H η,δ a little bit for technical reasons that will be cleared in the following. Let r > 1, and let us consider V given by (2.8)), equiped with the norm v η,r = v 1,2; ,η + v 1,r,Ω , and let K η,r denotes the closure of V for the norm • η,r . Notice that we naturally have

(2.21) K η,r → V η .
The tests vector fields set we consider is given by

(2.22) W η,δ = δ<q<r η K η,(q/2) ,
which is a subset of H η,δ .

Remark 2.5. We do not know wether, in general, W η,δ = H η,δ . However, as we shall see in the following section, we have W η,δ = V η for 0 < η < η c and δ 0 < δ < r η .

The fact that the set of vector field tests is labeled by a parameter δ implies that the notion of a weak solution depends on this parameter. This suggests the following definition of a weak solution.

Definition 2.1. We say that u

∈ V η is a δ-weak solution to Problem (2.4) if ∀ v ∈ W η,δ , (2.23) λ Ω u • v - Ω (u ⊗ u) : (∇v) T + Ω η ν(k)∇u : ∇v = Ω f • v. Remark 2.6. Let 2 ≤ δ 1 < δ 2 < r η . Then obvioulsy W η,δ 2 ⊂ W η,δ 1 .
Therefore any δ 1 -weak solution is a δ 2 -weak solution. However, it is not clear that a δ 2 -weak solution is a δ 1 -weak solution.

Theorem 2.3. Let η ∈ ]0; 1[, k ∈ L 1 (Ω), f ∈ L 2 (Ω) N , δ ∈ [2; r η [. Then Problem (2.4
) has a δ-weak solution obtained by approximation (SOLA).

Proof. We argue using viscous regularization, with the proof being divided into three steps. First, we introduce the viscous regularization we consider, and we obtain a uniform estimate in V η for the resulting equations. Then we extract subsequences by weak compactness and Theorem 2.2. Finally, we pass to the limit as the viscous parameter ε goes to 0.

Step 1. Approximations and estimate. We argue by singular perturbations. Let ε > 0.

According to standard results (see for instance in [START_REF] Chacòn-Rebollo | Mathematical and Numerical Foundations of Turbulence Models and Applications[END_REF]), we know the existence of u ε ∈ V 0 such that for any v ∈ V 0

(2.24) λ Ω u ε • v - Ω (u ε ⊗ u ε ) : (∇v) T + Ω η ν(k)∇u ε : ∇v + ε Ω ∇u ε : ∇v = Ω f • v.
Taking v = u ε as test, we get the following energy balance, since the transport term vanishes,

(2.25) λ Ω |u ε | 2 + Ω η ν(k)|∇u ε | 2 + ε Ω |∇u ε | 2 = Ω f • u ε .
In particular, as f ∈ L 2 (Ω) N , the familly (u ε ) ε>0 is bounded in V η .

Step 2. Extracting subsequences. From the bound in V η , we deduce the existence of a sequence (ε n ) n∈I N and u ∈ V η such that:

1) The sequence (u εn ) n∈I N weakly converges to u in V η ,

2) The sequence (u εn ) n∈I N weakly converges to u in

H 1-η 2 (Ω) N ,
3) The sequence (u εn ) n∈I N strongly converges to u in L r (Ω) N for all q < r η , where r η is given by (2.12),

4) The sequence (∇u εn ) n∈I N weakly converges to ∇u in W 1,p 0 (Ω) N ×N , for all p < p η .

Item 1) is straightforward. Items 2) 3) and 4) require additional comments. Indeed, let (u εn ) n∈I N which weakly converges to u in V η , as given by item 1. As the sequence (u εn ) n∈I N is bounded in H 1-η 2 (Ω) N by Theorem 2.1, we can extract a subsequence, still denoted (u εn ) n∈I N , which weakly converges to some v ∈ H 1-η 2 (Ω), and by the way in D (Ω). But as u εn → u in V η -weak, by (2.2) convergence in D (Ω) also holds. Therefore by the uniqueness of the limit, u = v. Items 3) and 4) stand by a similar argument.

Step 3. Passing to the limit in the equations. Let us take v ∈ V as test in (2.24). It is easily checked that

λ Ω u εn • v + Ω η ν(k)∇u εn : ∇v -→ n→+∞ λ Ω u • v + Ω η ν(k)∇u : ∇v, (2.26) Ω (u εn ⊗ u εn ) : (∇v) T -→ n→+∞ Ω (u ⊗ u) : (∇v) T . (2.27)
It remains to deal with the term ε n Ω ∇u εn : ∇v. As ∇v ∈ L ∞ (Ω), and in particular ∇v ∈ L q (Ω) N ×N for any q > (p η ) . Therefore by item 4),

Ω ∇u εn : ∇v -→ n→+∞ Ω ∇u : ∇v, hence ε n Ω ∇u εn : ∇v -→ n→+∞ 0.
In conclusion, u satisfies (2.23) for any v ∈ V. Let v ∈ W η,δ . There exists q ∈ ]δ; r η [ such that v ∈ K η,(q/2) . It follows the existence of a sequence (v n ) n∈I N of vector fields in V which converges to v in both V η and W 1,(q/2) 0

(Ω) N , which explains the reason of the special construction of W η,δ . Taking v n as test in (2.23), and passing to the limit when n → +∞ yields u satisfies (2.23) for any v ∈ W η,δ .

Remark 2.7. By De Rham Theorem, we know the existence of a pressure p ∈ L 2 loc (Ω) such that (2.4) holds in D (Ω).

Energy balance and strong convergence

In this section, we address the question of the strong convergence in V η of the approximations to Problem (2.23) constructed earlier. Our initial objective is to determine whether a δ-weak solution satisfies the energy balance for a suitable choice of δ and η. The following results hold.

Lemma 2.1. Let η ∈ ]0; η c [, where (2.28) η c = 1/2 if N = 2, 1/5 if N = 3.
Then there exists δ 0 ∈ ]2; r η [ such that for all δ ∈ ]δ 0 ; r η [, any δ-weak solution u of Problem (2.23) satisfies the energy balance (2.5).

Proof. The energy balance holds when we take u itself as test vector field. This is the case when

(2.29) W η,δ = V η , knowing already by that W η,δ ⊂ V η . According to Theorem 2.2, a function v ∈ V η satisfies ∇v ∈ L p (Ω) N ×N for p ∈ [1; p η [. Therefore, (2.29) is satisfied if r η 2 < p η ,
which yields the bound (2.28) after elementary calculations. Moreover, as [p → p ] is non-increasing and continuous, there exists δ 0 ∈]2; r η [, such that

r η 2 < δ 0 2 < p η .
Therefore, for δ 0 < q < r η we have q

2

< p η and we deduce from Theorem 2.2,

V η ⊂ W (q/2) 0 (Ω) N .
Therefore, by (2.2) v η,(q/2) ,Ω ≤ C v 1,2; ,η , which means V η → K η,(q/2) , hence V η = K η,(q/2) by (2.21), leading to (2.29).

Remark 2.8. After some calculus, we find for optimal value of δ 0 :

(2.30)

     δ 0 = 4 2 -η when N = 2, δ 0 = 12 5 -3η when N = 3.
We are now able to prove the following convergence result.

Theorem 2.4. Let η ∈ ]0; η c [, δ ∈ ]δ 0 ; r η [, and let (k ε ) ε>0 be a family in L 1 (Ω) which converges in L 1 (Ω) to some k. Let u ε ∈ V 0 such that for any v ∈ V 0

(2.31) λ Ω u ε • v - Ω (u ε ⊗ u ε ) : (∇v) T + Ω ( η + ε) ν(k ε )∇u ε : ∇v = Ω f • v.
Then there exists a sequence (ε n ) n∈I N such that the sequence (u εn ) n∈I N strongly converges in V η to a δ-weak solution u ∈ V η of Problem (2.4) which in addition satisfies the energy balance (2.5). Moreover, the sequence

( η ν(k εn )|∇u εn | 2 ) n∈I N converges strongly in L 1 (Ω) to η ν(k)|∇u| 2 .
Proof. We argue in two steps. First, following the proof of Theorem 2.3, we get the existence of a sequence (ε n ) n∈I N , such that the sequence (u εn ) n∈I N weakly converges in V η to a δ-weak solution u ∈ V η of Problem (2.4). Then we prove the strong convergence in V η by the energy method.

Step 1. The only addition to be made, is to check how to pass to the limit in the diffusion term. We first notice that the sequence (ε n ) n∈I N can be chosen such that the sequence (k εn ) n∈I N converges a.e. to k, by the inverse Lebesgue Theorem. Let v ∈ V, δ < q < r η . Then

| η/2 ν(k εn )∇v| ≤ ν M | η/2 ν(k εn )∇v| ∈ L 2 (Ω), by (2.
3), then the convergence in (2.33) also holds in L 2 (Ω) N ×N , by Lebesgue Theorem. In conclusion

Ω η ν(k εn )∇u εn : ∇v -→ n→+∞ Ω η ν(k)∇u : ∇v.
The rest is as in the proof of Theorem 2.3, the energy balance being satisfied by Lemma (2.28).

Step 2. It remains to prove the strong convergence. To do so we use the so called "energy method". We deduce from (2.32) and the same argument as above that

(2.34) η/2 ν(k εn )∇u εn -→ n→+∞ η/2 ν(k)∇u weakly in L 2 (Ω) N ×N .
Therefore,

(2.35)

Ω η ν(k)|∇u| 2 ≤ lim inf n→∞ Ω η ν(k εn )|∇u εn | 2 .
Now, we write the energy balances satisfied by u and u εn :

(2.36)

     λ Ω |u εn | 2 + Ω η ν(k εn )|∇u εn | 2 + ε n Ω |∇u εn | 2 = f , u εn λ Ω |u| 2 + Ω η ν(k)|∇u| 2 = f , u . As (u εn ) n∈I N converges to u strongly in L 2 (Ω) N , lim n→∞ Ω |u εn | 2 = Ω |u| 2 , lim n→∞ f , u εn = f , u .
Therefore, by (2.36), we have

(2.37) lim n→∞ Ω η ν(k εn )|∇u εn | 2 + ε n Ω |∇u εn | 2 = Ω η ν(k)|∇u|
η |∇u εn | 2 = Ω η |∇u| 2 ,
which combined with (2.32) and the L 2 -strong convergence of (u εn ) n∈I N to u, yields the strong convergence in V η .

TKE with a fixed velocity

In this section, we tackle a substantial challenge. We aim to obtain estimates in the spirit of Boccardo-Gallouët [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF] for the equation involving k. In addition to the typical difficulty arising from the source term in L 1 , we must confront the degeneracy of the diffusion coefficient at the boundary, introducing additional challenges.

Let β ∈ ]0; 1[, and D ∈ L 1 (Ω), 0 ≤ D a.e. in Ω.

We consider the following TKE equation, satisfied by k = k(x):

(3.1)    u • ∇k -div ( β µ(k)∇k) + k|k| 1/2 = D in Ω, k = 0 at Γ, Remark 3.1. Equation (3.1
) can be written as

u • ∇k -β µ(k)∆k -∇( β µ(k)) • ∇k + k|k| 1/2 = D.
Therefore as > 0 inside Ω, the strong maximum principle method used in [START_REF] Lewandowski | Existence and positivity results for the φ-θ and a modified k-two-equation turbulence models[END_REF] can be adapted to this case, so that any

C 2 -strong solution k to (3.1) is such that k > 0 in Ω.
Throughout this section, we assume Then for any

(3.3) β < 1 2N -1 = β c , q < q β = N (p β -1) = N (1 -β) 1 + β ,
where p β = 2 1+β , we have

(3.4) k 1,q;Ω + -1 k 3/2 0,1;Ω ≤ C = C(D, β).
Proof.

Step 1. A priori estimate over the sets {n ≤ k < n + 1}. Let n ∈ IN * , H n : IR → IR be the odd function given by (3.5)

H n : t →    0 if 0 ≤ t < n, t -n if n ≤ t < n + 1, 1 if t ≥ n + 1.
Notice that H n is continuous, piecewise C 1 , the derivative of which has a finite number of discontinuities and H(0) = 0. Therefore, Stampacchia's Theorem [START_REF] Stampacchia | Équations elliptiques du second ordre à coefficients discontinus[END_REF] applies, in particular, for any κ ∈ W 1,r 0 (Ω), H n (κ) ∈ W 1,r 0 (Ω), ∇H n (κ) = ∇κH n (κ) a.e and ∇κ = 0 over the level sets {κ = C} that are not of null measure.

Let k be a strong solution of (3.1). We take H n (k) as test function in (3.1) and integrate over Ω. Due to the boundary conditions, on one hand we have, (3.6)

Ω H n (k)u • ∇k = 0,
(see in [START_REF] Lewandowski | The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity[END_REF] for instance). On the other hand as H n is an odd function that satifies H n (t) ≥ 0 over IR + , then H n (k)k ≥ 0, which yields

(3.7) Ω H n (k)k|k| 1/2 ≥ 0.
According to Stampacchia Theorem, Stokes formula yields

(3.8) Ω β µ(k)∇k • ∇H n (k) ≤ Ω DH n (k) ≤ D, given that |H n | ≤ 1.
Let B n ⊂ Ω be defined for any n ∈ IN * by (recall that k > 0 in Ω),

B n = {x ∈ Ω / n ≤ k(x) < n + 1}.
We obtain by Stampacchia's Theorem

(3.9) ∇k • ∇H n (k) = |∇k| 2 1 Bn a.e. Ω. Inequality (3.8) yields, by (3.9), (3.10)
Bn

β |∇k| 2 ≤ D ν m .
The left-hand side of (3.8) can be transformed another way. Indeed, since H n (k) = 1 Bn a.e. Ω, we have

(3.11) ∇k • ∇H n (k) = |∇H n (k)| 2 1 Bn = |∇H n (k)| 2 a.e. Ω,
which yields by (3.9) (3.12)

|∇k| 2 1 Bn = |∇H n (k)| 2 a.e. Ω.
Thus, (

Ω β |∇H n (k)| 2 ≤ D ν m . 3.10) becomes (3.13)
Applying Theorem 2.2, we deduce from (3.13) that for any p < p β = 2 1+β , there exists a constant C = C(p, β, ν m , D) > 0 such that (3.14)

Bn |∇k| p = Ω |∇H n (k)| p ≤ C.
Step 2. Ladder process. We deduce now the W 1,p estimate (3.4) from (3.14) by adapting to this case the Boccardo-Gallouët technique [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]. To do so, Let p ∈ ]1; p β [, q ∈ ]1; p[. We get by Hölder's Inequality (3.15)

Bn |∇k| q ≤ Bn |∇k| p q/p |B n | 1-(q/p) ≤ C q/p |B n | 1-(q/p) .
Let n 0 ∈ IN * . On one hand we have (3.16)

n 0 -1 n=0 Bn |∇k| q ≤ n 0 C q/p |Ω| 1-(q/p) ≤ n 0 max(C; |Ω|).
On the other hand, let r ≥ 1, the definition of B n yields (3.17)

Bn |k| r ≥ n r |B n |.
We get by (3.15)

(3.18) +∞ n=n 0 Bn |∇k| q ≤ C q/p +∞ n=n 0 |B n | 1-(q/p) ≤ C q/p +∞ n=n 0 1 n r(p-q) p Bn |k| r p-q p .
Then the discrete Hölder Inequality yields

(3.19) +∞ n=n 0 Bn |∇k| q ≤ C q/p +∞ n=n 0 Bn |k| r p-q p +∞ n=n 0 1 n r(p-q) q q/p
.

Inequality
q p -q .

Let the two constants λ 1 (n 0 ) and λ 2 (n 0 ) be defined by

     λ 1 (n 0 ) = n 0 max(C; |Ω|) λ 2 (n 0 ) = C q/p +∞ n=n 0 1 n r(p-q) q q/p
Ω |∇k| q ≤ λ 1 (n 0 ) + λ 2 (n 0 ) Bn |k| r p-q p = λ 1 (n 0 ) + λ 2 (n 0 ) k r(p-q) p 0,r;Ω .
In order to obtain an inequality involving the norm k 1,q;Ω , the Sobolev embedding W 1,q 0 (Ω) → L r (Ω) must stand, which brings another condition on r:

(3.22) 1 ≤ r ≤ q = N q N -q .
For both conditions (3.20) and (3.22) to be satisfied, one needs to have (3.23) q p -q < N q N -q hence q < N (p -1) N -1 .

As this must be satisfied for any p < p β , we get the following bound for q, (3.24) q < N (p β -1) = q β . Condition (3.24) makes sense if and only if β is such that

(3.25) 1 < q β < p β hence 2 - 1 N < p β < N.
Since 0 ≤ β < 1, the condition p β < N is always satisfied. However,

p β > 2 - 1 N is equivalent to (3.26) β < 1 2N -1 =: β c .
Conditions (3.26) and (3.24) then yield the bounds (3.27)

β c = 1 3 and q β = 2 2 1 + β -1 if N = 2 β c = 1 5 and q β = 3 2 2 1 + β -1 if N = 3.
Therefore, if β < β c and q < q β , considering r satisfying (3.20) and (3.22) for all p < p β , q < q β , β < β c , Inequality (3.21) becomes

(3.28) k q 1,q;Ω ≤ λ 1 (n 0 ) + cλ 2 (n 0 ) k r(p-q) p 1,q;Ω .
As when 0 < β < β c , we verify γ = q β (p β -q β ) p β q β < 1, then we deduce from (3.28) that for q < q β and 0 < β < β c , (3.29) k 1,q;Ω ≤ C = C(β, ν m , D), a constant that blows up when q → q β or β → β c .

Step 3. L 3/2 (Ω, ρ -1 ) estimate. Let G ε : IR → IR be the odd function given by

(3.30) G ε : G ε (x) = x ε if 0 ≤ x < ε, G ε (x) = 1 if x ≥ ε.
We take G ε (k) as test function in (3.1). By the same argument as above,

Ω u • ∇k G ε (k) = 0, Ω β µ(k)∇k • ∇G ε (k) = Ω G ε (k) β µ(k)|∇k| 2 ≥ 0, hence, since 0 ≤ G ε (k) ≤ 1, Ω G ε (k)k 3 2 ≤ D 0,1;Ω . As k > 0 and > 0 in Ω, G ε (k)k 3 2 → k 3 2
a.e. in Ω. Then by Fatou's Lemma,

(3.31) 0 ≤ Ω k 3 2 ≤ D 0,1;Ω .
Notice that this last estimate does not require any restriction about β. Remark 3.4. We have assumed in Theorem 3.1 u ∈ C 0 (Ω) and k ∈ C 2 (Ω) for convenience. However such regularity is not crucial. Indeed, we obtain the same estimate if, instead of (3.1), we consider its following regularization, where ε > 0, (3.33)

   u • ∇k -div ( β µ(k)∇k) -∆k + k|k| 1/2 = D in Ω, k = 0 at Γ,
for a given u ∈ V 0 . We know the existence of a distributional solution k to this equation such that:

k ∈ r<N W 1,r 0 (Ω),
and such that ∀ n ∈ N, T n (k) ∈ H 1 0 (Ω) (see [START_REF] Lewandowski | The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity[END_REF]). For this solution, tests can be taken in W 1,∞ 0 (Ω), as well as functions of the form h(k)ϕ, where ϕ ∈ W 1,∞ 0 (Ω) and h : R → R is continuous, bounded, C 1 -piecewise, such that h is in L ∞ (R) with a finite number of discontinuities. This justifies that the entire procedure above, to obtain estimates (3.4), also applies to this regularized equation considering only this distributional solution.

In the following we set (3.34)

K β = 1≤q<q β W 1,q 0 (Ω).

Main result

Recall that η c is given in (2.28), β c in (3.3). According to the previous sections, the assumptions about the data are the following:

(4.1)    ν, µ : IR → IR are continuous and ∀ k ∈ IR, 0 < ν m ≤ ν(k), µ(k) ≤ ν M , f ∈ L 2 (Ω) N , λ > 0 0 < η < η c , β < β c . Definition 4.1. When (4.1) holds, we say that (u, k) ∈ V η × K β is a weak solution to the initial problem (1.4) if k ≥ 0 a.e in Ω, and if for all (v, ϕ) ∈ V η × D(Ω), λ Ω u • v - Ω (u ⊗ u) : (∇v) T + Ω η ν(k)∇u : ∇v = Ω f • v, (4.2) - Ω k u • ∇ϕ + Ω β µ(k)∇k∇ϕ + Ω k 3/2 ϕ = Ω D(u, k)ϕ, (4.3) where D(u, k) = η ν(k)|∇u| 2 .
Remark 4.1. As shown in section 2.3, the assumption about η makes sure that all the integrals in (4.2) are well defined. Moreover, by (2.5) and (3.2), D(u, k) ∈ L 1 (Ω). Therefore, by (3.4), it is guaranteed that all the integrals in (4.3) are well defined, except the tansport term about which we must take care. However, k ∈ L 3/2 (Ω) and as V η ⊂ H 1 loc (Ω) N , then for a given ϕ ∈ D(Ω), we have at least u ∈ L 3 (supp(ϕ)). Therefore, k u ∈ L 1 (supp(ϕ)), which makes well defined the integral Ω k u • ∇ϕ since ∇ϕ ∈ L ∞ (Ω). Definition 4.2. We say that (u, k) ∈ V 0 × (∩ q<N W 1,q 0 (Ω)) is a weak solution to the perturbed Problem (1.5) if k ≥ 0 a.e in Ω, and if for all (v, ϕ)

∈ V 0 × D(Ω), λ Ω u • v - Ω (u ⊗ u) : (∇v) T + Ω η ν(k)∇u : ∇v + ε Ω ∇u : ∇v = Ω f • v, (4.4) - Ω k u • ∇ϕ + Ω β µ(k)∇k • ∇ϕ + ε Ω ∇k • ∇ϕ + Ω k 3/2 ϕ + ε = Ω D(u, k)ϕ. (4.5)
By [START_REF] Lewandowski | The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity[END_REF], we already know:

Theorem 4.1. Let ε > 0. Assume that ν, µ : IR → IR are continuous, bounded and nonnegative, f ∈ V 0 , λ ≥ 0. Let 0 ≤ η, β ≤ 1. Then Problem (1.5) has a weak solution (u ε , k ε ) ∈ V 0 × (∩ q<N W 1,q 0 (Ω)).
We prove the following result:

Theorem 4.2. Assume that (4.1) holds. Then the initial Problem (1.4) has a weak solution obtained by approximation (SOLA). More precisely, let (u ε , k ε ) be a weak solution to the perturbed problem (1.5). Then there exists (ε n ) n∈I N that converges to 0 and (u, k) ∈ V η × K β a weak solution to the initial problem such that the sequence (u εn ) n∈I N strongly converges to u in V η and the sequence (k εn ) n∈I N weakly converges to k in W 1,q 0 (Ω) for all 1 ≤ q < q β . Proof. The proof is the synthesis of the sections 2 and 3. Passing to the limit in the fluid equation was already done in the proof of Theorem 2.3. Theorem 2.4 ensures the strong convergence of the source term in L 1 (Ω). We already know estimates for the TKE from Theorem 3.1 and Remark 3.4. Therfore, we summerize in a first step the sequence extraction process, and we procced in a second step to pass to the limit in the TKE equation.

Step 1. Extracting subsequences -Due to (4.1), particulary the choices of η and β, we know from step 1 of the proof of Theorem 2.3 and Theorem 3.1, that there exists (ε n ) n∈I N that converges to 0, such that the family (u εn ) n∈I N is bounded in V η , and the family (k εn ) n∈I N is bounded in W 1,q 0 (Ω) for any 1 < q < q β (recall that q β is given by (3.3)). Arguing as Step 2 of the proof of Theorem 2.3, based on a standard compactness argument combined to an uniqueness argument and Lebesgue inverse Theorem, we deduce that there exists (ε n ) n∈I N , u ∈ V η and k ∈ K β such that 1. (u εn ) n∈I N converges weakly to u in V η , 2. (u εn ) n∈I N converges weakly to u in W p 0 (Ω) N for all 1 < p < p η , 3. (u εn ) n∈I N converges strongly to u in L q (Ω) N for all q < r η , 4. (k εn ) n∈I N converges weakly to k in W q 0 (Ω) for all 1 < q < q β , 5. (k εn ) n∈I N converges strongly to k in L r (Ω) for all 1 < r < q β , 6. (k εn ) n∈I N converges a.e. to k.

Moreover, by Theorem 2.4 and from the proof of Theorem 2.3, we can add to the list that (u εn ) n∈I N converges strongly to u in V η , and that (u, k) satisfy (4.2). We also know that the sequence ( η ν(k εn )|∇u εn | 2 ) n∈I N converges strongly in L 1 (Ω) to η ν(k)|∇u| 2 . Finally, as V η ⊂ H 1 loc (Ω), by the same argument as in the proof of 2.3 based on the uniqueness of the limit, (ε n ) n∈I N can be chosen so that for any ω ⊂⊂ Ω, (u εn | ω ) n∈I N converges strongly to u| ω in L p (Ω) N , for all p < ∞ when N = 2, and p < 6 when N = 3.

Step 2. Passing to the limit in the TKE equation -Let ϕ ∈ D(Ω), and let ω ⊂⊂ Ω denotes the support of ϕ. From the above, we already know that By a usual argument [START_REF] Chacòn-Rebollo | Mathematical and Numerical Foundations of Turbulence Models and Applications[END_REF][START_REF] Lewandowski | The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity[END_REF], we also have

Ω β µ(k εn )∇k εn • ∇ϕ -→ n→+∞ Ω β µ(k)∇k • ∇ϕ.
Moreover, let q ∈ ]1; q β [ be fixed. By the weak convergence of (k εn ) n∈I N to k in W 1,q 0 (Ω) to k and D(Ω) ⊂ W 1,q 0 (Ω), As we were able to pass to the limit in the TKE equations, we have shown that (u, k) also satisfies (4.3), which concludes the proof.

A Appendix : Proof of Theorem 2.1

We start by giving an interpolation lemma.

Lemma A.1. Let Ω be a bounded Lipschitz domain, 0 < s < 1, and u ∈ D (Ω) such that ∇u ∈ H -s (Ω) N and u ∈ H -s (Ω). Then u ∈ H 1-s (Ω) and there exists a constant C > 0, which does not depend on u, be such that (A.1) u 1-s,2,Ω ≤ C ( u -s,2,Ω + ∇u -s,2,Ω ) .

( 2 .

 2 32) η/2 ∇u εn -→ n→+∞ η/2 ∇u weakly in L 2 (Ω) N ×N , while (2.33) η/2 ν(k εn )∇v -→ n→+∞ η/2 ν(k)∇v a.e. in Ω,because ν is a continuous function. Therefore, since

Remark 3 . 2 .Theorem 3 . 1 .

 3231 In the system we are looking at, D = D(u, k) = η µ(k)| ∇u| 2 , which satisfies (3.2) thanks to the energy balance (2.5). Assume u ∈ C 0 (Ω), and let k ∈ C 2 (Ω) be any strong solution to (3.1).

,

  with r satisfying(3.20). Consequently,(3.16) and(3.19) ensure the existence of two constants λ 1 (n 0 ) and λ 2 (n 0 ) such that λ 2 (n 0 ) -→

Remark 3 . 3 .

 33 Let T n be the truncation at height n. Then by taking T n (k) as test in (3.1), we see that T n (k) ∈ W 1,p 0 (Ω), uniformly in n, namely for any p < p β , (3.32) T n (k) 1,p;Ω ≤ C(D, ν m , β).

Ω

  D(u εn , k εn )ϕ -→ n→+∞ Ω D(u, k)ϕ.

  in dimension 2 or in dimension 3, we easily check that 3/2 < q β . Therefore, as| ω ≥ C ω > 0, k the same argument using u εn → u in L 3 (ω), Ω k εn u εn • ∇ϕ -→ n→+∞ Ω k u • ∇ϕ.

  .38) yields the strong convergence in L 2 (Ω) N ×N of ( η/2 ν(k εn )∇u εn ) n∈I N to η/2 ν(k)∇u, hence the strong convergence in L 1 (Ω) of the sequence ( η ν(k εn )|∇u εn | 2 ) n∈I N to η ν(k)|∇u| 2 . Moreover, by the standard argument based on the continuity of ν, (2.3) and the a.e. convergence of (k εn ) n∈I N to k, we get

								2 ,
	which combined with (2.35) leads to
	(2.38)	lim n→∞ Ω	η ν(k εn )|∇u εn | 2 =	Ω	η ν(k)|∇u| 2 , and lim n→∞	ε n	Ω	|∇u εn | 2 = 0.
	Therefore, combining (2.34) with (2(2.39) lim n→∞ Ω

We replace Du by ∇u for simplicity without loss of generality, thanks to the Korn inequality.

Still, for simplicity, we consider homogeneous boundary conditions instead of the usual nonlinear laws at the boundary given by the turbulence modeling framework.

SOlution Obtained by Approximations

This can be easily seen by reccaling that F ∈ H -1 (Ω) if and only if there exists f0, • • • , fN ∈ L 2 (Ω) be such that ∀ ϕ ∈ H 1 0 (Ω), F, ϕ = i (fi, ∂iϕ)2. Therefore we have, (F, G)-1 = i (fi, gi)2.

Proof. We start by recalling two inequalities. First, the definition of the H 1 norm reads for any v ∈ H 1 (Ω) (A.2) v 1,2,Ω = v 0,2,Ω + ∇v 0,2,Ω .

Then a result due to Nečas and proved in [START_REF] Duvaut | Inequalities in mechanics and physics[END_REF] ensures that for any v ∈ D (Ω) such that v ∈ H -1 (Ω) and ∇v ∈ H -1 (Ω) N , we have

In order to obtain an inequality for spaces with a noninteger exponent, we use the interpolation Theorem from [START_REF] Louis | Non-homogeneous boundary value problems and applications[END_REF]. To do so, let the spaces W 0 and W -1 be defined as the closure of D(Ω) respectively for the norms

Noticice that W 0 = H 1 0 (Ω). Remark A.1. The embedding Λ : W 0 → W -1 is continuous and dense. Moreover, because both W 0 and W -1 are Hilbert spaces, for all u, v ∈ W 0 , 4

As the interpolation space

so that the interpolation norm is defined by

Inequality (A.2) shows that the identity map, Id : W 0 → H 1 (Ω) is continuous, and similarly (A.3) yields that Id : W -1 → L 2 (Ω) is continuous. Then, the interpolation theorem yields

This means there exists a constant C > 0 such that for any u ∈ W 0 , W -1

hence (A.1) by using (A.5).

As L 2 (Ω) → H -s (Ω), we deduce from (A.1) the following.

Corollary A.1. Let u ∈ L 2 (Ω) be such that ∇u ∈ H -s (Ω). Then u ∈ H 1-s (Ω) and

Proof of Theorem 2.1.

Let η ∈]0; 1[ and Ω a lipschitz domain. Let u ∈ V η and ϕ ∈ D(Ω) N ×N .

(A.9)

Given that η < 1, Hardy inequality (see [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]) yields

where C > 0 depends neither on ϕ nor on u. Therefore, we obtain Therefore, we have indeed u ∈ H 1-η 2 (Ω) N , which concludes the proof.


Tariq Berrada

Camille Couprie

Karteek Alahari

Jakob Verbeek

Meta Fair

Guided Distillation for Semi-Supervised Instance Segmentation

Although instance segmentation methods have improved considerably, the dominant paradigm is to rely on fullyannotated training images, which are tedious to obtain. To alleviate this reliance, and boost results, semi-supervised approaches leverage unlabeled data as an additional training signal that limits overfitting to the labeled samples. In this context, we present novel design choices to significantly improve teacher-student distillation models. In particular, we (i) improve the distillation approach by introducing a novel "guided burn-in" stage, and (ii) evaluate different instance segmentation architectures, as well as backbone networks and pre-training strategies. Contrary to previous work which uses only supervised data for the burn-in period of the student model, we also use guidance of the teacher model to exploit unlabeled data in the burn-in period. Our improved distillation approach leads to substantial improvements over previous state-of-the-art results. For example, on the Cityscapes dataset we improve mask-AP from 23.7 to 33.9 when using labels for 10% of images, and on the COCO dataset we improve mask-AP from 18.3 to 34.1 when using labels for only 1% of the training data.

Introduction

The goal of instance segmentation is to extract a segmentation mask and a class label for all relevant object instances in an image. This generalizes both object detection, for which each instance is localized only with a bounding box, as well as semantic segmentation, in which different instances of the same class can be lumped together in the same mask. Although recent segmentation models have brought considerable improvements in performance to this task, see e.g. [START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF][START_REF] He | Mask R-CNN[END_REF], these methods require large amounts of images with pixel-level annotations in order to perform well. Consequently, the difficulty of manually producing precise instance segmentation maps for training forms a barrier for applying such models to settings where relevant labeled data sets are not readily available.

Semi-supervised learning can be used in label-scarce sit- , we achieve +15.7 mask-AP when using 1% of labels, for an AP of 34.0, which is more than what Polite Teacher achieved using 10 times more labels (30.8) on the COCO dataset.

uations by making use of unlabeled images to boost performance: for the unlabeled images the labeling process is avoided, while they still provide useful problem specific input data, see e.g. [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF][START_REF] Li | Semantic segmentation with generative models: Semi-supervised learning and strong out-of-domain generalization[END_REF][START_REF] Springenberg | Unsupervised and semi-supervised learning with categorical generative adversarial networks[END_REF][START_REF] Van Engelen | A survey on semisupervised learning[END_REF]. For semi-supervised instance segmentation, the "Noisy Boundaries" method [28] builds upon the detection-based Mask-RCNN model [START_REF] He | Mask R-CNN[END_REF], and leverages pixel-level pseudo labels on unlabeled images, while also making the training of the mask prediction head more robust to noise. It achieves comparable performance to Mask-RCNN while utilizing only 30% of labeled images. While providing an important gain in label efficiency, this approach relies on a fixed Mask-RCNN teacher model, which is trained in supervised mode from the available labeled images, which limits its performance in settings with few labeled images. The Polite Teacher approach [9] is currently state-ofthe-art for semi-supervised instance segmentation. It performs distillation on a CenterMask2 [START_REF] Lee | CenterMask: Real-time anchor-free instance segmentation[END_REF] model with a jointtraining framework, where the teacher is updated as an exponential moving average (EMA) of the student model. Low confidence thresholding on bounding-box and mask Figure 2. Results on the COCO dataset when using from 5% labeled data, with supervised training (top) and our semi-supervised approach (bottom). Our approach leads to more detected objects that are segmented with higher accuracy. Results best viewed by zooming in.

scores is used to filter out noisy pseudo labels. In our work, we also adopt a student-teacher training with EMA, with several important differences w.r.t. the Polite Teacher approach: we introduce a novel burn-in step in the distillation process, and adopt an improved Mask2Former segmentation architecture [START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF] in combination with Swin [START_REF] Liu | Swin transformer: Hierarchical vision transformer using shifted windows[END_REF] and DI-NOv2 [START_REF] Oquab | Dinov2: Learning robust visual features without supervision[END_REF] feature backbones.

To summarize, our contributions are the following :

• We develop a novel teacher-student distillation strategy to more effectively leverage unlabeled images. • We leverage vision transformer architectures for the first time in the context of semi-supervised instance segmentation. • We show largely improved performance in terms of mask-AP for both Cityscapes and COCO datasets across a wide range of labeled data budgets. Our implementation can be found on the project webpage1 .

Related work

In this section, we review instance segmentation models and their applications in the semi-supervised regime.

Segmentation models. Instance segmentation aims at identifying the semantic class of every pixel in the image, as well as associating each pixel with a specific instance of the object class. This contrasts with semantic segmentation which only aims at solving the first part. Instance segmentation is more challenging because it requires identifying each individual object instance, associating parts with the correct objects, and accurately determining the boundaries of individual objects. These difficulties compound with other challenges, such as variability in scale, color, lighting, occlusions, etc. Until recently, the state-of-the-art was held by detection based approaches such as Mask-RCNN [START_REF] He | Mask R-CNN[END_REF] and follow-ups such as ViTDet [17], with a component to detect object bounding-boxes and one for binary segmentation to isolate the object from background pixels. Further progress in transformers [START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF][START_REF] Cheng | Perpixel classification is not all you need for semantic segmentation[END_REF] allowed instance segmentation approaches to avoid a separate detection step. Additional gains have been brought by leveraging text-level supervision using CLIP and very large models [START_REF] Fang | EVA: Exploring the limits of masked visual representation learning at scale[END_REF]. To our knowledge, our work is the first one to explore the application of ViTs in the context of semi-supervised instance segmentation. Recently, MaskFormer [START_REF] Cheng | Perpixel classification is not all you need for semantic segmentation[END_REF] observed that mask classification is sufficiently general to solve both semantic and instance-level segmentation tasks. Their approach converts per-pixel classification into a mask classification model using a transformer-based set prediction mechanism. Mask2Former [START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF] provides further improvements and presents a universal segmentation model using the same mask classification mechanism, an efficient masked attention module, multi-scale feature pooling and other improved techniques for fast and efficient training.

Semi-supervised instance segmentation. Previous work has considered the use of pseudo labeling to leverage unlabeled data for semi-supervised instance segmentation [9,28]. A "teacher" instance segmentation model is used to annotate unlabeled images, which are then used by the "student" model in addition to labeled images for which groundtruth instance segmentations are available. A key factor for the success of such methods is the discrepancy between the teacher and student models, which ensures that the teacher provides a stable and strong training signal to the student by means of the pseudo labels. The first conditionstability-can be achieved by using a fixed teacher, or using a teacher with the same architecture as the student and updating it as an exponential moving average (EMA) from the student [9, [START_REF] Tarvainen | Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results[END_REF]28]. The second condition -strong training signal-can be met by using different data augmentations of the same image for the teacher and student model: using strong augmentations for the student and weaker augmentations for the teacher ensures high quality for the pseudo labels of the teacher and a useful training signal for the student being confronted with hard prediction tasks on strongly augmented training data.

Among the few existing semi-supervised instance segmentation approaches, Noisy Boundaries [28] performs distillation with a fixed Mask-RCNN teacher model trained from labeled data only. The student network is learning a noise-tolerant mask head for boundary prediction by leveraging low-resolution features. The Polite Teacher [9] approach is closer to our work and also uses EMA teacher updates. In our work we develop an improved burn-in stage, before starting the main semi-supervised training loop, in which we pre-train the student model from both labeled data as well as unlabeled data using pseudo labels from a fixed teacher model. In contrast, Polite Teacher pre-trains the student from supervised data only. Knowledge distillation. Our work employs a specific form of knowledge distillation (KD), where knowledge is transferred from the teacher model to the student by providing pseudo-labels for unlabeled samples in an online fashion, while at the same time updating the student progressively using EMA. Other approaches have explored teacher-free distillation approaches [START_REF] Li | Self-regulated feature learning via teacher-free feature distillation[END_REF], efficient N-to-1 matching of feature representations [START_REF] Liu | NORM: Knowledge distillation via N-to-one representation matching[END_REF], bridging the gap between online and offline KD methods with a lightweight shadow teacher module [START_REF] Li | Shadow knowledge distillation: Bridging offline and online knowledge transfer[END_REF], and teacher-dependent student architecture search [START_REF] Dong | DisWOT: Student architecture search for distillation without training[END_REF] for optimal knowledge flow. Such approaches are complementary to our work.

Method

Let D s = {(x 1 , y 1 ), ..., (x N , y N )} be a supervised dataset and D u = {x ′ 1 , ..., x ′ M } an unlabeled set of images, where x denotes images, y their segmentations, and M ≫ N. The goal is to make use of D u in order to improve the performance of the model using the available labels.

Below, in Sec. 3.1 we present model architectures, our improved distillation approach, and pseudo labeling strategy. In Sec. 3.2 we detail the losses and EMA teacher updates to train our models.

Architectures, distillation and pseudo-labeling

Model architecture. We follow the Mask2Former architecture [START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF] which achieves strong results across different (ii) The student is initialized from scratch, and trained on both labeled data and unlabeled data using pseudo labels generated by the pre-trained teacher. The teacher remains fixed during this step.

(iii) Copy the checkpoint of the student to the teacher, and then continue training on both labeled and unlabeled data, while using EMA updates for the teacher network from the student's weights.

segmentation tasks (semantic, instance, panoptic). For our study, we focus on the instance segmentation task. The model consists of a backbone for image-level feature extraction, a pixel decoder to gradually upsample image features to compute per-pixel embeddings and a transformer decoder that predicts N mask and class embeddings which generate N , possibly overlapping, binary masks via a dot product between the image-level features and the class embeddings.

Teacher-student distillation. Our training pipeline can be divided into three steps as illustrated in Fig. 3 : 1. Teacher pre-training: the teacher model, parameterized by θ t , is trained on annotated data only.

Improved burn-in:

The student model, parameterized by θ s , is initialized randomly, then trained on both labeled data and unlabeled data using pseudo labels provided by the pre-trained teacher model. The teacher model remains fixed during this stage. 3. Distillation stage: In this stage, we copy the student weights to the teacher, and then train the student on both labeled and unlabeled data as before. The teacher is updated using an EMA of the student's weights. Pseudo-label extraction. Mask2Former by default predicts a constant number of K = 100 masks, which we need to filter in order to obtain pseudo ground-truths for the unlabeled images. To do this, we follow a simple thresholding scheme that takes into account the predicted class probability and the predicted mask size. A predicted mask is selected as a pseudo label if (i) the maximum class probability is above the class threshold p c ≥ α C , and (ii) the size of the predicted mask is above the size threshold: H×W p=0 σ(ŷ (p) ) ≥ α S , with σ the sigmoid activation of the binary mask prediction.

Optimisation

In this section we introduce the loss functions used to train the models, and present our guided distillation ap- Loss. Following Mask2Former [START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF], we match predictions from the student and pseudo ground-truth instances from the teacher by constructing a cost matrix for bipartite matching similarly to DETR [START_REF] Carion | End-toend object detection with transformers[END_REF]. Once the matching is obtained, we use a weighted sum of cross entropy and Dice loss for the masks and cross-entropy loss for the class predictions. In order to minimize memory requirements, we follow PointRend's approach and only sample a limited number of points from the high resolution masks using importance sampling [START_REF] Kirillov | PointRend: Image segmentation as rendering[END_REF]. Furthermore, this loss is computed for every auxiliary output of the model as in [START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF]. Let x i be an image and y i = {(y k i , c k i )} k=1,...,n its associated ground-truth instances defined by the n binary masks

y k i and class indices c k i . The model predicts K candidate instances (ŷ k i , ĉk i ) 1≤k≤K = S m • G(x i ; θ)
, where G is the backbone and S m the segmentation head. Using bipartite matching, we obtain a matching between the ground truth and predictions (k, τ k ) 1≤k≤n , which is used to calculate the image-level loss for both masks and predicted classes:

L i s = 1 n n k=1 l CE (ŷ τ k i , y k i ) + λ D l D (ŷ τ k i , y k i ) +λ C l C (ĉ τ k i , c k i ), (1)
L s = 1 |B| i∈B L i s , (2)
where l CE is the cross-entropy loss for the class predictions, l C binary cross-entropy and l D the dice loss function, λ C and λ D are scaling parameters, and B is the training batch.

For the unsupervised loss, we follow the same procedure as the supervised setting. To obtain the pseudo labels, we filter the predictions of the teacher as described previously, and then define the pseudo labels by one-hot encoding both the mask and class predictions. Let x ′ j be an unlabeled image and y ′ j = (ŷ ′k j , ĉ′k j ) 1≤k≤K = S m • G(x ′ j ; θ t ) the predictions of the teacher, then pseudo-labels are defined as:

c ′k j = argmax c∈C c k j , y ′k j = σ(ŷ ′k j ) ≥ 0.5. (3)
Given the pseudo labels, we compute the unsupervised loss term L u analogous to how we compute the supervised loss L s in equations ( 1) and (2) above. Finally, we combine the supervised and unsupervised loss in a weighted sum as:

L = L s + λ u L u . (4)
Revisited burn-in stage. To ensure that the quality of the pseudo labels is high enough, existing distillation-based semi-supervised methods adopt a burn-in stage before utilizing the teacher in the training pipeline, e.g. as in Polite Teacher [9]. During this stage, the student is trained on labeled data only before copying its weights to the teacher. However, this is not ideal since the student model will be more likely to converge towards a local minimum induced by its initial training where only the limited labeled data was taken into account. Alternatively, we show that it is better to make use of the teacher model from the beginning of training. it less prone to over-fitting on the limited labeled training data. One important hyperparameter is the length of the burn-in stage, which we choose so that the EMA updates of the teacher start when the student's performance becomes comparable or superior to that of the pre-trained teacher.

We recap in Tab. 1 the different training strategies from the literature and highlight the specificity of our approach. Exponential moving average (EMA). During the second stage of training, the teacher model is updated as an EMA of the student's weights. This approach has been proven to stabilize training by providing more regular pseudo labels during the student's training [START_REF] Tarvainen | Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results[END_REF]. Let θ n s and θ n t be the student and teacher's weights respectively at iteration n. The update rule for the teacher can be written as:

θ n+1 t = α • θ n t + (1 -α) • θ n+1 s , ( 5
)
where α is the decay rate which regulates the contribution of the student's weights to the update in every iteration.

Experiments

Here we present our experimental setup in Sec. 4.1, and present our main results followed by ablations in Sec. 4.2.

Experimental setup

Datasets and evaluation metric. We use two datasets in our experiments. The Cityscapes dataset [START_REF] Cordts | The Cityscapes dataset for semantic urban scene understanding[END_REF] that contains 2,975 training images of size 1024 × 2048 taken from a car driving in German cities, labeled with 8 semantic instance categories. Following [28], we train models using 5%, 10%, 20% and 30% of the available instance annotations, and evaluate using the 500 validation images with public annotations. The different data splits are generated by randomly selecting a random subset from the training images. For the semi-supervised results we present with 100% annotated images, we use the additional 20k unlabeled images. The COCO2 dataset [START_REF] Lin | Microsoft COCO: common objects in context[END_REF] Implementation. We implement our approach in the De-tectron2 codebase [29]. By default, experiments are conducted on one machine with eight V100 GPUs with 32 GB RAM each. In some cases less GPUs were needed, e.g. only four GPUs are needed to train semi-supervised models with a ResNet-50 backbone on Cityscapes. For experiments using Swin-L, ViT-B or Vit-L backbones on COCO semi-supervised training we needed 16 GPUs across two machines to keep the same batch size. For our experiments with these large backbones, we enabled automatic mixed precision in order to reduce the memory footprint. In Appendix B we provide more details on the sizes, efficiency, and carbon footprint of our models. Hyper-parameters. Our hyper-parameters are set as follows. We set the EMA decay rate to α = 0.9996, the unsupervised loss weight to λ u = 2, the class threshold to α C = 0.7 and the size threshold to α S = 5. For the loss weights, we follow [START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF] and set λ D = 5 and λ C = 1. For Cityscapes, we train our models for 90k iterations, except for our semi-supervised model with the additional 20k images and 100% of labeled images used, in which case we found it beneficial to double the training length. For COCO, we train our models for 368k iterations. For all our models, we use AdamW with a learning rate of 10 -4 , weight decay of 0.05 and multiplier of 0.1 for the backbone updates, a batch size of 16 and early stopping. The number of iterations used for the burn-in period is given in Tab. 2.

Main results

Semi-supervised instance segmentation results. In Tab. 3c we compare our results on the Cityscapes dataset to these of Noisy Boundaries (NB) [28]. We provide results using different feature backbones, including the ImageNet-1k pre-trained ResNet-50 backbone used in NB. We observe large improvements over the latter using all backbones and percentages of labels used, with best results obtained using the larger SSL DINOv2 and the supervised Swin-L backbones. The DINOv2 backbone leads to best results (28.8 vs. 25.1 for Swin) in the most challenging setting where we used only 5% of the labeled data, while the Swin backbone Figure 5. Overview of our results on Cityscapes with 10% labeled data with R50 backbone. Top: Predictions of model trained using 5% of labeled data only. Bottom : Predictions of model trained using our semi-supervised method using 5% of labeled data. Without our supervision method, the model tends to merge different instances from the same semantic category (bounding boxes spanning multiple objects), the model also misidentifies certain objects by predicting a the wrong class or only a certain portion of the instance. Our method greatly helps in alleviating all these issues. improves by 0.5 to 1.3 points AP in the other settings. Notably, compared to NB we improve the AP by more than 10 points from 23.7 to 33.9 when using 10% of labels.

In Tab. 3d we report results for the COCO dataset, where we compare to Noisy Boundaries (NB) [28] and Polite Teacher (PT) [9] approaches. We similarly observe substantial improvements across all backbones and percentages of labeled data used. For example, improving the 22.3 AP of PT to 38.2 in the case of using 2% of the labels. In this case the Swin-L backbone yields (near) optimal results. Our results are better overall, but in particular for small amounts of labeled data, which makes the performance curve as a function of the amount of labeled data flatter, see also Fig. 1. For example, between the 1% and 10% label case, NB improves from 7.7 to 29.2 AP (+21.5), and PT improves from 18.3 to 30.8 (+12.5), where in our case with the Swin-L backbone the performance increase is 9.1 AP points (34.0 vs. 43.1).

On COCO, Wang et al.

[28] also report a semisupervised experiment using 100% of labeled data, and adding another 123k unlabeled COCO images. In this setting they obtain 38.6 AP, which is similar to the 38.2 AP that we attain using only 2% labeled data, so using roughly 50× less labeled images and twice less unlabeled images.

Comparison to supervised baselines. In Tab. 3a, 3b we present results obtained using supervised training only, to assess the improvements brought by our semi-supervised approach w.r.t. this baseline for different backbone architectures. We also compare to the supervised baselines using the Mask-RCNN and CenterMask2 segmentation models that underlie Noisy Boundaries and Polite Teacher, respectively.

Comparing the supervised and semi-supervised results for Cityscapes in Tab. 3a, 3c, and similarly for COCO in 3b, 3d, we observe that our approach shows consistent improvements over the supervised baseline across all experiments. Using the Swin-L backbone on Cityscapes in the 5% labeled data case, we improve the AP by 7.3 points from 17.8 to 25.1 AP, while for the DINOv2 backbone we improve it from 22.9 to 28.8 (+5.9 AP). For COCO at 1% labeled data, our semi-supservised approach improves the supervised result of 22.3 to 34.0 (+11.7 AP) using Swin-L, and from 30.2 to 34.1 (+3.9 AP) using the DINOv2 backbone. See Fig. 1 for a graph comparing our semi-supervised results to the supervised baseline as well as previous methods on COCO. We provide qualitative examples of instance segmentation using the supervised baseline and our approach on the COCO dataset in Fig. 2 and for Cityscapes in Fig. 5. In Appendix A we provide additional experiments for the COCO dataset using even smaller labeled training datasets.

Interpretation and ablation studies

In the following, unless specified otherwise, ablations are performed on Cityscapes with 5% labeled data and a ResNet-50 backbone. Results are reported on the validation set. Notably, these experiments allow to pinpoint the performance differences w.r.t. the state-of-the-art Polite Teacher approach (for which there is no public codebase), while using the same segmentation and backbone architecture. Revisited burn-in stage. In order to quantify the performance gains obtained by our revisited burn-in stage in isolation of other architectural choices, we compare it to the strategies used by Noisy Boundaries [28] and Polite Teacher [9]. See Tab. 1 for a comparison. We also consider a fourth variant, which is identical to our approach but setting the number of burn-in iterations to zero. Finally, we include the result when using supervised training only as a baseline result. The results in Tab. 4 show that our approach brings about +3.7 AP in comparison with the (standard) burn-in strategy used by Polite Teacher, and +4.1 AP over keeping the teacher fixed during training as in Noisy Boundaries. Dropping the burn-in stage all together gives worst results (16.9 AP), which underlines the importance of this burn-in stage in distillation approaches. Data augmentations.

One important component of teacher-student distillation is the asymmetry between the two models that ensures that the teacher yields useful pseudo-labels for the student [START_REF] Tarvainen | Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results[END_REF]. To fulfill this condition, the teacher and student receive differently augmented versions of the unlabeled image. For the student model, the augmentations consist of random sized crops and horizontal flipping. For the strong augmentations, we add texturalbased augmentations, including color jitter, random grayscale and blur. We show in Fig. 6 that this works better than the augmentations used in Polite Teacher [9] which include random cutout, since small instances are prone to be masked which results in inconsistencies in the pseudo-labels provided to the student. Without augmentations, the student model quickly diverges while using cutout underperforms compared to textural based augmentations only.

Weighting of unsupervised loss. We compare models trained with different values for the weight λ u for the unlabeled loss term on Cityscapes training with 10% of labels. The results in Tab. 5 show that even with a low λ u , the results are superior to the supervised baseline (λ u = 0), but the model benefits from putting more emphasis on the semi-supervised loss with larger λ u values. However, if this emphasis is too strong then the model tends to overemphasize the teacher's signal to the detriment of the supervised signal, we find it optimal to use λ u = 2.

Backbone pre-training. Most segmentation models rely on pre-trained backbones to extract features on which the segmentation head operates. We explore the effect of different datasets to pre-train the feature backbone. To reduce computational costs, we perform this exploration using a Swin-B architecture rather than the heavier Swin-L architecture we used in our main experiments. In Tab. 6 we report results obtained using ImageNet-1k and ImageNet-21k [START_REF] Deng | ImageNet: A large-scale hierarchical image database[END_REF] pre-training for supervised and semi-supervised instance segmentation. We find that ImageNet-21k pre-training consistently outperforms pre-training on the smaller ImageNet-1k. The performance gap is higher in the low annotation regime at 2% labeled where the improvement is of 4.0 and 4.1 AP points in the (semi)supervised setups. This difference reduces to 1.4 points in the supervised setting with 100% of labels, indicating a higher importance of the backbone pre-training in regimes with sparser labeling. SSL backbones. We quantify the performance of selfsupervised models trained on large scale datasets for our task. More specifically, we adapt ViT [7] models trained using DINOv2 [START_REF] Oquab | Dinov2: Learning robust visual features without supervision[END_REF] and compare the performance obtained with a similar model trained using the Deit [START_REF] Touvron | Going deeper with image transformers[END_REF] method on ImageNet-1K. We used similar ViT models, however the DINOv2 model has a patch token size of 14 while the standard ViT has a patch token size of 16. To adopt ViT models as backbones for our model, we adapt the feature representation of these models before the task-specific head. More precisely, we rearrange the patch tokens to their relative spatial coordinates then use bi-linear interpolation followed by a 1 × 1 convolution to produce a multiscale feature pyramid. Results are reported in Tab. 3b and Tab. 3d. We see that models trained with DINOv2 backbones show high performance and are comparable with Swin backbones. Such models are also more robust at very sparse regimes, yielding state-of-the-art results on COCO with 1% of labels and also on Cityscapes at 5%. However the model does not generalize as well to larger amounts labeled samples, and the gap in performance between Swin-L and DINOv2-L gets bigger the more labeled samples are available. This could be explained by the relatively simple adaptation of the ViT backbone to the segmentation head and the absence of a multi-scale feature pyramid. We also experiment with keeping the DINOv2 backbone fixed, and only training the segmentation head to evaluate the feature generalisation ability of the backbone. Finally, to validate whether the robustness at low annotation regimes is a property of the backbone pre-training or its architecture we report results of a Vit-B model trained on ImageNet-1k following Deit [START_REF] Touvron | Going deeper with image transformers[END_REF]. As reported in Tab. 7, we see that the Vit-B Deit pre-trained model is severely underperforming compared to DINOv2 pre-training, and performs on par with a ResNet-50 backbone despite being much larger. This indicates that the performance gains are more due to the large scale pre-training than the ViT's architecture. On the other hand, fixing the backbone and training the segmentation head results in mask-AP values lower by around 3 -4 points at 2% and 5 points at 100% supervision. This demonstrates that, while not ideal, training a segmentation head on top of frozen large scale pre-trained models such as in DINOv2 can be beneficial when having few annotated samples, especially if available GPU memory is limited.

Conclusions

In this paper we introduced Guided Distillation, a new distillation appraoch for semi-supervised instance segmentation. This novel approach, combined with recent advances in segmentation methods and vision feature backbones, outperforms previous state-of-the-art in instance segmentation by a large margin on the Cityscapes and COCO benchmarks. We found that the optimal pretrained feature backbone varied as a function of the amount of labeled training data: while Swin-L (IN-21k) was optimal for most experiments, we found the ViT-L (DINOv2) backbones to be more effective in the very low annotation regime.

In future work, we would like to test our approach for other prediction tasks such as object detection, to explore large scale SSL pretraining of transformer architectures that are better suited for dense prediction, and assess the robustness of our approach to domain shifts.

Figure 1 .

 1 Figure1. Compared with the state-of-the-art Polite Teacher method [9], we achieve +15.7 mask-AP when using 1% of labels, for an AP of 34.0, which is more than what Polite Teacher achieved using 10 times more labels (30.8) on the COCO dataset.

Figure 3 .

 3 Figure 3. Our semi-supervised training methodology consists of three steps. (i) The teacher network is trained on labeled data only.(ii) The student is initialized from scratch, and trained on both labeled data and unlabeled data using pseudo labels generated by the pre-trained teacher. The teacher remains fixed during this step. (iii) Copy the checkpoint of the student to the teacher, and then continue training on both labeled and unlabeled data, while using EMA updates for the teacher network from the student's weights.

Figure 4 .

 4 Figure 4. Workflow of our distillation stage.The student and teacher receive weakly and strongly augmented versions of the unlabeled image, respectively. The student predictions are filtered and one-hot encoded to obtain the pseudo-labels that serve as supervision for the student model. We use the same bipartite instance matching method as Mask2Former[START_REF] Cheng | Masked-attention mask transformer for universal image segmentation[END_REF]. The teacher is then updated using EMA.

Figure 6 .

 6 Figure 6. Effect of data augmentation on Cityscapes with 5% labeled data. Ours: Our default setup which includes color jitter, random grayscale and random blur. PT: Uses Ours plus cutout as in [9]. Same: Teacher and student receive weakly augmented images. Supervised: Model trained with labeled samples only.

Table 1 .

 1 To do this, we first train the teacher model on labeled data only and then use it to provide pseudo labels for training the student model on both labeled and unlabeled data. During this initial training stage, which we refer as our revisited burn-in stage, the teacher model stays constant and is therefore not influenced by the student predictions. This enables the student model to learn from more data, in particular in cases where very few labeled images are available in combination with many unlabeled images, making Comparing our proposed distillation approach to the ones previously used in Polite Teacher [9] and Noisy Boundaries[28].

			Polite	Guided Distillation	Noisy
			Teacher	(Ours)	Boundaries
	Init 0		Random	Pre-train T	Pre-train T
	Burn-in	Student Teacher	on D s -	on D s +D u Fixed
	Init 1		T ← S	T ← S	-
	Training	Student on D s +D u Teacher EMA	on D s +D u EMA	on D s +D u Fixed

Table 2 .

 2 has instance segmentations for 118,287 training images. We adopt the same Number of burn-in iterations used in our approach.

	Labels used 1% 2% 5% 10% 20% 30% 100%
	Cityscapes	-	-	15k 25k	30k	35k	50k
	COCO	20k 25k 30k 60k	-	-	-
	evaluation setting as [9], using 1%, 2%, 5% and 10% of
	labeled images for semi-supervised training and using the
	remaining training images as unlabeled. We use the same
	supervised/unsupervised splits as earlier work [9, 20]. Fol-
	lowing previous work [9,10,28], we use the standard mask-
	AP metric to evaluate instance segmentation quality.

Table 3 .

 3 Comparison of results on COCO and Cityscapes. Results for Mask-RCNN, CenterMask2, Polite Teacher and Noisy Boundaries are taken from [9] and [28]. For each backbone, we indicate the pre-training data, DINOv2 is SSL pre-trained on a dataset of 142M images.

Table 4 .

 4 Influence of burn-in stage on segmentation performance.

	Burn-in method	mask AP
	Supervised baseline	12.1
	No burn-in	16.9
	Fixed Teacher [28]	18.9
	Standard [9]	19.3
	Ours	23.0

Table 5 .

 5 Influence on unlabeled loss weight on validation performance, on Cityscapes dataset. Mask2Former, RN-50, 10% labels.

	λu	0	0.1	0.5	2	5
	AP 18.8 23.6 24.4 30.8 25.5
	Pre-training		Training		2%	100%
	ImageNet-1k	Supervised	26.3	46.8
	Imagenet-21k Supervised	30.3	48.2
	ImageNet-1k	Semi-supervised 32.4	-
	Imagenet-21k Semi-supervised 36.5	-

Table 6 .

 6 Backbone pre-training effect on COCO using a Swin-B.

Table 7 .

 7 Influence of backbone finetuning on training performance using DINOv2 backbone. COCO supervised training.

	Backbone Model	2%	100%
		Deit ViT-B	20.3	-
	Finetuned	DINOv2 ViT-B 30.0	46.9
		DINOv2 ViT-L 33.1	46.7
	Fixed	DINOv2 ViT-B 27.7 DINOv2 ViT-L 29.7	41.2 41.5

https://github.com/facebookresearch/GuidedDistillation

The COCO dataset, and all the experiments described in this paper have only been carried out for the purpose of writing this scientific paper and are not to be used in the context of any product or service.

Acknowledgements. Karteek Alahari is supported in part by the ANR grant AVENUE (ANR-18-CE23-0011).

Appendix

A. COCO experiments with fewer labels

We conducted additional experiments on the COCO dataset using fewer labeled training samples compared to earlier work: from 0.4% down to 0.1%. We report results for a supervised training baseline and our semisupervised approach in Tab. 8. In this extremely challenging, very sparsely labeled data regime, our approach shows marked improvements over the supervised baseline, e.g., +3.4 points for 0.1% labeled images using the DINOv2 backbone. Moreover, we find that in this regime the DI-NOv2 backbone is more effective than the Swin-L (IN-21k) backbone, e.g., in the 0.4% labeled data case, corresponding to less than 500 of image annotations: the DINOv2 backbone improves mask-AP by 8 points from 23.0 to 31.0, reaching similar performance as the previous state-of-theart approach Polite Teacher with 25 times less annotations (30.8 mask AP at 10% labeled data, see Tab. 3d. This is in line with our observation on the 5% labeled data case for Cityscapes in Tab. 3c, and 1% labeled data case for COCO in Tab. 3d. See Fig. 7 for an illustration of segmentations obtained with this model. after a few thousand iterations in sparse regimes. Therefore, in practice, it only takes a few dozen hours to train the models end-to-end.

Recent works in KD have explored alternative strategies where no separate teacher network is required, see e.g. [START_REF] Li | Self-regulated feature learning via teacher-free feature distillation[END_REF], with applications to image classification. Such alternatives present interesting directions of future work to improve the efficiency of semi-supervised methods. Model characteristics. We present the different characteristics of the models used. We report both the number of parameters and the FLOP count for each architecture used in our project in Tab. 9 as measured using count_flops function in the Detectron2 library. Comparing training protocols. Section 4.3 provides ablation for the training protocol, which compares the main changes in the distillation strategy with respect to PT while keeping the overall model architecture constant. Particularly, we :

1. Isolate the effect of our revisited burn-in stage.

2. Isolate the effect of student data augmentations (the teacher augmentations are the same between PT and ours).

In Tab. 4, we can see that using the standard burn-in stage as in PT reduces the AP by 3.7 points. In Fig. 6, we track the mask-AP evolution when using different data augmentations, we can see that our data augmentation yields better performance and more stable convergence that PT which additionally uses random cutout. Hence, both changes show improved performance with respect to the SOTA protocol while using the same underlying meta architecture, backbone, training epochs etc. This is evidence that our distillation protocol and revisited burn-in stage are both important factors for the improved performance, beyond the backbone and meta-architecture choices.

Estimation of carbon footprint. On COCO, it took 25 hours to train our ViT-L (DINOv2) model using 1% of annotations, and about 2 days to train a Swin-L (IN-21k) model using 10% of annotations. Trainings are approximately 2.5 times faster on Cityscapes. Given the same for-


Adrien Coatanea

N Nguyen-Trong

Christophe Fumeaux
email: christophe.fumeaux@adelaide.edu.au

Benjamin Potelon

Christian Person
email: christian.person@imt-atlantique.fr

Lightweight, Low-Profile 3-Port Antenna Using Additive Manufacturing Process

Keywords: 3D printing, additive manufacturing, technological process, MIMO, antenna

A 3D-printed 3-port antenna for MIMO communications is proposed. Its design relies on the combination of a cavity and annular slot structure. Originality of this works relies not only on its compactness and the method to excite the different radiating modes while preserving isolation, but also on its light weight obtained thanks to additive manufacturing process. The fabricated 3-port antennas, working at 5 GHz exhibit a low mass (less than 10 grams) and show a good agreement between measured and simulated electrical results.

I. INTRODUCTION

Compactness of radio-frequency front-ends is an essential feature in any communicating object (connected sensor, autonomous radio base station, …) or detection systems (onboard radar,..). It remains a major research and innovation topic, addressing different application domains such as space, health, defense, industry, or telecommunications. Of course, expectations regarding compactness must not be detrimental to functional performances, following requirements in terms of electrical losses, selectivity, radiation and associated efficiency to name a few.

In particular, because antennas have physical limitations in in terms of size versus functionality, size constraints are particularly challenging for antennas. MIMO antennas, which are highly required for enhancing data bit rate and minimizing sensitivity to propagation effects, are all the more affected by these requirements.

The integration of such multi-access ports antennas remains a great challenge for miniaturized systems such as WIFI terminals or pico-cells base-stations. Low-profile structures are highly expected for compactness and to avert implementation issues. Furthermore, as such devices are targeted to address mass-markets, maintaining low costs yet high performance is paramount.

In this context, the use of novel technological processes based on 3D-printing appears promising, as it opens the way to innovative designs without prohibitive fabrication costs. Amongst the numerous available additive manufacturing processes, metalized plastic appears a good candidate to antenna applications as it exhibits light weight, yet satisfactory electrical performance.

II. A CONCEPT OF ANTENNA WITH MULTIPLE ACCESS PORTS

A topology of low-profile 3D antenna is proposed; which relies on the combination of cavity and annular slot structures.

A. Generic structure: a combination of a cavity and a circular radiating slot structure

The generic structure is depicted in Fig. 1-a. It consists of a circular cavity resonating at a given frequency, combined with a radiating slot placed on the upper side of the cavity, at its periphery.

By considering the fields distribution within the circular cavity, maximum and minimum E-Fields values appeared for the fundamental TE11 modes, depending on the value of the circumference length with respect to the guided wavelength.

The antenna is coaxially fed with its excitation being ensured by a port located on the inner ground of the cavity. The feed consists of the vertical elongated central pin of a conventional coaxial cable or SMA connector, as shown in Fig. 1-b. The reflection coefficient of this single-mode ring slot antenna designed at 5.5 GHz is presented in Fig. 1-c.

B. Multi-mode antennas: an oversized radiating cavity with 3 independent access ports

In order to achieve a MIMO antenna, the 1.5λ circular resonance (TM11 mode) around 5 GHz is exploited. It leads to a slightly oversized cavity with respect to the one presented herein before. However, this mode, thanks to its null-E-fields regions, allows for three-mode excitation through the introduction of ports in these particular locations. A 0.58 x 0.52 x 0.05 λ0 3 antenna has been designed (Fig. 2-a) with a 10% bandwitdh (@10dB return loss). However, inter-ports coupling levels remain greater than 6 dB in this configuration. To minimize those parasitic couplings between access ports, vertical walls are added within the cavity (Fig. 2-b), without modifying external dimensions. Isolation between access ports can be reduced by adjusting the design dimensions, particularly the relative position and dimensions of such internal shielding walls. Simulated reflection and isolation S-parameters (obtained from HFSS™) are presented in Figs. 3 and4.

III. 3D DESIGN USING ADDITIVE MANUFACTURING

TECHNOLOGIES

Once the complete structure has been designed, fabrication is achieved thanks to the additive manufacturing process from ELLIPTIKA company [5]. It consists in 3D-printing the lowprofile antenna by using a SLA (Stereo Lithography Apparatus) Form3+ [6] machine from FormLabs™ with High Temp Resin [7]. These pieces are then fully metallized through copper electroless plating. In order to allow for a good flow of the electroless copper-based solution over all surfaces of the structure without impacting the electrical functionality, through holes are judiciously dimensioned and placed all over the structure.

Such a technological process offers almost endless design possibilities and materials with good mechanical and thermal properties. Furthermore, the resulting pieces are lightweight yet electrically performant.

Regarding feeding, two solutions are proposed: the first one (Prototype 1) consists in assembling SMA connectors directly onto the antenna, while the second one (Prototype 2) is based on transitions between a planar substrate (Printed Circuit Board -PCB) and the antenna, as shown in Fig. 5. In this latter case, a conventional substrate is considered (εr = 4,3) and the cavity is simply mounted and maintained to the PCB by screws to guarantee contacts. Electrical conductivity is insured thanks to conductive ink deposition.

Once the circuits are manufactured (Fig. 6 and7), the next step is to proceed with measurements. IV. MEASUREMENTS Both prototypes were measured using a calibrated 2-ports vector network analyser. As the devices are 3-port antennas, return losses and isolations were measured with the third port terminated with a 50-ohms load, as presented in Fig. 8.

Measurement results are shown in Fig. 9 (Prototype 1 with direct accesses) and Fig. 10 (Prototype 2 with PCB line feeds). The corresponding simulations are overlaid on these graphs to allow for direct comparisons. A fairly good agreement between measurements and simulations is observed.

The measured prototypes achieve an overlapped impedance bandwidth of about 6% @-10dB. For prototype 1, the isolation is about than -10dB, which can still be optimized by considering a dimensional optimization of the internal decoupling walls of the cavity. This will conduct to a resynchronisation between peak isolation values and perfect matching conditions. For prototype 2, the isolation is lower than -10dB, thanks to an improved accordance between isolation and matching conditions.

Finally, the circuits were weighted and their mass is 8.6 grams for the prototype 1 and 7.7 for the prototype 2. These masses do not include neither connectors nor the PCB substrate (in the case of the Prototype 2).

V. CONCLUSION In this paper, we have proposed a concept for designing low-profile multi-port antenna, dedicated to MIMO systems. The antenna is based upon a cylindrical shallow cavity with a circular radiating slot. The 3D capabilities of an additive manufacturing process were exploited to devise an approach to control coupling phenomena between the 3 co-integrated feeding ports.

Compact and lightweight prototypes have been implemented for experimental validation, with alternative solutions to incorporate excitations and inter-ports coupling structures. Experiments have been presented, with a satisfactory agreement with 3D EM simulations. We will also discuss during the conference about tuning capabilities to improve drastically isolation levels and bandwidth.

Fig. 1 .

 1 Fig. 1. Cavity backed ring slot antenna: (a) top view (gray=metal); (b) feeding; (c) reflection coefficient for the fundament TM11 mode

Fig. 2 .Fig. 3 .

 23 Fig. 2. Description of the structure based upon a circular resonant open-SIW Cavity with internal decoupling walls: (a) oversized cavity with peripherical slot; (b) introduction of internal walls for decoupling; (c) 3D view in transparency

Fig. 4 .

 4 Fig. 4. Multi-ports cavity backed ring slot antenna: isolation performances between the different ports

Fig. 6 .

 6 Fig. 6. Prototype 1: 3-port antenna prototype with direct SMA connectors excitations.

Fig. 8 .

 8 Fig. 8. Measurement set-up for the tests in the anechoic chamber

ACKNOWLEDGMENT

This work has been supported by the "Région Bretagne" in the context of the regional PhD Program ARED.


Jeanne Malet-Villemagne
email: jeanne.malet@universite-paris-saclay.fr

Laurent Evanno

Sandrine Denis-Quanquin

Claire Janoir

Thomas Candela

The anchoring of

INTRODUCTION

C. difficile is an anaerobic, motile and spore-forming bacterium, responsible for 15 to 25% of post-antibiotic diarrhea and 95% of pseudomembranous colitis. While its toxins are described to be the major virulence factors in C. difficile infections, there is an increasing interest in the role of non-toxin factors in pathogenesis and virulence. In many other pathogens, cell wall glycopolymers influence the virulence. In C. difficile, three major carbohydrates are described : the polysaccharide I (PSI), the polysaccharide II (PSII) and the lipoteichoic acid (LTA). In the synthesis model suggested before 1 , the polysaccharide II is synthetized in the cytoplasm, then exported to the surface by a flippase and finally anchored into the peptidoglycan by two proteins of the LytR family : LcpA and LcpB. The two lcp genes are supposed to be non-redundant and were previously disrupted by insertional inactivation 1 .

AIMS OF THE STUDY

The aims of the study were first to investigate the role of the polysaccharide II and its correct anchoring for growth, surface set-up and survival of C. difficile and also to develop an innovative technique to study essential genes in C. difficile.

RESULTS

CONCLUSION -PERSPECTIVES

Thanks to the construction of the single lcp mutants by allelic exchange, we confirmed the phenotypes previously observed 1 . The impossibility to isolate a double lcp mutant clone tends to confirm the essentiality of the PSII anchoring into the peptidoglycan. This was confirmed using to the conditional-lethal mutant of both lcp, which is completely unable to grow without induction of the supplementary copy of lcpB added in the chromosome. When lcpB expression is very low, we observe ellipsoid cells, unable to elongate. Our results show the critical role of polysaccharide II anchoring in growth, elongation, and correct surface set-up of C. difficile. Our technique provides new opportunities to study essential genes in C. difficile.

▪→

  Obtention of a conditional-lethal mutant ▪ Single and double mutation of the lcp genes → Modulation of the gene expression to allow survival and observe phenotype → Double lcp mutant is not able to grow without ATc → LcpA and LcpB are redundant → PSII anchoring is essential ➢ ∆lcpA and ∆lcpB : constructed by allele-coupled exchange technique ➢ ∆lcpA ∆lcpB : unsuccessful to obtain double mutant Only WT revertants found in PCR over 457 clones screened It suggests that LcpA and LcpB have redundant functions and the activity of at least one Lcp seems to be essential for survival CONDITIONAL-LETHAL MUTANT TECHNIQUE TO STUDY ESSENTIAL GENES MUTANTS ANALYSIS BY IMMUNOFLUORESCENCE MICROSCOPY AND WESTERN BLOT ▪ Single mutants PSII layer ∆lcpA mutant strain has a holed layer of PSII and ∆lcpB mutant strain has a smooth layer and deformed cells ▪ Double conditional-lethal mutant PSII layer ▪ Protein pTet lcpB 3 : ∆lcpAB Ptet lcpB + plcpA (ATc 0) 4 : ∆lcpAB Ptet lcpB (ATc 10) 5 : ∆lcpAB Ptet lcpB (ATc 50) 6 : ∆lcpAB Ptet lcpB + plcpA (ATc 50) Low induction of lcpB gene leads to ellipsoid cells, unable to elongate correctly → PSII anchoring is important for a correct bacterial elongation SlpA and Cwp66 proteins are found in the culture supernatant of the mutant 1 Chu M et al, Plos Pathogens, 2016 From Heloïse Coullon 20µm 2µm 20µm