Emmanuelle Itam
email: itamemmanuelle@gmail.com

Stephen Wornom
email: stephen.wornom@inria.fr

Bruno Koobus
email: koobus@math.univ-montp2.fr

Alain Dervieux
email: alain.dervieux@inria.fr

APPLICATION OF A HYBRID VARIATIONAL MULTISCALE MODEL TO MASSIVELY SEPARATED FLOWS

Keywords: LES, hybrid, dynamic, VMS, blunt body, cylinder, tandem

ABSTRACT

The simulation of high-Reynolds number massively separated flows is a challenging problem of prime interest in industry. Even in this context we have to investigate numerical tools which are able to work in a Large Eddy Simulation mode as well as RANS or the hybrid RANS-LES mode. It is then necessary to use an adapted numerical technology: in the present work, the spatial discretization is based on a mixed finite element/finite volume formulation on unstructured grids. The numerical dissipation of the upwind scheme is made of sixth-order space derivatives and corresponds to a fifth-order term in truncation error, in order to limit as far as possible the interactions between numerical and subgrid scale (SGS) dissipation, [START_REF] Camarri | A low-diffusion MUSCL scheme for LES on unstructured grids[END_REF]. The turbulence model is based on a hybridization strategy which blends a variational multiscale large-eddy simulation (VMS-LES) equipped with dynamic SGS models and a two-equation RANS model. Particular attention is paid to the VMS-LES approach used in this work. The separation between the large and the small resolved scales is obtained through a variational projection operator based on spatial average on agglomerated cells [START_REF] Koobus | A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes-application to vortex shedding[END_REF]. The dynamic procedure (Germano) allows the adaptation of the constant of the SGS model to the spatial and temporal variation of the flow characteristics, while the VMS formulation restricts the SGS model effects to the smallest resolved scales [START_REF] Moussaed | Impact of dynamic subgrid-scale modeling in variational multiscale large-eddy simulation of bluff body flows[END_REF]. The hybridization strategy uses a blending parameter, such that a VMS-LES simulation is applied in region where the grid resolution is sufficiently fine to resolve a significant part of the turbulence fluctuations, while a RANS model is used in the regions of coarse grid resolution [START_REF] Moussaed | Simulation of the flow past a circular cylinder in the supercritical regime by blending RANS and variational-multiscale LES models[END_REF]. For instance, near the body surface, it is used under two forms, either as a statistical model for the turbulent boundary layer or as a (thinner) wall law for the LES part of the flow. The capability of the proposed hybrid model to accurately predict the aerodynamic forces acting on a circular cylinder in the supercritical regime and tandem cylinders is investigated. It is shown that rather coarse grids can be used to obtain a reasonable prediction of important bulk coefficients.

INTRODUCTION AND MOTIVATIONS

This work takes place in a study of numerical simulations adapted to industrial problems equipped with turbulence models suited to the simulation of turbulent flows with massive separations and vortex shedding. Recent publications concerning this study can be found in [START_REF] Moussaed | Impact of dynamic subgrid-scale modeling in variational multiscale large-eddy simulation of bluff body flows[END_REF][START_REF] Moussaed | Simulation of the flow past a circular cylinder in the supercritical regime by blending RANS and variational-multiscale LES models[END_REF]. The numerical methods are of low order, applicable on unstructured tetrahedrizations, and involve numerical dissipation. The dissipation is made of sixth-order derivatives, and advective accuracy can be as high as fifth order on Cartesian region of the mesh. The main issue which then arises is the choice of turbulence modelling which can be combined with this numerical technology.

In order to address high Reynolds numbers, we have to consider RANS-LES hybridization as in Detached Eddy Simulation (DES) [START_REF] Spalart | Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach[END_REF][START_REF] Spalart | A new version of detached-eddy simulation, resistant to ambiguous grid densities[END_REF]. However, DES is not designed for computing subcritical flows for which pure LES is a natural approach. Therefore, it is important to have a LES mode as accurate as possible. For example, the shear layer between main flow and wake can suffer from the tendancy of a Smagorinsky-like model to apply a too strong filtering in these regions. Two devices are considered to improve the accuracy of the LES component. First, the time-space strength of the filter is controlled by a dynamic process. Second, the width of the filter is numerically controlled by using a variational multiscale formulation.

The present work focuses on the following items:

• The evaluation of dynamic and hybrid VMS-LES models on the prediction of vortex shedding flows

• The simulation of the flow around circular and tandem cylinders : these flows involve many features and difficulties encountered in industrial problems and have been studied with well documented benchmarks. They are also the first step before the computation of array of cylinders (offshore oil and gas industries, civil engineering, aeronautics)

NUMERICAL MODEL : MIXED-ELEMENT-VOLUME METHOD

The spatial discretization uses a mixed finitevolume finite-element formulation, with degrees of freedom located at nodes i of the tetrahedrization.

The finite volume part is integrated on a dual mesh built from medians in 2D (Figure 1) and median plans in 3D. The diffusive fluxes are evaluated by a finite-

Figure 1. Dual cell in 2D

element method, the convective fluxes by a finitevolume method, with a sixth-order dissipation. Time integration is performed by an implicit second-order backward difference scheme, allowing to address flows at various Mach numbers.

TURBULENCE MODELING: VMS-LES

Let us explain our VMS-LES approach in a simplified context. Assume the mesh is made of two meshes embedded, corresponding to a P 1continuous finite-element approximation space V h with the usual basis functions Φ i vanishing on all vertices but vertex i. Let be V 2h its embedded coarse subspace V 2h . Let V h be the complementary space: V h = V 2h ⊕ V h . The space of small scales V h is spanned by only the fine basis functions Φ i related to vertices which are not vertices of V 2h . We write the compressible Navier-Stokes equations as follows:

∂W ∂t + ∇ • F(W) = 0 where W = t (ρ, ρu, ρv, ρw, ρE)
W h = W i Φ i : ∂W h ∂t , Φ i + (∇ • F(W h ), Φ i ) = -τ LES (W h ), Φ i
In practise, embedding two unstructured meshes V h and V 2h is a constraint that one wishes to avoid. The coarse level is then built from the agglomeration of vertices/cells as sketched in Figure 2. It remains to define the model term τ LES (W h ). This term represents the SGS viscosity term, acting only on small scales W h , and computed from the small scale component of the flow field by applying either a Smagorinsky or a WALE SGS model, the constants of these models being eventually estimated by the Germano-Lilly dynamic procedure.

The main property of the VMS formulation is that the modeling of the unresolved structures is influencing only the small resolved scales, in contrast with, for example, the usual Smagorinsky model. This implies two main properties. First, the backscatter transfer of energy to large scales is not damped by the model. Second, the model does not produce an artificial viscous layer at no-slip boundaries, nor in shear layers.

TURBULENCE MODELING: HYBRID RANS/VMS-LES

The computation of massively separated flows at high Reynolds number on unstructured mesh can be addressed with three families of models. RANS is robust but lacks accuracy in flow regions with massive separation (as the flow around bluff-bodies). LES, and in particular our VMS-LES is more expensive than RANS, since a very fine resolutioncomparable to DNS-is required in boundary layers at high Reynolds number. Hybrid formulations combine RANS and LES in order to exploit the advantages of the two approaches. Our goal is to build and evaluate a hybrid RANS/VMS-LES. The central idea of the proposed hybrid VMS model is to correct the mean flow field obtained with a RANS model by adding fluctuations given by a VMS-LES approach wherever the grid resolution is adequate. In the above simplified case it writes:

∂W h ∂t , Φ i + (∇ • F(W h ), Φ i ) = θ τ RANS ( W ), Φ i -(1 -θ) τ LES (W h ), Φ i
where the blending function θ = tanh(ξ 2 ) is defined either by a prescription of the user (zonal option) or from ξ = µ S GS /µ RANS or ξ = ∆/l RANS , allowing to have most regions with θ = 0, VMS-LES, or θ = 1, RANS. The RANS is a k -ε proposed by Goldberg et al. in [START_REF] Goldberg | A wall-distance-free k -ε model with Enhanced Near-Wall Treatment[END_REF]. This model relies on a unique damping function

µ t = C µ f µ ρ k 2 ; f µ = 1 -e -A µ R t 1 -e -R 1/2 t max(1, ψ -1 )
with C µ = 0.09, A µ = 0.01, and where ψ = R 1/2 t /C τ , R t = k 2 /(νε) is a turbulent Reynolds number. The transport equations become:

                                 ∂ ρk ∂t + ∂( ρû j k) ∂x j = ∂ µ + ρµ σ k ∂k ∂x j ∂x j + R i j ∂û i ∂x j -ρε, ∂ ρε ∂t + ∂( ρεû j ) ∂x j = ∂ µ+ ρµ σ ∂ε ∂x j ∂x j + C 1 R i j ∂û i ∂x j -C 2 ρε + E τ T -1 τ
whereand ∼ denote the Reynolds and Favre average respectively. In the above equations, T τ , representing the realizable time scale, and the extra term E τ are defined by:

T τ = k max(1, ψ -1 ) ; E τ = ρA E V(εT τ ) 0.5 ξ.
For our applications, we keep the constants proposed in [START_REF] Goldberg | A wall-distance-free k -ε model with Enhanced Near-Wall Treatment[END_REF]:

C τ = 1.41 C 1 = 1.42 C 2 = 1.83
and the extra term E τ is computed with: A E = 0.3, V = max( √ k, (νε) 0.25 ) and ξ = max( ∂k ∂x i ∂τ ∂x i , 0), where τ = k/ε. The interest of this model is its ability to detect the near-wall region without using a distance-to-wall, an important advantage when our unstructured-mesh method is applied to complex geometries. Furthermore, this model is well suited to the simulation of adverse pressure gradient flows, and in particular when flow separation occurs.

WALL TREATMENT

The title of this section could also be the title of the previous one since today's use of hybrid models is mainly concentrated on the boundary layers. But some other issues need to be discussed for wall treatment (WT). First, we need to identify which kind of boundary behavior appears in the family of flow studied here.

First, we can distinguish at least three types of behavior and then three combinations:

(a)-Laminar WT of LES for subcritical flow regime: the upstream flow is not turbulent and the boundary layer remains laminar, the turbulence being limited to a part of the wake at some distance from the body. This is the situation of subcritical flows around smooth bodies like circular cylinders.

(b)-Turbulent wall modeling for LES: the upstream flow is turbulent, with a high Reynolds number, but can be represented -far from wall-by an (unsteady) LES model. This is the wall-modeled LES (WMLES), see [START_REF] Piomelli | Wall-layer models for Large-Eddy Simulations[END_REF]. In this case, a wall law can be applied as far as only a part of the usual model is used, typically for y + less than 40.

(c)-Fully turbulent WT for supercritical flows: typically, the upstream flow is not turbulent, but the boundary layer has transitioned and becomes fully turbulent. The RANS, with or without wall law, needs to be used with y + more than 40.

Further, a fully predictive model should involve the modeling of the switching from one WT to another one:

(a') Laminar to fully turbulent (b') Fully turbulent to WMLES (c') Relaminarisation.

We do not address in the present paper the problem of the prediction of these switchings. For (c'), a discussion can be found in [START_REF] Shur | A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities[END_REF].

As concerned the WT itself, and from the standpoint of prediction accuracy, resolution to the wall is compulsory for the study of flows around blunt bodies, but still too computer intensive for the study of industrial geometries like risers with thousands-diameter span.

In the present study, we have applied the following rule-of-thumb:

Wall law modeling will relies on the Reichardt formula, [START_REF] Reichardt | Vollstaendige Darstellung der turbulenten Geschwindigkeitsverteilung in glatten Leitungen[END_REF]. The interest of this model is to describe, through one expression, the three types of behavior of the turbulent boundary layer, viz, the laminar layer, the logarithmic one and the intermediate (buffer) layer.

U τ = τ p ρ ; U + = U U τ ; y + = ρU τ µ y
in which U, ρ, µ and y are averaged tangential velocity, density, molecular viscosity and normal distance to the wall respectively, and U τ denotes the friction velocity.

For (a) this choice is delicate. It can be applied for y + less than 20. For (b), it can be used for y + less than 40. For (c) the wall law will be used with matching inside the logarithmic part, y + more than 40.

Resolution to the wall for (a),(b),(c) will similarly be applied by choosing a mesh with first layer of points corresponding to y + = 1, and defining near the wall several thickness corresponding to several "protection zones" for the hybrid model: For (a) by either forcing the hybrid blending parameter to 1 for y + less than 20, or preferably computing the boundary layer as laminar.

For (b) by forcing the hybrid blending parameter to 1 for y + less than 40.

For (c) by forcing the hybrid blending parameter to 1 for y + less than 200.

APPLICATIONS : CIRCULAR CYLINDER -SUBCRITICAL REGIME

The flow around a cylinder is strongly dependant on the turbulence which exists in the inflow and the turbulence which is created along the wall boundary layer after the stagnation point. In case where the inflow involves no turbulence, turbulence is created on the wall for a sufficiently high critical Reynolds number, corresponding to the drag crisis. This Reynolds number is between 300, 000 and 500, 000. Under this number, the flow is subcritical and, in short the boundary layer flow is laminar. In this case, we put θ = 0 in our model, which means that the VMS-LES model is combined with a laminar boundary layer, an option which is justified when the turbulence arising is essentially wake turbulence.

We choose here a rather low Reynolds of 3900 (Mach number is 0.1) and analyse in which conditions a better prediction is obtained with a very coarse mesh (290, 000 vertices) and a medium mesh (1, 400, 000 vertices). The WALE SGS model is used, and we compare the association of VMS-WALE with dynamic limitation and without. Surprisingly, the effect on wake velocity is small in general, but yet rather notable at the limit of the wake, see Figures 3. For a higher Reynolds of 20, 000, we observed that the effect of the dynamic procedure is rather determinant for the proper prediction of the turbulent intensity in the wake, particularly in VMS, see Figure 4. Global coefficient are also slightly improved, see Table 1 where our results are compared with some LES computations [START_REF] Lee | A dynamical subgrid-scale eddy viscosity model with a global model coefficient[END_REF][START_REF] Kravchenko | Numerical studies of flow over a circular cylinder at Re=3900[END_REF][START_REF] Parnaudeau | Experimental and numerical studies of the flow over a circular cylinder at Reynolds number 3900[END_REF] and measurements data [START_REF] Parnaudeau | Experimental and numerical studies of the flow over a circular cylinder at Reynolds number 3900[END_REF][START_REF] Norberg | Effects of Reynolds number and low-intensity free-sream turbulence on the flow around a circular cylinder[END_REF][START_REF] Ong | The velocity field of the turbulent very near wake of a circular cylinder[END_REF].

CIRCULAR CYLINDER -SUPERCRITI-CAL REGIME

We address now the flow around a circular cylinder at a Reynolds of 1 million, higher than the critical one. We propose to compare the results of three options, (a) θ = 0, that is RANS, (b) θ = 1, that is VMS-WALE, (c) hybrid RANS/VMS-WALE. Moreover, we try to get reasonable results with a medium-sized mesh of 1.2 million vertices. The available experiments are not so many, [START_REF] Shih | Experiments on flow past rough circular cylinders at large Reynolds numbers[END_REF][START_REF] Gölling | Experimental investigations of separating boundary-layer flow from circular cylinder at Reynolds numbers from 10 5 up to 10 7 ; Three-dimensional vortex flow of a circular cylinder[END_REF][START_REF] Zdravkovich | Flow around circular cylinders[END_REF], and show some scatter. Computations are also difficult to find in the litterature, we have found [START_REF] Wang | Prediction of high Reynolds number the flow over a circular cylinder using LES[END_REF][START_REF] Catalano | A new reconstruction scheme for the computation of inviscid compressible flows on 3D unstructured grids[END_REF][START_REF] Kim | Prediction of unsteady loading on a circular cylinder in high Reynolds number flows using large eddy simulation[END_REF], which present a higher scatter and deviation to measurements (Table 2). Among options (a),(b),(c), we observe that the hybrid calculation is in a rather good accordance with the measurements. A snapshot of the vorticity field computed by the hybrid model is depicted in Figure 5, which shows that rather fine details of the flow are still captured with the medium-sized mesh used in our simulation.

Mesh

TANDEM CYLINDER

The last case concerns the flow around two cylinder in tandem, at Re=1.66 × 10 5 . This is a test case studied in an AIAA workshop, see [START_REF] Lockard | Summary of the Tandem Cylinder Solutions from the Benchmark Problems for Airframe Noise Computations-I[END_REF]. Among the conclusions of the workshop, adressing this case with LES was considered as a too difficult task with existing computers. Results with DES where much closer to measurements. We have computed the case with mesh of 2.59 million nodes. Two modeling options where considered: (a) RANS and (b) Hybrid VMS-LES-WALE dynamic. An idea of the impact of these options on the resulting flow is proposed in Figure 6, where an excessive damping of the flow is observed for the RANS computation. Some comparison of bulk coefficients is given in Table 3 in which the other computational figures are taken from [START_REF] Vatsa | Assessment of Hybrid RANS/LES turbulence models for areoacoustics applications[END_REF] and for Lockard and Aybay from [START_REF] Lockard | Summary of the Tandem Cylinder Solutions from the Benchmark Problems for Airframe Noise Computations-I[END_REF]. The experimental drag values were obtained by integrating outputs found in [START_REF] Neuhart | Measurements of the flowfield interaction between tandem cylinders[END_REF]. Drag for the first cylinder show a small scatter with the various hybrid calculations.

In a first hybrid computation, with wall law, option (c), thick turbulent boundary layer has been applied to the two cylinders. Pressure distribution is shown in Figure 7.

In a second hybrid simulation, integration to the wall is used. The thick turbulent boundary layer is restricted to the front part of the first cylinder. Option (b) is applied to the rest of cylinder 1 wall. Cylinder 2 is not equiped with a protection zone. The simulation around the second cylinder is more challenging as also illustrated by the pressure distributions (Figure 8).

CONCLUSION, PERSPECTIVES

The main effect of dynamic control of SGS is a significant reduction of the amount of SGS viscosity in VMS-LES. The impact is in VMS-LES smaller that for pure LES but still non-negligible. In a large part of the tested cases we observe that the dynamic procedure brings a sensible improvement of the agreement with reference data. Then we have presented a strategy for blending RANS and VMS-LES. The supercrit-

Figure 2 .

 2 Figure 2. Building the VMS coarse level

Figure 3 .

 3 Figure 3. Cylinder, Re=3900: Mean streamwise velocity profile at x/D = 1.06, 1.54 and 2.02 and a zoom of the x/D = 1.06 cut (finer grid).

Figure 4 .

 4 Figure 4. Cylinder, Re=20, 000: turbulence intensity in the axis of wake: Dynamic improves LES (top)and strongly improves VMS (bottom).

Figure 5 .

 5 Figure 5. Flow around a circular cylinder at Re=1 million: vorticity field computed by the hybrid RANS/VMS-LES model.

Figure 6 .

 6 Figure 6. Tandem of cylinders: Vorticity magnitude -Hybrid VMS versus RANS.

Figure 7 .

 7 Figure 7. Tandem of cylinders with wall law and hybrid model: Mean pressure coefficient distribution.

Figure 8 .

 8 Figure 8. Tandem of cylinders with integration to wall: Mean pressure coefficient distribution.

Table 1 . Bulk quantities for Re=3900 flow around a cylinder (VMS-WALE vs VMS-WALE dynamic)

 1

			C d	l r /D -C p b	St
	VMS	270K 0.96 1.06	0.94	0.22
	VMS dyn.	270K 0.97 1.08	0.93	0.22
	VMS	1.4M 0.94 1.47	0.81	0.22
	VMS dyn.	1.4M 0.94 1.47	0.85	0.22
	LES
	Lee(Min)	7.7M 0.99 1.35	0.89	0.209
	Lee(Max)	7.7M 1.04 1.37	0.94	0.212
	Kravchenko
	(Min)	0.5	1.04	1.	0.93	0.193
	(Max)	2.4M 1.38 1.35	1.23	0.21
	Parneaudeau
	(Min)	44M	-	1.56	-	0.207
	(Max)	44M	-	1.56	-	0.209
	Experiments
	Parneaudeau
	(Min)		-	1.51	-	0.206
	(Max)		-	1.51	-	0.210
	Norberg
	(Min)		0.94	-	0.83
	(Max)		1.04	-	0.93
	Ong(Min)		-	-	-	0.205
	Ong(Max)		-	-	-	0.215

Table 2 . Flow around a circular cylinder at Re=1 million: bulk coefficients.

 2

Table 3 . Tandem of cylinders: bulk coefficients.

 3

ACKNOWLEDGEMENTS

This work has been supported by French National Research Agency (ANR) through project MAID-ESC n o ANR-13-MONU-0010. HPC resources from GENCI-[CINES] (Grant 2010-x2010026386 and 2010-c2009025067) are also gratefully acknowledged.


Emmanuelle Itam
email: emmanuelle.itam@umontpellier.fr

Stephen Wornom
email: stephen.wornom@inria.fr

Bruno Koobus
email: koobus@math.univ-montp2.fr

Bruno Sainte-Rose
email: bruno.sainte-rose@lemma-ing.com

Alain Dervieux
email: alain.dervieux@inria.fr

Simulation of multiple blunt-body flows with a hybrid variational multiscale model

Keywords: LES, hybrid, dynamic, VMS, blunt body, cylinder, tandem

. The capability of the proposed hybrid model to accurately predict the aerodynamic forces acting on a circular cylinder in the supercritical regime and on tandem cylinders are investigated.

INTRODUCTION AND MOTIVATIONS

This work takes place in a study of a numerical simulation method suited to industrial problems, equipped with turbulence models adapted to the simulation of turbulent flows with massive separations and vortex shedding. Recent publications concerning this study can be found in [START_REF] Moussaed | Impact of dynamic subgrid-scale modeling in variational multiscale large-eddy simulation of bluff body flows[END_REF], [START_REF] Moussaed | Simulation of the flow past a circular cylinder in the supercritical regime by blending RANS and variationalmultiscale LES models[END_REF]. The numerical methods are of low order, applicable on unstructured tetrahedral meshes, and involve numerical dissipation. But this dissipation is made of sixth-order derivatives and very low, and advective accuracy can be as high as fifth order on Cartesian region of the mesh. The main issue which then arises is the choice of turbulence modelling which can combine well with this numerical technology.

In order to address high Reynolds number flows, we have to consider RANS-LES hybridization as in Detached Eddy Simulation (DES) [START_REF] Spalart | Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach[END_REF][START_REF] Spalart | A new version of detached-eddy simulation, resistant to ambiguous grid densities[END_REF]. However DES is not designed for computing subcritical flows for which pure LES is a natural approach. Now, it is important to have a LES mode as accurate as possible. For example, the shear layer between main flow and wake can suffer from the tendancy of a Smagorinsky-like model to reinforce the filtering in these regions. Two techniques are considered to improve the accuracy of the LES component. First, the time-space strength of the filter is controlled by a dynamic process. Second, the width of the filter is numerically controlled by using a variational multiscale formulation.

The present work focuses on the following items:

• The evaluation of dynamic and hybrid VMS-LES models on the prediction of vortex shedding flows

• The simulation of the flow around circular and square cylinders : these flows involve many features and difficulties encountered in industrial problems and are studied in well documented benchmarks. They are also the first step before the computation of array of cylinders (offshore oil and gas industries, civil engineering, aeronautics)

NUMERICAL MODEL

The spatial discretization is based on a mixed finite-volume finite-element formulation, with degrees of freedom located at nodes i of the tetrahedrization. The finite volume part is integrated on a dual mesh built in 2D from median (Figure 1), in 3D from median plans. The diffusive fluxes are eval-

Figure 1. Dual cell in 2D
uated by a finite-element method, whereas a finitevolume method is used for the convective fluxes. The numerical approximation of the convective fluxes at the interface of neighboring cells is based on the Roe scheme [START_REF] Roe | Approximate Riemann solvers parameters vectors and difference schemes[END_REF].

In order to obtain second-order accuracy in space, the Monotone Upwind Scheme for Conservation Laws reconstruction method (MUSCL) [START_REF] Leer | Towards the ultimate conservative scheme. IV : a new approach to numerical convection[END_REF] is used, in which the Roe flux is expressed as a function of reconstructed values of the discrete flow variable W h at each side of the interface between two cells. We refer to [START_REF] Camarri | A low-diffusion MUSCL scheme for LES on unstructured grids[END_REF] for details on the definition of these reconstructed values. We just emphasize that particular attention has been paid to the dissipative properties of the resulting scheme since this is a key point for its successful use in LES. The numerical (spatial) dissipation provided by this scheme is made of sixthorder space derivatives [START_REF] Camarri | A low-diffusion MUSCL scheme for LES on unstructured grids[END_REF] and this is concentrated on a narrow-band of the highest resolved frequencies. This is expected to limit undesirable damping by numerical dissipation of the large scales. Moreover, a parameter γ S directly controls the amount of introduced viscosity and can be explicitly tuned in order to reduce it to the minimal amount needed to stabilize the simulation.

Time integration uses an implicit second-order backward differencing scheme. In [START_REF] Camarri | A low-diffusion MUSCL scheme for LES on unstructured grids[END_REF], it is shown that when used in combination with moderate time steps, the time dissipation is also quite small.

TURBULENCE MODEL: VMS-LES

The Variational Multiscale (VMS) model for the large eddy simulation (LES) of turbulent flows has been introduced in [START_REF] Hughes | Large eddy simulation and the variational multiscale method[END_REF] in combination with spectral methods. In [START_REF] Koobus | A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes-application to vortex shedding[END_REF], an extension to unstructured finite volumes is defined. That method is adapted in the present work. Let us explain this VMS-LES approach in a simplified context. Assume the mesh is made of two embedded meshes, corresponding to a P 1 -continuous finite-element approximation space V h with the usual basis functions Φ i vanishing on all vertices but vertex i. Let be V 2h its embedded coarse subspace V 2h . Let V h be the complementary space:

V h = V 2h ⊕ V h .
The space of small scales V h is spanned by only the fine basis functions Φ i related to vertices which are not vertices of V 2h . We write the compressible Navier-Stokes equations as follows:

∂W ∂t + ∇ • F(W) = 0 where W = (ρ, ρu, ρE). The VMS-LES discretization writes for W h = W i Φ i : ∂W h ∂t , Φ i + (∇ • F(W h ), Φ i ) = -τ LES (W h ), Φ i .( 1
)
For a test function related to a vertex of V 2h , the RHS vanishes, which limits the action of the LES term to small scales. In practice, embedding two unstructured meshes V h and V 2h is a constraint that we want to avoid. The coarse level is then built from the agglomeration of vertices/cells as sketched in Figure 2: It remains to define the model term τ LES (W h ). This term represents the SGS stress term, acting only on small scales W h , and computed from the small scale component of the flow field by applying either a Smagorinsky or a WALE SGS model, [START_REF] Nicoud | Subgridscale stress modelling based on the square of the velocity gradient tensor[END_REF], the constants of these models being eventually evaluated by the Germano-Lilly dynamic procedure [START_REF] Germano | A dynamic subgridscale eddy viscosity model[END_REF][START_REF] Lilly | A proposed modification of the Germano subgrid-scale closure method[END_REF]. The main property of the VMS formulation is that the modeling of the unresolved structures is influencing only the small resolved scales, as described in Figure 3, in contrast with, for example the usual Smagorinsky model. This implies two main properties. First, the backscatter transfer of energy to large scales is not damped by the model. Second, the model is naturally unable to produce an artificial viscous layer at a no-slip boundary, or in shear layers.

TURBULENCE MODEL: HYBRID RANS/VMS-LES

The computation of massively separated flows at high Reynolds numbers on unstructured meshes Reynolds-Averaged-Navier-Stokes (RANS) is robust but shows accuracy problems in flow regions with massive separation (such as the flow around bluffbodies). LES, and in particular our VMS-LES is more expensive than RANS, since a very fine resolution -somewhat comparable to DNS-is required in boundary layers at high Reynolds numbers. Hybrid formulations combine RANS and LES in order to exploit the advantages of the two approaches. Our goal is to build and evaluate a hybrid RANS/VMS-LES. The central idea of the proposed hybrid VMS model is to correct the mean flow field obtained with a RANS model by adding fluctuations given by a VMS-LES approach wherever the grid resolution is adequate. Our hybrid model can be shortly derived as follows.

First, let us write the semi-discretization of the RANS equations :

∂ W h ∂t , Φ i + (∇ • F( W h ), Φ i ) = -τ RANS ( W h ), Φ i (2
) in which W h denotes the RANS variables.

An equation for the resolved fluctuations

W c h = W h -W h
, where W h denotes the VMS-LES variables that satisfy Eqs. (1), can then be derived by substracting Eqs. ( 1) and (2) :

∂W h c ∂t , Φ i + (∇ • F(W h ), Φ i ) -(∇ • F( W h ), Φ i ) = τ RANS ( W ), Φ i -τ LES (W h ), Φ i (3)
To identify the regions where the additional fluctuations need to be computed, i.e. the regions where the grid resolution is adequate for VMS-LES, a blending function θ is introduced, which smoothly varies between 0 and 1, and which allows to recover the RANS approach for θ = 1 and the VMS-LES model for θ = 0. Thus Eqs. (3) become

∂W h c ∂t , Φ i + (∇ • F(W h ), Φ i ) -(∇ • F( W h ), Φ i ) = (1 -θ) τ RANS ( W ), Φ i -τ LES (W h ), Φ i (4)
In order to avoid the solution of two systems of equations, Eqs (2) and ( 4) are recasted together as follows :

∂W h ∂t , Φ i + (∇ • F(W h ), Φ i ) = -θ τ RANS ( W ), Φ i -(1 -θ) τ LES (W h ), Φ i (5)
where W h denotes now the hybrid variables. The blending function θ = tanh(ξ 2 ) is defined either by a prescription of the user (zonal option) or from ξ = µ S GS /µ RANS or ξ = ∆/l RANS . The RANS component used in the proposed hybrid approach is the k -ε model proposed by Goldberg and Ota [START_REF] Goldberg | A k -ε Near-Wall Formulation for Separated Flows[END_REF]. The hybrid variables then write W h = (ρ h , ρ h u h , ρ h E h , ρ h k h , ρ h ε h ).

RESULTS

For the simulations presented in this section, characteristic based conditions are used at the inflow and outflow as well as the lateral surface. In the spanwise direction, periodic boundary conditions are applied. No-slip conditions are imposed on the walls for moderate Reynolds numbers and a wall law approach is used for higher Reynolds numbers. The freestream Mach number is set to 0.1 in order to make a sensible comparison with incompressible simulations in the literature. Preconditioning is used to deal with the low-Mach number regime. For all simulations, starting from a uniform flow and once the flow is established, statistics are computed by averaging in time for, at least, 30 vortex-shedding cycles.

CIRCULAR CYLINDER -SUBCRITICAL

The presented results are compared with other LES computations [START_REF] Lee | A dynamical subgrid-scale eddy viscosity model with a global model coefficient[END_REF][START_REF] Kravchenko | Numerical studies of flow over a circular cylinder at Re=3900[END_REF][START_REF] Parneaudeau | Experimental and numerical studies of the flow over a circular cylinder at Reynolds number 3900[END_REF] and with measurements [START_REF] Parneaudeau | Experimental and numerical studies of the flow over a circular cylinder at Reynolds number 3900[END_REF][START_REF] Norberg | Effects of Reynolds number and low-intensity free-sream turbulence on the flow around a circular cylinder[END_REF][START_REF] Ong | The velocity field of the turbulent very near wake of a circular cylinder[END_REF]. The flow around a cylinder is strongly dependent on the turbulence which exists in the inflow and the turbulence which is created along the wall boundary layer after the stagnation point. In the case where the inflow involves no turbulence, turbulence is created on the wall at a sufficiently high critical Reynolds number, corresponding to the drag crisis. This Reynolds number is between 300, 000 and 500, 000. Under this number, the flow is subcritical and, in short the boundary layer flow is laminar. In this case, we put θ = 0 in our model, which means that the VMS-LES model is combined with a laminar boundary layer, an option which is justified when the turbulence arising is essentially wake turbulence. We consider first a rather low Reynolds of 3900 and analyse in which conditions a better prediction is obtained with a very coarse mesh (290, 000 vertices) and a medium mesh (1, 400, 000 vertices). The WALE SGS is used, and we compare the association of VMS-WALE with dynamic limitation and without. The effect of dynamic limitation results in a global reduction of SGS viscosity, see for example the contours of SGS viscosity obtained at Reynolds 20, 000 which are depicted in Figure 5. Surprisingly, the effect on wake velocity is small in general, but yet rather notable at the limit of wake, see Figures 4. Global coefficients are also slightly improved, see Table 1.

A second example is the flow around the circular cylinder with Reynolds number of 20, 000. In Figure 5 we show the rather high variation between the VMS viscosity with and without the dynamic control. The impact of the dynamic procedure on the turbulence intensity along the axis of the wake is important as shown in Figure 6.

SQUARE CYLINDER

As a second example to show the interest in combining VMS and dynamic control, we computed a classical workshop test case [START_REF] Rodi | Status of large eddy simulation: results of a Workshop[END_REF]. The incompressible flow around a square cylinder at Reynolds 22, 000 and zero angle of attack is considered. The VMS-LES option is applied on a mesh of 1.2M vertices. The influence of the dynamic procedure on the evaluation of the bulk coefficients is quite small, as shown in Table 2 which also compares the current computation with LES results from [START_REF] Wang | Prediction of high Reynolds number the flow over a circular cylinder using LES[END_REF][START_REF] Kim | Prediction of unsteady loading on a circular cylinder in high Reynolds number flows using large eddy simulation[END_REF] and measurements from [START_REF] Shih | Experiments on flow past rough circular cylinders at large Reynolds numbers[END_REF][START_REF] Goelling | Experimental investigations of separating boundary-layer flow from circular cylinder at Reynolds numbers from 10 5 up to 10 7 ; Three-dimensional vortex flow of a circular cylinder[END_REF][START_REF] Zdravkovich | Flow around circular cylinders[END_REF]. Conversely the deviation to measurements of the mean velocity profiles is greatly reduced when the dynamic control is used, see Figures 7,8. Square cylinder, Re=22K: mean streamwise velocity at x/D = 1.

CIRCULAR CYLINDER -SUPERCRIT-ICAL

We now address the flow around a circular cylinder at a Reynolds 1 million, higher than the critical one. We propose to compare the results of three options, (a) θ = 0, that is RANS, (b) θ = 1, that is VMS-WALE, (c) hybrid RANS-VMS-WALE. Moreover, we try to get reasonable results with a medium-sized mesh of 1.2 million vertices. The available experiments are scarce, [START_REF] Shih | Experiments on flow past rough circular cylinders at large Reynolds numbers[END_REF][START_REF] Goelling | Experimental investigations of separating boundary-layer flow from circular cylinder at Reynolds numbers from 10 5 up to 10 7 ; Three-dimensional vortex flow of a circular cylinder[END_REF][START_REF] Zdravkovich | Flow around circular cylinders[END_REF], and show some scatter. Numerical results are found [START_REF] Wang | Prediction of high Reynolds number the flow over a circular cylinder using LES[END_REF][START_REF] Kim | Prediction of unsteady loading on a circular cylinder in high Reynolds number flows using large eddy simulation[END_REF]. They present a even higher scatter (Table 3). Among options (a),(b),(c), we observe that the hybrid calculation is in a rather good agreement with the measurements.

TANDEM CYLINDERS

The last case concerns the flow around two cylinders in tandem, at Re=1.66 × 10 5 . This test case was studied in an AIAA workshop, see [START_REF] Lockard | Summary of the Tandem Cylinder Solutions from the Benchmark Problems for Airframe Noise Computations-I[END_REF]. Among the conclusions of the workshop, adressing this case with LES was considered as a too difficult task with existing computers. Results of DES-based computations were much closer to measurements. We have computed the case with a mesh of 2.59M nodes. Two modeling options were considered: (a) RANS and (b) Hybrid VMS-LES-WALE dynamic. An idea of the impact of these options on the resulting flow is proposed in Figure 10. Some comparison of bulk coefficients is given in Table 4 in which the other computational figures are taken from [START_REF] Vatsa | Assessment of Hybrid RANS/LES turbulence models for areoacoustics applications[END_REF] and for Lockard and Aybay from [START_REF] Lockard | Summary of the Tandem Cylinder Solutions from the Benchmark Problems for Airframe Noise Computations-I[END_REF]. Experimental outputs are taken from [START_REF] Neuhart | Measurements of the flowfield interaction between tandem cylinders[END_REF]. Drag for the first cylinder shows a small scatter with the various hybrid calculations. The simulation around the second cylinder is more challenging as also illustrated by the pressure distributions (Figure 11).

CONCLUSION, PERSPECTIVES

The main effect of dynamic control of SGS viscosity is a significant reduction of the amount of SGS viscosity in VMS-LES. The impact in VMS-LES is smaller than for pure LES but still non-negligible. In a large part of the tested cases we observe that the dynamic procedure brings a sensible improvement to the agreement with reference data. Then a strategy for blending RANS and VMS-LES is presented. The supercritical flow around a circular cylinder is reasonably well predicted by the proposed VMS hybrid model. The tandem cylinder case shows at the same time some improvement and that further research is necessary.

Figure 2 .

 2 Figure 2. Building the VMS coarse level

Figure 3 .

 3 Figure 3. VMS principle

Figure 4 .

 4 Figure 4. Cylinder, Re=3900: Mean streamwise velocity profiles at x/D = 1.06, 1.54 and 2.02 and a zoom of the x/D = 1.06 cut (finer grid).

Figure 5 .

 5 Figure 5. Cylinder, Re=20,000: viscosity ratio µ S GS /µ -non-dynamic VMS versus dynamic VMS

Figure 7 .

 7 Figure 7.Square cylinder, Re=22K: mean streamwise velocity on the wake center-line.

Figure 8 .

 8 Figure 8. Square cylinder, Re=22K: mean streamwise velocity at x/D = 1.

Figure 9 .

 9 Figure 9. Flow around a circular cylinder at Re=1 million: vorticity field.

Figure 10 .

 10 Figure 10. Tandem cylinders: Vorticity magnitude -Hybrid VMS versus RANS.

Figure 11 .

 11 Figure 11. Tandem of cylinders: Mean pressure coefficient distribution and mean C d .

Table 1 .

 1 Bulk quantities for Re=3900 flow around a cylinder (VMS-WALE vs VMS-WALE dynamic)

		Mesh C d	l r /D -C p b	St
	VMS	270K 0.96 1.06 0.94	0.22
	VMS dyn.	270K 0.97 1.08 0.93	0.22
	VMS	1.4M 0.94 1.47 0.81	0.22
	VMS dyn.	1.4M 0.94 1.47 0.85	0.22
	LES
	Lee(Min)	7.7M 0.99 1.35 0.89 0.209
	Lee(Max)	7.7M 1.04 1.37 0.94 0.212
	Kravchenko
	(Min)	0.5	1.04	1.	0.93 0.193
	(Max)	2.4M 1.38 1.35 1.23	0.21
	Parneaudeau
	(Min)	44M	-	1.56	-	0.207
	(Max)	44M	-	1.56	-	0.209
	Experiments
	Parneaudeau
	(Min)		-	1.51	-	0.206
	(Max)		-	1.51	-	0.210
	Norberg
	(Min)		0.94	-	0.83
	(Max)		1.04	-	0.93
	Ong(Min)		-	-	-	0.205
	Ong(Max)		-	-	-	0.215

Table 2 .

 2 Square cylinder, Re=22, 000: Bulk coefficients (LES: Rodi ; DNS: Verstappen)

		Mesh C d	l r D	St	C L	C d
	VMS:
		1.2M 2.08 0.74 0.127 1.38 0.25
	dyn.	1.2M 2.06 0.82 0.128 1.28 0.24
	LES
	(Min) 110K 1.66 0.89 0.07 0.38 0.1
	(Max) 3.8M 2.77 2.96 0.15 1.79 0.27
	DNS	3.7M 2.1	-	0.133 1.22 0.21
	Exp.
	Lyn	2.1 0.88 0.133	-	-
	Luo	2.21	-	0.13 1.21 0.18

Table 3 .

 3 Flow around a circular cylinder at Rey=1 million: bulk coefficients.

Table 4 .

 4 Mesh C d Cyl. 1 C d Cyl. 2 Tandem cylinders: bulk coefficients (experimental coefficients are computed by integrating experimental pressure of Neuhart).

	Hybrid VMS dyn.	2.59M	0.64	0.38
	Numerical results
	Lockard (2011)	2M-133M 0.33-0.80 0.29-0.52
	DES Aybay (2010)	6.7M	0.64	0.44
	HRLES Vatsa (2010)	8.7M	0.64	0.45
	Experiments		0.64	0.31

ACKNOWLEDGEMENTS

This work has been supported by French National Research Agency (ANR) through project MAIDESC n o ANR-13-MONU-0010. HPC resources from GENCI-[CINES] (Grant 2010-x2010026386 and 2010-c2009025067) are also gratefully acknowledged.


Emilie Devijver

AN ℓ 1 -ORACLE INEQUALITY FOR THE LASSO IN MULTIVARIATE FINITE MIXTURE OF MULTIVARIATE GAUSSIAN REGRESSION MODELS

Keywords: 1991 Mathematics Subject Classification. 62H30. Key words and phrases. Finite mixture of multivariate regression model, Lasso, ℓ 1 -oracle inequality

.

Introduction

Finite mixture regression models are useful for modeling the relationship between response and predictors, arising from different subpopulations. Due to recent improvements, we are faced with high-dimensional data where the number of covariables can be much larger than the sample size. We have to reduce the dimension to avoid identifiability problems. Considering a mixture of linear models, an assumption widely used is to say that only a few covariates explain the response. Among various methods, we focus on the ℓ 1 -penalized least squares estimator of parameters to lead to sparse regression matrix. Indeed, it is a convex surrogate for the non-convex ℓ 0 -penalization, and it produces sparse solutions. First introduced by Tibshirani in [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] in a linear model Y = Xβ + ǫ, where X ∈ R p , Y ∈ R, and ǫ ∼ N (0, σ 2 ), the Lasso estimator is defined in the linear model by βLasso (λ) = argmin β∈R p ||Y -Xβ|| 2 2 + λ||β|| 1 , λ > 0.

Many results have been proved to study the performance of this estimator. For example, cite [START_REF] Bickel | Simultaneous analysis of Lasso and Dantzig selector[END_REF] and [START_REF] Efron | Least angle regression[END_REF], for studying this estimator as a variable selection procedure in the linear model. Note that those results need strong assumptions on the Gram matrix X t X, as the restrictive eigenvalue condition, that can be not fulfilled in practice. A summary of assumptions and results is given by Bühlmann and van de Geer in [START_REF] Van De Geer | On the conditions used to prove oracle results for the Lasso[END_REF].

One can also cite van de Geer in [START_REF] Van De Geer | The adaptive and the thresholded lasso for potentially misspecified models (and a lower bound for the lasso)[END_REF] and discussions, who precises a chaining argument to perform rate, even in a non linear case.

If we assume that (x i , y i ) 1≤i≤n arise from different subpopulations, we could work with finite mixture regression models. Indeed, the homogeneity assumption of the linear model is often inadequate and restrictive. This model was introduced by Städler et al., in [START_REF] Städler | ℓ 1 -penalization for mixture regression models[END_REF]. They assume that, for i ∈ {1, . . . , n}, the observation y i , conditionally to X i = x i , comes from a conditional density s ξ 0 (.|x i ) which is a finite mixture of K Gaussian conditional densities with proportion vector π, where

Y i |X i = x i ∼ s ξ 0 (y i |x i ) = K k=1 π 0 k √ 2πσ 0 k exp - (y i -β 0 k x i ) 2 2(σ 0 k ) 2
for some parameters ξ 0 = (π 0 k , β 0 k , σ 0 k ) 1≤k≤K . They extend the Lasso estimator by ŝLasso (λ) = argmin

s K ξ    - 1 n n i=1 log(s K ξ (y i |x i )) + λ K k=1 π k p j=1 |σ -1 k [β k ] j |    , λ > 0 (1)
For this estimator, they provide an ℓ 0 -oracle inequality satisfied by ŝLasso (λ), according to the restricted eigenvalue condition also, and margin conditions, which leads to link the Kullback-Leibler loss function to the ℓ 2 -norm of the parameters. Another way to study this estimator is to look after the Lasso for its ℓ 1 -regularization properties. For example, cite [START_REF] Massart | The Lasso as an ℓ 1 -ball model selection procedure[END_REF], [START_REF] Meynet | An ℓ 1 -oracle inequality for the lasso in finite mixture gaussian regression models[END_REF], and [START_REF] Rigollet | Exponential screening and optimal rates of sparse estimation[END_REF]. Contrary to the ℓ 0 -results, some ℓ 1 -results are valid with no assumptions, neither on the Gram matrix, nor on the margin. This can be achieved due to the fact that they are looking for rate of convergence of order 1/ √ n rather than 1/n. For finite mixture Gaussian regression models, we could cite Meynet in [START_REF] Meynet | An ℓ 1 -oracle inequality for the lasso in finite mixture gaussian regression models[END_REF] who gives an ℓ 1 -oracle inequality for another extension of the Lasso estimator, defined by ŝLasso (λ) = argmin

s K ξ    - 1 n n i=1 log(s K ξ (y i |x i )) + λ K k=1 p j=1 |[β k ] j |    , λ > 0. (2)
In this article, we extend this result to finite mixture of multivariate Gaussian regression models. We will work with random multivariate variables (X, Y ) ∈ R p × R q . As in [START_REF] Meynet | An ℓ 1 -oracle inequality for the lasso in finite mixture gaussian regression models[END_REF], we shall restrict to the fixed design case, that is to say non-random regressors. We observe (x i ) 1≤i≤n . Without any restriction, we could assume that the regressors x i ∈ [0, 1] p for all i ∈ {1, . . . , n}. Under only bounded parameters assumption, we provide a lower bound on the Lasso regularization parameter λ which guarantees an oracle inequality. This result is non-asymptotic: the number of observations is fixed, and the number p of covariates can grow. Remark that the number K of clusters in the mixture is supposed to be known. Our result is deduced from a finite mixture multivariate Gaussian regression model selection theorem for ℓ 1 -penalized maximum likelihood conditional density estimation. We establish the general theorem following the one of Meynet in [START_REF] Meynet | An ℓ 1 -oracle inequality for the lasso in finite mixture gaussian regression models[END_REF], which combines Vapnik's structural risk minimization method (see Vapnik in [START_REF] Vapnik | Estimation of Dependences Based on Empirical Data: Springer Series in Statistics[END_REF]) and theory around model selection (see Le Pennec and Cohen in [START_REF] Cohen | Conditional density estimation by penalized likelihood model selection and applications[END_REF] and Massart in [START_REF] Massart | Concentration inequalities and model selection[END_REF]). As in Massart and Meynet in [START_REF] Massart | The Lasso as an ℓ 1 -ball model selection procedure[END_REF], our oracle inequality is deduced from this general theorem, the Lasso estimator being viewed as the solution of a penalized maximum likelihood model selection procedure over a countable collection of ℓ 1 -ball models. The article is organized as follows. The model and the framework are introduced in Section 2. In Section 3, we state the main result of the article, which is an ℓ 1 -oracle inequality satisfied by the Lasso in finite mixture of multivariate Gaussian regression models. Section 4 is devoted to the proof of this result and of the general theorem, deriving from two easier propositions. Those propositions are proved in Section 5, whereas details of lemma states in Section 6.

Notations and framework

2.1. Finite mixture regression model. We observe n independent couples (x, y) = (x i , y i ) 1≤i≤n ∈ ([0, 1] p × R q ) n , with y i ∈ R q a random observation, realization of variable Y i , and x i ∈ [0, 1] p fixed for all i ∈ {1, . . . , n}. We assume that, conditionally to the x i s, the Y i s are independent and identically distributed with conditional density s ξ 0 (.|x i ), which is a finite mixture of K Gaussian regressions with unknown parameters ξ 0 . In this article, K is fixed, then we do not precise it with unknown parameters. We will estimate the unknown conditional density by a finite mixture of K Gaussian regressions. Each subpopulation is then estimated by a multivariate linear model. Detail the conditional density s ξ . For all y ∈ R q , for all x ∈ [0, 1] p ,

s ξ (y|x) = K k=1 π k (2π) q/2 det(Σ k ) 1/2 exp - (y -β k x) t Σ -1 k (y -β k x) 2 (3) ξ = (π 1 , . . . , π K , β 1 , . . . , β K , Σ 1 , . . . , Σ K ) ∈ Ξ = Π K × (R q×p ) K × (S ++ q ) K Π K = (π 1 , . . . , π K ); π k > 0 for all k ∈ {1, . . . , K} and K k=1 π k = 1 S ++ q
is the set of symmetric positive definite matrices on R q .

We want to estimate ξ 0 from the observations. For all cluster k ∈ {1, . . . , K}, β k is the matrix of regression coefficients, and Σ k is the covariance matrix in the mixture component k, whereas the π k s are the mixture proportions. For x ∈ [0, 1] p , we define the parameter ξ(x) of the conditional density s ξ (.|x) by

ξ(x) = (π 1 , . . . , π K , β 1 x, . . . , β K x, Σ 1 , . . . , Σ K ) ∈]0, 1[ K ×(R q ) K × (S ++ q ) K . For all k ∈ {1, . . . , K}, for all x ∈ [0, 1] p , for all z ∈ {1, . . . , q}, [β k x] z = p j=1 [β k ] z,j [x] j ,
and then β k x ∈ R q is the mean vector of the mixture component k for the conditional density s ξ (.|x).

2.2.

Boundedness assumption on the mixture and component parameters. Denote, for a matrix A, m(A) the modulus of the smallest eigenvalue of A, and M (A) the modulus of the largest eigenvalue of A. We shall restrict our study to bounded parameters vector ξ = (π, β, Σ), where π = (π 1 , . . . , π K ), β = (β 1 , . . . , β K ), Σ = (Σ 1 , . . . , Σ K ). Specifically, we assume that there exists deterministic positive constants A β , a Σ , A Σ , a π such that ξ belongs to Ξ, with (4) Ξ = ξ ∈ Ξ : for all k ∈ {1, . . . , K}, max z∈{1,...,q}

sup x∈[0,1] p |[β k x] z | ≤ A β , a Σ ≤ m(Σ -1 k ) ≤ M (Σ -1 k ) ≤ A Σ , a π ≤ π k .
Let S the set of conditional densities s ξ , S = s ξ , ξ ∈ Ξ .

2.3.

Maximum likelihood estimator and penalization. In a maximum likelihood approach, we consider the Kullback-Leibler information as the loss function, which is defined for two densities s and t by KL(s, t) = R q log s(y) t(y) s(y)dy if sdy << tdy; +∞ otherwise.

In a regression framework, we have to adapt this definition to take into account the structure of conditional densities. For the fixed covariates (x 1 , . . . , x n ), we consider the average loss function

KL n (s, t) = 1 n n i=1 KL(s(.|x i ), t(.|x i )) = 1 n n i=1 R q log s(y|x i ) t(y|x i ) s(y|x i )dy.
Using the maximum likelihood approach, we want to estimate s ξ 0 by the conditional density s ξ which maximizes the likelihood conditionally to (x i ) 1≤i≤n . Nevertheless, because we work with high-dimensional data, we have to regularize the maximum likelihood estimator. We consider the ℓ 1 -regularization, and a generalization of the estimator associated, the Lasso estimator, which we define by ŝLasso (λ) := argmin

s ξ ∈S ξ=(π,β,Σ)    - 1 n n i=1 log(s ξ (y i |x i )) + λ K k=1 q z=1 p j=1 |[β k ] z,j |    ;
where λ > 0 is a regularization parameter. We define also, for s ξ defined as in (3), and with parameters ξ = (π, β, Σ),

N [2] 1 (s ξ ) = ||β|| 1 = K k=1 p j=1 q z=1 |[β k ] z,j |. (5)

Oracle inequality

In this section, we provide an ℓ 1 -oracle inequality satisfied by the Lasso estimator in finite mixture multivariate Gaussian regression models, which is the main result of this article.

Theorem 3.1. We observe n couples (x, y) = ((x 1 , y 1 ), . . . , (x n , y n )) ∈ ([0, 1] p × R q ) n coming from the conditional density s ξ 0 , where ξ 0 ∈ Ξ, where

Ξ = ξ ∈ Ξ : for all k ∈ {1, . . . , K}, max z∈{1,...,q} sup x∈[0,1] p |[β k x] z | ≤ A β , a Σ ≤ m(Σ -1 k ) ≤ M (Σ -1 k ) ≤ A Σ , a π ≤ π k .
Denote by a ∨ b = max(a, b).

We define the Lasso estimator, denoted by ŝLasso (λ), for λ ≥ 0, by ŝLasso (λ) = argmin

s ξ ∈S - 1 n n i=1 log(s ξ (y i |x i )) + λN [2]
1 (s ξ ) ; ( 6)

with S = s ξ , ξ ∈ Ξ
and where, for ξ = (π, β, Σ),

N [2] 1 (s ξ ) = ||β|| 1 = K k=1 p j=1 q z=1 |[β k ] z,j |. Then, if λ ≥ κ A Σ ∨ 1 a π 1 + 4(q + 1)A Σ A 2 β + log(n) a Σ K n 1 + q log(n) K log(2p + 1)
with κ an absolute positive constant, the estimator (6) satisfies the following ℓ 1 -oracle inequality.

E[KL n (s ξ 0 , ŝLasso (λ))] ≤ (1 + κ -1 ) inf s ξ ∈S KL n (s ξ 0 , s ξ ) + λN [2] 1 (s ξ ) + λ + κ ′ K n e -1 2 π q/2 a π A q/2 Σ 2q + κ ′ K n A Σ ∨ 1 a π 1 + 4(q + 1)A Σ A 2 β + log(n) a Σ ×K 1 + A β + q a Σ 2
where κ ′ is a positive constant.

This theorem provides information about the performance of the Lasso as an ℓ 1 -regularization algorithm.

If the regularization parameter λ is properly chosen, the Lasso estimator, which is the solution of the ℓ 1penalized empirical risk minimization problem, behaves as well as the deterministic Lasso, which is the solution of the ℓ 1 -penalized true risk minimization problem, up to an error term of order λ.

Our result is non-asymptotic: the number n of observations is fixed while the number p of covariates and the size q of the response can grow with respect to n and can be much larger than n. The number K of clusters in the mixture is fixed.

There is no assumption neither on the Gram matrix, nor on the margin, which are classical assumptions for oracle inequality for the Lasso estimator. Moreover, this kind of assumptions involve unknown constants, whereas here, every constants are explicit. We could compare this result with the ℓ 0 -oracle inequality established in [START_REF] Städler | ℓ 1 -penalization for mixture regression models[END_REF], which needs those assumptions, and is therefore difficult to interpret. Nevertheless, they get faster rate, the error term in the oracle inequality being of order 1/n rather than 1/ √ n. The main assumption we make to establish the Theorem 3.1 is the boundedness of the parameters, which is also assumed in [START_REF] Städler | ℓ 1 -penalization for mixture regression models[END_REF]. It is needed to tackle the problem of the unboundedness of the likelihood (see [START_REF] Mclachlan | Finite Mixture Models[END_REF] for example). Moreover, we let regressors to belong to [0, 1] p . Because we work with fixed covariates, they are finite. To simplify the reading, we choose to rescale x to get ||x|| ∞ ≤ 1. Nevertheless, if we not rescale the covariates, and the regularization parameter λ bound and the error term of the oracle inequality depend linearly of ||x|| ∞ . The regularization parameter λ bound is of order (q 2 + q)/ √ n log(n) 2 log(2p + 1). For q = 1, we recognize the same order, as regards to the sample size n and the number of covariates p, to the ℓ 1 -oracle inequality in [START_REF] Meynet | An ℓ 1 -oracle inequality for the lasso in finite mixture gaussian regression models[END_REF]. A great attention has been paid to get a lower bound of λ with optimal dependence on p, which is the number of regressors, but we are aware that dependences in q and K may not be optimal. Indeed, even if roles of p and q are not symmetric, we can wonder if a dependence of order logarithm in q could be expected, which is not achieved here. For the number of components, a dependence in √ K could be envisaged, see [START_REF] Meynet | An ℓ 1 -oracle inequality for the lasso in finite mixture gaussian regression models[END_REF]. Those optimal rates are open problems.

Van de Geer, in [START_REF] Van De Geer | The adaptive and the thresholded lasso for potentially misspecified models (and a lower bound for the lasso)[END_REF], gives some tools to improve the bound of the regularization parameter to log(p) n . Nevertheless, we have to control eigenvalues of the Gram matrix of some functions (ψ j (x i )) 1≤j≤D 1≤i≤n , D being the number of parameters to estimate, where ψ j (x i ) satisfies

| log(s ξ (y i |x i )) -log(s ξ (y i |x i ))| ≤ D j=1 |ξ j -ξj |ψ j (x i ).
In our case of mixture of regression models, control eigenvalues of the Gram matrix of functions (ψ j (x i )) 1≤j≤D 1≤i≤n corresponds to make some assumptions, as REC, to avoid dimension reliance on n, K and p. Without this kind of assumptions, we could not guarantee that our bound is of order log(p) n , because we could not guarantee that eigenvalues does not depend on dimensions. In order to get a result with smaller assumptions, we do not use the chaining argument developed in [START_REF] Van De Geer | The adaptive and the thresholded lasso for potentially misspecified models (and a lower bound for the lasso)[END_REF]. Nevertheless, one can easily compute that, under restricted eigenvalue condition, we could perform the order of the regularization parameter to λ ≍ log(p) n log(n).

Proof of the oracle inequality

4.1. Main propositions used in this proof. The first result we will prove is the next theorem, which is an ℓ 1 -ball mixture multivariate regression model selection theorem for ℓ 1 -penalized maximum likelihood conditional density estimation in the Gaussian framework. Theorem 4.1. We observe (x i , y i ) 1≤i≤n with unknown conditional Gaussian mixture density s ξ 0 . For all m ∈ N * , we consider the ℓ 1 -ball S m = {s ξ ∈ S, N [START_REF] Boucheron | Concentration Inequalities: A Nonasymptotic Theory of Independence[END_REF] 1 (s ξ ) ≤ m} for S = {s ξ , ξ ∈ Ξ}, and Ξ defined by

Ξ = ξ ∈ Ξ : for all k ∈ {1, . . . , K}, max z∈{1,...,q} sup x∈[0,1] p |[β k x] z | ≤ A β , a Σ ≤ m(||Σ -1 k ||) ≤ M (Σ -1 k ) ≤ A Σ , a π ≤ π k . For ξ = (π, β, Σ), let N [2] 1 (s ξ ) = ||β|| 1 = K k=1 p j=1 q z=1 |[β k ] z,j |.
Let ŝm an η m -log-likelihood minimizer in S m , for η m ≥ 0:

- 1 n n i=1 log(ŝ m (y i |x i )) ≤ inf sm∈Sm - 1 n n i=1 log(s m (y i |x i )) + η m .
Assume that for all m ∈ N * , the penalty function satisfies pen(m) = λm with

λ ≥ κ A Σ ∨ 1 a π 1 + 4(q + 1)A Σ A 2 β + log(n) a Σ K n 1 + q log(n) K log(2p + 1)
for a constant κ. Then, if m is such that

- 1 n n i=1 log(ŝ m(y i |x i )) + pen( m) ≤ inf m∈N * - 1 n n i=1 log(ŝ m (y i |x i )) + pen(m) + η for η ≥ 0, the estimator ŝ m satisfies E(KL n (s ξ 0 , ŝ m)) ≤(1 + κ -1 ) inf m∈N * inf sm∈Sm KL n (s ξ 0 , s m ) + pen(m) + η m + η + κ ′ K n e -1 2 π q/2 A q/2 Σ 2qa π + κ ′ K n K A Σ ∨ 1 a π 1 + 4(q + 1) 2 A Σ A 2 β + log(n) a σ × 1 + A β + q a Σ 2 ;
where κ ′ is a positive constant.

It is an ℓ 1 -ball mixture regression model selection theorem for ℓ 1 -penalized maximum likelihood conditional density estimation in the Gaussian framework. Its proof could be deduced from the two following propositions, which split the result if the variable Y is large enough or not.

Proposition 4.2. We observe (x i , y i ) 1≤i≤n , with unknown conditional density denoted by s ξ 0 . Let M n > 0, and consider the event

T := max i∈{1,...,n} max z∈{1,...,q} |[Y i ] z | ≤ M n .
For all m ∈ N * , we consider the ℓ 1 -ball

S m = {s ξ ∈ S, N [2]
1 (s ξ ) ≤ m} where S = {s ξ , ξ ∈ Ξ} and

N [2] 1 (s ξ ) = ||β|| 1 = K k=1 p j=1 q z=1 |[β k ] z,j | for ξ = (π, β, Σ).
Let ŝm an η m -log-likelihood minimizer in S m , for η m ≥ 0:

- 1 n n i=1 log(ŝ m (y i |x i )) ≤ inf sm∈Sm - 1 n n i=1 log(s m (y i |x i )) + η m . Let C Mn = max 1 aπ , A Σ + 1 2 (|M n | + A β ) 2 A 2 Σ , q(|Mn|+A β )AΣ 2
. Assume that for all m ∈ N * , the penalty function satisfies pen(m) = λm with

λ ≥ κ 4C Mn √ n √ K 1 + 9q log(n) K log(2p + 1)
for some absolute constant κ. Then, any estimate ŝ m with m such that

- 1 n n i=1 log(ŝ m(y i |x i )) + pen( m) ≤ inf m∈N * - 1 n n i=1 log(ŝ m (y i |x i )) + pen(m) + η for η ≥ 0, satisfies E(KL n (s ξ 0 , ŝ m)1 T ) ≤(1 + κ -1 ) inf m∈N * inf sm∈Sm KL n (s ξ 0 , s m ) + pen(m) + η m + κ ′ K 3/2 qC Mn √ n 1 + A β + q a Σ 2 ;
where κ ′ is an absolute positive constant.

Proposition 4.3. Let s ξ 0 , T and ŝ m defined as in the previous proposition. Then,

E(KL n (s ξ 0 , ŝ m)1 T c ) ≤ e -1/2 π q/2 A q/2 Σ 2Knqa π e -1/4(M 2 n -2MnA β )aΣ .
4.2. Notations. To prove those two propositions, and the theorem, begin with some notations.

For any measurable function g : R q → R , we consider the empirical norm

g n := 1 n n i=1 g 2 (y i |x i ); its conditional expectation E(g(Y |X)) = R q
g(y|x)s ξ 0 (y|x)dy; its empirical process

P n (g) := 1 n n i=1 g(y i |x i );
and its normalized process

ν n (g) := P n (g) -E X (P n (g)) = 1 n n i=1 g(y i |x i ) - R q g(y|x i )s ξ 0 (y|x i )dy .
For all m ∈ N * , for all model S m , we define F m by

F m = f m = -log s m s ξ 0 , s m ∈ S m .
Let δ KL > 0. For all m ∈ N * , let η m ≥ 0. There exist two functions, denoted by ŝ m and sm , belonging to S m , such that

P n (-log(ŝ m)) ≤ inf sm∈Sm P n (-log(s m )) + η m ; (7) KL n (s ξ 0 , sm ) ≤ inf sm∈Sm KL n (s ξ 0 , s m ) + δ KL .
Denote by fm :=log ŝm

C Mn = max 1 aπ , A Σ + 1 2 (|M n | + A β ) 2 A 2 Σ , q(|M n | + A β )A Σ /2 . Assume that, for all m ∈ N * , pen(m) = λm, with λ ≥ κC Mn K n 1 + q log(n) K log(2p + 1) .
We derive from the two propositions that there exists κ ′ such that, if m satisfies

- 1 n n i=1 log(ŝ m(y i |x i )) + pen( m) ≤ inf m∈N * - 1 n n i=1 log(ŝ m (y i |x i )) + pen(m) + η; then ŝ m satisfies E(KL n (s ξ 0 , ŝ m)) = E(KL n (s ξ 0 , ŝ m)1 T ) + E(KL n (s ξ 0 , ŝ m)1 T c ) ≤(1 + κ -1 ) inf m∈N * inf sm∈Sm KL n (s ξ 0 , s m ) + pen(m) + η m + κ ′ C Mn √ n K 3/2 q 1 + (A β + q a Σ ) 2 + η + κ ′ e -1/2 π q/2 A q/2 Σ 2Knqa π e -1/4(M 2 n -2MnA β )aΣ .
In order to optimize this equation with respect to M n , we consider M n the positive solution of the polynomial

log(n) - 1 4 (X 2 -2XA β )a Σ = 0; we obtain M n = A β + A 2 β + 4 log(n) aΣ and √ ne -1/4(M 2 n -2MnA β )aΣ = 1 √ n .
On the other hand,

C Mn ≤ A Σ ∨ 1 a π 1 + q + 1 2 A Σ (M n + A β ) 2 ≤ A Σ ∨ 1 a π 1 + 4(q + 1)A Σ A 2 β + log(n) a Σ .
We get

E(KL n (s ξ 0 , ŝ m)) = E(KL n (s ξ 0 , ŝ m)1 T ) + E(KL n (s ξ 0 , ŝ m)1 T c ) ≤ (1 + κ -1 ) inf m∈N * inf sm∈Sm KL n (s ξ 0 , s m ) + pen(m) + η m + η +κ ′ K n e -1 2 π q/2 (qA Σ ) q/2 2qa π +κ ′ K n A Σ ∨ 1 a π 1 + 4(q + 1)A Σ A 2 β + log(n) a Σ ×K 1 + A β + q a Σ 2 .
4.4. Proof of the Theorem 3.1. We will show that there exists η m ≥ 0, and η ≥ 0 such that ŝLasso (λ) satisfies the hypothesis of the Theorem 4.1, which will lead to Theorem 3.1. First, let show that there exists m ∈ N * and η m ≥ 0 such that the Lasso estimator is an η m -log-likelihood minimizer in S m .

For all λ ≥ 0, if

m λ = ⌈N [2] 1 (ŝ(λ))⌉, ŝLasso (λ) = argmin s∈S N [2] 1 (s)≤m λ - 1 n n i=1 log(s(y i |x i )) .
We could take η m = 0. Secondly, let show that there exists η ≥ 0 such that

- 1 n n i=1 log(ŝ Lasso (λ)(y i |x i )) + pen(m λ ) ≤ inf m∈N * - 1 n n i=1 log(ŝ m (y i |x i )) + pen(m) + η.
Taking pen(m λ ) = λm λ ,

- 1 n n i=1 log(ŝ Lasso (λ)(y i |x i )) + pen(m λ ) = - 1 n n i=1 log(ŝ Lasso (λ)(y i |x i )) + λm λ ≤ - 1 n n i=1 log(ŝ Lasso (λ)(y i |x i )) + λN [2] 1 (ŝ Lasso (λ)) + λ ≤ inf s ξ ∈S - 1 n n i=1 log(s ξ (y i |x i )) + λN [2] 1 (s ξ ) + λ ≤ inf m∈N * inf s ξ ∈Sm - 1 n n i=1 log(s ξ (y i |x i )) + λN [2] 1 (s ξ ) + λ ≤ inf m∈N * inf s ξ ∈Sm - 1 n n i=1 log(s ξ (y i |x i )) + λm + λ ≤ inf m∈N * - 1 n n i=1 log(ŝ m (y i |x i )) + λm + λ;
which is exactly the goal, with η = λ. Then, according to the Theorem 4.1, with m = m λ , and ŝ m = ŝLasso (λ), for

λ ≥ κ A Σ ∨ 1 a π 1 + 4(q + 1)A Σ A 2 β + log(n) a Σ K n 1 + q log(n) K log(2p + 1) ,
we get the oracle inequality.

5. Proof of the theorem according to T or T c 5.1. Proof of the Proposition 4.2. This proposition corresponds to the main theorem according to the event T . To prove it, we need some preliminary results. From our notations, reminded in Section 4.2, we have, for all m ∈ N * for all m ′ ∈ M (m),

P n ( fm ′ ) + pen(m ′ ) ≤ P n ( fm ) + pen(m) + η ≤ P n ( fm ) + pen(m) + η m + η; E(P n ( fm ′ )) + pen(m ′ ) ≤ E(P n ( fm )) + pen(m) + η m + η + ν n ( fm ) -ν n ( fm ′ ); KL n (s ξ 0 , ŝm ′ ) + pen(m ′ ) ≤ inf sm∈Sm KL n (s ξ 0 , s m ) + δ KL + pen(m) + η m + η + ν n ( fm ) -ν n ( fm ′ ); (9)
thanks to the inequality [START_REF] Mclachlan | Finite Mixture Models[END_REF]. The goal is to bound -ν n ( fm ′ ) = ν n (-fm ′ ). To control this term, we use the following lemma.

|[Y i ] z | ≤ M n . Let C Mn = max 1 aπ , A Σ + 1 2 (|M n | + A β ) 2 A 2 Σ , q(|Mn|+A β )AΣ 2
and

∆ m ′ = m ′ log(n) K log(2p + 1) + 6 1 + K A β + q a Σ .
Then, on the event T , for all m ′ ∈ N * , for all t > 0, with probability greater than 1e -t , sup

f m ′ ∈F m ′ |ν n (-f m ′ )| ≤ 4C Mn √ n 9 √ Kq∆ m ′ + √ 2 √ t 1 + K A β + q a Σ Proof. Page 13
From ( 9), on the event T , for all m ∈ N * , for all m ′ ∈ M (m), for all t > 0, with probability greater than 1e -t ,

KL n (s ξ 0 , ŝm ′ ) + pen(m ′ ) ≤ inf sm∈Sm KL n (s ξ 0 , s m ) + δ KL + pen(m) + ν n ( fm ) + 4C Mn √ n 9 √ Kq∆ m ′ + √ 2 √ t 1 + K A β + q a Σ + η m + η ≤ inf sm∈Sm KL n (s ξ 0 , s m ) + pen(m) + ν n ( fm ) + 4 C Mn √ n 9 √ Kq∆ m ′ + 1 2 √ K 1 + K A β + q a Σ 2 + √ Kt + η m + η + δ KL , the last inequality being true because 2ab ≤ 1 √ K a 2 + √ Kb 2 . Let z > 0 such that t = z + m + m ′ . On the event T , for all m ∈ N, for all m ′ ∈ M (m), with probability greater than 1 -e -(z+m+m ′ ) , KL n (s ξ 0 , ŝm ′ ) + pen(m ′ ) ≤ inf sm∈Sm KL n (s ξ 0 , s m ) + pen(m) + ν n ( fm ) + 4 C Mn √ n 9 √ Kq∆ m ′ + 1 2 √ K 1 + K A β + q a Σ 2 + 4 C Mn √ n √ K(z + m + m ′ ) + η m + η + δ KL . KL n (s ξ 0 , ŝm ′ ) -ν n ( fm ) ≤ inf sm∈Sm KL n (s ξ 0 , s m ) + pen(m) + 4 C Mn √ n √ Km + 4C Mn √ n √ K(m ′ + 9q∆ m ′ ) -pen(m ′ ) + 4C Mn √ n 1 2 √ K 1 + K A β + q a Σ 2 + √ Kz + η m + η + δ KL .
Let κ ≥ 1, and assume that pen(m) = λm with

λ ≥ 4C Mn √ n √ K 1 + 9q log(n) K log(2p + 1)
Then, as

∆ m ′ = m ′ log(n) K log(2p + 1) + 6 1 + K A β + q a Σ , with κ -1 = 4C Mn √ n √ K 1 λ ≤ 1 1 + 9q log(n) K log(2p + 1) , we get that KL n (s ξ 0 , ŝm ′ ) -ν n ( fm ) ≤ inf sm∈Sm KL n (s ξ 0 , s m ) + (1 + κ -1 ) pen(m) + 4C Mn √ n 1 2 √ K 1 + K A β + q a Σ 2 + 4C Mn √ n 54 √ Kq 1 + K A β + q a Σ + √ Kz + η + δ KL + η m ≤ inf sm∈Sm KL n (s ξ 0 , s m ) + (1 + κ -1 ) pen(m) + 4C Mn √ n 27K 3/2 + 27 + 1/2 √ K 1 + K A β + q a Σ 2 + √ Kz + η m + η + δ KL .
Let m such that

- 1 n n i=1 log(ŝ m(y i |x i )) + pen( m) ≤ inf m∈N * - 1 n n i=1 log(ŝ m (y i |x i )) + pen(m) + η;
and

M (m) = {m ′ ∈ N * |P n (-log(ŝ m ′ )) + pen(m ′ ) ≤ P n (-log(ŝ m )) + pen(m) + η} . By definition, m ∈ M (m). Because for all m ∈ N * , for all m ′ ∈ M (m), 1 - m∈N * m ′ ∈M (m) e -(z+m+m ′ ) ≥ 1 -e -z (m,m ′ )∈(N * ) 2 e -m-m ′ ≥ 1 -e -z ,
we could sum up over all models. On the event T , for all z > 0, with probability greater than 1e -z ,

KL n (s ξ 0 , ŝ m) -ν n ( fm ) ≤ inf m∈N * inf sm∈Sm KL n (s ξ 0 , s m ) + (1 + κ -1 ) pen(m) + η m + 4C Mn √ n 27K 3/2 + 55q 2 √ K 1 + K A β + q a Σ 2 + √ Kz + η + δ KL .
By integrating over z > 0, and noticing that E(ν n ( fm )) = 0 and that δ KL can be chosen arbitrary small, we get

E(KL n (s ξ 0 , ŝ m)1 T ) ≤ inf m∈N * inf sm∈Sm KL n (s ξ 0 , s m ) + (1 + κ -1 ) pen(m) + η m + 4C Mn √ n 27K 3/2 + q √ K 55 2 1 + K A β + q a Σ 2 + √ K + η ≤ inf m∈N * inf sm∈Sm KL n (s ξ 0 , s m ) + (1 + κ -1 ) pen(m) + η m + 332K 3/2 qC Mn √ n 1 + A β + q a Σ 2 + η.

5.2.

Proof of the Proposition 4.3. We want an upper bound of E KL n (s ξ 0 , ŝ m)1 T c . Thanks to the Cauchy-Schwarz inequality,

E KL n (s ξ 0 , ŝ m)1 T c ≤ E(KL 2 n (s ξ 0 , ŝ m)) P (T c ).
However, for all s ξ ∈ S,

KL n (s ξ 0 , s ξ ) = 1 n n i=1 R q log s ξ 0 (y|x i ) s ξ (y|x i ) s ξ 0 (y|x i )dy = 1 n n i=1 R q log(s ξ 0 (y|x i ))s ξ 0 (y|x i )dy - R q log(s ξ (y|x i ))s ξ 0 (y|x i )dy ≤ - 1 n n i=1 R q log(s ξ (y|x i ))s ξ 0 (y|x i )dy.
Because parameters are assumed to be bounded, according to the set Ξ defined in (4), we get, with (β 0 , Σ 0 , π 0 ) the parameters of s ξ 0 and (β, Σ, π) the parameters of s ξ ,

log(s ξ (y|x i ))s ξ 0 (y|x i ) = log K k=1 π k (2π) q/2 det(Σ k ) exp - (y -β k x i ) t Σ -1 k (y -β k x i ) 2 × K k=1 π 0 k (2π) q/2 det(Σ 0 k ) exp - (y -β 0 k x i ) t (Σ 0 k ) -1 (y -β 0 k x i ) 2 ≥ log   K a π det(Σ -1 1 ) (2π) q/2 exp -(y t Σ -1 1 y + x t i β t 1 Σ -1 1 β 1 x i )   × K a π det((Σ 0 1 ) -1 ) (2π) q/2 exp -(y t Σ -1 1 y + x t i β t 1 Σ -1 1 β 1 x i ) ≥ log K a π a q/2 Σ (2π) q/2 exp -(y t y + A 2 β )A Σ × K a π a q/2 Σ (2π) q/2 exp -(y t y + A 2 β )A Σ .
Indeed, for u ∈ R q , if we use the eigenvalue decomposition of Σ = P t DP ,

|u t Σu| = |u t P t DP u| ≤ ||P u|| 2 ||DP U || 2 ≤ M (D)||P u|| 2 2 ≤ M (D)||u|| 2 2 ≤ A Σ ||u|| 2 2 .
To recognize the expectation of a Gaussian standardized variables, we put u = √ 2A Σ y:

KL(s ξ 0 (.|x i ), s ξ (.|x i )) ≤ - Ka π e -A 2 β AΣ a q/2 Σ (2A Σ ) q/2 R q log Ka q/2 Σ a π (2π) q/2 -A 2 β A Σ - u t u 2 e -u t u 2 (2π) q/2 du ≤ - a q/2 Σ Ka π e -A 2 β AΣ (2A Σ ) q/2 E log Ka π a q/2 Σ (2π) q/2 -A 2 β A Σ - U 2 2 ≤ - Ka q/2 Σ a π e -A 2 β AΣ (2A Σ ) q/2 log Ka π a q/2 Σ (2π) q/2 -A 2 β A Σ - 1 2 ≤ - Ka q/2 Σ a π e -A 2 β AΣ-1/2 (2π) q/2 A q/2 Σ e 1/2 π q/2 log Ka π e -A 2 β AΣ-1/2 a q/2 Σ (2π) q/2 ≤ e -1/2 π q/2 A q/2 Σ ;
where U ∼ N q (0, 1). We have used that for all t ∈ R, t log(t) ≥ -e -1 . Then, we get, for all s ξ ∈ S,

KL n (s ξ 0 , s ξ ) ≤ 1 n n i=1 KL(s ξ 0 (.|x i ), s ξ (.|x i )) ≤ e -1/2 π q/2 A q/2 Σ .
As it is true for all s ξ ∈ S, it is true for ŝ m, then

E(KL 2 n (s ξ 0 , ŝ m)) ≤ e -1/2 π q/2 A q/2 Σ .
For the last step, we need to bound P (T c ).

P (T c ) = E(1 T c ) = E(E X (1 T c )) = E(P X (T c )) ≤ E n i=1 P X (||Y i || ∞ > M n ) . Nevertheless, let Y x ∼ K k=1 π k N q (β k x, Σ k ), then, P (||Y x || ∞ > M n ) = R q 1 {||Yx||∞≥Mn} K k=1 π k 1 (2π) q/2 det(Σ k ) e - (y-β k x) t Σ -1 k (y-β k x) 2 dy = K k=1 π k R q 1 {||Yx||∞≥Mn} 1 (2π) q/2 det(Σ k ) e - (y-β k x) t Σ -1 k (y-β k x) 2 dy = K k=1 π k P X (||Y k x || ∞ > M n ) ≤ K k=1 q z=1 π k P X (|[Y k x ] z | > M n ) with Y k x ∼ N (β k x, Σ k ) and [Y k x ] z ∼ N ([β k x] z , [Σ k ] z,z ). We need to control P X (|[Y k x ] z | > M n ), for all z ∈ {1, . . . , q}. P X (|[Y k x ] z | > M n ) = P X ([Y k x ] z > M n ) + P X ([Y x,k ] z < -M n ) = P X U > M n -[β k x] z [Σ k ] z,z + P X U < -M n -[β k x] z [Σ k ] z,z = P X U > M n -[β k x] z [Σ k ] z,z + P X U > M n + [β k x] z [Σ k ] z,z ≤ e -1 2 Mn -[β k x]z √ [Σ k ]z,z 2 + e -1 2 Mn +[β k x]z √ [Σ k ]z,z 2 ≤ 2e -1 2 Mn -|[β k x]z | √ [Σ k ]z,z 2 ≤ 2e -1 2 M 2 n -2Mn |[β k x]z |+|[β k x]z | 2 [Σ k ]z,z
.

where U ∼ N (0, 1). Then,

P (||Y x || ∞ > M n ) ≤ 2Kqe -1 2 (M 2 n -2MnA β )aΣ ,
and we get

P (T c ) ≤ E n i=1 2Kqa π e -1 2 (M 2 n -2MnA β )aΣ ≤ 2Kna π qe -1 2 (M 2
n -2MnA β )aΣ . We have obtained the wanted bound for E(KL n (s ξ 0 , ŝ m)1 T c ). [START_REF] Massart | The Lasso as an ℓ 1 -ball model selection procedure[END_REF]. Some details 6.1. Proof of the Lemma 5.1. First, give some tools to prove the Lemma 5.1.

We define ||g|| n = 1 n n i=1 g 2 (y i |x i ) for any measurable function g. Let m ∈ N * . We have

sup fm∈Fm |ν n (-f m )| = sup fm∈Fm 1 n n i=1 (f m (y i |x i ) -E(f m (Y i |x i ))) .
To control the deviation of such a quantity, we shall combine concentration with symmetrization arguments. We first use the following concentration inequality which can be found in [START_REF] Boucheron | Concentration Inequalities: A Nonasymptotic Theory of Independence[END_REF]. Lemme 6.1. Let (Z 1 , . . . , Z n ) be independent random variables with values in some space Z and let Γ be a class of real-valued functions on Z. Assume that there exists R n a non-random constant such that sup γ∈Γ ||γ|| n ≤ R n . Then, for all t > 0,

P sup γ∈Γ 1 n n i=1 γ(Z i ) -E(γ(Z i )) > E sup γ∈Γ 1 n n i=1 γ(Z i ) -E(γ(Z i )) + 2 √ 2R n t n ≤ e -t .
Proof. See [START_REF] Boucheron | Concentration Inequalities: A Nonasymptotic Theory of Independence[END_REF].

Then, we propose to bound E sup γ∈Γ

1 n n i=1 γ(Z i ) -E(γ(Z i ))
thanks to the following symmetrization argument. The proof of this result can be found in [START_REF] Aw Van Der Vaart | Weak Convergence and Empirical Processes: With Applications to Statistics[END_REF]. Lemme 6.2. Let (Z 1 , . . . , Z n ) be independent random variables with values in some space Z and let Γ be a class of real-valued functions on Z. Let (ǫ 1 , . . . , ǫ n ) be a Rademacher sequence independent of (Z 1 , . . . , Z n ).

Then,

E sup γ∈Γ 1 n n i=1 γ(Z i ) -E(γ(Z i )) ≤ 2 E sup γ∈Γ 1 n n i=1 ǫ i γ(Z i ) .
Proof. See [START_REF] Aw Van Der Vaart | Weak Convergence and Empirical Processes: With Applications to Statistics[END_REF].

Then, we have to control E(sup γ∈Γ

1 n n i=1 ǫ i γ(Z i ) ).
Lemme 6.3. Let (Z 1 , . . . , Z n ) be independent random variables with values in some space Z and let Γ be a class of real-valued functions on Z. Let (ǫ 1 , . . . , ǫ n ) be a Rademacher sequence independent of (Z 1 , . . . , Z n ). Define R n a non-random constant such that

sup γ∈Γ ||γ|| n ≤ R n .
Then, for all S ∈ N * ,

E sup γ∈Γ 1 n n i=1 ǫ i γ(Z i ) ≤ R n 6 √ n S s=1 2 -s log(1 + N (2 -s R n , Γ, ||.|| n )) + 2 -S
where N (δ, Γ, ||.|| n ) stands for the δ-packing number of the set of functions Γ equipped with the metric induced by the norm ||.|| n .

Proof. See [START_REF] Massart | Concentration inequalities and model selection[END_REF].

In our case, we get the following lemma.

ǫ i f m (Y i |x i ) ≤ 18 √ K C Mn q √ n ∆ m ;
where ∆ m := m log(n) K log(2p + 1) + 6(1 + K(A β + q aΣ )).

Proof. Let m ∈ N * . According to Lemma 6.5, we get that on the event T ,

sup fm∈Fm ||f m || n ≤ R n := 2C Mn (1 + K(A β + q a Σ )).
Besides, on the event T , for all S ∈ N * , S s=1

2 -s log[1 + N (2 -s R n , F m , ||.|| n )] ≤ S s=1 2 -s log(2N (2 -s R n , F m , ||.|| n )) ≤ S s=1 2 -s log(2) + log(2p + 1) 2 s+1 C Mn qKm R n + S s=1 2 -s K log 1 + 2 s+3 C Mn qK R n a Σ 1 + 2 s+3 C Mn R n according to Lemma 6.7 ≤ S s=1 2 -s log(2) + log(2p + 1) 2 s+1 C Mn qKm R n + S s=1 2 -s K log 1 + 2 s+3 C Mn R n max(1, qK/a Σ ) 2 ≤ S s=1 2 -s log(2) + log(2p + 1) 2 s+1 C Mn qKm R n + 2(s + 3)K log(2)q/a Σ ≤ 2C Mn Kmq R n S log(2p + 1) + log(2) 1 + √ q a Σ √ 6K + 2 S s=1 2 -s √ s ≤ 2C Mn Kmq R n S log(2p + 1) + log(2) 1 + √ q a Σ √ 6K + √ q √ K √ 2e 2 - √ e because 2 -s √ s ≤ √ e 2 s
for all s ∈ N * . Then, thanks to the Lemma 6.3,

E sup fm∈Fm 1 n n i=1 ǫ i f m (Y i |x i ) ≤ R n 6 √ n S s=1 2 -s log[1 + N (2 -s R n , F m , ||.|| n )] + 2 -S ≤ R n 6 √ n 2C Mn Kmq R n S log(2p + 1) + log(2) 1 + q a Σ √ 6K + q a Σ √ K 2e 2 - √ e + 2 -S .
Taking S = log(n) log(2) to obtain the same order in the both terms depending on S, we could deduce that

E sup fm∈Fm 1 n n i=1 ǫ i f m (Y i |x i ) ≤ 12C Mn Kmq √ n log(2p + 1) log(n) log(2) + 2C Mn 1 + K A β + q a Σ log(2) √ n 1 + √ 6K + √ 2e 2 - √ 2e + 1 n ≤ 18C Mn Kmq √ n log(2p + 1) log(n) + 2 √ K √ n C Mn 1 + K A β + q a Σ log(2) 1 + √ 6 + √ 2e 2 - √ 2e + 1 ≤18 √ K √ n C Mn mq K log(2p + 1) log(n) + 6 1 + K A β + q a Σ .
It completes the proof.

We are now able to prove the Lemma 5.1.

sup fm∈Fm |ν n (-f m )| = sup fm∈Fm 1 n n i=1 (f m (y i |x i ) -E X (f m (Y i |x i ))) ≤ E sup fm∈Fm n i=1 f m (Y i |x i ) -E(f m (Y i |x i )) + 2 √ 2R n t n
with probability greater than 1e -t and where R n is a constant computed from the Lemma 6.5

≤ 2 E sup fm∈Fm n i=1 ǫ i f m (Y i |x i ) + 2 √ 2R n t n with ǫ i a Rademacher sequence, independent of Z i ≤ 2 18 √ K C Mn q √ n ∆ m + 2 √ 2R n t n ≤ 4C Mn 9 √ Kq √ n ∆ m + √ 2 t n 1 + K A β + q a Σ .
6.2. Lemma 6.5 and Lemma 6.7.

Lemme 6.5. On the event

T = max i∈{1,...,n} max z∈{1,...,q} |[Y i ] z | ≤ M n , for all m ∈ N * , sup fm∈Fm ||f m || n ≤ 2C Mn 1 + K A β + q a Σ := R n .
Proof. Let m ∈ N * . Because f m ∈ F m = f m =log sm s ξ 0 , s m ∈ S m , there exists s m ∈ S m such that f m =log sm s ξ 0 . For all x ∈ [0, 1] p , denote ξ(x) = (π, β 1 x, . . . , β K x, Σ) the parameters of s m (.|x). For all i ∈ {1, . . . , n},

|f m (y i |x i )|1 T = | log(s m (y i |x i )) -log(s ξ 0 (y i |x i ))|1 T ≤ sup x∈[0,1] p sup ξ∈Ξ ∂ log(s ξ (y i |x)) ∂ξ ||ξ(x i ) -ξ 0 (x i )|| 1 1 T ,
thanks to the Taylor formula. Then, we need an upper bound of the partial derivate. For all x ∈ [0, 1] p , for all y ∈ R q , we could write

log(s ξ (y|x)) = log K k=1 h k (x, y)
where, for all k ∈ {1, . . . , K},

h k (x, y) = π k (2π) q/2 det Σ k × exp   - 1 2   q z2=1   q z1=1 y z1 - p j=1 x j [β k ] z1,j   [Σ k ] -1 z1,z2     y z2 - p j=1 [β k ] z2,j x j     .
Then, for all l ∈ {1, . . . , K}, for all z 1 ∈ {1, . . . , q}, for all z 2 ∈ {1, . . . , q}, for all y ∈ R q , for all

x ∈ [0, 1] p , ∂ log(s ξ (y|x)) ∂([β l x] z1 ) = h l (x, y) K k=1 h k (x, y) - 1 2 q z2=1 [Σ l ] -1 z1,z2 ([β l x] z2 -y z2 ) ≤ q(|y| + A β )A Σ 2 ; ∂ log(s ξ (y|x)) ∂([Σ l ] z1,z2 ) = 1 K k=1 h k (x, y) × -h l Cof z1,z2 (Σ l ) det(Σ l ) - h l (x, y)(y z1 -[β l x] z1 )(y z2 -[β l x] z2 )[Σ l ] -2 z1,z2 2
≤ -Cof z1,z2 (Σ l ) det(Σ l ) + (y z1 -[β l x] z1 )(y z2 -[β l x] z2 )[Σ l ] -2 z1,z2 2
≤A Σ + 1 2 (|y| + A β ) 2 A 2 Σ ,
where Cof z1,z2 (Σ k ) is the (z 1 , z 2 )-cofactor of Σ k . We also have, for all l ∈ {1, . . . , K}, for all x ∈ [0, 1] p , for all y ∈ R q , ∂ log(s ξ (y, x)) ∂π l = h l (x, y)

π l K k=1 h k (x, y) ≤ 1 a π .
Thus, for all y ∈ R q , sup

x∈[0,1] p sup ξ∈ Ξ ∂ log(s ξ (y|x)) ∂ξ ≤ max 1 a π , A Σ + 1 2 (|y| + A β ) 2 A 2 Σ , q(|y| + A β )A Σ 2 = C y .
We have

C y ≤ A Σ ∧ 1 aπ 1 + q+1 2 A Σ (|y| + A β ) 2 . For all m ∈ N * , |f m (y i |x i )|1 T ≤ C yi ||ξ(x i ) -ξ 0 (x i )|| 1 1 T ≤ C Mn K k=1 (||β k x i -β 0 k x i || 1 + ||Σ k -Σ 0 k || 1 + |π k -π 0 k |).
Since f m and f 0 m belong to Ξ, we obtain |f m (y i |x i )|1 T ≤ 2C Mn (KA β + K q a Σ + 1)

and then sup fm∈Fm ||f m || n 1 T ≤ 2C Mn (KA β + K q a Σ + 1).

For the next results, we need the following lemma, proved in [START_REF] Meynet | An ℓ 1 -oracle inequality for the lasso in finite mixture gaussian regression models[END_REF].

Lemme 6.6. Let δ > 0 and (A i,j ) i∈{1,...,n} j∈{1,...,p} ∈ [0, 1] n×p . There exists a family B of (2p + 1) 1/δ 2 vectors of R p such that for all µ ∈ R p in the ℓ 1 -ball, there exists µ ′ ∈ B such that

1 n n i=1   p j=1 (µ j -µ ′ j )A i,j   2 ≤ δ 2 .
Proof. See [START_REF] Meynet | An ℓ 1 -oracle inequality for the lasso in finite mixture gaussian regression models[END_REF].

With this lemma, we can prove the following one:

) -ξ ′ (x i )|| 1 1 T ≤ C Mn K k=1 q z=1 |[β k x i ] z -[β ′ k x i ] z | + ||Σ k -Σ ′ k || 1 + |π k -π ′ k | .
Thanks to the Cauchy-Schwarz inequality, we get that Denote by

(f m (y i |x i ) -f ′ m (y i |x i )) 2 1 T ≤ 2C 2 Mn   K k=1 q z=1 |β k x i -β ′ k x i | 2 + (||Σ -Σ ′ || 1 + ||π -π ′ ||) 2   ≤ 2C 2 Mn   Kq K k=1 q z=1   p j=1 [β k ] z,j [x i ] j - p j=1 [β ′ k ] z,j [x i ] j   2 + (||Σ -Σ ′ || 1 + ||π -π ′ ||) 2
a = Kq K k=1 q z=1 1 n n i=1   p j=1 [β k ] z,j [x i ] j - p j=1 [β ′ k ] z,j [x i ] j   2 .
Then, for all δ > 0, if

a ≤ δ 2 /(4C 2 Mn ) ||Σ -Σ ′ || 1 ≤ δ/(4C Mn ) ||π -π ′ || ≤ δ/(4C Mn ) then ||f m -f ′ m || 2 n ≤ δ 2 .
To bound a, we write

a = Kqm 2 K =1 q z=1 1 n n i=1   p j=1 [β k ] z,j m [x i ] j - p j=1 [β ′ k ] z,j m [x i ] j   2

s ξ 0 8 ) 4 . 3 .

 0843 and fm :=log sm s ξ 0 . Let η ≥ 0 and fix m ∈ N * . We define the set M (m) byM (m) = {m ′ ∈ N * |P n (-log(ŝ m ′ )) + pen(m ′ ) ≤ P n (-log(ŝ m )) + pen(m) + η} . (Proof of the Theorem 4.1 thanks to the Propositions 4.2 and 4.3. Let M n > 0 and κ ≥ 36. Let

Lemme 5 . 1 .

 51 Let M n > 0. Let T = max i∈{1,...,n} max z∈{1,...,q}

Lemme 6 . 4 .

 64 Let m ∈ N * . Consider (ǫ 1 , . . . , ǫ n ) a Rademacher sequence independent of (Y 1 , . . . , Y n ). Then, on the event T ,

Lemme 6 . 7 .

 67 Let δ > 0 and m ∈ N * . On the event T , we have the upper bound of the δ-packing number of the set of functions F m equipped with the metric induced by the norm ||.|| n : N (δ, F m , ||.|| n ) ≤ (2p + 1) 4C

2 +

 2 ] z,j [x i ] j -p j=1 [β ′ k ] z,j [x i ] j   (||Σ -Σ ′ || 1 + ||ππ ′ ||) 2 .

  2 Mn K 2 q 2 m 2 /δ 2 1 + 8C Mn qK a Σ δ Let m ∈ N * and f m ∈ F m . There exists s m ∈ S m such that f m =log(s m /s ξ 0 ). Introduce s ′ m in S and put f ′ m =log(s ′ m /s ξ 0 ). Denote by (β k , Σ k , π k ) 1≤k≤K and (β ′ k , Σ ′ k , π ′ k )1≤k≤K the parameters of the densities s m and s ′ m respectively. First, applying Taylor's inequality, on the event[Y i ] z | ≤ M n ,we get, for all i ∈ {1, . . . , n},|f m (y i |x i )f ′ m (y i |x i )|1 T = | log(s m (y i |x i ))log(s ′ m (y i |x i ))|1 T

					K	1 +	8C Mn δ
	T =	max i∈{1,...,n}	max z∈{1,...,q}
	≤ sup x∈[0,1] p	sup ξ∈ Ξ	∂ log(s ξ (y i |x)) ∂ξ	||ξ(x i

K .

Proof.

|

and we apply Lemma 6.6 to [β k ] z,. /m for all k ∈ {1, . . . , K}, and for all z ∈ {1, . . . , q}. Since s m ∈ S m , we have q z=1 p j=1

≤ 1, thus there exists a family B of (2p + 1) 4C 2 Mn q 2 K 2 m 2 /δ 2 vectors of R p such that for all k ∈ {1, . . . , K}, for all z ∈ {1, . . . , q}, for all [β k ] z,. , there exists [β

Mn ). Moreover, since ||Σ|| 1 ≤ qK aΣ and ||π|| 1 ≤ 1, we get that, on the event T ,


Ludivine Leclercq

Ronan Modolo

François Leblanc

Ronan

Hybrid multi-grids simulations of Ganymede's magnetosphere

Hybrid multi-grids simulations of Ganymede's magnetosphere Ludivine Leclercq 1 , Ronan Modolo 1,2 , Francois Leblanc 1 1. LATMOS / IPSL / CNRS-INSU, Paris, France. 2. Université de Versailles St Quentin, Guyancourt, France.

1-Abstract

Ganymede is a unique object :

• Biggest moon of our solar system • Only satellite which has its own intrinsic magnetic field leading to the formation of a small magnetosphere • Magnetosphere of Ganymede is embedded in the Jovian magnetosphere : only known case of interaction between • two magnetospheres (Kivelson et al. 1996).

• Formation of Alfvèn wings

Interaction modeled by a 3D parallel multi-species hybrid model based on a CAM-CL algorithm (Mathews et al. 1994) :

• Generic model applied to Mars (Modolo et al. 2005;2006and 2012), Titan (Modolo et al. 2007, Modolo and Chauteur 2008) or Mercury (Richer et al. 2012). • Maxwell's equations are solved to compute the temporal evolution of electromagnetic field.

• Description of the the dynamics of the magnetospheric plasma and Ganymede's ionospheric plasma • Simulations results are presented and compared to magnetometer and particle observations obtained during G1 and G2 Galileo flybys.

2-Description of the model

∇ . B=0 ∇ ×E= -∂ B ∂ t ∇ ×B=μ 0 (J e +J i )+μ 0 ε 0 ∂ E ∂ t dv s dt = q s m s ( E+v s ×B)

  Hybrid formalism : • Ions have a kinetic description and are represented by macroparticles that have a mass, a charge and a weight (=number of physical particles that represents the macroparticle) • e -described as an inertialess fluid to ensure the quasi-neutrality of the plasma. Represented by a Maxwelian distribution. Maxwell's equations used in the model : Motion of the ions : Electron momentum equation : Method and Cyclic Leapfrog (CAM-CL) • Matthews et al. 1994 • Considerable reduction of time computation • Multiple ion species treated with only one pass through the • particle data

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Ion energy spectra during G2 flyby. Bottom: Galileo measurements. Top: Different species of our simulation (RUN 2)

Figure 4 :Figure

 4 Figure 4: Electron densityalong G1 (bottom) and G2 (top) trajectories. Observations in blue and simulation in red (RUN 2)