Fatima Zohra Kaddour

Emmanuelle Vivier

Mylene Pischella

Lina Mroueh

Philippe Martins

Energy efficiency of Opportunistic and Efficient Resource Block Allocation Algorithms for LTE Uplink Networks

Keywords: Energy Efficiency, Power control, Resource Block allocation, SC-FDMA, LTE

Actually, the energy efficiency in wireless network is the central concern of research. We propose in this paper, a new energy efficiency scheme which allocates the mobile's transmission power to users in function of the channel conditions on their allocated resource blocks. We focus on the energy efficiency of the Opportunistic and Efficient Resource Block Allocation (OEA) algorithm and its variant adapted to the Quality of Service (QoS) traffic, the QoS based OEA in LTE uplink network. The OEA and the QoS based OEA allocate the Resource Block (RB) to UEs efficiently and respecting the SC-FDMA constraints, such as, for one user, contiguous RB are allocated, and the same Modulation and Coding Scheme (MCS) are used over the whole allocated RBs. When the RB allocation is performed, the power control is applied to the mobile's transmission power considering the MCS used and the channel condition's. This energy efficiency allow users to achieve the same throughput than before the power control and do not affect the MCS selection established at the RB allocation step. This new scheme allow the transmission of a high number of bits per joule.

I. INTRODUCTION

Nowadays, the whole world of telecommunications and information communities is facing a more and more serious challenge with the increasing traffic requirements in current and the next generation mobile networks and the popularity of smart terminals. Then the energy consumption of wireless communication networks and the relevant global CO2 emission shows continuous growth for several years. It has been pointed out that currently 3% of the world-wide energy is consumed by the Information and Communication Technology (ICT) infrastructure that causes about 2% of the world-wide CO2 emissions. Energy costs to the mobile's operators a half of the operating expenses. Moreover, improving the energy efficiency as the resource efficiency is not only beneficial for the global environment but also makes commercial sense for telecommunication operators supporting sustainable and profitable business. Within the framework of Green Communications, a number of technical approaches are investigated in the literature. We focus on the energy efficient wireless transmission techniques on uplink 3rd Generation Partner Project (3GPP) Long Term Evolution (LTE) network. The 3GPP standard adopted, for the LTE networks, the Orthogonal Frequency Division Multiple Access (OFDMA) and the Single Carrier Frequency Division Multiple Access (SC-FDMA) for both downlink and uplink respectively. The relevance of the SC-FDMA on the uplink is that in addition to the OFDMA advantages, the SC-FDMA generates a low Peak to Average Power Ration (PAPR), by considering the whole allocated Resource Blocks (RB) as a single carrier. The reduction of the PAPR can be more than 25% compared to the OFDMA technique [START_REF]Lte-single carrier frequency division multiple access[END_REF]. This advantage not only leads to the decrease of the equalizer complexity and the cost of the mobile terminal by the same way, but also to the decrease of the mobile energy consumption. Using the SC-FDMA technique on the uplink is an encouraging start but it can not increase much the mobile battery life. Therefore, solving the battery life of the mobile's terminal becomes the central concern of the researchers. Works on this scope focus on: (i) maximizing the available energy and (ii) minimizing the energy consumption. The available energy can be increased by (a) the battery capacity improvement which is, unfortunately, not sufficient and is limited due to design aspect, and (b) using the surrounding energy sources, such as kinetic, thermal, and solar energy [START_REF] Joseph | Energy harvesting projects[END_REF]. The mobile's energy consumption can be minimized by first, optimizing the hardware energy consumption, such as choosing power efficient components and applying power management like performing sleep modes for inactive hardware [START_REF] Lioumpas | Uplink scheduling for machine-tomachine communications in lte-based cellular systems[END_REF] or the Discontinuous Reception (DRX) in idle mode [START_REF] Fowler | Study on power saving based on radio frame in lte wireless communication system using drx[END_REF]. The second solution to minimize the mobile's energy consumption is the adjustment of the mobile's parameter, like the brightness display and the processor speed for some applications. In the radio access network, the power consumption reduction is performed by a power control of the mobile's transmission power. Reduction on the mobile transmission power leads to low user's Signal to Interference plus Noise Ratio (SINR) and a low individual throughput. Therefore, the power control should take into account the required SINR which allows the User Equipement (UE) to use the same Modulation and Coding Scheme (MCS) and reach the same throughput. This study focuses on the energy efficiency of the mobile's terminals performed by a power control of the UE's transmission power considering the Resource Block (RB) allocation policy and the MCS used by each UE. We study the energy efficiency according to the Opportunistic and Efficient RB Allocation (OEA) algorithm proposed in our previous work [START_REF]Opportunistic and efficient resource block allocation algorithms for lte uplink networks[END_REF]. The paper is organized as follows. Section II present the system model. In Section III, we define the RB allocation algorithm used and the energy efficiency method. Simulation parameters are summarized in Section IV. Section V shows the numerical results and a comparison in terms of energy efficiency with other reference RB allocation algorithms. Finally, Section VI concludes the paper.

II. SYSTEM MODEL

We consider an uplink LTE network composed of 19 hexagonal cells. Each cell is provided with a tri-sectored eNodeB (eNB). We allocate at each sector a different bandwidth B (i.e. 1 × 3 × 1 frequency reuse pattern), corresponding to N RB available RBs. The functionalities of our RB allocation depending energy efficiency is divided into two entities: (i) the RB allocation, and (ii) the power control, as shown in Figure 1. The RB allocation entity, allocates the RBs to the UEs in function of their Channel Quality Identifier (CQI). One RB consists of 84 Resource Element (RE) corresponding to N RB sc = 12 subcarrier and N RB symb = 7 SC-FDMA symbol in the normal prefix cyclic case [START_REF]Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) radio transmission and reception[END_REF]. On the uplink, the CQI is evaluated by dint of the Reference Signals (RS) which are sent at each time slot in the 4th SC-FDMA symbol [START_REF] Nam | Evolution of reference signals for lte-advanced systems[END_REF]. The RS use 14% of the total number of RE. Then, the number of REs used for data transmission in one RB is equal to 72 and the real throughput (i.e. considering the MCS selection) that can be achieved by each user k can be computed as:

R k = c∈A k 72 T s BRE k | A k | (1)
where, T s is the time slot duration equivalent to 0. In case of frequency and time correlated fast fading and for a given RB, the SINR level of each UE is computed at each RE (i, j) (with 1 ≤ i ≤ N RB symb and 1 ≤ j ≤ N RB sc ), as:

SINR (i,j) k = P (i,j) k T x G t G r (θ k )Λ N + I (i,j) (2)
where Λ is the total channel gain, expressed as:

Λ = G c (r k ) A (i,j) f 2 A (i,j) s (3)
G c (r k ) is the path loss depending of r k , the distance between the user k and the eNB.

A (i,j) s
is the random shadowing over one RE which follows a log-normal distribution with parameter σ s and A (i,j) f is the time-frequency correlated coefficients fast fading.

P (i,j)
k T x is the transmission power of user k over one resource element. Since the power is equally divided over all the resource elements of one RB, P (i,j) k T x can be expressed as:

P (i,j) k T x = P k T x N U L symb N RB sc | A k | (4)
with P k T x the mobile's transmission power, set at its maximum before the power control (i.e. P k T x equal to P max ). The mobile's transmission antenna gain and the eNB antenna reception gain is denoted respectively as G t and G r . The eNB antenna reception gain G r depends on θ k , angle between UE k and eNB antenna boresight. N is the thermal noise in the subcarrier and I (i,j) is the ICI level at each resource element (i, j) obtained by Monte Carlo simulations.

The signal to interference plus noise ratio over one RB c of user k is computed using the mean instantaneous capacity method defined in [START_REF] Ramadas | Wimax system evaluation methodology[END_REF]. It is denoted the effective SINR (SINR eff k ) and is computed as follows:

SINR c eff k = 2 C k /N U L symb -1 (5)
with C k the theoretical Shannon capacity over the whole RB, computed as:

C k = 1 N RB sc N U L symb i=1 N RB sc j=1 log 2 1 + SINR (i,j) k (6)
The effective signal to interference plus noise ratio computed for each user k and over each RB c (SINR c eff k ) will be used as a metric for the RB allocation algorithm detailed in the next section.

III. RESOURCE BLOCK ALLOCATION DEPENDING ENERGY

EFFICIENCY

The LTE uplink radio resource management includes the RBs and the power allocation, which can be performed: (i) conjointly, or (ii) separately. The conjoint manner is proposed in [START_REF] Sokmen | Uplink resource allocation algorithms for single-carrier fdma systems[END_REF], where the Binary Integer Programming (BIP) is used to optimally minimize the total power expenditure subject to the rate constraints. The authors consider the contiguity constraint directed by the SC-FDMA technique but compute the total mobile throughput using the theoretical upper bound (i.e. the shannon capacity). The BIP is commonly proposed as an optimal solution for resource allocation in wireless network, but this method is NP hard. Since the radio resource management occurs at each transmission time interval (which is equal to 1 ms), using the optimal method becomes irrelevant, which motivates us to use a heuristic in allocating the RBs and the mobile's transmission power.

The proposed energy efficiency method allocates the mobile's transmission power after the RB allocation. The power control applied to the mobile's transmission power on the allocated RBs ensures the throughput maximization while minimizing the mobile's power consumption without disrupting the RB allocation, the Quality of Service (QoS) or the MCS selected. The two entities: (i) RB allocation and (ii) the power control, are described as follows:

A. RB allocation

For RBs allocation, we use the Opportunistic and Efficient RB Allocation (OEA) algorithms that we proposed in our previous work [START_REF]Opportunistic and efficient resource block allocation algorithms for lte uplink networks[END_REF]. We consider that there are N U E user able to transmit their data. Thus, we define K the set of users able to be scheduled

K = {1, • • • , k, • • • , N U E } and C the set of free RBs C = {1, • • • , c, • • • , N RB }.
The OEA algorithm aims to maximize the aggregate throughput of the network by allocating efficiently the resource blocks.

max N U E k=1 R k (7)
where R k is the total throughput of user k, over the whole allocated RBs and computed using Equation 1. The maximization problem is subject to: 1) the exclusivity of the allocated RBs:

N U E k=1 x c k (t) = 1 ∀c ∈ C (8)
2) the contiguity constraints:

x j k (t) = 0 ∀j > c+2 if x c k (t) = 1 and x c+1 k (t) = 0 (9) 3) the MCS robustness: BRE k = min c∈A k BRE c k (10)
where BRE c k is the number of bit per RE allowed for UE k on each allocated RB c.

The algorithm abides by the SC-FDMA constraints. It allocates exclusively adjacent RBs to the same UE (i.e. translated by Formulas 8 and 9) and respects the MCS robustness (i.e. Formula 10). To maximize the aggregate throughput, the algorithm uses the channel conditions the effective SINR experienced by each user over each RB as a metric. It searches first the pair (UE-RB) which maximizes the metric, then extends the allocation to the adjacent RBs. The final RBs expansion allocation is done if and only if the individual throughput of the UE increases. To adapt the OEA algorithm to the QoS required by users, we fix a maximum number of allocated RBs per UE which satisfy its QoS requirement. The RB allocation expansion is performed as long as this maximum number of allocated RBs per UE is not reached. More important step, which is neglected on the literature, is the update of the metric before each RBs allocation expansion. The update of the metric considering the update of the mobile's transmission power per RB (using Equation 4), allows us to compute the correct value of individual throughput, the parameter which is based on the expansion decision of the RB allocation.

B. Power control

Since each mobile user k has its set of allocated RBs A k , the second entity allocates them the appropriate transmission power which allows the UE to use the same MCS as before the power control and reach the same throughput. To use a definite MCS, the UE must experience in the whole allocated RB a SINR level higher than the minimum SINR range of the used MCS. To ensure the use of the same MCS by user k, we define an SINR target (SINR T g ) level which is an SINR margin ∆ SINR added to the minimum SINR range of the required MCS (SINR M CS,k ), as:

SINR T g = SINR M CS,k +∆ SINR (11)
Then, the new mobile's transmission power per RB P e k allocated to UE k and which allows it to use the target MCS and achieve the same throughput than the ones achieved before the power control, is expressed as:

P e,k = P k Tx | A k | SINR T g SINR eff k ,min (12)
where, SINR eff k ,min is the minimum effective SINR experienced by the UE k on the whole allocated RBs:

SINR eff k ,min = min c∈A k SINR c eff k (13)
The MCS robustness is guaranteed by the SINR eff k ,min based power control.

IV. SIMULATION PARAMETERS

To evaluate the performance of the RB allocation depending energy efficiency, we evaluate the mobile's energy consumption saving obtained thanks to power control. Our study focus on the RB and power allocation to one sector of the central cell users. We compare the OEA algorithm performance with the literature reference's RB allocation algorithms, such as: the Heuristic Localized Gradient Algorithm (HLGA), the Frequency Domain Packet Scheduling -Largest Metric value First algorithm (FDPS-LMF) and the Recursive Maximum Expansion algorithm (RME) proposed respectively in [START_REF] Al-Rawi | Opportunistic Uplink Scheduling for 3G LTE Systems[END_REF], [START_REF] Lee | Proportional Fair Frequency-Domain Packet Scheduling for 3GPP LTE Uplink[END_REF] and [START_REF] De Temiño | Channel-aware scheduling algorithms for sc-fdma in lte uplink[END_REF].

The simulation parameters are taken from the LTE standards. The performance evaluation is studied in low and high loaded network. Then, the total number of user per sector varies between 5 to 80 UEs while a bandwidth B = 5 MHz is allocated to each sector. This correspond to N RB = 25 available RBs per sector. We consider an infinite backlogged traffic, in which, for each user, there is always available data for transmission. The target Bit Error Rate (BER) is equal to 10 -6 and the throughput is computed using the MCS lookup table respecting the MCS robustness. The path loss is modeled by the Okumura Hata model [START_REF] Okumura | Field Strength and its Variability in VHF and UHF Land-Mobile Radio Service[END_REF] where the carrier frequency is set to 2.6 GHz, the eNB height is at 40 m and the UE height at 1.5 m. The remaining simulation parameters are summarized in Table I

V. PERFORMANCE EVALUATION

The objective of the proposed energy efficiency method is to allocate both RBs and the mobile's transmission power efficiently, which allows users to benefit from a higher throughput without wasting RB while radiating at a lower power and increase the mobile's battery life. Figure 2 shows the aggregate throughput of the central cell while varying the network's load (i.e. varying the number of UE from 5 to 80, which correspond to a number of UE per number of RBs ratio of 20% to 320%). Since the number of UEs per sector and the number of the served UE increase, the aggregate throughput per sector increase whatever the used RB allocation algorithms. We notice that the OEA algorithms reach its objective and maximizing the aggregate throughput. Its achieves more than 2 Mbps at low load network (i.e. when there are only 5 UE per sector) and 6.5 Mbps at hight load network (I.e. 80 UE per sector). The QoS based OEA achieve a lower aggregate throughput than the OEA, due to the low target throughput fixed at 300 kbps. Then, the QoS based OEA allocates an adequate number of RBs to UE which allow them to achieve the target throughput unlike the OEA algorithm, where the UE can be allocated all the RBs that maximize its individual throughput depending on its channel conditions even it takes all the available RBs allocated to the sector N RB . The HLGA and the FDPS-LMF achieve a lower aggregate throughput than QoS based OEA. Since their RB allocation policy is similar instead the allocation of the remaining RBs, we note a small difference between their total aggregate throughput. The aggregate throughput gap between HLGA and FDPS-LMF is notable at low load because their are more remaining RBs and the HLGA, by allocating the remaining RBs to users which satisfy the contiguity constrains whatever their SINR level, decrease the aggregate throughput.

The lower aggregate throughput is given by the RME since its allocate more RBs per UE. The HLGA, FDPS and RME achieve a lower aggregate throughput than the OEA and the QoS based OEA due to their negligence of the metric update at each RB allocation expansion and the MCS robustness, while computing the throughput by the theoretical upper bound capacity (i.e. Shannon capacity formula).

Figure 3 and 4 represent respectively, the energy efficiency of the data transmitted before and after the power control. The energy efficiency of the transmitted data in bit per joule is defined by the ratio between the number of data transmitted in one second and the energy consumed in one second to transmit this data. Before the power allocation, all the mobiles transmit at their maximum power P max . Since the OEA and the QoS based OEA achieve a higher throughput, then they achieve a higher energy efficiency. They reach respectively 3.5 10 6 and 2.4 10 6 bits/J at low load, while the RME, FDPS-LMF and the HLGA do not exceed 8 10 5 bits/J. At high load network, the energy efficiency decrease for the five algorithms to reach 7.5 10 5 bits/J by the OEA and 7 10 5 bits/J by the QoS based OEA, 5 10 5 bits/J for both HLGA and FDPS-LMF algorithms and 2 10 5 bits/J by the RME. Unlike the maximum transmission power case, the energy efficiency curves increase when the network load increase after applying the power control. The number of bits transmitted per one joule achieves 180Mbits and 170 Mbits for the QoS based OEA and the OEA respectively. The curves are not smoothed as before the power control curves, because the power allocation depend on the minimum SINR range of the used MCS which is not linear. Figure 5 shows the average mobile's transmission power in dBm. The thick line correspond to the maximum transmission power of the mobiles set to 21dBm (i.e. 125 mW). We notice that the RME do not decrease a lot the mobile's transmission power. The algorithms which allow a low mobile's transmission power is the HLGA, the FDPS-LMF and the QoS based OEA. This power reduction is explained by the number of RBs allocated to each UE. Since if | A k | is low, P k T x is high and the power control according to the SIN R T g is more efficient. In Figure 6, we represent the ratio of saved energy after the power control. The RME save from 2% to 25% of the mobile's transmission power according to the network load, where the QoS based OEA save from 28% to 65% of the mobile's transmission power, which allow it to increase more the mobile's battery life.

VI. CONCLUSION

In this paper we investigate the energy consumption problem of the wireless network and more specifically the energy efficiency on the LTE uplink network, which allow the increase of the LTE mobile's battery's life. We proposed a new energy efficiency scheme which depend on the resource block allocation algorithms. The proposed radio resource allocation allocates first the resource block to UE using, the opportunistic and efficient RB allocation algorithm and the Quality of Service based opportunistic and efficient RB allocation algorithm, to allocate the RB to the active users. The RBs are first, allocated to UEs according to their channel conditions and respecting the SC-FDMA constraints (i.e. the RB contiguity and the MCS robustness constraints); then it apply a power control to the mobile's transmission power which reduce the mobile's radiated power without affecting the MCS selection or the individual reached throughput. To evaluate the performances of the proposed energy efficiency scheme obtained by the OEA and the QoS based OEA, we compare the saved energy after the power control when the RME, FDPS-LMF and the HLGA algorithms are used for the RB allocation. The numerical results show that the OEA algorithms achieve the higher Energy efficiency and in case of QoS service, the QoS based OEA allow to save more than half of the mobile's energy. In future work, we aims to compare the proposed energy efficiency scheme with the optimal one, using the integer programming method.

Fig. 1 .

 1 Fig. 1. Energy efficiency scheme

Fig. 2 .

 2 Fig. 2. Aggregate throughput

Fig. 3 .Fig. 4 .

 34 Fig. 3. Average energy efficiency before the Power allocation

Fig. 5 .Fig. 6 .

 56 Fig. 5. Average mobile's transmission power per UE

  5 ms, BRE k denote the number of bits per RE allowed for UE k by the used MCS and A k is the set of RBs allocated to UE k by the RB allocation entity, with | A k | its cardinal. The MCS selection is based on the channel conditions (i.e. the SINR level experienced by each user in each RB c). The specificity of the SC-FDMA technique is that each UE should use the same MCS over the whole allocated RBs. Thus, the MCS selection is based on the minimum signal to interference plus

noise ratio experienced by the concerned UE on the whole allocated RBs (i.e. ∀c ∈ A k ).

  .

	Cellular layout	Hexagonal grid,19 tri-sector cells.
	Max/ Min UE-BS distance	1000 m/30 m
	Carrier frequency	2.6 GHz
	System bandwidth	B = 5 MHz per sector ⇒ N RB = 25
	FFT size	512
	Subcarrier spacing	15kHz
	Time slot duration	Ts = 0.5 ms
	Target Throughput	300 kbps
	the SINR margin ∆ SINR	0.3
	Radio channel gain	Okumura Hata for urban areas:
		Gc(r jc ) = 10 -a/10 * r a = 136.7 and b = 34.4. -b/10 jc	.
	BS antenna pattern	Gr(θ jc ) = -min[12 * (	θ jc θ 3dB	) 2 , β].
		θ 3dB = 70 • , β = 20 dB
	User power class	Pmax = 21 dBm (125 mW)
	User antenna gain	Gt = 0 dBi
	Rayleigh fading	coef corr = 0.5, UE velocity = 3 km/h
	Log-normal shadowing	σs = 5 dB
	MCS setting	QPSK 1/2, 2/3, 3/4
		16 QAM 1/2, 2/3, 3/4
		64 QAM 1/2, 2/3, 3/4


Alexey V Kavokin
email: alexey@phys.soton.ac.uk

Maria Vladimirova

Benoit Jouault

Timothy C H Liew

Jason R Leonard

Leonid V Butov

Exciton Spin Currents: Theory

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

An indirect exciton is a bound pair of an electron and a hole confined in spatially separated semiconductor layers. Long lifetimes of indirect excitons allow them to cool down to low temperatures below the temperature of quantum degeneracy. In coherent exciton gases the spin relaxation mechanisms linked with the scattering of particles are strongly suppressed. Recent experiments evidenced spontaneous coherence and condensation, phase singularities, spatial ordering, and polarization textures in a cold excitons gas [1,2]. We develop a theoretical model accounting for the spin-orbit interaction of electrons and holes, exchange splitting of exciton states and Zeeman effect. This model describes the polarization patterns observed experimentally, including helical patterns, four-leaf patterns, spiral patterns, and bell patterns, and confirms that the polarization textures at low temperature manifest ballistic propagation of cold excitons over the distances exceeding the exciton Bohr radius by 3 orders of magnitude and the exciton thermal De Broglie wavelength by more than an order of magnitude. The comparison of theory and experiment allows extracting the spin currents carried by electrons and holes bound to excitons. The spins tend to align along the effective spin-orbit field induced due to the Dresselhaus effect (Figure 1).

Figure 1 .

 1 Figure 1. Simulated in-plane exciton polarization (A), electron spin (B) and hole spin (C) patterns around a source of excitons (in the center of each image). (D) Schematic of exciton momentum (black arrows) and linear polarization (red lines) patterns. Schematic of effective magnetic fields given by the Dresselhaus spin-orbit interaction for electrons (E) and holes (F). Exciton polarization (G), electron spin (H) and hole spin (I) patterns in applied magnetic field B = 7 T. The lines (arrows) and the color visualize the orientation of the linear polarization (spin).


Keywords: AMS classification scheme numbers: 35R60, 35R30, 65M32 Inverse scattering problems, Linear Sampling Method, Factorization Method, Qualitative methods

come

A generalized formulation of the Linear Sampling Method with exact characterization of targets in terms of farfield measurements

Lorenzo Audibert, Houssem Haddar

A generalized formulation of the Linear Sampling Method with exact characterization of targets in terms of farfield measurements Lorenzo Audibert ♯ † Houssem Haddar ♯

Introduction

This work can be seen as a contribution to the development of so-called qualitative methods [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF][START_REF] Kirsch | The factorization method for inverse problems[END_REF]4] for solving inverse scattering problems for extended targets from fixed frequency multi-static data. More specifically, we introduce and analyze a new formulation of the so-called Linear Sampling Method (LSM) [START_REF] Colton | A simple method for solving inverse scattering problems in the resonance region[END_REF][START_REF] Colton | The linear sampling method in inverse electromagnetic scattering theory[END_REF], that we will refer to as Generalized Linear Sampling Method (GLSM), which is based on an new exact characterization of the targets shape in terms of the so-called farfield operator (at a fixed frequency). This characterization is based on constructing nearby solutions to the farfield equation as minimizing sequences of a special cost functional and uses two (complementary) factorizations of the farfield operator. The first one is the basic factorization used in the theoretical justification behind LSM and the second one is the one used by the factorization method (FM) [START_REF] Kirsch | Characterization of the shape of a scattering obstacle using the spectral data of the far field operator[END_REF][START_REF] Kirsch | The factorization method for inverse problems[END_REF]. This combination allows us for instance to require less restrictive assumptions than FM. It also turns out that one can establish a direct link between our method and FM for a special setting of GLSM and this also provides a direct link with the analysis in [START_REF] Arens | Why linear sampling works[END_REF][START_REF] Arens | The linear sampling method revisited[END_REF] justifying the use of LSM in some particular configurations. Although not directly inspired by them, the GLSM share some similarities with the so-called inf-criterion [START_REF] Kirsch | The factorization method for inverse problems[END_REF] or the formulation of this criterion in [START_REF] Nachman | On imaging obstacles inside inhomogeneous media[END_REF] as well as the probe method [START_REF] Ikehata | A new formulation of the probe method and related problems[END_REF][START_REF] Erhard | A numerical study of the probe method[END_REF]. The main idea behind our method is to explicitly construct the nearby solution of the LSM by adding to a standard least squares misfit functional a penalty term proportional to an appropriate norm of the associated Herglotz wave. Using the second factorization of the farfield operator (as used in FM), we express this term using the measured farfield operator. The main issues to address are first how to cope with the fact that the penalty term is compact and second how to address the case of noisy operators.

Here comes the role of the first factorization generally used for LSM. For more details we refer to the third section where the general formulation of the method is presented as well as the analysis for different configurations. In order to introduce the main ideas behind GLSM as well as a concrete application we choose to present the case of scalar inverse scattering form inhomogeneous inclusions. We show for this example how the method can be applied and we also indicate other possible straightforward applications (which are roughly speaking all cases where FM applies, or more generally where the inf-criterion and LSM apply).

The impact of our method on the numerical side is twofold. In fact, the analysis of GLSM for noisy farfield operators suggests a different indicator function for LSM than the one usually used. This new indicator function is similar to the one proposed in [START_REF] Arens | Why linear sampling works[END_REF] but contains an additional term that correctly fix the behavior of the indicator function outside the obstacle for noisy operators. The superiority of this new indicator function is demonstrated through some numerical results. The second alternative is to directly use the minimizing sequence constructed by GLSM, which is computationally more expensive but leads to better results for multi-connected objects. In fact the second numerical method can be used as a post-processing of the first one since from numerical experiments, we observed that only few iterations are needed to update the initial guess provided by LSM.

The article is organized as follows. In Section 2 a model problem is introduced to motivate GLSM after recalling the basis of the LSM and the factorization method. The theoretical foundation of the GLSM is given in Section 3. Section 4 provides an example of application of GLSM by completely treating the model problem introduced in Section 2 and indicating other possible applications. The last section (Section 5) is devoted to the introduction of two numerical algorithms issued from Section 3 along with validating numerical results and comparison with other algorithms.

A model problem and motivation for GLSM

In order to introduce the ideas and motivations behind the proposed new algorithm below, namely GLSM, we choose to present as a model problem the scalar inverse time harmonic scattering problem from inhomogenous targets. For a wave number k > 0, the total field solves the Helmholtz equation ∆u + k 2 nu = 0 in R d with d = 2 or 3 and with n ∈ L ∞ (R d ) denoting the refractive index such that the support of n -1 is equal to D with D a bounded domain with Lipschitz boundary and connected complement and such that ℑ(n) ≥ 0. We are interested in the cases where the total field is generated by plane waves, u i (θ, x) := e ikx•θ with x ∈ R d and θ ∈ S d-1 (the unit sphere) and we denote by u s the scattered field defined by

u s (θ, •) = u -u i (θ, •) in R d ,
which is assumed to be satisfying the Sommerfeld radiation condition,

lim r→∞ |x|=r ∂u s ∂r -iku s 2 ds = 0.
Our data for the inverse problem will be formed by noisy measurements of so called farfield pattern u ∞ (θ, x) defined by

u s (θ, x) = e ik|x| |x| (d-1)/2 (u ∞ (θ, x) + O(1/|x|))
as |x| → ∞ for all (θ, x) ∈ S d-1 ×S d-1 . The goal is to be able to reconstruct D from these measurements (without knowing n) using a new sampling algorithm. The foundation of this algorithm is inspired by the Linear Sampling Method and the Factorization Method that we shall briefly review here in the context of this special scattering problem. These methods are based on the farfield operator F : L 2 (S d-1 ) → L 2 (S d-1 ), defined by

F g(x) := S d-1 u ∞ (θ, x)g(θ)ds(θ). Let us define for ψ ∈ L 2 (D), the unique function w ∈ H 1 loc (R d ) satisfying      ∆w + nk 2 w = k 2 (1 -n)ψ in R d , lim r→∞ |x|=r ∂w ∂r -ikw 2 ds = 0. ( 1
)
By linearity of the forward scattering problem, F g is nothing but the farfield pattern of w solution of (1) with ψ = v g in D, where

v g (x) := S d-1 e ikx•θ g(θ)ds(θ), g ∈ L 2 (S d-1 ), x ∈ R d .
Now consider the (compact) operator H : L 2 (S d-1 ) → L 2 (D) defined by

Hg := v g | D , (2)
and the (compact) operator G : R(H) ⊂ L 2 (D) → L 2 (S d-1 ) defined by

Gψ := w ∞ ,
where w ∞ is the farfield of w ∈ H 1 loc (R d ) solution of (1) and where R(H) denotes the closure of the range of H in L 2 (D). Then clearly

F = GH.
The basis of the Linear Sampling Method is the following characterization of D in terms of the range of G. This characterization is based on the solvability of so called interior transmission problem defined by (u, v)

∈ L 2 (D) × L 2 (D) such that u -v ∈ H 2 (D) and          ∆u + k 2 nu = 0 in D, ∆v + k 2 v = 0 in D, (u -v) = f on ∂D, ∂ ∂ν (u -v) = g on ∂D, (3)
for given f ∈ H 1 2 (∂D) and g ∈ H -1 2 (∂D). We shall make the following assumption Hypothesis 1. We assume that k 2 ∈ R + and n ∈ L ∞ (D) are such that, ℑ(n) ≥ 0 and such that for all f ∈ H

1 2 (∂D) and g ∈ H -1 2 (∂D) problem (3) has a unique solution (u, v) ∈ L 2 (D) × L 2 (D) such that u -v ∈ H 2 (D).
It is well known for instance that if in addition, 1/(n -1) ∈ L ∞ (D) and ℜ(n -1) is positive definite or negative definite in a neighborhood of ∂D, then Hypothesis 1 is verified for all k ∈ R except a countable set without any finite accumulation point [START_REF] Sylvester | Discreteness of transmission eigenvalues via upper triangular compact operators[END_REF]. Defining φ z (x) := e -ikx•z , the main ingredient of LSM is the following.

Theorem 1. Under Hypothesis 1, φ z ∈ R(G) if and only if z ∈ D.
The proof of this theorem is rather straightforward using the important result of Lemma 1 (see [START_REF] Rynne | The interior transmission problem and inverse scattering from inhomogeneous media[END_REF]) and the fact that φ z is the farfield of Φ(•; z), the fundamental solution of the Helmholtz equation satisfying the Sommerfeld radiation condition.

Lemma 1. R(H) = {v ∈ L 2 (D); ∆v + k 2 v = 0 in D}.
From Theorem 1 one can deduce the following statement, which is the basic theoretical justification of the LSM. Theorem 2. Under Hypothesis 1, the operator F is injective with dense range. Moreover, the following holds.

• If z ∈ D then there exists g ǫ z such that F g ǫ z -φ z L 2 (S d-1 ) ≤ ǫ and lim sup ǫ→0 Hg ǫ z L 2 (D) < ∞. • If z / ∈ D then for all g ǫ z such that F g ǫ z -φ z L 2 (S d-1 ) ≤ ǫ, lim ǫ→0
Hg ǫ z L 2 (D) = ∞. This theorem thus suggests to use a nearby solution to F g ǫ z ≃ φ z for different sampling points z to obtain an indicator of D. Two problematic issues are then raised: the first one is that the indicator function (provided by the theorem) should be Hg ǫ z L 2 (D) which depends on D and the second one is that the theorem does not give explicit construction of g ǫ z . In practice, a Tikhonov regularization is usually used to build a nearby solution (as suggested by the first statement in Theorem 2) and g ǫ z L 2 (S d-1 ) is used in replacement of Hg ǫ z L 2 (D) . In [START_REF] Arens | Why linear sampling works[END_REF] it is proved, based on the Factorization method, that Tikhonov regularization provides the good solution as soon as ℑ(n) = 0 and in that case one can replace Hg ǫ z L 2 (D) with |Hg ǫ z (z)|. As it will be seen later, the proposed GLSM gives an alternative solution independent from the Factorization method (although inspired by this method) and more importantly that efficiently treats the case of noisy operator.

The idea behind GLSM is as simple as reconstructing a nearby solution of the LSM by using a least squares misfit functional with a penalty term that controls Hg ǫ z 2

L 2 (D)
. This is feasible thanks to the second factorization of the farfield operator, which is the starting point of the Factorization method. More precisely, for the case under consideration, since the farfield operator of w has the following expression ( [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF])

w ∞ (x) = - D e -iky.x (1 -n)k 2 (ψ(y) + w(y))dy, one simply has G = H * T ψ where H * : L 2 (D) → L 2 (S d-1
) is the adjoint of H given by

H * ϕ(x) := D e -iky.x ϕ(y)dy, ϕ ∈ L 2 (D), x ∈ S d-1 ,
and where T : L 2 (D) → L 2 (D) is defined by

T ψ := -k 2 (1 -n)(ψ + w), (4)
with w ∈ H 1 loc (R d ) being the solution of (1). Finally we get F = H * T H, which indicates that (F g, g) L 2 (S d-1 ) = (T (Hg), Hg) L 2 (D) . Therefore, if the operator T satisfies some appropriate coercivity property, the term (F g, g) L 2 (S d-1 ) would be equivalent to Hg ǫ z 2

L 2 (D) . One then can use (F g, g) L 2 (S d-1 ) as a penalty term and also as a criterion for building the indicator function. This is the starting point of GLSM. The detailed formulation and analysis of the method are given in the next section.

Theoretical Foundations of GLSM

In this section we shall give the theoretical foundations of the Generalized Linear Sampling Method. The general framework is given by the following assumptions. We shall denote by X and Y two (complex) reflexive Banach spaces with duals X * and Y * respectively and shall denote by , a duality product that refers to X * , X or Y * , Y duality. We consider two bounded linear operators F : X → X * and B : X → X * that are assumed to be bounded. Moreover we shall assume that the following factorizations hold

F = GH and B = H * T H (5)
where the operators

H : X → Y , T : Y → Y * and G : R(H) ⊂ Y → X * are bounded,
where R(H) the closure of the range of H in Y .

Formulation of GLSM for noise free measurements

Let α > 0 be a given parameter and φ ∈ X * . The GLSM (for noisy free measurements) is based on considering minimizing sequences of the functional J α (φ;

•) : X → R J α (φ; g) := α| Bg, g | + F g -φ 2 ∀g ∈ X. (6)
Indeed this functional has not a minimizer in general. However, since J α (φ; •) ≥ 0 one can define

j α (φ) := inf g∈X J α (φ; g). (7)
Then the first simple observation is the following.

Lemma 2. Assume that F has dense range. Then for all φ ∈ X * , j α (φ) → 0 as α → 0.

Proof. Since F has dense range, for a given ǫ > 0 there exists g ǫ such that F g ǫ -φ < ǫ 2 . Then one can choose α 0 (ǫ) such for all α ≤ α 0 (ǫ), α| Bg ǫ , g ǫ | < ǫ 2 so that j α (φ) < ǫ, which proves the claim.

The central theorem for noisy free GLSM is the following characterization of the range of G in terms of F and B.

Theorem 3. We assume in addition that

• G is compact and F = GH has dense range.

• T satisfies the coercivity property

| T ϕ, ϕ | > µ ϕ 2 ∀ ϕ ∈ R(H), (8)
where µ > 0 is a constant independent of ϕ. Let C > 0 be a given constant (independent of α) and consider for α > 0 and φ ∈ X * , g α ∈ X such that

J α (φ; g α ) ≤ j α (φ) + C α. ( 9
)
Then φ ∈ R(G) if and only if lim sup α→0 | Bg α , g α | < ∞ which is true if and only if lim inf α→0 | Bg α , g α | < ∞. Proof. • Assume that φ ∈ R(G). Then, by definition one can find ϕ ∈ R(H) such that Gϕ = φ. for α > 0, ∃g 0 ∈ X such that Hg 0 -ϕ 2 < α. Then by continuity of G, F g 0 -φ 2 < G 2 α.
On the other hand the continuity of T implies

| Bg 0 , g 0 | = | T Hg 0 , Hg 0 | ≤ T Hg 0 2 < 2 T (α + ϕ 2 )
From the definitions of j α (φ) and g α we have

α| Bg 0 , g 0 | + F g 0 -φ 2 > j α (φ) > J α (φ, g α ) -Cα.
We then deduce from the definition of J α and previous inequalities

α| Bg α , g α | ≤ J α (φ, g α ) ≤ Cα + 2α T (α + ϕ 2 ) + α G 2 .
Therefore lim sup

α→0 | Bg α , g α | < ∞. This also implies lim inf α→0 | Bg α , g α | < ∞.
• Assume that φ / ∈ R(G) and assume (by a contradiction argument) that lim inf α→0 | Bg α , g α | < ∞. Then, (for some extracted subsequence g α ) | Bg α , g α | < A for some constant A independent of α → 0. The coercivity of T implies that Hg α is also bounded. Since Y is reflexive, then one can assume that, up to an extracted subsequence, Hg α weakly converges to some ϕ in Y . In fact ϕ ∈ R(H) since the latter is a convex set. Since G is compact, we obtain that GHg α strongly converges to Gϕ as α → 0. On the other hand, Lemma 2 and the definition

J α (φ, g α ) imply that F g α -φ ≤ J α (φ, g α ) ≤ j α (φ) + Cα → 0 as α → 0. Since F g α = GHg α we obtain that Gϕ = φ which is a contradiction. We then conclude that if φ / ∈ R(G) then lim inf α→0 | Bg α , g α | = ∞. The latter also implies lim sup α→0 | Bg α , g α | = ∞.
As indicated in the previous section, the range of the operator G characterizes the inclusion D. Therefore this theorem would lead to a characterization of D in terms of the operators F and B. It also stipulates that an indicator function is given by | Bg α , g α | for small values of α. Let us note that the parameter α does not play the role of a regularization parameter, since for foreseen applications, the operator B is in general compact. However, constructing a sequence (g α ) satisfying [START_REF] Erhard | A numerical study of the probe method[END_REF] for fixed α > 0 may be viewed as a regularization of the minimization of J α (φ; •) that can be used for numerics. A different regularization procedure that would be more suited for noisy operators is introduced in the following subsection.

Let us finally remark that in most of the applications that we have in mind, taking B = F would be sufficient. In this particular case one can state the following straightforward corollary.

Corollary 1. Assume that G(ϕ) = H * T (ϕ) for all ϕ ∈ R

(H) and assume in addition that

• H is compact and F has dense range,

• T satisfies the coercivity property [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF].

Let C > 0 be a given constant (independent of α) and consider for α > 0 and φ ∈ X * , g α ∈ X such that

J α (φ; g α ) ≤ j α (φ) + C α. ( 10
)
Then φ ∈ R(G) if and only if lim sup α→0 | F g α , g α | < ∞ which is true if and only if lim inf α→0 | F g α , g α | < ∞.
The assumptions required in this corollary are weaker than the ones required by the Factorization method but are similar to those of so-called inf-criterion (See [START_REF] Kirsch | The factorization method for inverse problems[END_REF]). Indeed the main advantage of GLSM with respect to the inf-criterion (as it will explained in the numerical section) is that it leads to a more tractable numerical inversion algorithms. In some special configurations there is a direct link between GLSM and the factorization method as explained below.

We also remark that according to Lemma 2 the sequence (g α ) provides a nearby solution to F g ≃ φ satisfying

F g α -φ ≤ j α (φ) + C α.
The reader then easily observe from the proof that one obtain the same conclusion in Corollary 1 if we replace the indicator function | F g α , g α | by | φ, g α |. The latter criterion coincides with the one proposed in [START_REF] Arens | Why linear sampling works[END_REF] and has been analyzed in [START_REF] Arens | Why linear sampling works[END_REF] and [START_REF] Arens | The linear sampling method revisited[END_REF] based on the (F * F ) 1 4 method.

Link with the (F

* F ) 1 4 method
We found it useful to indicate a link between the GLSM and the first version of the factorization method, namely the so-called (F * F ) 1 4 -method [START_REF] Kirsch | Characterization of the shape of a scattering obstacle using the spectral data of the far field operator[END_REF]. This method applies when X is a Hilbert space with a scalar product denoted (, ), and F : X → X is compact, normal, injective and with dense range. Then it is shown that F can be factorized as

F = (F * F ) 1 4 J(F * F ) 1 4
with J : X → X a coercive operator. Among others, two possibilities are of interest:

• A first possibility is to apply the GLSM with B = F , H = (F * F ) 1 4 and G = (F * F ) 1 4 J. We then obtain that φ ∈ R((F * F ) 1 4 ) if and only if lim sup α→0 | F g α , g α | <
∞ where g α satisfies [START_REF] Erhard | A numerical study of the probe method[END_REF]. Therefore, whenever one can use the range of (F * F ) 1 4 to characterize the shape of the scattering object, one can also use GLSM with B = F to obtain a different characterization.

• Another (more informative) possibility is to apply GLSM with B = (F * F ) 1 2 . In this case, using the system (λ i , ψ i ) i≥1 of eigenvalues and eigenvectors of the normal operator F , we observe that

J α (φ; g) = α|((F * F ) 1 2 g, g)| + F g -φ 2 = α i |λ i ||(g, ψ i )| 2 + i (λ i (g, ψ i ) -(φ, ψ i )) 2 .
Hence, J α (φ; •) has a minimizer given by

g α = i λi (φ, ψ i ) α|λ i | + |λ i | 2 ψ i .
It is clear that this g α satisfies [START_REF] Erhard | A numerical study of the probe method[END_REF]. Let us now define

g FM α = i |λ i | 1 2 |λ i | + α (φ, ψ i )ψ i ,
which is the minimizer of the Tikhonov functional α g 2 + (F * F )

1 4 g -φ 2
. Then one observes that the GLSM indicator is nothing but

|((F * F ) 1 2 g α , g α )| = i |λ i |(φ, ψ i ) 2 (α + |λ i |) 2 = g FM α 2 .
We finally remark that one obtains similar link with the so called F # method (when it applies) and GLSM by taking B = F # and replacing F by F # in the GLSM setting. But when the F # method applies one can also apply GLSM with only B = F .

Regularized formulation of GLSM

As it will be clearer later, the previous formulation of GLSM has to be adapted to the case of noisy operators since in general a noisy operator B does not satisfy a factorization of the form (5) (with a middle operator satisfying a coercivity property similar to [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]). In order to cope with this issue we introduce a regularized version of J α which allows similar range characterization and where one both controls the noisy criteria and the noisy misfit term. Among several other options, it turned out that a convenient way to introduce this regularization is to consider for α > 0 and ǫ > 0 (that will later be linked with the noise level) and for φ ∈ X * , the functional

J ε α (φ; •) : X → R defined by J ε α (φ; g) = α(| Bg, g | + ε g 2 ) + F g -φ 2 . ( 11
)
Lemma 3. Assume that B is compact. Then for all α > 0, ǫ > 0 and φ ∈ X * the functional J ε α (φ; •) has a minimizer g ε α ∈ X. If we assume in addition that F has dense range, then lim

α→0 lim ε→0 J ε α (φ; g ε α ) = lim ε→0 lim sup α→0 J ε α (φ; g ε α ) = 0.
Proof. The existence of minimizer is clear: for fixed α > 0, ǫ > 0 and φ ∈ X * , any minimizing sequence (g n ) of J ε α (φ; •) is bounded and therefore one can assume that it is weakly convergent in X to some g ε α ∈ X. The lower semi-continuity of the norm with respect to weak convergence and the compactness property of B then imply

J ε α (φ; g ε α ) ≤ lim inf n→∞ J ε α (φ; g n ) ≤ inf g∈X J ε α (φ; g),
which proves that g ε α is a minimizer of J ε α (φ; •) on X. Now assume in addition that F has dense range. By Lemma 2, j α (φ) → 0 as α → 0. Showing that lim ε→0 J ε α (φ; g ε α ) = j α (φ) will then prove that lim

α→0 lim ε→0 J ε α (φ; g ε α ) = 0. We observe that J ε α (φ; g) = J α (φ; g) + αε g 2 (12)
and therefore |J ε α (φ; g) -J α (φ; g)| → 0 as ε → 0. For η > 0 one can choose g such that |J α (φ; g) -j α (φ)| ≤ η/2. For this g one then has for ε sufficiently small |J ε α (φ; g) -J α (φ; g)| < η/2. We obtain by triangular inequality that for ε sufficiently small J ε α (φ; g) ≤ j α (φ) + η. We now observe from the definitions of g ε α and j α and from [START_REF] Kirsch | Characterization of the shape of a scattering obstacle using the spectral data of the far field operator[END_REF], j α (φ) ≤ J α (φ; g ε α ) ≤ J ε α (φ; g ε α ) ≤ J ε α (φ; g), which proves the claim. We now prove lim ε→0 lim sup α→0 J ε α (φ; g ε α ) = 0. First consider g ε a minimizer on X of the Tikhonov functional ε 2 g 2 + F g -φ 2 and set

j ε = ε 2 g ε 2 + F g ε -φ 2
which goes to zero as ε goes to zero (classical result for Tikhonov regularization, see also Lemma 2 which is valid for any bounded operator B). We have that α ≤ ε, J ε α (g) ≤ ε 2 g 2 + F g -Φ 2 + α(|(Bg, g)|. For by taking the upper limit lim sup

α→0 J ε α (g ε α ) ≤ lim sup α→0 J ε α (g ε ) = j ε ,
which concludes the proof.

Theorem 4. Under the assumptions of Theorem 3 and the additional assumption that B is compact the following holds. If g ε α denotes the minimizer of J ε α (φ; •) (defined by ( 11)) for α > 0, ε > 0 and φ ∈ X * , then φ ∈ R(G) if and only if lim sup

α→0 lim sup ε→0 | Bg ε α , g ε α | < ∞ which is true if and only if lim inf α→0 lim inf ε→0 | Bg ε α , g ε α | < ∞.
Proof. The proof is similar to the proof of Theorem 3.

• Assume that φ = G(ϕ) for some ϕ ∈ R(H). We consider the same g 0 as in the first part of the proof of Theorem 3 (that depends on α but is independent from ε). Then we choose ε such that ε g 0 2 ≤ 1. Then

J ε α (φ; g ε α ) ≤ J ε α (φ; g 0 ) ≤ J α (φ; g 0 ) + α (13)
Consequently

α| Bg ε α , g ε α | ≤ J ε α (φ; g ε α ) ≤ α + 2α T (α + ϕ 2 ) + α G 2 which proves lim sup α→0 lim sup ε→0 | Bg ε α , g ε α | < ∞. • Assume φ / ∈ R(G) and assume that lim inf α→0 lim inf ε→0 | Bg ε α , g ε α | is finite. The coercivity of T implies that lim inf α→0 lim inf ε→0
Hg ε α 2 is also finite. This means the existence of a subsequence (α ′ , ε(α ′ )) such that α ′ → 0 and ε(α ′ ) → 0 as α ′ → 0 and

Hg ε(α ′ ) α ′ 2
is bounded independently from α ′ . On the other hand, the second part of Lemma 3 (namely the first limit), indicates that one can choose this subsequence such that J

ε(α ′ ) α ′ (g ε(α ′ ) α ′ ) → 0 as α ′ → 0 and therefore F g ε(α ′ ) α ′ -φ → 0 as α ′ → 0.
The compactness of G implies that a subsequence of GHg

ε(α ′ ) α ′
converges for some Gϕ in X * . The uniqueness of the limit implies that Gϕ = φ which is a contradiction.

In this theorem ε should be viewed as the regularization parameter (and not α which is rather used to construct an indicator function with a limiting process). As indicated by [START_REF] Kirsch | The factorization method for inverse problems[END_REF], this regularization parameter serves in the construction of the minimizing sequence of Theorem 3.

This theorem with regularization stipulates that a criterion to localize the target is given by | Bg ε α , g ε α | for small values of ǫ and α. The reader can easily see from the first part of the proof that the result holds true if we replace this by

(| Bg ε α , g ε α | + ε g ε α 2
). This latter criterion is more suited to the case of noisy measurements as indicated in the section below.

The GLSM for noisy data

In this section we will consider the case where there may be noise in the data. More precisely, we shall assume that one has access to two noisy operators B δ and F δ such that

F δ -F ≤ δ F and B δ -B ≤ δ B
for some δ > 0. We also assume in this section the operator, B, B δ F δ and F are compact. We then consider for α > 0 and φ ∈ X * , the functional

J δ α (φ; •) : X → R, J δ α (φ; g) := α(| B δ g, g | + δ B g 2 ) + F δ g -φ 2 ∀ g ∈ X, (14)
which coincides with a regularized noisy functional J ε α with a regularization parameter ǫ = δ B . According to Lemma 3 one can consider g δ α a minimizer of J δ α (φ; g). We first observe (similarly to in the second part of the proof of Lemma 3) Lemma 4. Assume in addition that F has dense range. Then for all φ ∈ X * ,

lim α→0 lim sup δ→0 J δ α (φ; g δ α ) = 0.
Proof. We observe that for all g ∈ X,

J δ α (φ; g) ≤ J α (φ; g) + (2αδ B + δ 2 F 2 ) g 2 . ( 15
)
Since (2αδ B + δ 2 F 2 ) → 0 as δ → 0, then as in the proof of Lemma 3, for any η > 0 (α fixed), one can choose g ∈ X such that for sufficiently small δ,

J δ α (φ; g) ≤ j α (φ) + η
Consequently, from the definition of g δ α ,

J δ α (g δ α ; φ) ≤ j α (φ) + η
This proves the claim, since j α (φ) → 0 as α → 0 (by Lemma 2).

Theorem 5. Assume that the assumptions of Theorem 3 and the additional assumptions of this subsection hold true. Let g δ α be the minimizer of J δ α (φ; •) (defined by ( 21)) for α > 0, δ > 0 and φ ∈ X * .

Then φ ∈ R(G) if and only if lim sup

α→0 lim sup δ→0 B δ g δ α , g δ α + δ B g δ α 2
< ∞ which is true if and only if

lim inf α→0 lim inf δ→0 B δ g δ α , g δ α + δ B g δ α 2 < ∞.
Proof. The proof of this theorem follows the lines of the proof of Theorem 4.

• Assume that φ = G(ϕ) for some ϕ ∈ R(H). We consider the same g 0 as in the first part of the proof of Theorem 3 (that depends on α but is independent from δ). Choosing δ sufficiently small such that we get

J δ α (φ; g δ α ) ≤ J δ α (φ; g 0 ) ≤ J α (φ; g 0 ) + α. (16)
Consequently

α | Bg δ α , g δ α | + δ B g δ α 2 ≤ J δ α (φ; g δ α ) ≤ α + 2α T (α + ϕ 2 ) + α G 2 ,
which proves lim sup

α→0 lim sup δ→0 B δ g δ α , g δ α + δ B g δ α 2 < ∞.
• Assume φ / ∈ R(G) and assume that lim inf

α→0 lim inf ε→0 B δ g δ α , g δ α + δ B g δ α 2 is finite. The coercivity of T implies that µ Hg δ α(δ) 2 ≤ | Bg δ α , g δ α | ≤ | B δ g δ α , g δ α | + δ B g δ α 2 .
Therefore lim inf

α→0 lim inf δ→0
Hg δ α 2 is also finite. This means the existence of a subsequence (α ′ , δ(α ′ )) such that α ′ → 0 and δ(α ′ ) → 0 as α ′ → 0 and Hg

δ(α ′ ) α ′ 2 is bounded independently from α ′ . One can also choose δ(α ′ ) such that δ(α ′ ) ≤ α ′ .
On the other hand Lemma 4 indicates that one can choose this subsequence such that J

δ(α ′ ) α ′ (g δ(α ′ ) α ′ ) → 0 as α ′ → 0 and therefore F δ g δ(α ′ ) α ′ -φ → 0 as α ′ → 0 and α ′ δ(α ′ ) g δ(α ′ ) α ′
2 → 0 as α ′ → 0. By a triangular inequality and δ(α ′ ) ≤ α ′ we then deduce that F g

δ(α ′ ) α ′ -φ → 0 as α ′ → 0. The compactness of G implies that a subsequence of GHg δ(α ′ ) α ′
converges for some Gϕ in X * . The uniqueness of the limit implies that Gϕ = φ which is a contradiction.

It is clear from the proof of the theorem that any strategy of regularization ε(δ) satisfying ǫ(δ) ≥ δ B and ǫ(δ) → 0 as δ → 0 would be convenient to obtain a similar result. From the numerical perspective this theorem indicates that a criterion to localize the object would be

| B δ g δ α , g δ α | + δ B g δ α 2
for small values of α. Indeed the theorem only says that this criterion would be efficient for sufficiently small noise. Building explicit link between the value of α and the noise level δ (in the fashion of a posteriori regularization strategies) would be of valuable theoretical interest but this seems to be challenging (due to the compactness of the operator B). One can see from the proof that adding the term δ B g δ α 2 is important to conclude when φ is not in the range of G. This means that this term is important for correcting the behavior of the indicator function outside the inclusion, which is corroborated by the numerical experiments below.

Some applications of GLSM

We turn back to our model problem and consider the notation and assumptions of Section 2. We shall apply GLSM with B = F . The central additional theorem needed for this case is the following coercivity property of the operator T . This theorem holds true under the following assumptions on the refractive index.

Hypothesis 2. We assume that n ∈ L ∞ (R d ), supp(n -1) = D, ℑ(n) ≥ 0 and there exists a constant

n 0 > 0 such that 1 -ℜ(n(x)) + ℑ(n(x)) ≥ n 0 for a.e. x ∈ D or ℜ(n(x)) -1 + ℑ(n(x)) ≥ n 0 for a.e. x ∈ D.
We recall that the values of k 2 ∈ R + for which Hypothesis 1 does not hold form a discrete set without finite accumulation point. The values k 2 ∈ R + for which Hypothesis 1 does not hold will be referred in the sequel as transmission eigenvalues. Theorem 6. Assume that Hypothesis 2 holds and that k 2 ∈ R + is not a transmission eigenvalue. Then the operator T defined by ( 4) satisfies the coercivity property ( 8) with X = X * = L 2 (D) and the operator H defined by [START_REF] Arens | The linear sampling method revisited[END_REF].

Proof. For the reader convenience we start by proving a useful (classical) identity related to the imaginary part of T . With ( , ) denoting L 2 (D) scalar product, for ψ ∈ L 2 (D) and w ∈ H 1 loc (R d ) solution of ( 1),

(T ψ, ψ) = -k 2 D (1 -n)(ψ + w)ψ dx. (17)
We remark that by elliptic regularity, w ∈ H 2 loc (R d ). Multiplying (1) with w and integrating by part over B R : a ball of radius R containing D,

k 2 D (1 -n)(ψ + w)w dx = - B R |∇w| 2 -k 2 n|w| 2 dx + |x|=R ∂w ∂r w ds.
The Sommerfeld Radiation condition indicates that lim

R→∞ ℑ |x|=R ∂w ∂r w ds = k S d-1 |w ∞ | 2 ds,
Therefore, taking the imaginary part then letting R → ∞ yields

k 2 ℑ D (1 -n)(ψ + w)w dx = D k 2 ℑ(n)|w| 2 dx + k S d-1 |w ∞ | 2 ds.
Consequently, decomposing (ψ + w)ψ = |ψ + w| 2 -(ψ + w)w, we obtain the important identity,

ℑ(T ψ, ψ) = D k 2 ℑ(n)(|ψ + w| 2 + |w| 2 )dx + k S d-1 |w ∞ | 2 ds. (18)
We are now in position to prove the coercivity property using a contradiction argument. Assume for instance the existence of a sequence ψ ℓ ∈ R(H) such that

ψ ℓ L 2 (D) = 1 and |(T ψ ℓ , ψ ℓ )| → 0 as ℓ → ∞.
We denote by w ℓ ∈ H 2 loc (R d ) solution of (1) with ψ = ψ ℓ . Elliptic regularity implies that w ℓ H 2 (D) is bounded uniformly with respect to ℓ. Then up to changing the initial sequence, one can assume that ψ ℓ weakly converges to some ψ in L 2 (D) and w ℓ converges weakly in H 2 loc (R d ) and strongly in L 2 (D) to some w ∈ H 2 loc (R d ). It is then easily seen (using distributional limit) that w and ψ satisfies (1), and since

ψ ℓ ∈ R(H) ∆ψ + k 2 ψ = 0 in D. (19)
Identity [START_REF] Sylvester | Discreteness of transmission eigenvalues via upper triangular compact operators[END_REF] and

|(T ψ ℓ , ψ ℓ )| → 0 implies that w ∞ ℓ → 0 in L 2 (S d-1
) and therefore w ∞ = 0. The Rellich theorem and unique continuation principle implies w = 0 outside D and consequently w ∈ H 2 0 (D). With the help of equation ( 19) we get that

u = w + ψ ∈ L 2 (D) and v = ψ ∈ L 2 (D) are such that u -v ∈ H 2 (D)
and are solution of the interior transmission problem (3) with f = g = 0. We then infer that w = ψ = 0. Identity [START_REF] Sorber | Complex optimization toolbox v1.0[END_REF] applied to ψ ℓ and w ℓ implies

|(T ψ ℓ , ψ ℓ )| ≥ k 2 D (1 -n)|ψ ℓ | 2 dx -k 2 D (1 -n)w ℓ ψ ℓ dx .
Therefore, since D (1 -n)w ℓ ψ ℓ dx → D (1 -n)wψdx = 0, and using the assumptions on n,

lim ℓ→0 |(T ψ ℓ , ψ ℓ )| ≥ k 2 n 0 /2 > 0,
which is a contradiction.

Let C > 0 be a given constant (independent of α) and consider for α > 0 and

z ∈ R d , g z α ∈ L 2 (S d-1 ) such that α|(F g z α , g z α )| + F g z α -φ z 2 ≤ j α (φ z ) + C α, (20)
where

j α (φ z ) = inf g∈L 2 (S d-1 ) α|(F g, g)| + F g -φ z 2 .
Combining the results of Theorems 6 and 1 and the first claim of Theorem 2, we obtain the following as a straightforward application of Corollary 1.

Theorem 7. Assume that Hypothesis 2 holds and that

k 2 ∈ R + is not a transmission eigenvalue. Then z ∈ D if and only if lim sup α→0 |(F g z α , g z α )| < ∞ which is true if and only if lim inf α→0 |(F g z α , g z α )| < ∞.
For applications, it is important to rather use the criterion provided in Theorem 5. Consider F δ : L 2 (S d-1 ) → L 2 (S d-1 ) a compact operator such that F δ -F ≤ δ, then consider for α > 0 and φ ∈ L 2 (S d-1 ), the functional J δ α (φ;

•) : L 2 (S d-1 ) → R, J δ α (φ; g) := α(|(F δ g, g)| + δ g 2 ) + F δ g -φ 2 ∀ g ∈ L 2 (S d-1 ). ( 21
)
Then as a direct consequence of Theorem 5, we have the following characterization of D.

(F δ g z α,δ , g δ α,δ ) + δ g z α,δ 2 < ∞.
The numerical algorithm associated with this theorem is given in next section. Let us note again as conclusion of this section that the results of Theorems 7 and 8 in fact apply whenever the so called F # method apply. For instance the result hold true for obstacle scattering with Dirichlet boundary conditions, Neumann boundary conditions or impedance boundary conditions [START_REF] Kirsch | The factorization method for inverse problems[END_REF][START_REF] Mathieu Chamaillard | Analysis of the factorization method for a general class of boundary conditions[END_REF]. One has just to remove assumption 2 and instead of excluding transmission eigenvalues, one has to exclude the resonant eigenfrequencies associated with the interior problem. One can also apply GLSM to cracks as a consequence of the work in [START_REF] Boukari | The Factorization method applied to cracks with impedance boundary conditions[END_REF]. For Maxwell's equations one can in principle also treat the inverse medium problem but the GLSM method does allow to treat (in its current form) the case of inverse obstacle scattering (i.e. for instance perfectly or imperfectly conducting obstacles).

Numerical algorithms issued from GLSM and validation

Minimizing J δ α (defined in equation 21) with B = F may be computationally expensive and not straightforward (see Section 5.2). Thus we first propose to use the indicator function of the GLSM with the solution of the LSM, which can be seen as a generalisation of [START_REF] Arens | The linear sampling method revisited[END_REF] in the case of noisy measurement. Then we introduce a second algorithm which is a post processing in the sense that it uses the solution of the LSM both to initialise the optimisation algorithm that minimize J δ α and to initialise the parameter α. In order to fix the ideas, we shall restric ourselves to the two dimensional case and will introduce the algorithms for the discrete version of GLSM. We identify S 1 with the interval [0, 2π[. In order to collect the data of the inverse problem we solve numerically (1) for N incident fields u i ( 2πj N , •), j ∈ {0...N -1} using the surface integral equation forward solver available in [START_REF] Haddar | Sampling 2d, Mars[END_REF]. The discret version of F is then the matrix

F N := (u ∞ ( 2πj N , 2πk N )) 0≤j,k≤N-1 .
We add some noise to the data to build a noisy far field matrix F δ N where (F δ N ) j,k = (F N ) j,k (1 + σN ij ) for σ > 0 and N ij an uniform complex random variable in [-1, 1] 2 . We denote Φ z,N ∈ C N , the vector defined by Φ z,N (j) = φ z ( 2πj N ) for 0 ≤ j ≤ N -1.

The use of GLSM as a new indicator function for the LSM

We introduce the Tikhonov regularized solution of the far field equation

g η,LSM z,N := argmin g N η g N 2 L 2 (S 1 ) + F δ N g N -Φ z,N 2
L 2 (S 1 ) ,
where the regularization parameter η is chosen using the Morozov discrepancy principle, i.e. η is defined as the unique solution of

F δ N g η,LSM z,N -Φ z,N L 2 (S 1 ) = δ g η,LSM z,N L 2 (S 1 )
.

Solving the same two equations with F # or (F * F ) 1 4 , depending of the nature of the scatter, instead of F will give the solution of the factorization method g η,FM z,N . To solve both the LSM and the FM equations we rely on the singular value decomposition of F δ N , which gives an explicit solution like in 3.2.

As proposed in [START_REF] Arens | The linear sampling method revisited[END_REF], [START_REF] Kirsch | Characterization of the shape of a scattering obstacle using the spectral data of the far field operator[END_REF] and [START_REF] Colton | A simple method for solving inverse scattering problems in the resonance region[END_REF], from these two problems three indicator functions can be computed:

I LSM (z) = 1 g η,LSM z,N L 2 (S 1 ) I HLSM (z) = 1 |Hg η,LSM z,N (z)
| = 1 |(Φ z,N , g η,LSM z,N ) L 2 (S 1 ) | I FM (z) = 1 g η,FM z,N L 2 (S 1 )
As shown in the previous sections, a fourth indicator function is relevant, namely

I GLSM (z) = 1 (F δ N g η,LSM z,N , g η,LSM z,N ) L 2 (S 1 ) + δ g η,LSM z,N 2
L 2 (S 1 )
This indicator is indeed motivated by GLSM. However let us note that since g η,LSM z,N is not the minimizer of J δ α (φ; •) (defined in equation ( 21)) the theory developed here does not apply for this indicator function (a last indicator function covered by the theory will be build in section 5.2 using a more computationally complex method). The numerical experiments presented below indicate in the same time that this indicator function provide results comparable to the Factorization method.

We will present two simulations: one where two ellipses have Dirichlet boundary conditions and one where n = 2 + 0.5i in one ellipse and 2 + 0.1i in the other. In both examples N = 100 and we will consider

F δ N -F N F N
= 0, 1 and 5%. Figures 1 and2 show the results of the four indicator functions. First we see that I HLSM is not robust to noise, the area outside the obstacle shows artefact where the indicator function is greater than inside the obstacle. This is an expected result since as stated at the end of 3.1 one can easily replace | F g α , g α | by | φ, g α |, which is not a valid indicator function in the presence of noise. Finally I LSM recover with less precision the border of the shape than I FM and I GLSM which exhibit comparable results.

minimizing J δ

α : a post-processing In order to apply Theorem 8, we should find the minimizer of J δ α (φ; •) (defined in equation ( 21)). There are two main difficulties in this theorem. First, we do not have an analytic solution of the minimizer thus we will rely on an optimisation algorithm and as already mentioned in 3.4 and second we do not have an a priori method to link α to the noise level. Because of the good performance of the Morozov discrepancy principle we look for an heuristic that stay close to this principle. Since we have α(|(F δ g, g)| + δ g 2 ) ≤ α( F δ + δ) g 2 , we choose:

α = η LSM F δ + δ
where η LSM is the parameter found when one applies the Morozov discrepancy principle to the Tikhonov formulation of the LSM.

Remark. The inequality |(F δ g, g)| ≤ F δ g 2 , we use to find the previous heuristic will reduce the strength of the penalty term compared to the Tikhonov-LSM. Moreover the fact that this inequality is coarser for eigenvector corresponding to small eigenvalue, means that the penality term will be smaller for points outside the obstacle. This is shown by figure 3, where we see that after the optimisation process the solution deviates from the Morozov discrepancy principles mainly outside the obstacle. Minimizing J δ α (φ; •) in C N is not an easy task since it is a not differentiable nor a convex cost functionnal. However we can hope that g η,LSM z,N will be close to a minimum which makes it worth to try a gradient method. As explained in [START_REF] Sorber | Unconstrained optimization of real functions in complex variables[END_REF] gradient method extended well for complex variable if one look at J δ α (φ; g N ) as a function of two variables, g N and ḡN , knowing that one can compute the gradient of J δ α with respect to g N :

∇ ḡN J δ α (φ; g N , ḡN ) := α(

F δ N g N • ḡN |F δ N g N • ḡN | F δ N g N + δg N ) + F δ * N (F δ N g N -Φ N )
where • is the standard scalar product between vector. We did not change the absolute value with a differentiable surrogate because with the initial guess we used it was not necessary, this is supported by the fact that for the unperturbed operator F the coercivity implies that |(F g, g)| is never zero when g is not zero. Finally to do the optimisation we use the non-linear conjugate gradient implemented in [START_REF] Sorber | Complex optimization toolbox v1.0[END_REF] with a modified Hestenes-Stiefel heuristic to update the direction descent, which is described in algorithm 1. We choose drastic stopping rules in order to ensure the convergence of the algorithm however we observe that convergence occurs before those stopping rules are satisfied. The design of a tailored method and set of parameters to minimize J δ α would be an interesting perspective for this work. The result of this optimisation performed for each z, gives us a new set: g α,GLSM z,N which ultimately creates a new indicator function:

I GLSMoptim = 1 (F δ N g α,GLSM z,N
, g α,GLSM z,N

) + δ g α,GLSM z,N Figures 4 and 5 show that this post processing increases the quality of the reconstruction especially in the space in-between the two scatters. Moreover figure 6 shows that the improvement on an isolated scattered, a kite of contrast n = 2 + 0.5i, is less impressive (i.e. we do not improve the reconstruction of the non-convex part of the kite).

Figure 1 .

 1 Figure 1. I HLSM (first line), I LSM (second line), I FM (third line) and I GLSM (forth line) applied to the Dirichlet scatters for 0, 1 and 5% of noise (from left to right)

Figure 2 .

 2 Figure 2. I HLSM (first line), I LSM (second line), I FM (third line) and I GLSM (forth line) applied to penetrable scatters for 0, 1 and 5% of noise (from left to right))

Figure 3 .

 3 Figure 3. F δ N g N -Φ z,Nδ g N after minimisation on the Dirichlet scatters with 5% of noise.

Figure 5 .Figure 6 .

 56 Figure 5. I GLSM (first line) and I GLSMoptim (second line) applied to the penetrable scatters for 1 and 5% of noise (from left to right)

  Theorem 8. Assume that Hypothesis 2 holds and that k 2 ∈ R + is not a transmission eigenvalue. For z ∈ R d denote by g z α,δ the minimizer of J δ α (φ z ; •) over L 2 (S d-1 ). Then z ∈ D if and only if lim sup

		α→0	lim sup δ→0	(F δ g z α,δ , g z α,δ ) + δ g z α,δ	2	< ∞ which is true if
	and only if lim inf α→0	lim inf δ→0

(2αδ B + δ 2 F 2 ) g 0

Algorithm 1 Minimizing J δ α for all z do g 0 = g η,LSM z,N and α = η LSM (z)

while g t+1 -g t ≤ 10 -10 g t or J δ α (g t+1 ) -J δ α (g t ) ≤ 10 -10 J δ α (g 0 ) or t < 200 do ∆g t = -∇ g J δ α (g t , ḡt )

) )

Remark. In the (less general) case where the F # method is valid, one could choose 21). We know that F δ # is a positive and self-adjoint operator then one can drop the absolute value in the definition of J δ α :


Sébastien Laurent

Xavier Leyronas

Frédéric Chevy

Momentum distribution of a dilute unitary Bose gas with three-body losses

Keywords: numbers: 05.20.Dd, 67.10.-j, 34.50.Cx

Using Boltzmann's equation, we study the effect of three-body losses on the momentum distribution of a homogeneous unitary Bose gas in the dilute limit where quantum correlations are negligible. We calculate the momentum distribution of the gas and show that inelastic collisions are quantitatively as important as a second order virial correction.

In the past few years, ultracold gases have become a unique tool for the experimental study of strongly correlated systems. In atomic vapours, strong interactions can be achieved either by trapping the atoms in an optical lattice or by using Feshbach resonances. While the first route has been very successful and has led to groundbreaking discoveries such as the observation of the Mott transition in both Bose [START_REF] Greiner | Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms[END_REF] and Fermi gases [START_REF] Schneider | Metallic and insulating phases of repuslsively interacting fermions in a 3D optical lattice[END_REF][START_REF] Jördens | A Mott insulator of fermionic atoms in an optical lattice[END_REF], Feshbach resonances could only be used to study strongly correlated Fermi gases. Indeed, despite interest in strongly correlated bosonic systems [START_REF] Cowell | Cold bose gases with large scattering lengths[END_REF][START_REF] Song | Ground state properties of cold bosonic atoms at large scattering lengths[END_REF][START_REF] Lee | Universality and stability for a dilute bose gas with a feshbach resonance[END_REF][START_REF] Borzov | Three-dimensional bose gas near a feshbach resonance[END_REF][START_REF] Li | Bose gases near unitarity[END_REF], the lifetime of the cloud of bosons near a Feshbach resonance is strongly reduced by the onset of three-body recombination towards deeply bound molecular states [START_REF] Roberts | Magnetic field dependence of ultracold inelastic collisions near a feshbach resonance[END_REF][START_REF] Inouye | Observation of feshbach resonances in a bose-einstein condensate[END_REF]. Recent experimental results suggested new routes to overcome this challenge and that it might be possible to quantitatively study the unitary Bose gas. First, it was demonstrated that at finite temperature the increase of the three-body loss rate scaling as a 4 actually saturates when a ≫ λ th , where λ th = h/ √ 2πmk B T is the thermal wavelength [START_REF] Rem | Lifetime of the bose gas with resonant interactions[END_REF][START_REF] Richard | Stability of a unitary bose gas[END_REF]. Moreover, recent experimental results from JILA demonstrated universal local dynamics of the momentum distribution of a unitary Bose gas towards a quasi-equilibrium state [START_REF] Makotyn | Universal dynamics of a degenerate unitary bose gas[END_REF] and have triggered several theoretical works on the dynamics of strongly correlated Bose gases near Feshbach resonances [START_REF] Sykes | Quenching to unitarity: Quantum dynamics in a 3d bose gas[END_REF][15][16].

The stability of the unitary Bose gas hinges on the following argument [START_REF] Li | Bose gases near unitarity[END_REF]: first, the three-body losses are characterized by a coefficient L 3 such that Ṅ = -L 3 n 2 N , where N is the total atom number and n is the particle density. This phenomenological law defines a characteristic loss rate γ 3 = L 3 n 2 . For a non-quantum degenerate gas, the cloud is brought back to equilibrium by elastic scattering at a characteristic rate γ 2 ≃ nσv, where σ is the scattering cross-section and v is the characteristic velocity of the atoms. At unitarity, the scattering cross-section follows a universal scaling σ = 8π/k 2 , where k is the relative wave-vector of two scattering particles. In the presence of losses, the system can be kept in a quasi-equilibrium state provided that the ratio γ 3 /γ 2 stays small. It was shown both theoretically and experimentally [START_REF] Rem | Lifetime of the bose gas with resonant interactions[END_REF][START_REF] Richard | Stability of a unitary bose gas[END_REF] that at unitarity the three body loss-rate is given by

L 3 ≃ 36 √ 3π 2 5 m 3 (k B T ) 2 (1 -e -4η ), ( 1
)
where η is a dimensionless parameter characterizing the probability of forming a deeply bound molecule at short distance [17]. Plugging Eq. ( 1) into the expression for γ 3 , we see that quasi-equilibrium can be achieved as long as (1 -e -4η )nλ 3 th is small, i.e. when the system is not too deeply in the quantum degenerate regime.

In this letter, we investigate the effect of 3-body losses on the momentum distribution of a unitary Bose gas. Our analysis is based on a semi-analytical resolution of Boltzmann's equation. Since Boltzmann's equation neglects all many-body correlations, our work is restricted to a low-phase space density regime where, as aforementioned, three-body losses can be treated perturbatively. We calculate the first correction to the momentum distribution and we compare it to the effect of two-body interactions. We show that in the dilute limit, both effects deplete the center of the momentum distribution proportionally to the phase-space density of the gas. Moreover, for realistic parameters, this depletion is dominated by three-body losses.

Consider a homogeneous Bose gas that we describe by a phase space density f (p). In the presence of losses, f is the solution of Boltzmann's equation that we write formally

∂ t f = I coll [f ] -L 3 [f ], (2)
where I coll and L 3 are non linear operators describing respectively the elastic collisions and the three-body losses. At low phase space density, we can neglect the bosonic stimulation and we have

I coll [f ](p 1 ) = d 3 p 2 d 2 ω ′ dσ dω ′ |p 2 -p 1 | m (f 3 f 4 -f 1 f 2 ) .
(3) Here, f α stands for f (p α ), (p 1 , p 2 ) (resp. (p 3 , p 4 )) are the incoming (outgoing) momenta satisfying energy and momentum conservation and dσ/dω ′ = 8 2 /|p 1p 2 | 2 is the differential scattering cross-section towards the outgoing solid angle ω ′ .

From [START_REF] Rem | Lifetime of the bose gas with resonant interactions[END_REF], the loss rate operator for a unitary Bose gas can be written as

L 3 [f ](p 1 ) = d 3 p 2 d 3 p 3 A 3 E 2 123 |φ(Ω 3 )| 2 f (p 1 )f (p 2 )f (p 3 ), (4) where
E 123 = (p 2 1 +p 2 2 +p 2 3 )/2m-(p 1 +p 2 +p 3 ) 2 /6m
is the energy in the center of mass frame of the three particles of momenta (p 1 , p 2 , p 3 ), A 3 = 2π 3 (k B T ) 2 L 3 and φ(Ω 3 ) is the hyperangular wave-function describing the angular structure of the Efimov trimers that we normalize by the condition

d 5 Ω 3 |φ(Ω 3 )| 2 = 1.
In absence of losses the system thermalizes to a distribution G solution of I col [G] = 0. For a classical gas, the solution of this equation is a Gaussian distribution G(n, E; p) = nλ 3 th e -βp 2 /2m /h 3 , where

β = 1/k B T and E = (G(p)p 2 /2m)d 3 p = 3nk B T /2 is the energy den- sity.
In the quasi-static regime γ 3 /γ 2 ≪ 1, three-body losses are small and we can use A 3 as an expansion parameter. Since for A 3 = 0 the system can reach a stationary thermal state, we expect the characteristic evolution time in the presence of losses to vary as A -1

3 and thus ∂ t must be considered to scale as A 3 . We write then f = f 0 + f 1 + ... where f j ∝ A j 3 . The expansion of Eq. ( 2) to first order in A 3 , yields

I coll [f 0 ] = 0 (5) ∂ t f 0 = I ′ coll [f 1 ] -L 3 [f 0 ]. (6)
where I ′ coll is the linearized collisional operator. According to (5), f 0 is a Maxwell-Boltzmann distribution. However, since the system loses particles by threebody recombination, its atom number and its energy vary with time. We therefore have f 0 (p, t) = G(n t , E t ; p). We then have in Eq. ( 6)

I ′ coll [f 1 ] = L 3 [f 0 ] + Ė∂ E G + ṅ∂ n G. (7)
Take f 1 (p, t) = G(n t , E t ; p)α(p, t). Eq. ( 7) then becomes

C[α] = 1 G L 3 [G] + Ė∂ E ln(G) + ṅ∂ n ln(G). (8)
with

C[α] = 1 G I ′ coll [Gα] (9) = d 3 p 2 d 2 ω ′ f 0 (p 2 ) dσ dω ′ |p 2 -p 1 | m × (α 3 + α 4 -α 1 -α 2 ) . ( 10
)
and

α k = α(p k ) for k = 1, • • • 4. The operator C is symmetric for the scalar product [18] α|α ′ = d 3 pG(p)α(p)α ′ (p). ( 11
)
Due to energy and particle number conservation, the kernel of C is spanned by α(p) = 1 and α(p) = p 2 . Finally, being a symmetric operator, its image is orthogonal to its kernel. To find the time evolution of the energy and the atom number, we project Eq. ( 8) on 1 and p 2 . Using the structure of the kernel of C, the collisional term vanishes and we obtain

ṅt = -1| 1 G L 3 [G] (12) Ėt = - p 2 2m | 1 G L 3 [G] . (13)
The explicit calculation of the rhs of these equations involves 9-dimensional integrals over the three momenta (p 1 , p 2 , p 3 ) in the three-body loss rate operator. This calculation can be performed analytically by introducing the momentum-space Jacobi coordinates (see supplemental material) and we finally obtain

ṅt = -L 3 n 3 (14) Ėt = - 5 9 EL 3 n 2 . ( 15
)
where we recover the usual formula for three-body losses, as well as the recombination heating discussed in [START_REF] Rem | Lifetime of the bose gas with resonant interactions[END_REF][START_REF] Richard | Stability of a unitary bose gas[END_REF].

To find α, we project Eq. ( 8) on the range of C (ie orthogonally to Span(1, p 2 )). We then have

C[α] = P 1 G L 3 [G] , (16)
where P is the orthogonal projector on Im(C), and where we used the fact that ln G is a linear combination of 1 and p 2 and thus lies in the kernel of C and P .

In the spirit of Chapman-Enskog's expansion, we expand α on a basis of orthogonal polynomials for the scalar product [START_REF] Rem | Lifetime of the bose gas with resonant interactions[END_REF]. Such a basis can be expressed in terms of the generalized Laguerre polynomials [19]

q k (p) = √ πk! 2nΓ(k + 3/2) L (1/2) k (βp 2 /2m) (17)
By definition, q 0 and q 1 lie in Ker(C) and as such will not contribute to the expansion. Take α(p) = k≥2 a k q k (p), where the coefficients a k are real numbers, Eq. ( 16) is then equivalent to the infinite set of linear equations

k ′ ≥2 a k ′ q k |C[q k ′ ] = q k | 1 G L 3 [G] , (18)
for k ≥ 2. In these equations, the coefficients q k |C[q k ′ ] can be calculated analytically to arbitrary order (see supplemental material), while the complex form of the Efimov wave-function allows only for a numerical calculation of the projection of the loss term on this polynomial basis. We solve this equation by truncating the indices (k, k ′ ) to a value k max . We observe in Fig.

k max »»a k max -a ¶ »» 2 ê »»a ¶ »» 2 FIG.
1: Convergence of the numerical solution of Eq. ( 18). We estimate the error on the solution using the norme α 2 = α|α and we compare the solution of Eq. ( 18) obtained by truncation at k = kmax with the "true" result corresponding to kmax = 15. convergence is very fast and that the first order result (k max = 2) gives the correct answer within a few percent accuracy.

In an experiment such as the one described in [START_REF] Makotyn | Universal dynamics of a degenerate unitary bose gas[END_REF], the cloud is not directly prepared in the quasi-static, strongly interacting state. Rather, the experimental sequence starts in the weakly interacting regime where losses can be neglected and the momentum distribution of the gas is gaussian. The magnetic field is then ramped quickly to unitarity where the system can relax towards the quasiequilibrium described above. To study the relaxation rate towards the quasi-static solution we write as before f = f 0 + f 1 with f 1 = f 1,qs + δf 1 , where f 1,qs is the quasi-static solution and δf 1 satisfies the initial condition δf 1 (p, t = 0) = -f 1,qs (p; t = 0), since at t = 0, f = f 0 . Expanding Boltzmann's Equation to first order in f 1 and using the properties of f 1,qs , we obtain for δf 1

∂ t δf 1 = I ′ coll [δf 1 ]. (19)
This equation shows that the relaxation towards the quasi-static regime is solely driven by two-body collisions and occurs at a rate ∼ γ 2 . This may seem paradoxical since one would rather expect the three-body characteristic rate ∼ γ 3 . However, as far as the phase-space density is concerned, the depletion of f at low momenta is quite small since the relative decrease of the peak momentum density is ∝ nλ 3 . Since 1/γ 3 is the time required to lose typically half the initial atom number, the dip should form on a time scale ≃ nλ 3 /γ 3 ≃ 1/γ 2 .

The three-body losses lead to a correction to the momentum distribution proportional to nλ 3 . This scaling is similar to the first virial correction, and one may wonder if the three-body losses might not mask the effects of twobody interactions. To clarify this point, we calculated the leading order corrections to the occupation number ρ(p) = h 3 f (p) using the scheme presented in [20]. In the virial expansion, the leading order term corresponds to the ideal Boltzmann gas. In the grand canonical ensemble, this term reads ρ (1) (p) = ze -βεp , where z is the fugacity and ε p = p 2 /2m. The next order term is the sum of two contributions. The first one corresponds to Bose's statistics and is simply ρ (2,a) (p) = z 2 e -2βεp while the second one is more involved and is due to interactions. Following [20], it is given by ρ (2,b)

(p) = 8πz 2 m Cγ ds 2πi +∞ 0 dP P 2 2π 2 e -βs √ -ms × e -β P 2 4m s + P 2 4m -p 2 2m -(P -p) 2 2m s + P 2 4m -p 2 2m -(P +p) 2 2m ( 20
)
where C γ is a Bromwich contour [21]. We note that this expression is simply twice that obtained for spin 1/2 fermions [20]. To convert this momentum distribution to the canonical ensemble, we use the virial expansion of the equation of state of the unitary Bose gas,

nλ 3 th = z + 2b 2 z 2 + ..., with b 2 = 9/4 √ 2.
We thus obtain

ρ(p) = nλ 3 th e -βεp + (nλ 3 th ) 2 ξ(λ th p/ ) -2b 2 e -βεp , (21)
where we took ρ (2)

(p) = ρ (2,a) (p) + ρ (2,b) (p) = z 2 ξ(λ th p/ ).
In Fig. 3, we compare the effect of 3-body losses with the virial corrections to the momentum distribution. We observe that for 7 Li, for which η = 0.2, the dip in the momentum distribution is dominated by three-body losses. From a quantitative point of view, the analysis presented above gives controlled results in the high temperature regime since at the lowest order in phase space density, we expect the two corrections (3-body losses and virial expansion) to be additive. From a more qualitative point of view, since at large temperature the ratio between 3-body losses and two-body collision rates is small, one may naively assume that the effect of the losses on the momentum distribution function to be superseded by quantum correlations effects (interactions and bosonic statistics). Surprisingly, we find on the contrary that they scale identically with nλ 3 th and that for typical values of the parameter η, the losses actually dominate.

If we decrease the temperature, the ratio γ 3 /γ 2 increases, and as a consequence we anticipate a more important role of 3-body losses. Therefore we think the phenomenon we discuss in the present work will become more important as the temperature decreases. As such, it may have an important role in the interpretation of the results presented in [START_REF] Makotyn | Universal dynamics of a degenerate unitary bose gas[END_REF]. One may argue that the experiments presented in this reference were obtained after a time short compared to the three-body lifetime. However, as shown above, even in the limit γ 2 ≫ γ 3 , the relaxation towards the quasi-static distribution driven by three-body recombination occurs on a time-scale 1/γ 2 ≪ 1/γ 3 .

Finally, even though our work is restricted to a homogeneous system while real experiments are usually performed in harmonic traps, we note that if the trapping frequency is small enough (as in the experiment of [START_REF] Makotyn | Universal dynamics of a degenerate unitary bose gas[END_REF]), the elastic and inelastic relaxation times can become shorter than the typical atomic diffusion time. In this case, both the elastic and inelastic dynamics occur locally, and can therefore be described using our formalism.

[

DERIVATION OF THE LOSS EQUATIONS

The coefficient 1|

1 G L 3 [G] can be written as 1| 1 G L 3 [G] = nλ 3 th h 3 3 d 3 p 1 d 3 p 2 d 3 p 3 A 3 E 2 123 |φ(Ω 3 )| 2 e -βEtot (22)
where E tot = (p 2 1 + p 2 2 + p 2 3 )/2m. We then define three new momentum variables which are conjugated to Jacobi coordinates in real space and verify

p 1 = P 3 - Π 1 a - aΠ 2 2 ( 23
)
p 2 = P 3 + Π 1 a - aΠ 2 2 ( 24
)
p 3 = P 3 + aΠ 2 . ( 25
)
with a = (4/3) 1/4 . The energy in the center of mass frame is then

E 123 = Π 2 /2µ with Π 2 = Π 2 1 + Π 2 2 and µ = m/ √ 3 while the total energy is E tot = P 2 G /6m + Π 2 /
Using the normalization condition on φ(Ω 3 ) we are left with Gaussian integrals which are straightforward to calculate. We then recover easily [START_REF] Sykes | Quenching to unitarity: Quantum dynamics in a 3d bose gas[END_REF].

To calculate

p 2 2m | 1 G L 3 [G]
we use the fact that it can be written as (28)

p 2 1 2m | 1 G L 3 [G] =
Therefore we can use the same change of variables to get rid of the hyperangular dependence and finally retrieve (15).

CALCULATION OF C

The coefficients q k |C[q k ′ ] can be expressed as follow q k |C[q k ′ ] = -n 2 πβ/m 3 A kk ′ , A = (A kk ′ ) being a matrix with purely numerical coefficients. Those coefficients can be calculated analytically to arbitrary order. As a "proof", all the coefficients to a value k max = 6 are shown below:

A =            

FIG. 2 :

 2 FIG.2:Color online. Deformation of the momentum distribution of a unitary Bose gas due to three-body losses. From top to bottom: nλ3 th (1e -4η ) = 0 (Blue, Boltzmann gas); nλ3 th (1e -4η ) = 0.05 (Orange) and nλ 3 th (1e -4η ) = 0.1 (Red).

3 M 2 f 1 HpLFIG. 3 :

 313 FIG.3: Correction to the Boltzmann gas: three-body losses vs interactions. The correction to Boltzmann's distribution is plotted for maximal three-body losses (η = ∞, red dashed line), η = 0.2, corresponding to7 Li (Orange dotted line). The blue solid line corresponds to the correction Eq. (21) due to Bose statistics and two-body interactions.

1 G L 3 [d 3 PE 2 123

 1332 2µ. The jacobian of such a change of variables is equal to one and we have the differential transformation d 3 p 1 d 3 p 2 d 3 p 3 = d 3 P G Π 5 where Πi = Π i /Π i and α = arctan(Π 1 /Π 2 ) ∈ [0; π/2]. It can be rewritten in terms of the hyperangular differential d 5 Ω 3 = 1/2sin 2 (α)dαd 2 Π1 d 2 Π2 . We thus obtain a new form for the integral 1| G Π 5 dΠd 5 Ω 3 A 3 |φ(Ω 3 )| 2 e -βEtot .

nλ 3 th h 3 3 d 3 p 1 d 3 p 2 d 3 p 3

 33 3 )| 2 e -βEtot .

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The authors thank A. Grier, C. Salomon, F. Werner and the ENS cold atom group for careful reading and helpful discussions. The authors acknowledge support from the ERC (Advanced grant Ferlodim and starting grant Thermodynamix), Institut Universitaire de France and Région Ile de France (IFRAF). FC thanks Laure Saint-Raymond for fruitful discussions.