Lionel Feugère
email: lionel.feugere@riseup.net.

Jung-Hee Seo

Umair Ismail

Rajat Gabriella Gibson

Rajat Mittal

Mittal 3d Audio-Visual

3D audio-visual recordings of mosquito wings for aeroacoustic simulation

Keywords: insect, flight-tone, 3D reconstruction, sound directivity, CFD

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

INTRODUCTION

The aerial courtship dance of mosquitoes has fascinated entomologists for more than two centuries [START_REF] De Riville | Mémoire sur l'accouplement des cousins[END_REF][START_REF] Knab | The swarming of Culex pipiens[END_REF]. The chase between female and male in mating swarms involves highly controlled variations in the frequency and intensity of flight-tones (i.e. sounds generated by the flapping wings) [START_REF] Gibson | Flying in tune: sexual recognition in mosquitoes[END_REF][START_REF] Warren | Sex recognition through midflight mating duets in Culex mosquitoes is mediated by acoustic distortion[END_REF][START_REF] Simões | A role for acoustic distortion in novel rapid frequency modulation behaviour in free-flying male mosquitoes[END_REF][START_REF] Pantoja-Sánchez | Precopulatory acoustic interactions of the new world malaria vector Anopheles albimanus (Diptera: Culicidae)[END_REF][START_REF] Somers | Hitting the right note at the right time: Circadian control of audibility in Anopheles mosquito mating swarms is mediated by flight tones[END_REF] with concurrent changes in flight speed and direction [START_REF] Puckett | Timefrequency analysis reveals pairwise interactions in insect swarms[END_REF][START_REF] Nakata | Auditory sensory range of male mosquitoes for the detection of female flight sound[END_REF], and enables detection of conspecifics [START_REF] Nakata | Auditory sensory range of male mosquitoes for the detection of female flight sound[END_REF][START_REF] Feugère | Behavioural analysis of swarming mosquitoes reveals high hearing sensitivity in Anopheles coluzzii[END_REF], possible display of fitness and transmission of mating interest [START_REF] Cator | Sizing up a mate: variation in production and response to acoustic signals in Anopheles gambiae[END_REF][START_REF] Aldersley | Female resistance and harmonic convergence influence male mating success in Aedes aegypti[END_REF]. However, despite over a century and a half of research [START_REF] Feugère | The role of hearing in mosquito behaviour[END_REF], significant knowledge gaps continue to exist in our understanding of this behavior. To decipher this courtship dance, entomologists must integrate acoustic, energetic and flight information for untethered, freeflying mosquitoes, but the tools that can provide these data have, so far, been limited [START_REF] Nakata | Auditory sensory range of male mosquitoes for the detection of female flight sound[END_REF][START_REF] Bomphrey | Smart wing rotation and trailing-edge vortices enable high frequency mosquito flight[END_REF]. In the present study, we combine the computational biomechanics and acoustics, and behavioral entomology, to generate data and insights into the biomechanics and physics of courtship-associated acoustic communication in mosquitoes. By combining computational modeling with biological assays, we generate frequency polar pattern of mosquitoes engaged in mating-swarm behaviour. The experiments were performed in July 2020 with tethered mosquitoes and in October 2021 for free-flying mosquitoes, at a time of the day corresponding to their circadium swarming phase (-1 h to + 3 h after the beginning of the 12 h dark cycle of the rearing room for the tethered mosquitoes and between 0.3 h and 1.3 h for the free-flying mosquitoes), in the presence of a light setup mimicking the sunset as in [START_REF] Feugère | Behavioural analysis of swarming mosquitoes reveals high hearing sensitivity in Anopheles coluzzii[END_REF]. For the tethered setup, a total of 13 males and 13 females were selected to be recorded based on their ability to beat their wings without interruption. Mosquitoes were tethered by warming bee wax on the tip of a thin wire which was kept right after on the top of their abdomen until the wax hardened. The other tip of the thin wire was attached to a syringe needle, which was fixed to a boom with adhesive putty. This let them beat their wing freely while being immobile and let the experimenter adjust their position. They were recorded in an individual configuration (i.e., no other mosquito around).

For the free-flying setup, 2-to 11-day-old males and females were released in the recording cage. Only individual males and females flying alone were selected for the analysis of this paper. A 15 cm × 15 cm grey acoustic foam was used as a visual marker for the free-flying mosquitoes to swarm around the audio-video recording spot (Fig. 1). In Culex quinquefasciatus, the swarming behaviour of a single or several mosquitoes consists of a loop trajectories next to a visual marker [START_REF] Gibson | Swarming behavior of the mosquitoe Culex pipiens quinquefasciatus: a quantitative analysis[END_REF]. When adding the microphone array above the rectangular acoustic foam, the swarm centre moved to the foam side which pointed toward the room lights, whatever the microphone array position, so the array was moved on this side to have the swarm centre at the centre of the array.

Audio/video recordings

High-speed cameras. Three high-speed cameras (FAST-CAM Mini AX200, Mini AX100 and Nova S6 ) were fitted with 105 mm macro lens (Zoom, Navitar) and an IR LED light (HAY-IR-70/30 or Bosch UFLED20-8BD) was located in front of each camera, with the mosquito at a distance of 20-22 cm from the camera lenses for the tethered setup and 15-18 cm for the free-flying setup (Fig. 1). Video resolutions were 384 × 384 pixels with 20k images per second for tethered mosquitoes and 512 × 512 pixels with 22.5k images per second for free-flying mosquitoes. For tethered mosquitoes, the focal points were set on the mosquito's left wing. The recording was triggered manually and images were recorded under the software provided by the camera (Photron FASTCAM Viewer v4).

Microphones. For the wing-tone recording, twelve omnidirectional miniature microphones (DPA 4060; frequency range of 0.02-20 kHz with a 3 dB boost at 8-20 kHz) were spread over a 9cm-radius circular array (30 • step) with a metal frame hung at the ceiling (Fig. 1) and plugged to a sound interface (MOTU Stage B16 runned on MacOSX 10.13.6). Sounds were recorded on a digital audio workstation (AudioDesk 4.01 for the tethered setup, Performer Lite 10.13 for the free-flying setup) at 24 bits, and sampling rate of 192 kHz for the tethered mosquitoes and 48 kHz for the free-flying mosquitoes. The cameras were located as far as possible from the preparation to limit sound reflections on the lens: the lenses were at a minimum distance of 11 cm from the closest microphone depending on the microphone and the array rotation (Fig. 1) for the tethered setup and a minimum distance of 6 cm for the free-flying setup (Fig. 1B,C). For the tethered setup, the microphone array was rotatable around the vertical axis. Four rotations were performed for each mosquito (0 • , 30 • , 60 • and 90 • ) but only two (0 • , 90 • ; Fig. 1A) were post-processed and included in the results.

Audiovisual synchronization. High-speed camera and sound recordings were synchronized with the camera's synch pulse sent from one of the camera to the sound interface. Synchronization sampling error was limited by the high-speed camera sampling period, i.e., 50 µs. Due to the speed difference between sound and light, at a given time, the recorded sound corresponded to the sound generated by the wing at a distance of 9 cm, i.e. 265 µs before.

Large-field-of-view cameras and calibration. Two cameras (Basler model-acA640-120gm for the tethered setup; acA2040-90umNIR for the fre-flying setup) were fitted with wide-angle lenses (Computar T3Z3510CS for the tethered setup, Fujinon HF6XA-5M for the free-flying setup) to capture a picture of the whole microphone-array from two different views. They were plugged to a personal computer (running on Windows 10) and recorded (Pylon Viewer 6.2). Camera calibration were performed simulta-neously for high-speed cameras and wide-angle cameras by the help of a set of ®Lego pieces, for each recording.

Temperature. Room temperature ranged between 25 -28 • C, measured with a logger (HH506RA, Omega Engineering, Inc) and a thermocouple Type T, IEC 584 Class 1 (Omega Engineering, Inc) just above the microphone array during each recording, with a total measurement accuracy error of ±0.9 • C.

Wing kinematics reconstruction

To reconstruct the 4D wing kinematics, which is required for the simulation, 11 landmark points on the mosquito wings were tracked from the high-speed camera recording data (Fig. 2a). For each mosquito recording, these points were semi-manually recorded using the DLTdv software [START_REF] Hedrick | Software techniques for two-and three-dimensional kinematic measurements of biological and biomimetic systems[END_REF] over ∼200 images (∼ 3-4 wingbeat periods) for each of the three high-speed cameras. The wing kinematics and deformations were extracted by applying the least-squarefitting surface method. In addition to the wing twisting, span wise and chord wise bending are also considered for the wing deformation. The out-of-plane wing deformation is fit into a second-order polynomial surface (Fig. 2b).

Physics based independent component analysis

The 3D sound data recorded from the experiment are processed by applying a novel physics-based independent component analysis. The method separates out the physically generated sound by using the free-space Green's function and thus is very effective to filter out background noise. Assuming the wing-tone sound pressure recorded by each microphone, p j , is generated by the physical, dipole source, F, and is contaminated by the statistically same background noise, , the physical dipole source and noise component in the frequency domain can be found by using the singular value decomposition (SVD) of the Green's function matrix:

F(ω) (ω) = [G j (ω) 1] + p j (ω) (1)
where G j is the free-space Green's function from the source (mosquito) location to the j-th microphone position, and + denotes a pseudo-inverse of the matrix. The physical wing-tone, p w,j can be then obtained by

[p ω,j (ω)] = [G j (ω)][F(ω)].
The processed 3D sound data showed a typical dipole sound pattern which is a wellknown characteristics of wing tone sounds.

Aerodynamic and aeroacoustic simulation

The CFD simulations presented herein are performed for single free-flying mosquitoes using wing kinematics re-constructed from the experimental recordings. These computations are performed in a reference frame affixed on the body of a translating mosquito. The mosquito wing, which is discretely represented via triangular mesh elements, is modelled by a zero-thickness membrane using a sharp-interface immersed-boundary method [START_REF] Mittal | A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries[END_REF] and its motion is prescribed using the reconstructed wing motion data. Further details on the reconstruction of the wing motion will be discussed elsewhere.

The anterior end of the mosquito is pointed in the -y direction. As a result, its nearly horizontal flapping stroke lies in the x -y plane (Fig. 3a.) and the direction of the vertical weight-supporting force is +z. The wing is immersed into a Cartesian volume grid of size L x × L y × L z = 5L s × 10L s × 10L s (L s is the spanwise length of the wing). The flow domain is divided into N x × N y × N z = 193 × 257 × 257 grid points. A uniform albeit fine mesh, with a grid size of 0.01L s , is employed in the section that covers the flapping volume and the region penetrated by flapping-induced vortices. Neumann boundary conditions for all three components of the velocity vector are applied at five out of the six faces of the flow domain. As the aerodynamic study investigates the flow physics and the vortical patterns induced by one flapping wing, a symmetry condition is instead imposed on the sagittal plane (x = 0). The computational time step is fixed at ∆t = 0.001/f 0 , which resolves one wing-beat cycle in 1000 time steps and ensures that the Courant-Friedrichs-Lewy condition is below 0.5. All CFD simulations are performed for five wing-beat cycles and only the results from the final cycle are used for analysis. However, it is important to note that the computed force/power levels and the examined coherent structures are virtually identical in cycles 2-5.

The aeroacoustic sound generated by the wing's flapping motion is predicted by employing the wellestablished formulation for the moving surface -the Ffowcs Williams and Hawkings (FW-H) equation. The FW-H equation has been widely used for the prediction of aeroacoustic sound from rotating blades, and also for the sound from flapping wings. We employ an integral formulation of the FW-H equation [START_REF] Kenneth | An analytical comparison of the acoustic analogy and kirchhoff formulation for moving surfaces[END_REF]. The surface pressure distributions on the wing obtained from the aerodynamic simulation are recorded at every time step, and the sound pressure is calculated by the FW-H equation from the surface pressure and velocity data. More details about the simulation methods can be found in [START_REF] Seo | Mechanism and scaling of wing tone generation in mosquitoes[END_REF].

RESULTS

To verify and validate the reconstruction, data processing, and simulation methods, the described methods are first applied to a tethered mosquito. The wing kinematics and deformation are reconstructed from the high-speed video recording data, and the aerodynamic simulation is performed. Within the tethered subclass, the wing-tone sounds at the micro-phone array locations as predicted by the FW-H equation are compared against the measured ones to validate the methods.

The recorded wing-tone sound, however, did not show a typical dipole pattern because of the background noise. At the distance of 9 cm, the wing-tone sound was as weak as about 20 dB at the peak and thus the signalnoise ratio was low. To extract the physical wing-tone sound, the physics based independent component analysis (ICA), and the processed sound data showed a typical dipole pattern which is well known characteristic of wing tone sound. The simulated wing-tone pattern compares very well with the recorded and processed data (Fig. 2d), though the recorded data showed a substantial cycle-tocycle variation which is about ± 10 dB. The present results showed that the wing-tone from individual mosquito can be predicted with reasonable accuracy by using the kinematics reconstruction and aerodynamic/acoustic simulations, and this enables one to investigate the wing-tone communications in courtship and mate chasing.

Within the free-flying subclass, wing kinematics are constructed for six individuals (three males and three females). These are then used to set up aerodynamic and aeroacoustic simulation for each specimen. Here we restrict our discussion to one representative male (M 1 ) (left wing) and one representative female (F 1 ) (right wing), both of which are performing a slow descent. Instantaneous snapshots of the wing surface along with the corresponding positions of the 11 points tracked in the digitizing exercise are presented in Fig. 3a. The objective is to highlight the quality of the spatio-temporal wing kinematics constructed for setting up the CFD simulations. Several parameters that categorize different aspects of these two individuals are provided in Tab. 1. While all 6 individuals translate at virtually uniform -albeit different -velocities, it is clear from video recordings that they are all in free flight. For instance, of the three males recorded, one is descending gradually whereas the other two are in a slow ascending flight. The choice of performing the CFD simulations without externally imposed free-stream velocity (U ∞ ) is justified by the small advance ratio J = U ∞ /v tip < 0.1 in Tab. 1. Despite the translating aspect of their flight, the very small levels of J imply that both individuals (M 1 and F 1 ) are roughly hovering. The temporal variation of lift and drag forces along the flapping cycle for M 1 and F 1 is presented in Fig. 3bc. The higher mean force levels for case F 1 are expected as it operates at a higher Reynold number (Re s ). The observation from these two individuals that peak lift occurs near the middle of the upstroke is consistent with the result by [START_REF] Bomphrey | Smart wing rotation and trailing-edge vortices enable high frequency mosquito flight[END_REF][START_REF] Seo | Mechanism and scaling of wing tone generation in mosquitoes[END_REF]. But this observation does not extend to all individuals tested. However, what is consistent across all free flying individuals simulated is the fact that mean lift produced during the downstroke (0 ≤ t/T ≤ 0.5) and upstroke (0.5 ≤ t/T ≤ 1.0) is essentially identical. Vortical structures that develop around the wing for case M 1 in a cross-spanwise plane at x/L s = 0.7 and at four key time instants in the flapping cycle are presented in Fig. 4. It is evident that peak lift during each half stroke coincides with the moment at which the trailing edge vortex (TEV) is around its largest size and closest to the trailing edge. Despite the apparent significance of the TEV, the major contributor to aerodynamic lift for free flying mosquitoes is the thin 'underside shear layer (USL)' that develops on the pressure side of the wing. The USL in turn feeds the TEV. In contrast to the TEV, a dominant leading edge vortex that would grow before shedding is not observed. This is not surprising given the shallow strokes and the rapid pitching component of the overall flapping motion.

The importance of the USL is confirmed by applying the Force Partitioning Method (FPM) [START_REF] Menon | Significance of the straindominated region around a vortex on induced aerodynamic loads[END_REF][START_REF] Ismail | Demystifying the Clap & Fling Mechanism: Insights from the Force Partitioning Method (FPM)[END_REF][START_REF] Aghaei-Jouybari | Contributions to pressure drag in rough-wall turbulent flows: Insights from force partitioning[END_REF] resulting flow field. The FPM offers a mathematical partitioning of the total pressure force on the wing surface into components due to vortical regions, added-mass effects, and viscous momentum diffusion. Perhaps more importantly, the FPM allows one to quantitatively identify dynamically important vortical regions within the aerodynamic field. Further details of the FPM are provided by [START_REF] Zhang | Centripetal acceleration reaction: An effective and robust mechanism for flapping flight in insects[END_REF]. Among the three constituent elements of the mean pressure-induced lift, the term due to the vortical regions, also referred to as the vortex-induced lift (VIL), is the largest contributor at 61% for case M 1 (Fig. 5a).

While the other two components also induce a positive lift in the mean sense, with lift due to viscous momentum diffusion being the dominant one at 32%, the contribution by the added-mass term has a negative correlation with the VIL term. It is worth highlighting that the addedmass term is independent of Re s , whereas the contribution by viscous momentum diffusion would decrease progressively with increasing Re s (i.e. higher wingtip velocity). Furthermore, there is non-negligible cycle-to-cycle variation in the 11 experimentally tracked points sited on the wing. This is accounted for by performing CFD simulations for every individual using two distinct sets of wing kinematics, where each set represents a different cycle of the flapping motion (Fig. 3b-c). The directivity pattern of the sound pressure level (SPL) at the fundamental (f 0 ) and superharmonic frequencies in the sagittal plane (x = 0), as predicted by the FW-H equation for M 1 , is presented in Fig. 5b. Apart from their symmetric bimodal shape, we see that the SPL radiates mostly in the horizontal and vertical directions at f 0 and f 1 frequencies, respectively. As noted by [START_REF] Seo | Mechanism and scaling of wing tone generation in mosquitoes[END_REF], the former is a consequence of strong temporal fluctuations of the drag force at f 0 , while the latter is a result of dominant temporal fluctuations of the lift force at f 1 .

ACKNOWLEDGMENTS

Figure 1 :

 1 Figure 1: Experimental set-up for tethered (A) and free-flying mosquitoes (B,C). High-speed cameras (h1, h2, h3) are directed toward the centre of a microphone array back-lighted by IR-lights, while stillcameras (s1, s2) record the space around the whole circular microphone array. A: Schematic top view of the tethered-mosquito setup with a rotating microphone array with fields of view of each highspeed camera in blue. B: Schematic top view of the free-flying setup, with fields of view of each highspeed camera in blue, net boundaries in dashed lines and square swarming marker in grey. C: Free-flying mosquito setup (without the net) with an acoustic foam used as a swarming mosquito marker.

  All experiments were performed with virgin Culex quinquefasciatus Say, 'Muheza' strain (separated by sex < 24h post-emergence). The colony was established at the Natural Resources Institute, University of Greenwich (UK) from eggs provided by the London School of Hygiene and Tropical Medicine, UK ,and kept in environmentally controlled laboratory rooms with a 12h:12h light:dark cycle, > 60% relative humidity and ∼ 24-26 • C.

Figure 2 :

 2 Figure 2: (a) Mosquito wing and 11 landmark points. (b) Reconstructed 3D wing model. (c) Aerodynamic simulation results for a representative tethered mosquito case. Instantaneous vortical structure visualized by the iso-surface of Q. (d) f 0 sound directivity pattern for a tethered specimen. Dashed line: simulation result; solid line: experimental measurements by the microphone array; measured sound pressure is processed by using the Green's function based ICA method.

Figure 3 :

 3 Figure 3: (a) An instantaneous snapshot of the wing during the flapping cycle upon construction from experimental data and for use in CFD simulations. The position of the 11 points tracked during the experiment at the same instant is also included using red cubes. Temporal variation of the pressure-induced lift, drag and side force along the flapping cycle for (b) M 1 and (c) F 1 . Thick and thin lines in (b, c) represent the results from two distinctly constructed wing kinematics that correspond to a different cycle in the recording flapping motion. Downstroke: 2.0 ≤ t/T ≤ 2.5; upstroke: 2.5 ≤ t/T ≤ 3.0.

  to the (a) t/T = 0.16 (b) t/T = 0.22 (c) t/T = 0.74 (d) t/T = 0.80

Figure 4 :

 4 Figure 4: Spanwise vorticity (ω x ) for case M 1 in a yz-plane at x/L s = 0.7. (a) Peak drag during downstroke, (b) peak lift during downstroke, (c) peak lift during upstroke and (d) peak drag during upstroke.

Figure 5 :

 5 Figure 5: (a) Temporal variation of different components of the FPM that constitute the total lift force along a flapping cycle of case M 1 . Normalization of the ordinate is by length of the wingspan L s and wingbeat frequency f 0 . (b) Directivity patterns of the SPL (in dB) at f 0 (top) and f 1 (bottom) for case M 1 .

  This research was funded by Human Frontier Science Program Research Grant No. RGP0038/2019. The authors would like to acknowledge Shahida Begum for providing mosquito eggs, Natalie Morley and Simon Springate for rearing mosquitoes and Tyson Hedrick for support on using the DLTdv digitalizing software.

Table 1 :

 1 A summary of several parameters that categorized the two cases discussed here: M 1 and F 1 . Lcf 0

	Parameter	M 1	F 1
	Wingbeat frequency f 0	750 Hz 450 Hz
	Reynolds number Re s = L 2 s f 0 ν	310	410
	Reduced frequency f v tip	0.12	0.08
	Advance ratio J = U ∞ /v tip	0.07	0.04
	Stroke amplitude (∆φ)	36 •	42 •
	Pitch (∆γ)	109 •	148 •
	Mean lift force F z	3.5 µN 9.1 µN
	Mean drag force F y	1.1 µN 90.0 µN
	Mean lift by upstroke	54 %	48 %

* =

Turin, Italy • 11 th -15 th September 2023 • Politecnico di Torino


The transformations of industrial brownfields into sustainable neighborhoods, a stepping-stone towards the sustainable city: the case studies of Ile-de-Nantes and Docksde-Seine

Varvara Toura, PhD Urban Studies, EHESS/Géographie-Cités, Paris, France valiatoura@gmail.com Supervisor: Prof. Marie-Vic Ozouf-Marignier; thesis defended on January 10, 2023.

The thesis questions the meaning of 'industrial heritage' in former industrial cities, both in terms of preservation of industrial heritage and with regard to current urban planning policies such as sustainable development and citizen engagement. Τhe thesis' subject was chosen by reflecting on existing studies, particularly in France and in French-speaking countries, on the question of redevelopment of former industrial cities and the conversion of industrial brownfields derived from the internal reorganization of cities and the economic dimension of sustainable development (works of geographers and economists such as those of Claude Chaline, Jean Claude Daumas, Corinne Luxembourg and Dalila Messaoudi). At the same time, published reports such as those of TICCIH (The international committee for the conservation of the industrial heritage) at international level and CILAC (Comité international pour la conservation du patrimoine industriel) in France, of Vincent Veschambre and of Maria Gravari-Barbas, address the question of industrial brownfields' heritage as a vector of local growth. On the other hand, publications in the field of urban studies in the 1980s and particularly after 2000 by geographers, economists and urban planners (I quote the works of Aurélien Boutaud, Jacques Theys, Cyria Emelianoff and Taoufik Souami) on the application of the concept of sustainable development at territorial level raise the question of integration of the three dimensions of sustainable development (economic development, protection of the environment, social equity) in programs for the reconversion of industrial brownfields. However, it should be noted that among the three dimensions, the one that is less addressed is the social dimension of sustainable development (there are references in the works of Camille Gardesse and Jodelle Zetlaoui-Léger on the involvement of inhabitants in sustainable urban projects), which in the case of industrial cities echoes their identity as working-class cities. The identity of industrial cities refers to the spatial organization as well as the social ties among the inhabitants. In this context, the thesis aims to answer the question of how industrial identity could be part of the new organization of deindustrialized cities and in particular of the so-called sustainable spatial planning models?

To answer this question I have chosen to analyze the thesis subject by using a top-down model in 4 spatial scales, the intermunicipality, the city, the district and the building in order to study, on the one hand, the policies and mechanisms of the transformation of industrial cities and the reconversion of brownfields (safeguarding of local heritage, economic development, reconstitution of nature within cities, citizen engagement) as well as the actors involved in these procedures (local elected officials, professional associations and inhabitants). I chose the scale of district, an intermediate spatial scale between the scales of city and building, for the study of case studies as it allows the study of the application of territorial policies (either on the preservation of heritage or on sustainable development) at local level, as well as the encounters with the actors involved, in particular inhabitants' associations, in the programs for the reconversion of industrial brownfields. The aim of these encounters was to study the interplay of actors and the form of citizen engagement.

The two case studies chosen for the thesis are considered as exemplary models in regard to the main questions addressed in the thesis (industrial identity, preservation of industrial heritage, economic development, environmental protection, and citizen engagement). On the one hand we have the reconversion project Ile-de-Nantes, a brownfield within the former industrial city of Nantes in the west of France, a city with a significant industrial heritage. The project functions as a vector of local economic and demographic growth. On the other hand, we have the reconversion project Docks-de-Seine, a brownfield located in the city of Saint-Ouen in the outskirts of Paris, a city characterized in the last 20 years by its urban transformations (spatial, economic, social), in which industrial activities have never ceased to exist. The project functions (as the first case study in Nantes) as a vector of local economic and demographic growth. The main objective of the case studies' choice was to examine contemporary territorial policies on the subject of reconversion of industrial brownfields which are located within and on the outskirts of cities, seeking to propose a methodological analysis framework for professionals and scholars (study of the subject of reconversion on different spatial scales, by considering the urban heritage and the economic, environmental, and social dimension of sustainable development). The two case studies show different approaches (methods, policies) on the subject of reconversion of industrial brownfields, which seems to lead to the sustainable development of former industrial cities. However, at the scale of city we should note that the development of Saint-Ouen continues to be based on industrial activities. This means that the safeguarding of the local industrial identity is based on the presence of industries and social housing for low-income populations. On the other hand, in Nantes, the safeguarding of industrial identity is based on the rehabilitation of former industrial buildings and the networks of professional associations and inhabitants who work on the questions of industrial heritage and community life at different scales through the consultation or active engagement of the inhabitants. We can see that even if the objective of the two projects seems to be the same (the sustainable development of cities), the local context (more favorable in Nantes than in Saint-Ouen)

and the evolution of the projects over time (an urban project in three phases in Nantes versus a project whose objectives have been the same since its first phase in Saint-Ouen) differentiate them.

The thesis' inscription in the field of social sciences (urban sociology, geography, political science), helps to introduce new contributions in relation to the existing literature on the questions of the social dimension of reconversion of industrial brownfields and the characteristics of this procedure. The works of French historians such as those of Annie Fourcaut show the working-class character of industrial cities which could be maintained in reconversion programs. For their part, the work of French sociologists such as those of Muriel Tapie-Grime on the social dimension of sustainable development reveal citizen engagement as a key element in the redevelopment of these sites. For my part, I emphasized the role and actions taken by inhabitants in the programs for the reconversion of industrial brownfields, which aim to maintain the working-class character of these places as well as to involve inhabitants in the production of their living environments. The existing literature on the form of reconversion is based primarily on one of its dimensions, for example the environmental dimension (I quote the works of Michel Letté) or the economic dimension such as the works of Lilas Demmou.

As for the reconversion of industrial brownfields, it is often linked to the rehabilitation of industrial buildings. The best-known works on this subject are those of Philippe Robert and Bernard Reichen. For my part, I have chosen to broaden the question of the reconversion of industrial brownfields and its characteristics by considering it as a procedure that refers to three spatial scales (city, district, building), which includes policies and practices of heritage protection and urban planning, and which refers both to material (such as buildings and industrial districts) and immaterial heritage (such as the social ties among the inhabitants).

The question of the preservation of local memory, which is often cited by institutional actors as a good practice in regard to the subject of the reconversion of industrial brownfields, has also its contradictions according to the two parameters analyzed in the thesis (preservation of industrial buildings, preservation of social fabric). The concept of industrial heritage is changing with the emergence of Critical Heritage Studies which consider heritage as an evolutionary process that responds to the needs of contemporary societies in terms of sustainable development, economy, or culture. The industrial heritage of the two case studies goes beyond the rehabilitation of certain industrial buildings and also includes the creation of industrial museums, publications, and exhibitions on the places of industrial production and the living conditions of former industrial workers as well as guided tours of industrial monuments. We can see, therefore, that industrial heritage contributes to the economic development of deindustrialized cities with the creation of new jobs and at the same time it offers the possibility of a collaborative economy with the direct engagement of local communities in the development process of their living environments.

The notions of citizen engagement and citizen know-how cited as fundamental subjects for the reappropriation of deindustrialized sites often encounter the question of spatial scale (city, district). In both case studies, I found that citizen engagement refers to the direct involvement of inhabitants in the redevelopment processes of the two sites, often mobilized by inhabitants' associations and activists working on the issues of preservation of industrial heritage and environmental protection (associations of former industrial workers in Nantes and the inhabitants' association Mon voisin des docks in Saint-Ouen). In addition, I have observed that urban planners and local elected officials have chosen to engage the inhabitants in the planning processes of small-scale urban projects, those of the district. The scale of city remains even for the two case studies a scale for big-scale urban projects and citizen engagement at this scale refers to the consultation of the inhabitants in the programming phase. I note, therefore, difficulties in generalizing participatory urban planning practices to the scale of city without the agreement of local elected officials and developers who refuse to delegate part of their power during the development of big-scale urban projects. Despite the difficulties and constraints for the generalization of participatory processes at city level, the study of the two projects showed that the two cities could benefit from micro-local experiences, as the inhabitants were involved in urban projects whose objectives go beyond the scale of district (preservation of industrial heritage, environmental protection). In addition, we must underline that in both case studies, citizen know-how refers to the desire of a certain number of inhabitants, activists, or members of inhabitants' associations, to involve people who do not participate often in participatory urban planning processes (for example immigrants and young workers), as well as creating an associative network to share their experiences beyond the district scale. We can see, therefore, that activism on the scale of the district as well as the experiences of micro-local urban projects, would contribute to the generalization of advanced participatory mechanisms on the scale of city, by including in this type of land use planning people with different profiles (age, ethnic origin, socio-professional category).

To go further on the subject of the reconversion of industrial brownfields and their redevelopment in a sustainable way, the thesis also underlined the risk of gentrification in former industrial cities as their working-class identity changes with the arrival of executives and the rejection towards the cities' periphery of the modest socio-professional categories. Can we continue to talk about industrial identity in the context of gentrified city? Policies such as the creation of social housing, the control of property prices and subsidies for access to housing could reverse this trend. However, we must ask ourselves under what conditions are such achievements possible while maintaining the industrial and workingclass character of these cities? It is the local political and social context that plays a major role in making decisions on the content of brownfields' reconversion programs. Not all cities have the means to allow such operations either because they lack funding or because they are in strong decline.

Voluntarism and political stability at local level, active associative networks, and favorable financing conditions (public-private partnership, public funds, European Union funds) make these achievements possible and at the same time differentiate them according to issues and orientations determined by different actors. In addition, we should note that a certain number of cities, especially at European level, find the opportunity to finance such operations through the organization of sporting (for example the Olympic Games in London in 2012 where the Olympic village was built on an industrial brownfield on the city's outskirts) or cultural events (for example the program of the European city of culture whose host city is often a deindustrialized one). We could, therefore, speak of the exceptional conditions and means (financing, media coverage) that lead to the realization of this type of operation.

However, we must also note the high cost of such operations, even in the case of large, deindustrialized cities, because the financing of such projects requires significant economic funds, at the risk of disfavoring other municipal actions (health, education, housing). In addition, these territorial policies are concentrated only on a certain number of city's districts, when the ideal would be to include them in municipal action programs that refer to the urban renewal of several poor city sectors. Future research on the subject of brownfield conversion could address the question of how good urban practices in the field of sustainable development would be generalized, especially in the cities of global South. However, we can see that there are 'negative' points that should be considered in the application of sustainable development projects in these cities, which are already subjected to strong social and spatial inequities, such as the rise in real estate prices and the rejection of the working classes towards the cities' outskirts. The aim is to help them develop legal frameworks and technical capacities, in order to formulate and implement their own models of sustainable cities while preserving the traces of their industrial past.

Figure 1 .

 1 Figure 1. Economic development on the site of Ile-de-Nantes through the conversion of industrial remnants into industrial museums and the construction of residential and office buildings. ©Author, 2023.

Figure 2 .

 2 Figure 2. Demographic evolution in Nantes between 1968 and 2017. ©Author, 2023.

Figure 3 .

 3 Figure 3. Economic development on the site of Docks-de-Seine through the conversion of industrial remnants into exhibition spaces and the construction of residential and office buildings. ©Author, 2023.

Figure 4 .

 4 Figure 4. Demographic evolution in Saint-Ouen between 1968 and 2017. ©Author, 2023.


MD Jérôme Dumortier
email: jerome.dumortier@chu-lyon.fr

Domitille Erard

François Villeret

François Faitot

Christophe Duvoux

Stéphanie Faure

Claire Francoz

Jean Gugenheim

Jean Hardwigsen

Jean-Baptiste Hiriart

Pauline Houssel-Debry

Arnaud Del Bello

Guillaume Lassailly

Claire Vanlemmens

Faouzi Saliba

Clara Altman

Marianne Latournerie

Sébastien Dharancy

Tarek Debs

Bariatric surgery and liver transplantation, here we are now: A French nationwide retrospective study

Keywords: Liver transplantation, MS: metabolic syndrome NAFLD: non-alcohol-related fatty liver disease liver transplantation, metabolic syndrome, NAFLD, obesity, bariatric surgery

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

Introduction

At the time of the growing obesity epidemic worldwide, especially in Western countries, liver transplantation (LT) and metabolic syndrome (MS) are closely linked. First, non-alcohol-related fatty liver disease (NAFLD) emerged as one of the leading indications for LT over the past decade, and disease recurrence after LT seems inevitable [START_REF] Villeret | The inevitability of disease recurrence after liver transplantation for NAFLD cirrhosis[END_REF]. In this context, MS usually worsens after LT. In addition, MS can appear after LT, related to immunosuppressive medications and weight gain, whatever was the initial liver disease leading to the indication for LT. Even if lifestyle modification and/or medications (such as antidiabetics) remain first-line in managing obesity, bariatric surgery (BS) is of great relevance due to its longer-lasting effect and efficacy. Therefore, the role of BS in the LT context has been explored more recently. In a recent review of the literature, Ahmed and coll. emphasized that the role of BS in LT patients remains an arena to be fully explored: the impact of BS and long-term outcomes are well understood in the non-transplant population however less is known for LT patients [START_REF] Ahmed | Bariatric surgery, obesity, and liver transplantation[END_REF]. Sleeve gastrectomy has shown to be more promising when compared to other procedures with regard to long-term weight loss without issues with malabsorption. Regarding BS in cirrhotic patients, there is a need for evidence-based management of hepatic failure post-BS.

Finally, the optimal timing of BS in the setting of LT remains controversial. These authors concluded that large multicenter randomized studies are needed to devise an optimal management plan and strategy for this patient population. Such studies will probably be difficult to implement. The aim of the present study was to describe all LT recipients who have a history of BS, before or after LT, in France.

Patients

We performed a retrospective review of all adult LT recipients before 2022, in all active and past adult LT French centers (n=21), with a history of BS. Were retrospectively collected data about pretransplant, transplant, and post-transplant characteristics. All data were analysed using SPSS version 23.0 (IBM, Armonk, NY). Data were described using a median with a range for continuous variables and a number with a percentage for categorical variables. Categorical variables were compared with the Chi-square test and quantitative variables were compared using Mann-Withney non-parametric test. A p value less than 0.05 was considered statistically significant.

The study population comprised 51 cases (50 patients), including 37 cases of BS before LT, and 14 after (Table 1). Eight patients had 2 successive BS procedures; 1 patient had both pre-and post-LT BS. There were three significantly different characteristics when comparing pre-and post-LT BS: patients were older (at the time of BS), presented more frequently arterial hypertension (at the time of LT), and the proportion of NAFLD as initial liver disease leading to LT was lower, in the post-LT group. Regarding pre-LT BS, in one case BS was complicated by liver failure leading to the rapid indication of LT; it was the single patient for whom the delay between BS and LT was less than 1 year; there was no patient who specifically underwent BS for the purpose of LT listing. Two patients (5.4%) died from surgical complications after LT in the group of pre-LT BS.

Discussion

The main result of the present nationwide study is that only a very small proportion of French LT patients have a history of BS; considering that 13,913 patients were alive after having undergone a LT in France by December 31, 2021, this represents 0.37% (data from the French Agence de la Biomédecine [3]). In addition, it consisted of pre-LT BS in the majority of the cases, which was performed without any relation to the subsequent LT (median delay 6.4 years). One explanation is the relatively low proportion of patients listed (first LT) for complications of NAFLD (with or without hepatocellular carcinoma (HCC)) in France, which is very recently increasing (13.6% in 2021, data from the French Agence de la Biomédecine [3]). Indeed, there is a rapidly growing literature on BS and LT, for patients with NAFLD. Timing for BS in LT patients and type of surgery (sleeve gastrectomy, Roux-en-Y gastric bypass, adjustable gastric banding) have to be better defined.

Theoretically, LT candidates can undergo BS while waiting for LT, if the waiting period is expected to be long (this is the case when the indication is HCC, in France, and more generally) and when liver function is preserved (it is also usually the case for HCC patients). A rare but potentially deadly complication after BS in the presence of cirrhosis is liver failure, with the need for an LT (we have one such patient in our cohort). A literature review described 14 studies, including 36 patients who underwent BS and had severe complications leading to liver failure (32 had an LT, while 4 died while on the waiting list for LT) [START_REF] Addeo | Liver transplantation for bariatric surgery-related liver failure: a systematic review of a rare condition[END_REF]. Interestingly, liver failure developed a median of 20 months after BS, which can be considered a long delay. Another open question is the impact of previous BS on the outcome of LT surgery. Ianelli and coll. compared 39 French LT patients with a history of BS, to 1798 obese LT patients without previous BS [START_REF] Iannelli | Clinical and Economic Impact of Previous Bariatric Surgery on Liver Transplantation: a Nationwide, Population-Based Retrospective Study[END_REF]. No significant difference was detected between the 2 groups regarding the rates of mortality, re-LT, re-hospitalization after LT, and costs of hospitalization and re-hospitalizations. Because patients with decompensated cirrhosis and obesity are considered at greater risk of accessing BS before LT, this situation makes a simultaneous (or with a short interval) sleeve gastrectomy and LT attractive. Interestingly, Safavi and coll. recently performed a systemic review examining the role of BS in LT patients; primary outcomes included patient and graft survival and secondary outcomes were post-operative morbidities after LT, length of hospital stay after LT, change in BMI and recurrence of steatosis [START_REF] Safavi | The role of bariatric surgery in liver transplantation: timing and type[END_REF]. This analysis included 162 patients who underwent both BS and LT and 1426 who had a LT alone. Four studies looked at patients with BS before LT (n=133) and one study looked at BS and LT being done simultaneously (n=26). The percentage of female patients in the BS + LT cohort was higher than in the LT alone cohort: 75% vs.

35% respectively. The most common indications for LT (when available) in the BS + LT cohort were NAFLD (n=76), viral hepatitis (n=39) and alcoholic liver disease (n=36), confirming that NAFLD is not exclusive, as observed in our population. Patients undergoing BS had comparable outcomes in terms of overall patient survival, graft survival, and postoperative morbidity compared to LT alone. When comparing BMI change in patients with prior vs. simultaneous BS+LT, no significant difference was found. Only the study with simultaneous BS + LT reported NAFLD recurrence, in which the BS + LT cohort showed a lower recurrence rate in the graft when compared to the LT-only cohort: 23.1% vs.

66.7%.

When conducting a dual (bariatric and transplantation surgery) psychosocial assessment, it seems prudent to understand the overlap and differentiation of specific psychosocial components that influence outcomes in these procedures [START_REF] Fipps | Psychosocial considerations in the combined bariatric surgery and organ transplantation population: a review of the overlapping pathologies and outcomes[END_REF]. In addition, a history of alcohol use disorders makes this assessment even more complex. While it has long been known that alcohol withdrawal can lead to massive weight gain, it has been reported more recently that patients with prior BS are at risk for developing alcohol use disorder, more frequently in females [START_REF] Lefere | Bariatric Surgery Patients Are at Risk for Alcoholic Liver Disease with Need for Liver Transplantation[END_REF]. It can be hypothesized that psychosocial concerns could explain in part the very low number of LT recipients who underwent bariatric surgery in our experience. Specific data on the psychosocial comorbidities were not available in our population; this is a limitation of your study, due to its retrospective nature.

In our cohort, none of the patients had planned BS before LT, or simultaneous BS+LT; 14 patients had BS after LT (median delay 5.1 years). BS after LT is a challenging task; as it can be technically difficult due to adhesions, the patient will be under immunosuppression, and the operation itself may alter the absorption of immunosuppressive medications, placing the graft at risk of rejection.

Nevertheless, BS can be performed after LT once the early postoperative period is over and the patient's clinical condition is stabilized, after 6 or 12 months for instance. In a comprehensive study, Rogers and coll. evaluated the pharmacokinetics of immunosuppressant drugs (tacrolimus, sirolimus, mycophenolic acid) in patients submitted to BS. They reported that people undergoing a transplant and who have BS should need a higher immunosuppressant medicine dose than non-BS patients, especially those who undergo bypass interventions [START_REF] Rogers | Pharmacokinetics of mycophenolic acid, tacrolimus and sirolimus after gastric bypass surgery in end-stage renal disease and transplant patients: a pilot study[END_REF]. As illustrated in our cohort, sleeve gastrectomy is therefore probably the first choice for surgery type, rather than other techniques such as Roux-en-Y gastric bypass, because of a lower risk of immunosuppressive drugs malabsorption, persistent accessibility of the gastric remnant and biliary tract, and its simplicity, without digestive anastomosis, and therefore a reduced risk of anastomosis-related complications [START_REF] Iannelli | Perioperative complications of sleeve gastrectomy: Review of the literature[END_REF]. In a single-center experience, Yemini and coll. reported 34 solid organ transplant recipients who underwent laparoscopic sleeve gastrectomy or laparoscopic Roux-en-Y gastric bypass. A successful weight loss occurred in the vast majority of patients (82%), comorbidities improved significantly but the surgical risk was found higher than in the regular BS population, and tacrolimus blood trough levels declined slightly but remained within the therapeutic range [START_REF] Yemini | Bariatric surgery in solid organ transplant patients: Long-term follow-up results of outcome, safety, and effect on immunosuppression[END_REF].

Finally, an open question is about the selection of obese patients who will benefit from BS.

Regarding the risk of severe NAFLD recurrence, we recently identified that independent significant risk factors for post-LT NAFLD-related advanced graft fibrosis were MS before LT, long-term use of cyclosporine, and grade ≥2 steatosis at 1 year [START_REF] Villeret | The inevitability of disease recurrence after liver transplantation for NAFLD cirrhosis[END_REF]. Of note, it must be emphasized that the majority of the patients reported here who underwent BS after LT was transplanted for an initial liver disease different than NAFLD, especially alcohol-related liver disease; this can be explained by a frequent massive weight gain and the development of de novo NAFLD in this particular population [START_REF] Dumortier | Non-alcoholic fatty liver disease in liver transplant recipients: another story of "seed and soil[END_REF].

In conclusion, the present French retrospective multicenter cohort results show that a very small part of LT recipients has a history of BS, especially after LT. Recommendations can be proposed as soon as the level of knowledge of risk factors for severe hepatic and extra-hepatic disease related to metabolic syndrome has significantly increased. This requires long-term follow-up of large cohorts.

Table 1 . Study population

 1 coronary artery disease, valvular heart disease, arrhythmia, stroke, obliterative arteriopathy ** last bariatric surgery type in cases of several successive procedures ***a single patient may have several causes of liver disease Abbreviations: NAFLD, non-alcohol-related fatty liver disease; HCV, Hepatitis C virus.

	Bariatric surgery	Bariatric surgery	p
	before LT	after LT
	(n=37)	(n=14)

*


N Fellmann

M Pasquier

C Blanchet-Scalliet

C Helbert
email: celine.helbert@ec-lyon.frnoe.fellmann

A Spagnol
email: adrien.spagnol@ifpen.fr

D Sinoquet

Sensitivity analysis for sets : application to pollutant concentration maps

In the context of air quality control, our objective is to quantify the impact of uncertain inputs such as meteorological conditions and traffic parameters on pollutant dispersion maps. It is worth noting that the majority of sensitivity analysis methods are designed to deal with scalar or vector outputs and are ill suited to a map-valued output space. To address this, we propose two classes of methods. The first technique focuses on pointwise indices. Sobol indices are calculated for each position on the map to obtain Sobol index maps. Additionally, aggregated Sobol indices are calculated. Another approach treats the maps as sets and proposes a sensitivity analysis of a set-valued output with three different types of sensitivity indices. The first ones are inspired by Sobol indices but are adapted to sets based on the theory of random sets. The second ones adapt universal indices defined for a general metric output space. The last set indices use kernel-based sensitivity indices adapted to sets. The proposed methodologies are implemented to carry out an uncertainty analysis for time-averaged concentration maps of pollutants in an urban environment in the Greater Paris area. This entails taking into account uncertain meteorological aspects, such as incoming wind speed and direction, and uncertain traffic factors, such as injected traffic volume, percentage of diesel vehicles, and speed limits on the road network.

Introduction

Air quality and dispersion of atmospheric pollutants are key aspects tackled by the emerging discipline of urban physics. Using computational fluid dynamics (CFD) as a tool, this branch of physics simulates pollutant dispersion to aid in urban planning and emergency decision-making. In [START_REF] Pasquier | A Lattice-Boltzmann-based modelling chain for traffic-related atmospheric pollutant dispersion at the local urban scale[END_REF], a complete modelling chain has been developed to simulate traffic-related pollutant dispersion at the local urban scale. The method combines a microscopic traffic simulator with a physical engine model to estimate realistic road emissions, which are used as input of a CFD code to model unsteady atmospheric dispersion and compute two-dimensional time-averaged concentration maps at ground level. A presentation of this application is displayed in figure 1. The modelling chain outputs two-dimensional concentration maps from a set of inputs associated with meteorological and traffic-related uncertainties. These five uncertain inputs are the incoming wind direction θ [radians] and intensity U ∞ [metres/second], together with the volume of traffic injected into the road network in the urban geometry, q [vehicles/hour], the proportion of diesel and petrol engines in the fleet β [-], and the speed limit over the network, ν max [kilometres/hour]. The modelling chain with the corresponding uncertain variables is displayed in figure 2 and the probability distributions associated with each variable are given in table 1 and plotted in figure 3.

To efficiently perform a global sensitivity analysis (SA) to compare the relative influence of these uncertain inputs on the output maps, we construct a metamodel of the entire modelling chain using a combination of principal component analysis and Gaussian process regression (GPR), known in the literature as POD-GPR [START_REF] Guo | Reduced order modeling for nonlinear structural analysis using Gaussian process regression[END_REF][START_REF] Marrel | Development of a surrogate model and sensitivity analysis for spatio-temporal numerical simulators[END_REF][START_REF] Nony | Reducedorder modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF]. The metamodel is trained on a Latin Hypercube Sampling (LHS) design of experiments of 150 expensive CFD simulations -each simulation requires about 7 hours of computational time on 960 cores on the high performance computing facility of IFP Energies Nouvelles -and validated on a set of 85 additional simulations. A global coefficient of determination, defined as in Nony et al. 2023, is computed to quantitatively evaluate the predictivity of the POD-GPR metamodel, which reaches about 86%. The lack of predictivity is related to residual noise in the numerical simulation results due to the difficult statistical convergence of the time-averaged concentration fields in CFD simulations.

Based on this cheap metamodel, we want to perform a global sensitivity analysis for concentration maps, which can be seen as instances of a functional output from D ⊂ R 2 to R. Initially, sensitivity analysis focused primarily on models with scalar inputs and outputs. Standard methods such as Morris analysis [START_REF] Morris | Factorial Sampling Plans for Preliminary Computational Experiments[END_REF]) and the Sobol' index [START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates[END_REF] were widely used to evaluate the variation of outputs in response to variations in input parameters. Morris analysis, for example, relies on sampling scenarios in which each parameter is modified one at a time to assess its effect on the model's output. However, as modelling and complex systems have evolved, the applications of sensitivity analysis have expanded to include more complex inputs and outputs. Inputs can be vectors, categorical data, time series and more. Outputs can be multi-dimensional, nonscalar or even probability distributions. To meet these needs, new advanced sensitivity analysis techniques have emerged. For example, Sobol' indices have been generalised for vectorial and functional outputs [START_REF] Gamboa | Sensitivity analysis for multidimensional and functional outputs[END_REF]. However, the latter uses a truncated expansion of the functional output to then reuse the proposed index on vector outputs, which implies a loss of information. In [START_REF] Perrin | Functional principal component analysis for global sensitivity analysis of model with spatial output[END_REF], the authors also propose a method to conduct a global sensitivity analysis in the case of spatial output through functional principal component analysis. Saint-Geours et al. 2014 also perform sensitivity analysis for spatial output by computing maps of sensitivity indices. In [START_REF] Fort | Global Sensitivity Analysis and Wasserstein Spaces[END_REF], the authors propose universal sensitivity indices that can be computed in any metric output space. The more recent indices based on the Hilbert-Schmidt Independence Criterion (HSIC) were also initially defined for scalar and vector inputs and outputs [START_REF] Gretton | Measuring Statistical Dependence with Hilbert-Schmidt Norms[END_REF][START_REF] Veiga | Global Sensitivity Analysis with Dependence Measures[END_REF], but are much more permissive and can be extended to arbitrary outputs as long as a kernel is defined on the output space. For example, HSIC-based indices have been used for probability distribution outputs in Da Veiga 2021.

The aim of this paper is to propose and compare different methods for carrying out a sensitivity analysis of a model that produces two-dimensional maps. We first adopt a pointwise approach, specifically a sensitivity analysis technique using Sobol' indices computed at each point of the map. The resulting Sobol' index maps can be interpreted directly or used to compute generalised Sobol' indices, similar to those proposed in [START_REF] Gamboa | Sensitivity analysis for multidimensional and functional outputs[END_REF], where the discretised map is considered as a vector. The other proposed approach analyses the spatial output without discretisation and instead aims to establish an index that quantifies the influence of each input on the overall spatial output. To achieve this, three new SA indices based on sets are introduced and adapted to maps. Each index is tested on the POD-GPR metamodel.

The paper is structured as follows. Section 2 proposes a pointwise method for performing sensitivity analysis on maps, where section 2.1 establishes essential notations and section 2.2 defines Sobol' indices at each point of the maps and proposes generalised indices that aggregate all pointwise indices. Section 3 is devoted to three new set-based SA indices adapted to maps. Section 3.1 provides the necessary notations for dealing with sets, while section 3.2 deals with sensitivity analysis based on random set theory. Section 3.3 adapts universal indices from [START_REF] Fort | Global Sensitivity Analysis and Wasserstein Spaces[END_REF] to set-valued outputs, and in section 3.4 we use indices based on kernels and the Hilbert-Schmidt independence criterion in the context of spatial outputs, as done in [START_REF] Fellmann | Kernel-based sensitivity analysis for (excursion) sets[END_REF]. Section 4 is devoted to a comparative study of the various sensitivity indices introduced. Finally, our conclusion (section 5) ties these sections together and summarises our main findings.

2 Pointwise sensitivity analysis for maps

Notations

Let us define a map-valued model Φ by

U -→ F(D, R) Φ : u → Φ u : x 1 , x 2 → Φ u (x 1 , x 2 ) with D ⊂ R 2 , U = U 1 × ... × U p ⊂ R p and F(D, R) the space of functions from D to R.
As in many sensitivity analysis framework, the inputs are assumed to be a random vector U = (U 1 , ..., U p ) ∈ U with independent components. It is defined on a probability space (Ω, F, P) with known distributions P U = p i=1 P U i . In the case of the POD-GPR metamodel of pollutant concentration maps, we have -U = [5; 10] × [-0.35; 1.75] × [100; 500] × [0; 1] × [30; 50] and the P U i are given in Table 1.

-D = [0; 550] × [0; 400] -Φ u (x 1 , x 2 )
is the pollutant concentration at position x 1 , x 2 ∈ D in the physical domain when the uncertain parameters are equal to u ∈ U.

Sobol' indices maps and generalized Sobol' indices

First, we propose to analyse the sensitivity of the concentration fields to the uncertain inputs by calculating the Sobol' indices of the spatial maps. In other words, for each x 1 , x 2 ∈ D, we compute the Sobol' indices of the output concentration Φ U (x 1 , x 2 ).

At each point x 1 , x 2 , from the ANOVA decomposition of u → Φ u (x 1 , x 2 ), a variance decomposition of Φ U (x 1 , x 2 ) can be obtained. The first-order Sobol' indices are defined from the first terms of the ANOVA decomposition, each of which depends on a single uncertain variable. The first-order index corresponding to the i-th uncertain variable is defined as

S i (x 1 , x 2 ) = Var[E[Φ U (x 1 , x 2 )|U i ]] Var[Φ U (x 1 , x 2 )] .
These indices indicate the proportion of the total variance that can be attributed to the individual contribution of each uncertain variable U i . Such first-order indices are computed using the standard partial variance estimator recalled in [START_REF] Saltelli | Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index[END_REF] and defined as

Var[E[Φ U (x 1 , x 2 )|U i ]] ≈ 1 N N j=1 Φ U ′(j) (x 1 , x 2 ) Φ U i(j) (x 1 , x 2 ) -Φ U (j) (x 1 , x 2 ) , (1)
where (U (j) , U ′(j) ) N j=1 are independent and identically distributed (iid) samples of (U , U ′ ), with U ′ being an independent copy of U . The matrix U i is the matrix U with the i-th column replaced by that of U ′ . The denominator Var[Φ U (x 1 , x 2 )] is estimated using the classical unbiased sample variance estimator. The indices are estimated for each location x

(j) 1 , x (j) 2 m j=1
of a discretised computational grid composed of m nodes to obtain twodimensional first-order sensitivity maps, as shown in Figure 4.

S gen i := m j=1 w j S i (x (j) 1 , x (j) 2 ) with w j = Var[Φ U (x (j) 1 , x (j)
2 )] Table 2 -Estimated generalised first-order Sobol' indices for each uncertain variable from two-dimensional sensitivity maps, using 2 12 sample points from Halton sequences.

m k=1 Var[Φ U (x (k) 1 , x (k
From the results shown in Figure 4 and 5 it can be seen that the wind angle and the rate of diesel engines in the fleet contribute to the major part of the variance of the output concentration field. The generalised indices reach approximately 55% and 32% for these two variables respectively, while the other indices fall below 10%, and in particular the value of S gen q is very close to zero, showing that the traffic volume q is not influential in this study. The two-dimensional Sobol' maps highlight the fact that the sensitivity to the two dominant variables varies depending on the location in the urban neighbourhood: thus, if we choose to focus our analysis on a specific part of the domain -e.g. a specific building such as a hospital or a primary school -we may end up with a different hierarchy among the variables. We also carried out a second analysis (the results of which are not shown here) by fixing θ and β to their mean values and evaluating the Sobol' maps for the three remaining variables. This showed that the wind intensity dominates over the speed limit (with a generalised index of about 70% for the former and 30% for the latter) 0.0 0.2 0.4

β ν max θ q U ∞ S gen i
Figure 5 -Estimated generalised first-order Sobol' indices for each uncertain variable from two-dimensional sensitivity maps, using 2 12 sample points from Halton sequences and 100 bootstrap resamples to construct 95% confidence intervals.

and the traffic volume remains negligible. We can also observe that the indices sum to about 95%, which means that there are almost no interactions between the inputs. Although this sensitivity analysis is performed point by point of the pollutant concentration map, it allows a first screening of the influencing variables and, in particular, generalised indices allow a quantitative ranking of the inputs. The next section presents another approach that considers the output as a subset of R 3 and performs a sensitivity analysis for such outputs.

New sensitivity analysis indices for sets

In this section we propose three types of indices designed to perform sensitivity analysis for sets, which we use on map-valued outputs. The first section 3.1 introduces the notations to see map-valued models as examples of set-valued models. Then, three sections are dedicated to introduce the three indices. In section 3.2, we propose new indices based on Vorob'ev's theory of random sets [START_REF] Molchanov | Theory of random sets[END_REF]. In section 3.

Notations

We want to perform a sensitivity analysis of a non-discretised map-valued model Φ.

To do this, we propose to consider Φ as a set-valued model Ψ defined by:

U -→ L(X ) Ψ : u → Γ u = {(x 1 , x 2 , c) ∈ D × C, c -Φ u (x 1 , x 2 ) ≤ 0}
where C = [c min , c max ] and L(X ) is the space of Lebesgue measurable subsets of X = D×C. Γ = Ψ(U ) is then a random set that depends directly on the inputs U i whose effect is to be measured.

For estimation purposes, X is discretised with a grid of m points denoted (x (1) , ..., x (m) ). This allows the volume λ(A) of A ⊂ X to be estimated as

λm (A) = 1 m m k=1 1 A (x (l) ).
Let also (U (i) , Γ (i) ), i = 1, ..., n be an independent and identically distributed (i.i.d.) sample of (U , Γ) with Γ = Ψ(U ).

Sensitivity analysis based on random set theory

Working with sets instead of scalars or vectors makes the study of random elements much more difficult. For example, expectations and variance are not easy to define. In Molchanov 2005, a complete theory of random sets is developed with definitions of expectations, medians and deviations of random sets. In particular, based on the Vorob'ev median and deviation, we propose to define indices inspired by Sobol' indices but adapted to random sets.

When dealing with a scalar output Y , first-order Sobol' indices S i can be defined by :

S i = 1 - E [Var[Y |U i ]]
Var Y .

Here we want to define similar indices, but for set-valued outputs. To do this, we propose to replace the variance with the Vorob'ev median deviation, which is a possible adaptation of the median deviation to random sets. This requires defining the Vorob'ev median and the Vorob'ev median deviation, but also defining a Vorob'ev conditional median deviation.

Definition 3.1 (Vorob'ev median). The Vorob'ev median Q 0.5 (Γ) of a random set Γ is defined as the set of points whose probability of being in Γ is at least 0.5, i.e.

Q 0.5 (Γ) = {x ∈ X , P(x ∈ Γ) ≥ 0.5}
Using the Vorob'ev median, the Vorob'ev Median Deviation is defined as the Vorob'ev deviation between the random set and its median. We also propose conditional version of the Vorob'ev Median Deviation.

Definition 3.2. The Vorob'ev Median Deviation of a random set Γ, denoted VMD(Γ), is defined by:

VMD(Γ) = E[λ(Γ∆Q 0.5 )].
where ∆ is the symmetric difference. The Vorob'ev conditional median deviation is the random variable VMD(Γ|U i ) defined by:

VMD(Γ|U i ) = E[λ(Γ∆Q i 0.5 )|U i ]
where Q i 0.5 is the Vorob'ev conditional median defined by:

Q i 0.5 = {x ∈ X , P(x ∈ Γ|U i ) ≥ 0.5}.
We can now define indices that describe the part of the Vorob'ev median deviation of a random set that is due to an input U i .

Definition 3.3. Using the previous definition, we define a sensitivity index S i V by:

S V i : = 1 - E [VMD(Γ|U i )] VMD(Γ) = 1 - E[λ(Γ∆Q i 0.5 )] E[λ(Γ∆Q 0.5 )]
This index quantifies the effect of U i on the output Γ, since it's zero if Γ and U i are independent (simply as Q i 0.5 = Q 0.5 if Γ and U i are independent). However, as first-order Sobol' indices, the converse doesn't hold. We show in Proposition 3.1 that it is between zero and one, as desired for sensitivity indices. The interpretation is close to a part of the total deviation explained by an input. However, there is no decomposition of the indices and they could sum to more than 1.

Proposition 3.1. 0 ≤ S V i ≤ 1 Proof. It is clear that S i V ≤ 1.
To show that it's non-negative, we use the tower property of the conditional expectation on the numerator and the denominator :

E[λ(Γ∆Q i 0.5 )] E[λ(Γ∆Q 0.5 )] = EE[λ(Γ∆Q i 0.5 )|U i ] EE[λ(Γ∆Q 0.5 )|U i ]
.

Then we show that E[λ(Γ∆Q i 0.5 )|U i ] ≤ E[λ(Γ∆Q 0.5 )|U i ] almost surely, using the fact that the Vorob'ev median minimises the deviation for any measurable sets (see Proposition 2.4 of [START_REF] Molchanov | Theory of random sets[END_REF]. Finally by taking the expectation we conclude that S V i is non-negative. The estimation of S V i is done by a double loop, since we are not aware of any cheaper estimator. Let U

(j) i , j ∈ (1, n) be n independent copies of a given input U i , the conditional median Q i 0.5 (U (j) i ) = {x ∈ X , P(x ∈ Γ|U i = U (j) i ) ≥ 0.5} is estimated by independently drawing another sample U (j,l)) -i , l ∈ (1, n) of U -i and evaluating the output sets Γ l (U (j) i ) = Ψ(U (j) i , U (j,l) -i ).
The estimator of the conditional Vorob'ev median can then be derived:

Qi 0.5 (U (j) i ) := {x ∈ X , 1 n n l=1 1 Γ l (U (j) i ) (x) ≥ 0.5} 0.0 0.1 0.2 0.3 0.4 θ U ∞ q β ν max S V i Figure 6 -Estimation of S i
V for each input with 1024 model evaluations and a confidence interval estimated with 100 bootstrap resamples.

Then S V i is estimated by:

SV i = 1 - n j=1 λm (Γ (j) ∆ Qi 0.5 (U (j) i )) n j=1 λm (Γ (j) ∆ Q0.5 ) . (2)
where

Γ (j) = Ψ(U (j) i , U (j)
-i ). The index is estimated by n = 32 model evaluations per loop, resulting in a total of n 2 = 1024 model evaluations. Confidence intervals are obtained by bootstrapping with 100 resamples. The results are shown in Figure 6.

As these indices do not have a decomposition, it is difficult to justify a comparison between the value of each index. The only possible interpretation is that an index far from zero means that the input has an effect on the output. Here, the intervals of β and θ do not include zero, which means that they have an effect. The other three, however, are close to 0, implying that they may not be influential. The confidence intervals are also wide, which means that in another application we might fail to detect influential inputs. To become significant, additional model evaluations or a different estimation method is required. However, methods such as pick and freeze or rank-based estimation do not appear to be applicable in this scenario, as the quantity being estimated is not strictly a conditional variance.

Universal sensitivity indices

In this second part, we propose to use universal indices from Fort et al. 2021. In the latter paper, the authors define indices that can be applied to any metric output space. Using them with set-valued outputs, we define the following indices: Definition 3.4.

S Univ i (γ, Q) = 1 0 Var E [λ(Γ∆γ a ) | U i ] dQ(a) 1 0 Var (λ(Γ∆γ a )) dQ(a)
.

where (γ a ) a∈[0,1] ⊂ X is a collection of test sets parameterised by a ∈ [0, 1] and Q is a probability distribution on [0, 1] to choose.

The idea behind these indices is to quantify the effect of inputs on multiple scalar transformations of the set-valued output, rather than on the set-valued output itself. Specifically, we measure this variability by the volume of the symmetric difference between the output set and a collection of test sets.

Similar to Sobol' indices, the contribution of each input and their interactions can be decomposed to rank the inputs. Indeed, the decomposition can be obtained by taking the Sobol-Hoeffding decomposition of Var (λ(γ∆γ a )) for each a and then summing for 0 to 1 with respect to the measure Q. An independence test is not accessible, mainly because the family of all transformations does not necessarily characterise the whole distribution of the set-valued output. However, screening can still be done, either by keeping only the inputs above a certain threshold, or by keeping the first inputs in the ranking.

After selecting the test sets and the law Q, these indices can be estimated in a similar way to Sobol's indices. We use a rank-based estimator, which is recalled in [START_REF] Fort | Global Sensitivity Analysis and Wasserstein Spaces[END_REF]. Plugging in the volume estimation, the estimator S Univ i of S Univ i is given by the ratio between

S num = 1 N a Na l=1 1 n n j=1 λm (Γ (j) ∆γ a l ) λm (Γ (N i (j)) ∆γ a l ) (3) - 1 N a Na l=1 1 n n j=1
λm (Γ (j) ∆γ a l ) 2 and,

S den = 1 N a Na l=1 1 n n j=1 λm (Γ (j) ∆γ a l ) 2 - 1 N a Na l=1 1 n n j=1 λm (Γ (j) ∆γ a l ) 2 , (4)
where (a l ) is an iid sample of the law Q and N i (j) is the index in the sample (U l i ) that comes after U j i when (U l i ) is sorted in ascending order (see [START_REF] Gamboa | Global sensitivity analysis: A novel generation of mighty estimators based on rank statistics[END_REF] for details). We compute the indices with n = 1000 model evaluations and N a = 100 test sets. To remain general, X is rescaled to [0, 1] 3 which is centered on x 0 = ( 1 2 , 1 2 , 1 2 ). We test four families of test sets defined on [0, 1] 3 :

-Centered balls:

γ a = B(x 0 , a) with Q ∼ U([0, 1 2 ]) -Centered squares: γ a = {x ∈ [0, 1] 3 , ||x -x 0 || ∞ ≤ a} with Q ∼ U([0, 1 2 ]) -Slides along the i-th dimension: γ i a = {x ∈ [0, 1] 3 , x i ≤ a} with Q ∼ U([0, 1]) -Vorob'ev quantiles: γ a = {x ∈ [0, 1] 3 , P(x ∈ Γ) ≥ a} with Q ∼ N ( 1 2 , σ
2 ) The slides are used with i = 3 to get levels of concentration test sets. Confidence intervals are obtained by bootstrapping with B = 100 resamples. We use bootstrap samples of size 0.8n = 800 without replacement because replacement introduces high bias in rank-based estimation. The intervals are adjusted by a correction factor. The results are displayed in Figure 7. The results are comparable when using centered balls, centered squares, and slices. β is the most dominant input with an index value of about 80%. The value of U ∞ is approximately 10%, and all other inputs have minimal impact. In contrast, when Vorob'ev quantiles are the test sets, β and θ are the most important inputs, and U ∞ also has some influence to a lesser extent. The output is clearly not influenced by either q or ν max . This difference shows that the choice of test sets is crucial, as it has a strong influence on the values of the indices. When the law Q of Vorob'ev quantiles is approximately 0.5, their geometry is similar to Γ's realisations. Therefore, changes in Γ can be accurately detected by measuring the volume of the symmetric difference. Conversely, it may not be fully possible to detect all variations of the random set by comparing realisations of Γ with simple sets as squares, cubes or slices.

HSIC-based sensitivity analysis

In this section, we propose to define kernel-based sensitivity indices with set-valued outputs, as the use of kernels makes such indices very permissive in the type of considered outputs (or inputs).

Sensitivity analysis, based on measures of dependence, consists in examining the influence of an input X i on an output Y by measuring their mutual dependence. To do this, a distance is calculated between the joint distribution P X i ,Y and the product of their marginal distributions P X i ⊗ P Y . If this distance is zero, then X i and Y are independent, meaning X i has no impact on Y . A frequently used measure for dependence is the Hilbert-Schmidt Independence Criterion (HSIC). It operates by selecting appropriate input and output kernels, and subsequently measuring the squared distance between the mean embeddings of P X i ,Y and P X i ⊗P Y in the corresponding Reproducing Kernel Hilbert Space (RKHS), see [START_REF] Gretton | Measuring Statistical Dependence with Hilbert-Schmidt Norms[END_REF] for a proper definition of the tools needed to define the HSIC. This calculation can be performed using just one sample, which is among the reasons why HSICs have gained popularity.

If a characteristic kernel is available on any space, meaning a kernel whose mean embedding in the RKHS is injective, then HSIC-based indices can be used. Consequently, their application is appropriate for sensitivity analysis of set-valued models. Fellmann et al. 2023 introduced a kernel k set defined on sets, which is proved to be characteristic. It is then used to define HSIC-based indices on sets. The latter definition requires to have -a characteristic kernel on L(X ). The proposed kernel in [START_REF] Fellmann | Kernel-based sensitivity analysis for (excursion) sets[END_REF] is

k set (γ 1 , γ 2 ) = exp - λ(γ 1 ∆γ 2 ) 2σ 2 ∀γ 1 , γ 2 ⊂ X ,
where σ 2 is a normalization constant. p characteristic kernels K i on the input space U i . As shown in Da Veiga 2021, an ANOVA-like decomposition of HSIC exists if the input kernels are ANOVA, i.e. they can be written as 1 + k i , where k i is a kernel whose induced RKHS consists of zero-mean functions. Sobolev kernels are the most commonly used kernels that fulfil these requirements. Furthermore, classic characteristic kernels can be modified to become ANOVA, as described in [START_REF] Ginsbourger | On ANOVA Decompositions of Kernels and Gaussian Random Field Paths[END_REF].

:= H set (U i , Γ) H set (U , Γ) , where H set (U i , Γ) = E [(K i (U i , U i ′ ) -1)k set (Γ, Γ ′ )] , H set (U , Γ) = E [(K(U , U ′ ) -1)k set (Γ, Γ ′ )] ,
with

K = p j=1 K j ,
and (U ′ , Γ ′ ) an independent copy of (U , Γ).

Since these indices are zero if and only if the considered input U i and the output Γ are independent, they are suitable for screening. Using this result, p-values associated with the independent tests can be calculated to screen the inputs. Besides, we choose the input kernels K i as ANOVA kernels so that the indices S Hset i have an ANOVA decomposition as done in [START_REF] Fellmann | Kernel-based sensitivity analysis for (excursion) sets[END_REF]. Subsequently, the inputs can be ranked by their impact based on each index's value.

The indices can be estimated using:

H set (U i , Γ) = 2 n(n -1) n j<l K i U (j) i , U (l) i -1 exp(- λ(X ) 2σ 2 λm (Γ (j) ∆Γ (l) )). (5)
We compute the indices for each of the five inputs using a sample size of n = 1000. The results for five different input ANOVA kernels are compared using the following kernels:

-the Sobolev kernel of order 1, with h > 0. The Sobolev kernel is already ANOVA, whereas four additional kernels need adaptations to become ANOVA (see [START_REF] Ginsbourger | On ANOVA Decompositions of Kernels and Gaussian Random Field Paths[END_REF] for the detailed transformation). Confidence intervals are estimated using bootstrap methods with B = 100 resamples. The value of σ is taken as the empirical median of λm (Γ (j) ∆Γ (l) ) for j < l. Figure 8 shows our results. We observe that the choice of the input kernel does not significantly affect the values of the indices: regardless of the kernel chosen, β and θ remain the most influential variables (with about 40% each) and the other three inputs the least influential (with about 10%). The input ranking can be used for screening purposes. For instance, one may decide to keep only the top three most influential inputs. However, using HSIC-based indices guarantees trustworthy screening by computing the p-values of independence tests between an input and an output. This is done in Table 3, where the p-values for each input are computed by asymptotic estimation [START_REF] Song | Supervised Feature Selection via Dependence Estimation[END_REF] for each input kernel. If the p-value exceeds 0.05, then the input has no influence on the output. Specifically, in this case, we observe that only q, for each kernel, has a p-value above 0.05. Additionally, a standard deviation is estimated for q through bootstrap analysis, which is given in the second column. We confirm that despite this deviation, the p-values of q remain above 0.05. For the remaining four inputs, p-values below 10 -16 were obtained, and standard deviations of the same scale indicate that these four inputs are influential. The chosen input kernel slightly modifies the p-values in this case, but it does not affect the screening results. However, in other test cases, the choice of kernel could impact the results, which β q θ U ∞ ν max k sob < 10 -16 0.74 ± 0.6 < 10 -16 < 10 -16 < 10 -16 k rbf < 10 -16 0.76 ± 0.6 < 10 -16 < 10 -16 < 10 -16 k exp < 10 -16 0.89 ± 0.5 < 10 -16 < 10 -16 < 10 -16 k 3/2 < 10 -16 0.82 ± 0.6 < 10 -16 < 10 -16 < 10 -16 k 5/2 < 10 -16 0.80 ± 0.6 < 10 -16 < 10 -16 < 10 -16 is the primary limitation of the HSIC-based indices. Some research has also been conducted to aid in kernel selection. For instance, in El Amri and Marrel 2021, the authors suggest selecting the kernel that maximises test power. Without further investigation into the impact of kernels, the Sobolev kernel is preferable as it does not require any choice of hyperparameters.

k sob (x, y) = 1+(x-1 2 )(y -1 2 )+ 1 2 [(x-y) 2 -|x-y|+ 1 6 ] 0.0 0.1 0.2 0.3 0.4 0.5 θ U ∞ q β ν max S H set i k sob k rbf k exp k 3/2 k 5/2

Comparison of the indices

In this section we compare the four indices recalled here :

-the generalised first-order Sobol' indices S gen i ; -the Vorob'ev based indices S V i ; -the universal indices S Univ i computed using Vorob'ev quantiles as test sets; -the HSIC-based indices S Hset i using the Sobolev kernel as ANOVA input kernel. Each index is computed with a budget of about 1000 model evaluations. S gen i are estimated with the pick and freeze estimator given in (1). S V i are estimated using a double loop method as described in (2), with 32 model evaluations in each loop, resulting in 1024 evaluations. The universal indices are evaluated using the rank-based estimation approach described in (3) and (4), with N a = 100 test sets. Lastly, the estimate for S Hset i is given by ( 5).

The results are shown in Figure 9. First we check that the four indices give similar results. The most influential inputs are θ and β, and the least influential are U ∞ , q and ν max . However, there are differences in the obtained results between the indices regarding screening and ranking. The HSIC is the only index that allows access to a reliable independence test, which categorizes q as non-influential in screening (according to the p-values given in Table 3). For the other three indices, a possible screening method is to select a threshold under which inputs will be deemed non-influential. For instance, if a 10% threshold is chosen, then U ∞ , q, and ν max would be considered negligible. However, with a smaller threshold, conclusions would be challenging to draw as some confidence intervals overlap. The four significant inputs, identified by HSIC-based tests, can be confidently ranked since their confidence intervals do not overlap. In contrast, the ranking of the remaining inputs varies for each index. Similar to S Hset i , S V i places θ and β at the same level. However, θ is much more influential than β according to the generalised Sobol' indices, and the opposite is true for the universal indices. The ranking is justified in theory for all indices except for S V i , which lacks any decomposition. Examining the confidence intervals, all four indices remain satisfactory with 1000 model evaluations, but the HSIC-based indices, and to a lesser extent the universal indices, have much less variability. To emphasise this disparity, we create an identical plot on Figure 10 with only 100 model evaluations (10 in each loop for S V i ). This time the HSICbased indices can be used as they have relatively small confidence intervals. The same conclusion in terms of screening and ranking can be reached with a single 100-sample. The universal indices can also still be used in this case, but their variability makes it possible to have non-positive indices. As for the other two indices, it is important to note that they have not yet converged, rendering them unusable with only 100 model evaluations.

The strengths and limitations of each index are summarised in Table 4.

Conclusion

In this paper, we aim to quantify the impact of inputs on a map-valued model, specifically a concentration map model for a realistic urban setting in the greater Paris region (France). To achieve this goal, we propose a sensitivity analysis of the model. As a starting point, we perform a pointwise sensitivity analysis on each point of the map. This produces maps of Sobol' indices and allows us to observe the influence of each input in each zone of the map. Generalised indices are also derived to synthesise the overall impact of each input. A second approach deals with the model as a whole. Three types of indices, defined for set-valued outputs, are introduced and used with the map-valued outputs. The first is based on quantities from random set theory. The second is based on universal indices that can be used with set outputs. Finally, the last relies on kernel-based sensitivity analysis and, in particular, on HSIC-based indices. A comparison of each index reveals that the HSIC-based index is the most efficient in terms of the number of model evaluations and can be used for both screening and ranking. However, it may not be the most suitable option if we are interested in examining the interactions between inputs. In such situations, universal indices and generalized Sobol' may be more appropriate.

The introduction of sensitivity indices for sets was initially devised for their application to maps. However, their utility extends beyond maps and can be effectively employed for any model with set-valued outputs, like for example sets of feasible points in mechanical engineering. While we specifically addressed the two-dimensional case to directly handle maps, it is also worth noting that theses indices can be used with X of any finite dimension.

Acknowledgment

This research was conducted with the support of the consortium in Applied Mathematics CIROQUO, gathering partners in technological research and academia in the development of advanced methods for Computer Experiments.

Figure 1 -

 1 Figure 1 -Concentration maps from CFD numerical simulations of pollutant dispersion at the urban scale (Pasquier et al. 2023). (a) Domain of interest, captured from OpenStreetMaps. (b) Time-averaged map of the logarithm of the concentration, denoted ⟨χ * ⟩, obtained from a run of the CFD code with realistic road emissions. The wind is coming from the left in this case.

Figure 2 -

 2 Figure 2 -Whole modelling chain designed in Pasquier et al. 2023 for the calculation of time-averaged concentration maps. The five uncertain parameters considered are indicated under each block.

Figure 4 -

 4 Figure 4 -Two-dimensional maps of pointwise first-order Sobol' indices for time-averaged pollutant concentration, estimated by quasi-Monte-Carlo using 2 10 sample points and Halton sequences.

  × 10 -1 4.47 × 10 -2 7.56 × 10 -4 3.22 × 10 -1 1.73 × 10 -2

  3 we adapt the universal indices of Fort et al. 2021 to sets. Finally, in Section 3.4, we present indices based on the Hilbert-Schmidt Independence Criterion (HSIC) in the context of set-valued output, as done in Fellmann et al. 2023.

Figure 7 -

 7 Figure 7 -Estimation of the universal indices S Univ i for each input and for four different test sets and N a = 100, with 1000 model evaluations. Confidence intervals are obtained with 100 bootstrap samples

Definition 3. 5 .

 5 With the previous kernels, HSIC-based indices on sets are defined by ∀i ∈ {1, ..., p}, S Hseti

Figure 8 -

 8 Figure 8 -Estimation of S Hset i for five input kernels, 1000 model evaluations. Confidence intervals are obtained by bootstrap with 100 resamples

Figure 9 -

 9 Figure 9 -Comparison of the four indices with a total budget of n = 1000 model evaluations. 100 bootstrap sample are used to estimate confidence intervals

Figure 10 -

 10 Figure 10 -Comparison of the four indices with a total budget of n = 100 model evaluations. 100 bootstrap sample are used to estimate confidence intervals

Table 3 -

 3 Asymptotic estimation of the p-values of HSIC-ANOVA-based independence test with five input kernel and 1000 model evaluations and 100 bootstrap samples.