C Michelot
email: candice.michelot@gmail.com

A Kato

T Raclot

K Shiomi

P Goulet

P Bustamante

Y Ropert- Coudert

Sea-ice edge is more important than closer open water access for foraging Adélie penguins: evidence from two colonies

Keywords: Pygoscelis adeliae, Incubation trip, Sea-ice edge, Colony, Diet, GPS, Stable isotopes

Sentinel species, like Adélie penguins, have been used to assess the impact of environmental changes, and their link with sea ice has received considerable attention. Here, we tested if Adélie penguins from 2 colonies in East Antarctica target the distant sea-ice edge or take advantage of closer open waters that are readily available near their colony. We examined the foraging behaviour of penguins during the incubation trips of females in 2016 and males in 2017, using GPS tracking and diet data in view of daily sea-ice data and bathymetry. In 2016-2017, sea-ice cover was extensive during females' trips but flaw leads and polynyas were close to both study sites. Sea ice receded rapidly during males' trips in [2017][2018]. Despite close open water near both colonies in both years, females and males preferentially targeted the continental slope and the sea-ice edge to forage. In addition, there was no difference in the diet of penguins from both colonies: all penguins fed mostly on Antarctic krill and males also foraged on Antarctic silverfish. Our results highlight the importance of the sea-ice edge for penguins, an area where food abundance is predictable. It is likely that resource availability was not sufficient in closer open water areas at such an early stage in the breeding season. The behaviours displayed by the penguins from both colonies were similar, suggesting a common behaviour across colonies in Terre Adélie, although additional sites would be necessary to ascertain this hypothesis.

INTRODUCTION

The climate is rapidly changing and this has important impacts on the Arctic and Antarctic, which are major regulators of the global climate [START_REF] Massom | Change and Variability in East Antarctic Sea Ice Seasonality[END_REF][START_REF] Hobbs | A review of recent changes in Southern Ocean sea ice, their drivers and forcings[END_REF]). In the last 30 yr, the annual sea-ice extent in Antarctica has increased by 1.5% decade -1 in response to atmospheric and oceanic warming [START_REF] Collins | Longterm Climate Change: Projections, Commitments and Irreversibility[END_REF][START_REF] Comiso | Positive Trend in the Antarctic Sea Ice Cover and Associated Changes in Surface Temperature[END_REF].

However, according to models, Antarctica will experience a major sea-ice extent loss by the end of the century [START_REF] Collins | Longterm Climate Change: Projections, Commitments and Irreversibility[END_REF][START_REF] England | Contrasting the Antarctic and Arctic Atmospheric Responses to Projected Sea Ice Loss in the Late Twenty-First Century[END_REF].

In polar regions, ecosystem dynamics are mostly driven by sea ice [START_REF] Massom | Antarctic sea ice change and variability -Physical and ecological implications[END_REF], as sea-ice covered areas play a key role in biogeochemical cycles and primary production and represent highly productive habitats [START_REF] Nicol | Ocean circulation off east Antarctica affects ecosystem structure and sea-ice extent[END_REF][START_REF] Arrigo | Phytoplankton dynamics within 37 Antarctic coastal polynya systems[END_REF][START_REF] Massom | Antarctic sea ice change and variability -Physical and ecological implications[END_REF]. More specifically, sea-ice ecosystem dynamics depend on seasonal change in sea-ice growth, extent and retreat. In this system, partially covered open water areas characterized by diffuse sea-ice conditions, called marginal ice zones (MIZ), represent areas of high primary production, as they are the first to be exposed to increasing light penetration in early spring [START_REF] Massom | The distribution and formative processes of latent-heat polynyas in East Antarctica[END_REF][START_REF] Arrigo | Phytoplankton dynamics within 37 Antarctic coastal polynya systems[END_REF].

Diatoms, which are at the base of the 'food chain of giants' [START_REF] Smetacek | The role of grazing in structuring Southern Ocean pelagic ecosystems and biogeochemical cycles[END_REF], develop in those areas and enhance krill and fish presence, which are then consumed by higher trophic level species. As the interface between open water and ice, the MIZ is found at the ice edge, but also at the borders of coastal polynyas and flaw leads. There are various ways of defining these structures; here, we follow the definitions of [START_REF] Massom | The distribution and formative processes of latent-heat polynyas in East Antarctica[END_REF], [START_REF] Smith | Chapter 13 -Polynyas and Climate Change: A View to the Future[END_REF] and [START_REF] Massom | Antarctic sea ice change and variability -Physical and ecological implications[END_REF]. Polynyas are large areas of open water that form, often recurrently, in the coastal zone and are characterized by persistent fast-ice. Flaw leads are long, linear, ephemeral water accesses that separate the persistent fast-ice from the packice, and are characterized by large drifting ice floes. Finally, the ice edge delimits the transition from the pack-ice to open sea and also presents highly productive conditions [START_REF] Massom | The distribution and formative processes of latent-heat polynyas in East Antarctica[END_REF][START_REF] Smith | Chapter 13 -Polynyas and Climate Change: A View to the Future[END_REF][START_REF] Massom | Antarctic sea ice change and variability -Physical and ecological implications[END_REF]. Despite physical and biogeochemical differences around Antarctica, the MIZ is important for predators' foraging activity around the continent [START_REF] Karnovsky | The Impact and Importance of Production in Polynyas to Top-Trophic Predators: Three Case Histories[END_REF][START_REF] Labrousse | Coastal polynyas: Winter oases for subadult southern elephant seals in East Antarctica[END_REF]. The occurrence of polynyas positively influences the presence of emperor penguin Aptenodytes forsteri breeding colonies [START_REF] Massom | The distribution and formative processes of latent-heat polynyas in East Antarctica[END_REF]) and the size of Adélie penguin Pygoscelis adeliae colonies [START_REF] Ainley | The Adélie Penguin: Bellwether of Climate Change[END_REF][START_REF] Arrigo | Phytoplankton dynamics within 37 Antarctic coastal polynya systems[END_REF]. In addition, during the winter when the sea-ice cover becomes a physical barrier, partially or totally ice-free areas are used as air-breathing access or as predictable foraging grounds for predators [START_REF] Karnovsky | The Impact and Importance of Production in Polynyas to Top-Trophic Predators: Three Case Histories[END_REF][START_REF] Labrousse | Coastal polynyas: Winter oases for subadult southern elephant seals in East Antarctica[END_REF]. In spring and summer, they can also be used as major transiting areas to rapidly reach profitable foraging grounds under extensive sea-ice conditions (Kato et al. 2002[START_REF] Clarke | Environmental conditions and life history constraints determine foraging range in breeding Adélie penguins[END_REF][START_REF] Widmann | Habitat use and sex-specific foraging behaviour of Adélie penguins throughout the breeding season in Adélie Land, East Antarctica[END_REF]. The dependence of predators on sea-ice conditions makes them particularly suitable as indicators of the state of the ecosystem. Being located in the higher levels of trophic chains, top predators integrate and respond rapidly to changes occurring at lower levels (e.g. emperor penguins, [START_REF] Jenouvrier | Effects of climate change on an emperor penguin population: analysis of coupled demographic and climate models[END_REF]; Adélie penguins, [START_REF] Cimino | Projected asymmetric response of Adélie penguins to Antarctic climate change[END_REF]; south polar skuas Catharacta maccormicki, [START_REF] Pacoureau | Population response of an apex Antarctic consumer to its prey and climate fluctuations[END_REF]. Central place foraging predators that breed on land and commute regularly between the land and the sea to feed during the breeding season [START_REF] Elliott | Centralplace Foraging in an Arctic Seabird Provides Evidence for Storer-Ashmole's Halo[END_REF]) are also easily observable. Together with information on dietary regime, the study of their foraging behaviour can inform ecosystem dynamics and resource availability in such a dynamic environment [START_REF] Frederiksen | Seabirds as environmental indicators: the advantages of combining data sets[END_REF][START_REF] Bost | The importance of oceanographic fronts to marine birds and mammals of the southern oceans[END_REF]).

Among the meso-predators, the Adélie penguin is often referred to as the 'bellwether' of the sea-ice ecosystem [START_REF] Ainley | The Adélie Penguin: Bellwether of Climate Change[END_REF]. This sentinel species is the most abundant seabird species in Antarctica, with more than 3.79 million pairs around the continent [START_REF] Lynch | First global census of the Adélie Penguin[END_REF]. Their foraging behaviour and breeding success are closely related to sea-ice conditions and dynamics during the winter but also during the summer period, as they time their reproduction to match the peak of food availability [START_REF] Ainley | The Adélie Penguin: Bellwether of Climate Change[END_REF][START_REF] Emmerson | Adélie penguin response parameters signal reduced prey accessibility: implications for predator-prey response curves[END_REF], Ropert-Coudert et al. 2018). They mainly feed on Antarctic and ice krill (Euphausia superba and E. crystallorophias), and also on Antarctic silverfish Pleuragramma antarctica [START_REF] Wienecke | Adélie penguin foraging behaviour and krill abundance along the Wilkes and Adélie land coasts, Antarctica[END_REF][START_REF] Ainley | The Adélie Penguin: Bellwether of Climate Change[END_REF][START_REF] Cherel | Isotopic niches of emperor and Adélie penguins in Adélie Land, Antarctica[END_REF]. The development and abundance of their main prey are highly dependent on the sea-ice dynamics throughout the winter [START_REF] Knox | Biology of the Southern Ocean[END_REF], Koubbi et al. 2010[START_REF] Meyer | The winter pack-ice zone provides a sheltered but food-poor habitat for larval Antarctic krill[END_REF]. During the winter, microalgae and bacteria attach underneath the sea ice where krill come to graze on them and reproduce [START_REF] Meyer | The winter pack-ice zone provides a sheltered but food-poor habitat for larval Antarctic krill[END_REF].

In spring, the ice breaks and makes krill-and subsequently, fish-available to penguins and other upper predators [START_REF] Knox | Biology of the Southern Ocean[END_REF][START_REF] Wienecke | Adélie penguin foraging behaviour and krill abundance along the Wilkes and Adélie land coasts, Antarctica[END_REF]. Adélie penguins' optimal growth rate and foraging activity are found under moderate sea-ice cover conditions and often associated with sea-ice concentration around 15-20%, i.e. diffuse sea-ice conditions found in the MIZ [START_REF] Ainley | The Adélie Penguin: Bellwether of Climate Change[END_REF][START_REF] Ballard | Foraging strategies of Adélie penguins: adjusting body condition to cope with environmental variability[END_REF][START_REF] Barbraud | Extreme ecological response of a seabird community to unprecedented sea ice cover[END_REF][START_REF] Guen | Reproductive performance and diving behaviour share a common sea-ice concentration optimum in Adélie penguins (Pygoscelis adeliae)[END_REF]. To examine their dependence on sea ice, the foraging behaviour of Adélie penguins has been studied at distant sites with contrasted sea-ice conditions [START_REF] Watanuki | Diving and foraging behaviour of Adélie penguins in areas with and without fast sea-ice[END_REF][START_REF] Watanuki | Parental mass change and food provisioning in Adélie penguins rearing chicks in colonies with contrasting sea-ice conditions[END_REF] or at a given colony but in years of different sea-ice conditions and/or at different breeding stages [START_REF] Wienecke | Adélie penguin foraging behaviour and krill abundance along the Wilkes and Adélie land coasts, Antarctica[END_REF][START_REF] Clarke | Environmental conditions and life history constraints determine foraging range in breeding Adélie penguins[END_REF][START_REF] Nesti | Diving behaviour of chick-rearing Adélie Penguins at Edmonson Point, Ross Sea[END_REF][START_REF] Erdmann | Characterization of winter foraging locations of Adélie penguins along the Western Antarctic Peninsula, 2001-2002[END_REF][START_REF] Guen | Reproductive performance and diving behaviour share a common sea-ice concentration optimum in Adélie penguins (Pygoscelis adeliae)[END_REF]. However, data on Adélie penguin foraging behaviour has, to date, only been obtained from a restricted number of colonies, mostly in relation to the relative ease of access of these sites to researchers.

In 2010, the intrusion of the B09B iceberg in Commonwealth Bay, East Antarctica, led to high fast-ice expansion in that region. Before this event, the Commonwealth Bay area was rarely covered by sea ice [START_REF] Clark | Effects of sea-ice cover on marine benthic communities: a natural experiment in Commonwealth Bay, East Antarctica[END_REF][START_REF] Wilson | The impact of the giant iceberg B09B on population size and breeding success of Adélie penguins in Commonwealth Bay, Antarctica[END_REF]. In recent years, episodes of extreme sea-ice extent coupled with rainfall in Terre Adélie (East Antarctica) led to massive breeding failures in the colony of Adélie penguins, as well as other seabirds, on the Ile des Pétrels [START_REF] Barbraud | Extreme ecological response of a seabird community to unprecedented sea ice cover[END_REF], Ropert-Coudert et al. 2018). The change in the icescape coupled with the occurrence of extreme weather events mean that the region and the resident animal populations are now facing new challenges. Populations on the Ile des Pétrels have been intensively studied over the past years, and the foraging behaviour of Adélie penguins breeding there has received considerable attention (Ropert-Coudert et al. 2001a,b, 2002a,b, 2004[START_REF] Cottin | Foraging strategies of male Adélie penguins during their first incubation trip in relation to environmental conditions[END_REF][START_REF] Widmann | Habitat use and sex-specific foraging behaviour of Adélie penguins throughout the breeding season in Adélie Land, East Antarctica[END_REF][START_REF] Guen | Reproductive performance and diving behaviour share a common sea-ice concentration optimum in Adélie penguins (Pygoscelis adeliae)[END_REF]. However, the foraging response to the new icescape situation by colonies located east and west of this well-known colony remains unknown.

In this context, and bearing in mind the potential role of polynyas and/or flaw leads noted above, the goal of our study was to test if Adélie penguins tend to target the distant seaice edge or if they take advantage of closer open water areas to optimize their foraging activity. We tested this on the first incubation trips of Adélie penguins from 2 colonies in East Antarctica: the regularly studied colony of Ile des Pétrels (near Dumont d'Urville French station, Terre Adélie) and that of Cap Bienvenue, 24 km east of the former (Fig. 1). We expected birds from the 2 colonies to target the MIZ from the closest available open water areas, especially under extensive sea-ice conditions, as seen in other colonies and with other species (Kato et al. 2002[START_REF] Cottin | Foraging strategies of male Adélie penguins during their first incubation trip in relation to environmental conditions[END_REF][START_REF] Labrousse | Coastal polynyas: Winter oases for subadult southern elephant seals in East Antarctica[END_REF].

MATERIALS AND METHODS

Study area and foraging data

During the breeding seasons of 2016 and 2017, Adélie penguin foraging data were collected at 2 locations in East Antarctica: a colony located on the Ile des Pétrels, near the Dumont d'Urville French research station (66.66° S, 140.01° E), and a colony on Cap Bienvenue (66.72° S, 140.52° E). These 2 colonies are separated by a stretch of coastline and the Astrolabe Glacier with no colonies between them (Fig. 1) and are surrounded by other distant colonies [START_REF] Barbraud | First census of breeding seabirds between Cap Bienvenue (Terre Adélie) and Moyes Islands (King George V Land), Antarctica: new records for Antarctic seabird populations[END_REF][START_REF] Lynch | First global census of the Adélie Penguin[END_REF]. About 15000 and 24100 breeding pairs of Adélie penguins are present in those 2 colonies respectively (C. Barbraud & K. Delord unpubl. data;counts made in November 2017 andDecember 2016 respectively) .

In 2016, a total of 16 females from the Ile des Pétrels colony were instrumented with GPS loggers before their first incubation trip, between 19 and 23 November. Three types of GPS devices were used: 11 CatTrack, 4 CatLog (Catnip Technologies), ca. 14 × 35 × 70 mm, 30 g and customized to be waterproof (see [START_REF] Widmann | Habitat use and sex-specific foraging behaviour of Adélie penguins throughout the breeding season in Adélie Land, East Antarctica[END_REF] for details), recording a location every 15 min; and 1 AxyTrek (TechnoSmart), ca. 8 × 35 × 70 mm, 28 g, recording 1 location min -1 . In addition, on 22 November 2016, 16 females from the Cap Bienvenue colony were equipped with CatTrack GPS loggers (same characteristics as above). For both sites, loggers were attached on the birds' back feathers with marine tape and mastic, and secured with cable-ties [START_REF] Wilson | Long-term attachment of transmitting and recording devices to penguins and other seabirds[END_REF]. Birds were released at their nest. Upon their return from a single trip at sea, equipped penguins were recaptured on their nest to retrieve the loggers, and blood samples (1-2 ml) were collected from the bird's foot vein into a syringe containing heparin to avoid blood clotting.

Similarly, in 2017, a total of 20 males from the Ile des Pétrels colony were instrumented with GPS loggers before their first incubation trip. A total of 10 CatTrack and 10 AxyTrek were deployed between 4 and 8 December. The AxyTreks were set to record locations every 1 min while the CatTracks were set to record locations every 15 min. On 6 December, 15 males from the Cap Bienvenue colony were instrumented with CatTrack, recording 1 point every 15 min. The same attachment and removal procedures as for the females were used.

On Ile des Pétrels, the nest and partner of the tagged birds were monitored from a distance every 3 h. Upon return of the tagged individual, the date and time were noted and the loggers removed. On Cap Bienvenue, we timed a second visit to the colony to coincide with the estimated return date of the tagged bird using the phenology of the Ile des Pétrels colony, as breeders are relatively well synchronized early in the season. In 2016, the second visit took place on 19 December, while in 2017, 3 visits to Cap Bienvenue took place on 25 and 30 December 2017 and 5 January 2018.

Environmental data

Environmental data were extracted for our study area, between 61-68° S and 134-147° E, and transformed into raster map with the 'raster' package in R v.3.5.1 [START_REF] Hijmans | Package 'raster[END_REF][START_REF] Core | R: a language and environment for statistical computing[END_REF], with a grid cell resolution of 6.25 km representing an area covering 516165.9 km².

We used bathymetric data from the finest available (at that time) resolution GVdem data set [START_REF] Beaman | A new high-resolution bathymetry model for the Terre Adélie and George V continental margin, East Antarctica[END_REF]; NASA, grid cell resolution: 0.001 × 0.001 arc-degree), which covers an area of 302476.4 km 2 between 63-69° S and 138-148° E, i.e. the majority of the study area used by birds except for the western part. To include the latter, we used the international bathymetric chart of the Southern Ocean (IBCSO) data set [START_REF] Arndt | The International Bathymetric Chart of the Southern Ocean (IBCSO) Version 1.0-A new bathymetric compilation covering circum-Antarctic waters[END_REF] grid cell resolution: 500 m, covering the whole Southern Ocean, based on a stereographic projection), averaged on the raster grid of the study area. The bathymetric features were extracted with the 'rasterToContour' function of the 'raster' package in R. The continental shelf was delimited between 0 and 549 m depth, with the shelf break between 500 and 549 m.

The continental slope between 550 and 3500 m was divided in 3 depth strata. The first stratum, between 550 and 1999 m, is characterized by a complex network of submarine canyons; the second, between 2000 and 2999 m, is characterized by the presence of deep-sea channels; and the third, between 3000 and 3499 m, has a lower gradient than the other strata.

The abyssal plain was considered at depths greater than 3500 m (Koubbi et al. 2010[START_REF] Beaman | A new high-resolution bathymetry model for the Terre Adélie and George V continental margin, East Antarctica[END_REF]) (see Fig. 2).

Daily sea-ice concentration data (Advanced Microwave Scanning Radiometer, AMSR-2, grid cell resolution of 6.25 × 6.25 km) were downloaded from the website of the University of Bremen (https://seaice.uni-bremen.de/data/amsr2/asi_daygrid_swath/s6250) from the date the first bird was equipped to the date of last logger recovery, for each season.

Open water features were extracted from the daily sea-ice concentration following the same procedure as for bathymetric features. The sea-ice edge was defined as the last 15% of sea-ice concentration cells before the open sea [START_REF] Stammerjohn | Opposing Southern Ocean climate patterns as revealed by trends in regional sea ice coverage[END_REF][START_REF] Massom | Antarctic sea ice change and variability -Physical and ecological implications[END_REF]. Polynyas were defined as any cells of open water that were totally surrounded by sea ice (>15%). Three recurrently observed polynyas were detected: the We calculated the nearest distance between each grid cell centroid and each colony using the 'gridDistance' function from the 'raster' package. We also calculated the nearest distance between each colony and each bathymetric and open water feature, using the 'dist2Line' function from the 'geosphere' package.

GPS data cleaning and processing

GPS tracks were processed using R v.3.5.1 (R Core Team 2017). Duplicate points corresponding to artefacts of recording (2 points recorded at the exact same location) and an excess of points at the colony before the departure and after the return of penguins were removed. Outlier points (e.g. locations with impossible date-time, or points with a Northern Hemisphere latitude), considered unrealistic, were also removed. Incomplete tracks from birds from Cap Bienvenue colony were removed from the analysis. For incomplete tracks from the Ile des Pétrels birds, a point at the colony, with the date and time of the bird's return noted from the regular visual checks of the nests, was virtually added. Using this approach, tracks for which more than 60% of the total trip duration was recorded were kept and completed with a straight line on their inward journey, between the last location recorded and the point added at their returning date-time to the colony. Doing so, the approximate distance travelled and potential habitat crossed during the birds' inward journey were taken into account in the analyses. GPS tracks were then resampled with a regular time step of 30 min, using the 'redisltraj' function of the R package 'adehabitatLT' [START_REF] Calenge | Analysis of Animal Movements in R: the adehabitatLT Package[END_REF], to have regular steps between locations and to avoid too many location interpolations, as some gaps between locations can occur during birds' dives. A total of 10 tracks in each colony were kept for the females' first incubation trips in 2016-2017 (see Table 1), and 14 and 10 tracks for males from Cap Bienvenue and Ile des Pétrels respectively in 2017-2018. Among the tracks kept, only 4 female tracks and 1 male track were incomplete and kept in the analysis. The missing part of their trips (which was interpolated) represented between 0.2 and 9.6% of the females' trip durations, and 20% of the male's trip.

The following parameters were defined for each bird: total distance travelled (km), total trip duration (h) and maximal distance to the colony (km). Heading of birds 24 h after their departure was calculated, relative to their colony and relative to the closest open water point (polynya, ice-edge, open sea or flaw leads), on the day of their departure, to assess if birds were targeting the nearest open water area available. The residence time, i.e. the number of locations in each cell per day, was used as a proxy for searching and diving activity [START_REF] Peron | Projected poleward shift of king penguins' (Aptenodytes patagonicus) foraging range at the Crozet Islands, southern Indian Ocean[END_REF][START_REF] Warwick-Evans | Time-inarea represents foraging activity in a wide-ranging pelagic forager[END_REF] and was related to environmental variables in the given cell.

Trophic niche

Analyses of the stable isotopes of nitrogen (δ 15 N) and carbon (δ 13 C) were used to estimate the trophic niche of Adélie penguins and the position of the prey they ingested.

Immediately after collection, the blood was centrifuged at 5000 rpm for 7 min to separate red blood cells from plasma that were both stored at -20°C. Plasma, with a turnover ranging from days to weeks, was used for examining diet during the incubation trip [START_REF] Hobson | Assessing Avian Diets Using Stable Isotopes I: Turnover of 13 C in Tissues[END_REF], Beaulieu et al. 2010[START_REF] Barquete | Stable isotope turnover in blood and claws: A case study in captive African Penguins[END_REF]. Plasma samples were first lyophilized for 48 h, powdered, weighed and then lipid-extracted. Indeed, as lipids are highly depleted in 13 C relative to other tissue components [START_REF] Deniro | Mechanism of carbon isotope fractionation associated with lipid synthesis[END_REF], they were removed using a chloroform-methanol solution (2:1 V:V derived from the Folch's procedure), at the LIttoral ENvironnement et Sociétés (LIENSs, La Rochelle, France). The samples were then encapsulated into tin capsules (0.2-0.4 mg) for stable isotope analysis trough isotope-ratio mass spectrometry. Isotopic analyses were performed with a Thermo Scientific Delta V Advantage mass spectrometer coupled to a Thermo Scientific Flash EA1112 elemental analyser. Results are presented in the usual δ notation relative to the deviation from standards (Vienna PeeDee Belemnite for δ 13 C and atmospheric N2 for δ 15 N), in parts per thousand (‰).

Replicate measurements of internal laboratory standards (acetanilide and peptone) indicated measurement errors <0.15‰ for both δ 13 C and δ 15 N values. The standard ellipse area corrected for small sample size (SEAC), an ellipse representing the probability of containing 40% of the data, and the Bayesian standard ellipses areas (SEAB) ('SIBER' package in R; [START_REF] Jackson | Comparing isotopic niche widths among and within communities: SIBER -Stable Isotope Bayesian Ellipses in R: Bayesian isotopic niche metrics[END_REF] were used to estimate the isotopic niches used by birds from the 2 colonies each year, with 2 × 10 5 replicates for quantifying the uncertainty and allowing robustness in our statistical analysis. We calculated the SEAC, and the area of overlap between the 40 and 95% isotopic niches from the SEAB using the maximum likelihood ellipses for the 2 groups.

Statistical analysis

The normality of trip duration, distance travelled, maximum distance, heading and isotopic ratios were tested with Shapiro-Wilks test. The equality of the variance was tested using Fisher's test. Finally, differences in the aforementioned parameters between colonies were tested using Student's t-test if normality and homogeneity of variances were respected, or Mann-Whitney test if not. Results are presented as means ± SD.

Two generalised linear mixed-effect models with a negative binomial distribution were built for each season, to compare the effects of both open water and bathymetric features on the residence time of birds. An interaction term was added to compare the effect of each feature on residence time according to the colony of origin. A random effect was added to the model to take into account the individual effect in the variability of the response variable. Models were built with the 'glmer.nb' function of the 'lme4' package in R [START_REF] Bates | Package 'lme4: Linear mixed-effects models using Eigen and S4[END_REF].

Statistical significance was defined at p ≤ 0.05. ANOVA was performed for each model to test the significance of the relationship among variables, with the 'Anova' function from the 'car' package. A Wald test of Type II was applied. Then, a post hoc test was performed to compare 2 by 2 the difference of the effect of features on bird residence time within each colony. It also allowed us to determine the significant difference between the 2 colonies for each feature. The post hoc tests were performed using the 'emmeans' function from the R package 'emmeans' [START_REF] Lenth | Emmeans: Estimated marginal means, aka least-squares means[END_REF].

Note that females and males were monitored in 2 different years and at 2 different times of the season (beginning of incubation for females in 2016, end of incubation for males in 2017), which precluded us from conducting a sex comparison of foraging parameters.

RESULTS

Sea-ice conditions

In 2016, the sea-ice extent was high during the females' first incubation trip (20 November to 22 December), extending on average 177.3 ± 36. There were no significant differences in trip duration, total distance covered during the trip and maximum distance to the colony between the 2 colonies for both females and males (Table 1). However, females from Ile des Pétrels headed north-west in their first day at sea (-25.20 ± 5.71°), whereas females from Cap Bienvenue headed significantly (p = 0.015) more north (7.17 ± 11.98°). Surprisingly, females from both sites did not head towards the nearest open water (Commonwealth Bay polynya in the north-east) or the nearest ice edge (also north-east) (Fig. 3A,B), except for one female from Cap Bienvenue that went in the direction of the closest edge point. Only 2 females from Cap Bienvenue crossed the Commonwealth Bay polynya before continuing their way to the ice edge.

Adélie penguin foraging behaviour

In 2017, males from the Ile des Pétrels headed towards the north-west (-18.08 ± 6.66°) on their first day at sea, while males from Cap Bienvenue headed north-east (32.01 ± 19.45°); this was significantly different between the 2 colonies (p < 0.001) (Fig. 3C,D). By doing so, the Ile des Pétrels males travelled in the direction of the closest open water point (the sea-ice edge). The date of 8 December was an exception, however: the Dumont d'Urville polynya appeared and was at that point the closest open water area from the colony, compared with the sea-ice edge which had moved north-east. Yet males departing on 8 December continued to head towards the north-west (Fig. 3C). All Cap Bienvenue males that started their trip on 6 December headed towards the north-east but the closest open water point was the sea-ice edge located north-west of the colony. On the next day, the sea-ice edge had moved, and the closest open water point was then located east of the colony (Fig. 3D). As males did not change their main direction of travel, they headed closer to the nearest open water area on that occasion, except for one male that went north-west, where the closest open water point had been located the day before (indicated on Fig. 3D). Before reaching the furthest point away from their respective colony, 6 males from the Ile des Pétrels and 2 from Cap Bienvenue crossed the Dumont d'Urville polynya. Although birds from a given colony did not leave on the same day, penguins of both sexes followed approximately similar directions, except for one male from Cap Bienvenue.

In 2016, females from both colonies spent similar residence times in each open water feature (Fig. 4A) and in each bathymetric feature (Fig. 4B). Within colonies, the Ile des Pétrels females had the lowest residence time in the Commonwealth Bay polynya, compared to the sea ice or at the ice edge (Fig. 4A, Table 2), and they favoured the deep regions of the continental slope between 3000 and 3500 m and the abyssal plain (Fig. 4B, Table 3). The residence time of Cap Bienvenue females was higher at the sea-ice edge and in the open sea than in any other open water features (Table 2). They also spent more time in the abyssal plain and the slope between 3000 and 3500 m and, to a lesser extent, the slope between 2000 and 2500 m (Table 3).

In 2017, the males from the Ile des Pétrels had a significantly higher residence time on sea ice (p = 0.036) and significantly lower residence time in open sea than males from Cap Bienvenue (Fig. 4C) but, like females, males from both colonies favoured the deep regions of the continental slope between 3000 and 3500 m and the abyssal plain (Fig. 4D). There was no difference in residence time between colonies for the other open water and bathymetric features (note that no bird from Cap Bienvenue used flaw leads). Within colonies, the Ile des Pétrels males had a significantly higher residence times at the sea-ice edge or on sea ice than in the open sea and the Dumont d'Urville polynya (Fig. 4C, Table 4), and less time at the continental shelf and the slope between 550 and 2000 m, with a clear preference for the abyssal plain (Fig. 4D, Table 5). Cap Bienvenue males spent a significantly higher residence time at the ice edge and in open water than on sea ice or in the Dumont d'Urville polynya (Fig. 4C, Table 4), and also significantly higher residence time in the bathymetry zones deeper than 2000 m (Fig. 4D, Table 5). In both colonies, males did not show any significant preference for the Commonwealth Bay and Dumont d'Urville polynyas, which were the least visited (Fig. 4C).

Trophic niche

No significant differences were detected between colonies with respect to carbon and nitrogen isotopic ratios, except for Cap Bienvenue males having significantly higher δ 13 C values than males from the Ile des Pétrels (Fig. 5, Table A1 in the Appendix). In 2016, SEAc values were 0.42 and 0.48‰ 2 (SEA = 0.37 and 0.43‰ 2 respectively) for females from Cap Bienvenue and Ile des Pétrels respectively. In 2017, values of SEAc were 0.81 and 0.33‰ 2 (SEA = 0.74 and 0.30 ‰ 2 ) for males from Cap Bienvenue and Ile des Pétrels respectively. For both sexes, both the SEAc (containing 40% of the data) of the 2 colonies or the SEAB (containing 95% of the data) overlapped (Fig. 5): the female SEAc and SEAB overlap areas represented 22 and 54% of the total area respectively; male SEAc and SEAB overlap areas represented 8 and 30% of the total area respectively. The Ile des Pétrels isotopic niche was greater than that of Cap Bienvenue for females in 2016 and the reverse for males in 2017. The δ 15 N values of female and male Adélie penguins were between that of the silverfish Pleuragramma antarctica (10.6 ± 0.3‰, according to [START_REF] Cherel | Isotopic niches of emperor and Adélie penguins in Adélie Land, Antarctica[END_REF] and that of the Antarctic and ice krill (Euphausia superba, 5.3 ± 0.5‰ and E. crystallorophias, 6.8 ± 0.7‰ respectively; Cherel 2008).

DISCUSSION

The foraging behaviour of female and male Adélie penguins from the Cap Bienvenue colony was monitored during the incubation period for the first time. The sea-ice habitat use and foraging activity of these birds were similar overall to that of birds from the well-studied Ile des Pétrels colony, 24 km away. Unexpectedly, Adélie penguins from both colonies did not target the closest available access to open water or zones with diffuse ice conditions when accessible. Instead, they headed to the distant ice edge and the continental slope regardless of the sea-ice extent.

Adélie penguins are central-place foragers, commuting trips between colonies and foraging habitats during their breeding season [START_REF] Ainley | The Adélie Penguin: Bellwether of Climate Change[END_REF]. The inherent prediction of the central place foraging theory is the occurrence of Storer-Ashmole's halo: central-place foragers tend to forage preferentially in close and suitable areas, leading to prey depletion around the breeding colony [START_REF] Elliott | Centralplace Foraging in an Arctic Seabird Provides Evidence for Storer-Ashmole's Halo[END_REF]). As such, we expected Adélie penguins to head towards the closest open water area border, where diffuse sea-ice conditions would allow them to dive and forage on abundant resources. Yet penguins did not target the polynyas that are present recurrently in the area or the leads which were opened during their trips, for food provisioning early in the season, nor did they specifically head towards them as transiting areas to reach the ice edge.

Note that because of the 15% concentration contour definition, the feature defined as 'sea ice' includes solid pack-ice, but also part of MIZ conditions which can extend farther than the 12.5 km buffer we applied around each open water area. Similarly, open sea or open water features (polynyas, flaw leads or sea-ice edge), i.e. concentration between 0 and 15% and the 12.5 km buffer, can also include diffuse MIZs [START_REF] Massom | Antarctic sea ice change and variability -Physical and ecological implications[END_REF].

Nonetheless, the icescape was very dynamic from one day to another, and it is thus difficult to evaluate how much the penguins anticipated changes occurring in their environment to find the most suitable areas at a given time.

During the late incubation period of 2017, polynyas and flaw leads often merged into open water, and this naturally explains why we noted few visits by males into those features. Some males briefly visited polynyas on their way to the slope (2 males from Cap Bienvenue and 6 from Ile des Pétrels), but did not specifically head towards those features when departing from the colony, as polynyas were mostly merged into open water, and the ice edge was the closest open water area. In contrast, in 2016 the Commonwealth Bay polynya remained open over the whole early incubation period, when sea-ice extent is often greater and more persistent than during the late incubation period. However, neither females from Cap Bienvenue nor the Ile des Pétrels headed to the nearest open water feature at the beginning of their trip. Only a small number of birds visited the polynyas while transiting to the sea-ice edge on their outward journeys (2 females from Cap Bienvenue in 2016 only), but they apparently did not spend time searching for food, as the low residence time suggests (although additional data would be needed to accurately infer the birds' behavioural state, such as hunting or resting). In fact, birds seemed to leave the colony following a fixed direction, independently of the icescape around the colony or departure date. This suggests a 'group effect', where birds may follow the specific direction taken by other birds leaving the colony before them, as seen in other seabird species [START_REF] Weimerskirch | Use of Social Information in Seabirds: Compass Rafts Indicate the Heading of Food Patches[END_REF][START_REF] Traisnel | Success comes with consistency in hard times: foraging repeatability relates to sex and breeding output in African penguins[END_REF]. In addition, primary production was highest in the diffuse ice located at the border of both the smaller ephemeral flaw leads and the larger persistent polynyas, resulting in limited availability of profitable areas for predators [START_REF] Massom | The distribution and formative processes of latent-heat polynyas in East Antarctica[END_REF][START_REF] Arrigo | Phytoplankton dynamics within 37 Antarctic coastal polynya systems[END_REF][START_REF] Labrousse | Coastal polynyas: Winter oases for subadult southern elephant seals in East Antarctica[END_REF]. During the winter, when the sea-ice extent is high, predators such as elephant seals Mirounga leonine and emperor penguins take advantage of open water accesses like polynyas and leads to feed [START_REF] Kirkwood | Seasonal change in the foraging ecology of emperor penguins on the Mawson Coast, Antarctica[END_REF][START_REF] Mcmahon | Winter distribution and abundance of crabeater seals off George V Land, East Antarctica[END_REF][START_REF] Labrousse | Coastal polynyas: Winter oases for subadult southern elephant seals in East Antarctica[END_REF]). As such, inter-specific competition can subsequently occur in those available but restricted foraging areas, leading to avoidance of those features by penguins. Although flaw leads can represent important transiting or foraging grounds, their ephemeral occurrence means they may not be targeted by predators due to their low predictability.

The extensive use of polynyas and leads by top predators during winter and early spring may substantially reduce prey availability in such restricted foraging grounds. At the beginning of the breeding season, when Adélie penguins go to sea for their first incubation trip, resources at the polynyas and leads MIZ may be nearly or completely exhausted. The primary production bloom generally occurs later in the season, at the beginning of the chickrearing period. When the incubation period starts, the development of phytoplankton, and consequently the concentration of prey such as krill or fish, may be low and therefore insufficient to sustain penguins [START_REF] Knox | Biology of the Southern Ocean[END_REF][START_REF] Massom | The distribution and formative processes of latent-heat polynyas in East Antarctica[END_REF], Beaulieu et al. 2010). Thus, penguins may forage near the polynyas only later in the season, during the critical chickrearing period. Indeed, during this part of their breeding cycle, Adélie penguins need to commute regularly to the nest to provision the chicks. As coastal polynyas are known to be highly dynamic and productive areas during the summer [START_REF] Massom | The distribution and formative processes of latent-heat polynyas in East Antarctica[END_REF][START_REF] Arrigo | Phytoplankton dynamics within 37 Antarctic coastal polynya systems[END_REF], Adélie penguins can use them as important foraging grounds for efficiently provisioning their chicks, especially when the sea-ice extent is high [START_REF] Ainley | The Adélie Penguin: Bellwether of Climate Change[END_REF][START_REF] Arrigo | Phytoplankton dynamics within 37 Antarctic coastal polynya systems[END_REF][START_REF] Clarke | Environmental conditions and life history constraints determine foraging range in breeding Adélie penguins[END_REF][START_REF] Widmann | Habitat use and sex-specific foraging behaviour of Adélie penguins throughout the breeding season in Adélie Land, East Antarctica[END_REF]. This idea finds support in the fact that most penguin colonies are located near coastal polynyas [START_REF] Massom | The distribution and formative processes of latent-heat polynyas in East Antarctica[END_REF][START_REF] Ainley | The Adélie Penguin: Bellwether of Climate Change[END_REF][START_REF] Arrigo | Phytoplankton dynamics within 37 Antarctic coastal polynya systems[END_REF][START_REF] Ainley | Birds and Antarctic sea ice[END_REF].

Incubating Adélie penguins from Lützow-Holm Bay (Kato et al. 2002) and Béchervaise Island [START_REF] Clarke | Environmental conditions and life history constraints determine foraging range in breeding Adélie penguins[END_REF], 2 other colonies of East Antarctica, also targeted the ice edge but headed straight to the polynya as a way to travel faster to the ice edge, as suggested by the start of diving activity and time elapsed in the trips in the former study, and by analysis of satellite tracking in the latter. Further studies coupling productivity around the polynya during the whole breeding season, associated with Adélie penguins' habitat selection for foraging activity during the chick-rearing period could be useful to assess of the importance of the polynya during that critical stage.

During the incubation, penguins must replenish their body reserves and preferentially head to the distant sea-ice edge, where prey concentration may be more predictable. The ice edge coincides with the continental slope and is a food-enriched area, where primary production first occurs between spring and summer [START_REF] Knox | Biology of the Southern Ocean[END_REF][START_REF] Massom | The distribution and formative processes of latent-heat polynyas in East Antarctica[END_REF][START_REF] Nicol | Ocean circulation off east Antarctica affects ecosystem structure and sea-ice extent[END_REF]. It represents an area of predictable food availability and abundance for penguins: the upwelling along the slope allows nutrients to be brought near the surface layer and, coupled with light, benefits the development of plankton [START_REF] Wienecke | Adélie penguin foraging behaviour and krill abundance along the Wilkes and Adélie land coasts, Antarctica[END_REF][START_REF] Ainley | The Adélie Penguin: Bellwether of Climate Change[END_REF], Koubbi et al. 2010). Following the fast during pre-laying and egg production for females and the incubation fast of males, birds may decide to head directly to zones where they anticipate prey to be available, as is seen in other (even tropical) seabird species [START_REF] Weimerskirch | Does Prey Capture Induce Area-Restricted Search? A Fine-Scale Study Using GPS in a Marine Predator, the Wandering Albatross[END_REF]).

The sea-ice edge and the slope correspond to the habitat range of Antarctic krill, one of the main prey of Adélie penguins, whereas Antarctic silverfish are generally found in more coastal waters where they feed on ice krill [START_REF] Cherel | Isotopic niches of emperor and Adélie penguins in Adélie Land, Antarctica[END_REF], Koubbi et al. 2010, 2011). Taking into account the δ 13 C gradient between inshore and offshore Antarctic waters [START_REF] Cherel | Isotopic niches of emperor and Adélie penguins in Adélie Land, Antarctica[END_REF], Beaulieu et al. 2010), the carbon isotopic ratios of Adélie penguin in our study suggest that females forage in more oceanic waters, as do males from the Ile des Pétrels, whereas Cap Bienvenue males forage in more neritic waters. However, our δ 13 C values encompass a small range and do not highlight high variations in the habitat of Adélie penguins' foraging activity.

Our nitrogen isotopic ratios were higher than the ca. 9‰ δ 15 N measured in adult whole blood samples and the 8.4-8.7‰ δ 15 N in the red blood cells of incubating males [START_REF] Cherel | Isotopic niches of emperor and Adélie penguins in Adélie Land, Antarctica[END_REF][START_REF] Cottin | Foraging strategies of male Adélie penguins during their first incubation trip in relation to environmental conditions[END_REF]. They also were only slightly lower than the isotopic ratio of Adélie penguin chicks of 10.2 ± 0.8‰ (whole blood analysis; Cherel 2008). Our samples consisted of delipidated plasma, for which δ 15 N values are about 0.8‰ higher than in the red blood cells (but not δ 13 C), as shown in king Aptenodytes patagonicus and gentoo penguins Pygoscelis papua [START_REF] Xavier | Sexual and individual foraging segregation in Gentoo penguins Pygoscelis papua from the Southern Ocean during an abnormal winter[END_REF][START_REF] Cherel | Stable isotopes document the winter foraging ecology of king penguins and highlight connectivity between subantarctic and Antarctic ecosystems[END_REF]. With this in mind, the difference in δ 15 N values between krill and Adélie penguins in our results corresponds to the natural enrichment factor of about 3.4‰ between trophic levels [START_REF] Post | Using stable isotopes to estimate trophic position: models, methods, and assumptions[END_REF], suggesting that these penguins were feeding principally on Antarctic krill, with males probably feeding on a small proportion of Antarctic silverfish and ice krill leading to an increase in their δ 15 N values. The results for ecological niche are thus consistent with the Adélie penguins' main prey habitats: penguins fed on lower trophic level species in oceanic waters (lower δ 15 N and δ 13 C) and on higher trophic level species in more coastal waters (higher δ 15 N and δ 13 C). The persistent ice edge near the slope may have favoured the development of the trophic food web in 2016, with high concentrations of Antarctic krill. Females heading to the ice edge located near the slope may have benefited from a high concentration of Antarctic krill in this area. In 2017, the greater sea-ice retreat meant that incubating males had access to a large array of possibilities, including Antarctic krill in the ice edge and continental slope waters but also ice krill and Antarctic silverfish in more neritic waters. Although Adélie penguins favoured the ice edge and open, deep water on the continental slopes, they also spent some time in flaw leads located on the continental shelf, where they may have found other prey such as Antarctic silverfish. Isotopic niches from females from both colonies overlapped, and despite significant difference in δ 13 C values, males from both colonies shared a similar isotopic niche.

The low variation in δ 15 N values highlighted the low diversity in prey consumed at both sites.

Moreover, our results are consistent with other studies revealing the intermediate generalist foraging strategy of Adélie penguins, where they feed predominantly on krill during the incubation period but also on fish during the chick-rearing period, both in East Antarctica [START_REF] Wienecke | Adélie penguin foraging behaviour and krill abundance along the Wilkes and Adélie land coasts, Antarctica[END_REF][START_REF] Cottin | Foraging strategies of male Adélie penguins during their first incubation trip in relation to environmental conditions[END_REF], and the Peninsula [START_REF] Herman | Seasonal consistency and individual variation in foraging strategies differ among and within Pygoscelis penguin species in the Antarctic Peninsula region[END_REF].

The Commission for the Conservation of Antarctic Marine Living Resources (CCAMLR) selected the Adélie penguin as a sentinel species in Antarctica to assess ecosystem variation, and through their krill consumption, to manage krill fisheries [START_REF] Southwell | Large-scale population assessment informs conservation management for seabirds in Antarctica and the Southern Ocean: a case study of Adélie penguins[END_REF]. Several well-studied colonies are thus used as outposts around the Antarctic but often little is known of the situation in neighbouring colonies, apart from occasional direct or satellite counts [START_REF] Fretwell | Penguins from space: faecal stains reveal the location of emperor penguin colonies[END_REF][START_REF] Lynch | First global census of the Adélie Penguin[END_REF]. For instance, the Ile des Pétrels Adélie penguin colony was recently affected by 2 massive breeding failures [START_REF] Barbraud | Extreme ecological response of a seabird community to unprecedented sea ice cover[END_REF], Ropert-Coudert et al. 2018) but there is no information on what happened 24 km away in the Adélie penguin colony at Cap Bienvenue. Was the breeding success equally affected in Cap Bienvenue? Did the birds there adjust their foraging activity to respond to the extensive sea ice differently? Here, our data suggest that foraging activity is consistent across these 2 colonies, at least during the incubation phase. The isotopic niche shared by both colonies and the low variation in prey diversity, in accordance with other studies, highlights the importance of their critical foraging habitat, i.e. diffuse sea-ice zones offering abundant resources such as Antarctic krill but also access to Antarctic silverfish and ice krill. This is of course only a first attempt at examining consistency across colonies of the Terre Adélie region. In a context where it is logistically difficult to extend long-term monitoring studies on the foraging and demographic activities of species across several remote sites, our results highlight similar foraging behaviour in 2 colonies and suggest that one colony could be used for describing others nearby. While we cannot extrapolate further, this gives some evidence to suggest common measures of conservation that the CCAMLR could put forward in the region to protect the different Adélie penguin colonies and their habitats. and 95% Bayesian standard ellipses areas (dashed line) for each colony. Individual values are given as small circles with the same colour code (see also Table A1). Mean Adélie penguin prey isotopic ratios (±SD) are indicated in black, following [START_REF] Cherel | Isotopic niches of emperor and Adélie penguins in Adélie Land, Antarctica[END_REF].

Supplementary material

Table S1. Carbon (δ 13 C), nitrogen (δ 15 N) and carbon:nitrogen (C/N) isotopic ratios for Adélie penguin females (2016-2017) and males (2017-2018) in both colonies. No significant difference was detected between colonies for their δ 15 N values within each year (females: 9.5 ± 0.3 and 9.21 ± 0.4‰ for Cap Bienvenue and Ile des Pétrels respectively, p > 0.05; males: 10.3 ± 0.6 and 9.91 ± 0.4‰ for Cap Bienvenue and Ile des Pétrels respectively, p > 0.05). No difference was detected for females δ 13 C values between the 2 colonies (25.1 ± 0.4 and -24.9 ± 0.3‰ for Cap Bienvenue and the Ile des Pétrels), but the δ 13 C values were significantly higher for males from Cap Bienvenue than for those from the Ile des Pétrels in 2017 (-24.5 ± 0.4 and -25.0 ± 0.3 respectively, p = 0.005).

  Dumont d'Urville polynya in the vicinity of the Ile des Pétrels (around the centroid located 66.1° S, 139.6° E, slightly west from the colony), the Commonwealth Bay polynya (around the centroid located 66.2° S, 142° E, east of Cap Bienvenue) and the Mertz polynya (around the centroid located 66.6° S, 147.2° E and opening around the Mertz glacier). Finally, flaw leads with a surface equivalent or larger than the smallest polynya recorded during the season were extracted with 50% of sea-ice concentration to account for larger portion of ice-free areas (Videos S1 & S2). Each cell was assigned a binary value: 0 if the cell corresponded to an open water fraction (flaw lead/polynya or open sea beyond the ice-edge) or 1 otherwise. To ascertain that birds were genuinely on sea ice with limited access to open water, and to account for the MIZ around each open water feature, we attributed a value of 0 to cells located in a 12.5 km buffer (2 rows of cells) around the contour of each open water feature.

  2 and 176.1 ± 33.0 km from the Ile des Pétrels and Cap Bienvenue colonies respectively. The large polynya of the Commonwealth Bay (1794.0 ± 1006.0 km 2 ) was 63.9 ± 5.4 km from the Cap Bienvenue colony and, like the Mertz polynya, remained present during the time the females were at sea. In contrast, the Dumont d'Urville polynya did not open; the closest open water to the Ile des Pétrels-Commonwealth Bay polynya-was 80.9 ± 3.9 km away. Females from both sites had access to flaw leads opening along the shelf edge during their trips (Videos S1, S3 & S4).In 2017, sea-ice conditions varied greatly over the incubation period of the males (4-29 December): large open water areas were already available at the start of the male incubation trip period, followed by a marked sea-ice retreat during which polynyas alternated with open sea. Sea-ice extent was on average 127.8 ± 89.0 km from Cap Bienvenue and 131.9 ± 83.6 km from the Ile des Pétrels over the study period. When it appeared (1 d only during the period), the 1057.2 km 2 Commonwealth Bay polynya was 91.4 km east of Ile des Pétrels, and 81.2 km east of Cap Bienvenue. During the 15 d it was opened, the Dumont d'Urville polynya was located on average 52.5 ± 20.4 km west of Cap Bienvenue and 39.4 ± 14.7 km of the Ile des Pétrels. Its area ranged from 577.2 to 4460.3 km 2 during this period, with an averaged area of 2841.6 ± 1136.9 km 2 (Videos S2, S5 & S6).

Females

  left the Ile des Pétrels between 20-24 November 2016 and returned between 9-22 December 2016 (Figs. 2A, Video S3). Females left Cap Bienvenue between 22-25 November 2016 and returned between 7-8 December 2016 (Figs. 2B, Video S4). In total, 11 functional GPS units were recovered in each colony, but only 20 tracks with 60% or more of the trip covered (10 in each colony) were kept for analyses. Males left the Ile des Pétrels between 4-8 December 2017 and returned between 18-24 December 2017 (Figs. 2C, Video S5). Males left Cap Bienvenue on 6-7 December 2017 and returned between 20-29 December 2017 (Figs. 2D, Video S6).

Figure 1 .

 1 Figure 1. Global view of Antarctica (left) with the Terre Adélie sector (red triangle) and a close-up (right) of the coastline in Terre Adélie (black polygon) within our study area (larger dashed-blue polygon), showing the 2 studied Adélie penguin colonies: Ile des Pétrels in green, and Cap Bienvenue in orange. Note that the colony colour-code is used in all figures. Other surrounding colonies are indicated with black dots.

Figure 2 .

 2 Figure 2. GPS-tagged Adélie penguin tracks, showing female movements in 2016 from (A) Ile des Pétrels and (B) Cap Bienvenue, and male movements in 2017 from (C) Ile des Pétrels and (D) Cap Bienvenue (see Fig. 1 for colony locations). Black lines: bathymetric limit of the continental shelf and slope.

Fig. 3 .

 3 Fig. 3. Directions and distance taken by Adélie penguins (solid lines) from (A,C) the Ile des Pétrels, and (B,D) Cap Bienvenue after a 24 h trip (females and males departed their colonies between 20-28 Nov 2016 and 4-8 Dec 2017 respectively), together with the direction of the closest ice edge (dotted navy lines) and, when opened and closer to the colonies, the closest polynya or leads (dotted brown lines) from each colony on the day of the birds' departure. Note that in (C), the ice edge moved east of the Iles des Pétrels colony on 8 December and a polynya opened (indicated in red). Note that in (D), the ice edge moved on 7 December, and one male from Cap Bienvenue took the opposite direction.

Figure 4 .

 4 Figure 4. Average (±CI) residence time in each (A,C) open water feature and (B,D) topographic feature for Adélie penguin (A,B) females and (C,D) males from the Ile des Pétrels colony (green bars) and Cap Bienvenue colony (orange bars). DDU: Dumont d'Urville polynya; CB Commonwealth Bay polynya. Significant differences between colonies are indicated as *p < 0.05; ***p < 0.001. Red crosses: residence time was not calculated for the specific feature (feature not present or not visited by penguins). Differences between each feature within colony are not indicated on the figure for clarity reasons but results can be found in Tables 2-4.

Figure 5 .

 5 Figure 5. Mean (±SD) carbon (δ 13 C) and nitrogen (δ 15 N) isotopic ratios for Adélie penguin (A) females and (B) males from the Ile des Pétrels (green) and Cap Bienvenue (orange) colonies, and representation of the 40% standard ellipse area corrected for small sample size (solid line)

Table 1 .

 1 Number of Adélie penguin GPS tracks analysed, mean trip duration, total trip distance and maximum distance to colony for females from the Ile des Pétrels colony in 2016 and males from the Cap Bienvenue colony in 2017, and result of the Student's t-tests comparing trip parameters between colonies.

	Sex: year	Colony	Trips	Trip duration (d) Mean ± SD p-value	Total distance (km) Mean ± SD p-value	Maximum distance (km) Mean ± SD p-value
	Females: 2016	Ile des Pétrels	10	21.28 ± 4.90	0.260	779.18 ± 148.78	0.747	237.07 ± 35.53	0.936
		Cap Bienvenue	10	19.00 ± 3.78		748.20 ± 170.70		235.57 ± 46.38
	Males: 2017	Ile des Pétrels	10	17.06 ± 2.22	0.268	751.72 ± 91.76	0.285	270.92 ± 37.53	0.099
		Cap Bienvenue	14	16.01 ± 2.25		810.00 ± 168.00		242.59 ± 41.24

Table 2 .

 2 Pairwise comparison of open water feature effects on residence time of Adélie penguin females from each colony in 2016, with the estimates, standard error (SE) and p-value corresponding to the significance of the effect of one feature on the other. Significant differences

	between open water feature effects are indicated as *p < 0.05; **p < 0.01; ***p < 0.001. Each reference open water feature is compared to the
	other levels. CB polynya: Commonwealth Bay polynya.
	Reference	Compared	Ile des Pétrels			Cap Bienvenue
	feature	feature	Estimate	SE	p-value	Estimate	SE	p-value
	Ice edge	Open sea	0.166	0.124	0.943	-0.158	0.105	0.888
		Sea ice	0.065	0.054	0.972	0.194	0.054	0.012*
		Flaw leads	0.052	0.074	0.999	0.356	0.084	0.001***
		CB polynya	0.496	0.101	<0.001***	0.835	0.100	<0.001***
	Open sea	Sea ice	-0.102	0.121	0.998	0.353	0.102	0.020*
		Flaw leads	-0.114	0.131	0.997	0.514	0.121	<0.001***
		CB polynya	0.330	0.146	0.416	0.993	0.129	<0.001***
	Sea ice	Flaw leads	-0.012	0.0363	1.000	0.161	0.075	0.497
		CB polynya	0.431	0.096	<0.001***	0.640	0.096	<0.001***
	Flaw leads	CB polynya	0.167	0.145	0.979	0.479	0.116	0.001**

Acknowledgements. This study was approved by the ethics committee of the French Polar Institute Paul Emile Victor (IPEV). We are grateful for all financial and logistic support: the IPEV (program 1091, resp. Y.R.C. & T.R.), the WWF-UK through R. Downie, the Zone Atelier Antarctique et Subantarctique -LTER France of the CNRS, the Terre Australes et Antarctiques Françaises (TAAF). This study is a contribution to the program SENSEI funded by the BNP Paribas Foundation. We thank M. Brault-Favrou and G. Guillou from LIENSs for stable isotope analysis help and measurements. The IUF (Institut Universitaire de France) is acknowledged for its support to P.B. as a Senior Member. We also thank 2 reviewers for their help in improving an earlier version of the manuscript.

Table 3. Pairwise comparison of bathymetric feature effects on residence time of Adélie penguin females from each colony in 2016, with the estimates, standard error (SE) and p-value corresponding to the significance of the effect of one feature on the other. Significant differences between bathymetric feature effects are indicated as *p < 0.05; **p < 0.01; ***p < 0.001. Each reference bathymetric feature is compared to the other levels.


Yohan Chatelain

Eric Petit

Pablo De Oliveira Castro

Ghislain Lartigue

David Defour

Automatic exploration of reduced floating-point representations in iterative methods

With the ever-increasing need for computation of scientific applications, new application domains, and major energy constraints, the landscape of floating-point computation is changing. New floatingpoint representation formats are emerging and there is a need for tools to simulate their impact in legacy codes. In this paper, we propose an automatic tool to evaluate the effect of adapting the floating point precision for each operation over time, which is particularly useful in iterative schemes. We present a backend to emulate any IEEE-754 floating-point operation in lower precision. We tested the numerical errors resilience of our solutions thanks to Monte Carlo Arithmetic and demonstrated the effectiveness of this methodology on YALES2, a large Combustion-CFD HPC code, by achieving 28% to 67% reduction in communication volume by lowering precision.

Introduction

Representing infinite real numbers on a finite machine format exposes complex trade-off: IEEE-754 floating-point (FP) single and double are widely used standardized formats which have empirically proved their efficiency. Today's numerical algorithms conveniently rely on double precision FP format. Numerical formats outside IEEE-754 single and double FP are largely unexplored in hardware and software design for HPC.

With new application domains such as machine learning, FP arithmetic used in general purpose processor is entering a new burst of evolution as the IEEE-754 single and double FP are not necessarily the best choices for these problems. Novel hardware and software solutions are explored: variable precision formats (e.g. on FPGAs [START_REF] De Dinechin | Designing custom arithmetic data paths with FloPoCo[END_REF]), new vector instructions such (e.g. Intel Vector Neural Network Instruction), novel FP representations (e.g. BFloat [START_REF] Das | Mixed precision training of convolutional neural networks using integer operations[END_REF], posits [START_REF] Gustafson | Beating floating point at its own game: Posit arithmetic[END_REF], fpanr [START_REF] Defour | Fp-anr: A representation format to handle floating-point cancellation at run-time[END_REF]), and libraries for approximate computing [START_REF]Intel vml[END_REF].

In this paper we propose a methodology to explore alternative representations of hardware and software FP. Our first application is the widely used Computational Fluid Mechanics (CFD) solver YALES2 operated in many recent HPC simulations [START_REF] Legrand | A multi-grid framework for the extraction of large-scale vortices in large-eddy simulation[END_REF][START_REF] Benard | Large-eddy simulation of wind turbines wakes including geometrical effects[END_REF][START_REF] Boulet | Modeling of conjugate heat transfer in a kerosene/air spray flame used for aeronautical fire resistance tests[END_REF][START_REF] Benard | Mesh adaptation for large-eddy simulations in complex geometries[END_REF][START_REF] Guedot | Design of implicit high-order filters on unstructured grids for the identification of large-scale features in large-eddy simulation and application to a swirl burner[END_REF].

Variable precision can be harnessed at the application, compiler, and architecture levels. Our exploration goes beyond traditional mixed precision (single and double format). Due to hardware memory constraints of general purpose processors, the target format can only be represented with 8,16,32 or 64 bits. However, accuracy can be lowered to any given precision to limit the cost of computation and energy. Finally, to go further in the optimization, one may use architectural support, like FPGAs, where precision is fine-tuned up to the bit level [START_REF] De Dinechin | Designing custom arithmetic data paths with FloPoCo[END_REF].

The contributions of this paper are:

-An empirical methodology integrated in Verificarlo [START_REF] Chatelain | Veritracer: Context-enriched tracer for floating-point arithmetic analysis[END_REF][START_REF] Denis | Verificarlo: Checking floating point accuracy through monte carlo arithmetic[END_REF] to automatically lower precision in iterative algorithms while maintaining an user defined accuracy criterion. -A VPREC backend to Verificarlo to emulate variable FP representations with a [START_REF] Anzt | Adaptive precision in block-jacobi preconditioning for iterative sparse linear system solvers[END_REF][START_REF] Das | Mixed precision training of convolutional neural networks using integer operations[END_REF]-bits exponent and [0, 52]-bits pseudo-mantissa. -Integration of the temporal dimension in the exploration of FP precision.

Precisions are not fixed for a variable or code section, but adapt as time passes.

-A validation step based on stochastic arithmetic [START_REF] Sohier | Confidence Intervals for Stochastic Arithmetic[END_REF] to increase robustness to rounding and cancellation errors. -An evaluation of the validity and the performance gains of the proposed methodology on an actual implementation demonstrating runtime savings in an industrial use case.

Motivating example

Let us consider as a simple example the Newton-Raphson method which finds the root x of a real-valued function f such that f (x ) = 0. Starting from an initial guess x 0 , the method iteratively computes x k+1 = x k -f (x k )/f (x k ) until the relative error between two successive iterations is below a given threshold. At iteration k, the error k and the number of significant digits s β k in base β are defined as:

k = x -x k+1 x s β k = -log β ( k )
The speed of convergence of this method is quadratic [START_REF] Higham | Accuracy and stability of numerical algorithms[END_REF], which means that the number of significant digits doubles at each iteration. Table 1 shows the evolution of x k and s k when computing the inverse of π (f (x) = 1/x -π) with a stopping threshold set to 10 -15 . The insignificant digits are highlighted in gray.

In the first iterations, most digits are incorrect, hinting that a low precision for the evaluation of f is enough. We used the methodology proposed in this paper and described in section 3 to explore the impact of lowering the precision with the VPREC backend for Verificarlo presented in section 3.1.

Our methodology automatically finds for each iteration k, a reduced precision p k ; the p k are chosen such that the overall convergence speed is not degraded. Convergence speed Since the first seven iterations require less than 24 bits of precision, they can be executed in single precision (IEEE-754 binary32). To validate the solution found by VPREC, we run a mixed-precision version of the Newton-Raphson scheme where the first seven iterations use the binary32 format. We note that the convergence speed and final result are almost identical to the full binary64 version. While more efficient, the solution found by VPREC has only been validated for a given input data set and its resiliency has to be examined as discussed in section 4.2.

s 2 k IEEE s 2 k VPREC

Exploring variable precision

In large computational scientific codes, basic numerical computation functions are used many times in various places. Each call of the function may require a different numerical precision. Furthermore, as seen in section 2, the same code section may require a different precision over time. To explore the temporal dimension of numerical precision, we use two components presented in the remaining of this section:

-A variable precision backend for Verificarlo: VPREC, detailed in section 3.1; -A heuristic optimizer to automatically minimize the precision configuration, while ensuring accuracy and convergence as described in section 3.2. Verificarlo [START_REF] Denis | Verificarlo: Checking floating point accuracy through monte carlo arithmetic[END_REF] is an open-source tool for FP interposition whose workflow is described in figure 2. The LLVM front-end replaces every FP operations by a call to the Verificarlo interface. After compilation, the program is dynamically linked against various backends [START_REF] Chatelain | Veritracer: Context-enriched tracer for floating-point arithmetic analysis[END_REF][START_REF] Denis | Verificarlo: Checking floating point accuracy through monte carlo arithmetic[END_REF]. Interposing FP operations at the compiler level allows to capture the compiler optimization effects on the generated FP operation flow. Furthermore, it reduces the interposition overhead by optimizing its integration with the original code.

Verificarlo and the VPREC backend

The VPREC backend simulates FP formats that can fit into the IEEE-754 double precision format, avoiding the complex engineering process to implement a shadow memory [START_REF] Serebryany | Addresssanitizer: A fast address sanity checker[END_REF].

As illustrated in figure 3, the current implementation of VPREC allows modifying the bit length of the exponent r ∈ [START_REF] Anzt | Adaptive precision in block-jacobi preconditioning for iterative sparse linear system solvers[END_REF][START_REF] Das | Mixed precision training of convolutional neural networks using integer operations[END_REF] p ∈ [0, 52]. The explored format can be converted back and forth to double without loss of precision. At each instrumented floating-point operation, VPREC rounds the operands in (r, p), converts them to double, performs the operation in double precision, rounds the result using the Round Ties to Even mode, and stores it as a 64-bit number. This process presents two advantages:

-VPREC operands can be converted to double to use hardware operators to perform VPREC operations with low overhead, including special values (subnormal numbers, NaN and ±∞) related to the user defined format (r, p) -After rounding, converting the result back to double enables graceful degradation if some external libraries are not instrumented.

The user should note that in some rare cases, our implementation suffers from double rounding issue [START_REF] Boldo | When double rounding is odd[END_REF], when the first rounding done by the hardware at 53 bits impacts the second rounding to the required precision p. In practice, these cases, which occurs for numbers close to the midpoint of two p bits floating-point numbers, have no significant impact on our experiments.

The VPREC backend requires a single execution of the program and Verificarlo supports MPI. Therefore, we observe a reasonable overhead on full scale parallel applications ranging from 2.6× to 16.8× for very FP intensive codes.

Piecewise constant exploration heuristic

Let consider an iterative program with n iterations. VPREC simulates the effect of using FP numbers represented on a p k -bit mantissa and r k -bit exponent at iteration k. From now on, the sequence of values [(p 0 , r 0 ), . . . , (p n , r n )] represents one VPREC run called a configuration.

When the precision is too low, the program execution fails: either it does not converge or it produces wrong results. We are only interested in valid configurations that preserve convergence and accuracy according to user knowledge. In our experiment (sec. 4), the validation function checks that the number of iterations and the final results are within a threshold of some reference values.

Reducing the precision can be seen as introducing numerical errors terms in the computation. Determining how and where to distribute the errors across the iterations is an optimization problem with a large number of valid configurations. Exploring the whole search-space is too costly on any non-trivial application, we have to rely on heuristics.

We propose the piecewise search heuristic with the three following design principles:

1. Configurations where the precision changes slowly over time are preferable to configurations which quickly oscillate between a low and high precision. 2. Precision lowering should be distributed among all iterations. 3. Early iterations are generally more robust to error. Therefore we foster configurations with lower precision in early iterations compared to late iterations. To enforce the first and second principles, we follow a top-down approach. The solution is modeled by a piecewise constant function that is progressively refined. Figure 4 illustrates the first three steps of the piecewise approach on the Newton-Raphson problem. For the sake of simplicity, we are solely focusing on lowering the mantissa size p k . However, the method can simultaneously deals with the mantissa and exponent size. Initially, the piecewise constant function has a single domain (marked 0) that spawns all the iterations [0, n). The valid lowest precision, corresponding to 40, is found by dichotomy on the precision domain between 1 and 53. In the second step, the domain is split in two subdomains (marked 1 and 2). To enforce the third principle, the precision in the left domain (marked 1) is lowered in a greedy manner, while keeping the maximal precision found in the previous step for the right domain (marked 2). Once the lowest precision for domain 1 is identified, we lower the precision in domain 2. This ends the second step and produces the blue piecewise function. This process continues recursively. This approach guarantees by construction that the piecewise function at step i is an upper bound of the function at step i + 1 and progressively refines the solution following the first principle. The exploration is breadth-first to evenly distribute the reduction in precision, according to the second principle.

Large scale study validation on YALES2

YALES2 is a parallel CFD library that aims at solving the unsteady Navier-Stokes equations in the low-Mach number approximation for multiphase and reactive flows [START_REF] Moureau | Design of a massively parallel CFD code for complex geometries[END_REF]. It efficiently handles unstructured meshes with several billions of elements, enabling the Direct Numerical Simulation of laboratory and industrial configurations. A projection method [START_REF] Chorin | Numerical solution of the navier-stokes equations[END_REF][START_REF] Pierce | Progress-variable approach for large-eddy simulation of nonpremixed turbulent combustion[END_REF] enforces the mass-conservation constraint on the flow thanks to the resolution of a Poisson equation at each timestep. In the HPC context, this is usually achieved with Krylov methods and we focus here on the Deflated Preconditioned Conjugate Gradient (DPCG) [START_REF] Nicolaides | Deflation of conjugate gradients with applications to boundary value problems[END_REF][START_REF] Malandain | Massively parallel simulation of low-Mach number turbulent flows[END_REF][START_REF] Malandain | Optimization of the deflated conjugate gradient algorithm for the solving of elliptic equations on massively parallel machines[END_REF] implemented in YALES2. In this algorithm, a coarse grid is built from the fine mesh by merging a fixed number of elements together in super-cells (this procedure is conceptually similar to the multi-grid approach [START_REF] Dendy | Black box multigrid[END_REF]). The general principle of deflation is the following. The coarse grid is used to converge the low frequency eigenmodes of the solution which represent the long range interactions of the Poisson equation. This requires much less work than performing a classical CG on the fine mesh which is only used to obtain the remaining high frequencies in a small number of iterations.

Numerically, the deflated operator is solved in iterdef iterations using a usual PCG method such that the convergence criteria convcrit is met. The solution is then expanded and injected into the main PCG loop on the finer grid. This whole process is repeated until the maxnorm of the fine grid residual is below a threshold. In both CG solvers (on coarse and fine grids), the system is preconditioned by the inverse of the diagonal. In all experiments, we constraint the total number of iterations performed, iterdef k , by the algorithm to be below 1% of additional iterations compared to the original.

The representative use-case we focus on is the PRECCINSTA burner [START_REF] Lartigue | Experimental and numerical investigation of self-excited combustion oscillations in a scaled gas turbine combustor[END_REF][START_REF] Benard | Large-eddy simulation of the lean-premixed preccinsta burner with wall heat loss[END_REF]. It is a well-known lab-scale burner used to validate combustion CFD solvers. We use 3 different mesh sizes of 1.75 million, 40 million and 870 million of tetrahedral elements. For all configurations, the super-cell size in the coarse grid was set to 500 elements. We set the max norm of residual convergence criteria to 10 -8 .

To reduce the search space, we consider two sets of functions. The first set, named deflated, is the set of functions used on the coarse grid to solve the deflated operator. The second set, named all, contained all the functions used to solve the fine grid operator.

Adaptive precision algorithm experiment on DPCG

In this section, we apply our VPREC tool on the 1.75M mesh case to explore valid variable precision implementation. In order to use true single precision, we statically set the exponent range to 8-bits in VPREC exploration.

Figure 5 shows the result of the exploration. In both graphics the x-axis represents the number of iterations on the fine grid. In the top plot, the right y-axis represents, on the same scale, the norm maxnorm of the residual between two successive iterations and the convergence criterion convcrit of the deflated operator. The left y-axis represents iterdef , the number of iterations on the and the entire code (right). On both plots, we can see that our reduced precision solution follows the reference IEEE convergence profile. deflated operator. In the bottom plot, the y-axis represents the virtual precision used to compute the given iteration on x-axis.

Figure 5 (left) shows that less than 23 bits of precision are required for the deflated operator on the 1.75 million elements mesh, with an average precision of 18 bits. Therefore, the deflated operator can be computed with binary32, resulting in a mixed-precision implementation detailed in section 4.3.

Figure 5 (right) shows the results of the proposed explorations on both coarse and fine grid at the same time. We notice that, contrary to the deflated experiment, the required precision increases over iterations. This is expected because the solver needs more and more precision to converge as it refines the solution. Surprisingly, the required precision drops at iteration 50 from 34 bits to 21 bits.

We cannot yet explain this sudden drop, more investigations are needed.

Validating resiliency to round-off errors

In the previous sections, we demonstrated that YALES2's DPCG converges with a lower precision format. This result is only valid with the particular rounding mode used by VPREC and is sensitive to the input dataset. In a realistic setup, small rounding errors may occur when performing FP operations with a different software representation and/or hardware.

Monte Carlo Arithmetic (MCA) is a stochastic method to model round-off errors by artificially introducing noise within computations and performing Monte Carlo sampling. For the theoretical underpinnings, readers may refer to [START_REF] Denis | Verificarlo: Checking floating point accuracy through monte carlo arithmetic[END_REF][START_REF] Parker | Monte carlo arithmetic: exploiting randomness in floating-point arithmetic[END_REF]. MCA is able to simulate rounding errors at a given virtual precision. We use MCA as a second step of our VPREC analysis to find a configuration that is resistant to round-off errors. We model this process as a Bernoulli trial. We run 29 MCA samples to simulate the effect of rounding errors. If any one of the samples fail to converge, we conclude that the solution is not robust to round-off errors. On the other hand, if all the 29 samples converge we conclude, thanks to the confidence intervals introduced in [START_REF] Sohier | Confidence Intervals for Stochastic Arithmetic[END_REF], that the probability of convergence in the presence of round-off errors is 90% with a 0.95 confidence level.

Figure 6 shows that the VPREC solution found in the previous section is robust and converges for all the samples. Since the solution is very close to the binary32 precision, our objective is to achieve a robust binary32 configuration. The binary32 constant-precision configuration represented by the light red envelop in figure 6 converges in 57 to 63 iterations in the presence of round-off errors for all samples. This demonstrates that it is possible to safely rewrite the coarse grid operator of DPCG in binary32.

Evaluating mixed-precision version

The deflated operator of DPCG can be computed within the binary32 format for most iterations as shown in previous sections. To validate the results, we compiled a mixed-version of YALES2 where the deflated operator can be executed either in binary32 or binary64 format.

We evaluated the mixed precision version on the three different grids of PRECCINSTA and 10 -9 convergence criteria. We limit the exploration algo-rithm to double and single precision since we are not running on variable precision hardware.

We use CRIANN cluster constituted of 366 bisocket Intel Xeon E5-2680 nodes and Intel Omnipath interconnect. We gather statistics using Intel IPM interface for Intel MPI.

As predicted by our methodology, the computation converges and all versions satisfy all accuracy constraints on the results. However, we noticed that larger experiments require extra initial double precision iterations on the deflated grid. For examples, respectively two and four extra double precision iterations are necessary for the 40M and 870M mesh. This is coherent with the observations of Cools et al. [START_REF] Cools | Analysis of rounding error accumulation in Conjugate Gradients to improve the maximal attainable accuracy of pipelined CG[END_REF] about the importance of being precise in the first iteration of a CG recurrence:

We noticed as well, that on these larger cases it is necessary to switch for the deflated grid from single to double precision when the deflated convergence criteria is difficult to reach with single precision ∼ 10 -8 . This effect did not appear on the smaller case with 500 elements per group. Our hypothesis is that the granularity difference between the two grids level is larger on the small mesh and therefore the small errors on the coarse grid iteration are less impacting on the fine grid iterations [START_REF] Malandain | Massively parallel simulation of low-Mach number turbulent flows[END_REF][START_REF] Malandain | Optimization of the deflated conjugate gradient algorithm for the solving of elliptic equations on massively parallel machines[END_REF].

We measure a 28% to 67% reduction in the communication volume. The energy gain can be estimated to be linearly related to this volume gain with the simple model proposed in [START_REF] Anzt | Adaptive precision in block-jacobi preconditioning for iterative sparse linear system solvers[END_REF].

Since DPCG is mostly bounded by communication latency, the performance gain is limited when the number of processor grows for a given size falling from 28% speedup to -2% slowdown on critical strong scaling experiments. However, according to these results and end-user usage of the code, the expected speedup for daily usage will be in the 10% range.

Related works and Background

Many tools and strategies have been developed for lowering precision in codes. For HPC purposes, the challenge is to have fast and scalable tools for addressing real world applications. A comparison with our methodology is presented in table 2. Most of the tools focus on the spatial dimension while we investigate the temporal dimension as well. In addition, most of them focus on the mixedprecision exploration while we provide a more in-depth analysis by working at a bit level. Evaluating the resiliency to rounding errors is only proposed by Verificarlo and Promise although FlexFloat [START_REF] Ho | Efficient floating point precision tuning for approximate computing[END_REF] and fpPrecisionTuning [START_REF] Tagliavini | A transprecision floating-point platform for ultralow power computing[END_REF] propose statistic optimization according to input data ranges. However, they require the re-implementation of code to adopt the specific libraries.

Daisy [START_REF] Darulova | Sound mixed-precision optimization with rewriting[END_REF], Herbie [START_REF] Panchekha | Automatically improving accuracy for floating point expressions[END_REF] and STOKE [START_REF] Schkufza | Stochastic optimization of floating-point programs with tunable precision[END_REF] are optimizing precision or accuracy by rewriting formulas. Most of them provides high level of guarantees, however they all face scalability issues. Some authors propose adaptive schemes for specific linear algebra algorithms. Anzt et al. [START_REF] Anzt | Adaptive precision in block-jacobi preconditioning for iterative sparse linear system solvers[END_REF][START_REF] Haidar | Harnessing gpu tensor cores for fast fp16 arithmetic to speed up mixed-precision iterative refinement solvers[END_REF] propose an adaptive precision version of the Block-Jacobi preconditioner. Authors store data at low precision by truncating bits while computations remain in double precision. The change of format is guided by the condition number and the data range. The authors estimate energy gains with predictive models with the underlying hypothesis that the cost depend linearly on the bit length of the data. These methods are because they use mathematical properties of numerical schemes for adapting precision over iterations. However, the authors focused on small program sections based on their knowledge at high engineering cost. Therefore, their results are restricted to a class of specific algorithms unlike our method which provides a broader exploration tool. Of course, educated developers are still required to take the final decision to use lower precision provided by our VPREC tool.

Conclusion

Reducing communication volume and computation cost is important to reach exascale computing. Tailoring the precision to the requirements of the application offers consequent savings in performance and energy. We presented a methodology to automatically and finely adapt the precision over time for numerical iterative schemes. Our methodology goes beyond mixed-precision approaches by exploring precision configurations at bit level. The method explores the precision requirements over time, and therefore chooses an optimal precision for each application phase. To guarantee the accuracy of the results, we validate the robustness of our solutions to rounding errors with the help of stochastic arithmetic. Finally, our experiments show that the methodology handles large HPC codes like the Combustion-CFD solver YALES2. For YALES2, our approach shows that lowering the precision is viable and achieves 28% to 67% reduction in the communication volume, lowering the energy and runtime cost.

Figure 1

 1 Figure1shows on the left the p k 's value found by VPREC. The plot at right compares the convergence speed of the standard binary64 representation and the VPREC configuration. Both versions converge within the 10 -15 threshold in nine iterations.

Fig. 1 .

 1 Fig. 1. (left) Precision found with VPREC for Newton-Raphson. (right) Both the standard IEEE-754 binary64 version and the VPREC low precision configuration converge to 10 -15 in nine iterations.

Fig. 2 .

 2 Fig. 2. General Verificarlo exploration workflow with VPREC backend

Fig. 3 .

 3 Fig.3. By setting r = 5 and p = 10, VPREC simulates a binary16 embedded inside a binary32. Opaque bits represent the new exponent range and precision available. The sign remains the same.

Fig. 4 .

 4 Fig. 4. First three steps of the piecewise constant search heuristic for the Newton-Raphson problem. The constant piecewise functions of step 0,1 and 2 are represented in green, blue and black.

Fig. 5 .

 5 Fig.5. Adaptive precision searching on YALES2's DPCG with the deflated part (lef t) and the entire code (right). On both plots, we can see that our reduced precision solution follows the reference IEEE convergence profile.

Fig. 6 .

 6 Fig.6. Resiliency of VPREC and binary32 configurations. In red the IEEE maxnorm convergence for reference. Blue envelop shows the 29 MCA samples for the previously found VPREC configuration. Green envelop shows the 29 MCA samples for the binary32 configuration. All samples converge, showing the resiliency of both configurations.

Table 1 .

 1 (left)Convergence speed of Newton-Raphson for the computation of the inverse of π using the IEEE-754 binary64 format (right). The stopping threshold is 10 -15 . Highlighted digits in gray are non significant.

	k x k+1	s 10 k	s 2 k
	0 0.0690266447076745 0.11 0.37 1 0.1230846130203958 0.21 0.70 2 0.1985746566605835 0.43 1.43 3 0.2732703639721015 0.84 2.79 4 0.3119369815109966 1.79 5.95 5 0.3181822938100336 3.40 11.3 6 0.3183098350392471 6.79 22.6 7 0.3183098861837825 13.6 45.2 8 0.3183098861837907 15.6 51.8 9 0.3183098861837907 15.6 51.8	double newton ( double x0 ) { double x_k , x_k1 = x0 , b = PI ; do { x_k = x_k1 ; x_k1 = x_k *(2 -b * x_k ); } while ( fabs (( x_k1 -x_k )/ x_k ) >= 1e -15); return x_k1 ; }

Table 2 .

 2 Comparisons of the different tools for exploring precision reduction.

	Tool	Localization	Mixed prec.	Variable prec.	Round. Error	Automatic
	Precimonious [34]	Spatial
	Blame-Analysis [33]	Spatial		1
	Promise [16]	Spatial
	CRAFT [23]	Spatial		2
	fpPrecisionTuning [21]	Spatial
	FlexFloat [38]	Spatial
	Verificarlo (this paper) Temporal

Acknowledgments We thank Exascale Computing Research Lab supported by CEA, Intel, and UVSQ. This work has been granted access to the HPC resources of CINES under the allocation 20XX-A0031010295 made by GENCI and the computing resources of CRIANN (Normandy, France).