Carlos Cuadrado-Collados

Georges Mouchaham

Luke Daemen

Yongqiang Q Cheng

Anibal J Ramirez-Cuesta

Himanshu Aggarwal

Alexander Missyul

Mohamed Eddaoudi

Youssef Belmabkhout
email: youssef.belmbkhout@um6p.ma

Joaquin Silvestre-Albero

Alexander Missyul

Aleksandr Missyul

Quest for an Optimal Methane Hydrate Formation in the Pores of Hydrolytically Stable Metal-Organic Frameworks

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

INTRODUCTION

Nowadays, in spite of the tremendous progress in the discovery and development of highly porous metal-organic frameworks (MOFs), only few works on methane (CH 4 ) storage fulfilled the department of Energy of USA (DoE) target (gravimetric (0.5 g/g) and volumetric (263 v/v)), albeit at 258 K [START_REF] Alezi | MOF Crystal Paving the Way to Gas Storage Needs: Aluminum-Based soc-MOF for CH 4 , O 2 , and CO 2 Storage[END_REF] . Nevertheless, this target is still hardly achievable at relatively higher temperatures. Obviously, the main challenges are mainly governed by the progress in materials science, particularly with regards to the discovery of 3D highly porous networks with high control on pore confinements and functionalities.

In general, MOFs, porous organic polymers (POPs) and activated carbon (AC) materials are the main families of materials in the pool potentially capable to achieve high methane storage in the adsorbed phase form in dry conditions, although at pressures as high as 8.0-10.0 MPa. [START_REF] Peng | Methane Storage in Metal-Organic Frameworks: Current Records, Surprise Findings, and Challenges[END_REF][START_REF] Casco | High-Pressure Methane Storage in Porous Materials: Are Carbon Materials in the Pole Position?[END_REF][START_REF] Tian | A Sol-Gel Monolithic Metal-Organic Framework with Enhanced Methane Uptake[END_REF][START_REF] Connolly | Tuning porosity in macroscopic monolithic metal-organic frameworks for exceptional natural gas storage[END_REF][START_REF] Jia | Extremely Hydrophobic POPs to Access Highly Porous Storage Media and Capturing Agent for Organic Vapors[END_REF] A potential alternative approach to enhance methane storage is the socalled solid methane storage, i.e. the storage of methane in crystalline hydrate structures similar to those found in nature. [START_REF] Sloan | Fundamental Principles and Applications of Natural Gas Hydrates[END_REF] Although that bulk hydrates suffer from a low yield, due to the limited gas-liquid interface, the experience made on AC showed that confining the CH 4 hydrates assembly in nanoporous solids allows to increase the water-to-hydrate conversion at relatively low pressure, while shortening the nucleation time and growth kinetics. [START_REF] Casco | Methane Hydrate Formation in Confined Nanospace Can Surpass Nature[END_REF][START_REF] Cuadrado-Collados | Freezing/Melting of Water in the Confined Nanospace of Carbon Materials: Effect of an External Stimulus[END_REF][START_REF] Siangsai | Investigation on the Roles of Activated Carbon Particle Sizes on Methane Hydrate Formation and Dissociation[END_REF][START_REF] Celzard | Optimal Wetting of Active Carbons for Methane Hydrate Formation[END_REF] Although this first development step for CH 4 storage via CH 4 hydrates in porous materials was encouraging, the nucleation process is still not understood.

Due to their high modularity in terms of porous structure and surface chemistry, MOFs stand as model versatile platform to shed light on the CH 4 hydrates nucleation process in relation to specific pore size and shape in porous media, with the ultimate objective to promote gas hydrate formation. [START_REF] Zhou | Introduction to Metal-Organic Frameworks[END_REF][START_REF] Kaskel | The Chemistry of Metal-Organic Frameworks: Synthesis, Characterization and Applications[END_REF][START_REF] Eddaoudi | Systematic Design of Pore Size and Functionality in Isoreticular MOFs and Their Application in Methane Storage[END_REF][START_REF] Furukawa | Ultrahigh Porosity in Metal-Organic Frameworks[END_REF] This opportunity could be a key in understanding the methane hydrates formation in porous materials, in general, and in ordered-crystalline porous materials, in particular. Early stage studies on solid CH 4 storage was carried out on a plethora of porous materials including MOFs, but to the best of our knowledge no report showed gas hydrates growth inside the cavities of MOFs. [START_REF] Casco | Paving the Way for Methane Hydrate Formation on Metal-Organic Frameworks (MOFs)[END_REF][START_REF] Mu | A Novel Method to Improve the Gas Storage Capacity of ZIF-8[END_REF] In fact, it was reported that highly hydrophilic mesoporous MOFs (e.g., MIL-100(Fe)) promote strong water-network interactions, with the associated structural blocking. Conversely, hydrophobic MOFs (e.g., ZIF-8) promote gas hydrate formation exclusively in the external surface, inner cavities remaining fully available for gas physisorption. Thus, promoting the formation of gas hydrates within MOFs cavities is a challenge yet to be addressed and early stage studies are of high importance to delineate the potential of CH 4 hydrates technology. Accordingly, prior to evaluating the performance of this technology using MOFs vs other CH 4 storage approaches, standing at different development stages, it is necessary to shed light on the structural properties of MOFs vs CH 4 hydrates nucleation potential. In this regard, the use of the adequate MOF, showing appropriate pores' geometry and chemical texture, remains crucial and need to be rationally selected, taking into consideration the often-conflicting factor of hydrolytical stability. The understanding of the nucleation mechanism of isolated, uniform and homogeneously distributed MH nanocrystals within MOF cavities will allow to find pathways to improve methane storage capacity over a narrow pressure window (defined by the formation and dissociation pressures), compared to conventional physisorption processes (i.e., a larger working capacity under less demanding pressure swinging conditions).

Recent studies have shown that reticular chemistry via the design of suitable organic building block allowed the discovery of porous network MOFs with an exceptional water adsorption capacity (up to 200 wt.%). [START_REF] Abdulhalim | A Fine-Tuned Metal-Organic Framework for Autonomous Indoor Moisture Control[END_REF][START_REF] Abtab | Reticular Chemistry in Action: A Hydrolytically Stable MOF Capturing Twice Its Weight in Adsorbed Water[END_REF] The preferential location of the adsorbed water in the form of cluster filling in the inner cavities, the adequate size of these cavities (similar to the MH unit crystal), and the associated strong adsorption potential, anticipates a priori a large potential as platforms to promote the nucleation of isolated gas hydrate nanocrystals. To this end, two hydrolytically stable MOFs, Y-shp-MOF-5 and Cr-soc-MOF-1, have been selected and evaluated for their CH 4 storage performance at high-pressure both under dry adsorption process conditions and water pre-loaded (adsorbed) samples with different amounts of water.

Y-shp-MOF-5 (Figure 1a) is a hybrid microporous highly connected rare-earth-based MOF characterized by a relatively intermediate water uptake (up to 0.45 g/g) under a narrow relative humidity range uniquely suitable for moisture control in confined spaces. [START_REF] Abdulhalim | A Fine-Tuned Metal-Organic Framework for Autonomous Indoor Moisture Control[END_REF] The Y-shp-MOF-5 structure can be viewed as a hexagonal close packing of nonanuclear molecular building blocks, thus forming uniform triangular 1D channels with an opening of 1.2 nm along the c axis. Apparent BET surface area is 1550 m 2 /g and the total pore volume is 0.63 cm [START_REF] Casco | High-Pressure Methane Storage in Porous Materials: Are Carbon Materials in the Pole Position?[END_REF] /g (see Figures S1-S3 for further details).

Cr-soc-MOF-1 (Figure 1b) is built of chromium trimers and large tetratopic organic building blocks and offers an extraordinary CH 4 and water adsorption behavior, in addition to a remarkable chemical stability. [START_REF] Abtab | Reticular Chemistry in Action: A Hydrolytically Stable MOF Capturing Twice Its Weight in Adsorbed Water[END_REF] Cr-soc-MOF-1 has an apparent surface area above 4500 m 2 /g, and an exceptional water uptake as high as 1.95 g/g, i.e. nearly 200 wt.%. Cr-soc-MOF-1 also possess both cuboidal cages and 1D channels with a pore diameter of ~ 1.5 nm, ~ 1.7 nm, respectively (see Figures S4-S6 for further details). In addition to the hydrolytic stability, the selection of these two MOFs has been motivated by their extraordinary water adsorption capacity accompanied with clusterization phenomenon, together with the dimension of their pores, rather similar (Y-shp-MOF-5) or slightly above (Cr-soc-MOF-1) the unit crystal size of the methane hydrates (1.2 nm). The main goal would be to shed light on the growth of isolated MH nanocrystals using the different controlled confinement of these two MOFs and evaluate the potential of using each individual cavity of the MOFs studied as a nanoreactor to this end.

RESULTS AND DISCUSSION

High-pressure methane isotherms at 275 K. To evaluate the effect of the water content in the hydrate formation process, high-pressure methane sorption measurements were performed on the two MOFs before and after loading water at different R w (see Supporting Information). Figure 2 shows the excess methane uptake for (a) Y-shp-MOF-5 and (b) Cr-soc-MOF-1 samples, both at dry and different R w , associated to different water pre-loading conditions. As expected, dry MOFs exhibit a type I-like adsorption isotherm with a maximum CH 4 uptake around 12 wt.% (0.12 g/g) at 6-7 MPa, for Y-shp-MOF-5, and a larger amount, up to 24 wt.%, at 7-8 MPa for Cr-soc-MOF-1.

After measuring the dry isotherms, Y-shp-MOF-5 was prehumidified starting at R w = 0.3 (60% load) followed by progressive pressurization with CH 4 . The main purpose of progressive loading at R w = 0.3 was to incorporate water first exclusively at the Y 3+ open metal sites, as primary adsorption sites at low relative humidity. [START_REF] Abdulhalim | A Fine-Tuned Metal-Organic Framework for Autonomous Indoor Moisture Control[END_REF] Below 4 MPa, the threshold pressure for methane hydrate formation, the CH 4 adsorption isotherm resembles the original type I, i.e. cavities are partially accessible. Above this pressure some anomalies can be noticed for R w = 0.3 (mainly at around 6 MPa). Further increase in R w (R w = 0.5; 100% load) to saturate the Y-shp-MOF-5, followed by a subsequent CH 4 pressurization led to a clear alteration in the CH 4 isotherm shape from the original dry baseline, with a marked hysteresis loop between 5 and 10 MPa. This result clearly reflects some nucleation (upon adsorption) and dissociation (upon desorption) of some confined hydrates. In this particular case, we conclude that the presence of a significant adsorption capacity for methane at low pressures denotes the presence of a significant uptake due to pure physisorption, very probably due to the displacement of water from inner cavities by high-pressure methane. In a final attempt to promote the MH formation, Y-shp-MOF-5 was oversaturated with R w = 0.75 (150% load). However, under these water pre-loaded conditions, the blocking effect is more significant at low pressures, although associated later with a significant hydrate formation at about 3-4 MPa. From all the above-mentioned results and observations, it was found that the cavity dimension of Y-shp-MOFs (1.2 nm) is not favorable to accommodate both water and CH 4 in conditions to allow in turn favorable conditions for MH formation. This is clearly seen for the CH 4 uptake at 10 MPa located below the dry CH 4 baseline for all water pre-loaded experiments. A similar experimental approach was implemented for the Cr-soc-MOF-1 (Figure 2b). In this case, the amount of water pre-loaded ranged from under-saturation to over-saturation conditions in similar/close loading percentages as Y-shp-MOF-5. Undersaturated and saturated samples (R w = 0.5 (25% load) and 2 (100% load), respectively) exhibit a complete blocking of the porosity by water with a poor methane uptake over the whole pressure range evaluated. These results anticipate that when the cavities are partially or totally filled with water, methane cannot penetrate the inner cavities and/or interact with water to make hydrate crystals. Interestingly, the scenario changes completely under slight oversaturation conditions. Indeed, the Cr-soc-MOF-1 sample pre-loaded with 3 g/g (150% load) exhibits a complete blockage behavior toward methane below 4.5 MPa. However, above this threshold pressure, the methane uptake exhibits two consecutive jumps in a short pressure range to reach an immediate saturation of 33 wt. % at pressures slightly above 6

MPa. Markedly, this abrupt increase in the methane uptake and the shift observed in the desorption branch, associated with a hysteresis loop, are strong evidences for the promotion of gas hydrates formation, with a high water-to-hydrate yield. Compared to previous hydrates confined in activated carbon materials, hydrates confined in MOFs exhibit some peculiar characteristics. While nucleation of gas hydrates in the external surface or in large cavities of carbon materials takes place at 3-4 MPa, larger pressures are required for crystals grown in their narrower cavities. [START_REF] Casco | Methane Hydrate Formation in Confined Nanospace Can Surpass Nature[END_REF] Based on these premises, the shift in the threshold pressure in Cr-soc-MOF-1 to 4.5-5 MPa is considered as a clear indication of the preferential nucleation in the inner cavities (nucleation on the external surface can be excluded, except the small step at 4.5 MPa, most probably associated with the MH nucleation at the pore mouth). In addition, the uniform jump between 4.5 MPa and 5.5 MPa suggests that hydrate nanocrystals must be quite uniform, their formation and dissociation taken place in a narrow pressure window. This observation has tremendous implications for a potential application of these materials in methane storage devices since the system can be charged and discharged under less demanding pressure swinging conditions, i.e., it exhibits an ideal working capacity profile. Last but not least, it is important to highlight that the total uptake at 10 MPa constitutes a 50% improvement compared to the dry material (up to 0.33 g/g), this value being among the best reported in the literature so far for carbon materials and MOFs. [START_REF] Peng | Methane Storage in Metal-Organic Frameworks: Current Records, Surprise Findings, and Challenges[END_REF][START_REF] Casco | High-Pressure Methane Storage in Porous Materials: Are Carbon Materials in the Pole Position?[END_REF] A further increase in the amount of water preadsorbed to R w = 4 becomes detrimental and seems to limit the process of MH growth. Although a significant MH formation is preserved in the pressure window 3.5-4.5 MPa, the total yield remains quite limited. This last result is another evidence of optimal formation of CH 4 hydrate at R w =3 for Cr-soc-MOF-1. It is important to mention that the purpose of imposing an excess of water surrounding the material while progressively preforming CH 4 pressurization is to trigger the formation of hydrate and drive stoichiometry in favor to natural hydrate formation (1 CH 4 •5.75 H 2 O).

The intensive experimental work carried out and summarized above confirm that Y-shp-MOF-5 with a pore size similar to the methane hydrate crystal does not promote significantly the water-to-hydrate formation process; most probably due to the steric restrictions in narrow cavities. However, slightly larger cavities, such as those present in Crsoc-MOF-1, accompanied by the appropriate amount of water, can promote hydrate nucleation with a high yield.

Further analysis of the CH 4 storage data allow to access the stoichiometry driving the methane hydrate formation/dissociation at 10 MPa and R w = 3. In fact, assuming that the whole amount of water participates in the hydrate formation process, gives a value of 1 CH 4 •8.0 H 2 O. Compared to natural hydrates (1 CH 4 •5.75 H 2 O), gas hydrates confined in MOFs exhibit some excess of water, most probably reflecting some limitations in the water-to-hydrate yield. One can assume that 2/3 of the water (2 g/g) are converted to gas hydrate, thus fitting the stoichiometry of the natural hydrates, i.e., 1CH 4 •5.8H 2 O, whereas 1/3 of water remains unchanged or displaced. The 1/3 residual water left in the pores scenario, could most probably be caused by direct interaction of water with the MOF inner walls (water not prone to make hydrates; yield MH66%). The presence of a surface layer of non-freezable water could explain the obtained results. Such explanation is in close agreement with the case of gas hydrates grown in silica. [START_REF] Bai | Microsecond Molecular Simulations of the Kinetic Pathways of Gas Hydrate Formation from Solid Surfaces[END_REF][START_REF] Liang | Crystal Growth Simulations of Methane Hydrates in the Presence of Silica Surfaces[END_REF] Water displacement at increasing CH 4 pressure could be the most probable scenario to reach the equilibrium of 1CH 4 •5.8H 2 O relevant to natural hydrate formation.

Inelastic neutron scattering (INS) experiments.

A potential technique to assess the presence of gas hydrates, even when they are confined in narrow cavities, is inelastic neutron scattering (INS). INS is a suitable technique to evaluate the dynamic of atoms, especially vibrational and rotational modes. Due to the large neutron scattering cross section of hydrogen, INS is ideal to probe not only structural dynamics but also rotational tunneling of entrapped and isolated CH 4 molecules. In a first approach, the INS spectra were measured on the desolvated MOFs in order to identify the main rotational and vibrational modes of the parent structure before proceeding with the methane hydrate evaluation.

Figure 3 shows the INS spectra of the bare (a) Y-shp-MOF-5 and (b) Cr-soc-MOF-1 samples measured at 5 K in the low energy transfer region. The spectra for both samples in the low & mid-energy regions are included in the supporting information (Figures S7 &S8). The low energy transfer region contains the terahertz spectroscopic modes and reveals framework dynamics, whereas the mid-energy region contains the deformation and bending modes from the linker. In the specific case of the Y-shp-MOF-5, the INS spectra were also simulated using CP2K calculations (see Figure 3a). INS spectrum of the Y-shp-MOF-5 shows two main contributions at ~8 meV and ~50 meV, together with small contributions at ~24 and ~35 meV. As shown in Figure 3a, CP2K calculations are able to reproduce the two main peaks. The contribution at ~8 meV is associated to the ring spinning/flapping and the one at ~50 meV is due to the ring twisting (see animations in Supporting Information).

The low energy transfer region describes a similar spectrum for Cr-soc-MOF-1 with main contributions at ~4.5, 8, 15 and 50 meV. The similarity between both INS spectra is due to the presence of similar characteristics in the organic linkers used for both metal-organic frameworks (1,2,4,5-tetrakis(4carboxyphenyl)benzene-H 4 BTEB and 3,3", 5, 5"-tetrakis(4carboxyphenyl)-p-terphenyl-H 4 TCPT, for Y-shp-MOF-5 and Cr-soc-MOF-1, respectively). Consequently, assignment of low energy contributions coincides with Y-shp-MOF-5. Unfortunately, CP2K phonon calculations of Cr-soc-MOF-1 could not be completed within reasonable time mainly due to its much larger unit cell (1000 atoms). Regarding the midenergy region, Figure S7 shows a good fitting between the experimental and predicted spectra. As described above, this region contains the vibrational and librational modes of the organic linker. INS spectra of the Y-shp-MOF-5 (Figure S9) does not allow to categorize the methane hydrate formation. Only after a background subtraction of the D 2 O-loaded MOF and the adsorbed but unreacted CH 4 background (Figure 4a), the low energy transfer region allows to identify peaks at E = 0.96 meV (7.98 cm -1 ), E = 2.3 meV (18.54 cm -1 ) and E = 3.3 meV (26.61 cm -1 ), attributed to the rotational transitions from the ground state to the first and second excitation states, J = 0  J = 1, (J = rotational quantum number), J = 1  2, and J = 0  2, respectively. These contributions perfectly fit with the ones observed for natural hydrates. [START_REF] Gutt | Quantum Rotations in Natural Methane-Clathrates from the Pacific Sea-Floor[END_REF] These rotational transitions of the methane molecule are in agreement with an almost free rotor (E = 1.31 meV) due to their entrapped & isolated nature (interaction between methane molecules in different cages must be rather weak). However, the tinny signal observed for Y-shp-MOF-5 clearly reflects a low waterto-hydrate yield, in close agreement with the high-pressure measurements described and discussed in detail above. Compared to previous measurements shown in the literature, Y-shp-MOF-5 shows an additional contribution at 0.5 meV, most probably due to the defective nature of these confined nanocrystals.

Conversely, the INS data analysis demonstrated completely different results for Cr-soc-MOF-1 (Figure 4b). Under saturation conditions and 5 K, the D 2 O-loaded MOF pressurized with CH 4 exhibits well-defined methane hydrate contributions at E = 1.00 meV, E = 2.3 meV and E = 3.4 meV (see arrows), thus reflecting a large conversion yield. These results constitute a clear experimental proof about the role of Cr-soc-MOF-1 as a host structure to promote optimal condition for the nucleation and growth of confined hydrates. In contrast, the same conditions, these processes are limited in Y-shp-MOF-5 due to the presence of a smaller size cavities. Synchrotron X-ray powder diffraction measurements (SXRPD). In order to evaluate further the observed methane hydrates growth, Y-shp-MOF-5 and Cr-soc-MOF-1 have been evaluated at the MSPD station at the ALBA synchrotron (Spain) under operando conditions (λ = 0.4135 Å). Figure 5 shows the SXRPD patterns for both samples before and after dosing high-pressure methane (due to the characteristics of the experimental setup, the maximum testing pressure was 4.0 MPa). As it can be observed, in the absence of methane both samples exhibit the characteristic peaks of the studied MOF framework together with some additional peaks (once at 263 K) due to the nucleation and growth of hexagonal ice crystals (I h ). These peaks can be clearly assignable at 2 theta = 6.0, 6.4 and 6.9º. Upon pressurization with CH 4 up to 4.0 MPa and 263 K, the SXRPD patterns did not show significant changes for Y-shp-MOF-5 sample, whereas a small contribution attributed to methane hydrate formation (sI structure) can be appreciated for Cr-soc-MOF-1 at 2 theta = 7.4 º. These findings are in good agreements with the behavior of the high-pressure isotherms (adsorption branch) below 4.0 MPa (under these conditions methane hydrate formation is rather limited for both samples as seen in Figure 2). To overcome the set-up pressure limitation, we deliberately attempted shifting of the methane hydrate formation to lower than 4.0 MPa by decreasing the temperature to 223 K. Indeed, in such conditions, Figure 5 shows unambiguously the promotion of the sI structure (main contributions at 2 theta = 7.2, 7.4 and 8.2º), the extend of the water-to-hydrate process being significant for Cr-soc-MOF-1. However, the same experiment in Y-shp-MOF-5 sample only gives rise to a low contribution (see marked region), thus corroborating previously demonstrated evidence using INS. In other words, these results constitute an additional experimental evidence that Cr-soc-MOF-1 is able to promote the methane hydrate formation with a high yield. These crystals have the sI structure, whereas under similar experimental conditions Y-shp-MOF-5 promotes preferentially the formation of hexagonal ice (only a very small fraction of the ice can be converted to methane hydrate). Rietveld refinement calculations confirm that upon pressurization with CH 4 additional molecules are accommodated in the cavities of Cr-soc-MOF-1 pre-loaded with H 2 O. Assuming that these additional molecules correspond to methane (unfortunately SXRPD data does not allow to distinguish between water and methane inside cavities), the confined hydrates will have a stoichiometry of 1CH 4 •4.4H 2 O. The calculated stoichiometry for the confined hydrates is rather similar to the natural system, although with some water deficiency, most probably associated with the presence of confined, non-freezable, water. In addition, these calculations for Cr-soc-MOF-1 predict the presence of additional ice (24%) and methane hydrates (13%) in the external surface and/or at the pore mouth, in agreement with the excess of water pre-loaded (oversaturation conditions).

Phase diagram for the confined methane hydrates.

Knowing the pressure threshold for the methane hydrate nucleation and growth, and the dissociation pressure, both values from the high-pressure isotherms, these two processes can be compared with the phase diagram for bulk hydrates (Figure S11). [START_REF] Borchardt | Illuminating Solid-Gas Storage in Confined Spaces -Methane Hydrate Formation in Porous Model Carbons[END_REF][START_REF] Handa | Thermodynamic Properties and Dissociation Characteristics of Methane and Propane Hydrates in 70-A Radius Silica Gel Pores[END_REF] As expected, under confined conditions the nucleation and growth process does not follow thermodynamics, i.e. higher pressures are required to promote methane hydrate formation at a given temperature (metastable conditions). As shown in Figure S11, there is a clear trend between the required pressure for nucleation and the pore size of the host structure, independently of the nature of the nanoporous system, i.e., activated carbon or metal-organic framework. In fact, Cr-soc-MOF-1 and Y-shp-MOF-5 follow the expected tendency for their pore size with the pressure threshold to promote nucleation growing from Cr-soc-MOF-1 to Y-shp-MOF-5 (these values correspond to the formation of the first nuclei, independently of the obtained yield). On the contrary, experimental results show that hydrate dissociation is indeed under equilibrium conditions, i.e., the pressure threshold is independent of the host structure (chemical nature and porous structure).

In conclusion, the combination of high-pressure adsorption measurements, INS and SXRPD experiments allowed to shed light on the role of pore size control for promoting gas hydrates formation in the pores of MOFs. It was found that Crsoc-MOF-1 combines a proper porous structure (cuboidal cages and 1D channels with an average pore size of ca. 1.6 nm) and surface chemistry to promote the nucleation of isolated methane hydrates with a sI structure. We hypothesize that high water-to-hydrate yield requires slight oversaturation conditions which will in turn promote nucleation at the pore mouth (small step at 4.5 MPa). Further maintaining high CH 4 pressure will allow to propagate nucleation to the inner cavities as inferred from the drastic jump at 5.0 MPa. Under these conditions and assuming the stoichiometry of the natural hydrates, a conversion around 66% can be estimated. The final uptake at 10.0 MPa constitutes a 50% improvement compared to the dry material. Most importantly, this excess uptake (0.33 g/g) can be loaded and released within a narrow working capacity pressure window (only 2.5 MPa). In contrast, the absence of these characteristics in Y-shp-MOF-5 precludes the formation of confined methane hydrates with an insignificant yield.

This work represents a highly encouraging step in the development of solid methane storage in MOFs in particular and ordered porous materials in general. Further work will be dedicated to explore isostructural hydrolytic MOFs, with a larger pore sizes, for solid methane storage (SMS).

Figure 1 .

 1 Figure 1. Crystalline structures of a) Y-shp-MOF-5 and b) Crsoc-MOF-1. Yellow spheres highlight the triangular channels of Y-shp-MOF-5 and the cuboidal cages of Cr-soc-MOF-1. Color code: Y, blue; Cr, green; O, red; C, grey. H atoms and counterions are not represented for sake of clarity.

Figure 2 .

 2 Figure 2. High-pressure excess methane adsorption (full symbols)/desorption (empty symbols) isotherms at 275 K and under different water pre-loaded conditions (Rw corresponds to gram of H 2 O per gram of MOF; stands for the ratio of water loaded per gram of sample) for samples (a) Y-shp-MOF-5 and (b) Cr-soc-MOF-1.

Figure 3 .

 3 Figure 3. INS spectra of (a) Y-shp-MOF-5 and (b) Cr-soc-MOF-1 at 5 K in the low energy region.

Figure 4 .

 4 Figure 4. Inelastic neutron scattering (INS) spectra for (a) Y-shp-MOF-5 loaded with D 2 O R w =0.5 and (b) Cr-soc-MOF-1 loaded with D 2 O R w = 2 at 5 K and under high-pressure methane (6 MPa).

Figure 5 .

 5 Figure 5. Synchrotron X-ray powder diffraction (SXRPD) patterns for (a) Y-shp-MOF-5 loaded with H 2 O R w =0.75 and (b) Cr-soc-MOF-1 loaded with H 2 O R w = 3 at different temperatures before and after dosing high-pressure methane (4 MPa).

ACKNOWLEDGMENT G.M, M.E and Y.B thank Aramco sponsored research fund (contract. 66600024505). We would like also to acknowledge the support by King Abdullah University of Science and Technology. J.S.A would like to acknowledge financial support from the MINECO (MAT2016-80285-p), Generalitat Valenciana (PROMETEOII/2014/004), Oak Ridge beam time availability (Project IPTS-20859.1) and Spanish ALBA synchrotron (Project 2020014008).

ASSOCIATED CONTENT

Supporting Information. Experimental and modelling details are included in the Supporting Information. Furthermore, INS spectra for the bare MOFs and the materials are D 2 O loading and pressurization with CH 4 at 6.0 MPa are also included.

The Supporting Information is available free of charge on the ACS Publications website.

AUTHOR INFORMATION Corresponding Author

Author Contributions

All authors proofread and approved the final version of this manuscript for submission.

Notes

No competing financial interests have been declared.


Laurent Berenguer

Damien Tromeur-Dervout
email: damien.tromeur-dervout@univ-lyon1.fr

SPARSE AITKEN-SCHWARZ WITH APPLICATION TO DARCY FLOW

Keywords: Restricted Additive Schwarz, Domain Decomposition Methods, acceleration of convergence, Darcy flow

This paper focuses on the acceleration of the Schwarz method by the Aitken's acceleration of the convergence technique with taken into account the special structure of the error operator. This allows an enhancement of the building of the low rank space approximating the solution at the gathered interfaces of all subdomains computed by singular values decomposition of the sequence of iterated solutions presents in the Aitken-Schwarz technique. The new method Sparse-Aitken Schwarz method builds low rank spaces associated to each subdomain's interfaces. Comparisons between Aitken-Schwarz and Sparse-Aitken Schwarz results obtained on a 3D Darcy flow application show the improvement by using the special structure of the propagation error operator.

INTRODUCTION

Schwarz domain decomposition methods is nowadays widely used to solve linear problems of the form Ax = b because it is well suited for parallel computing. Indeed, it is based on the splitting of the global problem into subproblems. Artificial boundary conditions arise from the decomposition of the domain into subdomains. Then, the Schwarz method consists of the solution of subproblems, and the update of the artificial boundary conditions. In practice, this implies local communications between neighboring subdomains. The main drawback of Schwarz domain decomposition methods is the slow convergence, that depends on the nature of the problem, the geometry of the subdomains and the overlap. There exist several method for the acceleration of convergence, they are all based on the solution of small problem coupling all subdomains. The classical Aitken-Schwarz method [START_REF] Garbey | On some Aitken-like acceleration of the Schwarz method[END_REF](A-S: the dash is to avoid the confusion with Additive Schwarz ) consists in the approximation of the interface problem [START_REF] Tromeur-Dervout | Meshfree Adaptative Aitken-Schwarz Domain Decomposition with application to Darcy Flow[END_REF], and its solution via the Aitken's formula. This idea was first given in [START_REF] Garbey | On some Aitken-like acceleration of the Schwarz method[END_REF] with the restriction to have problem with separable operator and regular mesh at the interfaces in order to use Fourier approximation for building the acceleration. This technique use the pure linear convergence of the Schwarz method i.e the error propagation operator does not depend ot he itretation. The pure algebraical version of the acceleration of Schwarz by Aitken's technique (for non separable operator and independent of the mesh at the artificial interface) was given in [START_REF] Tromeur-Dervout | Meshfree Adaptative Aitken-Schwarz Domain Decomposition with application to Darcy Flow[END_REF] and the first massively parallel implementation was studied in [START_REF] Berenguer | Aitken's acceleration of the schwarz process using singular value decomposition for heterogeneous 3d groundwater flow problems[END_REF]. This technique uses the Singular Values Decomposition (SVD)of the matrix gathering the iterated Schwarz solutions at all artificial interfaces of the domain decomposition to build a low rank approximation of the searched converged solution. In this talk, we take into account the sparsity structure of the of error propagation operator to build low rank approximations of the solution associated to individual artificial interface. The resulting method named the Sparse Aitken-Schwarz (SA-S) method shows better results of convergence and good parallel efficiency on 3D Darcy flow problem.

Numerical acceleration of the Schwarz method

A domain of n unknowns is split in N overlapping subdomains. The ith subdomain has n i if the overlap is included, or n i if the overlap is excluded. In that case,

n = Σ N -1 i=0 n i . Let R i ∈ R n i ×n (respectively R i ∈ R n i ×n
) be the restriction operator of a global vector to the ith subdomain, including the overlap (respectively setting to 0 the components of the overlap). The additive Schwarz method with Dirichlet boundary conditions on the artificial boundary conditions can be written as the Richardson process:

x k+1 = x k + M -1 RAS b -Ax k (1)
with the matrix M -1 RAS is the Restricted Additive Schwarz (RAS) preconditioner [START_REF] Cai | A restricted additive Schwarz preconditioner for general sparse linear systems[END_REF]:

M -1 RAS = N -1 i=0 R T i R i AR T i -1 R i = N -1 i=0 R T i A -1 i R i . (2)
If x ∞ is the exact solution of the linear system Ax = b, and x k the solution at the kth Schwarz iteration, then subtracting two Schwarz iterations:

x k -x ∞ = I -M -1 RAS A x k-1 -x ∞ (3)
which shows the purely linear convergence. This property still holds if we consider only the artificial interfaces. Let R Γ be the operator that restrict a vector to the artificial interface. The restriction of Eq.(1) to the interface is

R Γ M -1 RAS AR T Γ I-P R Γ x y = R Γ M -1 RAS b c . ( 4
)
where the matrix

P := R Γ I -M -1 RAS A R T Γ is the error propagation operator since e k+1 = P e k if e k = y k -y ∞ is the error on the interface at iteration k. If the matrix [y n Γ -y n Γ -1 , • • • , y 1 -y 0 ]
is not singular,the error propagation operator P can be computed as:

P = [y n Γ +1 -y n Γ , . . . , y 2 -y 1 ] [y n Γ -y n Γ -1 , . . . , y 1 -y 0 ]
-1 . The solution y ∞ can be computed as:

y ∞ = (I -P ) -1 y n Γ +1 -P y n Γ . (5)
In practice, it may not be possible to use the exact acceleration for 2D and 3D problems because it requires n Γ + 1 Schwarz iterations, where n Γ is the number of unknowns on the artificial interface. Then, the acceleration is approximated in a low-dimensional space U and an approximated propagation error operator P .

Approximation of the interface problem

In order to approximate the exact solution y ∞ ∈ R n Γ on Γ, we replace the error propagation operator P ∈ R n Γ ×n Γ of (5) by a low-rank approximation of it. This approximation is P = U U T P U U T where U ∈ R n Γ ×l is a matrix with orthonormal columns. The interface problem solved by the Aitken's formula is (I -P )y ∞ = c with c = R Γ M -1 RAS b. The Aitken's acceleration is computed in span(U ), the space spanned by the columns of U .

In the ideal case, that is to say if y ∞ ∈ span(U ), then

I -U U T P U U T y ∞ = y ∞ -U U T P y ∞ = U U T c. ( 6
)
where we used that y ∞ is such that y ∞ = P y ∞ + c. One can avoid the computation of U U T c if there exists an integer q such that y q and y q-1 ∈ span(U ). In such cases

U U T c = U U T (y q -P y q-1 ) = U U T y q -U U T P U U T y q-1 . (7)
In practice y ∞ / ∈ span(U ), but we will form the matrix U such that y ∞ -U U T y ∞ is small. Furthermore, the matrix P will be an approximation of U U T P U U T . Finally, the y i are neither exact, since the local problems can be solved with an iterative Krylov method. For all these reasons, the interface problem becomes:

I -P y ∞ = y k -P y k-1 (8)
where y ∞ is an approximation of y ∞ . It is sometimes possible to find a matrix U a priori, for example when y ∞ can be written in the Fourier space with a few modes. Nevertheless we will consider only the case of a matrix U computed a posteriori from q + 1 traces [y q , . . . , y 0 ] coming from q Schwarz iterations. The computation of the matrices U and P will be detailed in the next section. Eq.( 8) require the inversion of an n Γ × n Γ matrix, which may not be possible for 3D problems. Eq.( 5) can be written as (9) that requires the inversion of a matrix of size l << n Γ .

y ∞ ≈ y ∞ = U I -P -1 U T y q -P U T y q-1 (9)
According to the Woodbury matrix identities, the following equalities hold:

I -U P U T -1 = I -U I -P -1 -1 U T = I -U I -I -P -1 U T . (10)
Using the fact that y q and P y q-1 belong to span(U ), we can write:

y ∞ := I -P -1 y q -P y q-1 = I -U I -I -P -1 U T y q -P y q-1 = I -U U T y q -P y q-1 =0 +U I -P -1 U T y q -P y q-1 = U I -P -1 U T y q -U T P U U T y q-1 = U I -P -1 U T y q -P U T y q-1 .
The Aitken's acceleration is given in Algorithm 1. The step 3 of this algorithm is the restriction of the Schwarz iterations to the interface, which is implemented as in Eq.(1). Each iteration of the step 3 requires the solution of the local problems and the exchange of the artificial boundary conditions. It has been considered that the operator P was Algorithm 1 Approximated Aitken's Acceleration Require: y 0 an initial guess 1: repeat 2:

for i = 1 . . . q do 3:

y i ← P y i-1 + c //Schwarz iterations 4:
end for

5:

Compute a matrix U with orthonormal columns such that y q and y q-1 ∈ span(U )

6:

Compute P an approximation of U T P U 7:

y 0 ← U I -P -1
U T y q -P U T y q-1 8: until convergence approximated from q + 1 successive Richardson's iterations. In step 5 the U is computed with the SVD of the matrix [y 0 , . . . , y q ] = U ΣV T . So far, the approximation P of P given by U P U T was a full matrix but in fact the matrix P can be very sparse. We propose a new methods, called sparse Aitken-Schwarz, to approximate the Aitken's acceleration that preserves the null blocks of the matrix P corresponding to independent subdomains. The operator P can actually be computed from q arbitrary vectors and their images after one Schwarz iteration.

A space spanned by the last traces

We discuss the low-rank space in which the approximation of the matrix P is computed from the matrix Y = [y 0 , • • • , y q ] ∈ R n Γ ×(q+1) that contains q + 1 consecutive traces. The singular value decomposition (SVD) [START_REF] Gene | Matrix computations[END_REF] of the matrix Y gives Eq.( 11), where U is an n Γ × n Γ matrix with orthonormal columns, and S is an n Γ × (q + 1) diagonal matrix such that S ii = σ i , 1 ≤ i ≤ (q + 1), and the matrix V ∈ R (q+1)×(q+1) is orthogonal.

Y = U S V T (11)
In practice, we use an economical version of the SVD that computes only the q + 1 first columns of U required to write Eq.( 12), where U i and V i are the ith last columns of U of V, and σ i is the iit largest singular value.

Y = q+1 i=1 σ i U i V T i ( 12
)
The truncation of the SVD of Y to its l largest singular values gives the matrix Y of rank l minimizing the Frobenius norm ||Y -Y || F . This truncated SVD is given in Eq.( 13).

Y = l i=1 σ i U i V T i ( 13
)
In the following, we will denote by U the matrix corresponding to the l first columns of U associated to the l largest singular values, that is to say the significant ones.

Approximation of the error propagation error

Since the restriction of the Schwarz iterations to the interfaces y k = P y k-1 + c, after q iterations we can write: y qy q-1 , . . . , y 2y 1 = P y q-1y q-2 , . . . , y 1y 0 (14) then :

U T [y qy q-1 , . . . , y 2y 1 ] = U T P [y q-1y q-2 , . . . , y 1 -

y 0 ] = U T P U U T [y q-1 -y q-2 , . . . , y 1 -y 0 ] +U T P I -U U T [y q-1 -y q-2 , . . . , y 1 -y 0 ] . (15)
Because the SVD is truncated to its l largest singular values, there is

I -U U T y q-1 -y q-2 , . . . , y 1 -y 0 2 ≤ 2σ l+1 . (16)
Furthermore, if the Schwarz method converges, then P 2 < 1. If we neglect U T P I -U U T [y q-1y q-2 , . . . , y 1y 0 ], we have the following approximation of U T P U :

U T P U ≈ P := U T y qy q , . . . , y 2y 1 U T y q-1y q-2 , . . . , y 1y 0 + (17

)
where E + is the Moore-Penrose pseudoinverse of the matrix E, which is equal to E -1 if E is invertible. Finally, the low-rank approximation of the matrix P is is P = U P U T . When the Schwarz method converges, the q + 1 last traces of the Schwarz process

[y 0 , • • • , y q ] converges toward [y ∞ , • • • , y ∞ ]. Then, the matrix U converges toward [y ∞ / y ∞ 2 , 0, • • • , 0] and the limit of U U T y ∞ -y ∞ 2 is 0.
The Aitken's acceleration was introduced in a general subspace spanned by the q + 1 traces, that is the Krylov subspace K q+1 . The Richardson process accelerated by the Aitken's formula cannot converge faster than the GMRES method without restart.

We propose to take advantage of the sparse structure of the matrix P , that is equivalent to compute the acceleration in a subspace larger than the Krylov subspace associated to the q + 1 iterations. The idea is to by approximating independently each block of the operator P associated to each subdomain. This can be related to the sparse approximation of the Schur complement of the matrix A discussed in [START_REF] Giraud | Sparse approximations of the Schur complement for parallel algebraic hybrid linear solvers in 3D[END_REF] and to the local approximation of the Dirichlet-to-Neumann map proposed in [START_REF] Nataf | A coarse space construction based on local Dirichlet-to-Neumann maps[END_REF].

So far, the approximation of

P = R Γ (I -M -1 RAS A)R T Γ ,
given by U P U T was a full matrix but the matrix P can be very sparse. This sparsity can be explained by the fact that the solution at one subdomain interface depends only on the solution of the neighboring subdomains.

We first consider only the case of two subdomains.

For two subdomains, let denote by v i 0 and v i 1 the solutions at the interface of the two subdomains at the ith iteration. We also denote by R Γ 0 and R Γ 1 the restriction operators to these two interfaces. Then, v i 0 = R Γ 0 x i . The purely linear convergence of the Schwarz process can be written as

v i+1 0 -v i 0 v i+1 1 -v i 1 = R Γ I -M -1 RAS A R T Γ v i 0 -v i-1 0 v i 1 -v i-1
1 where the matrix P = R Γ (I -M -1 RAS A)R T Γ can be decomposed P = 0 P 0 P 1 0 .

Let e n 0 = v i+1 0 v i 0 and e i 1 = v i+1 1 v i 1 then:

P 0 e 0 1 , . . . , e q-1 1 = e 1 0 , . . . , e q 0 and P 1 e 0 0 , . . . , e q-1 0 = e 1 1 , . . . , e q 1 .

In order to approximate the acceleration in low dimensional space, we compute independently the SVD of the trace of each interface.

U i Σ i V T i = [v 0 i , . . . , v q+1 i ] for i = 0, 1. Then P 0 := U T 0 [e 1 0 , . . . , e q 0 ] U T 1 e 0 1 , . . . , e q-1 1 -1 ≈ U T 0 P 0 U 1 P 1 := U T 1 [e 1 1 , . . . , e q 1 ] U T 0 e 0 0 , . . . , e q-1 0 -1 ≈ U T 1 P 1 U 0 . (19)
The approximation preserving the diagonal null blocs of the matrix P is

P ≈ 0 U 0 P 0 U T 1 U 1 P 1 U T 0 0 = U 0 0 0 U 1 U × 0 P 0 P 1 0 P × U 0 0 0 U 1 T U T . ( 20
)
The Aitken's acceleration is computed after q iterations, the dimension of the Krylov subspace is q + 1. The matrix U has l ≤ 2q + 2 columns linearly independent, spanning a subspace larger than the Krylov subspace if l > q + 1. The Aitken's acceleration in which each blocs of P has been approximated independently is more suitable for parallel computing because there is one SVD per interface instead of one global SVD.

Implementation details and results

The Schwarz method is often implemented using one processor per subdomain. Even this can be relevant when the Schwarz method is used to precondition the Krylov iteration, it does not allow us to achieve a competitive solver. Generally speaking, Schwarz solver are not as efficient as classical solvers such as parallel Krylov solver. Nevertheless, these classical solvers require numerous communications between the process, and their extensibility reach a limit. Then it is relevant to use Schwarz method when we have to solve a problem that cannot be solved using a classical solver. The Aitken-Schwarz methods allows us to couple multiple instances of a classical parallel solver.

Two levels of parallelism

The first level corresponds to the parallel solver chosen for the subdomains problems. The second level is the Schwarz method, implemented using MPI, that couples these local linear solvers. The Aitken's acceleration has implemented as follows:

One processor of each interface gathers all traces and compute their SVD. We could have chosen to compute the SVD in parallel to avoid this bottleneck, but their relative precision than sequential algorithms (i.e. DGESVD from Lapack).

The acceleration in the low rank space in computed on a single processor and redistributed to the interfaces. This step requires global synchronization between all subdomains.

An illustration of the two levels of parallelism is given in Figure 1. The Schwarz method requires communications only between the processors that handles a part of the artificial interfaces. If the decomposition of the mesh is regular, as in Figure 1, then each processor communicate with one and only one other processor during the exchange of boundary conditions.

The problem to be solved

The groundwater flow in saturated media can be modelled using the Darcy's laws and the conservation of mass that gives Eq.( 21), where u is the hydraulic head K(x, y, z) is the permeability field.

∇. (K(x, y, z) ∇u) = 0 in Ω u = α, on Γ L , u = β, on Γ R , ∂u ∂n = 0, on ∂Ω \ (Γ 1 ∪ Γ 2 ) (21)
The domain Ω is a parallelepiped, with two Dirichlet boundary conditions on the left Γ L and right Γ R wall, and homogeneous Neumann boundary conditions on the other walls. The problem is discretized using a standard 7 points stencil on a regular grid.

The permeability field can be deterministic or generated randomly using a log-normal distribution which has two parameters: λ the correlation length and σ the standard deviation.

L. Berenguer and D. Tromeur-Dervout topology and boundary conditions regular data distribution scheme over processors We first compare the numerical behavior of the classical Aitken-Schwarz method and the sparse approach. In order to highlight the numerical noise arising from the classical method, we consider the following problem:

Schwarz domain decomposition

1 st level parallelism overlap overlap parallel Krylov solvers 2 nd level parallelism Ω 0 Ω 1 Ω 2 Γ L Γ R Γ L Γ R Γ L Γ R no f low f ixed head f ixed head no f low
The domain is Ω = [0, 1] × [0, 1] × [0, 30].
The mesh size is 64 × 64 × 1920, distributed over 120 cores.

The domain is decomposed in 5 subdomains, and the overlap is 4 .

We consider the homogeneous Poisson problem: K(x, y, z) = 1.

The Dirichlet boundary conditions are 1.0 on the left and 10.0 on the right.

Each subproblem is solved with a relative tolerance of 10 -8 by a preconditioned Krylov method.

The initial guess is x 0 = 0 and the absolute tolerance for the global residual is 10 -7 . This 3D problem is actually very close to 1D problem because the successive solution of the Schwarz methods are constant in xy cuts if the subproblems are solved exactly.

The standard deviation of the solution (the values has been centered in 0) is plotted in Figure 2. This shows that the classical Aitken's acceleration produces a numerical noise on the interface, then this numerical noise is damped by the Schwarz iterations. In the case of sparse Aitken-Schwarz, the standard deviation remains lower than 10 -8 after the first acceleration. tolerance is obtained after the acceleration for the sparse approach, but not for the classical one. The second acceleration for the classical method is inefficient because the numerical noise has not been damped: there are not enough Schwarz iterations between the two accelerations. The traces of the Schwarz methods are not exactly constant because of the tolerance of the Krylov method. This affects the columns of the matrix U . This noise is amplified and propagated in the classical approach because the approximation of P is dense.

We now consider a domain Ω = [0, 1] × [0, 1] × [0, 15] discretized in 128 × 128 × 1920 points. The permeability field is given by: K(x, y, z) = 10 2×sin(πx)×sin(πy)×sin (πz) .

(

) 22
The number of subdomains in increased, keeping constant the number of processors (i.e. the number of processors per subdomain is decreased). Table 1 shows the number of Schwarz iteration and the computational times of the Aitken-Schwarz method. This computational times does not include the setup of the matrix and the output of the final solution. Let us remark that the sparse Aitken-Schwarz gives lower times. Furthermore, the difference increases with the number of subdomains. This can be explained by the fact that the analytical matrix P become sparser and sparse when the number of subdomains increases. Let remark that the best computational are obtained for 2 subdomains. Indeed, the combination of GMRES and Hypre is very efficient for solving this problems on a small scale. The solution of the first local linear system required 14.6 seconds for the two subdomains case, and 11.2 for 15 subdomains. In other words, the solution time of the local linear systems by GMRES with multigrid preconditioning is multiplied by 1.3 when the size of the problem is multiplied by 7.5. These solution times included the computation of the coarse grids, that is done only once.

L. Berenguer and D. Tromeur-Dervout

Weak scaling

In order to test the weak scaling of our implementation, we set the size of one subdomain to 512 × 512 × 256, and we increase the number of subdomains. Nine Schwarz iterations are computed before the acceleration, and one after. Table 2 shows the computational times and their repartition. The total number of Schwarz iterations is 10 for all considered sizes of meshes. This means that the required tolerance b -Ax 2 / b 2 < 10 -5 has been reached after the first acceleration. The computational times could have been reduced for 2 subdomains, performing the acceleration after 7 or 8 Schwarz iterations. In the other cases, 9 is the optimal number of Schwarz iterations.

Only the time of the Aitken-Schwarz method has been considered. In particular, the output of the final solution is not included of time measurements. The code has been run twice, and the averages are given. The maximum difference between two identical runs is 12 seconds. The FMGRES method preconditioned by BoomerAMG of Hypre requires a lot of communications, and the duration of these communications depends on the load of the network.

The computational time is spent in the local solver mostly. Let also remark that the number of Krylov iterations may vary from one domain to another. The time spend in the local solver given in Table 2 included these idle times.

The time for the Aitken's acceleration included the singular value decomposition and the communications. This time increases when the number of subdomain increases, because the acceleration requires global synchronizations, and the dimension of the low-rank space increases with respect of the number of interfaces.

The computational time of the Schwarz exchanges is small, but in increases with respect of the number of subdomains. For this particular problem, the total number of communications increases when the number of subdomain is increased, but not the size of each communication.

The remaining time is mostly due to the computation of the global residual, that involves a global synchronization.

Figure 1 :

 1 Figure 1: Example of two levels of for a 2D problem, the center part shows the domain decomposition, the lower part shows the data distributions over 18 processors.

Figure 2 :

 2 Figure 2: Standard deviation of the solution at the interfaces for the homogeneous Poisson problem.

Figure 3 :

 3 Figure 3: Residuals of the additive Schwarz method for the homogeneous Poisson problem.

Table 1 :

 1 Number of Schwarz iterations and computational times.

	subdomains	Traces	A-S	SA-S
	2	9	18 (63.9s)	9 (42.7s)
	5	9	120 (353.8s)	18 (61.0s)
	10	19	n.c	19 (66.3s)
	15	19	n.c	19 (65.1s)
	A-S: classical Aitken-Schwarz , SA-S: sparse Aitken-Schwarz
	Mesh distributed over 120 cores.
	Dirichlet B.C. of 1.0 on the left and 10.0 on the right.
	Subproblems solved by GMRES preconditioned by Hypre, relative tolerance of 10 -10 .
	Stopping criterion: relative tolerance of 10 -8 for the residual norm.
	n.c : non-convergence after 200 Schwarz iterations.

Table 2 :

 2 Repartition of the computational time of SA-S(9) Dirichlet b.c. of 1.0 on the left and 10.0 on the right. 8 overlapping points between each subdomains. Subproblems solved by FGMRES preconditioned by Hypre with a relative tolerance of 10 -12 .

	Subd.	Cores	Time (s)	Local solution	Aitken	Exchanges	Remaining
	2 4 8 16	512 1024 2048 4096	752 811 828 817	99.444% 99.051% 98.548% 98.063%	0.123% 0.186% 0.168% 0.208%	1.54 × 10 -3 % 2.86 × 10 -3 % 4.35 × 10 -3 % 5.00 × 10 -3 %	0.431% 0.761% 1.280% 1.724%

Acknowledgements

This work has been supported by the French National Agency of Research (project ANR-MN2012-H2MNO4), and the région Rhône-Alpes. Authors also thank the Center for the Development of Parallel Scientific Computing (CDCSP) of the University of Lyon 1 for providing us with computing resources.