Ajinkya Powar
email: ajinkya.powar@ensait.fr

Anne Perwuelz

Nemeshwaree Behary

Vinh Le

Thierry Hoang

Carmen Aussenac

Loghin

Sergiu Stelian

Jinping Maier

Guoqiang Guan

Chen

Le Vinh Hoang

Thierry Aussenac

Carmen Loghin

Stelian Sergiu Maier

Jinping Guan

Guoqiang Chen

Investigation into the color stripping of the pigment printed cotton fabric using the ozone assisted process: A study on the decolorization and characterization

Keywords: Decolorization, Pigment Blue 15, pigment printed, cotton, ozonation, Box-Behnken design

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Introduction

Pigment printing is one of the oldest and easiest printing methods, which is also simple to apply. Pigment printing accounts for more than 80% of the printed based goods due to merits like ease of near final print at the printing stage itself, versatility, print quality, applicability to almost every kind of fiber or mixture and no washing processes required after fixation. [START_REF] El-Molla | Development of ecofriendly binders for pigment printing of all types of textile fabrics[END_REF][START_REF] Yaman | Improvement fastnesses and color strength of pigment printed textile fabric[END_REF] In pigment printing, binders are used to fix the insoluble pigments onto the fabric surface because pigments do not have any type of bonding with the fibrous polymer. [START_REF] Christie | The chemistry of colour application[END_REF][START_REF] Saxena | Challenges in sustainable wet processing of textiles[END_REF] and without the binder, the pigments can be readily washed away.

Sometimes, misprinted fabrics are found in mass production with digital printing due to unfamiliar errors such as lost or clogged nozzles. Such misprinted fabrics are reused by employing a color stripping process. [START_REF] Kıcık | Application of ozone gas for the stripping of fabric ink-jet-printed with reactive dyes[END_REF] Similarly, chemical recycling of cotton textiles into regenerated fibers requires removal of impurities like colorants, additives and other finishing agents from the cotton fabric. These impurities may represent major technical hurdles in maintaining the continuity of chemical recycling. Hence it is necessary to study the removal of such impurities from textiles. [START_REF] Wedin | Preparation of viscose fibres stripped of reactive dyes and wrinkle-free crosslinked cotton textile finish[END_REF] While studying the color management in the circular economy, researchers have reported the need to study the decolorization of the pigment printed goods since pigment printing is also one of the common ways to color the cellulosic textiles by digital printing or either by the traditional screen printing. [START_REF] Määttänen | Colour management in circular economy: decolourization of cotton waste[END_REF] It is difficult to remove the phthalocyanine pigments from textiles because they are deposited on textiles with resinuous and binders which fix them firmly and to some extent protect these pigments from attack by chemicals. The removal of phthalocyanine pigment colors from textiles in a substantially complete manner has been achieved by treating them in a formulation containing an aliphatic quaternary compound together with caustic soda and sodium hydrosulfite. Also the process involved high temperature and treatment time. For satisfactory removal a two-step process was found desirable in the patent. [START_REF] Cook | Process for removal of pigments from printed textiles and agents useful therein[END_REF] Hence, researchers are focusing their attention to find ecofriendly technologies to replace the conventional chemical processes followed in the textile industry. [START_REF] Agrawal | The performance of cutinase and pectinase in cotton scouring[END_REF][START_REF] Agrawal | Wax removal for accelerated cotton scouring with alkaline pectinase[END_REF][START_REF] Chatha | A biological approach for color-stripping of cotton fabric dyed with C.I. reactive black 5 using fungal enzymes from solid state fermentation[END_REF][START_REF] Preša | Bioscouring and bleaching of cotton with pectinase enzyme and peracetic acid in one bath[END_REF][START_REF] Arumugam | Enzymatic treatment of fibers for nonwovens[END_REF] Among various alternatives, ozone is an attractive choice due to its high oxidation potential (2.07 V), low pH and temperature used as compared to the conventional chemical based stripping methods. [START_REF] Arooj | A pilot-scale application of ozone to bleach raw cotton fabric using various additives[END_REF] Numerous applications of ozone have been reported in textile wet processing industry such as bleaching of cotton, [START_REF] Perincek | An investigation in the use of ozone gas in the bleaching of cotton fabrics[END_REF] jute, [START_REF] Perincek | Ozone bleaching of jute fabrics[END_REF] angora rabbit, [START_REF] Perincek | Ozone treatment of Angora rabbit fiber[END_REF] and silk. [START_REF] Sargunamani | A study on the effects of ozone treatment on the properties of raw and degummed mulberry silk fabrics[END_REF] Apart from this, Ozone has also been used to decolorize textile effluents, [START_REF] Sevimli | Ozone treatment of textile effluents and dyes: effect of applied ozone dose, pH and dye concentration[END_REF][START_REF] Sevimli | Decolorization of textile wastewater by ozonation and Fenton's process[END_REF][START_REF] Sancar | Decolorization of different reactive dye wastewaters by O3 and O3/ultrasound alternatives depending on different working parameters[END_REF][START_REF] Carrière | Decolorization of textile dye solutions[END_REF] for reduction clearing treatment of disperse dyed polyester, [START_REF] Eren | Simultaneous afterclearing and decolorisation by ozonation after disperse dyeing of polyester[END_REF][START_REF] Eren | Surface trimer removal of polyester fibers by ozone treatment[END_REF] and also in the color stripping of the reactive dyed textiles. [START_REF] Powar | Application of ozone treatment for the decolorization of the reactive-dyed fabrics in a pilot-scale process-optimization through response surface methodology[END_REF][START_REF] Eren | Colour stripping of reactive-dyed cotton by ozone treatment[END_REF] Experimental design methodology is a good strategy that makes simultaneous variation of all experimental variables feasible. Thanks of the statistical analysis of the generated data, remarkable information is provided on the interactions among the experimental variables. From here, the number of tests and the required time would be reduced, leading to a considerable reduction in the overall required cost.

Box-Behnken experimental design was used to optimize the decolorization of the pigment printed textiles and detailed study was discussed with regards to the effects of various process parameters like pH, ozone dosage, and the treatment time on the overall performance of the decolorization process. The decolorized fabric was characterized in terms of color stripping %, mechanical properties as well as the analytical tests such as scanning electron microscopy (SEM), X-ray photoelectron spectroscopy (XPS).

However, there is no work reported on the ozone application for the decolorization of the pigment printed textiles. In this study, we report for the first time the use of an ozone assisted process at pilot scale for the removal of color from a pigment printed cotton fabric.

Experimental

Materials

A 100% cotton woven fabric (well scoured and bleached quality) of 150 GSM was used throughout the study. UNISPERSE BLUE G, C.I. Pigment Blue 15 from Huntsman, a phthalocyanine pigment was used for the pigment printing experiments (Figure 1). UNISPERSE BLUE G pigment (Huntsman); APPRETAN N 9210, an acrylic ester copolymeric binder from Clariant; and LYOPRINT ® PT-RV NEW (Huntsman) a thickener were of commercial grade. All of other chemicals namely, sodium hydroxide, phosphoric acid used were of the reagent grade.

Method

The pigment paste was prepared as per the formulation described in Table 1:

For printing treatment, a Johannes Zimmer Type Magnetic System Laboratory machine was used. Printing speed was 4 m/min. For drying and curing, a Mathis Laboratory finishing line was used and the printed samples were dried and cured at 160°C for 3 min. Printing was applied to rectangular fabrics (10 × 30 cm). A simple and plain rectangular bar design pattern was selected for the printing of the fabric (Refer Figure 2).

Ozone experimental setup. The pilot scale set up consists of the ozonator, which produces ozone gas of the required concentration from pure oxygen (>99.5 wt%, Messer, France) by Corona discharge (Ozat CFS3-2G, Ozonia, France). It also includes an ozone analyzer (BMT, Germany), injection or suction pump and the residual ozone destroyer. Ozone gas was introduced into the reactor by the venturi system. The combined use of a liquid circulation pump and a venturi system allowed, the transfer of the ozone gas to the treatment liquid. The flow rate of ozone gas was 0.3 Nm 3 /h, and the flow rate of the water pump was set to 2 m 3 /h in the reactor assembly for all experiments. Figure 3 shows the experimental set-up for ozone treatment used. [START_REF] Powar | Application of ozone treatment for the decolorization of the reactive-dyed fabrics in a pilot-scale process-optimization through response surface methodology[END_REF] Procedure for ozone treatment of pigment printed cotton. A total of 34 g of pigment printed cellulosic textile (two slides of A4 size) was placed in the ozonation reactor. The decolorization process was configured by varying the process parameters such as pH values (3, 5, and 7), ozone concentration (40, 100, 160 g/Nm 3 ozone) and ozone exposure times (20, 70, 120 min). The pH of the ozonation treatment was adjusted using the various concentrations of sodium hydroxide and/or phosphoric acid (laboratory grade). Reverse osmosis water was used for all the experiments and the total volume of liquor used was 30 l for each experiment. After the completion of the ozonation treatment all the samples were rinsed with cold water and then air dried.

Characterization

Color stripping (%) measurements. The determination of the color intensity (K/S value) on the printed and the color stripped cotton fabrics were carried out using a Konica Minolta CM3600d spectrophotometer (Konica Minolta Inc., Tokyo, Japan). The K/S values for the printed and color stripped cotton fabrics were evaluated based on the Kubelka-Munk equation [START_REF] Lab | The Kubelka-Monk Theory and K / S[END_REF] at maximum absorption wavelength (λmax = 580 nm) of the pigment. For the color stripped samples, the K/S values were measured at 10 different places on each sample and the average K/S value was determined. The color stripping percentage was calculated using the following formula [START_REF] Uddin | Effects of reductive stripping of reactive dyes on the quality of cotton fabric[END_REF] :

Stripping percentage K S value of dyed sample K S value of stripped s = - a ample K S value of dyed sample             ×100
Figure 4 shows the K/S spectral curve of the printed cotton fabric with a main peak at 580 nm and two other peaks at 360 and 680 nm, respectively.

Color analysis by the CIELAB method. The CIELAB method was applied to analyze the color of the decolorized samples. The CIELAB color space also known as CIE L*a*b* was defined by the International Commission on Illumination [START_REF] Powar | Application of ozone treatment for the decolorization of the reactive-dyed fabrics in a pilot-scale process-optimization through response surface methodology[END_REF] (CIE) in 1976. In the color space CIELAB, L* corresponds for the lightness (100 = white, 0 = black), a* to the red-green coordinate from (negative sign = green, positive sign = red) and b* to the yellow-blue coordinate from (negative sign = blue, positive sign = yellow). [START_REF] Powar | Application of ozone treatment for the decolorization of the reactive-dyed fabrics in a pilot-scale process-optimization through response surface methodology[END_REF] Tensile strength testing. Tensile strength properties were evaluated using a MTS Criterion Model 43 automated testing system according to the International standard NF EN ISO 13934-1; 2013. The tensile properties were calculated only in one direction (warp wise). The mechanical properties of the treated and untreated textile fabric were obtained by repetition of the test atleast five times and the average values were taken.

Surface analysis using SEM. The changes in the surface morphology of selected fabric samples, was observed using the scanning electron microscope (SEM) Model JEOL JEM-1400 Plus, Japan. Before the measurements, all the samples were coated with carbon on JEOL JEE-4X vacuum evaporator for 5 min.

XPS analysis. All the samples were analysed with a Kratos Axis ultra DLD instrument, equipped with an Al Kα monocromatized X-ray source. The analysed area for each sample was about 700 × 300 µm.

Box-Behnken design for experimental plan

To optimize the process conditions for color stripping by ozone, to examine the effects of combined conditions (pH, Ozone content, and reaction time) and to better understand how co-effects influence the quality of the end-product especially in the color stripping and the tensile strength loss, a statistical model can be applied. Response surface methodology (RSM) is a powerful approach that can be handled to test several parameters employing a minimum number of experimental trials. Generally, it includes a collection of mathematical and statistical procedures that are operated to build an experimental design that can analyze the effects of parameters on the response variable to determine the optimized response. In this work, the Box-Behnken design (BBD), a RSM method, was used for the optimization of process variables because it is very advantageous and efficient. It can determine the factors of the quadratic model, detect lack of fit of the model, and looks more desirable if the points are at the midpoints of edges of the process and at the center. Therefore, in the present investigation, the main purpose was to optimize the color stripping of printed cotton to access the effect of process conditions (pH, ozone content, and reaction time) in conserving the mechanical characteristics. A series of experiments were carried out with values of these three variables (pH; ozone content and reaction time) included in the suitable range (lower, middle, and upper levers) (Table 2).

Some variables such as pH, the ozone concentration and reaction time which could affect the color stripping efficiency were selected to be investigated by a Box-Behnken design. The responses can be modeled by a second-order polynomial equation as the following equation: Y a a x a x a x a x x a x x a x x a x a x

= + + + + + + + + + 0 1 1 2 2 3 3 1 2 1 2 1 3 1 3 23 2 3 11 1 2 22 2 2 a a x 33 3 2
Where Y is the color stripping percent as a function of x 1 (pH), x 2 (concentration of ozone) and x 3 (reaction time) variables. a 0 is the intercept, a 1 , a 2 , a 3 to a 11 ; a 22 and a 33 are the regression coefficients.

The number of experiments (N) based on the color stripping by ozone is defined by the following expression:

N K K C = - ( )+ = + = ( ) 2 1 2 3 3 1 4 16 0 - * *
Where K denotes the number of variables and C 0 denotes the number of replication at the center point. In this study, K and C 0 were set at 3 and 4, respectively. Hence, 16 experiments had to be done to perform a Box-Behnken design. [START_REF] Box | Some new three level designs for the study of quantitative variables[END_REF][START_REF] Goupy | Introduction aux plans d'expériences. 5e éd. Paris: Toutes les techniques nécessaires à la conduite d'une étude[END_REF] The levels of each variable are listed in the Table 2.

Y represents the predicted response, a 0 is the constant term, a i is the linear coefficient, a ii is the quadratic coefficient, and a ij is the interaction coefficient. A statistical analysis of variance (ANOVA) based on Box-Behnken ) test, were used to determine the model significance. An F-test was also applied to determine the significance of the effects. The F test was used to compare statistical models that have been fitted to a data set in order to identify the model that gives the best fit to the population from which the data were obtained. Box-Behnken design is a cubic, independent quadratic and rotatable design, with the treatment combinations at the midpoints of the edges of a multidimensional cube without embedded factorial or fractional factorial design and is used for fitting second-order response surfaces. The Box-Behnken matrix design for the experimental plan of our decolorization experiments is as per Table 3.

Results

Evaluation of color stripping of the printed fabrics treated with different experiments (E1-E16

). The Figure 5 show the K/S spectral curves of the fabrics decolorized using the 16 different experimental conditions of ozone treatment. It can be observed that with the variation in the experimental conditions performed, we can see a substantial decrease in the K/S values and hence an enhancement of the decolorization.

For the experiments (E1-E7), it can be seen that the K/S values at maximum absorbance wavelength(580 nm) is still higher, with the K/S values still greater than 4 and the decolorized samples are still blue (Figure 5 E2, E4, E7). Smaller quantities of ozone concentration along with lower treatment time or higher pH value are not sufficient to obtain a good color stripping.

The K/S values at 360 nm of the stripped samples are between 0.5 and 13.9, which are higher than that of the unprinted cotton sample (K/S = 0.15) but not that lower than the pigment printed reference sample (K/S = 14). This could be related to the presence of degraded pigment molecule. Similar trend was observed for K/S values at 680 nm. The results and experimental conditions show that the complete color removal from pigment printed goods is not easy and requires the use of process conditions involving acidic pH, huge ozone concentration and longer treatment times. The same can be observed through the photograph swatches of the ozone treated samples: it is difficult to obtain clear white fabric even after the extreme conditions of ozonation process (Figure 6).

Color analysis of the color stripping samples. Sample photographs in Figure 6 show, the color stripping results obtained for parameters selected in the designed experimental plan.

The CIE Lab methodology was additionally used to carry out a colorimetric analysis of all fabric samples (Table 4).

Referring to the results listed in (Table 3), it can be clearly observed that the L* values are higher in case of experiment E12 as compared to experiments E2 and E4, and the standard fabric, which confirms that the shade of the fabric is getting lighter. However, the a* values are not closer to that of the unprinted fabric and the fabric turns out to be more on the greener side. Similar observation was found in case of b* values, which corresponds to the fabric getting less bluer as compared to the pigment printed fabric. When stripped samples subjected to selected experiments (E8, E11, and E12) are compared to the unprinted cotton fabric, it can be seen that there is still some part of degraded pigment present on the fabric.

Box-Behnken results and optimization. The result analytics of characteristics of printed textile including the Color stripping (%) and Tensile strength loss (%) may vary depending on the operating conditions, such as pH, ozone concentration and reaction time. In this study, the effect of pH, ozone concentration and reaction time were evaluated using linear regression and Solver Functions in the Microsoft Office Excel. A three-factor three level Box-Behnken experimental design was utilized to investigate the correlation between the combined effects of the individual parameters and both the responses.

In Table 5, the Box-Behnken matrix, the experimental results from all the tested combinations of factors and the corresponding responses for each run are polynomial equations obtained showing the empirical relationships For the designed experimental model, the color stripping percentage was calculated and the tensile strength was measured along with the coefficient of variation determined for both the parameters (Table 5). We can see the effects of different process parameters on the decolorization of the pigment printed goods. However, there is deviation in the results obtained which may be attributed to the heterogeneity of the treatment. It can be observed that there is decrease in the tensile strength with an increase in color stripping.

The responses modeled by second-order polynomial equations have the following particular forms:

For the Color Stripping (CS): For the Tensile Strength Loss (TSL): 6 and7 respectively.

Y CS x x x x x x x x ( )= - + + + + - 82
Y TSL x x x x x x x x ( )= - + + - + + 19
When analyzing ANOVA results, a large value of F with a small value of p (i.e. p < 0.05) show that the model is statistically significant. Among the ANOVA results reported in Tables 6 and7, the (Prof > F) value was found to be <0.05 for the Color Stripping and Tensile Strength Loss, with F value of 18.60 and 12.37, respectively, indicating a significant model fit. The F-test gave a low probability value, which also indicated the high significance of the model for both response. Moreover, the high coefficients of determination (R2) of 0.965 and 0.949 for the Color Stripping and Tensile Strength Loss responses further indicated a good correlation between the measured and predicted responses.

In this study, the independent parameters of pH (x1), ozone concentration (x2) and reaction time (x3), interaction effect between the pH and reaction time (x1x3) and the second-order effect of pH (x12), reaction time (x32), were significant parameters for the response of Color Stripping with p-value < 0.05, as shown in the Table 5. Regarding the Tensile Strength Loss response, the ozone concentration (x2) and reaction time (x3), were highly significant parameters. However, the rest of the terms did not show any significant impact.

Response surface result analysis. From the response surface plots A, B, and C in Figure 7, it can be seen that the pH of the ozonation process had a considerable impact on the decolorization of the pigment printed goods. As seen in the response surface plots, the acidic pH range showed good results in terms of the decolorization while the neutral pH showed adverse effects on the decolorization. Also the ozone concentration and the treatment time are important favorable conditions for the maximum decolorization. Severe process conditions like acidic pH, higher ozone concentration and higher reaction time were required for the efficient decolorization of the pigment printed goods.

From the response surfaces A, B and C in Figure 8, shows that the mechanical properties like tensile strength loss increased with an increase in all the process parameters (ozone concentration, and time).

Optimized process conditions and validation of model.

Based on the Color Stripping (CS) and for the Tensile Strength Loss (TSL) the responses modeled by secondorder polynomial equations have been presented before. Thanks of Solver function in Excel, the optimal value or operating's parameters were found targeting maximum Color Stripping and minimum of Tensile Strength Loss for a formula in one cell called the objective cell, subjected to certain constraints of limits, (e.g. x 1 < 1; x 1 > -1 it means that the pH is between 3 and 7, x 2 < 1; x 2 > -1 it means that the ozone concentration is between 40 and 160 g/Nm3 and x 3 < 1; x 3 > -1 it means that the reaction time is between 20 and 120 min). The weight of CS and TSL were taken part in 50%/50%. This means that the Solver works with a group of cells called decision variables that were used in computing the formulas in the objective and constraint cells. Solver adjusts the values in the decision variable cells to satisfy the limits on constraint cells and produces the waiting results for the objective cell.

If the conditions; Color Stripping must be greater than 80%; and Tensile Strength Loss must be lower than 20% are set up, then the optimal conditions are pH 4, with ozone concentration 40 gO 3 /Nm 3 ; and reaction time: 102 min.

Characterization of the stripped fabrics. In this study, the pigment printed samples subjected to the different ozonation process parameters were chosen as the representative fabrics to study the decolorization phenomenon. In addition to that, pigment printed fabric and blank cotton fabric were selected as reference to study the effect of ozone on the decolorization process (Table 8). SEM Images. Figure 9 shows SEM images of ozone treated samples compared with the untreated blank cotton and pigment printed cotton. SEM micrograph of unprinted cotton fabric shows the clear surface. When we look at the SEM image of the pigment printed cotton fabric, it can be clearly observed that the fibers are coated with a polymeric binder film, which helps in the binding and fixation of the pigment particles onto the fabric samples. SEM micrographs of the decolorized samples with various ozonation process conditions, confirmed the presence of the binder deposition on the fiber surface of the printed samples. This shows that even after extreme ozonation it is difficult to remove the polymeric binder coating from the pigment printed fabric.

XPS analysis to study the surface chemistry. On the survey XPS spectra, four peaks are observed corresponding to the electrons C1s of the C atoms (285/290 eV), O1s of O atoms (531/536 eV), to N1s of N atoms (398 eV), to Cu2p3/2 of Cu atoms (932/935 eV). The percentage of each species is calculated by the percent area of each peak (Table 9).

During the ozone treatment for the selected samples, the oxygen content increases due to the oxidation processes that occurs in presence of ozone. This treatment breaks the covalent bonds of the binder and the pigment, while introducing oxygen in the chain.

Copper and Nitrogen atoms are attributed to the pigments, thus there is a very small amount of those species and the XPS is not very accurate. However, it is obvious that the ozonation treatment decreases the amount of Copper and Nitrogen. This could be attributed either to the removal of the pigment due to chain scissions of the acrylic binder or to the oxidation of the pigment molecule by the ozonation treatment. Bigambo et al. [START_REF] Bigambo | Surface chemical and colorimetric analysis of reactively dyed cellulosic fabric. The effect of ISO 105 CO9 laundering and the implications for waste cellulosic recycling[END_REF] have studied the effect of oxidative bleaching on the reactive dyed goods which is in line with results observed in our study.

Although the discoloration of the sample is very weak (23.1%) with the E2 treatment at pH 7 the degree of oxidation seems similar to that observed with the E13 treatment at pH 5 and longer time with better results of color stripping (83.7%). It can also be observed that E11 and E12 experiments result in identical stripping, while the O 3 amount is much higher with E11. This could be attributed to the complete removal of the copper during E12 experiment. Again, the effect of the acidic pH for this experiment could be one explanation. As a conclusion, the XPS results confirm the interest of ozonation at lower pH, with a preferential oxidation of the pigment and a better discoloration of the samples.

Discussion

Oxidation by ozone is considered to be a powerful technique for decolorization of the reactive dyes by the destruction of the chromophoric system. There may be occurrence of two possible reaction mechanisms: direct reaction involving molecular ozone attack and the indirect reaction consisting of the free radical mechanism. Both of these reaction mechanisms have been found to exist simultaneously during the ozonation reaction process. [START_REF] Sharma | Studies on degradation of reactive red 135 dye in wastewater using ozone[END_REF] Recent studies for color removal from pigment wastewater have demonstrated 90% color removal using the ozone. The ozone dosage for color removal from pigment waste water was higher (4-5 times) than the dye waste water so as to obtain the same color removal efficiency using identical conditions. The high dosage of ozone required is probably explained by the presence of the low nucleophilic chromophores of the color imparting organics in the pigment waste water and the higher alkalinity. [START_REF] Lee | Removal of color from wastewater using various ozonation techniques[END_REF] The color removal in pigment printed goods is in agreement with this study of color removal from the pigment waste water, we were able to achieve the decolorization of about 90% and more, but the process conditions deployed were harsh (huge ozone concentration and treatment time; acidic pH range). May be due to the cross-linking of the binder at the fabric surface.

Studies on the decolorization of the pigment wastewater with ozone have shown that 85% decolorization rate was achieved. [START_REF] Saastamoinen | Chemical stability of printing ink colourants[END_REF] Ozonation was more successful at low pH (pigment dissolved in H 2 SO 4 , pH < 1), than at higher pH (pigment dissolved in ethanol, pH = 6). The reason could be related to the unselective hydroxyl groups caused by higher pH or poor solubility. [START_REF] Sevimli | Ozone treatment of textile effluents and dyes: effect of applied ozone dose, pH and dye concentration[END_REF][START_REF] Saastamoinen | Chemical stability of printing ink colourants[END_REF][START_REF] Ciardelli | Ozone treatment of textile wastewaters for reuse[END_REF][START_REF] Ma | Quantitative prediction of direct and indirect dye ozonation kinetics[END_REF][START_REF] Demirev | Ozonation of two acidic azo dyes with different substituents[END_REF][START_REF] Liu | Evaluation of selected operational parameters for the decolorization of dye-finishing wastewater using UV/ozone[END_REF] Also, the studies have shown that the decolorization rate increases with the increase in the ozone dose. This study assumes the possibility of the complete decolorization of the Pigment Blue 15 could be carried out with the higher amount of the ozone dosages. [START_REF] Saastamoinen | Chemical stability of printing ink colourants[END_REF][START_REF] Tosik | Dyes color removal by ozone and hydrogen peroxide: some aspects and problems[END_REF] However, the experimental conditions (e.g. the Pigment Blue 15 dissolved in concentrated H 2 SO 4 ), were based on unrealistic scenarios and impossible for practical applications. Considering the decolorization efficiency, lower pH values were often more efficient, the reason being selective direct ozone reactions targeting chromophoric bonds in the colored materials. [START_REF] Adams | Effect of pH and gas-phase ozone concentration on the decolorization of common textile dyes[END_REF] In a study related to the bleaching of cotton fabrics with ozone, highest DP loss was observed in case of ozone bleached fabrics as compared to the treatment with sodium hypochlorite and hydrogen peroxide. Also, it was found that in case of ozone bleaching, the DP of cotton was decreased largely based on the pH of the fabric. Hydrocellulose starts to form with an increase in the acidity of the fabrics and hence the DP decreases. Likewise, decrease in the DP values, the physical properties of fabrics like breaking strengths were also found to be decreased at the neutral and acidic pH values. [START_REF] Perincek | An investigation in the use of ozone gas in the bleaching of cotton fabrics[END_REF] While discussing the use of ozone in the textile industry, the author has stated that the strength loss is very high in the case of ozonated fabric, and research to minimize the damage due to the ozone are continued. [START_REF] Perincek | Ozone: new tendency in textile finishing[END_REF] While studying the parameters affecting dry and wet ozone bleaching of denim fabric, it was found that ozone slightly damaged the denim fabrics by reducing the tensile strength. Moreover, the ultimate tensile strengths decreased with an increase in the ozone concentration and the treatment time. The loss in the strength could be attributed to the damage induced by the ozone due to the hydrolysis of the glycosidic bonds. [START_REF] Hmida | Study of parameters affecting dry and wet ozone bleaching of denim fabric[END_REF] In our case, similar results were obtained with respect to the losses in mechanical properties of the printed goods.

There is a hypothesis in our study which may suggest that the main reaction of decolorization of the pigment printed goods occurred via oxidation of the binder with chain breaks (degradation) -so it allowed pigments to be removed -almost partially.

Conclusion

This paper proposes an ozone assisted process for the decolorization of the pigment printed cellulosic textiles. Trials were performed with the selected pigment (C.I. Pigment Blue 15) printed fabric based on the copper phthalocyanine chemistry. The results indicate that with the current experimental conditions, pH 5, ozone concentration of 100 gO 3 /Nm 3 and the exposure time of 120 min yielded the best decolorization results. Decolorization of 90% and above was obtained for the samples with ozone treatment time of 120 min at acidic pH range. The current experimental investigation indicates that this Ozone based process has potential to be proposed for the industrial scale up in the textile industry and it would be interesting to propose this study on a wide range of pigment dyed/printed goods for decolorization. SEM analysis suggests that it is not easy to remove the binder film and/or pigment decolorization completely. XPS analysis reveals the decolorization of the pigments with the oxidative ozone treatment. Further studies need to be implemented to enhance the process for complete removal of the pigment, it would also be interesting to investigate the pigment stripped fabric for chemical recycling of textiles. In the current study we have focused on the decolorization of only one pigment type CI Pigment Blue 15. It would be interesting to apply this decolorization technique to a wider range of pigment colorants and observe their behavior since each colorant has its individual characteristics and has to be separately evaluated.

Figure 1 .

 1 Figure 1. Structure of C.I. Pigment Blue 15.

Figure 4 .

 4 Figure 4. K/S spectrum of the blue pigment printed fabric.

  and the independent variables in terms of coded factors for both Color Stripping (CS) and Tensile Strength Loss (TSL).

Figure 5 .

 5 Figure 5. K/S versus Wavelength graph for all the color removal experiments carried with ozone under varying conditions.

Figure 6 .

 6 Figure 6. Images of the small swatch of the standard printed fabric and the ozone assisted treatment samples with the (selected) various experimental parameters: (a) Standard printed fabric, (b) E2, (c) E4, (d) E7, (e) E8, (f) E11, (g) E12, and (h) E16.

Figure 7 .

 7 Figure 7. 3D response surface diagrams showing the effects of ozone concentration and treatment time on the color stripping of the pigment goods at different pH values.(color of the graphs are related to color stripping %): (a) pH 3, (b) pH 5, and (c) pH 7.

Figure 8 .

 8 Figure 8. 3D response surface diagrams showing the effect of pH, [O3], and reaction time on the tensile strength loss %: (a) pH = 3, (b) pH = 5, and (c) pH = 7.

Figure 9 .

 9 Figure 9. SEM micrographs before and after ozone treatment. The scale bar of the images is 50 µm: (a) blank fabric, (b) pigment printed fabric, (c) sample treated E11, (d) sample treated E12, (e) sample treated E13, and (f) sample treated E2.

Table 1 .

 1 Components of the pigment printing paste.

	Printing paste components	g/kg
	Pigment colorant	30
	Binder	80
	Thickener	40
	Water	850
	Total	1000

Figure 2. Pigment printed cotton fabric. Figure 3. Pilot

ozonation (1: ozone generator; 2: ozone analyzer; 3: venturi injection system; 4: circulation pump; 5: filter; 6: dissolved ozone analyzer and pH meter).

Table 2 .

 2 Range and levels of parameter in Box-Behnken experimental design.

	Parameter	Factors Levels
			-1	0	+1
	pH	x 1	3	5	7
	Ozone concentration (g/Nm 3 )	x 2	40	100	160
	Reaction time (min)	x 3	20	70	120

design was performed using Design Expert to determine the fitness, suitability, and significance of the model coefficient. It is important to bring off ANOVA to examine the significance and the fit of the second-order polynomial equation. Various statistical parameters provided by Linear Regression and Solver Function of Excels of Microsoft Office (ANOVA), lack of fit test, and multiple determination coefficients (R 2

Table 3 .

 3 Box-Behnken matrix used.

	Run	Coded level
	Actual level of factors	factors
	pH [O 3 ] g/Nm 3 Reaction time x1	x2 x3

Table 4 .

 4 Colorimetric values (L*, a*, b*, dE*) for the printed, unprinted cotton and stripped printed cotton subjected to color stripping experiments E2, E4, E7, E8, E11, E12, E16.

	Description	L*(D65)	a*(D65)	b*(D65)	dE*ab(D65)	dE CMC (l:c)(D65)
	Reference pigment printed fabric	31.54	-6	-36.21	-	-
	Unprinted cotton fabric	83.61	-0.14	0.19	63.81	64.97
	E2	32.42	-12.91	-25.88	12.47	8.36
	E4	40.08	-14.58	-19.78	20.64	16.4
	E7	46.97	-17.26	-15.75	28.31	25.05
	E8	59.03	-14.63	-12.49	37.73	37.93
	E11	63.51	-16.08	-6.69	45	44.73
	E12	65.37	-13.07	-9.59	44.02	45.43
	E16	46.68	-15.03	-16.28	26.86	23.8

Table 5 .

 5 Box-Behnken matrix and the experimental results.

	Run	Actual level of factors		Response
		pH	[O 3 ] g/Nm 3	Reaction	Color	CV %	Tensile	CV %	Tensile strength
				time	stripping (%)		strength (N)		loss (%)
	E1	5	40	20	48.51	16	340	6	4
	E2	7	100	20	23.1	14	342	7	3
	E3	5	160	20	69.19	17	313	6	11
	E4	3	100	20	64.91	21	329	6	7
	E5	3	40	70	74.47	28	306	8	13
	E6	7	40	70	56.89	14	310	4	12
	E7	7	160	70	78.97	27	257	7	27
	E8	3	160	70	91.56	40	244	7	31
	E9	5	40	120	86.63	26	269	4	24
	E10	7	100	120	83.93	26	224	7	36
	E11	5	160	120	94.22	41	215	12	39
	E12	3	100	120	94.92	45	231	3	34
	E13	5	100	70	83.72	18	300	4	15
	E14	5	100	70	81.64	24	260	6	26
	E15	5	100	70	83.8	22	289	6	18
	E16	5	100	70	79.9	22	287	10	18

Table 6 .

 6 ANOVA results of the quadratic model for color stripping (%).

	Source	Coefficient	Error type	Statistic t	p-value Prob > F
	Constant	82.27	2.78	29.62	9.82E-08
	x 1 (pH)	-10.37	1.96	-5.28	1.87E-03
	x 2 (ozone concentration)	8.43	1.96	4.29	5.13E-03
	x 3 (reaction time)	19.25	1.96	9.80	6.50E-05
	x 1 x 2	1.24	2.78	0.45	6.70E-01
	x 1 x 3	7.70	2.78	2.77	3.23E-02
	x 2 x 3	-3.27	2.78	-1.18	2.83E-01
	x 1	2	-7.36	2.78	-2.65	3.81E-02
	x 2	2	0.57	2.78	0.20	8.45E-01
	x 3	2	-8.19	2.78	-2.95	2.56E-02

Table 7 .

 7 ANOVA results of the quadratic model for tensile strength loss (%).

	Source	Coefficient	Error type	Statistic t	p-value Prob > F
	Constant	19.25	2.05	9.38	8.31E-05
	x 1 (pH)	-0.88	1.45	-0.60	5.68E-01
	x 2 (ozone concentration)	6.88	1.45	4.74	3.19E-03
	x 3 (reaction time)	13.50	1.45	9.31	8.71E-05
	x 1 x 2	-0.75	2.05	-0.37	7.27E-01
	x 1 x 3	1.50	2.05	0.73	4.92E-01
	x 2 x 3	2.00	2.05	0.97	3.67E-01
	x 1 2	1.00	2.05	0.49	6.43E-01
	x 2 2	0.50	2.05	0.24	8.16E-01
	x 3 2	-0.25	2.05	-0.12	9.07E-01

  [START_REF] Saastamoinen | Chemical stability of printing ink colourants[END_REF]

Table 8 .

 8 Selected samples for characterization.

	Experiment	pH	[O 3 ] g/Nm 3	Reaction	Color
	number			time	stripping %
	E11	5	160	120	94.2
	E12	3	100	120	94.2
	E13	5	100	70	83.7
	E2	7	100	20	23.1

Table 9 .

 9 XPS spectra for selected samples with Surface elemental composition.

	Sample	Surface atomic composition
		(at. %)
		C	O	N	Cu
	Pigment printed cotton fabric	88.6	9.7 1.6 0.2
	Sample treated with expt E11	80.1	19.8 0	0.1
	Sample treated with expt E12	81.1	18.9 0	0
	Sample treated with expt E13	83.2	16.6 0	0.1
	Sample treated with expt E2	82.0	17.4 0.5 0.1

Acknowledgements

The authors would sincerely thank the UniLaSalle, Beauvais platform for their help and support in the ozone experiments. The author would like to acknowledge Christian Catel from GEMTex Lab at ENSAIT (France) for his help in the laboratory experiments.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research has been conducted in the framework of Erasmus Mundus Joint Doctorate Project -Sustainable Management and Design for Textiles (SMDTex), which is financed by the European Commission.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.


Albert Cohen

Giovanni Migliorati

Near-optimal approximation methods for elliptic PDEs with lognormal coefficients

 for best n-term approximation by Hermite polynomials, under the same minimial assumptions on the Gaussian random field. This is in contrast with the current state of the art results for the stochastic Galerkin method that suffers the lack of coercivity due to the lognormal nature of the diffusion field. Numerical tests with b as the Brownian bridge confirm our theoretical findings.

Introduction 1.Elliptic PDEs with lognormal coefficients

The theoretical and numerical treatment of parametric PDEs with a large number of deterministic or stochastic parameters has been the object of intensive investigation in recent years, see [START_REF] Cohen | Approximation of high dimensional parametric PDEs[END_REF] and references therein. One principal objective is to design numerical approximation methods that are immune to the curse of dimensionality in the sense that approximation rates can be established even in the case of countably many variables.

The present work is devoted to design and study concrete numerical methods that provably achieve this goal in the case of elliptic PDEs with lognormal coefficients. While our analysis can be applied to other models, we focus on the main following prototype: given a bounded Lipschitz domain D ⊂ R d with d = 1, 2, 3 and a function f ∈ L 2 (D), we consider the elliptic partial differential equation

-div(a∇u) = f, in D, (1.1)
completed with homogeneous Dirichlet boundary conditions:

u = 0, on ∂D. (1.2)
We assume that the diffusion coefficient a is a random function defined as a = exp(b), (1.3) where b is a centered Gaussian process defined on D, with covariance function

C b (x, x ) := E(b(x)b(x )), x, x ∈ D.
(1.4)

The motivation for considering such a diffusion model is for instance the description of sedimental media with high contrast in ground water modeling. We are interested in parametric expansions of the form a = a(y) = exp(b(y)) with b(y) = j≥1 y j ψ j ,

where y = (y j ) j≥1 is a sequence of independent standard Gaussian random variables and the functions ψ j : D → R are given. An instance of the expansion (1.5) is the Karhunen-Loève (KL) expansion b = j≥1 ξ j ϕ j using the L 2 (D)-orthonormal eigenfunctions (ϕ j ) j≥1 of the integral operator with kernel C b and defining ψ j := λ j ϕ j with λ j := E(ξ 2 j ). Other expansions may be considered as well, possibly more relevant and/or advantageous than KL for our numerical purposes, as recently shown in [START_REF] Bachmayr | Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients[END_REF].

The sequence y ranges over the unbounded domain

U := R N ,
sometimes referred as the parameter set. In this paper we work with the usual product measure space given by (U, B(U

), dγ) = (R N , B(R N ), dγ), (1.6)
where B(U ) = B(R N ) denotes the Σ-algebra generated by the Borel cylinders and dγ the tensorized Gaussian probability measure on U . For any given y ∈ U such that b(y) ∈ L ∞ , Lax-Milgram theory allows us to define the solution u(y) in the space V := H 1 0 (D) through the variational formulation D a(y)∇u(y)∇v

= D f v, v ∈ V. (1.7)
Note that for each y the solution u(y) is a function x → u(x, y) of the spatial variable x ∈ D, and the gradient ∇ and integration over D in the above formulation are with respect to this variable, which we often omit to lighten the notation. The standard Lax-Milgram a-priori estimate tell us that u(y

) V ≤ C exp( b(y) L ∞ ), (1.8)
where C := f V = f H -1 ≤ C P f L 2 and C P is the Poincaré constant of D. The variational formulation extends to any f ∈ V , replacing the right side integral by the duality bracket f, v .

Approximation results

Let us recall some results from [START_REF] Bachmayr | Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients[END_REF][START_REF] Cohen | Multivariate Approximation in Downward Closed Polynomial Spaces[END_REF][START_REF] Hoang | N-term Galerkin Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs[END_REF] on the analysis and approximation of the solution map y → u(y), induced by the model (1.1)- (1.4). Such results involve the size properties of the functions ψ j . More precisely, in [START_REF] Bachmayr | Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients[END_REF][START_REF] Cohen | Multivariate Approximation in Downward Closed Polynomial Spaces[END_REF] these size properties are described in the mildest form through conditions of the type sup

x∈D j≥1 ρ j |ψ j (x)| = K < ∞, (1.9)
where ρ = (ρ j ) j≥1 is a given sequence of positive number with prescribed growth. This type of condition has the merit of taking into account the support properties of the functions ψ j , in contrast to conditions on the summability of the ψ j L ∞ which were initially proposed in [START_REF] Hoang | N-term Galerkin Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs[END_REF]. The solution map is only defined on the set

U 0 := {y ∈ U : b(y) ∈ L ∞ }.
Note that this set is equal to U in the case of finitely many variable, that is, when ψ j = 0 for j > J.

However, in both finitely or countably many variable cases, the solution map is not uniformly bounded. We thus study this map in the Bochner spaces

V k = L k (U, V, dγ) = L k (U, dγ) ⊗ V,
of functions from U to V := H 1 0 (D) which are strongly measurable and such that y → u(y) V belongs to L k (U, dγ).

One sufficient condition established in [START_REF] Charrier | Strong and weak error estimates for elliptic partial differential equations with random coefficients[END_REF] for membership in V k for all finite k is that the covariance function C b has C ε Hölder smoothness for some ε > 0. The following result from [START_REF] Bachmayr | Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients[END_REF] gives an alternative criterion in terms of a very mild growth conditions on the sequence (ρ j ) j≥1 such that (1.9) holds.

Theorem 1.1 Assume that there exists a sequence (ρ j ) j≥1 of positive numbers such that (1.9) holds and j≥1 exp(-ρ 2 j ) < ∞. Then U 0 has full measure and E(exp

(k b(y) L ∞ )) < ∞ for all k < ∞. In turn the map y → u(y) belongs to V k for all k < ∞.
Under the above assumptions, the solution map is in particular in V 2 and may therefore be expanded in Hermite series

u = ν∈F u ν H ν (y), H ν (y) = j≥1 H ν j (y j ), u ν := U u(y)H ν (y) dγ, (1.10)
where F denotes the set of finitely-supported sequences of nonnegative integers, and (H k ) k≥0 is the sequence of univariate Hermite polynomials. These polynomials are defined by

H k (t) := (-1) k √ k!g(t) d k dt k (g(t)) , g(t) = 1 √ 2π exp(-t 2 /2),
and normalized according to R |H k (t)| 2 g(t)dt = 1. Note that (H ν ) ν∈F is an orthonormal basis of L 2 (U, dγ) and that the Hermite coefficients u ν are elements of V .

Truncation of the Hermite series to a finite set Λ ⊂ F yield a approximation u Λ = ν∈Λ u ν H ν that belongs to the V -valued polynomial space

V Λ := V ⊗ P Λ , where P Λ := span {H ν : ν ∈ Λ}. By Parseval's equality u -u Λ V 2 = ν / ∈Λ u ν 2 V 1/2 .
For any given n, the set Λ of cardinality #(Λ) = n that minimizes the above tail bound is the set Λ * n of indices that corresponds to the n largest u ν V . This set is not necessarily unique and it is not accessible in practice. It is however possible to easily access a family of suboptimal sets (Λ n ) n≥1 for which u -u Λn V 2 satifies a decay estimate of the form n -s under certain growth conditions on the weights (ρ j ) j≥1 such that (1.9) holds. This is based on two main results from [START_REF] Bachmayr | Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients[END_REF]. The first result establishes a weighted 2 estimate for the u ν V .

Theorem 1.2 Let r ≥ 1 be an integer. Assume that there exists a positive sequence ρ = (ρ j ) j≥1 such that j≥1 exp(-ρ 2 j ) < ∞ and such that j≥1

ρ j |ψ j (x)| = K < C r := ln 2 √ r , x ∈ D. (1.11)
Then, one has

ν∈F ξ ν u ν 2 V < ∞, (1.12)
where

ξ ν := ν ∞ ≤r ν ν ρ 2 ν = j≥1 r l=0 ν j l ρ 2l j , ρ ν := j≥1 ρ ν j j , ν ν := j≥1 ν j ν j , (1.13)
with the convention that k l = 0 when l > k. The constant bounding this sum depends on f V , j≥1 exp(-ρ 2 j ) and on the difference C r -K.

The truncation strategy now consists in taking for Λ n the set corresponding to the n smallest weights ξ ν , so that

u -u Λn V 2 ≤ Cξ -1/2 n+1 , (1.14)
where C is the value of the weighted sum in (1.12) and (ξ k ) k≥1 is the increasing rearrangement of the (ξ ν ) ν∈F . In order to further bound the right-hand side in (1.14), we use a second result that relates the q summability of the ρ -1 j and ξ

-1/2 ν
where the ξ ν are the weights in (1.13).

Theorem 1.3 For any 0 < q < ∞ and any integer r ≥ 1 such that q > 2 r , we have

(ρ -1 j ) j≥1 ∈ q (N) =⇒ (ξ -1/2 ν ) ν∈F ∈ q (F).
Since for any positive decreasing sequence α = (α k ) k≥1 ∈ q , one has α k ≤ Ck -1/q with C = α 1/q q , the combination of the two above results yields the following convergence bound.

Theorem 1.4 If (1.11) holds with (ρ -1 j ) j≥1 ∈ q (N) for some 0 < q < ∞, then u -u Λn V 2 ≤ Cn -s , s := 1 q ,
where Λ n is a set corresponding to the n largest ξ

-1/2 ν
, with ξ ν given by (1.13).

Remark 1.5 Since renormalization of the sequence ρ does not affect the fact that (ρ -1 j ) j≥1 ∈ q (N), we may replace (1.11) by the condition (1.9) without a specific constraint on K. However the construction of the set Λ n requires to define the ξ ν using a normalization of ρ j such that (1.11) holds.

Remark 1.6 One of the advantage of the above results is that condition (1.9) takes into account the spatial localization properties of the functions ψ j , in contrast to the analysis firstly proposed in [START_REF] Hoang | N-term Galerkin Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs[END_REF] that uses conditions on the norms ψ j L ∞ (D) . This is particularly effective in the case of Gaussian fields b that can be described by multiresolution expansions of wavelet type, such as the Brownian bridge and the general class of multivariate Matérn processes as discussed in [START_REF] Bachmayr | Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients[END_REF].

One elementary yet important observation, see Lemma 4 from [START_REF] Cohen | Multivariate Approximation in Downward Closed Polynomial Spaces[END_REF], is that for any integer r ≥ 1, the sequence (ξ ν ) ν∈F is monotone increasing, that is

ν ≤ ν =⇒ ξ ν ≤ ξ ν ,
where ν ≤ ν means that νj ≤ ν j for all j. This implies that the Λ n may be chosen so to have the following property: a set Λ is downward closed if

ν ∈ Λ and ν ≤ ν =⇒ ν ∈ Λ.
This property plays an important role in the development of numerical methods for parametric and stochastic PDEs like (1.1), and we use it for those developed in the present paper. We say that the polynomial space P Λ := span {H ν : ν ∈ Λ} is downward closed whenever the set Λ is downward closed. Notice that, in this setting, the H ν can be replaced by the monomials y → y ν or more generally any tensorized polynomial basis G ν .

Numerical methods

Once given the sets Λ n , the calculation of the truncated Hermite series u Λn requires the coefficients u ν for all ν ∈ Λ n . These coefficients belong to the space V , and in general they cannot be computed exactly. The main purpose of the present paper is the development and analysis of numerical methods for the approximation of these coefficients. We aim at methods that provably achieve the convergence rate that can be established for the ideal approximation u Λn , under the same assumption (1.9) with (ρ -1 j ) j≥1 ∈ q , and at reasonable computational cost. One prominent numerical method for parametric elliptic problem is the stochastic Galerkin method, which is based on a variational formulation in both space and parametric variable. In the present case, the most natural such formulation is obtained by integrating the spatial variational formulation (1.7) on the parametric domain U with respect to the measure γ: find u in V 2 such that

A(u, v) = L(v), v ∈ V 2 ,
where the bilinear and linear forms A and L are defined by

A(u, v) := U D
a(y)∇u(y)∇v(y) dγ(y) = U ×D a(x, y)∇u(x, y).∇v(x, y) dx dγ(y),

L(v) := U D f v(y) dγ(y) = U ×D f (x)v(x, y) dx dγ(y).
The Galerkin method then amounts to restricting the above variational formulation to a subspace of V 2 such as the polynomial space V Λn . However this approach leads to a generally ill-posed problem, due to the degeneracy of the lognormal diffusion coefficient a(x, y) which is positive but may attain arbitrarily small or large values. As a consequence the bilinear form A is neither continuous nor coercive on V 2 , thus preventing the use of Lax-Milgram theory and Cea's lemma. More precisely, the quadratic form associated to the bilinear form A is given by This y-dependent norm and the fixed V -norm satisfy the equivalence

A(u, u) = U u(y)
exp (-b(y) L ∞ ) v 2 V ≤ v 2 a(y) ≤ exp (+ b(y) L ∞ ) v 2 V , v ∈ V, (1.15)
with sharp constants that depend on y and are not uniformly controlled over U . Therefore A(u, u) is not equivalent to u 2 V 2 . Several alternatives have been developed in the literature in order to circumvent this obstacle. One approach first proposed in [START_REF] Galvis | Approximating infinity-dimensional stochastic Darcy's equations without uniform ellipticity[END_REF] is to modify the choice of spaces in the formulation and establish proper inf-sup conditions. Another approach introduced in [START_REF] Gittelson | Stochastic Galerkin discretization of the log-normal isotropic diffusion problem[END_REF] is based on the use of an auxilliary Gaussian measure dγ and corresponding formulation

Ã(u, v) = L(v),
where à and L are defined similarly as A and L with γ replaced by γ. This second approach is used in [START_REF] Hoang | N-term Galerkin Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs[END_REF] where approximation rates for the Galerkin method on appropriate polynomial spaces are established. This however requires conditions of the type (j ψ L ∞ ) j≥1 ∈ p for some p ≤ 1.

It is not clear if the methods in [START_REF] Galvis | Approximating infinity-dimensional stochastic Darcy's equations without uniform ellipticity[END_REF][START_REF] Gittelson | Stochastic Galerkin discretization of the log-normal isotropic diffusion problem[END_REF][START_REF] Hoang | N-term Galerkin Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs[END_REF] may yield the same convergence rate that can be established for the ideal approximation u Λn , under the mild assumption (1.9) with (ρ -1 j ) j≥1 ∈ q . In the present paper we adopt a different strategy based on weighted least squares that allows us to achieve this goal by computationally feasible methods.

Outline of the paper

As a first step, discussed in §2, we proceed to a discretization in the spatial variable x, by replacing for each y ∈ U the solution u(y) ∈ V defined in (1.7) by its discrete counterpart u h (y) ∈ V h defined by the Galerkin method

D a(y)∇u h (y)∇v h = D f v h , v h ∈ V h .
Here V h ⊂ V is a finite-dimensional approximation space, such as a finite element space. Our main result in this section gives convergence bounds as h → 0 for the V 2 error between the solution map y → u(y) and its discrete counterpart y → u h (y). These bounds are established by classical elliptic regularity estimates when the random fields b and a are sufficiently smooth, and extended to arbitrarily rough fields by interpolation theory arguments. We illustrate these bounds for Gaussian fields b described by multiresolution expansions, already mentionned in Remark 1.6.

In §3, we discuss the computation of the polynomial approximation by the discrete weighted least-squares method: we compute particular snapshots u(y i ) for i = 1, . . . , m, up to space discretization in V h , where y i ∈ U are randomly sampled according to some measure dµ, and obtain the approximation by minimizing a criterion of the form

m i=1 w(y i ) u(y i ) -v(y i ) 2 V , over v ∈ V Λn .
Here, we exploit the results from [START_REF] Cohen | Optimal weighted least-squares methods[END_REF], which show that a proper choice of the weight function w and sampling measure dµ provides optimal error estimates in expectation, with m of the same order as n up to a logarithmic factor. In §4, we discuss the convergence rates guaranteed by our error analysis, for particular Gaussian processes b of practical interest: univariate Brownian processes and multivariate Matérn covariances. The chosen expansion (1.5) is of wavelet type, building up on the results in [START_REF] Bachmayr | Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients[END_REF]. With such choices, it is shown that the obtained rates for spatial and parametric approximation are the same and proportional to the smoothness parameter δ > 0 appearing in the covariance function for Matérn processes. This is in contrast with the use of standard Karhunen-Loeve basis for which the provable parametric rate is lesser than the spatial rate.

Finally numerical tests are presented in §5 for the weighted least-squares method, for the Brownian processes truncated up to various parametric dimensions.

Space discretization

In this section, we introduce and study a semi-discretization of the solution u to the parametric elliptic problem (1.1), with respect to the spatial variable x ∈ D. For this purpose, we denote by (V h ) h>0 a family of closed finite-dimensional subspaces of V of dimension

n h := dim(V h ), where n h → +∞ as h → 0.
For any y ∈ U such that b(y) ∈ L ∞ (D), we define u h (y) ∈ V h as the Galerkin approximation to u(y):

D a(x, y)∇u h (x, y)∇v h dx = D f v h dx, v h ∈ V h . (2.1)
Note that for a given y, each query of the approximate solution map

y → u h (y),
amounts to solving an n h × n h linear system. Similar to the exact solution map y → u(y), the approximate solution map satisfies the following measurability and integrability property under mild growth conditions on the sequence (ρ j ) j≥1 such that (1.9) holds.

Proposition 2.1 If the assumptions of Theorem 1.1 hold, then the map y → u h (y) is measurable and belongs to V k for all k < ∞.

By the previous proposition, the approximate solution map y → u h (y) is in V 2 so that we can expand it into multidimensional Hermite series

u h (y) = ν∈F u ν,h H ν (y),
where the coefficients u ν,h belong to V h . Let Λ be a given subset of F with finite cardinality n. We introduce the subspace

V Λ,h := V h ⊗ P Λ of V 2 , which has dimension n h × n.
In §3 we construct an approximation u Λ,h ∈ V Λ,h to the solution map by the weighted least-square method, which can be viewed as a vehicle to obtain discretizations in the parametric variable. The error estimate in V 2 between u and u Λ,h may be bounded according to

u -u Λ,h V 2 ≤ u -u h V 2 + u h -u Λ,h V 2 , (2.2)
where the first and second terms on the right-hand side represents the spatial and parametric discretization errors, respectively. The estimation of the second term depends on the chosen discretization method in the parametric variable, and it is discussed in §3. Here, we concentrate on the first error term u -u h V 2 . The rate of approximation which is achievable for this error term depends on the smoothness of the Gaussian field b.

Smooth Gaussian field

Let us introduce the spatial regularity space

W := {v ∈ V : ∆v ∈ L 2 (D)}, equipped with the norm v W := ∆v L 2 .
We assume that the spaces V h possess the following approximation property with respect to the space W : there exist C > 0 and t > 0 such that min

v h ∈V h v -v h V ≤ Cn -t h v W , v ∈ W, h > 0. (2.3)
It is well known that W coincides with V ∩ H 2 (D) with equivalent norms when the domain D is either convex or has C 1,1 boundary. Therefore, in this case, taking for (V h ) h>0 the P 1 Lagrange finite elements associated to a regular family of quasi-uniform conforming simplicial partitions (T h ) h>0 where h is the mesh-size of T h , we may use results from classical finite-element approximation theory [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF] min

v h ∈V h v -v h V ≤ Ch|v| H 2 , v ∈ H 2 (D), h > 0, (2.4)
to conclude that (2.3) hold with rate t = 1 d . It is also well known that this rate can be maintained in the case where D is a nonconvex polygon or polyhedron provided that the spaces V h are enriched by appropriate local refinement near the re-entrant corner or edges.

In order to study the error term u -u h V 2 , we need to study the integrability of y → u(y) in the sense of the Bochner space

W k = L k (U, W, γ),
which is obviously smaller than V k . We begin by observing that for the nonparametric equation (1.1), the solution has W -regularity when in addition to being elliptic, the coefficient a belongs to W 1,∞ (D). Indeed, the equation then writes

∆u = a -1 (∇a • ∇u -f ),
and since it was assumed that f ∈ L 2 (D), one has

u W ≤ a -1 min ( ∇a L ∞ u V + f L 2 ) ≤ a -1 min ( ∇a L ∞ a -1 min f V + f L 2 ), (2.5)
where

a min = min x∈D a(x) > 0. If a = exp(b) for some b ∈ L ∞ , this gives u W ≤ exp( b L ∞ )( ∇b L ∞ exp(2 b L ∞ ) f V + f L 2 ). (2.6) Note that f V ≤ C P f L 2
where C P is the Poincaré constant. This leads to the following result that gives a condition ensuring L k -integrability of u(y) W with respect to the Gaussian measure dγ.

Proposition 2.2 If the Gaussian field b belongs to W 1,∞ (D) almost surely with E( ∇b k L ∞ ) < ∞ for all k < ∞, then u ∈ W k for all k < ∞.
Proof: Using (2.6), we may write for any y such that b(y

) ∈ W 1,∞ (D), u(y) W ≤ exp( b(y) L ∞ ) ∇b(y) L ∞ exp(2 b(y) L ∞ ) f V + f L 2 .
By application of Cauchy-Schwarz inequality and using Theorem 1.1, we find that u ∈ W k with

u W k ≤ E( ∇b 2k L ∞ ) 1 2k E(exp(6k b(y) L ∞ )) 1 2k f V + f L 2 E(exp(k b(y) L ∞ )) 1 k ,
where the terms on the right are controlled from the assumptions. Indeed, if the Gaussian field b belongs to W 1,∞ (D) almost surely, this implies that b induces a Gaussian measure on the separable Banach space C 0 (D). Then, by Fernique's Theorem (see Theorem 2.8.5 in [START_REF] Bogachev | Gaussian measures[END_REF]), we find that

E(exp(k b ∞ )) < ∞ for all k < ∞. Finally, by assumption, E( ∇b k L ∞ ) < ∞ for all k < ∞.
As an immediate consequence, we obtain a convergence rate of the spatial discretization error for regular Gaussian fields.

Theorem 2.3 If the assumptions of Proposition 2.2 hold, then there exists C > 0 such that, for any h > 0, we have

u -u h V 2 ≤ Cn -t
h , where t is the approximation rate for functions in W , as in (2.3).

Proof: Using the norm equivalence (1.15) and the fact that the Galerkin approximation u h (y) is the orthogonal projection of u(y) onto V h in the sense of the a(y) norm, we obtain

u(y) -u h (y) 2 V ≤ exp ( b(y) L ∞ ) u(y) -u h (y) 2 a(y) ≤ exp ( b(y) L ∞ ) min v h ∈V h u(y) -v h 2 a(y) ≤ exp (2 b(y) L ∞ ) min v h ∈V h u(y) -v h 2 V .
By the approximation property (2.3), this yields

u(y) -u h (y) 2 V ≤ C exp (2 b(y) L ∞ ) n -2t h u(y) 2 W .
Therefore, by Cauchy-Schwarz inequality, we find

u -u h 2 V 2 ≤ Cn -2t h E(exp(4 b(y) L ∞ ) 1/2 u 2 W 4
, which is the claimed estimate up to a change in the constant C.

We end this subsection by discussing the validity of the additional assumption in Proposition 2.2, namely that the Gaussian field b belongs to W 1,∞ (D) almost surely and E( ∇b k L ∞ ) < ∞ for all k < ∞. Standard conditions for this to hold can be stated in terms of the smoothness of the covariance function C b . Roughly speaking this function should be a bit smoother than C 2 along the diagonal x = x . In the simple case of one-dimensional second order stationary stochastic processes where C b (x, x ) = c b (x -x ), such a result (see Section 9 in [START_REF] Cramer | Stationary and related stochastic processes: Sample function properties and their applications[END_REF]) reads, for some a > 3 and < +∞,

c b (r) = c b (0) - r 2 2 + O r 2 | log(|r|)| a ,
when r tends to 0.

Based on this type of criterion we find that b is almost surely in C 1 (D) and induces a Gaussian measure on this separable Banach space. Then Fernique's Theorem (Theorem 2.8.5 in [START_REF] Bogachev | Gaussian measures[END_REF]) implies that b(y) C 1 belongs to L k (U, dγ) for all k < ∞ and therefore E( ∇b k L ∞ ) < ∞ for all k < ∞. Many relevant random fields are however excluded from this analysis since their sample path are not regular enough. This is typically the case for Brownian processes and their multivariate extensions and for the Matérn Gaussian fields with small smoothness parameter, that are discussed in §6. In the next subsection, we refine our analysis in order to include these relevant rough Gaussian fields.

Rough Gaussian field

The previous analysis assumes the fact that the Gaussian field b is smooth in the sense that ∇b exists almost surely and is in L ∞ (D). However, the Gaussian field b is usually not a regular field and its sample paths possess some Hölderian regularity with index strictly less than 1. Thus, we need to perform a discretization in the spatial variable taking into account the roughness of the sample paths of b. First of all, we introduce some notation. For 0 < α < 1, we denote by C α (D) the Banach space of Hölder continuous functions whose norm is given by:

b C α := b L ∞ + sup x 1 ,x 2 ∈D x 1 =x 2 |b(x 1 ) -b(x 2 )| |x 1 -x 2 | α .
We denote as well by | • | C α the Hölderian seminorm defined by:

|b| C α := sup x 1 ,x 2 ∈D x 1 =x 2 |b(x 1 ) -b(x 2 )| |x 1 -x 2 | α .
In the following, we use the real interpolation method in order to find the suitable scale of functional spaces in which the solution to the PDE belongs, in the case of such rough fields, as well as the spatial approximation results for such spaces. We recall briefly the definition of interpolation spaces, see [START_REF] Bergh | Interpolation spaces, an introduction[END_REF] for a general treatment. For general Banach spaces X and Y which are compatible, we define the K-functional of any elements g of X + Y by:

K(g, t, X, Y ) := inf{ g X X + t g Y Y : g X ∈ X, g Y ∈ Y, g = g X + g Y }, t ≥ 0,
For any 0 < θ < 1 and for any 0 < q ≤ ∞, we define the interpolation space [X, Y ] θ,q as the set of functions of X + Y such that the following quantity is finite

g [X,Y ] θ,q = g θ,q :=      +∞ 0 (t -θ K(g, t, X, Y )) q dt t 1 q , q < ∞, sup t>0 t -θ K(g, t, X, Y ), q = ∞.
In the case where Y ⊂ X which is of interest for our present discussion, the K-functional also writes K(g, t, X, Y

) := inf{ g -g Y X + t g Y Y : g Y ∈ Y }, t ≥ 0.
In particular, one has K(g, t, X, Y ) ≤ g X and the above interpolation norm may be equivalently defined by taking the integral or supremum on t ∈ [0, 1]. We shall use the particular well-known case of interpolation spaces: if D is a bounded Lipschitz domain then one has

C α (D) = [L ∞ , W 1,∞ ] α,∞ ,
with equivalent norms. As a substitute to the estimates (2.5) and (2.6) in W for smooth fields, the following result gives estimates in the interpolation space

W α := [V, W ] α,∞ .
for the solution to the elliptic problem (1.1) when the fields a and b are in C α (D).

Theorem 2.4 Assume that D is a bounded Lipschitz domain of R d , that a ∈ C α (D) with a min := min x∈D a(x) > 0 and that f ∈ L 2 (D). Then, the solution u to the elliptic problem (1.1) belongs to W α and,

u W α ≤ C a -1 min + a 1 α C α a -1+α α min f L 2 . (2.7)
For a of the form a = exp(b) with b ∈ C α , one has

u W α ≤ C exp( b L ∞ ) + (1 + 2 b C α ) 1 α exp 2 + α α b L ∞ f L 2 .
(2.8)

The constants C in the above estimate depend on α and D only.

Proof: We make use of a standard stability result for second order elliptic PDEs. If u and ũ are two weak solutions to (1.1) with diffusion coefficients a and ã, respectively, and with the same data f , then one has

u -ũ V ≤ a -ã L ∞ (a min ãmin ) -1 f V , (2.9)
where a min and ãmin are the minimal values of a and ã on D. This is easily checked by equating the variational formulations for both solutions and taking v = u -ũ as a test function.

Since C α (D) = [L ∞ , W 1,∞ ] α,∞ with equivalent norms, for any ε > 0 there exists a ε ∈ W 1,∞ (D) such that a -a ε L ∞ ≤ C a C α ε α , (2.10)
and

|a ε | W 1,∞ ≤ C a C α ε α-1 , (2.11)
where the constant C only depends on α and D.

Moreover, if ε ≤ ε 0 = (a min /(2 a C α C)) 1/α , we have a ε (x) ≥ a min 2 , x ∈ D. (2.12)
For such values of ε, we denote by u ε the solution to (1.1) with a ε as diffusion coefficient. We note that u ε belongs to W and satisfies

u -u ε V ≤ 2a -2 min C a C α f V ε α ,
by (2.11) combined with (2.9), as well as

u ε W ≤ 2a -1 min (C a C α 2a -1 min ε α-1 f V + f L 2 ),
by (2.11) combined with (2.5). We replace f V by f L 2 up to a multiplication of C by the Poincaré constant, and use u ε to estimate K(u, t, V, W ) for the relevant range 0 < t ≤ 1, by writing

K(u, t, V, W ) ≤ 2C a C α a -2 min ε α + 2ta -1 min (C a C α 2a -1 min ε α-1 + 1) f L 2 .
Thus, for any 0

< ε < ε 0 ∧ ε 1 , with ε 1 = 2C a C α a -1 min 1/(1-α) K(u, t, V, W ) ≤ C 2 a C α a -2 min ε α + 8ta -2 min a C α ε α-1 f L 2 .
For t ≤ 1, we take ε = t(ε 0 ∧ ε 1 ) in the previous inequality to obtain,

K(u, t, V, W ) ≤ C 2 a C α a -2 min (ε 0 ∧ ε 1 ) α + 8a -2 min a C α (ε 0 ∧ ε 1 ) α-1 t α f L 2 . Using ε 0 ∧ ε 1 ≤ ε 0 and ε 0 ∧ ε 1 ≥ ε 0 ε 1 /(ε 0 + ε 1 )
, we arrive at

t -α K(u, t, V, W ) ≤ C a -1 min + a 1 α C α a -1+α α min f L 2 ,
which gives (2.7). Finally, when a = exp(b) with b ∈ C α , we use a

C α ≤ exp( b L ∞ )(1 + 2 b C α ) together with a -1 min ≤ exp( b L ∞ ) to obtain (2.8).
Remark 2.5 The interpolation argument used in the proof of the above result also allows us to treat the case of data f that are rougher than L 2 , by using the stability estimate

u -ũ V ≤ a -ã L ∞ (a min ãmin ) -1 f V + 1 a min f -f V .
in place of (2.9) when u and ũ are associated with different data f and f . Using the fact that

H -1+α = [H -1 , L 2 ] α,2 ⊂ [H -1 , L 2 ] α,∞
, the same argument shows that the solution u belongs to W α when a ∈ C α and f ∈ H -1+α . In the case where W coincides with V ∩ H 2 (D) (for example, when D is either convex or has C 1,1 boundary), we find that

W α = [V, V ∩ H 2 (D)] α,∞ = V ∩ [H 1 (D), H 2 (D)] α,∞ = V ∩ B 1+α,2 ∞ (D),
where B 1+α,2 ∞ is the usual Besov space, which contains all Sobolev spaces H 1+β for β < α. The fact that a ∈ C α and f ∈ H -1+α implies that u ∈ H 1+β for β < α is proved in [START_REF] Hackbusch | Elliptic differential equations -theory and numerical treatment[END_REF] by different techniques, see Theorem 9.19 therein.

In order to obtain an estimate on the quadratic error u -u h V 2 similar to the one obtained in Theorem 2.3, we introduce the Bochner type space

W α k = L k (U, W α , γ),
for 0 < α < 1 and for all k < ∞. We now state a result that comes as a substitute to Proposition 2.2 in the case of rough fields.

Proposition 2.6 Assume that b is a Gaussian field which belongs to C α (D) almost surely and such that

E( b k C α ) < ∞ for all k < ∞. Then, u ∈ W α k for all k < ∞.
Proof: Based on Theorem 2.4, for all y such that b(y) ∈ C α (D),

u(y) W α ≤ C exp( b(y) L ∞ ) + (1 + 2 b(y) C α ) 1 α exp 2 + α α b(y) L ∞ f L 2 .
Since the Gaussian field b induces a Gaussian measure on the separable Banach space C 0 (D) Fernique's theorem (see Theorem 2.8.5 in [START_REF] Bogachev | Gaussian measures[END_REF]) implies that E(exp(k b(y) L ∞ )) < ∞ for all k < ∞. Moreover, by assumption E( b k C α ) < ∞ for all k < ∞. Then, Cauchy-Schwarz inequality concludes the proof of the proposition. Now, we notice that functions in the space W α can be approximated at a certain rate. Indeed, the approximation property (2.3) for the space W and the sequence (V h ) h>0 induces a similar property for the space W α with the same sequence (V h ) h>0 , namely min

v h ∈V h v -v h V ≤ Cn -αt h v W α , v ∈ W α , h > 0.
(2.13) This is readily checked by applying the interpolation inequality

T [X,Y ]α,∞→X ≤ T α Y →X T 1-α
X→X , with X = V and Y = W to T = I -P V h where P V h is the V -orthogonal projection onto V h . This leads to the following approximation result.

Theorem 2.7 If the assumptions of Proposition 2.6 hold, then there exists C > 0 such that, for any h > 0, we have

u -u h V 2 ≤ Cn -αt
h , where t is the approximation rate for functions in W , as in (2.3).

Proof: Similar to the proof of Theorem 2.3, we write

u(y) -u h (y) 2 V ≤ exp (2 b(y) L ∞ ) min v h ∈V h u(y) -v h 2 V .
Using the approximation property (2.13) applied to u(y) we obtain

u(y) -u h (y) 2 V ≤ C exp (2 b(y) L ∞ ) n -2tα h u(y) 2
W α , and by Cauchy-Schwarz inequality

u -u h 2 V 2 ≤ Cn -2tα h E(exp(4 b(y) L ∞ ) 1 2 u 2 W α 4 .
(2.14)

This concludes the proof of the theorem.

We close this subsection by discussing conditions that ensure that the Gaussian field b belongs to the Hölder space C α (D) almost surely, with bounded moments

E( b k C α ) < ∞, k < ∞, (2.15)
as needed in Proposition 2.6. We first observe that the boundedness of all moments holds provided that the Gaussian field b belongs to C α+ε (D) almost surely for some ε > 0. Indeed, this implies that b belongs to the space C α 0 (D), which is defined by the condition lim

x 1 →x 2 |b(x 1 ) -b(x 2 )| |x 1 -x 2 | α = 0.
This space is a separable Banach space, endowed with the same norm as C α (D). Therefore Fernique's theorem implies that the moment bound (2.15) holds. There are several criteria which ensure that b belongs to the Hölder space C α (D). A standard probabilistic one is Kolmogorov's continuity theorem (see e.g. Theorem 2.1 [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]), that reads on the covariance C b (x, x ) := E(b(x)b(x )) of the Gaussian random field: if for some K > 0 and β > α,

C b (x, x) + C b (x , x ) -2C b (x, x ) ≤ K|x -x | 2β , x, x ∈ D, (2.16)
then b admits a continuous modification which belongs to C α (D) almost surely. In particular, this also implies that b belongs to C α+ε for some ε > 0 sufficiently small which thus implies the moment bounds (2.15). Note that (2.16) holds in particular when the covariance function C b has Hölder smoothness C 2β (D × D). We apply this criterion in §6 to treat the case of Brownian and Matérn processes.

Weighted least-squares method

For a given set Λ n ⊂ F with n = #(Λ n ), the weighed least-squares methods allows us to construct an approximation of the solution map u in V Λn from a finite number of evaluations of u: we are given m evaluations u 1 , . . . , u m of u at the points y 1 , . . . , y m ∈ U , that is,

u i = u(y i ), i = 1, . . . , m, (3.1)
and define the weighted least-squares estimator u W n ∈ V Λn of u as

u W n := argmin v∈V Λn m i=1 w i v(y i ) -u i 2 V , (3.2)
where w i ≥ 0 are positive weights. The properties of stability and accuracy of this estimator depend on the distribution of the samples y 1 , . . . , y m , on their number m and on the choice of the weights.

Optimal weights and sampling measure

Following the analysis of weighted least squares in [START_REF] Cohen | Optimal weighted least-squares methods[END_REF], we introduce the nonnegative weight function y → w(y) := n

ν∈Λn |H ν (y)| 2 , (3.3)
for a given Λ n associated to V n , where we recall that H ν are orthonormal in L 2 (U, dγ). Using the function (3.3) we define the probability measure

dµ := w -1 dγ = 1 n ν∈Λn |H ν (y)| 2 dγ. (3.4)
The y 1 , . . . , y m used in (3.2) are independent samples drawn from dµ. The evaluations of the function (3.3) at the selected points

w i := w(y i ),
give the weights in (3.2). By expanding the estimator u W n over the orthonormal basis we obtain

u W n = ν∈Λn v ν H ν ,
where v := (v ν ) ν denotes the collection of the coefficients. The calculation of the estimator u W is equivalent to solving the normal equations

Gv = d, (3.5)
where G and d are defined element-wise as

G ν, ν := 1 m m i=1 w i H ν (y i )H ν (y i ), d ν := 1 m m i=1 w i u i H ν (y i ). (3.6)
The solution to the weighted least-squares always exists and is unique if and only if G is nonsingular. In order to avoid the situation where G is ill-conditionned, we consider the conditioned least-squares estimator u C n introduced in [START_REF] Cohen | Optimal weighted least-squares methods[END_REF], that is defined as

u C n := u W n , if G -I 2 ≤ 1 2 , 0, otherwise.
The choice of the particular weight function (3.3) and sampling measure (3.4) is crucial, in that it yields stability and optimal accuracy of the method when the number m of sample to be evaluated is larger than n only by a logarithmic factor. This is expressed by the following result, which is an adaptation of Theorem 13 in [START_REF] Cohen | Multivariate Approximation in Downward Closed Polynomial Spaces[END_REF] to the Hermite polynomial case.

Theorem 3.1 For any real s > 0 and any integer n ≥ 1, if m satisfies

n ≤ κ m ln m , with κ = κ(s) := 1 -ln 2 2 + 4s , (3.7)
and the samples y 1 , . . . , y m are independent and drawn from (3.4) then the estimator u C n satisfies

E( u -u C n 2 V 2 ) ≤ (1 + β(m)) min v∈V Λn u -v 2 V 2 + 2 u 2 V 2 m -2s , (3.8)
with β(m) := 4κ ln m → 0 as m → +∞.

This allows us to obtain convergence estimates in expectation for the solution to (1.1).

Theorem 3.2 Let (ρ j ) j≥1 be a sequence that satisfies the assumptions of Theorem 1.4, and let Λ n be sets corresponding to the n largest ξ

-1/2 ν
, with ξ ν given by (1.13). Then the estimator u C n built with m samples drawn from (3.4) under (3.7) with s := 1/q satisfies

E( u -u C n 2 V 2 ) ≤ C 1 + 4κ(2s) ln n + 2 u 2 V 2 n -2s , (3.9)
where the constant C is the same as in Theorem 1.4.

The convergence estimate (3.9) shows that, in the ideal situation where exact evaluations of u are available, the approach based on weighted least squares achieves in expectation the optimal convergence rate n -s for the approximation error in V 2 of the solution to the lognormal PDE of interest.

In practice, the exact evaluation u i = u(y i ) is not available and is replaced by its finite element approximation u i h := u h (y i ). Following the approach outlined in §2, we thus apply the weighted least-squares approximation to the map y → u h (y), and denote by u C h,n ∈ V h ⊗ P Λn the resulting estimator of u h . By the same argument, we reach the following analog of Theorem 3.2. Theorem 3.3 Let (ρ j ) j≥1 be a sequence that satisfies the assumptions of Theorem 1.4, and let Λ n be sets corresponding to the n largest ξ -1/2 ν , with ξ ν given by (1.13). Then the estimator u C h,n built with m samples u i h := u h (y i ) drawn from (3.4) under (3.7) with s := 1/q satisfies

E( u h -u C h,n 2 V 2 ) ≤ C 1 + 4κ(2s) ln n + 2 u h 2 V 2 n -2s , (3.10)
where the constant C is the same as in Theorem 1.4.

By Cauchy-Schwarz inequality, the estimate in the above theorem implies

E( u h -u C h,n V 2 ) ≤ Cn -s ,
with s := 1/q. Combining this result together with Theorem 2.3 or with Theorem 2.7, we obtain an estimate on the total error,

E( u -u C h,n V 2 ) ≤ C(n -αt h + n -s ) (3.11)
where C > 0 and α is equal to 1 if the Gaussian field b is smooth (Theorem 2.3) or is equal to the global Hölder regularity index of b if it is rough (Theorem 2.7).

Truncation to a finite number of random variables

At this point, it is worth to observe that u h (y i ) still cannot be evaluated, due to the infinite number of coordinates of y i that prevents the practical generation of the samples from (3.4). This difficulty can be overcome by using a proper truncation of the coordinates of y i . For some finite J to be fixed further, we truncate y i by setting to 0 its coordinates y i j for all j > J. The resulting samples y 1,J , . . . , y m,J can be directly generated from the probability measure

dµ J := 1 n ν∈Λn 1≤j≤J |H ν j (y j )| 2 dγ J , (3.12)
From (1.9), it is readily checked that b(y) -b J (y) L ∞ → 0 when J → ∞ for any y such that sup

j≥1 |y j | ρ j < ∞.
Then, for such y, using the a-priori estimate for u h (y) V , we obtain

u h (y) -u h,J (y) V ≤ 2 f V exp 3K sup j≥1 |y j | ρ j b(y) -b J (y) L ∞ (3.15)
where K is as in (1.9). We bound the term b(y) -b J (y) L ∞ by writing, for all x ∈ D,

|b(x, y) -b J (x, y)| ≤ sup j≥1 y j log(1 + j) j≥J+1 log(1 + j)|ψ j (x)| ≤ K log(1 + J) ρ J sup j≥1 y j log(1 + j) , (3.16)
where the second inequality uses the assumption that

√ log(1+j) ρ j j≥1
is non-increasing.

In order to pursue, we prove that sup

j≥1 y j √ log(1+j)
is bounded by a quantity C(y) which admits moments of any order with respect to the Gaussian measure dγ. We follow closely the proof of Lemma 1 in [START_REF] Ayache | Rate optimality of wavelet series approximations of fractional Brownian motion[END_REF]. With j := log(1 + j), for all j ≥ 1, let b > 0 be a deterministic constant to be chosen later. By standard estimates,

|y j |≥b j dγ(y) ≤ C (j + 1) b 2 2
.

Choosing b > √ 2, Borel-Cantelli lemma ensures the existence of a subset Ũ ⊂ U with full measure such that for all y ∈ Ũ , there exists j 0 (y) ≥ 1 such that |y j | ≤ b log(1 + j), j ≥ j 0 (y). Thus, for j ≥ 1

|y j | ≤ C(y) log(1 + j), y ∈ Ũ , (3.17)
with C(y) = b log(2) max(D(y), 1). Combining (3.15), (3.16) and (3.17), for y ∈ Ũ

u h (y) -u h,J (y) V ≤ 2K f V log(1 + J) ρ J C(y) exp 3Ksup j≥1 y j ρ j . (3.18)
From the results in [START_REF] Bachmayr | Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients[END_REF], we know that (1.9) with j≥1 exp(-ρ 2 j ) < +∞ implies that

U exp p sup j≥1 y j ρ j dγ(y) < ∞,
for all p > 0. In addition, the proof of Lemma 1 in [START_REF] Ayache | Rate optimality of wavelet series approximations of fractional Brownian motion[END_REF], shows that for p ≥ 1 and b > 2 √ 2,

Ũ |D(y)| p dγ(y) < ∞.

Therefore (3.14) follows by integration of (3.18) after squaring, and by application of Cauchy-Schwarz inequality.

Let us observe that by a proper reordering of the variables, we may always assume that √ log(1+j) ρ j j≥1

is non-increasing, so that the above proposition can be applied. In addition, if (ρ -1 j ) j≥1 ∈ q (N), we may write

log(1 + J) ρ J q ≤ J -1 J j=1 log(1 + j) ρ j q ≤ J -1 (log(1 + J)) q (ρ -1 j ) j≥1 q q .
Combining the previous results, we conclude this section with the following error estimate for the weighted least-squares method with space discretization and truncation of the variables. Theorem 3.6 Let (ρ j ) j≥1 be a sequence that satisfies the assumptions of Theorem 1.4, and let Λ n be sets corresponding to the n largest ξ -1/2 ν , with ξ ν given by (1.13). Then the estimator u C h,n,J built with m samples y 1 J , . . . , y m J drawn from (3.12) under (3.7) and with the observation model

u i h,J = u h (y i J ) satisfies E( u -u C h,n,J V 2 ) ≤C n -αt h + n -s + (log(1 + J)) 2 J -s ,
where C > 0, s = 1/q, and where α is equal to 1 if the Gaussian field b is smooth (Theorem 2.3) or is equal to the global Hölder regularity index of b if it is rough (Theorem 2.7).

From the above estimate, a reasonable strategy is to choose the truncation level J of the same order as n, up to logarithmic factors.

Examples of random fields

The analysis in §2 and §3 shows that for both Galerkin and weighted least-squares methods, the total error of approximation is split into two terms, resulting from the spatial and parametric discretization, respectively. The spatial error term is controlled by the Hölderian regularity of the sample path of b, while the parametric error term is controlled by the size properties of the functions ψ j in the representation of the Gaussian random field b.

In the following subsections we compute more explicitly the exponents appearing in (3.11) for relevant Gaussian field: Brownian bridge on [0, 1] and Gaussian fields with Matérn covariances. As to the functions (ψ j ) j≥1 we consider wavelet representations adapted to such fields, as studied in [START_REF] Bachmayr | Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients[END_REF]. We show that, for these relevant examples, the spatial discretization and parametric rate exponents coincide. This leads us to error estimates in terms of the total number of degrees of freedom n dof := n h n, that describes the approximant u C h,n . Here, we do not discuss anymore the third error term due to variable truncation that appears in Theorem 3.6, which is specific to the weighted least-squares method, and can always be absorbed in the second term by taking J slightly greater than n.

Brownian bridge

In this subsection, we consider Brownian type processes. For simplicity, we focus on Brownian bridge but a similar analysis can be performed for Brownian motion. First, recall that a Brownian bridge on [0, 1] is the centered Gaussian process starting from 0 with covariance function given by

C b (x 1 , x 2 ) := x 1 ∧ x 2 -x 1 x 2 , x 1 , x 2 ∈ [0, 1].
It is well known that, by application of Kolmogorov's continuity theorem, for any δ < 1/2, there exists a continuous modification such that its sample paths have C δ Hölder regularity on [0, 1] almost surely. In addition, one has E( b k C δ ) < ∞ for all k < ∞. Then, Theorem 2.7 applies with α = 1/2 -ε for any 0 < ε < 1/2, and t = 1 since we work here in spatial dimension d = 1. This leads to the spatial discretization error

u -u h V 2 ≤ Cn -( 1 2 -) h , h > 0.
It is well known that Brownian bridge admits several series expansion. As discussed in [START_REF] Bachmayr | Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients[END_REF], the most relevant one for the purpose of polynomial approximation is in terms of the Schauder basis

b(x) := l≥0 2 l -1 k=0 y l,k ψ l,k (x), (4.1)
where the y l,k are independent standard normal random variables. The functions ψ l,k are defined by

ψ l,k (x) := 2 -l 2 ψ(2 l x -k), x ∈ [0, 1] (4.2)
with ψ(x) = max{1/2 -|x -1/2|, 0}. This is also known as the Levy-Ciesieslki representation, and can be rewritten as j≥1 y j ψ j , by enumeration of all functions from coarser to finer scales, i.e. j = 2 l + k.

Concerning the parametric discretization error, as observed in [START_REF] Bachmayr | Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients[END_REF], from the decay and localization properties of the Schauder basis functions, one has

l≥0 2 l -1 k=0 2 lβ |ψ l,k (x)| < ∞, x ∈ [0, 1] for all β < 1/2, or equivalently j≥1 j β |ψ j (x)| < ∞, x ∈ [0, 1]. (4.3)
This shows that the assumptions of Theorem 1.4 are satisfied for all q > 1/β > 2, that is, q can be made arbitrarily close to 2. Thus for any 0 < ε < 1/2, Theorem 3.3 gives the estimate

E( u h -u C h,n V 2 ) ≤ Cn -( 1 2 -ε) , (4.4)
for the weighted least-squares approximation. The rate of decay in terms of n h and n coincide for the spatial and parametric error terms. This suggests to take n = n h , which leads, for any ε > 0 to a global error estimate n

-( 1 4 -ε) dof
in terms of the total number of degrees of freedom.

Stationary Gaussian Matérn fields

In this subsection, we consider the centered Gaussian Matérn fields on a general domain D ⊂ R d . The covariance function for such fields is given by

E(b(x 1 )b(x 2 )) = C δ, (|x 1 -x 2 |), x 1 , x 2 ∈ D, with C δ, (r) = 2 1-δ Γ(δ) √ 2δr δ K δ √ 2δr
where , δ > 0, K δ is the modified Bessel function of the second kind and Γ is the gamma function.

We restrict our analysis to small parameters values δ < 1, that correspond to rough fields, but a similar one can be performed for δ ≥ 1. Based on series expansion of the modified Bessel function

K when r → 0 + , |1 -C δ, (r)| ≤ Cr 2δ
for some positive constant C only depending on δ and . Then, for all (x 1 , x 2 ) ∈ D × D,

2(1 -Eb(x 1 )b(x 2 )) ≤ C|x 1 -x 2 | 2δ
for some C > 0 only depending on δ and . Thus, Kolmogorov's continuity theorem implies that the centered Matérn Gaussian random fields admit continuous modifications which belong to C δ (D) almost surely for any δ < δ. Then, Theorem 2.7 applies with α = δ-ε for any 0 < ε < δ. Moreover, when W is equal to V ∩ H 2 (D), t is equal to 1/d leading to the following spatial discretization error,

u -u h V 2 ≤ Cn -( δ d -ε) h , h > 0.
Based on wavelet expansion obtained in [START_REF] Bachmayr | Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients[END_REF] for the Gaussian Matérn fields, it is possible to compute explicitly a parametric discretization rate. For this purpose, let (ψ λ ) λ∈I be the wavelet basis defined in Section 4 of [START_REF] Bachmayr | Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients[END_REF] where λ is a scale-space index with |λ| denoting the scale parameter and where

I
E( u h -u C h,n V 2 ) ≤ Cn -( δ d -ε) ,
for the weighted least-squares approximation. We note that the rate of decay in terms of n h and n coincide for the spatial and parametric error terms. This suggests to take n = n h , which leads for any ε > 0 to a global error estimate n -( δ 2d -ε) dof in terms of the total number of degrees of freedom.

Numerical illustration with the Brownian bridge

In this section we numerically verify the error estimate (4.4) for the weighted least-squares method. The estimator u C h,n is calculated using m = 3 n ln n random samples. The index set Λ n is chosen by taking the n smallest ξ ν in (1.13) with r = 1, as an application of Theorems 1.2, 1.3, 1.4. The weights ξ ν are calculated by (4.3) from the sequence (ρ i ) i≥1 defined as for any β ∈ (0, 1 2 ), that satisfies (1.11) as a finite sum up to 2 L+1 -1 and with an additional factor 1 2 . The set Λ n produced with this construction therefore depends on the value of β: large values of β promote higher degrees along the first levels of the Schauder basis, and small values of β generate more isotropic spaces. If the Schauder basis ψ i is rescaled as τ ψ i then (1.11) still holds with the rescaled sequence of weights (τ -1 ρ i ) i . Rescaling with τ < 1 reduces the variance of the random field (4.1), and also promotes higher degrees along the first levels of the Schauder basis. It is worth to notice that the convergence estimate (4.4) requires β < 1 2 , but similar polynomial spaces as those generated with any value of β (even larger than 1 2 ) can actually be generated taking a smaller value of β and rescaling the Schauder basis by a constant. Therefore, in the following we only verify that the numerical convergence rates agree with (4.4), but we do not advocate a precise value of β to use in the computations, for example the one giving the fastest convergence rate. We test τ = 1 and β = 1 8 , β = 1 4 , β = 1 2 . Figure 5.1 shows the growth of the largest (component of the) indices of the Hermite polynomials associated to the random variables y 0,0 , y 1,0 , y 2,0 , y 3,0 that multiply the first four levels of the Schauder basis ψ 0,0 , ψ 1,0 , ψ 2,0 , ψ 3,0 in (4.1). An example of the interactions between different levels of the Schauder basis is shown in Figure 5.2 when β = 1 2 . Notice that anisotropy is between components associated to different levels of the Schauder basis, e.g. between ψ 0,0 and ψ 1,0 as shown in Figure 5.2-top-left, but not between components associated to the same level, e.g. between ψ 1,0 and ψ 1,1 as in Figure 5.2-bottom-center, whose sections remain isotropic regardless of the value of β. For the numerical validation of the estimate (4.4) we approximate 8 produce a staircase convergence plot, with a substantial reduction of the error when a relevant group of important indices is activated. With higher values of β like β = 1 4 or β = 1 2 the convergence plot approaches the optimal asyptotic rate n -1/2 predicted by (4.4).

E( u h -u C h,n V 2 ) ≈ E( u C h,ref -u C h,n V 2 ) ( 5

  2 a(y) dγ(y), where v 2 a(y) := D a(y)|∇v| 2 .

  Now, we may define the random variables τ (y) := min{j ≥ 1 : |y j | ≤ b log(1 + j)}, and D(y) := sup 1≤j≤τ (y) |y j |.

  denotes the set of these indices with #{λ ∈ I : |λ| = l} ∼ 2 dl for l ≥ 0. The following crucial localization property is ensured by Corollary 4.3 of [3] sup x∈D |λ|=l |ψ λ (x)| ≤ C2 -δl , l ≥ 0 for some constant C > 0 independent of l. Then Corollary 1.3 in [3] can be applied and parametric rates s strictly less than δ/d are achievable. Setting ρ λ = 2 |λ| with λ ∈ I, for any 0 < ε < δ/d, Theorem 3.3 gives the estimate

Figure 5 . 1 :

 51 Figure 5.1: Largest indices in Λ associated to ψ 0,0 , ψ 1,0 , ψ 2,0 , ψ 3,0 : β = 1 8 (left), β = 1 4 (center), β = 1 2 (right).

. 1 )

 1 by replacing u h with a reference solution u C h,ref = ν∈Λ ref u C h,ref,ν H ν , which is computed as a weighted least-squares estimator of u h with n ref := #(Λ ref ) = 5 × 10 3 and m ref = 20 n ref ln n ref random samples. The expectation in the right-hand side of (5.1) is estimated as a Monte Carlo average with m runs.

Figure 5 . 2 : 2 .Figure 5 .

 5225 Figure 5.2: Some sections of the set Λ n with n = 4000 and β = 1 2 .

Figure 5 . 3 :

 53 Figure 5.3: Monte Carlo averages of uC h,ref -u C h,n V 2 with m = 5: β = 1 8 , 1 4 , 1 2 , τ = 1.

* This research was supported by the Institut Universitaire de France; the ERC Adv grant BREAD.

where γ J is the J-dimensional Gaussian measure. We thus now evaluate u i h,J = u h (y i J ).

The application of the weighted-least square method to these data yields an approximation u C h,n,J

to the function u h,J defined in a similar way as u h by (2.1), however with diffusion coefficient a replaced by

The following result is obtained by the exact same arguments as those leading to Theorem 3.3.

Theorem 3.4 Let (ρ j ) j≥1 be a sequence that satisfies the assumptions of Theorem 1.4, and let Λ n be sets corresponding to the n largest ξ

, with ξ ν given by (1.13). Then the estimator u C h,n,J built with m samples y 1 J , . . . , y m J drawn from (3.12) under (3.7) and with the observation model

where s = 1/q and the constant C is the same as in Theorem 1.4.

In order to estimate the total error u -u C h,n,J V 2 , we write

The first term is estimated by Theorem 2.3 or Theorem 2.7, and the last one by Theorem 3.4. We are thus left with estimating the second term u h -u h,J V 2 that describes the error when the variables y j are put to 0 for j > J. This is the object of the following result.

Proposition 3.5 Let (ρ j ) j≥1 be a sequence of positive real numbers such that (1.9) holds and that j≥1 exp(-ρ 2 j ) < +∞ . Assume in addition that

is a non-increasing sequence.

Then, for any J ≥ 1,

with C 3 > 0.

Proof: Since u h and u h,J are the solutions to (2.1) with diffusion coefficients a = exp(b) and a J = exp(b J ), respectively, we derive by substraction of the two variational formulations the standard stability estimate

Using |e x -e y | ≤ |x -y|(e x + e y ) for any x, y ∈ R, u h (y) -u h,J (y) V ≤ exp ( b J (y) L ∞ ) b(y) -b J (y) L ∞ u h (y) V exp( b(y) L ∞ ) + exp( b J (y) L ∞ ) .