Marion Salomé

François Mariotti
email: francois.mariotti@agroparistech.fr.

Alison Dussiot

Emmanuelle Kesse-Guyot

Jean-François Huneau

Hélène Fouillet

Jean- François Huneau

Prof François Mariotti

Plant-based meat substitutes are useful for healthier dietary patterns when adequately formulated -an optimization study in French adults (INCA3)

Keywords: Plant-based meat substitutes, Diet optimization, Healthy dietary patterns, Nutrient fortification

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Introduction

Plant-based meat substitutes are products that aim to mimic the appearance and use of meat products and are seeing strong commercial development [START_REF] Boukid | Plant-based meat analogues: from niche to mainstream[END_REF][START_REF] Choudhury | Commercialization of Plant-Based Meat Alternatives[END_REF]. These products are considered as a good way to replace meat in a dish without disrupting the meal pattern [START_REF] Schösler | Can we cut out the meat of the dish? Constructing consumer-oriented pathways towards meat substitution[END_REF] so they could assist in the transition towards more plant-based diets that are being advocated for health and environmental reasons [START_REF] Willett | Food in the Anthropocene: the EAT-Lancet Commission on healthy diets from sustainable food systems[END_REF][START_REF] Clark | Multiple health and environmental impacts of foods[END_REF]. Indeed, as well as their considerable environmental impacts [START_REF] Notarnicola | Environmental impacts of food consumption in Europe[END_REF][START_REF] Crippa | Food systems are responsible for a third of global anthropogenic GHG emissions[END_REF][START_REF] Gerber | Tackling climate change through livestock: a global assessment of emissions and mitigation opportunities[END_REF], current dietary patterns in western countries are not healthy, favoring the development of non-communicable diseases [START_REF] Afshin | Health effects of dietary risks in 195 countries, 1990-2017: a systematic analysis for the Global Burden of Disease Study 2017[END_REF]. In this context, modeling studies have been used to identify healthier dietary patterns that could assist in defining food-based dietary guidelines [START_REF]Scientific Report of the 2020 Dietary Guidelines Advisory Committee: Advisory Report to the Secretary of Agriculture and the Secretary of Health and Human Services[END_REF][START_REF]A modelling system to inform the revision of the Australian guide to healhy eating[END_REF]. For instance, the most recent French dietary guidelines are based on an optimization method for modeling dietary patterns that could favor long-term health, based on epidemiological relationships between the consumption of different food groups and the risk of noncommunicable diseases [START_REF]Actualisation des repères du PNNS : révision des repères de consommations alimentaires[END_REF][START_REF] Mariotti | Perspective: Modeling Healthy Eating Patterns for Food-Based Dietary Guidelines-Scientific Concepts, Methodological Processes, Limitations, and Lessons[END_REF][START_REF]Actualisation des repères du PNNS : étude des relations entre consommation de groupes d'aliments et risque de maladies chroniques non transmissibles[END_REF]. As with many dietary guidelines, consuming plant-based foods (vegetables, fruits, whole-grain products, legumes) is to be encouraged while consuming sweet and salty foods, red meat and processed meat should be reduced [12 ,15,16].

However, the potential role for plant-based meat substitutes in modeling healthy diets has rarely been considered [START_REF] Mertens | Potential Impact of Meat Replacers on Nutrient Quality and Greenhouse Gas Emissions of Diets in Four European Countries[END_REF]. Indeed, there are concerns whether plant-based substitutes could be part of a healthy diet, as their impacts on health have been little studied and many appear to be ultra-processed [START_REF] Hu | Can Plant-Based Meat Alternatives Be Part of a Healthy and Sustainable Diet?[END_REF][START_REF] Wickramasinghe | The shift to plant-based diets: are we missing the point?[END_REF]. Furthermore, their nutritional composition and impacts on diet may vary depending on the type of meat substituted (red meat and/or poultry) and of the type of substitute and ingredients used [START_REF] Fresán | Meat Analogs from Different Protein Sources: A Comparison of Their Sustainability and Nutritional Content[END_REF][START_REF] Salomé | Substituting Meat or Dairy Products with Plant-Based Substitutes Has Small and Heterogeneous Effects on Diet Quality and Nutrient Security: A Simulation Study in French Adults (INCA3)[END_REF]. Plant-based meat substitutes generally contain low amounts of total and saturated fats and high amounts of fiber when compared to meat, but they may also contain high amounts of sodium [START_REF] Fresán | Meat Analogs from Different Protein Sources: A Comparison of Their Sustainability and Nutritional Content[END_REF][START_REF] Curtain | Plant-Based Meat Substitutes in the Flexitarian Age: An Audit of Products on Supermarket Shelves[END_REF], while the meat products they are intended to replace are important contributors of indispensable nutrients such as iron and zinc [START_REF] Cocking | The role of meat in the European diet: current state of knowledge on dietary recommendations, intakes and contribution to energy and nutrient intakes and status[END_REF][START_REF] Phillips | Commonly consumed protein foods contribute to nutrient intake, diet quality, and nutrient adequacy[END_REF].

The objective of this study was therefore to investigate whether plant-based meat substitutes could be a potential nutritional lever to support healthier dietary patterns, by effectively replacing meat while maintaining a diet close to the observed one regarding other food groups. We explored the degree to which the availability of plant-based meat substitutes in the food repertoire could be useful when modeling healthy eating patterns, and whether this would depend on the nutrient composition of the meat substitute.

To achieve this, we used multi-criteria optimization to identify modeled diets that would be nutritionally adequate and healthier while remaining close to observed diets, according to a series of scenarios where two distinct meat substitutes with or without theoretical fortification in iron and zinc, were made available into the food repertoire to potentially replace meat and also possibly other food groups. The first meat substitute corresponded to an average substitute having the average nutrient composition of meat substitutes available on the market, while the second meat substitute was a substitute for which the recipe has been previously optimized in terms of its nutrient content [START_REF] Salomé | The potential effects of meat substitution on diet quality could be high if meat substitutes are optimized for nutritional composition-a modeling study in French adults (INCA3)[END_REF]. This mainly pulse-based substitute was made from minimally processed ingredients and designed to maximize the diet quality (as assessed using the PANDiet score) when replacing meat, under technological and nutritional constraints to ensure formulation feasibility and nutrient security, respectively [START_REF] Salomé | The potential effects of meat substitution on diet quality could be high if meat substitutes are optimized for nutritional composition-a modeling study in French adults (INCA3)[END_REF].

Methods

Input of dietary data

The data used for this study were extracted from the French Individual and National Study on Food Consumption Survey 3 (INCA3) conducted in 2014-2015. The INCA3 survey is the most recent representative cross-sectional survey of the French population and its method and design have been fully described elsewhere [START_REF] Dubuisson | The Third French Individual and National Food Consumption (INCA3) Survey 2014-2015: method, design and participation rate in the framework of a European harmonization process[END_REF]. Men aged 18-64y (n=564) and women aged 18-54y (n=561), not identified as under-reporters as previously described [START_REF] Henry | Basal metabolic rate studies in humans: measurement and development of new equations[END_REF][START_REF] Black | Critical evaluation of energy intake using the Goldberg cut-off for energy intake:basal metabolic rate. A practical guide to its calculation, use and limitations[END_REF], were included in the present study; the final sample contained 1125 adults.

Dietary data were collected by professional interviewers assisted by a dietary software from three (two weekdays and one weekend day) unplanned, non-consecutive, 24h dietary recalls spread over a 3-week period. Portion sizes were estimated using validated photographs [START_REF] Dubuisson | The Third French Individual and National Food Consumption (INCA3) Survey 2014-2015: method, design and participation rate in the framework of a European harmonization process[END_REF] and the nutrient contents of different food items came from the 2016 food composition database operated by the French Information Centre on Food Quality (CIQUAL) [START_REF]ANSES-CIQUAL French food composition table version[END_REF]. Mixed foods were broken down into ingredients and then gathered into 45 food groups (Supplemental Table 1), by using a food grouping with an appropriately high level of detail for modeling healthier, more plant-based diets as in a previous study by our group [START_REF] Dussiot | Modeled healthy eating patterns are largely constrained by currently estimated requirements for bioavailable iron and zinc -a diet optimization study in French Adults[END_REF]. For each sex, the nutrient content of each food group was calculated as the mean nutrient content of food items constituting the food group weighted by their mean intake by the sex considered, as previously described [START_REF] Dussiot | Modeled healthy eating patterns are largely constrained by currently estimated requirements for bioavailable iron and zinc -a diet optimization study in French Adults[END_REF].

Multi-criteria optimization of the diet under constraints

We used a multi-objective, non-linear mathematical optimization program to identify the modeled diet (i.e., the modeled consumptions of the 45 food groups) with a maximal health profile and minimal deviation from the observed diet (taking account of cultural acceptability and inertia), under constraints that would ensure adequate nutrient intakes and remain within current consumption limits. In this context, we investigated the relative value of introducing different plant-based meat substitutes into the food repertoire. Each modeled diet was characterized by the optimized consumptions of the 45 food groups plus that of the meat substitute(s) added to the food repertoire.

Optimization was performed using the OPTMODEL procedure of SAS software version 9.4 (SAS Institute Inc., Cary, NC, USA.). Optimization was implemented at the population level but in men and women separately, because of their differences in nutritional requirements and observed consumptions. The optimized diets of men and women were then gathered to derive optimized diets for the overall population by weighting each population by its relative distribution in the study population.

Objective function

The optimization procedure consisted in maximizing the health profile of the modeled diet (Healthy Dietary Pattern criterion, HDP) while minimizing its deviation from the current diet (Diet Departure criterion, DD), as previously described [START_REF] Mariotti | Perspective: Modeling Healthy Eating Patterns for Food-Based Dietary Guidelines-Scientific Concepts, Methodological Processes, Limitations, and Lessons[END_REF][START_REF] Dussiot | Modeled healthy eating patterns are largely constrained by currently estimated requirements for bioavailable iron and zinc -a diet optimization study in French Adults[END_REF].

The HDP criterion set as its objective dietary recommendations based on epidemiological studies that had assessed associations between the consumption of different food groups and the risk of chronic diseases [START_REF]Actualisation des repères du PNNS : étude des relations entre consommation de groupes d'aliments et risque de maladies chroniques non transmissibles[END_REF][START_REF] Dussiot | Modeled healthy eating patterns are largely constrained by currently estimated requirements for bioavailable iron and zinc -a diet optimization study in French Adults[END_REF]. Its aim was thus to limit the consumption of red meat, processed meat and sweetened beverages while promoting the consumption of whole cereal products, vegetables and fruits, in accordance with current French recommendations [START_REF]Actualisation des repères du PNNS : révision des repères de consommations alimentaires[END_REF]. HDP was thus expressed and maximized as:

Max 𝐻𝐻𝐻𝐻𝐻𝐻 = � 3 𝑖𝑖=1 � Opt(i) P95(i) � -� 3 j=1 � Opt(j) Max(j) �
Where i denotes the food groups to be increased (vegetables, fruits and whole cereal products), j denotes the food groups to be decreased (red meat, processed meat and sweetened beverages), Opt(i) and Opt(j) are the optimized consumptions of food groups i and j, respectively (in g/d), P95(i) is the current 95 th percentile of consumption of food group i (in g/d) and Max(j) is the upper limit of consumption of food group j (in g/d). This latter value was of 71, 25 and 263 g/d respectively for red meat, processed meat and sweetened beverages, as defined by the French Agency for Food, Environmental and Occupational Health & Safety (ANSES) [START_REF] Mariotti | Perspective: Modeling Healthy Eating Patterns for Food-Based Dietary Guidelines-Scientific Concepts, Methodological Processes, Limitations, and Lessons[END_REF].

The DD criterion was designed to minimize deviation from the observed diet in order to account for dietary cultural acceptability and inertia. It was defined as the sum of the squares of the differences between observed and optimized food group consumptions standardized by their observed standard deviations [START_REF] Dussiot | Modeled healthy eating patterns are largely constrained by currently estimated requirements for bioavailable iron and zinc -a diet optimization study in French Adults[END_REF]. DD was thus expressed and minimized as:

Min 𝐻𝐻𝐻𝐻 = � 𝑛𝑛 𝑘𝑘=1 � Obs(k) -Opt(k) SD(k) � 2
Where k is the number of food groups, Obs(k) and Opt(k) are respectively the observed and optimized consumptions of food group k (in g/d) and SD(k) is the current standard deviation of the consumption of food group k. Some food groups were however considered to be easily interchangeable because they are usually consumed in the same way: this concerned the intra-category substitutions within fruits groups (fresh/dry/processed fruits), bread groups (refined/whole grains), other starches groups (refined/whole grains), vegetable fats groups (low/high in alpha-linolenic acid), sweetened beverages groups (sodas/fruits juices), fish groups (fat/lean fishes), fresh dairy products groups (natural/sweetened) and soups/bouillon groups. Most meat groups (beef and veal/pork and other meats/poultry) were also considered to be substitutable between each other and with plant-based meat substitutes, based on the rationale that these products are designed to replace meat in a meal [START_REF] Schösler | Can we cut out the meat of the dish? Constructing consumer-oriented pathways towards meat substitution[END_REF]. In the DD criterion, the food groups within each easily substitutable category were grouped together so that only variations in their sums (and not their individual variations) would be considered, and that such intra-category substitutions were allowed without cost.

Finally, the optimization procedure consisted in minimizing a whole objective function combining the two objective criteria: min Obj = min (DD -HDP) in order to both minimize DD and maximize HDP.

Constraints

Nutritional constraints were applied during optimization to ensure adequate nutrient intakes in the men and women populations (Supplemental Table 2). These constraints were based on the most recent reference values from ANSES [START_REF]AVIS de l'Anses relatif à « Actualisation des références nutritionnelles françaises en vitamines et minéraux[END_REF]. Because the absorption of iron and zinc is dependent on dietary factors, the requirements were based on bioavailable iron and zinc calculated from the dietary intake using equations that predict their absorption (notably taking into account the limiting effect of phytate) [START_REF] Hallberg | Prediction of dietary iron absorption: an algorithm for calculating absorption and bioavailability of dietary iron[END_REF][START_REF] Armah | A Complete Diet-Based Algorithm for Predicting Nonheme Iron Absorption in Adults[END_REF][START_REF] Miller | A Mathematical Model of Zinc Absorption in Humans As a Function of Dietary Zinc and Phytate[END_REF][START_REF]Scientific Opinion on Dietary Reference Values for zinc[END_REF], as detailed in a previous study by our group [START_REF] Dussiot | Modeled healthy eating patterns are largely constrained by currently estimated requirements for bioavailable iron and zinc -a diet optimization study in French Adults[END_REF]. Previous work had demonstrated that current recommendations regarding bioavailable iron and zinc are very constraining when trying to model healthier diet, but that allowing more flexibility would increase the prevalence of iron deficiency [START_REF] Dussiot | Modeled healthy eating patterns are largely constrained by currently estimated requirements for bioavailable iron and zinc -a diet optimization study in French Adults[END_REF]. Therefore, in this work, nutritional constraints for bioavailable iron and zinc were based on current intakes so as to avoid any increase in the prevalence of iron and zinc deficiency in the modeled diets. A sensitivity analysis was nevertheless conducted, by comparing the results obtained when using the current recommendations versus the current intakes as lower bounds for bioavailable iron and zinc (Supplemental Method 1).

Dietary constraints were applied to the food groups included in the HDP criterion for which the consumption needed to be limited: red meat intake (beef and veal/pork and other meats/offal) was limited at 71g/d, processed meat intake at 25g/d and sweetened beverages intake (sodas/fruits juices) at 263 mg/d (corresponding to the average portion size) [START_REF] Mariotti | Perspective: Modeling Healthy Eating Patterns for Food-Based Dietary Guidelines-Scientific Concepts, Methodological Processes, Limitations, and Lessons[END_REF]. Acceptability constraints were also applied to keep the food group intakes within the range of observed intakes, by bounding each food group intake between its 5 th and 95 th percentile of observed consumption in men and women separately (Supplemental Table 3), with the exception of food groups for which a dietary constraint with an upper limit was already defined (red meat, processed meat and sweetened beverages). In addition, most food groups that were considered to be easily substitutable in the DD criterion were grouped together for the acceptability constraints. Additional constraints were also applied to restrict the introduction of meat substitutes so that the total intake of meat and meat substitutes would not exceed the 99 th percentile of the initial meat consumption (Supplemental Table 3).

Plant-based meat substitutes in modeling scenarios

In order to study the potential of plant-based meat substitutes to model healthier diets, two types of plant-based meat substitute, which were not initially consumed in the observed diets, were made available for addition to the food repertoire of men and women during the diet optimization procedure. In the observed diets, the "Substitutes of animal products" food group denoted both the meat and dairy substitutes initially consumed. In the modeled diets , the intake of this group was kept constant and equal to its low observed value (<5g/d), in order not to interfere with the newly made available substitutes being tested. As explained above, beef and veal, pork and other meat and poultry groups could be replaced with meat substitutes without cost in the DD criterion.

The first substitute was an average meat substitute that corresponded to the mean nutrient composition of 43 plant-based meat substitutes (Supplemental Table 4), the detailed nutrient compositions of which were extracted from French databases [START_REF]ANSES-CIQUAL French food composition table version[END_REF][START_REF]ANSES-CIQUAL French food composition table version[END_REF][START_REF]Table de composition des aliments de l'étude NutriNet-Santé (NutriNet-Santé Study Food Composition Database[END_REF]. The second substitute was an optimized meat substitute whose ingredients had previously been optimized to improve the diet quality of French adults when replacing meat products with this substitute [START_REF] Salomé | The potential effects of meat substitution on diet quality could be high if meat substitutes are optimized for nutritional composition-a modeling study in French adults (INCA3)[END_REF]. Its ingredient composition is shown in Table 1. The nutrient compositions of the average and optimized meat substitutes are shown in Table 2.

We also tested the same meat substitutes after their theoretical fortification in zinc and iron to reach 30% or 50% of the Nutrient Reference Values (NRVs) [START_REF]Regulation (EU) No 1169/2011 of the European Parliament and of the Council of 25 October 2011 on the provision of food information to consumers[END_REF]. The value of 30% was chosen because it is the minimum value required for the nutrition claim 'High in' and 'increased in'. The value of 50% was selected as a significantly higher value than 30%

that would yet not exceed commonly used levels in the field of fortification. Without fortification, the average and optimized substitutes initially contained 2.4 and 4.4 mg/100g of iron and 1.4 and 2.4 mg/100g of zinc, respectively (total content and not bioavailable content). With fortification at 30% of the NRVs, the average and optimized substitutes contained 4.2 and 4.4mg/100g of iron, respectively (because the optimized substitute already contained more than 30% of the iron NRV) and both contained 3 mg/100g of zinc.

With fortification at 50% of the NRVs, both substitutes contained 7 mg/100g of iron and 5 mg/100g of zinc.

Two different options regarding the introduction of meat substitutes were investigated: the average substitute only could be introduced into the food repertoire (AS scenarios) or both the average and optimized substitutes could be introduced, at different proportions (OS scenarios). These two options were implemented with or without the two levels of substitute fortification, thus producing a total of six modeling scenarios. We therefore identified modeled diets involving possible introduction of the average substitute that was non-fortified (AS) and fortified with zinc and iron at 30% (AS-30%) and 50% (AS-50%) of the NRVs, and modeled diets involving the possible introduction of both substitutes, either non-fortified (OS) and fortified for zinc and iron at 30% (OS-30%) and 50% (OS-50%) of the NRVs.

Analysis of the relative influence of nutritional constraints and food group contributions to limiting nutrients

The dual values associated with each constraint during optimization represent what would be the potential gain in objective function in case of a one-unit relaxation in the limiting bound (lower or upper) of the considered constraint. In order to compare the relative influence of the nutritional constraints on modeled diets, all constraints were standardized by the value of their limiting bound. This resulted in standardized dual values reflecting the potential gain in objective function if the limiting bound was relaxed by 100%, and enabled classification of the nutritional constraints from the most limiting (highest absolute standardized dual value) to the least limiting (lowest value, close or equal to zero), as previously explained [START_REF] Dussiot | Modeled healthy eating patterns are largely constrained by currently estimated requirements for bioavailable iron and zinc -a diet optimization study in French Adults[END_REF].

For the most limiting nutrients in the different modeled diets (i.e. nutrients with the most limiting constraints) we studied the contributions of different food groups to that nutrient intake in each modeled diet of all scenarios.

Results

Modeled diets including non-fortified plant-based meat substitutes

By construction, compared to the observed diets, all the modeled diets were both nutrientadequate (because the nutritional constraints ensured adequate intakes for all nutrients, except for bioavailable iron and zinc for which levels were only constrained to be greater or equal to current intakes) and healthier (because of the optimization of the HDP criterion based on food group consumptions). Without iron and zinc fortification, when only the average meat substitute was made available into the repertoire (scenario AS), it was virtually not introduced into the diet (1g/d); when both substitutes could be introduced into the repertoire (scenario OS), only the optimized meat substitute was introduced into the diet, in substantial quantities (90g/d) (Figure 1). Compared to the observed diet, both scenarios led to healthier dietary patterns with more vegetables, fruits and whole grain products and less refined grain products, processed meats and sweetened beverages.

There was also a higher intake of seafood, dairy products, eggs, potatoes and vegetable fat and a lower intake of animal fat (Supplemental Table 5). The HDP gain (defined as the difference in HDP values between the modeled and observed diets) was similar under both scenarios (2.6 with AS and 2.4 with OS) but the DD was lower under the OS scenario (2.3) than the AS scenario (3.9), meaning a higher similarity with the observed diet for the OS scenario (Figure 2).

In terms of meat consumption (Figure 3), under the AS scenario that hardly introduced the average meat substitute, the consumption of poultry doubled (+97%) and that of red meat remained unchanged compared to the observed diet. In contrast, under the OS scenario that significantly integrated the optimized substitute, poultry was entirely removed and red meat was moderately reduced (-20%). Furthermore, under both scenarios, there were considerable disparities regarding the different types of red meat:

offal systematically increased markedly (respectively +180% and +160% under the AS and OS scenarios) and beef and veal consumption increased moderately (respectively +39% and +11%) while that of pork and other meats were entirely removed (Supplemental Table 6).

There were also some differences in meat pattern consumptions between sexes (Supplemental Table 6). In men, red meat decreased similarly under both scenarios (-10%) to reach its upper limit (71g/d) without going below, processed meat was removed and poultry increased under the AS scenario (+203%) but was removed in the OS scenario. In women, red meat increased under the AS scenario (+17%) but decreased under the OS scenario (-38%); poultry remained stable under the AS scenario but was removed in the OS scenario, and processed meat decreased under both scenarios (-73%

under AS and -53% under OS). Both the HDP gain and DD values were higher in men than in women (Supplemental Table 7).

Influence of iron and zinc fortification on modeled diet composition and HDP and DD values Iron and zinc fortification (at 30 or 50% of the NRV) led to a higher introduction of meat substitutes in dietary patterns (Figure 1). While the average meat substitute was hardly introduced into the AS scenario, it reached 62g/d under the AS-30% scenario and 69g/d under the AS-50% scenario. However, it was never introduced into diets when both substitutes were available in the repertoire because the optimized meat substitute was always preferred and introduced in quantities that increased in line with its fortification level (106g/d under the OS-30% scenario and 115g/d under the OS-50% scenario).

Introducing fortified meat substitutes led to greater reductions in meat intake, especially with the optimized meat substitute (Figure 3). Reductions in red meat consumption ranged from -19% and -56% under AS-30% and AS-50%, respectively, to -43% and -91% under OS-30% and OS-50%, respectively. While pork and other meats were always removed and offal increased, a reduction in beef and veal was only seen under the AS-50%, OS-30% and OS-50% scenarios. Processed meat was reduced but not entirely removed (-83% to -91%) and the reduction in poultry ranged from -72% to -100% (Supplemental Table 6). Red meat reductions were generally higher in women than in men but offal increased more in women, while processed meat persisted in women but not in men.

Changes to other food groups in the modeled diets were similar to scenarios with nonfortified substitutes.

Introducing fortified meat substitutes increased the HDP gain while slightly decreasing the DD (Figure 2). The HDP gain was however slightly more important and the DD always largely lower with the optimized meat substitute than with the average meat substitute (Supplemental Table 7).

For sensitivity analysis, we tested an alternative optimization procedure where the nutritional constraints on bioavailable iron and zinc were based on recommended intakes rather than observed intakes (Supplemental Tables 8 and9). In this case, the reduction in red meat was much lower and only effective under the AS-50% and OS-50% scenarios.

This was especially marked in women with "high" iron requirements (20% of women, see Supplemental Method 1) where red meat intake systematically increased to reach its upper limit (71g/d). For other women with "low-to-medium" iron requirements, red meat intake also increased, except under the OS-50% scenario. For men, modeled changes to meat intake were quite similar but less marked when using recommended intakes rather than observed levels of bioavailable iron and zinc as nutritional constraints.

Identification of limiting nutritional constraints in the different optimization scenarios

The nutritional constraints identified as limiting were those that prevented the identification of healthier but acceptable dietary solutions. The limiting nutrients thus identified were sources of conflicts between nutrient adequacy and the health value of the dietary pattern.

Whatever the scenario, the most limiting nutritional constraints in both men and women were for bioavailable zinc, sodium, saturated fatty acids and vitamin A (Supplemental Table 7). Some nutritional constraints were more limiting in women than in men (iodine and sugars excluding lactose). The constraint on bioavailable iron was not limiting, but its influence could be revealed by relaxing the zinc constraint (Supplemental Table 7).

Indeed, the constraints on bioavailable zinc and iron worked together by involving similar dietary changes, and the constraint on bioavailable zinc was so predominant that it masked the influence of the bioavailable iron constraint. The constraint on bioavailable zinc was more limiting in men than in women and the constraint was less limiting when meat substitutes were fortified at 50% of the NRV in iron and zinc. The constraint on sodium was more limiting in scenarios involving the average meat substitute, and constraints on fiber, ALA and vitamin B6 were only limiting under these scenarios.

Contributions of food groups to nutrient intakes under different optimization scenarios

We investigated the contribution of each food group to the intake of the most limiting nutrients (i.e. nutrients with the most limiting constraints) and of the nutrients where meat (red meat, poultry and processed meat) and/or meat substitute contributed significantly (>10% of nutrient intake in the observed diet or a modeled diet) (Supplemental Figure 1).

In the observed diets, meat was an important contributor to bioavailable iron (43%), bioavailable zinc (39%), vitamin B6 (27%) and vitamin A (23%) intakes. Meat substitutes gradually replaced the contribution of meat to bioavailable iron and zinc intakes according to the degree of their fortification. The optimized meat substitute was also effective in compensating for the contribution of meat to vitamin B6 intake. However, meat substitutes were not able to compensate for the important contribution of processed meat to vitamin A intake, and this compensation tended to be achieved by an increase in offal under all scenarios. The contribution of meat to linoleic acid (LA) intakes in observed diets (15%) was completely compensated for in modeled diets by meat substitutes (which became important contributors, at between 17% and 25%). For sodium and saturated fatty acids, where the observed intakes were above upper levels and needed to be lowered in the modeled diets, meat substitutes contributed little to intakes, except for the average meat substitute which made a substantial contribution to sodium intake (~13-14%).

Finally, meat did not contribute to intakes of ALA, vitamin C and fiber in observed diets, but meat substitutes became important contributors of these nutrients as they were introduced into the diet. In particular, the optimized meat substitute added substantial quantities of ALA, fiber and vitamin C to the modeled diets (with contributions of 43%, 36% and 20%, respectively, under the OS-50% scenario).

Discussion

In this diet optimization study, we have shown that plant-based meat substitutes could offer good nutritional levers for the design of nutrient-adequate and healthier dietary patterns if they have been formulated to convey the relevant nutrients and are fortified with iron and zinc. We have indeed shown that the average meat substitute currently available is ineffective in rearranging the diet in order to improve its quality, whereas a nutritionally-designed meat substitute [START_REF] Salomé | The potential effects of meat substitution on diet quality could be high if meat substitutes are optimized for nutritional composition-a modeling study in French adults (INCA3)[END_REF] could largely be used as a lever for healthier patterns. Fortifying any substitute with iron and zinc increased its usefulness as a nutritional lever.

As we aimed to explore the nutritional usefulness of plant-based substitutes in a general way covering distinct alternative uses, various dietary changes were allowed, within a food category (such as partial to total replacement of any type of meat with plant-based meat substitutes) and also between food categories, even though the latter were hindered by the search for a healthier dietary pattern that would remain close to current eating habits.

Without fortification, the average meat substitute was not introduced to a significant degree in the modeled diet, showing that it had a nutrient composition insufficient to act as a nutritional lever. Fortification with iron and zinc enabled its introduction, along with important reductions in red meat, processed meat and poultry intakes. Overall, our findings are in line with those of Mertens et al. who reported that theoretical fortifications in iron and certain vitamins (B1, B2, B3 and B12) led to larger quantities of meat substitutes and smaller quantities of meat in modeled diets compared to those involving non-fortified meat substitutes [START_REF] Mertens | Potential Impact of Meat Replacers on Nutrient Quality and Greenhouse Gas Emissions of Diets in Four European Countries[END_REF]. In contrast however, in our different scenarios, group B vitamins were generally not identified as nutrients of concern (except for vitamin B6 and B12, to a lower extent, in some scenarios in women), because sufficient intakes were provided by higher than observed consumptions of fish and dairy products. In addition to the low iron and zinc content of the average meat substitute, introduction of the average substitute may have been limited by its relatively high sodium content, sodium intake was initially excessive in the observed diets. High amounts of sodium in plant-based meat substitutes have been highlighted [START_REF] Curtain | Plant-Based Meat Substitutes in the Flexitarian Age: An Audit of Products on Supermarket Shelves[END_REF], and in view of the detrimental effects of excessive sodium intake [START_REF] Afshin | Health effects of dietary risks in 195 countries, 1990-2017: a systematic analysis for the Global Burden of Disease Study 2017[END_REF], a reduction in the sodium content of plant-based meat substitutes is critical to their nutritional value.

Under scenarios where both substitutes were added to the repertoire, the optimized meat substitute was always more useful to build healthier diets, even when fortifications led to the same amounts of iron and zinc in the average and optimized substitutes. Indeed, when compared to the average meat substitute, the optimized meat substitute was introduced in higher quantities, along with a much higher reduction in red meat intake. This meat substitute had previously been optimized to improve global nutrient adequacy, containing ingredients rich in nutrients that are currently insufficiently consumed [START_REF] Salomé | The potential effects of meat substitution on diet quality could be high if meat substitutes are optimized for nutritional composition-a modeling study in French adults (INCA3)[END_REF]. In this regard, it is not surprising that it proved to be useful when modeling healthier diets, but it ultimately contrasted dramatically with the average available market substitute.

Like its low sodium content, the high contents in iron and zinc (before fortification) of the optimized meat substitute were of considerable value. The nutritional requirements for bioavailable zinc and the sodium intake limit were indeed the most limiting constraints which had the greatest impact on the composition of the modeled diets, whatever the scenario. Another advantage of the optimized substitute was its higher content in fiber and ALA, the intake of which was also identified as limiting when using the average meat substitute.

Interestingly, with both the optimized meat substitute and the average meat substitute, iron and zinc fortifications were necessary to elicit a substantial reduction in or near elimination of red meat. Iron and zinc absorptions are generally lower with plant-based than animal-based foods, particularly because the amount of phytate is a very limiting factor regarding their absorption. This hampers the use of plant-based products to compensate for these nutrients that are largely provided by animal products [START_REF] Hunt | Bioavailability of iron, zinc, and other trace minerals from vegetarian diets[END_REF][START_REF] Van Vliet | Plant-Based Meats, Human Health, and Climate Change[END_REF].

Fortification in iron and zinc does indeed appear necessary to level the nutritional composition of beef, but not poultry, when modeling meat substitutes [START_REF] Van Mierlo | A model for composing meat replacers: Reducing the environmental impact of our food consumption pattern while retaining its nutritional value[END_REF].

In addition, we found that introducing the consumption of meat substitutes did not result in consistent changes to meat consumption across meat types. While pork, poultry and processed meat were readily reduced, or even removed, a reduction in beef required the fortification of meat substitutes. By contrast, the consumption of offal increased under all scenarios (up to ~7 g/d in general), starting from a very low observed intake (3 g/d) while remaining at a level considered acceptable at the population level (far below 25 g/d, the 95 th percentile of observed consumption). Offal is very rich in nutrients [START_REF] Williams | Nutritional composition of red meat[END_REF] and we can assume that while the optimization procedure drives towards a reduction in total red meat, some preference is given to the most nutrient-dense subcategory within red meat.

One important finding of this study was that although contents in zinc and iron were critical characteristics of a meat substitute to favor meat reduction, the composition in other nutrients was also important when modeling healthy dietary patterns. Indeed, one advantage of plant-based meat substitutes is that they can contribute to an intake of nutrients other than those provided by meat, although the nutritional composition of plantbased meat substitutes varies markedly depending on their ingredients [START_REF] Fresán | Meat Analogs from Different Protein Sources: A Comparison of Their Sustainability and Nutritional Content[END_REF][START_REF] Bohrer | An investigation of the formulation and nutritional composition of modern meat analogue products[END_REF]. When formulating meat substitutes, the choice of ingredients may tend to be driven more by a willingness to reproduce the appearance and taste of meat [START_REF] Boukid | Plant-based meat analogues: from niche to mainstream[END_REF][START_REF] Kyriakopoulou | Plant-Based Meat Analogues[END_REF], as this is an important element in consumer acceptance [START_REF] Weinrich | Opportunities for the Adoption of Health-Based Sustainable Dietary Patterns: A Review on Consumer Research of Meat Substitutes[END_REF], rather than nutritional quality, consideration of which appears to be limited to mainly the macronutrient and particularly protein content [START_REF] Bohrer | An investigation of the formulation and nutritional composition of modern meat analogue products[END_REF][START_REF] Kyriakopoulou | Plant-Based Meat Analogues[END_REF]. Conversely, a nutritionally-optimized meat substitute as the one used in this study, although the prototype had acceptable orosensory characteristics [START_REF] Salomé | The potential effects of meat substitution on diet quality could be high if meat substitutes are optimized for nutritional composition-a modeling study in French adults (INCA3)[END_REF], should rather be emphasized on the basis of its nutritional value.

Therefore, if plant-based meat substitutes are appropriately designed from a nutritional point of view (especially in terms of zinc and iron) they could be useful to favor healthier dietary patterns, by supplying the nutrients that come from both meat and plant ingredients while limiting deviation from observed diets, that is, limiting the extent of other dietary changes. During this study, we considered that replacing meat with a meat substitute would not count as a deviation from the structure of observed diets (as was done for wholegrain products replacing refined grains), because meat substitutes are designed to be consumed in the same way as meat [START_REF] Schösler | Can we cut out the meat of the dish? Constructing consumer-oriented pathways towards meat substitution[END_REF][START_REF] Elzerman | Consumer acceptance and appropriateness of meat substitutes in a meal context[END_REF]. However, replacing meat with meat substitutes was not the only change to the modeled diets, and this substitution will not be sufficient to achieve healthy and sustainable diets [START_REF] Mertens | Potential Impact of Meat Replacers on Nutrient Quality and Greenhouse Gas Emissions of Diets in Four European Countries[END_REF]. Indeed, if the initial dietary pattern is unhealthy, using a meat substitute will not be sufficient to reach an healthier diet [START_REF] Hu | Can Plant-Based Meat Alternatives Be Part of a Healthy and Sustainable Diet?[END_REF],

which actually requires major direct (intra-category changes between easily substitutable foods) and indirect (inter-category changes consisting a remodeling of the overall dietary pattern) changes. Here, we found that the utility of appropriate and nutritionally-designed meat substitutes stemmed from their dual effects on enhancing the health value of the diet and reducing the need for other indirect dietary changes deemed less culturally acceptable. This was because the better intrinsic nutritional quality of such optimized meat substitutes induces less need for changes to the consumption of other food groups in order to meet nutritional requirements.

Meat substitutes with a good nutrient package can be integrated in large quantities and widely replace meat, but this may raise other questions regarding potential adverse effects. Many plant-based meat substitutes are ultra-processed [START_REF] Boukid | Plant-based meat analogues: from niche to mainstream[END_REF][START_REF] Wickramasinghe | The shift to plant-based diets: are we missing the point?[END_REF][START_REF] Bohrer | An investigation of the formulation and nutritional composition of modern meat analogue products[END_REF] and they would replace foods that are generally minimally processed (except processed meat). An excessive intake of ultra-processed foods can be expected to have adverse effects on health [START_REF] Monteiro | Ultra-processed foods, diet quality, and health using the NOVA classification system[END_REF][START_REF] Lane | Ultraprocessed food and chronic noncommunicable diseases: A systematic review and meta-analysis of 43 observational studies[END_REF][START_REF] Chen | Consumption of ultraprocessed foods and health outcomes: a systematic review of epidemiological studies[END_REF]. Meat substitutes may therefore appear to be effective for the design of healthier dietary patterns if the aim is to stay close to observed diets, but may be ineffective if the goal is to promote the consumption of minimally processed foods that are currently little consumed (such as legumes and nuts). In the latter case, it would certainly be necessary to deviate to a greater extent from the observed diets. Of note, since the optimized meat substitute is made of simple unprocessed or minimally-processed ingredients (coco bean, yellow sweet pepper, flaxseed, etc.), it can be prepared at home as a plant-based patty to replace meat, but, alternatively, its ingredients can be included separately as foods in an overall diet with less meat [START_REF] Salomé | The potential effects of meat substitution on diet quality could be high if meat substitutes are optimized for nutritional composition-a modeling study in French adults (INCA3)[END_REF]. The advantage of its formulation as a meat substitute is to offer an easy replacement for meat.

This study has some limitations. First, as the average meat substitute we took the mean of 43 meat substitutes that have very diverse nutritional compositions (some of them containing egg or cheese), but using an average substitute was a robust scenario taking into account overall the diversity of the food offers while avoiding using a large number of scenarios which are more difficult to interpret together. Second, the optimization procedure at the population level did not take account of the diversity of diets, which may be important when modeling healthier diets [START_REF] De Gavelle | The Initial Dietary Pattern Should Be Considered when Changing Protein Food Portion Sizes to Increase Nutrient Adequacy in French Adults[END_REF]. Third, in the absence of adequate specific data, the cultural acceptability of dietary changes was only theoretically and imperfectly considered, by searching for the smallest combined deviations in food group consumptions while bounding them between their observed 5 th and 95 th percentiles.

Finally, most of the results presented in this study were obtained by considering observed levels of bioavailable zinc and iron rather than the recommended levels of intake as a nutritional constraint. Zinc and iron references are very constraining and limit the modeling of healthy diets [START_REF] Dussiot | Modeled healthy eating patterns are largely constrained by currently estimated requirements for bioavailable iron and zinc -a diet optimization study in French Adults[END_REF] but removal of these constraints might have led to much lower intakes of iron and zinc. As a compromise we therefore chose not to degrade the current situation and to model diets that supplied at least the same levels of absorbed iron and zinc as the observed diets. Using recommendations rather than observed intakes led to the same conclusions, except that fortifications with iron and zinc appeared to be much more necessary.

To conclude, the standard meat substitutes that are currently available are of little use when modeling healthy dietary patterns unless they are fortified with iron and zinc, and they constitute a potential vector for sodium. By contrast, a meat substitute that had been nutritionally designed proved to act as an efficient lever to improve observed diets, but iron and zinc fortifications were required in order to drastically reduce the consumption of red meat. Finally, plant-based meat substitutes may be useful to implement healthier dietary patterns involving less red meat consumption, provided that they contain enough iron and zinc. Formulation of meat substitutes should consider their nutritional quality and choose ingredients that are both low-sodium, nutrient-dense and especially rich in iron and zinc. Fortification must also be considered when seeking to design products which aim to fully replace red meat. We have thoroughly and rigorously evaluated how the composition of meat substitutes affects the nutritional adequacy of the modeled diets in which they are incorporated, while designing these diets to find the best trade-off between their long-term health value and their proximity to current diets. However, their cultural acceptability was only theoretically and imperfectly assessed. Further studies are also needed to jointly assess impacts on other dimensions of sustainability, such as diet cost and environmental footprints.

Table 1 .

 1 Ingredient content of the optimized meat substitute[START_REF] Notarnicola | Environmental impacts of food consumption in Europe[END_REF]. Consumptions of different food groups in observed and modeled diets according to different optimization scenarios in the whole population. Obs, observed diet. AS, AS-30%, AS-50%, modeled diets with the possible introduction of the average meat substitute: not fortified, fortified with iron and zinc at 30% and 50% of the NRVs, respectively. OS, OS-30%, OS-50%, modeled diets with the possible introduction of average and optimized meat substitutes: not fortified, fortified with iron and zinc at 30% and 50% of the NRVs, respectively. RG, refined grains; WG, wholegrains. Water and hot beverage consumptions are not shown, for clarity. Miscellaneous foods include Soups, Bouillons, Sweet products or sweet and fatty products, Substitutes of animal products, Salt, Condiments and Aromatic herbs, spices except salt groups. Details about food grouping are given in Supplemental Table1and consumptions of food groups or categories not shown here are given in Supplemental Tables 5, 6, 8 and 9. Healthy dietary pattern (HDP) gain (difference in HDP values between modeled and observed diets) vs Diet Departure (DD) for the different modeled diets. AS, AS-30%, AS-50%, modeled diets with the possible introduction of the average meat substitute: not fortified, fortified with iron and zinc at 30% and 50% of the NRVs, respectively. OS, OS-30%, OS-50%, modeled diets with the possible introduction of average and optimized meat substitutes: not fortified, fortified with iron and zinc at 30% and 50% of the NRVs, respectively. Consumption of red meat (beef & veal, pork & other meats, offal), processed meat, poultry and meat substitutes in the observed and modeled diets. Obs, observed diet. AS, AS-30%, AS-50%, modeled diets with the possible introduction into the repertoire of the average meat substitute: not fortified, or fortified with iron and zinc at 30% or 50% of the NRVs, respectively. OS, OS-30%, OS-50%, modeled diets with the possible introduction of average and optimized meat substitutes: not fortified, or fortified with iron and zinc at 30% or 50% of the NRVs, respectively.

	Ingredient

Acknowledgments and statement of authors' contributions to the manuscript

The authors would like to thank the Nutritional Epidemiology Research Team (EREN) at Université Sorbonne Paris Nord, France for providing the nutritional compositions of plantbased substitutes extracted from the food composition table of the NutriNet-Santé Study.

MS, HF and FM: designed the research; HF: conducted the research; HF, MS and FM analyzed the data; AD, J-FH and EK-G provided methodological support; MS wrote the first draft of the manuscript and all authors provided critical comments on the manuscript. MS, HF and FM had primary responsibility for the final content and all authors have read and approved the final manuscript.

The authors report that no funding was received for this study.

Author disclosures: MS's PhD fellowship has been funded in part by a research contract with Terres Univia, the French Interbranch organization for plant oils and proteins. FM was the scientific leader of this contract.


Hichem Arioui
email: hichem.arioui@univ-evry.fr

Lamri Nehaoua

Hicham Hadj-Abdelkader

Robust Structure from Motion observer: Input to State Stability approach

Keywords: Nonlinear Observer, Structure from Motion, Linear Matrix Inequalities

The authors present a novel nonlinear Thau-Luenberger observer for estimating Structure from Motion using a calibrated camera. Accurate reconstruction of the 3D structure of a scene relies on precise estimation of the camera's translational and angular velocities, which can be challenging for cameras on mobile platforms. The proposed observer aims to estimate these velocities robustly in the presence of measurement noise, with stability characterized through Input to State Stability analysis and Lyapunov theory. The stability conditions are determined using optimization techniques based on Linear Matrix Inequalities. The performance of the proposed approach is validated through simulation and experimental data, demonstrating its effectiveness in recovering the depth of tracked features and its robustness against disturbances.

INTRODUCTION

The 3D structure of a scene can be reconstructed from the depth information of features in images captured by a moving camera. This process, known as Structure from Motion (SfM), remains a significant area of research in computer vision. Currently, proposed solutions for SfM can be categorized as either offline, also referred to as batch, or online approaches [START_REF] Oliensis | A Critique of Structure-from-Motion Algorithms[END_REF]. In batch schemes, the structure is estimated by processing all the information acquired from an image sequence using nonlinear optimization, commonly referred to as Bundle Adjustment [START_REF] Agarwal | Bundle Adjustment in the Large[END_REF], or matrix factorization [START_REF] Wang | Guide to Three Dimensional Structure and Motion Factorization[END_REF]. However, these methods are computationally intensive and pose challenges for real-time applications. Additionally, they often require subsequent iterative or online methods to ensure convergence. Online methods formulate SfM as a differential equation, where the image dynamics are derived from point sets extracted from a continuous image sequence. Estimates are obtained using recursive filtering methods or deterministic observers. Filtering techniques, such as the Kalman Filtering, Extended Kalman Filter (EKF) [START_REF] Civera | Structure from Motion using the Extended Kalman Filter[END_REF], Unscented Kalman Filter [START_REF] Omari | Metric visual-inertial navigation system using single optical flow feature[END_REF], and Particle Filter [START_REF] Pupilli | Real-Time Visual SLAM with Resilience to Erratic Motion[END_REF], have been shown to be efficient methods for estimating 3D structure, and have been developed in the context of Visual-Monocular Simultaneous Localization And Mapping (VM-SLAM). As a result, the distinction between SfM and VM-SLAM is ambiguous, depending on the field where these terms are used: SfM is more popular in the computer vision community, whereas VM-SLAM is gaining increased interest in robotics research [START_REF] Chhaya | Monocular reconstruction of vehicles: Combining SLAM with shape priors[END_REF]. However, most filtering techniques do not guarantee convergence, and their performance relies on an accurate a priori noise model. Furthermore, their linearized dynamics do not sufficiently approximate highly nonlinear systems. To overcome these limitations, different approaches have been proposed, relying on the design of nonlinear deterministic observers with theoretical stability guarantees. For example, in [START_REF] Lowe | Distinctive Image Features from Scale-Invariant Keypoints[END_REF], feature point depth estimation is addressed using an active strategy to optimize the convergence rate of the error by acting on the camera translational velocity. Similarly, geometric primitives such as lines [START_REF] Oliensis | A Critique of Structure-from-Motion Algorithms[END_REF], spherical and cylindrical targets [START_REF] Zhang | Robust Bundle Adjustment for Structure from Motion[END_REF], and moments [START_REF] Agarwal | Bundle Adjustment in the Large[END_REF] have been introduced in the context of active SfM. Active SfM typically require a priori knowledge of camera velocity for accurate reconstruction of the 3D structure of a scene. Camera velocity can be precisely measured and controlled using methods such as differential kinematics, joint encoders, and motors when the camera is mounted on the end effector of a fixed base robotic manipulator. However, this becomes considerably more challenging for mobile or flying robots, as they are subject to significant dynamics and non-holonomic constraints [START_REF] Spica | Contributions to active visual estimation and control of robotic systems[END_REF]. To address this issue, several approaches have been proposed, including obtaining partial measurements/estimations of camera linear/angular velocity [START_REF] Keshavan | An Analytically Stable Structure and Motion Observer Based on Monocular Vision[END_REF], or using rotation-invariant features to mitigate the effects of disturbances on angular velocity. In this work, a nonlinear deterministic observer is designed for feature depth estimation. The stability conditions derived are formulated in terms of LMI constraints. The robustness of the observer to the estimation error uses the Input-to-State (ISS) stability paradigm to minimize the measurement perturbations of the camera velocity input and unreliable feature tracking. The performance of the proposed scheme is validated through experimental results using a mobile platform [START_REF] Spica | Contributions to active visual estimation and control of robotic systems[END_REF]. Additionally, the performance is compared with the recent work presented in [START_REF] Tahri | Brunovsky's Linear Form of Incremental Structure From Motion[END_REF] which exploits rotationally invariant features to design a robust active vision scheme based on two transformations: the first transformation efficiently decouples the nonlinear model, whereas the second transforms the decoupled model into Brunovsky's linear form.

The paper is organized as follows: Section 2 summarizes the vision model expressed here in quasi-Linear Parameter Varying (qLPV) form and discusses the observability conditions. Section 3 addresses the design of the non-linear observer to estimate the depth information. The simulation and experimental results demonstrating the performance of the presented work are evaluated in section 4. Finally section 5 summarizes the conclusions of this work.

MODEL DESCRIPTION

This work adopts the following notations: matrices are represented in bold uppercase X and vectors in bold lowercase x . The symmetric matrix property is expressed as X = X ⊺ and (.) -1 is the inverse of a square matrix. X ≻ 0 (resp. X ≺ 0) means that X is a positive definite (resp. negative definite) matrix. 0 and I denote to null and identity matrices of appropriate dimensions.

[.]× is the skew-symmetric matrix operator. (.) 2 and (.)

∞ are the Euclidean and infinity norms respectively. Lastly, x is the representation of x in homogeneous coordinates. This section introduces the Takagi-Sugeno transformation utilized for observer synthesis. Subsequently, an analysis of the uniform observability conditions is presented.

Camera-Object Relative Motion Model

Consider a 3D point p of coordinates defined in the camera frame F c as c p = (X c Y c Z c ) ⊺ ∈ R 3 and defined in the world frame F w as w p = (X Y Z 1) ⊺ . Its projection into a 2D point on the image plane with normalized homogeneous image coordinates, given by m = (x y 1) ⊺ . The projection is expressed by the pinhole equations:

x = X c Z c y = Y c Z c (1)
The pose of F c with respect F w is defined in terms of the camera extrinsic parameters, encapsulated by the homogeneous transformation matrix T wc .

T wc = R t 0 1 (2)
where R ∈ SE (3) is the rotation matrix F c between and F w and t ∈ R 3 represents transalational displacement between F c and F w The inverse of T wc , denoted by T cw = T -1 wc takes the following form:

T cw = R ⊺ -R ⊺ t 0 1 (3)
Then, the mapping coordinates from F w to F c is given by: c p = T cw w p (4) Time derivative of (4) to obtain the velocity of p in F c :

c ṗ = Ṫcw w p + T cw w ṗ (5) = Ṫcw T cw c p + T cw w ṗ
which can be written as: c ṗ = Vc p + T cw w ṗ (6) where w ṗ is the homogeneous representation of the absolute velocity of p in F w , and Vc = Ṫcw T cw c , called twist, is an element of the Lie group SE (3) representing the velocity of F w with respect to F c in F c . Hence, the homogeneous matrix representation of Vc is given by:

Vc = Ṙ⊺ R -R ⊺ ṫ 0 1 (7) = [ω] × -υ 0 1 (8)
with υ = (υx υy υz) ⊺ and ω = (ωx ωy ωz) ⊺ are the instantaneous translational and angular velocity of the camera frame origin. Thus, ( 6) is equivalent to:

c ṗ = T cw w ṗ -υ + [ω] × p 0 . ( 9
)
with w ṗ representing the translational velocity of p in F w which is always assumed to be zero since the feature is considered fixed in the scene. Expanding (9), the following matrix is obtained where u = υ ⊺ ω ⊺ ⊺ is the measured spatial velocity of the camera motion:

c ṗ = ⎛ ⎜ ⎝ -1 0 0 0 -Z c Y c 0 -1 0 Z c 0 -X c 0 0 -1 -Y c X c 0 ⎞ ⎟ ⎠ u ( 10
)
The motion of the feature m is related to the velocity of the camera by an interaction matrix which itself depends on the current value of the features and other 3D information (depth). From the equations ( 1) and ( 10), one can deduce that the time derivative of the image point m is linked to the camera spatial velocity u by the following interaction matrix [START_REF] Tahri | Rotation free active vision[END_REF]:

ṁ = -1 Z c 0 x Z c x y -(1 + x 2 ) y 0 -1 Z c y Z c (1 + y 2 ) -x y -x u (11)
The dynamics of the depth information 1 Z denoted by χ is deduced from the equation ( 10):

χ = 0 0 χ 2 y χ -x χ 0 u (12)
Let x = (m ⊺ , χ) denote the state vector where m = (x y) ⊺ is a measurable vector, and χ = 1 Z c is the unmeasurable 3D information to be estimated. A moving camera observing a scene induces a motion in the image plane and hence in the features. Point correspondences (the measurable variables) in an image sequence can be computed using feature-tracking techniques [START_REF] Bay | SURF: Speeded Up Robust Features[END_REF].

T-S Representation

The objective of this work is to design a nonlinear observer which robustly estimates the depth information χ during the camera motion in the presence of measurement noise. For this purpose, a new system representation of the state vector is adopted x expressed as a qLPV model:

ẋ = A(x, u) x + B(y) ω y = Cx (13)
where:

A(x, u) = ⎛ ⎜ ⎝ 0 0 -v x + xv z 0 0 -v y + yv z 0 0 χv z + yw x -xw y ⎞ ⎟ ⎠ B(y) = ⎛ ⎜ ⎝ xy -(1 + x 2 ) y 1 + y 2 -xy -x 0 0 0 ⎞ ⎟ ⎠
and y represents the output of the system:

C = 1 0 0 0 1 0
Furthermore, the qLPV system is transformed into TS form with unmeasured premise variables. Following the sector nonlinearity approach [START_REF] Kazuo | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], the TS model of the system is formulated as:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ = r ∑ i=1 µ i (x) A i x + B(y) ω y = Cx (14)
where A i ∈ R 3×3 , B ∈ R 3×3 and x ∈ R 3 is the state vector. The weighting functions µ i , i = 1, . . . , r satisfy the following convex sum property:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 ⩽ µ i ⩽ 1 r ∑ i=1 µ i = 1 (15)
with r is the number of the sub-models defined by the following three system nonlinearities:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ h 1 = -v x + xv z h 2 = -v y + yv z h 3 = χv z + yw x -xw y (16)
These terms are assumed to be bounded, so that the TS model obtained is a weighted sum of 8 sub-models r = 2 3 corresponding to the nonlinear model on the compact set.

OBSERVER DESIGN

This section develops the estimation of the unmeasured state by considering the given polytopic observer:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ẋ = 8 ∑ i=1 µ i (x) A i x + L i (y -ŷ) + B ω ŷ = Cx (17)
The estimated state and output vector are respectively denoted by x and ŷ. A state estimator is developed for the system in equation ( 13) under the following assumptions.

Assumption 1. We consider tha a frozen values of parameters [START_REF] Sename | Linear Parameter Varying systems: from modelling to control[END_REF]:

(1) the state vector and the system inputs are bounded.

(2) the system is structurally uniformly observable.

The aim is to compute the gain L i to minimize the following estimation error. e(t) = x(t) -x(t) (18) The time dependency is omitted for brevity. The dynamics model of the error is:

ė = ẋ - ẋ = 8 i=1 µ i x) A i x + B ω - 8 i=1 µ i (x) A i x + B ω + L i (y -ŷ) = 8 i=1 µ i (x)A i x - 8 i=1 µ i (x) A i x + L i (y -ŷ) (19) it follows: ė = 8 i=1 µ i (x)(A i -L i C)e + 8 i=1 (µ i (x) -µ i (x))A i x = A e e + ∆(x, x) (20
) To satisfy the asymptotic stability of (19), let the following assumptions hold true [START_REF] Bergsten | Thau-Luenberger observers for TS fuzzy systems[END_REF]:

(1) the system A e is Hurwitz.

(2) ∆(x, x) is a vanishing disturbance i.e: ∆(x, x) → 0 when x → x Theorem 1. Assuming that the system nonlinearities are bounded, the error dynamics in ( 19) are ISS with respect to ∆(x, x) with minimized ISS gain φ 2 :

(e(t)) 2 < φ 1 (e(0)) 2 e -σ 2 t + φ 2 (∆(x, x) ∞ (21)
Thus, there exists a positive definite symmetric matrix P ∈ R 3×3 , matrices F i ∈ R 3×3 , positive scalars η and such that the LMIs constraints (22c),( 22d) and (22e) are satisfied. The resulting observer gains are given by L i = P -1 F i . If it exists a symmetric positive definite matrix P ∈ R 3×3 with the minimal and maximal eigenvalues denoted by X 1 and X 2 , matrices F i ∈ R 3×3 such that the following LMI conditions for a given σ, α, β and X 1 hold ∀ i = 1, . . . , 8: min γ s.t. (22a)

X 1 I ≤ P (22b) A ⊺ i P + C ⊺ F ⊺ i + PA i + F i C + σ P P P -γ I < 0 (22c) βP PA i + F i C A T i P + C T F T i βP > 0 (22d) A T i P + C T F T i + PA i + F i C + 2αP < 0 i = 1, . . . , 8 (22e) with: F i = PL i , φ 1 = X2 X1 , φ 2 = γ αX1
, and X 1 I ⩽ P ⩽ X 2 I. Proof 1. Lyapunov theory is used to investigate the stability of the error dynamics in (19) by considering the following quadratic function:

V = e T Pe P = P T > 0 (23) Given that the Lyapunov function is positive definite, it is bounded in the following manner:

X 1 (e(t)) 2 ⩽ V (t) ⩽ X 2 (e(t)) 2 (24)
where X 1 and X 2 are the minimum and maximum eigenvalues of the matrix P. By taking the time derivative of the Lyapunov function and replacing the estimation error (19) derived previously, one obtains:

V = ė⊺ Pe + e ⊺ P ė (25)
= (e ⊺ A e + ∆ ⊺ (x, x))Pe + e ⊺ P(A e e + ∆(x, x))

= e ⊺ (A e ⊺ P + PA e )e + ∆ T (x, x)Pe + e T P∆(x, x) which is equivalent to:

V (t) = e ⊺
V (t) < V (0) e -σt + γ t 0 e -σ(t-s) t ∆(s) 2 ds (28) ⩽ V (0) e -σt + γ σ ∆(s) 2 ∞
including the bounds of the Lyapunov function in (24) yields:

e(t) 2 < X 2 X 1 e(0) 2 e -σt + γ αX 1 ∆(t) 2 ∞ (29)
which is equivalent to:

(e(t)) < X 2 X 1 e(0) e -σ 2 t + γ σX 1 (∆(t) ∞ (30)
This inequality proves the ISS stability of the system (19) with respect to the disturbance ∆ and with minimal ISS gain ensured by the condition (22a) defined by φ 2 = γ (σX 1 ). The pole placement of (A i -L i C), i = 1, . . . , 8 is selected to in order to maximize the convergence rate of the estimation. The eigenvalues are assigned in particular regions defined by the combination of a disk of radius β centered at (0,0) and the half-plane delimited by α:

S(α, β) = {z ∈ C, R(z) < -α, z < β} (31)
The following constraints define the chosen pole clustering:

βP (A i -L i C) (A i -L i C) T P βP > 0 (32a) (A i -L i C) T P+P(A i -L i C)+2αP < 0 i = 1, . . . , 8 (32b)
To obtain feasible LMIs, the variables must appear linearly in the constraints. Hence, the following change of variables is performed: F i = PL i The pole clustering in inequality ( 32) is equivalent to the following and this ends the proof.

βP PA i + F i C A T i P + C T F T i βP > 0 (33a) A T i P + C T F T i + PA i + F i C + 2αP < 0 i = 1, . . . , 8 (33b)
The feasibility of the LMIs in ( 22) after imposing the parameters α, β, σ and X 1 ensures the stability of the error dynamics in ( 19) with improved convergence rate.

VALIDATION RESULTS

The performance of the proposed observer is validated against simulation and experimental results. The proposed approach is compared to the active Incremental Structure from Motion (ISfM) scheme proposed in [START_REF] Tahri | Brunovsky's Linear Form of Incremental Structure From Motion[END_REF] in the presence of strong measurement noise on the camera velocity. The latter observer was chosen for comparison due to its robust performance.

Simulation Results

Concerning the simulation results, three points defined with respect to the initial camera frame by the following coordinates were considered:

P 0 = ⎛ ⎜ ⎝ -1 .9 0.7 0.2 0.9 0.8 1.6 1.1 1.2 ⎞ ⎟ ⎠ (34)
The Root Mean Squared Error (RMSE) is used to compare the performance of the proposed work with the results obtained in [START_REF] Tahri | Brunovsky's Linear Form of Incremental Structure From Motion[END_REF] to quantify the performance of our estimator in the presence of measurement noise and modeling error where κ is the parameter to be estimated and n is the number of error samples in a given interval.

RMSE = 1 n n i=1 (κ -κ) 2 i (35)
The RMSE of the depth estimation error in the steadystate performance for both approaches is compared in order to conclude which observer delivers more robust performance with respect to camera velocity uncertainties. Hence, both the gains of both observers were tuned to have approximately the same rate convergence and the steady state was chosen to start at 0.5 seconds. The initial values of all estimated parameters were set to zero for the comparison test. The following four cases were considered:

Case 1 : Noise-free measurements.

Case 2 : The controlled angular velocity is affected by a measurement noise of zero mean and variance of 0.001 using the random number block function in Simulink. Case 3 : Both translational and angular velocities are affected by measurement noise of zero mean and variance of 0.001. Case 4 : Both translational and angular velocities are affected by measurement noise of zero mean and variance of 0.003.

Observer

Observer in [START_REF] Tahri | Brunovsky's Linear Form of Incremental Structure From Motion[END_REF] Proposed Observer

Case 1 1.4781 × 10 -5 6.5898 × 10 -4 Case 2 5.0537 × 10 -3 0.0654 Case 3 0.2529 0.0854 Case 4 0.4247 0.1549

Table 1.

From the RMSE table, it can be seen that both observers perform almost identically for noise-free system. RMSE results obtained with the observer in [START_REF] Tahri | Brunovsky's Linear Form of Incremental Structure From Motion[END_REF] in the second case show a lower value implying that it performs better in the presence of measurement noise on the angular velocity due to extracted features being rotation invariant. However, when the observer performance is evaluated for strong measurement noise in cases 3 and 4, one notices that the proposed observer has favourable noise rejection properties. Thus, our observer delivers better performance in the presence of measurement noise.

An Experimental Case Study

To validate the proposed approach experimentally, point depth estimation was performed on images captured from a Kinect camera mounted on a TurtleBot mobile robot platform. The 2D feature considered in this experiment is the center of a detected and tracked circle on the image. The estimate is compared to the ground truth collected using the indoor positioning tool VICON. After calibrating the VICON Motion Capture system, the Kinect fixed on the turtleBot and the center of the image circle were recognized by the VICON Tracker system (version 3.7.) using mounted markers. The ground truth (VICON) date was published to the ROS network and saved as .bag files created with the rosbag utility. Images were published at 30Hz while VICON readings were updated at 100 Hz. Let v p c be the position of the marker attached to the circle and v p k the position of the Kinect camera with respect to the VICON-fixed frame. The aim is to build a ground truth consisting of successive circle center positions with respect to the camera frame during camera motion. Hence, the position of the marker p c v can be expressed in the camera frame with coordinates p c k as follows:

p c k = c R v -1 p c v ⊺ -c R v -1 p k v ⊺ (36)
where: c R v is the rotation matrix from the VICON-fixed frame to the camera frame.

The feature considered here is the center of the circle on the image. The imposed parameters used in the optimization problem were chosen in order to minimize ISS gain between disturbances ∆(x, x) and the estimation error e while satisfying all LMI constraints. If the optimization problem did not permit a numerical solution, the imposed parameters σ and X 1 were relaxed further and the optimization problem was re-run under these new LMI constraints. The choice of predefined parameters of the pole clustering is crucial: High values will result in a rapid convergence of the estimation error. However, the observer becomes more sensitive to measurement noise.

Ultimately, there was a compromise in the selection of the imposed parameters between the accuracy of the estimate and the feasibility of the LMIs. After extensive testing, the following parameter set was used for feasibility reasons: σ = 10, X 1 = 10 -5 . After exploring different regions, the pole clustering was taken in the region defined by β = 30 and α = 5 which resulted in a faster convergence rate and reduced sensitivity to measurement noise. The LMIs conditions in ( 22) are solved using YALMIP giving:

γ = 1.3171 10 -6 P = 10 -9 ⎛ ⎜ ⎝ 0.2193 -0.0002 -0 -0.0002 0.2193 -0 -0 -0 0.0004 ⎞ ⎟ ⎠
The observability of (A i , C) can be calculated from ( 14) and it is shown that the eight sub-models of the TS system are observable. Note that this does not necessarily guarantee the true observability of the nonlinear system, only that it satisfies the observability criterion used here. The experimental results show the satisfactory performance of the observer. In Fig. 2, the true and estimated depth Z k c converge after 4 seconds. Fig. 1 illustrates the behavior of the real and estimated depth information χ for different initial values of χ. In summary, the theoretical claims have been validated against extensive simulations and experimental results. Non-holonomic constraints and non-negligible dynamics strongly perturb estimation of the camera velocity, particularly the translational component. Thanks to the ISS property of the proposed observer, the impact of these disturbances is confined to a spherical region characterized by φ 2 . Finally, the experimental results are highlighted in the video link https://t.ly/ub5K.

CONCLUSIONS

This paper presents a nonlinear observer design for structure and motion identification reconstruction using a calibrated monocular camera. Sufficient conditions were provided to ensure the uniform observability of the vision system modeled as a qLPV system. The convergence of the estimation error was investigated based on the ISS property and Lyapunov analysis and the observer gains were computed by solving the derived LMI constraints.

The observer was implemented in MatLab and compared with a recently proposed rotation-invariant feature-based active Incremental Structure from Motion scheme [START_REF] Tahri | Brunovsky's Linear Form of Incremental Structure From Motion[END_REF] against simulation data and experimental results. An interesting use case of the proposed approach would be navigation tasks for mobile robots: a robot could apply this approach to scene features while exploring an environment to create a depth map.

  ⊺ P e + γ ∆ ⊺ ∆ where σ is a real positive constant.Recalling the relations8 ∑ i=1 µ i (x)(A i -L i C) = A e andF i = PL i , then if the feasibility of the LMIs expressed in (22c) ∀i = 1, . . . , 8 holds, this infers the following bounds of the Lyapunov function derivative: V (t) < -σ e ⊺ P e + γ ∆ ⊺ ∆ (27) By integrating the previous inequality (27), one obtains:

Fig. 1 .

 1 Fig. 1. Simulation results of the estimated χ and real χ depth information for the following initial values: (a) χ = 0, (a) χ = 1 and (c) χ = 10.

Fig. 2 .

 2 Fig. 2. Simulation results of the estimated Ẑ and real Z k c depth for the following initial values of χ: (a) χ = 0, (a) χ = 1 and (c) χ = 10.


Konstantin Brenner
email: konstantin.brenner@univ-cotedazur.fr

On global and monotone convergence of the preconditioned Newton's method for some mildly nonlinear systems

Introduction

Let 𝛽 be a diagonal mapping from R 𝑁 to itself and let 𝐴 be an 𝑁 × 𝑁 real matrix. Given some 𝑟 ∈ R 𝑁 , we are interested in solving numerically the flowing system of nonlinear equations 𝛽(𝑢) + 𝐴𝑢 = 𝑟.

Such mildly nonlinear systems with only a diagonal nonlinearity are commonly found in the context of geophysical flow and transport modeling, where they result from the discretization of nonlinear evolutionary PDEs. They arise, for example, from Richards' and porous medium equations, or, alternatively, from the models of reactive transport involving equilibrium adsorption. This contribution is concerned with the global convergence analysis of preconditioned Newton's methods applied to [START_REF] Baluev | On the abstract theory of Chaplygin's method[END_REF]. Nonlinear preconditioning is an increasingly popular technique that may drastically improve the robustness and convergence rate of linearization schemes such as Newton's method. As in the case of linear problems, the nonlinear preconditioning consists in replacing the original system by an equivalent one that can be solved more efficiently. Since more than twenty years variety of nonlinear preconditioning methods has been proposed, including Schwarz-inspired methods ASPIN [START_REF] Cai | Nonlinearly preconditioned inexact Newton algorithms[END_REF], MSPIN [START_REF] Liu | Field-split preconditioned inexact Newton algorithms[END_REF], [START_REF] Kern | Preconditioned iterative method for reactive transport with sorption in porous media[END_REF] and RASPEN [START_REF] Dolean | Nonlinear preconditioning: How to use a nonlinear Schwarz method to precondition Newton's method[END_REF], as well as the nonlinear versions of FETI-DP [START_REF] Klawonn | Nonlinear FETI-DP and BDDC methods[END_REF] and BDDC [START_REF] Klawonn | Nonlinear BDDC methods with approximate solvers[END_REF].

Nonlinear preconditioning appears to be particularly efficient in the application to models of subsurface flow and reactive transport [START_REF] Kern | Preconditioned iterative method for reactive transport with sorption in porous media[END_REF], [START_REF] Klemetsdal | A numerical study of the additive Schwarz preconditioned exact Newton method (ASPEN) as a nonlinear preconditioner for immiscible and compositional porous media flow[END_REF], where the failures and lack of robustness of nonlinear solvers is one the major factors limiting the reliability of the simulation codes. In those applications, the major benefit seems to result in the form of an extended convergence region, which, in case of time-dependent problems, allows for larger time steps [START_REF] Klemetsdal | A numerical study of the additive Schwarz preconditioned exact Newton method (ASPEN) as a nonlinear preconditioner for immiscible and compositional porous media flow[END_REF].

The present work aims to contribute to the theoretical analysis of nonlinear preconditioning methods, which remains relatively unexplored. We extend the previous work [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF], concerned with the Jacobi-Newton method, and cover the nonlinear counterparts of some popular linear preconditioners based on multi-splitting of the system. Our analysis includes, in particular, nonlinear preconditioning by block Jacobi or RAS methods. We prove that, under appropriate assumptions discussed below, the one-level RAS-Newton method (or RASPEN [START_REF] Dolean | Nonlinear preconditioning: How to use a nonlinear Schwarz method to precondition Newton's method[END_REF]) applied to (1) exhibits global and essentially monotone convergence. The analysis of this method is carried out in the framework on nonlinear multi-splitting methods [START_REF] Frommer | Parallel nonlinear multisplitting methods[END_REF], [START_REF] Frommer | An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms[END_REF], and extends to other methods such as, for example, block Gauss-Seidel.

As an alternative to nonlinear preconditioning, we study a simpler two-step scheme alternating the nonlinear multi-splitting and the standard Newton linearization steps. The two-step multi-splitting/Newton scheme enjoys the same global and monotone convergence properties as the full preconditioned method. We note that in the context of the RAS approach, such scheme has been proposed in [START_REF] Cai | Inexact Newton methods with restricted additive Schwarz based nonlinear elimination for problems with high local nonlinearity[END_REF] under the name of NKS-RAS method. It turns out that for simple splitting methods, like (block) Jacobi or Gauss-Seidel, the preconditioned Newton's method is equivalent to the former two-step approach.

Our convergence analysis relies on the Monotone Newton Theorem [START_REF] Baluev | On the abstract theory of Chaplygin's method[END_REF], [START_REF] Ortega | Iterative solution of nonlinear equations in several variables[END_REF]; and requires two major assumptions on the system (1), namely the concavity of the nonlinear map involved in [START_REF] Baluev | On the abstract theory of Chaplygin's method[END_REF] and the assumption that the Jacobian of the system has a nonnegative inverse. More specifically, we will assume the following ( 𝐴 1 ) For each 0 ≤ 𝑖 ≤ 𝑁, the functions 𝛽 𝑖 ∈ 𝐶 1 (R) are monotone and concave. ( 𝐴 2 ) For any 𝑢 ∈ R 𝑁 , the matrix 𝛽 (𝑢) + 𝐴 is an M-matrix.

We wish to stress that the assumptions ( 𝐴 1 ) and ( 𝐴 2 ) are quite sub-optimal and aim to improve reader's experience at the expense of sharpness. For example, the generalizations of ( 𝐴 1 ) can be performed along the following lines. First, one can relax the regularity assumption; clearly piecewise regular functions 𝛽 𝑖 would do. Secondly, the derivative of 𝛽 𝑖 need not to be bounded, or alternatively 𝛽 𝑖 need not be defined over R, such case has been treated in [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF]. As a matter of fact, we believe that the analysis presented here can be extended to 𝛽 𝑖 being merely maximal monotone and concave (in some appropriate sense). We also note that the analysis presented below applies to 𝛽 convex instead on concave. The explicit concavity assumption is motivated the the applications to porous media flow models that we have in mind. Similarly, the assumption ( 𝐴 2 ) can be relaxed by allowing positive off-diagonal elements in the Jacobian, assuming, for example, that 𝐴 is nonsingular and 𝐴 -1 ≥ 0.

Before moving any further, let us recall some basic properties of the system (1):

Proposition 1 (Existence and uniqueness of solution)

Let 𝐹 (𝑢) = 𝛽(𝑢) + 𝐴𝑢. Under the assumptions ( 𝐴 1 ) and ( 𝐴 2 ), the mapping 𝐹 -1 is well defined on R 𝑁 and is convex.

Next, we state the global version of the Monotone Newton Theorem, for which we refer to [START_REF] Ortega | Iterative solution of nonlinear equations in several variables[END_REF].

Theorem 1 (Global Monotone Newton Theorem)

Let F : R 𝑁 → R 𝑁 be continuous, Gâteaux differentiable and concave. Suppose that F (𝑢) has a nonnegative inverse for all 𝑢 ∈ R 𝑁 , and assume that F (𝑢) = 0 has a solution. Then, for any 𝑢 0 ∈ R 𝑁 , the sequence

𝑢 𝑛+1 = 𝑢 𝑛 -F (𝑢 𝑛 ) -1 F (𝑢 𝑛 ), 𝑛 ≥ 0 satisfies F (𝑢 𝑛 ) ≤ 0 and 𝑢 𝑛 ≤ 𝑢 𝑛+1 ≤ F -1 (0) for all 𝑛 ≥ 1.
If, in addition, there exists an invertible 𝑃 ∈ M(𝑁) such that F (𝑢) -1 ≥ 𝑃 ≥ 0 for all 𝑢 ∈ R 𝑁 , then the sequence 𝑢 𝑛 converges to F -1 (0).

In view of Theorem 1 and Proposition 1 one can deduce that Newton's method applied to the system (1) converges regardless of the initial guess. Unfortunately, depending on the stiffness of the function 𝛽, this convergence may become arbitrarily slow as pointed out in [START_REF] Brenner | Improving Newton's method performance by parametrization: the case of the Richards equation[END_REF] and [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF]. While the lack of robustness with respect to the shape of 𝛽 can be addressed by diagonal Jacobi preconditioning [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF], the efficiency of the Jacobi-Newton method for systems resulting from the discretization of degenerate PDEs is still controlled by the mesh size, which motivates the use of nonlinear preconditioning of domain decomposition type.

Having in mind the application to the overlapping domain decomposition, we introduce in Section 2 the preconditioning technique based on the nonlinear multisplitting of (1). We prove that the preconditioned system satisfies Theorem 1 and, therefore, Newton's method is unconditionally convergent. In Section 3 we present the numerical results based on a discretized porous media equation [START_REF] Vázquez | The porous medium equation: mathematical theory[END_REF] and using some variants of nonlinear RAS method including RASPEN and RAS/Newton twostep methods (NKS-RAS method from [START_REF] Cai | Inexact Newton methods with restricted additive Schwarz based nonlinear elimination for problems with high local nonlinearity[END_REF]).

Nonlinear multi-splitting method

In this section we present the nonlinear preconditioning procedure inspired by the linear multi-splitting methods [START_REF] O'leary | Multi-splittings of matrices and parallel solution of linear systems[END_REF], [START_REF] Frommer | Parallel nonlinear multisplitting methods[END_REF].

Let (𝑃 𝑖 , 𝑄 𝑖 ) 𝑖=1,...,𝐾 be a finite of family matrices such that 𝐴 = 𝑃 𝑖 -𝑄 𝑖 . We denote 𝑀 𝑖 (𝑢) = 𝛽(𝑢) + 𝑃 𝑖 𝑢 and 𝑁 𝑖 (𝑢) = 𝑄 𝑖 𝑢 + 𝑟. If 𝑀 𝑖 (𝑢) admits an inverse defined on R 𝑁 , then one can reformulate the original problem (1) as

F 𝑖 (𝑢) := 𝑢 -𝑀 -1 𝑖 (𝑁 𝑖 (𝑢)) = 0. (2)
Let (𝐸 𝑖 ) 𝑖=1,...,𝐾 be a family of nonnegative diagonal matrices such that 𝐾 𝑖=1 𝐸 𝑖 = 𝐼. Multiplying (2) by 𝐸 𝑖 and summing over 𝑖 we obtain the system

F (𝑢) := 𝐾 ∑ 𝑖=1 𝐸 𝑖 F 𝑖 (𝑢) = 𝑢 - 𝐾 ∑ 𝑖=1 𝐸 𝑖 𝑀 -1 𝑖 (𝑁 𝑖 (𝑢)) = 0. (3)
Clearly, the solution of the original system satisfies (3). The proposition below states that F is concave and that F (𝑢) has nonnegative inverse for all 𝑢, which implies, in particular, that F is inverse isotone, and, therefore, the solution to (3) is unique.

Proposition 2 Assume that for all 𝑢 ∈ R 𝑁 and all 𝑖, the splitting

𝐹 (𝑢) = 𝑀 𝑖 (𝑢) -𝑄 𝑖
is weakly regular and that 𝑀 𝑖 (𝑢) is an M-matrix. Then, the mapping F (𝑢) from (3) is a concave bijection from R 𝑁 to R 𝑁 , and, for all 𝑢 ∈ R 𝑁 , the matrix F (𝑢) is an M-matrix satisfying F (𝑢) -1 ≥ 𝐼.

Proof: In view of Proposition 1 the mappings 𝑀 -1 𝑖 are well defined on R 𝑁 and are convex, which implies that F is concave since 𝐸 𝑖 ≥ 0. Let us show that for all 𝑢 ∈ R 𝑁 , the matrix F (𝑢) is an M-matrix satisfying F (𝑢) -1 ≥ 𝐼. We begin with the following spectral bound, which is the founding stone for the analysis of the multi-splitting methods (see [START_REF] O'leary | Multi-splittings of matrices and parallel solution of linear systems[END_REF])

𝜌 ∑ 𝐸 𝑖 𝑀 𝑖 (𝑢) -1 𝑄 𝑖 < 1. ( 4
)
Let

𝑢 𝑖 = 𝑀 -1 𝑖 (𝑁 𝑖 (𝑢)), we have F (𝑢) = 𝐼 -𝑖 𝐸 𝑖 𝑀 𝑖 ( 𝑢 𝑖 ) -1 𝑄 𝑖 . Let 𝑤 ∈ R 𝑁 be a component-wise maximum of the vectors 𝑢 𝑖 ; that is (𝑤) 𝑘 = max 𝑖 ( 𝑢 𝑖 ) 𝑘 . Since 𝑢 ↦ → 𝑀 𝑖 (
𝑀 𝑖 ( 𝑢 𝑖 ) -1 𝑄 𝑖 = 𝑀 𝑖 (𝑤) -1 𝑄 𝑖 + 𝑀 𝑖 ( 𝑢 𝑖 ) -1 𝑀 𝑖 (𝑤) -𝐼 𝑀 𝑖 (𝑤) -1 𝑄 𝑖 ≤ 𝑀 𝑖 (𝑤) -1 𝑄 𝑖 and 0 ≤ 𝑖 𝐸 𝑖 𝑀 𝑖 ( 𝑢 𝑖 ) -1 𝑄 𝑖 ≤ 𝑖 𝐸 𝑖 𝑀 𝑖 (𝑤) -1 𝑄 𝑖 . It follows from (4) that 𝜌( ∑ 𝑖 𝐸 𝑖 𝑀 𝑖 ( 𝑢 𝑖 ) -1 𝑄 𝑖 ) ≤ 𝜌 ∑ 𝑖 𝐸 𝑖 𝑀 𝑖 (𝑤) -1 𝑄 𝑖 < 1,
which implies in turn that F (𝑢) -1 ≥ 0. Clearly the off-diagonal part of F (𝑢) is nonpositive, implying that it is M-matrix; moreover, since F (𝑢) ≤ 𝐼, we deduce that F (𝑢) -1 ≥ 𝐼.

Based on Proposition 2 one shows that the mapping F satisfies the assumptions of Theorem 1. In addition, we consider the following multi-splitting/Newton two-step scheme: Given 𝑢 0 ∈ R 𝑁 , compute for all 𝑛 ≥ 0

𝑢 𝑛 = ∑ 𝑖 𝐸 𝑖 𝑀 -1 𝑖 (𝑁 𝑖 (𝑢 𝑛 )) (5)
and

𝑢 𝑛+1 = 𝑢 𝑛 -𝐹 ( 𝑢 𝑛 ) -1 (𝐹 ( 𝑢 𝑛 ) -𝑟) . (6)
We note that (5) can be interpreted as a step of a quasi-Newton method applied to (3), where the matrix F (𝑢) -1 has been replaced by its subinverse 𝐼. It can be shown that ( 5)-( 6) leads again to a globally convergent scheme. Remarkably enough, in the case of a simple splitting, like (block) Jacobi or Gauss-Seidel, the two-step scheme is equivalent to the preconditioned Newton's method.

Proposition 3 Let 𝐴 = 𝑃-𝑄 be some splitting such that the inverse of 𝑀 = 𝛽(𝑢)+𝑃𝑢 is well defined and 𝑀 (𝑢) is non-singular for all 𝑢 ∈ R 𝑁 . Then, the two-step scheme and the preconditioned Newton's generate the same iterates.

Proof: Let 𝑢 = 𝑀 -1 (𝑁 (𝑢)), we remark that F (𝑢) = 𝑢 -𝑢 and F (𝑢) = 𝐼 -𝑀 ( 𝑢) -1 𝑄 = 𝑀 ( 𝑢) -1 𝐹 ( 𝑢). Therefore, the update generated by the preconditioned Newton's method starting from 𝑢 is given by

𝛿 prec (𝑢) = 𝑀 ( 𝑢) -1 𝐹 ( 𝑢) -1 ( 𝑢 -𝑢) . (7)
Now, let us consider the update generated by the two-step method ( 5)-( 6). We have 𝛿 two-step (𝑢) = 𝑢 -𝑢 -𝐹 ( 𝑢) -1 𝐹 ( 𝑢). We remark that 𝐹 ( 𝑢) = 𝑀 ( 𝑢) -𝑁 ( 𝑢) = 𝑁 (𝑢) -𝑁 ( 𝑢), and using linearity of 𝑁, we deduce that 𝐹 ( 𝑢) = 𝑄(𝑢 -𝑢). Therefore,

𝛿𝑢 two-step = 𝐼 + 𝐹 ( 𝑢) -1 𝑄 ( 𝑢 -𝑢) = 𝐹 ( 𝑢) -1 𝑀 ( 𝑢) ( 𝑢 -𝑢),
which, in view of ( 7), provides 𝛿 two-step (𝑢) = 𝛿 prec (𝑢).

Numerical experiment

We now proceed with the numerical experiment that illustrates the performance of block Jacobi-Newton, RASPEN and the two-step RAS/Newton methods applied to the system resulting from the discretization of a degenerate parabolic equation. The (block) Jacobi-Newton consists of applying Newton's method to the system of the form (2) obtained from a simple splitting 𝐴 = 𝑃 -𝑄, where 𝑃 is a (block) diagonal part of 𝐴. On the other hand, RASPEN can be expressed as Newton's method applied to the system (3) resulting from a particular multi-splitting (wee refer to [START_REF] Frommer | An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms[END_REF] for further details). Using same multi-splitting, the RAS/Newton method is given by ( 5)-( 6). The test case considered here is similar to the one presented in [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF] to which we refer for more detailed discussion. In brief, we are interested in the algebraic system resulting from the implicit in time discretization of the porous media equation [START_REF] Vázquez | The porous medium equation: mathematical theory[END_REF]. More specifically, focusing on a single step (of length 𝜏) of the backward Euler time integration scheme, we consider the system of the form (1) resulting from the finite difference discretization of the following boundary value problem

𝛽(𝑢) -𝛽(𝑢 𝑖𝑛𝑖 ) = 𝜏𝜕 2 𝑥 𝑥 𝑢 𝑥 ∈ (0, 1), 𝜕 𝑥 𝑢(0) = -𝑞, 𝜕 𝑥 𝑢(1) = 0, (8)
where 𝛽(𝑢) = 𝑢 1/𝑚 with 𝑚 > 1. We consider the following set of parameters: 𝑚 = 10, 𝑞 = 1, 𝜏 = 0.5, and 𝛽(𝑢 𝑖𝑛𝑖 ) = 10 -6 . The problem (8) is discretized using 𝑁 = 100 or 400 degrees of freedom, and the vector 𝑢 𝑖𝑛𝑖 is used as the initial guess by the iterative methods under consideration. We note that, since the derivative of 𝛽 is unbounded at the origin, one may consider the change of variable in [START_REF] Frommer | An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms[END_REF]. For example, using 𝛽(𝑢) as the new unknown will improve the performance of the straightforward Newton's method [START_REF] Brenner | Improving Newton's method performance by parametrization: the case of the Richards equation[END_REF]. Unfortunately, the modified system is no longer concave and, therefore, the monotone convergence is lost; moreover, compared to the splitting-based preconditioning, the convergence of Newton's method applied to the modified system turns out to be slower [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF]. The solutions of the porous media equation are characterized by the finite speed of propagation of the support. Qualitatively this behavior persists even for strictly positive but small initial data. For the discrete counterpart of the elliptic problem (8) the latter property is reflected in the performance of Newton's method. Typically, and unless some Schwarz-type preconditioning is performed, the solution fronts resulting from Newton's method can cross at most one degree of freedom at time. For the Jacobi-Newton method, this behavior is illustrated by Figure 1. The left sub-figure exhibits the final position of the solution front and some iterates of the method. The right sub-figure reports the convergence history of the method for two values of the mesh size. The numerical performance is characterized by two very distinct regimes: a very fast near-solution convergence is preceded by a long period of a slow error decrease. As a matter of fact, the length of the convergence plateau is proportional to the number of the degrees of freedom 𝑁 𝑓 that has to be crossed by the solution front, and can be expressed as 𝜎 𝑁 𝑓 , where 𝜎 is the cost of propagating the front trough one degree of freedom. As shown in [START_REF] Brenner | Improving Newton's method performance by parametrization: the case of the Richards equation[END_REF] and [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF], the parameter 𝜎 of the standard Newton's method can become arbitrarily large depending on the coefficient 𝑚 and the initial data. In contrast, the Jacobi-Newton method [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF] appears to be virtually independent of 𝑚 and can handle general nonnegative initial data.

Nevertheless, the efficiency of the latter method is still dependent on 𝑁 𝑓 and thus on the discretization. More precisely, the right side of Figure 1 reflects the convergence of the Jacobi-Newton for 𝑁 = 100 and 400 degrees of freedom. By scaling the iteration count by 𝑁 we observe that the performance of the method is essentially controlled by 𝑁 𝑓 . The vertical black line positioned at 𝑁 𝑓 /𝑁 reflects the final location of the front. The scaled convergence curves are almost identical, while the total iteration count measured for the Jacobi-Newton method is of 71 for 𝑁 = 100 and 271 for 𝑁 = 400. The dependency of the mesh size can be removed by means of the nonlinear domain decomposition. We report on Figure 2 the convergence history of RASPEN and RAS/Newton (RAS/N) methods for 5 equally sized sub-domain with the relative overlap of 0.1. In addition, we consider the case of the minimal algebraic overlap, denoted RASPEN(0), corresponding to preconditioning based on the block Jacobi method. In this numerical experiment none of the considered methods appear to exhibit any substantial dependency on the mesh size. Unsurprisingly, the overlap seems to be beneficial for the convergence of both RASPEN and RAS/N. While being slightly less efficient than RASPEN, the two-step RAS/N method still appears as a competitive alternative. The convergence of the nonlinear RAS method, applied as a solver instead of being used as a preconditioner, is not reported here, but roughly speaking, the nonlinear RAS method is as inefficient as the linear one.

Conclusion

We have analyzed a family of preconditioned Newton methods based on the nonlinear multi-splitting approach in application to mildly nonlinear systems resulting from the discretization of some degenerate evolutionary PDEs such as porous media or Richards' equation. Based on the Monotone Newton Theorem we show that the preconditioned method is globally convergent. The current result extends our previous analysis [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF] to the one-level RASPEN method [START_REF] Dolean | Nonlinear preconditioning: How to use a nonlinear Schwarz method to precondition Newton's method[END_REF]. In addition, for the preconditioning based on a single nonlinear splitting, including the method presented in [START_REF] Brenner | On the monotone convergence of Jacobi-Newton method for mildly nonlinear systems[END_REF], the preconditioned Newton's method is equivalent to a simpler to implement predictor-corrector scheme. The numerical experiment based on discrete porous media equation shows that the performance of block Jacobi-Newton, RASPEN and RAS/Newton methods is essentially independent of the mesh size.

Fig. 1 :

 1 Fig. 1: Left: the solution for 𝑁 = 100 (black) and the iterates 𝛽(𝑢 𝑛 ) of the Jacobi-Newton method for 𝑛 = 10, 20, . . . 70. Right: convergence history of the Jacobi-Newton method for 𝑁 = 100 (blue) and 400 (red), the iteration count is scaled by the size of the discrete system; the error is measured in 𝑙 ∞ norm. The vertical black line positioned at 𝑁 𝑓 /𝑁 indicate the location of the solution front.

Fig. 2 :

 2 Fig. 2: Convergence history of preconditioned Newton's method for 𝑁 = 100 (blue) and 400 (red); the error is measured in 𝑙 ∞ norm.