Mathilde Yousefi

Stéphanie 2023 Fleck

Keywords: Human-centered computing → Interaction design; Collaborative and social computing Hybrid collaborative learning, Emotional presence, Tangible interface, Affective computing Apprentissage collaboratif hybride, Présence émotionnelle, Interface tangible, Informatique affective

Incarnation de la présence émotionnelle grâce à une interface tangible en situation d'apprentissage collaboratif hybride : état de l'art

Embodying emotional presence through a tangible interface in a hybrid collaborative learning situation: state of the art

INTRODUCTION

Ces dernières années ont été marquées par une accélération des changements de modalités de travail amenant, notamment, à davantage d'interactions à distance. Exacerbées durant la récente pandémie, ces formes d'interactions concernent, aujourd'hui, aussi bien les situations personnelles (e.g., échanges familiaux longue distance), les situations professionnelles (e.g., télétravail) que les situations d'éducation et formation (e.g., apprentissage à distanceonline, en condition hybride ou blended learning -modèle d'apprentissage couplant présentiel et distanciel). Malheureusement, dans le cadre éducatif, comme l'attestent les nombreux travaux conduits notamment lors de la pandémie, ces distanciations sociales entraînent des conséquences négatives sur les ressentis des apprenants (e.g., stress, démotivation, perte d'efficacité personnelle, émotions négatives) [START_REF] Acosta-Gonzaga | Students' Emotions and Engagement in the Emerging Hybrid Learning Environment during the COVID-19 Pandemic[END_REF][START_REF] Leyrit | Les conditions de travail à distance et le stress ressenti par les étudiants en France pendant la période de confinement[END_REF][START_REF] Mercier | Facteurs de protection et modélisation du bien-être universitaire des étudiants en formation à distance[END_REF]. De plus, lorsque des situations pédagogiques collaboratives sont menées en distanciel, la présence sociale, i.e., la capacité d'un individu à se comporter de manière appropriée et efficace dans les interactions sociales, essentielle à toute collaboration [START_REF] Murphy | Recognising and promoting collaboration in an online asynchronous discussion[END_REF], est mise en difficulté [START_REF] Girardet | Collaboration au moyen du numérique dans des travaux de groupe : perceptions d'étudiants et étudiantes universitaires en temps de pandémie[END_REF]. Ces activités pédagogiques collectives peuvent également générer une multitude d'états affectifs (e.g., stress, ennui, isolement, frustration, plaisir, neutre, [START_REF] Ali Akber Dewan | Engagement detection in online learning : a review[END_REF]) qui deviennent alors difficiles à percevoir et donc à réguler par l'enseignant (tutelar regulation), par l'apprenant lui-même (self-regulation) mais aussi par les autres apprenants (Socialy-shared regulation) [START_REF] Hadwin | Self-regulation, co-regulation, and shared regulation in collaborative learning environments[END_REF][START_REF] Järvelä | Contemporary perspectives of regulated learning in collaboration[END_REF][START_REF] Panadero | Effects of self-assessment on self-regulated learning and self-efficacy : Four meta-analyses[END_REF]. Face à l'enjeu, ce travail en cours correspond à la première étape d'un projet visant, à terme, la conception d'une nouvelle génération d'interfaces utilisateur tangibles (TUI), qui, couplées à une solution d'affective-computing, visent à rendre la présence émotionnelle perceptible aux groupes de travail et aux enseignants malgré la distance.

En vue de nourrir la conception de TUI dites affectives au sein d'une approche de recherche fondée sur la conception ou design-Based Reseach (ou DBR -voir [START_REF] Sasha | Design-based research : Clarifying the terms[END_REF][START_REF] Reimann | Design-based research[END_REF][START_REF] Istenic Starcic | Design-based research on the use of a tangible user interface for geometry teaching in an inclusive classroom[END_REF][START_REF] Wang | Design-based research and technology-enhanced learning environments[END_REF]), cet article propose une revue de littérature adossée sur une première sélection de 15 articles. Après avoir présenté le contexte spécifique de ces travaux, nous discuterons dans une perspective de DBR les résultats ciblant deux questions principales :

• Q1 -Quelles informations recueillir et présenter aux participants ?

• Q2 -Comment matérialiser au mieux ces informations via une interface tangible ?

ÉLÉMENTS DE CONTEXTE

L'apprentissage collaboratif à distance ou en condition hybride

Dans le cadre spécifique des dispositifs de formation mobilisant l'apprentissage collaboratif, la qualité des processus interpersonnels est une variable déterminante. En effet, ce type de situation pédagogique demande que les individus travaillent collectivement pour atteindre un objectif commun défini [START_REF] Dillenbourg | Collaborative Learning : Cognitive and Computational Approaches[END_REF]. Les élèves bénéficient du partage d'expériences dès lors que les interactions sont régulées et structurées [START_REF] Dillenbourg | The evolution of research on collaborative learning[END_REF][START_REF] Hutchins | Distributed cognition[END_REF]. Ces régulations impliquent des interactions constantes avec les connaissances, les pairs et l'enseignant. Cela nécessite également une synchronisation opérationnelle pour soutenir la progression de la tâche, la coordination et le contrôle conjoints. De plus, pour être optimale, la collaboration exige une forte conscience situationnelle (i.e., la capacité de comprendre et de réagir de manière appropriée aux stimuli environnementaux dans un contexte donné) ainsi qu'un climat d'apprentissage positif et motivant [START_REF] Liesbeth Kj Baartman | Evaluating assessment quality in competence-based education : A qualitative comparison of two frameworks[END_REF][START_REF] Mircea | Importance and Opportunities of Sentiment Analysis in Developing E-Learning Systems through Social Media[END_REF][START_REF] Heras | Responsible research and innovation indicators for science education assessment : how to measure the impact[END_REF]. En distanciel, tout cela demande donc la structuration d'une "communauté d'enquête" (Community of Inquiry -CoI voir [START_REF] Rienties | Measuring and understanding learner emotions : Evidence and prospects[END_REF][START_REF] Stenbom | Emotional presence in a relationship of inquiry : The case of one-to-one online math coaching[END_REF]), pour laquelle plusieurs états de présence des participants sont reconnus comme nécessaires [START_REF] Garrison | Researching the community of inquiry framework : Review, issues, and future directions[END_REF].

• La présence cognitive. Elle est basée sur la construction de représentations partagées et sur une représentation cohérente de la situation, du problème par les participants [START_REF] Hernández-Sellés | Computer-supported collaborative learning : An analysis of the relationship between interaction, emotional support and online collaborative tools[END_REF][START_REF] Roschelle | The construction of shared knowledge in collaborative problem solving[END_REF]. Cela demande d'exprimer et échanger des connaissances, un engagement dans une pensée critique et la résolution de problèmes [START_REF] Garrison | The first decade of the community of inquiry framework : A retrospective[END_REF]. Elle contribue à la construction d'un « référentiel commun » [START_REF] Resnick | Socially shared cognition[END_REF].

• La présence sociale. Elle s'apparente à une projection des caractéristiques personnelles des étudiants dans le groupe. Elle façonne le point de départ de l'activité collaborative et permet une socialisation entre les membres.

Elle participe au sentiment de faire communauté, améliore la communication et l'entraide [START_REF] Trespalacios | Developing online sense of community : Graduate students' experiences and perceptions[END_REF].

• La présence pédagogique, ou présence enseignante, consiste pour l'enseignant à créer un environnement favorable à l'apprentissage et non seulement au travail. Cela comprend, notamment, la communication d'objectifs d'apprentissage clairs, des retours explicites et au bon moment. L'enseignant a donc un rôle clé dans ces situations distanciées. Le cadre donné au cours de l'activité permet d'éviter les sentiments d'isolement et les interactions non productives, dégradant l'expérience d'apprentissage [START_REF] Garrison | Exploring causal relationships among teaching, cognitive and social presence : Student perceptions of the community of inquiry framework[END_REF]. Les bonnes interactions entre enseignants et étudiants sont donc primordiales [START_REF] Lee | Examining the relationship among student perception of support, course satisfaction, and learning outcomes in online learning[END_REF].

Au-delà de la qualité de l'ingénierie pédagogique, tout cela demande des participants (étudiants et enseignant) qu'ils puissent établir et maintenir des relations entre eux au sein de l'environnement d'apprentissage en ligne ou hybride.

Aujourd'hui, ces aspects sont potentiellement soutenus par des outils couramment proposés par les plateformes tels que les chats, la visioconférence, les feedbacks ou les espaces partagés de travail synchrone/asynchrone [START_REF] Richardson | Social Presence in Relation to Students' Satisfaction and Learning in the Online Environment : A Meta-analysis[END_REF][START_REF] Rourke | Assessing social presence in asynchronous text-based computer conferencing[END_REF]. Or, une quatrième dimension, plus récemment ajoutée au modèle de CoI, est peu ou pas prise en compte par les systèmes actuellement déployés en formation : la présence émotionnelle [START_REF] Cleveland-Innes | The influence of emotion on cognitive presence in a case of online math coaching[END_REF]. La présence émotionnelle fait référence à l'état affectif perceptible et identifiable par les participants durant l'expérience d'apprentissage [START_REF] Tyler | Facets of emotional awareness and associations with emotion regulation and depression[END_REF][START_REF] Rieffe | Emotion awareness and internalising symptoms in children and adolescents : The Emotion Awareness Questionnaire revised[END_REF]. Cette dernière est directement influencée par les trois précédentes et inversement [START_REF] Mircea | Importance and Opportunities of Sentiment Analysis in Developing E-Learning Systems through Social Media[END_REF][START_REF] Rienties | Measuring and understanding learner emotions : Evidence and prospects[END_REF]. L'absence de régulation émotionnelle peut mener à de la frustration, des interactions non productives et donc une expérience d'apprentissage dégradée [START_REF] Goetz | Academic emotions from a social-cognitive perspective : Antecedents and domain specificity of students' affect in the context of Latin instruction[END_REF][START_REF] Järvenoja | Supporting groups' emotion and motivation regulation during collaborative learning[END_REF][START_REF] Pekrun | Academic emotions[END_REF].

C'est donc un aspect essentiel en contexte en ligne ou hybride. Or, il est compliqué d'être conscient des émotions des autres à distance.

Pour cette raison, il apparaît essentiel de chercher à rendre tangible l'état émotionnel des participants au sein des environnements de travail collaboratif hybrides pour améliorer le sentiment de présence émotionnelle malgré un contexte de distanciation sociale.

La tangibilisation des émotions en contexte éducatif

La conception d'interactions en lien avec les affects est un domaine de recherche en plein développement. En Interaction Humain-Machine (IHM), de nombreux travaux témoignent du potentiel des interfaces utilisateurs tangibles (TUI) pour soutenir l'apprentissage, la collaboration ou une expérience d'apprentissage positive (e.g., [14, 21-23, 38, 50, 56, 65, 66].

Cependant, ces derniers se centrent quasi systématiquement sur les apprentissages en face à face, notamment car les TUI sont favorables aux activités de manipulation directe. Les études ciblant les supports tangibles pour l'apprentissage en distanciel ou contexte hybride ou ciblant la tangibilisation d'émotions personnelles ou collectives (i.e, emotion awareness) via des TUI sont encore peu développées (e.g., [START_REF] Maria | EmoTEA : Teaching Children with Autism Spectrum Disorder to Identify and Express Emotions (Interaction '19)[END_REF][START_REF] Molinari | Soutenir les stratégies volitionnelles et améliorer l'expérience des étudiants en formation à distance. Quels potentiels pour le design tangible ? Distances et médiations des savoirs[END_REF][START_REF] Woodward | Emoecho : A tangible interface to convey and communicate emotions[END_REF]). Plusieurs travaux antérieurs ont exploré des supports tangibles interactifs dans le but de permettre la communication d'émotions, mais ils visaient essentiellement à rendre possible à l'utilisateur l'expression d'émotions, par exemple, pour interagir avec un avatar au sein d'un scénario de storytelling [START_REF] Paiva | SenToy : an affective sympathetic interface[END_REF], aider à percevoir [START_REF] Roseway | BioCrystal : An Ambient Tool for Emotion and Communication[END_REF], ou apprendre à exprimer son état émotionnel [START_REF] Angelini | Towards an Anthropomorphic Lamp for Affective Interaction[END_REF]. A notre connaissance, seul

Nimio [START_REF] Brewer | Nimio : an ambient awareness device[END_REF] correspond à une interface tangible ambiante pensée pour la collaboration. Cependant, elle ne matérialise qu'une part de la présence sociale des collaborateurs.

Face au manque de travaux répondant spécifiquement à nos questions, nous nous sommes donc concentrés à identifier les facteurs favorables à la conception d'une TUI permettant d'incarner la présence émotionnelle. En effet, certaines questions se posent. La première que nous explorons ici est de savoir que montrer. La seconde est comment montrer cette information par le biais d'une interface qui soit non-intrusive pour, notamment, faire que l'incarnation de la présence émotionnelle soit sans ambiguïté dans son contenu.

MOYENS ET MÉTHODES

Dans le cadre de cette étude en cours, il s'agit donc de mieux identifier les moyens pertinents pour matérialiser l'état émotionnel au sein d'un groupe de participants. Les principaux champs de notre recherche concernent donc principalement les sciences de l'éducation, mais aussi les sciences du numérique (affective computing, IHM, CSLC ou computer-supported collaborative learning). Nous nous sommes également appuyés sur les champs de recherche du marketing où nous nous sommes principalement intéressés à la sensorialité des produits.

Pour effectuer cet état de la littérature, les différents termes utilisés étaient "collaborative learning" et "emotional awareness", "social awareness" mais aussi "colors", "emotions", "tangible interface" et "affective computing". , 6, 7, 8, 11, 15). Deux études sont, elles, des présentations de prototypes peu de participants, mettant des limites certaines aux résultats obtenus et confirmant le besoin d'études empiriques dans ce domaine. De plus, on note que les modèles théoriques reliés aux états émotionnels ne sont pas toujours invoqués.

Limites

Beaucoup d'études sur l'apprentissage collaboratif portent sur les modalités de travail en distanciel où chaque étudiant est seul face à son ordinateur. Aucune ne s'intéresse spécifiquement au contexte d'apprentissage hybride, qui peut présenter des difficultés supplémentaires comme notamment une asymétrie de l'information et un manque de communication entre les groupes d'élèves en présentiel avec les autres groupes à distance.

RÉSULTATS ET DISCUSSION

Fiabilité de l'incarnation

Comme pour tout feedback, que ce soit en IHM [START_REF] Limerick | The experience of agency in human-computer interactions : a review[END_REF] ou en éducation [START_REF] Perrenoud | La part d'évaluation formative dans toute évaluation continue[END_REF], rendre compte de l'état émotionnel de manière tangible exige une certaine fiabilité et fidélité dans l'incarnation des données d'entrée (inputs) pour faire sens aux participants (i.e., être compréhensible et pertinente).

L'étude 1 - [START_REF] Cohn | Automated Face Analysis for Affective Computing[END_REF]) montre à travers une revue de littérature que les expressions faciales, qui portent souvent de fortes significations, peuvent être utilisées de manière fiable pour la reconnaissance automatique des émotions. Mais selon l'étude 13 [START_REF] Wang | A systematic review on affective computing : emotion models, databases, and recent advances[END_REF], renforcée par [START_REF] Feldman | Emotional Expressions Reconsidered : Challenges to Inferring Emotion From Human Facial Movements[END_REF][START_REF] Wang | A review of emotion sensing : categorization models and algorithms[END_REF] dans une optique de conception d'une incarnation basée sur des couleurs, il serait intéressant de contrôler l'ambiguïté de certaines pour éviter de potentiellement biaiser les utilisateurs. Un axe intéressant serait également de jouer sur l'évolution de l'apparence de l'interface. En conditions expérimentales (N=30 participants), les auteurs de l'étude 6 [START_REF] Lin | Feeling Colours : Crossmodal Correspondences Between Tangible 3D Objects, Colours and Emotions[END_REF] se sont servis du paradigme Bouba/Kiki pour trouver des associations de différentes modalités (croisées) entre la forme (anguleuse ou ronde), la complexité (nombre de protubérances) et la couleur d'un objet en trois dimensions. Ils montrent que les émotions calmes, plaisantes et associées à un fort contrôle étaient liées à des teintes bleues, alors qu'à l'inverse, les émotions agitées, déplaisantes et le manque de contrôle, étaient liées à des teintes rouges (résultats concordants avec ceux de l'étude 11 [START_REF] Roseway | BioCrystal : An Ambient Tool for Emotion and Communication[END_REF]). Leurs résultats ont aussi montré que les formes fortement anguleuses et complexes étaient également associées aux teintes rouges alors que les formes faiblement anguleuses et peu complexes étaient plutôt associées aux teintes bleues. Une limite à cette étude est que les deux associations étaient mesurées séparément, il n'y donc pas de mesure d'un potentiel effet combiné.

Dans la mesure où les apprenants ressentent le plus souvent une émotion neutre (voir supra), l'interface pourrait envoyer un feedback seulement lorsque la mesure physiologique atteint des valeurs définies, et ne montrer qu'une couleur ou autre information "neutre" (comme les nuances de beiges selon l'étude 4) le reste du temps. Une autre possibilité pour lever l'ambiguïté liée à la compréhension d'une teinte ou d'une forme serait sans doute d'ajouter des caractéristiques anthropomorphiques à l'objet. Effectivement, les études 7 et 15 ([55, 76]) démontrent une préférence pour les interfaces disposant de caractéristiques anthropomorphiques (regards, voix) qui pourraient donc favoriser une incarnation "humaine". L'étude 7 souligne également le besoin d'avoir un mode "privé" ouvrant la question de l'éthique de l'exposition publique. Enfin, à notre connaissance, les travaux autour des feedbacks haptiques pour incarner des émotions restent à explorer.

La distribution de l'information entre les participants

Dans le contexte hybride, plusieurs groupes d'élèves en présentiel peuvent se trouver dans le même site, d'autres groupes ailleurs, et leur enseignant se trouver également sur un autre campus. Il nous semble donc important de questionner à qui fournir l'interface. L'étude 12 [START_REF] Hernández-Sellés | Computer-supported collaborative learning : An analysis of the relationship between interaction, emotional support and online collaborative tools[END_REF] propose une étude exploratoire autour du design d'interfaces par des étudiants.

Ces objets tangibles avaient pour ambition d'être utilisés dans des contextes de vidéoconférences. Les prototypes produits proposent d'optimiser la présence sociale et la présence émotionnelle notamment via la régulation des tours de paroles et de l'ambiance. Ceci met en évidence l'importance de ces dimensions pour les étudiants. Malheureusement, ces solutions n'ont pas fait l'objet de validations. Il n'est donc pas possible de connaître leur effet sur la dynamique d'apprentissage.

  Nous avons sélectionné des études issues des bases de données ScienceDirect, Web of Science et PsycArticles. Les premiers critères de sélection étaient la date de publication postérieure à 2012, le type d'utilisateurs (notre étude s'intéressant aux adultes en formation). Nous avons ainsi trouvé une centaine d'articles. Puis nous avons affiné la sélection en nous concentrant spécifiquement sur les contextes d'apprentissage ou professionnel, et sur les articles porteurs d'éléments pour l'incarnation de la présence émotionnelle. Les principaux résultats des articles retenus sont ensuite discutés à la lumière d'autres travaux (n>60) dans une perspective orientée design pour la conception d'une nouvelle génération de TUI affective.

Fig. 1 .

 1 Fig. 1. Évolution des dates de publication des différentes études sélectionnées pour ce travail (Voir Annexes pour la description détaillée des 15 articles).15 articles résultent de cette sélection. L'essentiel a moins de 5 ans (voir 1), ce qui confirme l'intérêt très récent autour de ces questions. Les études discutées sont référencées par ordre alphabétique dans les Tables 1 et 2 en annexes. Dans ces tables, nous mettons en avant leur lien avec les deux questions soulevées en introduction, les modèles théoriques utilisés ainsi que les limites identifiées de ces travaux. Ces travaux retenus sont des états de la littérature conséquents (N°1, 13, 14), des enquêtes portant sur les représentations personnelles (N°3, 4, 9) d'étudiants ou grand public à l'échelle internationale, des études expérimentales (N°5,[START_REF] Feldman | Emotional Expressions Reconsidered : Challenges to Inferring Emotion From Human Facial Movements[END_REF][START_REF] Tyler | Facets of emotional awareness and associations with emotion regulation and depression[END_REF][START_REF] Boehner | How emotion is made and measured[END_REF][START_REF] Cleveland-Innes | The influence of emotion on cognitive presence in a case of online math coaching[END_REF][START_REF] Antonio R Damasio | L'Ordre étrange des choses : La vie, les sentiments et la fabrique de la culture[END_REF]. Deux études sont, elles, des présentations de prototypes

  de nos expériences antérieures et de la situation dans laquelle le déclencheur émotionnel survient. De plus, il peut varier en intensité et en durée, voire perdurer même en l'absence des déclencheurs. C'est donc un phénomène complexe et multifacette, fondamentalement intrinsèque au sujet et situationnel.Des travaux semblent ainsi encore nécessaires pour rendre tangible une telle complexité. Un nouveau challenge repose selon l'étude 13 sur le fait d'apprendre aux IA à détecter les émotions masquées, simulées/fausses. Dans notre cas, nous pensons également nécessaire de discriminer les émotions non liées à la situation dans laquelle la personne est placée. Par exemple, une personne arrivant triste à cause d'un événement extérieur pourrait ne pas arriver à éprouver du plaisir lors de la tâche d'apprentissage, sans que la tâche en soit la cause, questionnant ainsi l'incarnation de la ligne de base à utiliser (qu'est-ce qu'un état moyen ?). Les travaux de l'étude 11[START_REF] Roseway | BioCrystal : An Ambient Tool for Emotion and Communication[END_REF] alertent aussi sur le fait que les seules réponses physiologiques ne retranscrivent pas totalement la perception personnelle des sujets. Les auteurs ont ici proposé un dispositif ambiant lumineux pour matérialiser à un utilisateur son état émotionnel en situation de travail en se basant sur le modèle Pleasure-Arousal. Les auteurs ont préalablement collecté des indices physiologiques afin de créer une baseline émotionnelle. En parallèle, les participants étaient invités à exprimer leur état émotionnel sur le modèle PAD tout au long de la journée. Or, de manière intéressante, les données auto-rapportées montrent une différence significative avec les données physiologiques captées. Ainsi, les sujets évaluent leur état comme étant plus énergique (axe "arousal") et plus positif (axe "pleasure") que ce que les mesures physiologiques rapportaient, traduisant un possible biais.Malheureusement, l'influence de la manière de retranscrire au sujet l'information sur son état émotionnel n'est pas questionnée dans ces études.Pour designer une interface qui facilite la prise de conscience émotionnelle du groupe, nous pouvons nous appuyer sur différents travaux qui ont interrogé les utilisateurs potentiels et prenant en compte les formes, les couleurs, les caractéristiques de l'interface. Selon l'étude 4 de[START_REF] Järvenoja | Supporting groups' emotion and motivation regulation during collaborative learning[END_REF] qui s'intéresse aux liens entre couleurs et émotions à travers 30 pays (N=4883 répondants), les associations les plus universelles, malgré les variations culturelles, incluent le noir (tristesse,

	fournir une trop importante typologie d'émotions engendrerait sans doute une forte charge mentale chez les utilisateurs
	et pourrait influencer l'expérience d'apprentissage [4]. Les résultats de ces études tendent à plaider plutôt pour un
	modèle multidimensionnel et pour la conception d'un artefact permettant de matérialiser des évolutions émotionnelles
	marquées. De plus, les émotions positives étant dominantes lors d'une tâche d'apprentissage en groupe, il pourrait être
	plus facile d'indiquer des fluctuations au sein d'une tendance en lien par exemple avec les modèles tridimensionnels
	[63]. Cependant, le faible nombre de sujets limite les résultats. Il nous apparaît donc important de poursuivre les travaux
	autour de cette question.
	4.3 Les caractéristiques de l'interface
	Il n'est ici pas clair s'il faut privilégier des retours unidimensionnels (e.g., indiquer une succession d'états émotionnels
	sur la base d'une typologie d'émotions) ou multidimensionnels (e.g., indiquer une tendance entre plusieurs états). En
	effet, en situation d'apprentissage, les émotions se succèdent à des temporalités variables, voire se superposent et
	peuvent être ambivalentes, comme, par exemple, lorsque vous faites face à une tâche intéressante, mais difficile. De plus,

, associer la reconnaissance faciale à d'autres sources est à privilégier pour appréhender au mieux un état émotionnel. En effet, les humains expriment leurs états émotionnels à travers des informations multimodales. Aussi, l'incarnation tangible devrait également pouvoir prendre en considération et rendre compte de tous les indices possibles (e.g., réponses physiologiques émotionnelles, comportements, tons de la voix, expressions faciales et gestes corporels). Sur la base de l'étude 13

[START_REF] Wang | A systematic review on affective computing : emotion models, databases, and recent advances[END_REF]

, coupler une TUI à des solutions d'affective computing en vue de fournir des données d'entrée multimodales en temps réel apparaît envisageable. Les récentes avancées en ML et DL permettent de fusionner des bases de données physiques (reconnaissances faciales, données textuelles, audio) et physiologiques (ECG, EEG). Cet état de l'art fondé sur près de 400 articles statue également sur la performance des modèles d'analyses affectives multimodaux améliorant la robustesse de la reconnaissance émotionnelle et autorisant aujourd'hui l'application en condition réelle. La reconnaissance des émotions grâce à des capteurs physiologiques de type électroencéphalogramme, électrocardiogramme semble, selon nous, complexe en condition éducative car très invasive. Des solutions portables (voir

[START_REF] Cecchi | Physical stimuli and emotions : EDA features analysis from a wrist-worn measurement sensor[END_REF][START_REF] Kamišalić | Sensors and functionalities of non-invasive wrist-wearable devices : A review[END_REF]

) devraient être privilégiées. Les IA d'affective computing se fondent sur de grandes bases de données pré-enregistrées auxquelles différentes émotions sont associées

[START_REF] Wang | A review of emotion sensing : categorization models and algorithms[END_REF]

. Les modèles utilisés invoquent soit des modèles discrets

[START_REF] Jessica | Four models of basic emotions : A review of Ekman and Cordaro, Izard, Levenson, and Panksepp and Watt[END_REF]

fondés sur les émotions de base tel le modèle de Ekman (joie, tristesse, colère, dégoût, peur, surprise), mais aussi des modèles multidimensionnels tels le modèle PAD ou tridimentionnel qui ajoutent la perspective temporelle

[START_REF] Alexander T Latinjak | The underlying structure of emotions : A tri-dimensional model of core affect and emotion concepts for sports[END_REF][START_REF] Mehrabian | Pleasure-arousal-dominance : A general framework for describing and measuring individual differences in temperament[END_REF]

. Ce dernier nous apparaît le plus pertinent. En effet, comme l'indiquent

[START_REF] Boehner | How emotion is made and measured[END_REF][START_REF] Cosnier | Psychologie des émotions et des sentiments[END_REF][START_REF] Antonio R Damasio | L'Ordre étrange des choses : La vie, les sentiments et la fabrique de la culture[END_REF]

, l'état émotionnel d'une personne est une réaction subjective à des stimuli internes ou externes (e.g., des événements, des personnes, des artefacts ou des idées). Il correspond autant à une réponse physiologique qu'à une représentation mentale consciente, et donne une valeur (positive/neutre/négative) sous l'influence

4.2 La spécificité des états émotionnels en situation éducative

Pour identifier les typologies d'incarnations possibles, il convient de cerner la diversité des informations émotionnelles à fournir (outputs). L'étude 14

[START_REF] Yadegaridehkordi | Affective computing in education : A systematic review and future research[END_REF]

met en évidence les 20 principaux états émotionnels utilisés dans les études sur l'informatique affective ; à savoir l'ennui, la colère, l'anxiété, le plaisir, la surprise, la tristesse, la frustration, la fierté, l'espoir, le désespoir, la honte, la confusion, le bonheur, le naturel, la peur, la joie, le dégoût, l'intérêt, le soulagement et l'excitation. Les auteurs de l'étude 2

[START_REF] Duo | An E-learning System based on Affective Computing[END_REF]

affirment que 4 états émotionnels se succèdent durant les tâches d'apprentissage : la surprise, la confusion, le désespoir et enfin la confiance en soi (notamment, après avoir réussi). Cependant, ils n'adossent leurs affirmations sur aucune donnée ou travail antérieur. Si l'on se fonde sur les études en contexte éducatif, Dewan et al. (2019) [4] et D'Mello et al. (2013) [20] indiquent, par exemple, que les émotions les plus éprouvées dans le contexte d'apprentissage sont la frustration, l'ennui, l'engagement, le stress, la confusion, mais aussi le plaisir, le gain d'apprentissage et la neutralité. L'étude 5 -[40]) cherche à mesurer subjectivement la conscience des émotions grâce à une grille d'auto-évaluation chez 11 participants (8 femmes et 3 hommes) d'un âge moyen de 24 ans.

Ces derniers devaient reporter 4 fois au cours de la journée les émotions qu'ils ont éprouvées selon une liste restreinte proche de celle des auteurs précédents : anxiété, ennui, confusion, curiosité, plaisir, engagement, frustration, surprise et neutralité. Ils ont également mené des entretiens rétrospectifs afin de connaître les causes présupposées de ces émotions.

Selon leurs résultats, les étudiants éprouveraient davantage d'émotions négatives que positives (anxiété, frustration et ennui). Les causes seraient le plus souvent liées à la tâche d'apprentissage plutôt qu'aux résultats (reçus ou attendus). L'émotion "neutre" a été rapportée plus souvent en situation d'apprentissage individuel qu'en situation collective qui induirait des émotions positives avec une plus forte valence. Ces résultats s'inscrivent dans la continuité des résultats de l'étude 11 qui montrent que les salariés ressentent majoritairement un état émotionnel neutre, ou éprouvent de l'ennui, du stress ou de la frustration au cours de leurs activités. Les ressentis positifs étaient eux principalement suscités par les interactions sociales. peur, haine), le rouge (amour, colère), le rose (amour, joie, plaisir), le gris (tristesse, déception), le jaune (joie), le orange (joie, amusement) et le blanc (soulagement). Ces résultats soulignent cependant l'ambiguïté dans la représentation émotionnelle de certaines couleurs. La couleur rouge, par exemple, est le plus souvent liée à la colère, mais plus encore à l'amour. D'autres ne sont pas représentatives d'une émotion particulière (les nuances de beiges, de marrons. . .). Ainsi,

Table 1 .

 1 [START_REF] Molinillo | Exploring the impacts of interactions, social presence and emotional engagement on active collaborative learning in a social web-based environment[END_REF] (Etude 9 -[START_REF] Molinillo | Exploring the impacts of interactions, social presence and emotional engagement on active collaborative learning in a social web-based environment[END_REF]) ont présenté un modèle de liens existants entre différents concepts soutenant l'apprentissage actif dans une modalité de travail en distanciel. Les auteurs s'appuient sur les "apprenantenseignant" et "apprenant-apprenant" et déterminent leurs liens et leur importance dans un contexte d'apprentissage collaboratif. Leurs résultats démontrent que les interactions entre étudiants n'auraient pas d'effet direct sur l'apprentissage actif. Selon les auteurs, les facteurs clés d'un apprentissage actif (par ordre d'importance) sont la présence sociale, l'engagement émotionnel et l'interaction entre enseignants et étudiants. L'étude 3[START_REF] Hernández-Sellés | Computer-supported collaborative learning : An analysis of the relationship between interaction, emotional support and online collaborative tools[END_REF] s'appuie elle sur les concepts soutenant l'apprentissage collaboratif. Leurs hypothèses sont que i) les interactions sociales entre étudiants (apprenants) et enseignants améliorent les interactions entre étudiants dans le groupe, ii) les interactions entre étudiants améliorent l'apprentissage collaboratif et iii) le soutien émotionnel du groupe, iv) les outils collaboratifs en ligne améliorent les interactions entre étudiants et le soutien émotionnel du groupe et v) le soutien émotionnel du groupe améliore l'apprentissage collaboratif. Les résultats de l'étude valident toutes les hypothèses. Les résultats des études 3 et 9 mettent donc en avant le rôle primordial de l'enseignant. S'il n'est pas certain que les étudiants aient besoin de la TUI, il apparaît nécessaire que l'enseignant soit l'un des premiers destinataires des informations liées à l'état émotionnel au sein des groupes de travail collaboratif. Il est alors question du timing. En effet, l'étude 8[START_REF] Molinari | Soutenir les stratégies volitionnelles et améliorer l'expérience des étudiants en formation à distance. Quels potentiels pour le design tangible ? Distances et médiations des savoirs[END_REF] met en évidence que le temps peut également apparaître comme un facteur d'anxiété ou d'ennui et propose des objets tangibles passifs pour soutenir la régulation du temps et la gestion de la motivation (notamment en mettant en avant les progrès Les résultats de ces différents travaux mettent en évidence le caractère émergent des problématiques liées à l'incarnation des états émotionnels. Cependant, des éléments tels l'utilisation de modèles multimodaux tant en entrée qu'en sortie, l'incarnation de valences plutôt que d'un état de base, et une conception permettant d'optimiser la présence enseignante semblent des pistes privilégiées pour la conception d'interfaces tangibles affectives. Cependant, les différentes questions que nous nous sommes posées ont à leur tour soulevé des problématiques. La suite de ces travaux vise donc à répondre aux zones de flou restantes, notamment les questions éthiques reliées jamais abordées par les études recensées. En effet, faut-il montrer des états individuels ? Les utilisateurs pourraient ne pas apprécier l'accès à leurs données personnelles, ou se sentir obligés de simuler telle ou telle émotion par biais de désirabilité sociale. Si l'on matérialise une tendance de groupe, comment indiquer qu'un seul individu marginal peut avoir besoin de soutien sans risque de stigmatisation.Aussi, des études supplémentaires restent à réaliser. Au-delà de nourrir nos travaux, nous espérons que cet état de l'art sera profitable aux chercheurs intéressés par la matérialisation de la présence émotionnelle en situation d'apprentissage. Liste détaillée des études, incluant la nature des études conduites, si elles invoquent un modèle théorique lié aux émotions, les principaux résultats en lien avec les deux questions de notre étude et les limites identifiées.

	et tâches réalisés) dans un contexte d'apprentissage à distance. De plus, si l'enseignant a un rôle clé, il faudrait qu'il
	soit alerté dès que son intervention est requise (conflit, confusion, abandon). Effectivement, comme le montre l'étude 3
	[33], il se peut qu'il y ait des divergences au sein du groupe, tous les individus n'ayant pas les mêmes motivations, ou
	bien personne n'ayant pris la direction du groupe. L'intérêt de l'interface serait alors de capter et de retranscrire ces
	variations dans l'état émotionnel du groupe à l'enseignant. L'interface pourrait aussi inclure une dimension temporelle
	en rendant compte d'un cheminement émotionnel accompli jusque-là.
	5 CONCLUSION

Table 2 .

 2 Suite de la liste détaillée des études, incluant la nature des études conduites, si elles invoquent un modèle théorique lié aux émotions, les principaux résultats en lien avec les deux questions de notre étude et les limites identifiées.

(N°2 & 12). Il est important de noter que la plupart des sessions de test ou expérimentales sont faites avec parfois très


Amina Benaceur
email: benaceur@mit.edu.

Reducing sensors for transient heat transfer problems by means of variational data assimilation

Keywords: 2020 Mathematics Subject Classification. 65K10 PBDW, model reduction, data assimilation, sensor reduction, heat transfer

We propose a contribution that combines model reduction with data assimilation. A dedicated Parametrized Background Data-Weak (PBDW) approach has been introduced in the literature so as to combine numerical models with experimental measurements. We extend the approach to a time-dependent framework by means of a POD-greedy reduced basis construction. Since the construction of the basis is performed offline, the algorithm addresses the time dependence of the problem while the time stepping scheme remains unchanged. Moreover, we devise a new algorithm that exploits offline state estimates in order to diminish both the dimension of the online PBDW statement and the number of required sensors collecting data. The idea is to exploit in situ observations in order to update the best-knowledge model, thereby improving the approximation capacity of the background space.

Introduction

State estimation is a task in which the quantity of interest is the "true" state u true of a physical system over a space or space-time domain of interest. However, numerical prediction based on a given mathematical model may be deficient due to limitations imposed by available knowledge. In other words, the mathematical model can only take anticipated or parametric uncertainty into account. A more accurate prediction requires the incorporation of experimental observations in order to accommodate unanticipated or non-parametric uncertainty.

For this reason, data assimilation methods have been widely explored in the literature. The goal of these methods is to use a priori information to deduce the best mathematical model, while using available experimental data to produce the most accurate approximation of a physical system. Many data assimilation methods involve the minimization of a cost function, such as least-squares type methods, designed to compute the mismatch between the model approximation and given observations. For instance, we mention the Stable Least Squares approximation [START_REF] Cohen | On the stability and accuracy of least squares approximations[END_REF][START_REF] Cohen | Correction to: On the stability and accuracy of least squares approximations[END_REF] and the linearized Structured Total Least Squares [START_REF] Moor | Structured total least squares and L 2 approximation problems[END_REF]. As opposed to statistical techniques [START_REF] Wikle | A Bayesian tutorial for data assimilation[END_REF][START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF], variational data assimilation relies more heavily on the mathematical model. Considering the adjoint method [START_REF] Nassiopoulos | Fast three-dimensional temperature reconstruction[END_REF][START_REF] Waeytens | Inverse computational fluid dynamics: Influence of discretization and model errors on flows in water network including junctions[END_REF], the goal is to reconstruct a physical state through the minimization of a cost function in order to optimize the parameters of the model with respect to experimental data. However, this method is only based on the precision of the mathematical model and mainly uses experimental data to correct the input, but not the final state. The Proper Orthogonal Decomposition (POD) [START_REF] Kunisch | Galerkin proper orthogonal decomposition methods for parabolic problems[END_REF] has also been exploited in a data assimilation perspective, such as for the Gappy-POD [START_REF] Willcox | Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition[END_REF][START_REF] Everson | Karhunen-Loeve procedure for gappy data[END_REF], which was introduced to handle incomplete data-sets, or as in [START_REF] Mons | Data assimilation-based reconstruction of urban pollutant release characteristics[END_REF] where data assimilation is enhanced using the POD method. We also mention other existing methods such as the 3D-VAR [START_REF] Lorenc | A global three-dimensional multivariate statistical interpolation scheme[END_REF] and the 4D-VAR [START_REF] Kärcher | Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation[END_REF] for variational data assimilation. One of the drawbacks of many variational data assimilation methods is their computational intrusivity, which means that at any stage, computational procedures need to access the A. Benaceur model in order to perform their calculations. Intrusivity is very inconvenient in many contexts, for instance when using industrial high-fidelity black-box solvers. Hence, non-intrusive or partially-intrusive options can be valuable. The methods used in [START_REF] Willcox | Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition[END_REF][START_REF] Mons | Data assimilation-based reconstruction of urban pollutant release characteristics[END_REF] are less intrusive. Nevertheless, they have not been applied with reduced order models.

The Parameterized-Background Data-Weak (PBDW) formulation for variational data assimilation is a data-driven reduced order modeling approach that was initially devised in [START_REF] Maday | A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics[END_REF] so as to merge prediction by model with prediction by data, without being intrusive. It can represent the physics of the state using a sophisticated model, and applies non-intrusive and non-iterative real-time variational data assimilation employing Reduced Basis (RB) methods [START_REF] Hesthaven | Certified reduced basis methods for parametrized partial differential equations[END_REF][START_REF] Quarteroni | Reduced basis methods for partial differential equations[END_REF], with correction of model error. The PBDW relies on the knowledge of some particular solutions to the parameterized model, and some measurements over the physical state to be approximated. Unlike classical design-of-experiment approaches [START_REF] Franceschini | Model-based design of experiments for parameter precision: State of the art[END_REF], the PBDW emphasised state estimation rather than parameter estimation. The PBDW approach has been developed in order to estimate the true state u true for several configurations of a physical system. Supposing that the true state u true depends on some unknown parameter ω in an unknown parameter set Θ that represents the unanticipated uncertainty, the goal is to account for the dependency of the true state u true (ω) on uncertain parameters by means of the sole knowledge of data. In this paper, whenever the context is unambiguous, the parameter ω is dropped.

The formulation combines a so-called "best-knowledge" (bk) model represented by a parametrized partial differential equation (PDE) and experimentally observable measurements. The use of data in the PBDW approach is fundamental not only to reconstruct the quantities of interest, but also to correct the possible bias in the mathematical bk model. The PBDW approach provides the following attractive features:

• The PBDW variational formulation simplifies the construction of a priori error estimates which can guide the optimal choice of the experimental observations.

• The PBDW formulation uses a bk model that accommodates anticipated uncertainty associated with the parameters of the model in a computationally convenient way. This bk model is typically built using model-order reduction techniques.

Note that the PBDW formulation does not explicitly include the equations of the bk model, but only a finite collection of solutions to the bk model. Thus, another important feature of the PBDW approach is its non-intrusivity. In fact, once a bk space has been generated, we no longer need the bk model. The PBDW approach has been subject to active research in recent years. It has been used for several applications. For instance, [START_REF] Taddei | Simulation-based classification; a model-order-reduction approach for structural health monitoring[END_REF] applies the PBDW for structural health monitoring; [START_REF] Hammond | PBDW: A non-intrusive Reduced Basis Data Assimilation method and its application to an urban dispersion modeling framework[END_REF] proposes a non-intrusive PBDW with application to urban dispersion modeling frameworks; and [START_REF] Argaud | Sensor placement in nuclear reactors based on the generalized empirical interpolation method[END_REF][START_REF] Argaud | Stabilization of (G)EIM in presence of measurement noise: application to nuclear reactor physics[END_REF] exploit the generalized empirical interpolation method [START_REF] Maday | A generalized empirical interpolation method: application of reduced basis techniques to data assimilation[END_REF][START_REF] Maday | The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation[END_REF][START_REF] Maday | Convergence analysis of the generalized empirical interpolation method[END_REF] in a data interpolation perspective. As a further step towards efficient industrial implementation, [START_REF] Maday | PBDW state estimation: noisy observations; configuration-adaptive background spaces; physical interpretations[END_REF] develops a PBDW approach based on noisy observations and [START_REF] Maday | Adaptive PBDW Approach to State Estimation: Noisy Observations; User-Defined Update Spaces[END_REF] introduces an adaptive PBDW approach with a user-defined update space. Moreover, [START_REF] Galarce | State estimation with nonlinear reduced models[END_REF] tackles the case of nonlinear problems with an unsteady application. Besides, the issue of noisy observations is also addressed in [START_REF] Gong | Pbdw method for state estimation: error analysis for noisy data and nonlinear formulation[END_REF]. Further work regarding the selection of experimental observation functionals has been conducted in [START_REF] Taddei | Validation by Monte Carlo sampling of experimental observation functionals[END_REF], whereas [START_REF] Taddei | A localization strategy for data assimilation; application to state estimation and parameter estimation[END_REF] proposes a localization procedure that treats the uncertainty related to boundary conditions. Finally, [START_REF] Cohen | Optimal reduced model algorithms for data-based state estimation[END_REF] proposes optimal choices for the reduced models and [START_REF] Binev | Greedy Algorithms for Optimal Measurements Selection in State Estimation Using Reduced Models[END_REF] addresses optimal measurements in state estimation using reduced models.

The PBDW approach was devised in [START_REF] Maday | A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics[END_REF] for steady problems. In this paper, we propose, as initiated in [START_REF] Benaceur | Model reduction for nonlinear problems in thermics and mechanics[END_REF], an extension of the PBDW approach to time-dependent state estimation. Two main contributions to the standard PBDW approach are presented:

• We build appropriate background spaces for the time-dependent setting using the POD-greedy algorithm [START_REF] Haasdonk | Convergence rates of the POD-greedy method[END_REF]. Here, the bk-model is approximated in space using the Finite Element Method (FEM). The POD-greedy procedure has been combined with FEM in previous work [START_REF] Li | Multiscale model reduction for fluid infiltration simulation through dual-continuum porous media with localized uncertainties[END_REF][START_REF] Haasdonk | A Reduced Basis Method for the Simulation of American Options[END_REF][START_REF] Benaceur | A Progressive Reduced Basis/Empirical Interpolation Method for Nonlinear Parabolic Problems[END_REF].

• We propose a modified offline stage so as to alleviate its computational cost which can be sizeable in a time-dependent setting. The new offline stage allows for better computational efficiency owing to a smaller online system. Moreover, it achieves substantial cost savings associated with data collection since it diminishes the number of observation sensors needed online. Note that this modified offline stage can also be applied to a steady framework.

This paper is organized as follows. Section 2 sets the notation and introduces the PBDW setting in the stationary framework as introduced in the literature. The reader familiar with this material can directly skip to Section 3 which extends the PBDW approach to the time-dependent framework. In Section 4, we discuss the offline stage and present the suggested algorithm in detail. Finally, in Section 5, we illustrate our method by some numerical results. We focus on standard parabolic problems exemplified by the linear and nonlinear heat problem. We expect similar behaviour if a moderate advection reaction term is added to the model. The more delicate question of dominant low-order perturbations is left for future work. The main idea and results presented in this work are inspired from the PhD thesis of the author [START_REF] Benaceur | Model reduction for nonlinear problems in thermics and mechanics[END_REF].

Basic notation and stationary PBDW

We consider a spatial domain (open, bounded, connected subset) Ω ⊂ R d , d ≥ 1, with a Lipschitz boundary. We introduce a Hilbert space U composed of functions defined over Ω. The space U is endowed with an inner product (•, •) and we denote by • the induced norm; U consists of functions {w : Ω → R | w < ∞}. To fix the ideas, we assume that H 1 0 (Ω) ⊂ U ⊂ H 1 (Ω), and we denote the dual space of U by U . The Riesz operator R U : U → U satisfies, for each ∈ U , and for all v ∈ U, the equality (R U ( ), v) = (v). For any closed subspace Q ⊂ U, the orthogonal complement of Q is defined as

Q ⊥ := {w ∈ U | (w, v) = 0, ∀v ∈ Q}.
Finally, we introduce a parameter set P ⊂ R p , p ≥ 1, whose elements are generically denoted by µ ∈ P.

The first source of information we shall afford ourselves in the PBDW approach is a so-called "bestknowledge" (bk) mathematical model in the form of a parametrized PDE posed over the domain Ω (or more generally, over a domain Ω bk such that Ω ⊂ Ω bk ). Given a parameter value µ in the parameter set P, we denote the solution to the bk parametrized PDE as u bk (µ) ∈ U. Then, we introduce the manifold associated with the solutions of the bk model M bk := {u bk (µ) | µ ∈ P} ⊂ U. In ideal situations, the true solution u true is well approximated by the bk manifold, i.e., the model error bk mod (u true ) := inf

z∈M bk u true -z , ( 2.1)
is very small. We introduce nested background subspaces Z 1 ⊂ . . . ⊂ Z N ⊂ . . . ⊂ U that are generated to approximate the bk manifold M bk to a certain accuracy. These subspaces can be built using various model-order reduction techniques, for instance, the Reduced Basis method. Note that the indices of the subspaces conventionally indicate their dimensions. To measure how well the true solution is approximated by the background space Z N , we define the quantity bk N (u true ) := inf z∈Z N u truez . The background space is built so that bk N (u true ) -→ N →+∞ bk mod (u true ). Moreover, we introduce the reduction error bk red,N := sup u∈M bk inf z∈Z N inf u -z , which encodes the loss of accuracy caused by solving the bk model in the N -dimensional background space Z N . For later purposes, we introduce Π Z N (u true ) as the closest point to u true in Z N . Note that Π Z N is the U-orthogonal projection onto Z N . The background space Z N can be interpreted as a prior space that approximates the bk manifold which we hope approximates well the true state u true . As previously alluded to, u true rarely lies in M bk in realistic engineering study cases.

The remainder of this Section is a brief review of the PBDW method in the stationary case.

A. Benaceur

Unlimited-observations statement

Let us first describe an ideal situation. The unlimited-observations PBDW statement reads: find

(u * N , z * N , η * N ) ∈ U × Z N × U such that (u * N , z * N , η * N ) = arginf u N ∈ U z N ∈ Z N η N ∈ U η N 2 , (2.2a) subject to (u N , v) = (η N , v) + (z N , v), ∀v ∈ U, (2.2b)
(u N , φ) = (u true , φ), ∀φ ∈ U. (2.2c) A direct consequence of (2.2b) is that u * N = z * N + η * N , ( 2.3)
whereas (2.2c) implies that u * N = u true . The so-called "deduced background estimate" z * N can only represent anticipated uncertainty. Since the bk mathematical model of a physical system is often deficient, one cannot realistically assume that the state estimate u * N of u true lies completely in the bk manifold (or in the background space Z N ). Therefore, the so-called "update estimate" η * N is meant to cure the deficiency of the bk model by capturing unanticipated uncertainty. In other words, the key idea of the PBDW statement (2.2) is to search for the smallest correction to the bk manifold. The following result is proved in [START_REF] Maday | A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics[END_REF].

The Euler-Lagrange saddle-point problem associated with the PBDW statement (2.2) reads: find

(z * N , η * N ) ∈ Z N × U such that (η * N , q) + (z * N , q) = (u true , q), ∀q ∈ U, (η * N , p) = 0, ∀p ∈ Z N , (2.4)
and set u * N = z * N + η * N .

Observable space

The evaluation of the right-hand side (u true , q) in (2.4) requires the full knowledge of the true state u true which is unrealistic. In practice, one can only afford a limited number of experimental observations of the true state u true . In the present setting, the experimental observations are interpreted as the application of prescribed observation functionals obs m ∈ U for all m ∈ {1, . . . , M } such that the mth experimental observation is given by obs m (u true ) ∈ R, ∀m ∈ {1, . . . , M }. One can consider any observation functional that renders the behavior of some physical sensor.

Generally, we introduce an experimentally observable space U M := Span{q m } 1≤m≤M ⊂ U, where q m := R U ( obs m ) is the Riesz representation of obs m ∈ U , for all m ∈ {1, . . . , M }. The experimental observations of the true state satisfy u true , q m = obs m (u true ), for all m ∈ {1, . . . , M }. Hence, for all q ∈ U M such that q = M m=1 α m q m and (α m ) 1≤m≤M ∈ R M , the inner product u true , q can be deduced from the experimental observations as a linear combination of the M available observations:

u true , q = M m=1 α m u true , q m = M m=1 α m obs m (u true ).
(2.5)

Limited-observations statement

Let us now describe the PBDW statement in the case of limited observations. Henceforth, we make the crucial assumption that

Z N ∩ U ⊥ M = {0}, (2.6)
Reducing sensors by means of variational data assimilation which is meant to ensure the well-posedness of the PBDW statement with limited observations [START_REF] Benaceur | Model reduction for nonlinear problems in thermics and mechanics[END_REF]. The limited-observations PBDW statement reads: find

(u * N,M , z * N,M , η * N,M ) ∈ U × Z N × U such that (u * N,M , z * N,M , η * N,M ) = arginf u N,M ∈ U z N,M ∈ Z N η N,M ∈ U η N,M 2 , (2.7a) subject to (u N,M , v) = (η N,M , v) + (z N,M , v), ∀v ∈ U, (2.7b) (u N,M , φ) = (u true , φ), ∀φ ∈ U M . (2.7c)
As above, (2.7b) implies that the limited-observations state estimate

u * N,M satisfies u * N,M = z * N,M + η * N,M . (2.8)
One can show (e.g., by introducing the Lagrangian) that the limited-observations problem (2.7) is equivalent to the limited-observations saddle-point problem: find

(z * N,M , η * N,M ) ∈ Z N × U M such that (η * N,M , q) + (z * N,M , q) = (u true , q), ∀q ∈ U M , (2.9a) (η * N,M , p) = 0, ∀p ∈ Z N , ( 2.9b)
and define u * N,M according to (2.8). The linear system (2.9) is well posed under assumption (2.6). Remark 2.1 (Perfect background space). The choice of the background space Z N and of the observable space U M may lead to several specific configurations. In particular, the background space Z N is said to be perfect if the reduction error vanishes, i.e., u true ∈ Z N . In this case, the pair (u true , 0) ∈ Z N × U M is the unique solution to (2.9). Hence, the state estimate u * N,M also belongs to Z N and the update estimate satisfies η * N,M = 0.

Algebraic formulation

We now present the algebraic formulation of the limited-observations PBDW statement. We first introduce an N -dimensional approximation space U N of the infinite-dimensional space U as well as discrete approximation spaces Z N ⊂ U N and U M ⊂ U N of the subspaces Z N and U M , respectively. These spaces are built using finite elements [START_REF] Ern | Theory and practice of finite elements[END_REF]. We assume that the size of the mesh is small enough so that the N -dimensional space discretization delivers High-Fidelity (HF) approximations within the requested level of accuracy. To alleviate the notation, we have dropped the superscript N ; hence, the discrete FEM spaces are denoted Z N and U M instead of Z N N and U N M , but we still keep the notation U N for the FEM-discretization space. Then, we introduce a basis for the background space Z N := Span{ζ n } 1≤n≤N . The update space is spanned by the Riesz representations of the observation functionals in U N , i.e., U M := Span{q m } 1≤m≤M , where q m ∈ U N , for all m ∈ {1, . . . , M }. The highfidelity (HF) discretization of the saddle-point problem (2.9) is:

Find (z * N,M , η * N,M ) ∈ Z N × U M such that (η * N,M , q) + (z * N,M , q) = (u true , q), ∀q ∈ U M , (2.10a) (η * N,M , p) = 0, ∀p ∈ Z N . (2.10b)
The solution of (2.10) is then searched under the form

z * N,M = N n=1 z n ζ n , and η * N,M = M m=1 η m q m , (2.11)
and we introduce the component vectors

z * N,M := (z n ) 1≤n≤N ∈ R N and η * N,M := (η m ) 1≤m≤M ∈ R M .
We also introduce the basis matrices Z N ∈ R N ×N and U M ∈ R N ×M whose column vectors are the A. Benaceur components of the functions {ζ n } 1≤n≤N and {q m } 1≤m≤M respectively in the basis of U N . In algebraic form, the FEM-discretized saddle-point problem (2.10) reads: find

(z * N,M , η * N,M ) ∈ R N × R M such that A B B T 0 η * N,M z * N,M = obs M 0 , ( 2.12)
with the matrices

A = (q m , q m ) 1≤m,m ≤M ∈ R M ×M , B = (ζ n , q m ) 1≤m≤M,1≤n≤N ∈ R M ×N , (2.13)
and the vector of observations obs M = obs m (u true ) 1≤m≤M ∈ R M . In practice, the matrices A and B are computed using the algebraic formulas A = U M MU M , and B = Z N MU M , where M is the Gram matrix of the inner product in U. Thus, solving (2.12) allows for a straightforward reconstruction of the components of the state estimate in the basis of U N as follows:

u * N,M = Z N z * N,M + U M η * N,M .
(2.14)

Time-dependent PBDW

Consider a finite time interval I = [0, T ], with T > 0. To discretize in time, we consider an integer

K ≥ 1, we define 0 = t 0 < • • • < t K = T as (K + 1
) distinct time nodes over I, and we set

K tr = {1, . . . , K}, K tr = {0} ∪ K tr and I tr = {t k } k∈K tr .
This section aims at deriving a state estimate for a time-dependent solution.

Remark 3.1 (Initial condition). In the present setting, we choose not to solve the PBDW statement for the initial time node k = 0. It is straightforward to consider a setting where the initial time node is also included.

Unlimited-observations statement

In this ideal setting, we assume that u true ∈ C 0 (I; U). Here, the observations are considered unlimited in both space and time. The time-dependent unlimited-observations PBDW statement reads: for each

k ∈ K tr , find (u k, * N , z k, * N , η k, * N ) ∈ U × Z N × U such that (u k, * N , z k, * N ,η k, * N ) = arginf u N ∈U z N ∈Z N η N ∈U η N 2 , (3.1a) subject to (u N , v) = (η N , v) + (z N , v), ∀v ∈ U, (3.1b) (u N , φ) = (u k,true , φ), ∀φ ∈ U. (3.1c)
where u k,true := u true (t k , •).

Lemma 3.2 (Solution).

For each k ∈ K tr , the solution to (3.1) is given by

u k, * N = u k,true , z k, * N = Π Z N (u k,true ), η k, * N = Π Z ⊥ N (u k,true ). (3.2) Proof. From (3.1c), we infer that u k, * N = u k,true . Next, we deduce from (3.1b) that u k,true = z k, * N + η k, * N . Since (3.1a) is a minimization of η k, * N , it follows that z k, * N = Π Z N (u k,true ). Thus, η k, * N = Π Z ⊥ N (u k,true ).

Reducing sensors by means of variational data assimilation

The Euler-Lagrange saddle-point problem associated with the time-dependent PBDW statement (3.1) reads:

for each k ∈ K tr , find (z k, * N , η k, * N ) ∈ Z N × U such that (η k, * N , q) + (z k, * N , q) = (u k,true , q), ∀q ∈ U, (3.3a) (η k, * N , p) = 0, ∀p ∈ Z N . (3.3b)
The unlimited-observations state estimate is then

u k, * N = z k, * N + η k, * N , ∀k ∈ K tr . (3.4)
The saddle-point problem (3.3) is purely geometric and does not include any explicit reference to the bk model. The unique link to the bk model is through the background space Z N . Therefore, the PBDW approach is applicable to a wide class of engineering problems. Moreover, the non-intrusivity of (3.3) simplifies its implementation. In practice, non-intrusivity means that computational procedures do not need to access the bk-model in order to perform their calculations.

Limited-observations statement

We now weaken the regularity assumption on the true state and only assume that u true ∈ L 1 (I; U). We introduce the time-integration intervals

I k = [t k -δt k , t k + δt k ], ∀k ∈ K tr , ( 3.5)
where δt k > 0 is a parameter related to the precision of the sensor (ideally,

δt k < min(t k+1 -t k , t k -t k-1 )
with obvious adaptation if k=K). Then, for any function v ∈ L 1 (I; U), we define the time-averaged snapshots

v k (x) := 1 |I k | I k v(t, x) dt ∈ U, ∀k ∈ K tr . (3.6)
The evaluation of the right-hand side (u k,true , q) in (3.3) requires the full knowledge of the true state u k,true , which is unrealistic. In practice, one can only afford a limited number of experimental observations of the true state u k,true . Therefore, we consider observation functionals that render the behavior of given sensors. We use the same observation functionals as in the time-independent setting [START_REF] Maday | A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics[END_REF], but we let them act on the time-averaged snapshots of the true solution, i.e., we consider k,obs m

(u true ) := obs m (u k,true ), ∀m ∈ {1, . . . , M }, ∀k ∈ K tr . (3.7)
For instance, if the sensors act through local uniform time integration, we have k,obs m

(u true ) = 1 |R m | Rm u k,true (x) dx = 1 |R m | 1 |I k | Rm I k u true (t, x) dxdt, ( 3.8)
whereas if the sensors act through integration against a Gaussian, we have

k,obs m (u true ) = 1 2πr 2 m Rm u k,true (x) exp -(x -x c m ) 2 2r 2 m dxdt, = 1 |I k | 1 2πr 2 m I k Rm u true (x) exp -(x -x c m ) 2 2r 2 m dxdt.
(3.9) Generally, we introduce the time-independent observable space U M ⊂ U such that

U M = Span{q 1 , . . . , q M }.
(3.10)

Remark 3.3 (Sensors)

. The most convenient configuration to collect observations in industrial contexts is to measure the quantities at user-defined space-time locations. In actual practice, sensors do not take pointwise measures but localized ones. A sensor collects the data that is enclosed in a small A. Benaceur area centered at the sensor location. Hence, equation (3.8) means that the sensor returns a measurement that is equal to the space-time averaged quantity we are collecting. The setup defined by equation (3.9) means that the sensor returns a Gaussian-like measurement over a centered space-time domain. Moreover, the observation functionals are Riesz representors of any type of physical data available for the user, and not necessarily pointwise or pointwise-like (integrations over small patches) observations.

Remark 3.4 (Functionals). The functionals obs m in (3.7) are fixed over time. If they change over time, the space U M in (3.10) will change at every time step. Another possibility is to have a larger space U M with optional information in specific directions (corresponding to physical locations of the sensors) at each time step as soon as stability is granted. Yet, such an approach can suffer from continuity issues due to the change of directions with available measurements between time steps.

The observation functionals in U are then defined as k,obs m (u true ) = (u k,true , q m ), ∀m ∈ {1, . . . , M }, ∀k ∈ K tr .

(3.11)

Note that, for fixed sensor locations, the computational effort to compute the Riesz representations of the observation functionals is time-independent and is incurred only once so that the experimental observations of the true state satisfy k,obs m

(u true ) = u k,true , q m = 1 |I k | I k obs m (u true (t, •))dt, ∀m ∈ {1, . . . , M }, ∀k ∈ K tr .
(3.12)

Hence, for all q ∈ U M such that,

q = M m=1 α m q m , (3.13)
the inner product u k,true , q is deduced from the experimental observations as follows:

u k,true , q = 1 |I k | I k M m=1 α m u true (t, •), q m dt = 1 |I k | M m=1 α m I k obs m (u true (t, •))dt. (3.14)
Henceforth, we make the crucial assumption that

Z N ∩ U ⊥ M = {0}, (3.15)
which is also equivalent to

β N,M := inf w∈Z N sup v∈U M (w, v) w v ∈ (0, 1], (3.16)
where β N,M is the so-called stability constant (the reader can refer to [START_REF] Benaceur | Model reduction for nonlinear problems in thermics and mechanics[END_REF] for a proof). Assumption (3.15) can be viewed as a requirement to have enough sensors (note that Z N ∩ U ⊥ = {0}). Under this assumption, the limited-observations PBDW statement reads: for each k ∈ K tr , find

(u k, * N,M , z k, * N,M , η k, * N,M ) ∈ U × Z N × U such that (u k, * N,M , z k, * N,M , η k, * N,M ) = arginf u N,M ∈U z N,M ∈Z N η N,M ∈U η N,M 2 , (3.17a) subject to (u N,M , v) = (η N,M , v) + (z N,M , v), ∀v ∈ U, (3.17b)
(u N,M , φ) = (u k,true , φ), ∀φ ∈ U M . (3.17c)

Reducing sensors by means of variational data assimilation

The limited-observations saddle-point problem associated with (3.17) reads: for each k ∈ K tr , find

(z k, * N,M , η k, * N,M ) ∈ Z N × U M such that (η k, * N,M , q) + (z k, * N,M , q) = (u k,true , q), ∀q ∈ U M , (3.18a) (η k, * N,M , p) = 0, ∀p ∈ Z N , ( 3.18b)
and the limited-observations state estimate is

u k, * N,M = z k, * N,M + η k, * N,M , ∀k ∈ K tr . (3.19)
Remark 3.5 (Pointwise measurements). For simplicity of implementation, assuming that u true ∈ C 0 (I; U), one may consider pointwise measurements in time, i.e.,

u k,true , q m = obs m (u true (t k , •)), ∀m ∈ {1, . . . , M }, ∀k ∈ K tr . (3.20)
The assumption (3.20) is typically reasonable for a sensor of small precision δt k .

In algebraic form, the limited-observations PBDW statement reads: for each k ∈ K tr , find

(z k, * N,M , η k, * N,M ) ∈ R N × R M such that A B B T 0 η k, * N,M z k, * N,M = k,obs M 0 , ( 3.21)
with the matrices

A = (q m , q m ) 1≤m,m ≤M ∈ R M ×M , B = (ζ n , q m ) 1≤m≤M,1≤n≤N ∈ R M ×N , ( 3.22)
and the vector of observations Remark 3.7 (Choice of spaces). The subspaces Z N and U M must be chosen carefully. In fact, we want a small angle between the spaces Z N and U M in order to increase the stability constant β N,M , but we need some overlap between the spaces Z ⊥ N and U M to improve the approximation capacity of Z ⊥ N ∩ U M . We refer the reader to [START_REF] Maday | A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics[END_REF] for a detailed a priori error analysis and to [START_REF] Benaceur | Model reduction for nonlinear problems in thermics and mechanics[END_REF] for an overview on the influence of the spaces Z N and U M on the accuracy of the results. We highlight that using multiple time-dependent bases did not appear to be beneficial in our numerical experiments. In particular, this choice leads to higer storage requirements and the need for higher time sampling.

k,obs M = obs m (u k,true ) 1≤m≤M ∈ R M . ( 3
A. Benaceur

Offline stage

In this section, we discuss the offline stage. Our main goal is to address the construction of the background space Z N .

Background space construction via POD-greedy

Suppose that we have computed a set of high-fidelity (HF) trajectories

S = (S k ) k∈K tr = (u k (µ)) µ∈P tr k∈K tr , ( 4.1)
where u k (µ) := u(µ)(t k , •), for all k ∈ K tr . If we were to consider the PBDW statement (3.17) for each k ∈ K tr as an independent steady PBDW statement, we would be working with the time-dependent background spaces

Z k N k = POD(S k , pod ), ∀k ∈ K tr , (4.
2) where the procedure POD is introduced in [START_REF] Kunisch | Galerkin proper orthogonal decomposition methods for parabolic problems[END_REF]. However, this strategy is not convenient since the sizes N k of the background spaces Z k N k would depend on k. Since the observable space U M is fixed, the same non-homogeneity between time nodes would also arise in the stability constant β N k ,M . Thus, we propose to apply a POD-greedy algorithm [START_REF] Haasdonk | Convergence rates of the POD-greedy method[END_REF][START_REF] Haasdonk | Reduced basis method for finite volume approximations of parametrized linear evolution equations[END_REF] in order to build a time-independent background space Z N that will be used for all k ∈ K tr . The advantage is that the PBDW matrices A and B and the stability constant β N,M remain unchanged regardless of the discrete time node. The offline stage using the POD-greedy algorithm is summarized in Algorithm 1.

Algorithm 1 Offline stage via POD-greedy

Input : S and pod .

Q init : an initial set of Riesz representations for the observations.

Background space construction via state estimation

We now devise a new algorithm in the context of time-dependent PBDW to perform the offline stage.

Main idea

Here, the construction of the background space Z N , the choice of the observation space U M and the PBDW matrices are modified. The key idea of the new procedure is to precompute the PBDW state estimates of the parameters in the training set P tr during the offline stage. The background space is then deduced from these PBDW state estimates. The benefit is that the newly created background space incorporates data-based knowledge. The modified offline stage of the PBDW for time-dependent problems is described in Algorithm 2. Within the modified offline stage, we use the so-called "Greedy stability maximization" (S-Greedy) algorithm (considered in [START_REF] Maday | A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics[END_REF]) in line 6 in order to identify the least stable mode and then take the best measurement. The algorithm uses an input space Z N that results from a POD-greedy procedure so that Z 1 contains the dominant mode, and so forth. The S-greedy algorithm selects the observations progressively. Thus, the enrichment of the observable

Reducing sensors by means of variational data assimilation

space U M stops once the minimum stability β min is reached. The procedure S-greedy is described in Algorithm 3 below. Altogether, the modified offline stage in the proposed algorithm offers four major Algorithm 2 Modified offline stage of the time-dependent PBDW Input : P tr , K tr , S, pod , init pod and β min . Q init : an initial set of Riesz representations of observations. (1) Improved background space: Since the background space Z N is built using both the bk model and the observations, it is expected to have better approximation capacities of the true state.

(2) Reduced number of online observations: In line 6 of Algorithm 2, we select each new data point so as to maximize the stability constant β N,M . Thus, the observations that will be used during the online stage are mainly needed only for stability and not for accuracy.

(3) Reduced dimension of the online PBDW statement: Since the number of observations is significantly reduced, the modified PBDW matrices are of smaller size compared to the matrices of the standard PBDW. Thus, using the modified offline algorithm, the online PBDW formulation is solved faster.

(4) Reduced storage cost: Owing to the reduced number of online measurements, the dimensions of the observable space U M and of the matrices A and B are smaller, whence the storage gain.

Regarding computational efficiency, the modified procedure consists of more steps than in the standard PBDW. However, all the additional steps of the algorithm are performed offline. As for all reducedorder modeling techniques, the goal of the algorithm is to further improve the online efficiency. Hence, the computational savings brought by the new PBDW formulation come, in our opinion, at a reasonable offline price. Indeed, the resolution of the (online) standard PBDW statement for each parameter µ ∈ P tr has a reduced computational cost. The only relevant additional computational cost incurred offline is related to the second POD-greedy algorithm (cf. line 5 of Algorithm 2). We believe this computational effort remains acceptable.

Remark 4.1 (Least stable mode). Line 6 of Algorithm 3 may return several infima. Among these infima, we select a function whose norm in U is maximal.

Remark 4.2 (Steady setting)

. In a time-dependent framework, the computational savings induced by the modified offline stage are substantial, in particular because of the influence of the time steps. However, Algorithm 3 can be applied in the steady setting as well. Compute the least stable mode and the associated supremizer

w inf ∈ arginf w∈Z N sup v∈U m-1 (w, v) w v
, and v sup = Π U m-1 (w inf ).

7:

Identify the least well-approximated vector q m = argsup

q∈Q init |(q, w inf -v sup )|.

8:

Set U m := Span{U m-1 , q m }.

9:

Compute the stability constant β N,m .

10:

m = m + 1. 11: end while 12: M := m. Output : U M .

Practical aspects

As the locations of the sensors are fixed in many industrial applications, the offline optimized strategy does not always assume the freedom to place the sensors. In fact, the sensor placement is an option in the procedure to maximize accuracy, but the overall strategy proves efficient even for fixed sensor locations as illustrated in the applications of Section 5. To be more precise, the initial choice of the Riesz representations of observations Q init considered as an input in Algorithm 2 can be left to the user depending on their industrial constraints, and must not necessarily be performed using the S-Greedy algorithm. The same reasoning can be applied when updating the observable space in line 6 of Algorithm 2. However, we emphasize that the final set of observations must be a subset of the initial set Q init .

Most practical applications in the literature are cases for which measurement values must come from experiments that were performed offline during a one-shot data collection step, meaning implicitly that the user is allowed to work with the same number of sensors throughout the entire experiment. On the contrary, the optimized offline procedure also handles practical cases for which the sensor placement may be reduced within the offline stage. This context is related to real case scenarii in which a first data collection campaign has been performed at a first occasion with insufficient results. When the experimental data collection needs to be redeployed for further investigation of a physical system, the optimized offline stage diminishes substantially its cost by taking into account the previous data, complemented by a less expensive renewed sensor placement. This can typically be the case in industrial research and development engineering units, such as that which hosted the author of the paper during the elaboration of this work. Yet, the optimized offline procedure can still be used for the usual one-shot data collection framework. In this case, the benefit of the strategy is to further diminish the size of the online system, thereby improving the storage and resolution performances of the online stage.

Reducing sensors by means of variational data assimilation

Numerical results

In this section, we illustrate the above developments on test cases related to the heat equation. The goal is to illustrate the computational performance of our algorithms. In all our test cases, we consider a twodimensional setting based on the plate illustrated in the left panel of Figure 5.

1 with Ω = (-2, 2) 2 ⊂ R 2 .
We use a finite element (FE) [START_REF] Ern | Theory and practice of finite elements[END_REF] subspace U N ⊂ U = H 1 (Ω) consisting of continuous, piecewise affine functions in order to generate HF trajectories. The FE subspace U N is based on a mesh that contains N = 6561 nodes. The experimental data is generated synthetically and the observation subsets {R m } 1≤m≤M are uniformly selected over the plate as illustrated in the left panel of Figure 5.1. Regarding the implementation, the HF computations use the software FreeFem++ [START_REF] Hecht | New developments in FreeFem++[END_REF], whereas the reduced-order modeling and the PBDW-related algorithms have been developed in Python.

Physical model problem

We apply the above methodology to the following parabolic PDE: For many values of the parameter µ ∈ P, find u(µ) :

I × Ω → R such that          ∂u(µ) ∂t -∇ • (D(µ)∇u(µ)) = 0, in I × Ω, u(µ)(t = 0, •) = u 0 , in Ω, Boundary conditions, on I × ∂Ω, (5.1)
where u 0 = 293.15K (20 o C). We will supplement (5.1) with two types of boundary conditions:

(1) Linear heat equation: We apply a homogeneous Neumann boundary condition on ∂Ω 0 and a non-homogeneous Neumann boundary condition on ∂Ω n , i.e.,

       -D(µ) ∂u(µ) ∂n = 0, on I × ∂Ω 0 , -D(µ) ∂u(µ) ∂n = φ e , on I × ∂Ω n , (5.2)
with

φ e = 3K•m•s -1 , ∂Ω 0 = (-2, 2) × {2} ∪ {2} × (-2, 2), and ∂Ω n = (-2, 2) × {-2} ∪ {-2} × (-2, 2
). Thus, the resulting problem (5.1)-(5.2) is linear. Note that ∂Ω 0 consists of the upper and right sides of the plate and ∂Ω n consists of its lower and left sides, so that ∂Ω = ∂Ω 0 ∪∂Ω n .

A. Benaceur

(2) Nonlinear heat equation: We apply Stefan-Boltzmann boundary conditions on ∂Ω, i.e., -D(µ) ∂u ∂n = σε(u 4 -u 4 r ), on I × ∂Ω, (

where u r = 303.15K (30 o C) is an enclosure temperature, σ = 5.67 × 10 -8 W.m -2 .K -4 is the Stefan-Boltzmann constant and ε = 3.10 -3 is the emissivity. The Stefan-Boltzmann boundary condition is nonlinear and so is the resulting problem (5.1)-(

In what follows, the background spaces Z N will be generated by solving either the linear PDE (5.1)-(5.2) or the nonlinear PDE (5.1)-( 5.3) with a uniform diffusivity function D(µ) such that for all x ∈ Ω,

D(µ)(x) = D uni (µ)(x) := µ1 Ω (x).

Synthetic data generation

We synthesize the data by first synthesizing a true solution and then applying to it the linear functionals by means of their Riesz representations in the observable space U M . In order to synthesize the true solution, we consider a "true model" based on the bi-material plate (cf. right panel of Figure 5.1) where we choose a fixed internal diffusivity D int = 1 and define, for each µ ∈ P, the diffusivity function

D(µ) as D(µ)(x) = D syn (µ)(x) := µD int 1 Ωext (x) + D int 1 Ω int (x), for all x ∈ Ω, where Ω int = (-1, 1) 2 and Ω ext = (-2, 2) 2 \ (-1, 1) 2 , so that Ω = Ω int ∪ Ω ext and Ω int ∩ Ω ext = ∅.
The synthetic true solutions are then defined as the solutions of (5.1) for all µ ∈ P, with either the linear boundary condition (5.2) or the nonlinear boundary condition (5.3).

Test configurations

In order to investigate the PBDW formulation, we perform test cases on two distinct configurations:

(1) Perfect model: The bk model is said to be perfect when bk mod (u true (ω)) = 0, for every ω ∈ Θ (see (2.1)) (we recall that ω represents the unanticipated uncertainty). In this situation, u true (ω) ∈ M bk for all ω ∈ Θ. Although the model is perfect, some discrepancies between the HF solutions and the measurements might arise from model-order reduction since M bk = Z N (cf. Figure ??). Note that this scenario seldom occurs in engineering situations. This test configuration is meant to assess the accuracy of the PBDW formulation when the observable space U M scarcely has additional information compared to Z N .

(2) Imperfect model: The bk model is said to be imperfect when the modeling error does not vanish. In this situation, there exists at least one (and in general many) ω ∈ Θ such that bk mod (u true (ω)) = 0, i.e., u true (ω) / ∈ M bk . Consider for instance the plates in Figure 5.1. If the true solution is generated synthetically using the bi-material plate, an example of an imperfect bk model can be the one for which we solve the same PDE that has generated the true states without accounting for the difference in diffusivity between the subdomains of the plate.

Background space construction via POD-greedy

In this section, four test cases are considered to study the PBDW approach.

• Test case (a): Linear perfect.

• Test case (b): Linear imperfect.

• Test case (c): Nonlinear perfect.

• Test case (d): Nonlinear imperfect. we show the HF temperature profiles for the model problem (5.1)-(5.2) over the homogeneous plate at the end of the simulation, i.e., for t K = 10s and for two parameter values. We recall that these solutions will be used as true solutions for the perfect linear case. As the simulation time evolves, the energy related to the flux φ e propagates through the plate which is progressively heated. Moreover, the overall temperature is higher for smaller values of the parameter µ than for larger values. As physically expected, the thermal diffusion over the plate is stronger for larger values of µ than for smaller values.

Test case (a): Linear perfect model. We consider the case of a perfect bk model for which the diffusivity is uniform over the entire domain Ω. Thus, the true solutions correspond to the HF computations of the bk model. The resulting trajectories are reduced using the POD-greedy algorithm. For instance, for a tolerance value pod = 10 -2 , the background space Z N is composed of N = 5 modes. Regarding observations, the initial set Q init is obtained using M = Card(Q init ) = 121 sensors that are uniformly placed over the plate (see Figure 5.1). Using both the background space Z N and the observable space U M (Q init ), we build the offline matrices A and B. During the online stage, we estimate the state u * N,M for every parameter µ in the training set P tr . Using the weighted H 1 -norm, the state estimation relative H 1 -error e k (µ) defined as

e k (µ) := u k,true (µ) -u k, * N,M (µ) H 1 (Ω) u k,true (µ) H 1 (Ω)
, ∀µ ∈ P, (5.4) is displayed in Figure 5.3 as a function of the value of the parameter µ for several values of pod . In this first configuration, one can notice that the error decreases for smaller tolerances pod , i.e., with the dimension N of the background space Z N . However, the right panel of Figure 5.3 shows a starting increase in the relative H 1 -error e k (µ) for pod = 10 -6 and an oscillatory behavior of the relative H 1 -error e k (µ). Although counter-intuitive in the reduced-basis context, this phenomenon is due to the deterioration of the stability constant β N,M . This observation confirms the claims made in Remark 3.7. stability constant β N,M is degraded when increasing the dimension N of the background space Z N . Moreover, the enrichment of Z N does not add relevant modes anymore (in terms of associated singular values). For the sake of comparison, we enrich the observable space U M such that M = 676 and plot the relative H 1 -errors e k (µ) for the same values of pod in Figure 5.6. Our interpretation is confirmed since the stability issues do not arise anymore for pod = 10 -4 . Owing to the increase of M , the stability decrease with respect to N is somewhat compensated. Finally, the bottom-right panel of Figure 5.6 shows the beginning of an error increase. Using the same reasoning as above, we conclude that more observations are needed for pod = 10 -6 . Finally, we visualize the stability constant β N,M as a function of the number of observations M in Figure 5.7. The left panel of the figure shows a single curve for clarity, whereas the right panel includes curves for several values of the tolerance pod (note that the two panels do not use the same rule). As expected, for a constant value of N , the more the observations, the better the stability. For a number of observations M = 3000, the PBDW formulation is perfectly stable (or close to) for all the considered values of pod .

Nonlinear case

Here, we consider the PDE (5.1)-(5.3) with u r = 303.15K, σ = 5.67×10 -8 W.m -2 .K -4 and ε = 3.10 -3 . Except for the parameter interval P = [0.1, 2], the set P tr = {0.1i, 1 ≤ i ≤ 20} and the time step ∆t k = 0.1, all the other numerical data remain the same as for the linear test case from the previous section. the training set P tr . In Figure 5.9, we display the state estimation relative H 1 -error e k (µ) defined in (5.4) as a function of the value of the parameter µ for several values of pod . In contrast to the linear case, the error always decreases for smaller tolerances pod , i.e., with the dimension N of the background space Z N . However, we expect that, for some very small tolerance value (e.g. pod such that N > M ), the stability issues mentioned above would arise again.

A. Benaceur

Test case (d): Nonlinear imperfect model. This test case investigates a nonlinear imperfect bk model for which the HF bk solutions and the true solutions are respectively displayed in the left and right panels of Figure 5.8. The temperature profile for the true solution over the bi-material plate at the end of the simulation, i.e., at t K = 10s clearly shows a different behavior at the boundaries of the inner material. Regarding the PBDW state estimation, Figure 5.10 shows the relative H 1 -error e k (µ) defined in (5.4) using M = 121 observations to build the observable space U M . For pod = 10 -4 , Z N is spanned by N = 7 vectors. Notice that the error vanishes for µ = 0.25 since this configuration is equivalent to a perfect bk model. We notice that the relative H 1 -error e k (µ) increases because the stability constant decreases. Figure 5.11 visualizes the relative H 1 -error e k (µ) for a higher number of observations M = 676. We observe that augmenting the dimension of the observable space U M cures the stability issues. Also, the errors are lower owing to the higher number of observations. Finally, A. Benaceur estimates in order to build the modified background space (cf. line 5 of Algorithm 2). For a tolerance pod = 5.10 -2 , the POD-greedy algorithm selects four modes. Then, we build the observable space U M using M = 121 uniformly distributed sensors (the optimal choice can be made using the S-Greedy algorithm, see Algorithm 3).

Conclusion and perspectives

We have presented a time-dependent extension of the PBDW approach with a modified offline stage that suits both the steady and time-dependent cases. As regards the time-dependent setting, numerical tests on both linear and nonlinear cases assess the efficiency of the method for well chosen dimensions of the bk space Z N and of the observable space U M . The test cases show that augmenting the dimension N of Z N is counter-productive starting from a certain rank due of the deterioration of the stability constant β N,M . In such cases, increasing the dimension of the observable space U M is an alternative that restores a good stability of the problem. However, the measurements are expensive to obtain in engineering scenarios. In this context, a modified offline stage is introduced so as to reduce the number of final observations that are required within the online stage. The numerical performances obtained produce accuracy levels that are comparable to the standard method. A promising application is the assessment of the proposed methodology for other types of time-dependent problems as well as for three-dimensional industrial cases. Another interesting research direction is the inclusion of noise in the time-dependent framework.

. 23 )Remark 3 . 6 (

 2336 Offline/online procedure: Since several realizations u k,true (ω) of the true state are considered, an offline/online procedure can be employed to solve(3.21). During the offline stage, one precomputes the RB functions (ζ n ) 1≤n≤N and the Riesz representers (q m ) 1≤m≤M leading to the matrices A ∈ R M ×M and B ∈ R N ×M once and for all. Then, during the online stage, for each new set of observations corresponding to a new realization of the true state u k,true (ω), all that remains to be performed is to form the vector of observations k,obs M (ω) and to retrieve the deduced background estimate z k, * N,M (ω) and the update estimate η k, * N,M (ω) by solving the (N + M )-dimensional linear problem(3.21). The PBDW state estimate u k, * N,M (ω) is then computed using(3.19). PBDW matrices). Notice that the PBDW matrices A and B are time-independent; only the right-hand side in (3.21) depends on k.

1 : 2 : 3 :

 123 Compute Z N := POD-greedy(S, pod ). Set U M := Span{Q init }. Compute the matrices A and B using Z N and U M . Output : Z N , U M , A and B.

1 : 3 : 4 :

 134 Compute Z init N init := POD-greedy(S, init pod ). 2: Set U init M init := Span{Q init }. Compute the matrices A init and B init using Z init N init and U init M init . Estimate the state u k, * (µ) for all (µ, k) ∈ P tr × K tr . 5: Compute Z N := POD-greedy({u k, * (µ)} µ∈P tr ,k∈K tr , pod ). 6: Compute U M := S-Greedy(P tr , K tr , N , {Z n } N n=1 , β min , Q init ). 7: Compute the matrices A and B using Z N and U M . Output : Z N , U M , A and B. advantages:

Algorithm 3 S 1 : 2 : 3 : 4 : 5 :

 312345 -Greedy: Stability-maximization algorithm Input : N , Z N and β min ∈ (0, 1].Q init : an initial set of Riesz representations of the observations. Choose a random q 1 ∈ Q init . Set U 1 := Span{q 1 }. Compute the stability constant β 1,1 using Z 1 and U 1 . Set m := 2. while β N,m-1 < β min or m < M do 6:

Figure 5 . 1 .

 51 Figure 5.1. Computational domain and mesh with N = 6561. The little black squares are observation subsets {R m } 121 m=1 . Left: Mono-material plate. Right: Bimaterial plate.

Figure 5 . 2 .

 52 Figure 5.2. Test cases (a) and (b) : HF solutions for the bk model with Neumann boundary conditions. Left: µ = 1 in D uni (values from 17.3 o C to 118.3 o C). Right: µ = 20 in D uni (values from 30.7 o C to 43.3 o C).

Figure 5 . 3 .Figure 5 . 4 .Figure 5 . 5 .

 535455 Figure 5.3. Test case (a): Relative H 1 -error e k (µ) for some time nodes k ∈ K tr . Left: pod = 10 -2 (N = 5). Middle : pod = 10 -4 (N = 10). Right: pod = 10 -6 (N = 15).

Figure 5 . 6 .

 56 Figure 5.6. Test case (b) : Relative H 1 -error e k (µ) for some time nodes k ∈ K tr and M = 676. Top left: pod = 10 -2 (N = 5). Top right: pod = 10 -3 (N = 7). Bottom left: pod = 10 -4 (N = 10). Bottom right: pod = 10 -6 (N = 15).

Figure 5 . 7 .

 57 Figure 5.7. Test case (b) : Stability constant β N,M as a function of M . On the right panel, the values of N are respectively 3, 5, 7, 10, 13, 15 for the values of pod in decreasing order.

Figure 5 . 8 .

 58 Figure 5.8. Test cases (c) and (d) : Left: HF solution for the bk model (values from 17.80 o C to 18.25 o C). Right: Synthetic true solution using a bi-material plate (values from 17.90 o C to 18.23 o C).

Figure 5 . 9 .

 59 Figure 5.9. Test case (c) : Relative H 1 -error e k (µ) for some time nodes k ∈ K tr and M = 121. Left: pod = 10 -2 (N = 3). Middle: pod = 10 -4 (N = 7). Right: pod = 5.10 -6 (N = 11).

Figure 5 . 10 .

 510 Figure 5.10. Test case (d) : Relative H 1 -error e k (µ) for some time nodes k ∈ K tr and M = 121. Left: pod = 10 -2 (N = 3). Middle: pod = 10 -4 (N = 7). Right: pod = 10 -6 (N = 11).

Figure 5 . 11 .

 511 Figure 5.11. Test case (d) : Relative H 1 -error e k (µ) for some time nodes k ∈ K tr and M = 676. Left: pod = 10 -2 (N = 3). Middle: pod = 10 -4 (N = 7). Right: pod = 5.10 -6 (N = 11).

Figure 5 .

 5 Figure 5.12 shows the stability constant β N,M as a function of the number of observations M . The behavior is quite similar to the linear case. Hence, the nonlinear character of the problem does not influence the overall features of the PBDW statement. This observation corroborates the independence with regard to the bk model.

Figure 5 . 12 .

 512 Figure 5.12. Test case (d) : Stability constant β N,M . On the right panel, the values of N are respectively 2, 3, 5, 7, 11 for the values of pod in decreasing order.

Figure 5 .Figure 5 . 15 .

 5515 Figure 5.15. Test case (e) : Relative H 1 -error e k (µ) for the state estimate as a function of the time nodes during the online stage. The various curves correspond to the different values of µ. Left: for all µ ∈ P tr . Right: for all µ ∈ P verif .

Figure 5 . 16 .

 516 Figure 5.16. Test case (e) : Absolute H 1 -norms of the contributions z * N,M and η * N,M as a function of the time nodes. The various curves correspond to the different values of µ.

Acknowledgements

This work is partially supported by Electricité De France (EDF) and a CIFRE PhD fellowship from ANRT. The author is thankful to A. T. Patera, A. Ern and V. Ehrlacher for proof-reading and stimulating discussions.

Reducing sensors by means of variational data assimilation

Background space construction via state estimation

We now illustrate the performances of Algorithm 2 for the following linear imperfect case:

• Test case (e): We consider a simulation duration T = 4s and a time step ∆t = 0.1s. Test truths are synthesized with an internal diffusivity D int = 0.2.

As opposed to the previous section, we choose a non-parametric bk model based on an HF computation for µ = 0.5. The resulting unique trajectory is then reduced using a POD algorithm, which is equivalent to a POD-greedy for a single trajectory (cf. line 1 of Algorithm 2). For a tolerance value init pod = 10 -2 , we obtain a background space Z init N init composed of N init = 34 modes. As regards observations, the initial set