Viviane Durand-Guerrier
email: viviane.durand-guerrier@umontpellier.fr

Melih Turgut²
email: melih.turgut@ntnu.no

TWG3: Teaching and learning of linear and abstract algebra, logic, reasoning and proof

INTRODUCTION

In this group, there were 10 oral communications and 4 posters that were organised in two thematics: linear and abstract algebra, presented in the first parallel session; logic, reasoning, and proof presented in the second parallel session. The list of papers and posters is in an annexe of the document. The group comprised 19 participants from various countries. Each paper was allocated a 15-minute presentation, followed by 5-minute discussion with the audience. During the two first discussion sessions, a slot was devoted to the presentation of posters linked to the theme of the corresponding parallel session. For the discussion following each of both sessions, we split into three non-thematic subgroups around two papers or a paper and one or two posters. Previously to the conference, for each paper, two registered participants were invited to prepare a few slides to act as critical friends during the conference. These slides have been used during the discussion sessions in small groups. This was followed by a collective discussion.

During the third discussion session, we decided to split into two thematic subgroups: Linear and abstract algebra; Logic, reasoning, and proof, to discuss the papers and posters linked to the thematic and let emerge first ideas. Seven attended the subgroup on linear and abstract algebra, and 12 attended the subgroup on logic, reasoning, and proof. During the fourth session, we go on working in thematic subgroups to reflect collaboratively on the new ideas that have emerged during the conference. Each subgroup has prepared a report on its theme that has been presented in the closing session.

TRENDS AND PERSPECTIVES IN LINEAR AND ABSTRACT ALGEBRA

In this section, we provide a summary of presentations and discussions regarding linear and abstract algebra concluding with a brief synthesis of emerging topics/issues and questions, and further research directions.

The main topics in the papers and posters presented

There were four main topics in the papers and posters; Eigentheory (Piori and Lyse-Olsen & Fleischmann, Wawro & Thompson), vector spaces (Can, Aguilar & Trigueros), Gauss algorithm (My Hahn) and computational thinking (Turgut). Regarding eigentheory, we underline the following three main themes:

• Semiotic analysis of signs (including gestures) produced by students during a collective activity aimed at making sense of eigenvectors and eigenvalues.

• Students' conceptions of eigenvectors and eigenvalues ○ Representations and formal elements used in students' descriptions ○ Focus on task design for individual learning activities

• Student reasoning about eigenequations (or not) in quantum mechanics Piroi explores eigentheory teaching and learning processes in her continuing PhD research and provided the preliminary findings. The focus of the presented paper was an investigation of students' collective meaning-making processes within the lens of the theory of objectification, as a sociocultural theory. The paper described an activity that was created especially to support these processes of objectification. University engineering freshmen worked collaboratively to rethink eigentheory principles and rules while working in small groups. The usage of various semiotic resources by students, as well as how they relate to one another and how they have evolved, are then discussed. Under the same topic, Lyse-Olsen & Fleischmann examine students' understanding of eigenvectors at an early stage of their linear algebra instruction.

Students' various explanations of eigenvectors are examined in relation to the mathematical objects they choose to depict (algebraic, geometric, or abstract representations), as well as the formalism they employed. Students nevertheless demonstrated their ability to switch between many representations and descriptions and produce unique concept images, even when the modes of description that were presented to them appeared to impact their own choice of description.

Wawro & Thompson's poster focused on student reasoning in quantum mechanics regarding matrix equations as eigenequations. Wawro & Thompson particularly explored how students would be able to distinguish between eigenequations and (quantum mechanics) matrix equations, and how this connects to their justifications for eigentheory in both mathematical and quantum contexts.

Regarding the Gauss algorithm, My Hahn presented research about constructing online cloze style (fill-in-the-blank with drop-down menu) questions to help students improve their understanding of and ability to use mathematical language. Examples given from discussing solution processes for systems of linear equations were provided. The paper discussed the preliminary findings of the analysis as well as Steinbring's epistemological triangle as a potential analytical tool for (aforementioned) comprehension processes.

Regarding vector spaces, Can, Aguilar & Trigueros focused on a teaching strategy for the learning of the concept of vector space using non-standard binary operations with a diversity of sets to promote student reflection on the vector space axioms more generally. The design is based on the APOS theory using its ACE cycle as a didactic approach, and a group of engineering students solved the provided task. Using sets and binary operations that aren't typically covered in a first course in linear algebra encouraged students to think critically about the axioms that define vector spaces.

Regarding computational thinking (CT), Turgut presented an emerging framework for integrating CT into teaching and learning linear algebra. The framework refers to three teaching principles of linear algebra, theory of instrumental genesis and CT. The paper presents a vignette in terms of GeoGebra's specific tools, functions and commands to teach the system of linear equations within the lens of CT.

Discussions, emerging topics and questions

The variety of theoretical and analytical frameworks available to support and guide the various research goals and the flexible way of using these frameworks were one of the main points that were discussed in the group. Presenters referred to different theoretical/conceptual lenses, such as APOS theory, the three teaching principles of linear algebra, theory of objectification, multimodal paradigm and semiotic bundle, knowledge in pieces, symbolic forms, modes of representation/thinking and (levels of) formalism. The broad spectrum of lenses not only raised discussion about the theoretical/conceptual frameworks in themselves, but also brought us to elaborate on how they should be used, the possibility of selecting the topic as a point of departure, and the role of emerging frameworks about eigentheory in research.

The second point that the group discussed was the role of task design in our research. For example, where to start and which has a priority; (i) designing to overcome students' epistemological issues, (ii) designing innovative teaching-learning environments, or (iii) both in a synchronised way. The group discussed the function of guiding/orienting frameworks in task design too, like the role of Realistic Mathematics Education theory. The group also discussed the emerging role of CT [START_REF] Wing | Computational thinking[END_REF] as a (possible) mediator context to create mathematical meanings of linear algebra topics (and also in other STEM fields), like exploring the system of linear equations and Gram-Schmidt diagonalisation etc.

A third point was about aspects and nature of formalism and representations in linear algebra. The role of representations, shifting between them (i.e. semiotic registers, in the sense of Duval) and how this could inform researchers to design tasks were discussed. The instructor's role was at the heart of discussion on some occasions, with a particular role to make a shared discussion at the end of each teaching episode.

Future directions for next INDRUM conferences

The following points have emerged after our thematic group discussions regarding linear and abstract algebra. The first point was about the role and purpose of theoretical/conceptual frameworks in our work. The group underlined that some frameworks provide orientation (for example in task design) and a better understanding of the observed phenomenon, but such frameworks come with some constraints. The second point was about the integration with computer science (e.g., CT, computer graphics etc.), and (possibility of) digital assessment of linear and abstract algebra in particular. The third point was the contribution of linear and abstract algebra to the professional development pre-service and in-service mathematics teachers.

TRENDS AND PERSPECTIVES IN LOGIC, REASONING AND PROOF

In this section, we first present the main issues that emerge from the papers presentations; then we present the main elements that emerged from the discussion, namely the new trends and the perspective.

The main issues in the papers and posters presented

The main issues in the papers presented in this thematic were: Multi proofs analysis as a means to foster conceptualisation (Viviane Durand-Guerrier); Comparison of various proof assistants at the interface between mathematics and computer sciences (Evmorfia Iro Bartzia, Antoine Meyer & Julien Narboux); the support of CAS (Kinga Szücs); the use of cloze test and multichoice questionnaire (My Hahn); students reasoning on visual words problems (Francesco Beccuti); the generic power of proofs in number theory (Véronique Battie); refutation beyond counter-examples (Alon Pinto and Jason Cooper); linguistic issues (Dimitri Lipper, Thomas Karavi & Angeliki Mali). During the discussion sessions, two main issues have emerged: mathematical and epistemological needs and overview of analysis criteria.

Mathematical and epistemological needs

There is a necessity to go beyond the illusion of transparency of mathematical knowledge [START_REF] Artigue | Épistémologie et didactique [Epistemology and didactic[END_REF] for strengthening the a priori analysis in a didactic perspective. The question is "how to do this?". Relying on the research experience of some of the participants, we list the following practices: write down our own proofs, individually and then sharing with our research team and beyond, including researchers and practitioners who specialise in the mathematical domain at stake; making a review of existing proofs in the literature as well in mathematics education as in domain-specific mathematics; analysing historical proofs; considering philosophical perspective; analysing experts' practises; considering implicit and explicit norms among various communities. The discussion showed that the introduction of proof assistants and computer scientist methods raised new epistemological questions or calls for revisiting more classical ones. This appears as a challenge for research on proof and proving in mathematics education (Hanna, De Villiers & Reid, 2019).

Overview of proof analysis criteria

In the literature and in the participants' research experience and practice, there are various types of analysis criteria that are used in research on proof and proving, both for designing, and for a priori and a posteriori analysis, depending on the goal and the research questions. Nevertheless, we have tried to provide a non-exhaustive list as the first contribution to a collective state-of-the-art that the group considered worthwhile.

When the criteria were explicitly discussed in a paper presented in the TWG, we mention the name of the author(s) in brackets.

The first group of criteria refer to language, level of formalisation and semiotic register. A second group concerns norms and expectations, depending on the audience. A third group addresses the epistemological and pragmatic issue of the type of proof expected or provided, formal proof, operational proof, experimental proof (Dimitri Lipper, Thomas Karavi & Angeliki Mali) and Proof schemata: procedural versus conceptual, meaning versus ritual etc.

The fourth group of criteria concerns the proof structure analysis: organising and operative dimensions (Veronique Battie), data, hypothesis and introduction of objects (Viviane Durand-Guerrier), direct or indirect proof (Alon Pinto & Jason Cooper, visualisation (Francisco Beccuti). The fifth group of criteria refers to the precise analysis of a particular proof: steps in the proving process, logical validity, modes of inference, clarity, modularities, encapsulation, etc.

Perspectives for next INDRUM conferences

The first one is the relevance of revisiting epistemological questions in light of the increasing use of proof assistants in both research in mathematics and in mathematics education. The second one aims to deepen the logical dimension of analysis for the teaching and learning of proof and proving, which have already been shown to be relevant, but remain underrepresented in many research on proof and proving. The third one concerns the necessity of going on designing and implementing activities aiming at improving proof and proving skills at university, taking into consideration the specificity of the condition and constraints in university mathematics education. Another issue consists in exploring more systematically the possible contribution of proof and proving to address university mathematics students' difficulties, depending on the mathematical domain.

Finally, we consider that introducing specific work on proof and proving in

University teacher training would be valuable, to initiate a change in the way proof is taught in general at university: moving from proof made in front of students, to students' proof elaboration and analysis.


Asma Zbitou

MD Emmanuel Desandes
email: emmanuel.desandes@univ-lorraine.fr

Sandra Guissou

Coralie Mallebranche

Brigitte Lacour

Thyroid cancers in children and adolescents in France: Incidence, survival and clinical management over the 2000-2018 period AUTHORS

Introduction: Thyroid cancer is the first cause of endocrine malignancy among children. Over the past decades, an increase in the incidence rates (IR) has been observed around the world.

Our study aimed to describe epidemiology, therapeutic management and survival rates of children and adolescents with thyroid cancer in France.

Methods:

A population-based study was conducted between 2000 and 2018 in children and adolescents less than 17 years with a diagnostic of thyroid cancer using the French National Registry of Childhood Cancers (RNCE). Age standardized incidence rates (ASR) and survival analysis were performed.

Results: A total of 774 thyroid cancers were included : 579 papillary (PTC), 83 follicular (FTC), and 111 medullary carcinomas (MTC). PTC are more frequent in females and in adolescents whereas MTC mainly concerned children, mostly with a familial predisposition.

Almost all patients underwent thyroidectomy, completed for most patients with PTC and FTC by radioiodine therapy. Between 2000-2018, thyroid cancers IR in children fluctuated between 1.3 and 3.2 per million, without any significant trend. The median follow-up time was 11.3 years in children, and 5.7 years in adolescents. The 5year-OS was greater than 98.5%.

Conclusions:

The RNCE plays a critical role in surveillance, and tells us where we are in the efforts to better understand thyroid cancers in pediatric and adolescent population. Considering the very favorable survival, a stratification should be proposed between cases at low risk and cases at high risk of relapse, in order to consider a strategy of therapeutic de-escalation in the most favorable cases.

INTRODUCTION

Pediatric thyroid cancer is rare and counts for 1 to 3% of pediatric cancers. However, it represents the first cause of endocrine malignancy among children. Thyroid cancers in children have a good prognosis, especially papillary carcinoma (PTC), despite their more advanced nature at diagnosis than in adults and the higher risk of recurrence [START_REF] Al-Qurayshi | A National Perspective of the Risk, Presentation, and Outcomes of Pediatric Thyroid Cancer[END_REF][START_REF] Ștefan | Differentiated Thyroid Cancer in Children in the Last 20 Years: A Regional Study in Romania[END_REF][START_REF] Wang | Prognostic analysis of recurrence in children and adolescents with differentiated thyroid cancer[END_REF][START_REF] Huang | Clinicopathological characteristics and prognosis of thyroid cancer in northwest China: A population-based retrospective study of 2490 patients[END_REF]. Differentiated thyroid carcinoma (DTC) is the most frequents (more than 90%) with an age-standardized incidence rate (ASR) of 6.7 (95%CI: [6.5-6.8]) cases per million person-years in the United States [START_REF] Bernier | Trends in pediatric thyroid cancer incidence in the United States, 1998-2013[END_REF].

Rates increase during the second decade and particularly after the age of 15 years [START_REF] Paulson | Thyroid Cancer in the Pediatric Population[END_REF][START_REF] Dermody | Pediatric thyroid cancer: An update from the SEER database 2007-2012[END_REF].

Medullary thyroid carcinoma (MTC) is more common in children aged 0 to 4 years [START_REF] Dermody | Pediatric thyroid cancer: An update from the SEER database 2007-2012[END_REF].

Globally, girls are more affected with a sex ratio reported at 4.4:1 among children and young adults aged 0-19 years and up to 6:1 among the 15-19-year age group [START_REF] Paulson | Thyroid Cancer in the Pediatric Population[END_REF][START_REF] Dermody | Pediatric thyroid cancer: An update from the SEER database 2007-2012[END_REF].

Over the past decades, an increase in the incidence of pediatric thyroid cancers has been observed around the world. Gupta et al reported an increase, between 1998 and 2012, in adolescents and young adults in more than 33 countries for women and 22 for men, among the 41 studied [START_REF] Gupta | International Trends in the Incidence of Cancer Among Adolescents and Young Adults[END_REF]. In the United States, an overall increase in the incidence rate (IR) between 1973 and 2013 was reported in people aged under 20 with a rate going from 0.5 (95%CI: [0.3-0.7]) to 1.1 (95%CI: [0.9-1.4]) per 100,000 person-years [START_REF] Qian | Pediatric Thyroid Cancer Incidence and Mortality Trends in the United States, 1973-2013[END_REF]. This increase concerned more particularly PTC and was more marked between 2006 and 2013 with an average annual percentage change in IR of 9.6% (95%CI: [5.1-14.2]) [START_REF] Qian | Pediatric Thyroid Cancer Incidence and Mortality Trends in the United States, 1973-2013[END_REF]. In France, a similar increase was observed among children aged 0-14 years between 1978 and 2001, through general departmental registers survey who have nevertheless recorded a low number of cases [START_REF]Surveillance sanitaire en France en lien avec l'accident de Tchernobyl -Bilan actualisé sur les cancers thyroïdiens et études épidémiologiques en cours en 2006[END_REF].

Between 2000 and 2008, according to data from the French National population-based Registry of Childhood Cancers, the incidence of thyroid cancers remained stable [START_REF]Evolution de l'incidence du cancer de la thyroïde en France métropolitaine -Bilan sur 25 ans[END_REF].

The causes of this increase in the occurrence of thyroid cancers remain controversial. The improvement in the recording of cases over time could be an explanation [START_REF]Surveillance sanitaire en France en lien avec l'accident de Tchernobyl -Bilan actualisé sur les cancers thyroïdiens et études épidémiologiques en cours en 2006[END_REF]. On the other hand, the improvement of imaging techniques, which are more sensitive, and the evolution in the diagnostic approach of thyroid nodules, particularly in children, making it possible to detect localized tumors earlier, could explain an overdiagnosis of thyroid cancers [START_REF] Bernier | Trends in pediatric thyroid cancer incidence in the United States, 1998-2013[END_REF][START_REF] Qian | Pediatric Thyroid Cancer Incidence and Mortality Trends in the United States, 1973-2013[END_REF][START_REF] Vaccarella | Worldwide Thyroid-Cancer Epidemic? The Increasing Impact of Overdiagnosis[END_REF][START_REF] Chen | Children and thyroid cancer: Interpreting troubling trends[END_REF].

However, the observation of the significant increase in cancers with an advanced stage and large tumors, as reported among the 10-19-year age group in the United States, could show a real increase in incidence, potentially linked to environmental exposures or to yet unknown risky lifestyles [START_REF] Bernier | Trends in pediatric thyroid cancer incidence in the United States, 1998-2013[END_REF][START_REF] Qian | Pediatric Thyroid Cancer Incidence and Mortality Trends in the United States, 1973-2013[END_REF][START_REF] Goldenberg | We cannot ignore the real component of the rise in thyroid cancer incidence[END_REF].

The increasing IR of thyroid pediatric cancers observed throughout the world requires careful epidemiological monitoring of this malignancy, in particular through the updating of national epidemiological data. Thus, our study aimed to describe epidemiology, incidence, therapeutic management, and survival rates of children and adolescents aged 0 to 17 years and diagnosed with thyroid cancer between 2000 and 2018.

MATERIALS AND METHODS

Study population

The present study used data from the French National population-based Registry of Childhood Cancers (RNCE) recording cases of solid pediatric tumors for children (0-14 years) since 2000 and extending to adolescents (15-17 years) since 2011. All incident cases of thyroid cancer (group XI.b according to the third version of the International Classification of Childhood Cancer (ICCC-3) occurring in children and adolescents (0-17 years) and diagnosed in France between 2000 and 2018 were included [START_REF] Steliarova-Foucher | International Classification of Childhood Cancer[END_REF]. Lacour et al have previously described the methodology of case registration for the RNCE [START_REF] Lacour | Incidence of childhood cancer in France: National Children Cancer Registries, 2000-2004[END_REF].

Data collection

The registration form included child and adolescent characteristics, information on patient medical and family history, clinical data at diagnosis, initial treatment, patient pathway from diagnosis to treatment, vital status and last contact date.

Using the third edition of the International Classification of Disease for Oncology (ICDO-3), histology was tabulated as: papillary carcinomas (8260/3, 8340/3-8344/3, 8350/3), follicular carcinomas (8290/3, 8330/3, 8335/3, 8337/3) and medullary carcinomas (8345/3) [START_REF] Frizt | International Classification of Diseases for Oncology[END_REF]. Stage at diagnosis were standardized according to the UICC TNM classification of malignant tumors, Eight edition (TNM8) [START_REF] Brierley | TNM classification of malignant tumours[END_REF]. Vital status was obtained at the time of analysis by active search in medical records and by matching the RNCE files and the central population register (RNIPP) sorted by date and place of birth to obtain the mention of death and the date of death.

Statistical Analysis

Age standardized IR per million children were calculated using direct standardization and world standard population aged 0-14 years as reference [START_REF] Parkin | Cancer Incidence in Five Continents Volume VIII[END_REF]. Crude incidence rate (CIR) were calculated for adolescents aged from 15 to 17 years. Methods based on Poisson distribution were used to specify the 95% confidence interval (95%CI) for ASR and CIR [START_REF] Dobson | Confidence intervals for weighted sums of Poisson parameters[END_REF]. The average annual percent change (APC) was considered to test for a time trend in ASR and CIR. APC was derived from the slope parameter estimate of a linear regression model considering the natural logarithm of the ASR as a function of the calendar year [START_REF] Ries | Cancer Incidence and Survival among Children and Adolescents: United States SEER Program 1975-1995[END_REF]. To study overall survival, the time origin was the date of diagnosis. The date for last follow-up was March 1, 2021. The end point of interest was the occurrence of death, all causes combined. The median length of follow-up was calculated using the reverse Kaplan-Meier estimator [START_REF] Shuster | Median follow-up in clinical trials[END_REF]. Follow-up (<5 and ≥5 years) according sex, age group and histological group over the 2000-2015 period was analyzed. 5year overall survival (5y-OS) was calculated using the Kaplan-Meier estimator [START_REF] Kaplan | Non-parametric estimation from incomplete observation[END_REF]. Survival curves were compared using log-rank test [START_REF] Mantel | Evaluation of survival date and two new rank order statistics arising in its consideration[END_REF][START_REF] Peto | Design and analysis of randomized clinical trials requiring prolonged observation of each patient. II. analysis and examples[END_REF].

RESULTS

Patient characteristics (

Patient medical history (Table 2)

A total of 88 (11.4%) patients had a family relative with thyroid cancer history, mostly (n=73, 83.0%) a first-degree member family (parents, brothers or sisters). Familial predisposition mainly concerned patients with MTC at diagnosis (n=81, 93.1%; p<0.01). Predispositions of patients with MTC were mostly Multiple Endocrine Neoplasia type 2 (MEN2) with 79 (74.5%) MEN2A syndromes, 20 (18.9%) MEN2B, 3 (2.8%) MEN 2 (type not specified), 3 (2.8%) RET mutation (syndrome type not specified) and 1 (0.9%) Familial Medullary Thyroid Carcinoma (FMTC). PTC predispositions were: APC (Adenomatous Polyposis Coli) gene mutation (n=3), Cowden syndrome (n=1) and RET mutation (n=1). Among patients having FTC, 1 (1.2%) had a Cowden syndrome.

A thyroiditis history was observed in 32 patients (4.1%), all with PTC. Among them, 12 (37.5%) had a Basedow disease, 11 (34.4%) a Hashimoto thyroiditis, and for 9 (28.1%) the autoimmune thyroid disease was not specified. A total of 20 (2.6%) patients with PTC had a neck or total body irradiation history, all for treating a previous cancer. Among them, 11 (55%) had irradiation before 5 years and 9 (45%) at 5 years or more. An history of cancer was observed in 37 patients (4.8%): one cancer for 34 patients (91.9%), two cancers for 2 patients (5.4%) and three cancers for one patient (2.7%). The most frequent primary cancer was leukemia ( 12patients). Among the 37 patients, just one patient had a known predisposition (APC gene mutation).

Tumor characteristics (Table 2)

Overall, 74.9% of patients had PTC, 10.7% FTC and 14.4% MTC. Patients having MTC were younger compared to patients having PTC or FTC (mean age (SD) respectively: 7.4 years (4.9), 13.6 years (3.0), 13.2 years (2.6); p<0.01). Most tumors were unilateral or median for DTC compared to MTC (PTC: 75.0% and FTC: 90.3% vs MTC: 28.8%, p<0.01) and they were more frequently unifocal (PTC: 58.0% and FTC: 81.9% vs MTC: 25.2%, p<0.01). Majority of MTC were smaller (≤10 mm) compared to DTC (MTC: 77.5% vs PTC: 16.2% and FTC: 2.4%, p<0.01). FTC and MTC were significantly mostly located compared to PTC (% of N0: 90.4%,84.7%, and 39.9%, respectively) which tended to be more regional. PTC were more frequently metastatic compared to others carcinoma (PTC: 8.5% vs FTC: 1.2% and MTC: 2.7%). For DTC, metastases were mainly located in the lung (93.9%) and more rarely in the thoracic lymph nodes (2.0%). The sites of metastases in MTC were more varied: pancreas, liver, lung, bones and thoracic lymph nodes.

Disease management (Table 2)

Therapeutic care

Most patients underwent thyroid surgery, except 2 patients having metastatic MTC. The majority of them had a total thyroidectomy. Partial surgery was more frequent in patients having FTC (7.2%; p<0.01). Two-stage surgery (thyroidectomy in two stage or second stage for neck dissection) was observed for 27.6% of patients (not specified for 10 children). Majority of one stage surgery were performed in patients with MTC or PTC, respectively: 97.2% and 71.5% whereas children with FTC underwent more frequently surgery in two stages (65.1%, p<0.01).

During surgery, cervical dissection was performed more frequently in patients having PTC and MTC compared to the ones presenting FTC (respectively: 79.1% and 72.1% vs 56.6%; p<0.01).

Moreover, patients having PTC or MCT had more often a largest dissection (bilateral neck and central compartment) compared with the ones with FTC. Most of the patients having a DTC had complementary treatment by Radioactive Iodine Therapy (PTC: 83.1% and FTC: 80.7% vs MTC: 0%, p<0.01). Only 5 patients (3 with regional or metastatic MTC, one with a metastatic PTC and one with an insular carcinoma) had other treatments as targeted treatment, external radiotherapy or chemotherapy.

Care system

Patients with MTC had mostly the first medical contact with a university hospital practitioner specialized in children diseases (p<0.01). For DTC, the first contact was more various and patients were mostly seen at diagnosis in an adult unit (p<0.01). Differentiated carcinomas were firstly managed by endocrinologists and nuclear medicine physicians compared to MTC which were more frequently managed by endocrinologists and pediatricians (p<0.01). For surgery, similar results were observed on patient's journey.

Incidence

Age-specific incidence rates (IR) per million by sex for thyroid cancers over the 2000-2018 period are presented in Figure 1. IR are close for both genders between 0 and 7 years and below 2 per 1,000,000 person-year. For females, there is a dramatical increasing of IR with age from 2.1 [95%CI: 1.2-3.5] for 8 years to 35.0 [95%CI: 28.7-42.3] per 1,000,000 person-year for 17 years. Therefore, the sex ratio is 5:1 in the adolescent group. Between 0 and 6 years, IR are higher for MTC compared to DTC (Figure 2). For PTC, there is a strong increasing of IR with age, from 0.5 [95%CI: 0.2-1.1] for 7 years to 18.7 [95%CI: 15.4-22.4] per 1,000,000 personyear for 17 years.

Between 2000 and 2018, there was no significant evolution in thyroid cancers IR in the children age group (Figure 3). All tumor types confounded (DTC and MTC), the rates fluctuated between 1.3 per 1,000,000 person-year in 2016 and 3.2 in 2005 (APC: -1.1%; 95%CI: [-5.7; +3.8]; p=0.66). For DTC, it fluctuated between 1.1 per 1,000,000 person-year in 2003-2004 and 2.5 in 2005 (APC: +0,1%/year; 95%CI: [-1.4; +1.7]; p=0.89).

Overall survival (Table 3)

Overall, 14.2% of cases were lost to follow-up. Over the 2000-2015 period, 92.3% of cases (n=550/596) had a follow-up more than 5 years after diagnosis. No significant differences were observed for the follow-up (<5 years or ≥5 years) when comparing by sex, age group and histological group. Over the 2000-2018 period, the median follow-up time (Interquartile range) was 11.3 (6.6-15.7) years in children, and 5.7 (3.3-7.9) years in adolescents. The 5y-OS was greater than 98.5%. For children diagnosed during the 2000-2010 period, we observed higher survival rates for PTC (5y-OS:100%) and MTC (5y-OS: 97.1; 95%CI: [88.9-99.3]) compared to FTC (5y-OS: 92.4%; 95%CI: [73.0-98.1]). Overall survival did not differ according to metastasis status in both age groups or periods. No significant differences were observed when comparing children and adolescent group over the 2011-2018 period.

DISCUSSION

This study describes the incidence, the clinical management and the overall survival of pediatric thyroid cancers, in France, between 2000 and 2018, based on national population-based cancer registry data.

As described in literature [START_REF] Bernier | Trends in pediatric thyroid cancer incidence in the United States, 1998-2013[END_REF][START_REF] Paulson | Thyroid Cancer in the Pediatric Population[END_REF][START_REF] Dermody | Pediatric thyroid cancer: An update from the SEER database 2007-2012[END_REF], our study confirms that DTC, compared to MTC, are more frequent and affect mostly females with an increased incidence after the age of 10 years. It is generally agreed that the clinical course of PTC differs between children and adults. Children usually present with larger tumors and a greater incidence of lymph node metastasis and lung metastasis. This is the case in our study with a majority of tumors of 10 to 40 mm, 50% of lymph node involvement and 8.5% of lung metastasis. FTC were mostly located and unifocal.

They are reported to be less aggressive than PTC implying a less advanced disease at diagnosis and fewer cases of recurrence [START_REF] Chan | Pediatric Thyroid Cancer[END_REF][START_REF] Francis | Management Guidelines for Children with Thyroid Nodules and Differentiated Thyroid Cancer[END_REF]. Concerning risk factors, MTC were mostly associated with a predisposition. MTC are known to be hereditary in 95% of cases, occurring because of germline RET mutations and resulting in MEN syndrome (particularly MEN2A, more rarely MEN2B) or FMTC [START_REF] Paulson | Thyroid Cancer in the Pediatric Population[END_REF][START_REF] Francis | Management Guidelines for Children with Thyroid Nodules and Differentiated Thyroid Cancer[END_REF]. Some PTC were also rarely associated with genetic mutations as reported in literature like APC or Cowden syndromes [START_REF] Francis | Management Guidelines for Children with Thyroid Nodules and Differentiated Thyroid Cancer[END_REF]. Radiation exposure are an environmental risk factor of DTC and particularly PTC [START_REF] Paulson | Thyroid Cancer in the Pediatric Population[END_REF][START_REF] Francis | Management Guidelines for Children with Thyroid Nodules and Differentiated Thyroid Cancer[END_REF]. In our study, we observed that all patients who had a neck or total body irradiation history for a prior malignancy presented a PTC at diagnosis. Total thyroidectomy is the recommended treatment in PTC pediatric cancers because of the risk of bilateral and multifocal disease and the increased risk of secondary tumor [START_REF] Paulson | Thyroid Cancer in the Pediatric Population[END_REF][START_REF] Chan | Pediatric Thyroid Cancer[END_REF][START_REF] Francis | Management Guidelines for Children with Thyroid Nodules and Differentiated Thyroid Cancer[END_REF]. Total thyroidectomy is also recommended for MTC which can be aggressive tumors [START_REF] Filetti | Electronic address: clinicalguidelines@esmo.org, Thyroid cancer: ESMO Clinical Practice Guidelines for diagnosis, treatment and follow-up †[END_REF]. According management guidelines, we observed that almost all of the PTC and MTC patients underwent total thyroidectomy with frequent lymph nodes dissection. Patients having FTC, compared to other patients, underwent more frequently partial surgery with less lymph node dissection performed. Indeed, FTC is known to be generally an unifocal tumor with rare spreads regional lymph nodes involvement [START_REF] Francis | Management Guidelines for Children with Thyroid Nodules and Differentiated Thyroid Cancer[END_REF]. In addition to surgery, radioiodine-131 (I 131 ) therapy is recognized to be an effective treatment in the prevention of recurrence in case of DTC [START_REF] Paulson | Thyroid Cancer in the Pediatric Population[END_REF][START_REF] Francis | Management Guidelines for Children with Thyroid Nodules and Differentiated Thyroid Cancer[END_REF].

We observed that more than 80% of patients presenting a DTC underwent this additional treatment. This proportion testifies of the widespread use of this treatment. However, this therapy present potential side effects which can be acute (tissues damage as xerostomia, stomatitis, ocular dryness, ...) or develop on long-term basis (second malignancies, pulmonary fibrosis, ...), increasing overall mortality [START_REF] Francis | Management Guidelines for Children with Thyroid Nodules and Differentiated Thyroid Cancer[END_REF]. Thus, these data invites physicians to better identify patients who should benefit from this treatment, as reminded by the American Thyroid Association [START_REF] Francis | Management Guidelines for Children with Thyroid Nodules and Differentiated Thyroid Cancer[END_REF]. Concerning care pathway entry, we observed that patients with MTC were mostly followed in pediatric unit of University hospital due to the earlier diagnostic and the hereditary character of the disease requiring specific management by specialized physicians. In comparison, for patients with DTC, the type of health care services, the unit and the practitioner specialty were varied. Patient care in the adult units remains the more frequent, explained by the fact that patients are older at diagnosis. These results show the diversity of care pathway taken by patients. This fact complicates the identification of pediatric cases for the registry, even more for DTC that are mainly followed in adult units implying to continue to strengthen links with the national network partners.

We observed that thyroid cancer IR were similar and low (below 2 per 1,000,000 person-year) for both genders between 0 and 7 years. However, after 8 years IR increased dramatically for females with a 4:1 Female/Male ratio by 16 years of age and 5:1 by 17. Paulson et al reported a nearly 6:1 F/M ratio by 15-19 years of age [START_REF] Paulson | Thyroid Cancer in the Pediatric Population[END_REF]. Conversely, IR are higher for MTC before 6 years, fluctuating between 0.3 [95%CI: 0.1-0.7] and 0.8 [95%CI: 0.4-1.4] per 1,000,000 personyear. During 2000-2004 period, the French ASR for all thyroid carcinomas confounded for children under 15 years was 1.9 per 1,000,000 person-year [START_REF] Lacour | Incidence of childhood cancer in France: National Children Cancer Registries, 2000-2004[END_REF]. The age-standardized rate per 1,000,000 person-year observed in France during 2000-2018, in the children group was 1.3 for 15-19 years [START_REF] Bernier | Trends in pediatric thyroid cancer incidence in the United States, 1998-2013[END_REF]. For MTC, the ASR per 1,000,000 person-year observed in France during 2000-2018, in children was 0.5 [95%CI: 0.3-0.7] and the IR was 0.5 [95%CI: 0.2-0.9] for the adolescent group. MTC are known to be rare, explaining these low rates. Over the last years, increased IR for thyroid cancer were observed worldly. In the US, IR increased from 1973 to 2013 (mean APC: 2.54; 95%CI: 1.69-3.40) for children under 20 years [START_REF] Qian | Pediatric Thyroid Cancer Incidence and Mortality Trends in the United States, 1973-2013[END_REF]. Same phenomenon was observed in South Korea [START_REF] Gupta | International Trends in the Incidence of Cancer Among Adolescents and Young Adults[END_REF][START_REF] Lee | Trends in Pediatric Thyroid Cancer Incidence, Treatment, and Clinical Course in Korea during 2004-2016: A Nationwide Population-Based Study[END_REF]. We did not observe similar trend in our study as there was no significant evolution in IR in children over 2000-2018 in France. Improvement of imaging techniques or overdiagnosis may be associated to the increasing IR observed worldly [START_REF] Bernier | Trends in pediatric thyroid cancer incidence in the United States, 1998-2013[END_REF][START_REF] Qian | Pediatric Thyroid Cancer Incidence and Mortality Trends in the United States, 1973-2013[END_REF][START_REF] Vaccarella | Worldwide Thyroid-Cancer Epidemic? The Increasing Impact of Overdiagnosis[END_REF][START_REF] Chen | Children and thyroid cancer: Interpreting troubling trends[END_REF]. This improvement in diagnostic procedures might have occurred in France prior to the period of our study, thus explaining the absence of an observed increase in incidence.

Finally, our data confirm, as reported in literature, that pediatric thyroid cancer has an excellent prognosis, as the 5-year overall survival was superior to 98.5% [START_REF] Al-Qurayshi | A National Perspective of the Risk, Presentation, and Outcomes of Pediatric Thyroid Cancer[END_REF][START_REF] Ștefan | Differentiated Thyroid Cancer in Children in the Last 20 Years: A Regional Study in Romania[END_REF][START_REF] Wang | Prognostic analysis of recurrence in children and adolescents with differentiated thyroid cancer[END_REF][START_REF] Huang | Clinicopathological characteristics and prognosis of thyroid cancer in northwest China: A population-based retrospective study of 2490 patients[END_REF]. The challenge in these tumors is to limit the long-term effects such as second cancers linked to irradiation. We will be able to study this soon by linking the RNCE cases with the medico-administrative databases to get information about late effects or second cancers [START_REF] Poulalhon | Data Resource Profile: The French Childhood Cancer Observation Platform (CCOP)[END_REF].

Our study presents several limitations. First, 14.2% of cases were lost to follow-up due to missing birth data (8.5%). Indeed, place of birth is required for linking the registry data to the RNIPP and this information were sometimes missing, especially in adolescent population.

However, over the 2000-2015 period, the major part of the cohort (92.3%) had a follow-up more than 5 years after diagnosis allowing us to have a mid-term view of patient medical history. Then, detection of death predictors was limited due to the low occurrence of the death event.

But our study also has some notable strengths. First, we studied a large case series as this study used data from a national pediatric cancer registry. The registry relies on a wide network of national partners. Indeed, cases are identified actively within the hematology-oncology and pediatric surgery departments. In complementary, the registry collaborates with the medical information departments of University Hospital Centers and cancer centers, and with the general cancer registries, allowing to pursue an exhaustive collection of cases. The confirmation and documentation of cases are carried out by the network of clinical research associates of the registry. Then, for this study, to update the quality of our data, for the 774 cases, tumors histological type was verified through the reading of each pathology report and coded and classified using the ICD-O-3 by a single physician. Also, all data regarding tumoral extension were standardized according the TNM8 classification by the same specialist.

CONCLUSION

The RNCE, by providing robust demographic, diagnostic, clinical and therapeutic data on thyroid cancers over a 19-year period, plays a critical role in surveillance, and tells us where we are in the efforts to better understand these cancers in pediatric and adolescent population. Contrary to some data observed in other countries, we did not observe a significant increase in incidence since 2000. In France, pediatric thyroid cancers are treated as recommended with primary surgery and radioactive iodine in most cases. The prognosis of these cancers is very favorable with a global survival greater than 98%. In view of this very favorable survival, a stratification should be proposed between cases at low risk and cases at high risk of relapse, in order to consider a strategy of therapeutic de-escalation in the most favorable cases. This would limit the adverse effects of treatment, particularly in the case of I 131 irradiation.

[

  95%CI: 1.1-1.6] for PCT 13.4 [95%CI: 11.8-15.1] for adolescent group. Our results are close to the United States (US) rates. Indeed, in a study based on 39 US cancer registries during 1998-2013, Bernier et al, reported for PTC an ASR of 0.5 per million person-year for children under 10 years, 4.7 for 10-14 years and 18.44 for 15-19 years [5]. For FTC, the ASR per 1,000,000 person-year observed in France during 2000-2018, in the children group was 0.2 [95%CI: 0.2-0.4] and the IR was 1.3 [95%CI: 0.8-1.9] for the adolescents. Bernier et al reported for FTC an ASR of 0.04 per million person-year for children under 10 years, 0.6 for 10-14 years and 1.6

Figure 1 Figure 2 Figure 3

 123 Abbreviations: 95%CI, 95% confidence interval; 5y-OS, five-year overall survival

a

  Period of thyroid cancer diagnosisAbbreviations: 95%CI, 95% confidence interval; ASR, age specific incidence rate; CIR, crude incidence rate; SD, standard deviation

a

  Period of thyroid cancer diagnosis b p-value logrank test Results in bold translate a p-value<0.05 Abbreviations: 95%CI, 95% confidence interval; 5y-OS, five-year overall survival

Table 1 )

 1

	Between 2000 and 2018, 774 patients, aged 0-17 years, with thyroid cancer were included: 490
	(63.3%) aged 0-14 years (as children group) and diagnosed from 2000 to 2018 and 284 (36.7%)
	aged 15-17 years (as adolescent group) and diagnosed from 2011 to 2018. Mean age at
	diagnosis was 10.6 (3.6) years for children and 16.2 (0.8) years for adolescents. Over the 2011-
	2018 period, adolescents were more frequently female compared to children (81.3% vs 67.5%;
	p<0.01) and had more frequently PTC (88.4% vs 67.5%; p<0.01) and less frequently follicular
	carcinoma (FTC) (8.4% vs 15.7%; p<0.01) and MTC (3.2% vs 16.8%; p<0.01). One male
	patient was diagnosed with a Spindle epithelial tumor with thymus-like differentiation
	(SETTLE) at 12 years. He presented a metastatic relapse 10 months after diagnosis and died 6
	years after diagnosis. This patient was excluded from analysis and in following results, as this
	tumor type is extremely rare.

Table 1

 1 Patient socio-demographic characteristics, thyroid cancer histological type and incidence rates per million children by age group(France, 2000(France, -2018) )

	a Period of thyroid cancer diagnosis
	Abbreviations: 95%CI, 95% confidence interval; ASR, age specific incidence rate; CIR, crude
	incidence rate

Table 2

 2 Patient and tumor characteristics,

Table 1

 1 Patient socio-demographic characteristics, thyroid cancer histological type and incidence rates per million children by age group(France, 2000(France, -2018) )

Table 2

 2 Patient and tumor characteristics, treatment and disease management by histological type(France, 2000(France, -2018) )

		Papillary	Follicular	Medullary	p-value a
		n=579	n=83	n=111
		n (%)	n (%)	n (%)
	PATIENT CHARACTERISTICS
	Gender
	Female	449 (77.6)	60 (72.3)	52 (46.8)
	Male	130 (22.4)	23 (27.7)	59 (53.2)	<0.01
	Age (years)
	[0 ; 5[	7 (1.2)	0 (0.0)	37 (33.3)
	[5 ; 10[	58 (10.0)	7 (8.4)	32 (28.8)
	[10 ; 15[	263 (45.4)	52 (62.7)	33 (29.7)
	[15 ; 18[	251 (43.4)	24 (28.9)	9 (8.1)	<0.01
	Patient medical history
	Patient relatives having thyroid
	cancer history
	No	539 (93.1)	78 (94.0)	18 (16.2)
	Yes	26 (4.5)	4 (4.8)	58 (52.3)
	Unknown	14 (2.4)	1 (1.2)	35 (31.5)	<0.01
	Patient relatives having thyroid
	cancer predisposition
	No	560 (96.7)	82 (98.8)	17 (15.3)
	Yes	5 (0.9)	1 (1.2)	81 (73.0)
	Unknown	14 (2.4)	0 (0.0)	13 (11.7)	<0.01
	Patient Thyroid cancer
	predisposition
	No	560 (96.7)	82 (98.8)	4 (3.6)
	Yes	7 (1.2)	1 (1.2)	106 (95.5)
	Unknown	12 (2.1)	0 (0.0)	1 (0.9)	<0.01
	Patient Thyroiditis history
	No	536 (92.6)	83 (100.0)	111 (100.0)
	Yes	32 (5.5)	0 (0.0)	0 (0.0)
	Unknown	11 (1.9)	0 (0.0)	0 (0.0)	<0.01
	Patient having Neck or total body
	irradiation history
	No	548 (94.7)	83 (100.0)	111 (100.0)
	Yes	20 (3.5)	0 (0.0)	0 (0.0)
	Unknown	11 (1.9)	0 (0.0)	0 (0.0)	<0.01
	TUMOR CHARACTERISTICS
	Tumor laterality
	Bilateral	132 (22.8)	7 (8.4)	75 (67.6)
	Unilaterial or median	434 (75.0)	75 (90.3)	32 (28.8)
	Unknown	13 (2.3)	1 (1.2)	4 (3.6)	<0.01
	Multifocal
	Yes	219 (37.8)	11 (13.3)	79 (71.2)
	No	336 (58.0)	68 (81.9)	28 (25.2)
	Unknown	24 (4.2)	4 (4.8)	4 (3.6)	<0.01
	Size of tumor (mm) b
	[0 ; 10]	94 (16.2)	2 (2.4)	86 (77.5)
	]10 ; 20]	176 (30.4)	10 (12.1)	8 (7.2)
	]20 ; 40]	220 (38.0)	48 (57.8)	6 (5.4)
	> 40	68 (11.7)	22 (26.5)	2 (1.8)
	Unknown	21 (3.6)	1 (1.2)	9 (8.1)	<0.01

a Chi-square test b Size of the largest lesion c Post-surgical data or clinical data when post-surgical data is unknown d Genetic, Internal medicine, Hepato-Gastro-Enterology, Pneumology

Table 3

 3 Five-year overall survival by age group and period of thyroid cancer diagnosis(France, 2000(France, -2018) )

ACKNOWLEDGMENTS

The authors are grateful to the numerous investigators who have meticulously contributed to the active search for information in clinical departments. The authors would also like to express their gratitude to the heads of the Medical Information Departments of all the University Hospitals and Comprehensive Cancer Centers and the Regional Departments of the Medical Control Service of Health Insurance.

FUNDING INFORMATION

This work was supported by the "French National Cancer Institute", and the "French Public Health Agency".

AUTHORS' CONTRIBUTION

All named authors have participated in the study to a sufficient extend to be named as authors.

All authors were involved in the study design and analysis plan. All authors critically reviewed the study results, commented on the drafts, and approved the final manuscript.

ETHICAL APPROVAL

This study was performed at the RNCE, CRESS Team

DECLARATION OF COMPETING INTEREST

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Ronan Dupont
email: ronan.dupont@umontpellier.fr

Megan Cook
email: megan.cook@umontpellier.fr

Frédéric Bouchette
email: frederic.bouchette@umontpellier.fr

Bijan Mohammadi
email: bijan.mohammadi@umontpellier.fr

Samuel Meulé
email: meule@cerege.fr

Opti-Morph, a new platform for sandy beach dynamics by constrained wave energy minimization

Keywords: Hydro-morphodynamics, Optimization, Model validation, Coastal, Variational approach, Energy minimization, Optimal transport, Waves

This paper focuses on a new approach to describe coastal morphodynamics, based on optimization theory, and more specifically on the assumption that a sandy seabed evolves in order to minimize a waverelated function, the choice of which depends on what is considered the driving force behind the coastal morphodynamic processes considered. The numerical model derived from this theory uses a gradient descent method and allows us to account for physical constraints such as sand conservation in basin experiments. Hence, the model automatically adapts to either basin or open sea settings and only involves two hyper-parameters: sand abrasion and the critical angle of repose. The Opti-Morph model is illustrated on a flume configuration. Comparison of the resulting seabed with experimental data as well as the results of the widely distributed coastal morphodynamic software XBeach demonstrates the potential of a model by wave energy minimization.

Introduction

Optimization theory is the study of the evolution of a system while searching systematically for the minimum of a function derived from physical properties of the system. In this paper, we have applied this approach to coastal dynamics, with our primary objective to simulate the interactions between the waves and seabed. Using mathematical optimization theory [START_REF] Isèbe | Design of passive defense structures in coastal engineering[END_REF][START_REF] Isebe | Optimal shape design of defense structures for minimizing short wave impact[END_REF][START_REF] Isèbe | Shape optimization of geotextile tubes for sandy beach protection[END_REF][START_REF] Bouharguane | 34 095 Montpellier: Low complexity shape optimization & a posteriori high fidelity validation[END_REF][START_REF] Bouharguane | Minimization principles for the evolution of a soft sea bed interacting with a shallow[END_REF][START_REF] Mohammadi | Optimal dynamics of soft shapes in shallow waters[END_REF][START_REF] Mohammadi | Extreme scenarios for the evolution of a soft bed interacting with a fluid using the value at risk of the bed characteristics[END_REF][START_REF] Mohammadi | Optimal dynamics of soft shapes in shallow waters[END_REF][START_REF] Cook | Optimal port design minimizing standing waves with a posteriori long term shoreline sustainability analysis[END_REF], we have designed a model that describes the evolution of the seabed while taking into account the coupling between morphodynamic and hydrodynamic processes. This study focuses on a theoretical and numerical approach to the modeling of this coupling, based on the assumption that the seabed adapts to minimize a certain wave-related function. The choice of this function determines the driving force behind the morphological evolution of the seabed. This optimization problem is subjected to a certain number of constraints, allowing for a more accurate description of the morphodynamic evolution.

This study is accompanied by the development of a numerical hydromorphodynamic model, which has the advantages of being fast, robust, and of low complexity. The model was given the name Opti-Morph.

The paper starts with a description of the simple hydrodynamic model used to calculate the driving forces behind the morphodynamic processes. Then, we provide a description of the morphodynamic model (Opti-Morph) based on wave-energy minimization. With the purpose of validating Opti-Morph, we compare the results of the numerical simulation with that of experimental data acquired in a flume experiment. We also compared the model to another nearshore hydro-morphodynamic model, XBeach [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF], to see how it fares against existing hydro-morphodynamic models, XBeach being considered to be quite a reputable model in the coastal dynamic community [START_REF] Zimmermann | Longshore transport and sedimentation in a navigation channel at blankenberge (belgium)[END_REF][START_REF] Bugajny | Modelling morphological changes of beach and dune induced by storm on the southern baltic coast using xbeach (case study: Dziwnow spit)[END_REF][START_REF] Williams | Modelling storm responses on a high-energy coastline with xbeach[END_REF].

State of the Art

Numerical models of morphodynamic processes are seen as a valuable tool for understanding and predicting the evolution of the sediment and morphology over time in coastal areas. Different morphodynamic models exist in the literature, ranging from empirical models [START_REF] De Vriend | A morphological behaviour model for the outer delta of mixed-energy tidal inlets[END_REF][START_REF] Gravens | An approach to modeling inlet and beach evolution[END_REF][START_REF] Kana | Mesoscale sediment transport at southeastern u.s. tidal inlets: conceptual model applicable to mixed energy settings[END_REF][START_REF] Ruessink | The behaviour of nearshore bars on the time scale of years: A conceptual model[END_REF] to process-based models. The latter can be sorted into several categories, such as i) profile evolution models [START_REF] Larson | Sbeach: Numerical model for simulating storminduced beach change[END_REF][START_REF] Larson | Sbeach: Numerical model for simulating storm-induced beach change[END_REF][START_REF] Nairn | Deterministic profile modelling of nearshore processes. part 2. sediment transport and beach profile development[END_REF], which use only cross-shore transport, ii) 2D morphological models [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF][START_REF] Fleming | Application of sediment transport model[END_REF][START_REF] Latteux | Harbour design including sedimentological problems using mainly numerical technics[END_REF][START_REF] Coeffe | Modelling of sea-bed evolution under waves action[END_REF][START_REF] Yamaguchi | Numerical simulation on the change of bottom topography by the presence of coastal structures[END_REF][START_REF] Watanabe | Numerical prediction model of three-dimensional beach deformation around a structure[END_REF][START_REF] Maruyama | A simulation system of near-shore sediment transport for the coupling of the sea-bottom topography, waves and currents[END_REF][START_REF] Wang | A time-dependent nearshore morphological response model[END_REF][START_REF] Johnson | Identification of some relevant processes in coastal morphological modelling[END_REF][START_REF] Nicholson | Intercomparison of coastal area morphodynamic models[END_REF], which use depth-averaged wave and current equations to model the sediment transport while neglecting the vertical variations of wave-derived parameters, as well as iii) 3D and quasi-3D models [START_REF] Roelvink | Design and development of delft3d and application to coastal morphodynamics, 1st international conference, hydroinformatics 94[END_REF][START_REF] Lesser | Development and validation of a three-dimensional morphological model[END_REF][START_REF] Roelvink | Morphological modelling of keta lagoon case[END_REF][START_REF] Briand | Sediment transport in the surf zone: A quasi 3-d numerical model[END_REF][START_REF] Zyserman | Modelling morphological processes in the vicinity of shore-parallel breakwaters[END_REF][START_REF] Ding | Development and validation of a quasi-threedimensional coastal area morphological model[END_REF][START_REF] Droenen | Quasi-three-dimensional modelling of the morphology of longshore bars[END_REF], which determine the sediment evolution using both horizontal and vertical variations of the wave-derived parameters.

The Opti-Morph model described in this paper is based on optimal control. In the past, the use of optimization theory has primarily been used in the design of coastal defense structures, whether in the design of ports and offshore breakwaters [START_REF] Isebe | Optimal shape design of defense structures for minimizing short wave impact[END_REF][START_REF] Isèbe | Shape optimization of geotextile tubes for sandy beach protection[END_REF].

Optimal control has already been considered for the modeling of shallow water morphodynamics, based on the assumption that the seabed acts as a flexible structure and adapts to a certain hydrodynamic quantity [START_REF] Bouharguane | 34 095 Montpellier: Low complexity shape optimization & a posteriori high fidelity validation[END_REF][START_REF] Mohammadi | Optimal dynamics of soft shapes in shallow waters[END_REF]. These studies were based on somewhat theoretical developments with no direct relationship with real case studies. Our objectives in this work is to produce a physically robust numerical morphodynamic model based on optimal control and to validate it using numerical data from well established morphodynamics software and also basin experiments.

Hypotheses

Opti-Morph is based on a certain number of assumptions. Since the model is based on the minimization of a cost function, certain hypotheses must be made regarding the choice of this function. This function, which originates from a physical quantity, must be directly linked to the elevation of the seabed. At present, we set the quantity to be minimized as the energy of shoaling waves. This implies that the seabed reacts to the state of the waves by minimizing the energy of shoaling waves. Other assumptions assess the behavior of seabed and originate from general observations. Sediment transport is influenced by the orbital velocity of water particles [START_REF] Soulsby | Calculating bottom orbital velocity beneath waves[END_REF], which leads to greater sediment mobility in shallower waters. Another natural observation concerns the slope of the seabed, which cannot be overly steep without an avalanching process occurring [START_REF] Reineck | Depositional Sedimentary Environments; with Reference to Terrigenous Clastics[END_REF]. Last, in an experimental flume configuration, the quantity of sand must remain constant over time, with no inflow or outflow of sand to alter the sand stock.

Theoretical Developments

Modeling Framework

For the sake of simplicity, we present the principle of morphodynamics by optimization in a one-dimensional setting. This enables us to compare the numerical results based on this theory with experimental flume data. However, no assumptions were made regarding the dimension of the problem, and as a result, it is straightforward to extend this theory to a two-dimensional configuration.

We consider a coordinate system composed of a horizontal axis x and a vertical axis z. We denote Ω := [0, x max ] the domain of the cross-shore profile of the active coastal zone, where x = 0 is a fixed point in deep water where no significant change in bottom elevation can occur, and x max is an arbitrary point at the shore beyond the shoreline, as shown by Figure 1. The elevation of the seabed is a one-dimensional positive function, defined by: ψ : Ω × [0, T ] × Ψ → R + where [0, T ] is the duration of the simulation (s) and Ψ is the set of physical parameters describing the characteristics of the seabed. In order to model the evolution over time of ψ and given the assumption that the seabed ψ changes over time in response to the energy of shoaling waves, a description of the surface waves is needed.

Hydrodynamic Model

The literature on hydrodynamic models is vast [START_REF] Murray | Reducing model complexity for explanation and prediction[END_REF]. However, our main focus in this work is a) on the morphodynamic part of the approach and b) on providing evidence of the ability of optimization to perform robust morphodynamic prediction even under weakly constrained hydrodynamics. So we present the procedures with a hydrodynamic model as simple as possible, that is based on the linear wave theory [START_REF] Dean | Coastal processes with engineering applications[END_REF], a very basic shoaling equation and some geometrical breaking parameter. It has the advantage of being easy to differentiate compared to more sophisticated models that would need automatic differentiation [START_REF] Mohammadi | Optimal dynamics of soft shapes in shallow waters[END_REF][START_REF] Hascoet | Tapenade user's guide[END_REF] or huge additional numerical developments. This numerical implementation has a significantly short run-time as shown by the convergence results of the section 4.1. This model has the advantage of expressing wave height as an explicit function of the seabed, which leads to rapid calculations of the morphodynamic model.

Let h be the depth of the water from a mean water level h 0 (cf. Figure 1). Ocean waves, here assumed monochromatic, are characterized by phase velocity C, group velocity C g , and wave number k, determined by the linear dispersion relation [START_REF] Isèbe | Design of passive defense structures in coastal engineering[END_REF], where σ is the pulsation of the waves and g is the gravitational acceleration.

σ 2 = gk tanh(kh) (1)
We define Ω S as the time-dependent subset of Ω over which the waves shoal and Ω B the subset of Ω over which the waves break, cf. Figure 1. Munk's breaking criterion [START_REF] Munk | The solitary wave theory and its application to surf problems[END_REF] enables us to define Ω S (t) = x ∈ Ω, H(x,t) h(x,t) < γ and Ω B (t) = x ∈ Ω, H(x,t) h(x,t) ≥ γ , where γ is a wave breaking index. Then we have

H(x, t) = H 0 (t)K S (x, t) (2)
The height of the waves H over the cross-shore profile is inspired by the shoaling equation [START_REF] Isebe | Optimal shape design of defense structures for minimizing short wave impact[END_REF], where H 0 (t) is the deep water wave height and K S is a shoaling coefficient, given by

K S = 1 2 C 0 C g 1 2 (3)
where C 0 is the deep water wave velocity, and:

n = C C g , C = C 0 tanh(kh), C g = 1 2 C 1 + 2kh sinh(2kh) . (4)
Instead of considering that waves depend solely on offshore wave height H 0 , this model suggests that shoaling waves are decreasingly influenced by seawards waves. The greater the distance, the less effect it has on the present wave height. As such, we introduce a weighting function w. Assuming that the maximal distance of local spatial dependency of a wave is denoted d w , the weighting function over the maximal distance d w is given by w : [0, d w ] → R + such that w(0) = 1 , w(d w ) = 0 and decreases exponentially.

Equation (2) for shoaling wave height becomes equation [START_REF] Bouharguane | Minimization principles for the evolution of a soft sea bed interacting with a shallow[END_REF], where H w 0 is defined by [START_REF] Mohammadi | Optimal dynamics of soft shapes in shallow waters[END_REF].

H(x, t) = H w 0 (x, t)K S (x, t) (5)
H w 0 (x, t) = 1 x x-X w(x -y)dy x x-X w(x -y)H(y)K(y)dy (6)
Equation ( 5) applies only to the shoaling, nearshore-dependent waves of Ω S , significant wave height over the cross-shore profile H : Ω → R + is defined by [START_REF] Mohammadi | Extreme scenarios for the evolution of a soft bed interacting with a fluid using the value at risk of the bed characteristics[END_REF], where α(x) = x d w over [0, d w ] to allow a smooth transition between offshore and nearshore-dependent waves.

H(x, t) =            [(1 -α(x))H 0 (t) + α(x)H w 0 (x, t)] K S (x, t) if x ∈ Ω S and x < d w H w 0 (x, t)K S (x, t) if x ∈ Ω S and x ≥ d w γh(x, t) if x ∈ Ω B (7)

Morphodynamic Model by Wave Energy Minimization

The evolution of the seabed is assumed to be driven by the minimization of a cost function J. Recalling the hypotheses made in Section 1.2 , the shape of the seabed is determined by the minimization of the potential energy of shoaling waves, for all t ∈ [0, T ]:

J(ψ, t) = 1 16 ΩS ρ w gH 2 (ψ, x, t)dx [J.m -1 ] (8)
where H denotes the height of the waves over the cross-shore profile, ρ w is water density (kg.m -3 ), and g is the gravitational acceleration (m.s -2 ). In order to describe the evolution of the seabed, whose initial state is given by ψ 0 , we assume that the seabed ψ, in its effort to minimize J, verifies the following dynamics:

ψ t = Υ Λ d ψ(t = 0) = ψ 0 (9)
where ψ t is the evolution of the seabed over time [m.s -1 ], Υ is the abrasion of sand (m.s.kg -1 ), Λ is the excitation of the seabed by the water waves, and d is the direction of the descent, which indicates the manner in which the seabed changes. The approach involves two parameters with clear physical interpretation. The first Υ takes into account the physical characteristics of the sand and represents the mobility of the sediment. At the present time, we consider Υ to be a measure of sand mobility expressed in m.s.kg -1 . Further explanation of the nature of this parameter will be given at a later stage of the model's development. The second parameter Λ is a local function which represents the influence of the water depth on the seabed and is defined using the term describing the vertical variation of the classic velocity potential of the linear wave theory [START_REF] Soulsby | Calculating bottom orbital velocity beneath waves[END_REF]:

φ : Ω × [0, h 0 ] -→ R + (x, z) -→ cosh(k(x)(h(x) -(h 0 -z))) cosh(k(x)h(x)) (10)
In unconstrained circumstances, for instance, if a total sand volume constraint does not need to be enforced, we set d = -∇ ψ J, which indicates a direction for local minimization of J with regards to ψ. The calculation of ∇ ψ J is described in Appendix A.1. However, constraints are added to the model to incorporate more physics and deliver more realistic results. Driving forces behind the morphological evolution of the seabed are described by the minimization of the cost function J. Secondary processes are expressed by constraints. In the interest of simplicity, we have adopted two physical constraints though more can be introduced if necessary. The first concerns the slope of the seabed. Depending on the composition of the sediment, the slope of the seabed is bounded by a grain-dependent threshold M slope [START_REF] Dean | Coastal processes with engineering applications[END_REF]. This is conveyed by the following constraint on the local bathymetric slope:

∂ψ ∂x ≤ M slope (11)
The dimensionless parameter M slope represents the critical angle of repose of the sediment, and varies between 0.2 and 0.6 [START_REF] Al-Hashemi | A review on the angle of repose of granular materials[END_REF].

A second example concerns the sandstock in the case of an experimental flume. This constraint states that the quantity of sand in a flume must be constant over time, as given by ( 12), contrarily to an open-sea simulation where sand can be transported between the onshore and the offshore zones [START_REF] Hattori | Onshore-Offshore Transport and Beach Profile Change[END_REF][START_REF] Quick | Onshore-offshore sediment transport on beaches[END_REF].

Ω ψ(t, x)dx = Ω ψ 0 (x)dx ∀t ∈ [0, T ] ( 12
)
This constraint is necessary for verifying and validating the numerical model with physical simulations.

Numerical Application

In this section, we present the numerical results produced by the Opti-Morph model. For validation purposes, the resulting seabed is compared to experimental data acquired during a flume tank experiment. We also conduct a comparative analysis between the physical seabed, the seabed produced by Opti-Morph and the seabed produced by XBeach, with the aim of assessing how Opti-Morph holds up against existing hydro-morphodynamic models. A brief description of the experiment is provided, as well the XBeach model.

Description of the Experiment

The experimental observations have been collected as part of the COPTER project and a series of laboratory wave-flume experiments were performed in order to investigate the morphodynamic impact of introducing solid geotextile tubes to the Hatzuk (Israel) seafloor [START_REF] Bouchette | Coastal defense strategy along hatzuk beach (northern tel aviv, israel). insights from the copter physical experimentation with moveable bed[END_REF]. We use the data collected without tubes to describe the natural evolution of the seabed over time.

A glass flume measuring 36 m long, 0.55 m wide and 1.3 m deep is equipped with a wave-maker and gauges measuring the height of the water. Artificial particles are placed inside the flume representing the mobile sea bottom and an ultrasonic gauge is used to measure the sedimentary topography.

The experimental seabed, described in Figure 1 is subjected to a series of 30-minute storm climates, among which a typical moderate storm event (at the scale of the flume) with a significant wave height and period of H s = 135 mm and T s = 2.5 s. Time and length scale ratio are set to 1/3 and 1/10 respectively to that of the field.

XBeach Model

XBeach is an open-source process-based model developed by Deltares, UNESCO-IHE, and Delft University of Technology to simulate the hydromorphodynamic processes in coastal areas [START_REF] Roelvink | Modelling storm impacts on beaches, dunes and barrier islands[END_REF][START_REF] Zimmermann | Longshore transport and sedimentation in a navigation channel at blankenberge (belgium)[END_REF][START_REF] Bugajny | Modelling morphological changes of beach and dune induced by storm on the southern baltic coast using xbeach (case study: Dziwnow spit)[END_REF][START_REF] Williams | Modelling storm responses on a high-energy coastline with xbeach[END_REF].

In brief, XBeach uses four interconnected modules to model near-shore processes [START_REF] Daly | Low frequency waves in the shoaling and nearshore zone a validation of xbeach[END_REF]. The two hydrodynamic modules consist of the short wave module and the flow module. The first is based on wave action equations [START_REF] Holthuijsen | a prediction model for stationary, short crested waves in shallow water with ambient current[END_REF], and incorporates breaking, dissipation [START_REF] Roelvink | Dissipation in random wave groups incident on a beach[END_REF], and wave current interactions, while the latter is governed by shallow water equations [START_REF] Andrews | An exact theory of nonlinear waves on a lagrangian-mean flow[END_REF][START_REF] Walstra | Calculation of wavedriven currents in a 3d mean flow model[END_REF]. One of the two morphodynamic modules is the sediment transport module based on the equilibrium sediment concentration equation [START_REF] Soulsby | Dynamics of Marine Sands[END_REF] and a depth-averaged advection-diffusion equation [START_REF] Galappatti | A depth-integrated model for suspended sediment transport[END_REF]. The other is the morphology module which concerns seabed transformations such as the evolution of the seabed and avalanching.

In order to configure the XBeach model for the experimental flume setting, we refer to the XBeach user manual [START_REF] Roelvink | Xbeach model -description and manual[END_REF]. The domain Ω is defined over 32 m with a uniform subdivision of 320 cells. The incoming wave boundary condition is provided using a JONSWAP wave spectrum [START_REF] Hasselmann | Measurements of wind-wave growth and swell decay during the joint north sea wave project (jonswap)[END_REF], with a significant wave height of H m0 = 0.015 m and a peak frequency at f p = 0.4s -1 . The breaker model uses the Roelvink formulation [START_REF] Roelvink | Dissipation in random wave groups incident on a beach[END_REF], with a breaker coefficient of γ = 0.4, a power n = 15, and a wave dissipation coefficient of 0.5. These parameters were calibrated using the hydrodynamic data produced during the physical flume experiment. Concerning sediment parameters, the D50 coefficient is set as 0.0006, and the porosity is 2650 kg.m -3 . No other parameters such as bed friction or vegetation were applied. The model is set to run for a period of 1800 s, as a short-term simulation.

Hydrodynamic Validation

This section is devoted to the comparison of the two numerical hydrodynamic models to the experimental wave data obtained in the experimental flume of Section 3.1. Mean wave height profiles were calculated over the short-term storm simulation, for both Opti-Morph and XBeach, and compared to the mean wave height of the experimental model. The latter was calculated using the measures taken by the gauges of the flume.

Figure 2 shows that the hydrodynamic module of both Opti-Morph (red) and XBeach (blue) are both comparable with respect to the experimental measurements (green) excluding, as is often the case, the second point at x = 6 m. XBeach demonstrates a close qualitative fit over the 10-22 m section of the flume, whereas Opti-Morph excels at the coast (21-27 m), with a near-perfect fit with the experimental data. Despite the simplicity of the hydrodynamic model used by Opti-Morph, the resulting wave height is of the same order of magnitude over the cross-shore profile than that measured during the flume experiment, which indicates that the resulting seabeds are comparable with regard to the forcing energy driving the morphodynamic response.

Numerical Results of the Morphodynamic Simulations

The Opti-Morph model was applied to the configuration of the COPTER experiment of Section 3.1, and the resulting beach profile is shown by the red profile, in Figure 3.A. The main observation is the decrease of 2.5 cm in height of the sandbar, at x = 9 m. We observe a slight decrease of the seabed adjacent to the wave-maker, and a slight increase at the plateau, situated at 15-25 m.

No mobility is observed at the coast.

When comparing the results provided by Opti-Morph (red), with that of XBeach (blue) and the experimental data (green), as shown on Figure 3.A, we observe that the red seabed profile provided by the Opti-Morph model shows a general quantitative agreement when compared to the experimental data, as does the XBeach morphological module. In fact, both models produce profiles close to the experimental data over the plateau located at 15-25 m from the wave-maker (Fig. 3.C). At the shore, Opti-Morph matches the experimental data whereas XBeach shows a vertically difference of up to 3 cm at x = 27 m (Fig. 3.D). Discrepancies on the part of both models occur in the area surrounding the tip of the sandbar, as both Opti-Morph and XBeach fail to predict the shoreward shift of the sandbar (Fig. 3.B); the experimental data show that the height of the sandbar remains unchanged with regards to the initial profile. Both sandbars have a height of 0.375 m; however, the sandbar resulting from the experimental simulation has moved towards the coast, an occurrence that neither numerical model was able to predict.

As such, this new model based on wave-energy minimization shows potential when compared to XBeach, in the case of short-term simulations.

Discussion

Robustness analysis of the convergence in time and space of the hydrodynamic model

We computed a reference OPTIMORPH simulation using a very small time step of 0.05 s which is much smaller than what is usually used in hydromorphodynamic simulations. The simulation was performed with the original bathymetric profile of the COPTER experiment and the forcings of the wave maker.

This simulation provides a reference computed sea bed ψ ref (t f , x) at some given time t f .We would like to see the convergence toward this reference solution of various other OPTIMORPH simulations with different decreasing time steps. From this series of simulations, we quantify a residual error with L 2 norm as

E L 2 =||ψ ref -ψ|| L 2 in [m]
. We performed 10 simulations with time steps ranging in [0.05; 160] s and we get the results described in figure 4.a). In order to analyze the convergences in space and time, we choose, respectively, a reference time step of ∆t = 3 s and a spatial step size ∆x = 1 m. ∆t = 3 s corresponds to the kind of time steps we would like to use in simulations. But, we will use larger spatial resolution in practice. The results in figure 4 show first order (illustrated by the continuous line) convergence rates in both time and space.

To understand why a time step of 3 seconds is interesting for computing efficiency, it is useful to look at the CF L stability condition analysis for the shallow-water Saint-Venant model. The analysis provides a typical upper bound for the time step of the form:

∆t = min i ∆x 2 max i (|u i ± √ gh i |) = ∆x 2 max i (|u 0 ± √ gh 0 |) ,
where subscript i indicates the mesh node which means that the minimum is taken over all the nodes of the mesh. In our situation, it correspond to the off-shore position (subscript i = 0). Typicals values in our simulation are: u 0 = 10 m.s -1 , ∆x = 1 m, h = 0.55 m and g = 9.81 m.s -2 . This gives us ∆t = 0.04 s, which is about two orders of magnitude smaller than our reference time step of ∆t = 3 s. In addition, the costs of one iteration of the Saint-Venant and Optimorph models are comparable.

Parameter Robustness Analysis

One of the advantages of the Opti-Morph model is the low number of morphodynamic hyper-parameters required. At the present time, Opti-Morph requires two hyper-parameters: the mobility parameter Υ and the maximal slope parameter M slope . Here, an assessment on these parameters is conducted. In Figure 3.E, three simulations were performed in identical settings with changes made solely to the mobility parameter. Initially, this parameter Υ has a value of 5 × 10 -6 , m.s.kg -1 . Figure 3.E shows no significant difference despite a 50% increase (Υ = 7.5 × 10 -6 m.s.kg -1 ) (orange) or decrease (Υ = 2.5 × 10 -6 m.s.kg -1 ) (light blue) of Υ with regard to the baseline seabed profile (black). Similar conclusion can be deduced for the maximal slope parameter M slope , whose reference value here is 0.2. The corresponding parameter of XBeach is wetslp, described in the XBeach manual as the critical avalanching slope under water, and is also set to 0.2. In Figure 3.F, we observe little difference between the reference seabed (black), the seabed resulting from a 50% increase (M slope = 0.3) (orange) and the seabed resulting from a 50% decrease (M slope = 0.1) (light blue). The only apparent discrepancy can be found at x = 28 m, where the seabed is at its steepest, and therefore the sand slope constraint is more prone to be active. The reduction of the critical angle of repose results naturally in a less steep slope. The robustness of Opti-Morph in relation to both the mobility parameter and the slope parameter, despite a significant increase or decrease of their value, is apparent. Further simulations show that the robustness of these parameters is not specific to this particular flume configuration, but can be observed regardless of the initial configuration.

Long-term Simulations

This section is devoted to the long-term behavior of Opti-Morph, the main question being, is this numerical model capable of creating an equilibrium state after being subjected to a great number of repeated events. Five forcing scenarios, lasting either 2 or 6 days, were applied to the same initial seabed in the same parametric configuration. The current Opti-Morph code is in Python.

Typically, using time-steps of 1 s simulating a day of forcing requires about 1.5 hours on a 2GHz PC computer. Each time iteration gathering the steps presented in this paper requires therefore about 63 ms. Regarding the section 4.1, we could use 3 s time-step and divide the simulation time by 3. An analysis of the resulting seabeds is performed as well as their behavior throughout the simulation. The latter is achieved through a comparative study of four timeseries, focusing on: (1), the vertical evolution of seabed elevation at the tip of the sandbar; (2), the vertical evolution of seabed elevation at a point of the plateau; [START_REF] Isèbe | Shape optimization of geotextile tubes for sandy beach protection[END_REF], the distance between the wave-maker and the onset of the seabed; and (4), the location of the shoreline position.

Fig. 5 Long-term simulation of Opti-Morph. A. Forcing wave height for scenario 1, composed of several long-term events over a 2-day period. B. Forcing wave height for scenario 2, composed of numerous short-term events over a 2-day period. C. Forcing wave height for scenario 3, composed of several long-term events over a 6-day period. D. Forcing wave height for scenario 4, composed of numerous short-term events over a 6-day period. E. Forcing wave height for scenario 5, composed of few long-term events over a 6-day period. F. Seabeds resulting from the different forcing scenarios produced by Opti-Morph. Two points of interest have be identified: P1 located at x = 9.3 m and P2 located at x = 20.1 m. G. Evolution of the distance, devoid of sediment, between the wave-maker (located at x = 0 m) and the seabed (WM-S), regarding forcing scenarios 3, 4, and 5. H. Vertical evolution of seabed elevation at P1, driven by the 6-day forcing scenarios 3, 4, and 5. I. Vertical evolution of seabed elevation at P2, driven by the 6-day forcing scenarios 3, 4, and 5. J. Evolution of shoreline position, driven by the 6-day forcing scenarios 3, 4, and 5.

Applying Opti-Morph over a longer time-series leads to the results of Figure 5. The two 2-day forcing scenarios are shown in Figures 5.A and 5.B. In both cases, we observe that the resulting seabeds of Figure 5.F are subjected to the destruction of the sandbar and have a tendency to evolve progressively towards an equilibrium beach profile [START_REF] Engineers | Coastal Engineering Manual, Engineer Manual[END_REF]. Simulations over a 6-day period were conducted to confirm this tendency. These scenarios are depicted in Figures 5.C, 5.D, and 5.E, and the resulting seabeds given in Figure 5.F show once again the destruction of the sandbars, the elevation of the plateau, and erosion at the shoreline. Furthermore, all three tend towards an equilibrium state. This is confirmed by the four time-series analysis presented in Figures 5.G, 5.H, 5.I, and 5.J. The vertical elevation of the seabed at both points P1 and P2 show initial variations over the first 2 days: a decrease in the case of P1 (cf. Figure 5.H) and an increase in the case of P2 (cf. Figure 5.I). However, both studies show a stabilization of the seabed elevation over the last 4 days of the 6-day period. Similar conclusions can be drawn regarding the length of the zone containing no sediment adjacent to the wave-maker (cf. Figure 5.G). An initial increase between 2 and 3 meters can be observed, with stability achieved in the later stages of the simulations. Finally, Figure 5.J shows the evolution of the shoreline position. Initially found at x = 28.3 m, all scenarios provoke a retreat of the shoreline: 0.4 m in scenario 3, 0.3 m in scenario 4, and 2 m in scenario 5. The shorelines of the latter two converge, whereas scenario 3 shows an abrupt advance of the shoreline at day 5, with an attempt to return back to its stable state of x = 30 m. The seabed has been flattened, the sandbar has been destroyed and erosion can be observed at the coast [START_REF] Grasso | Experimental simulation of shoreface nourishments under storm events: A morphological, hydrodynamic, and sediment grain size analysis[END_REF]. This tendency to evolve towards an equilibrium state [START_REF] Dean | Coastal processes with engineering applications[END_REF] is consistent with the choice of morphogenic and constant storm-like forcing conditions.

The comparisons made between the two 2-day simulations and the three 6-day simulations, in this given configuration, also reveal the little influence heritage has on the morphodynamic response. Both scenarios 1 and 2 have a comparable cumulative incoming wave energy density E = 1 16 T 0 ρgH 2 0 dt of 0.0591 J.m -2 . The resulting seabeds evolve towards similar profiles (reduction of the sandbar, increase of elevation of the plateau, and erosion at the coast), despite two different forcing conditions. Similar conclusions can be hydrodynamic quantities H, this sensitivity can be expressed using the chain rule:

∇ ψ J = ∂ ψ J + ∂ H J ∂ ψ H (A1)
where ∂ ψ J = ∂J ∂ψ . ∂ ψ H requires the linearization of the hydrodynamic model, and ψ is a parametric representation of the bathymetry.

A.2 Gradient of the Wave Height with respect to the Bathymetry

This section is devoted to the calculation of the gradient of the wave height H, given by [START_REF] Mohammadi | Extreme scenarios for the evolution of a soft bed interacting with a fluid using the value at risk of the bed characteristics[END_REF], with regards to the seabed elevation ψ and denoted ∂ ψ H.

Being as h = h 0 -ψ, the derivation of the third line of [START_REF] Mohammadi | Extreme scenarios for the evolution of a soft bed interacting with a fluid using the value at risk of the bed characteristics[END_REF] with regards to ψ is immediate. The calculation of the gradient of the first line of ( 7) is analogous to that of the second. It remains to differentiate the second line of ( 7) with regards to ψ. Observing that the chain rule yields for all x, t ∈ Ω S × [0, T ] with x ≥ d w ,

∂ ψ H(x, t) = H w 0 (x, t)∂ ψ K S (x, t) + ∂ ψ H w 0 (x, t)K S (x, t), (A2)

and that the term ∂ ψ H w 0 (x, t) can be determined iteratively, using ∂ ψ H 0 = 0, it remains to determine ∂ ψ K S (x, t). Injecting the definitions of n, C and C g , given in (4), yields K S = tanh(kh) 1 + 2kh sinh(2kh)

1/2 . ( A3
)
For the sake of simplicity, let U = tanh(kh) 1 + 2kh sinh(2kh) and X = kh.

Equation (A3) becomes

∂ ψ K S = - 1 2 U -3/2 ∂ ψ U, (A4)
and we have

∂ ψ U = ∂ ψ X 2 cosh 2 (X) -X sinh(2X) cosh 4 (X) , (

Fig. 1

 1 Fig. 1 Diagram of a cross-shore profile in the case of an experimental flume.

Fig. 2

 2 Fig. 2 Comparison of mean wave height over a storm simulation. The green points correspond to the mean wave height provided by the gauges of the flume experiment. The mean wave height determined by Opti-Morph (red) and XBeach (blue) also appears. The nonzero wave height beyond the shoreline as presented by XBeach is due to wave set-up, which Opti-Morph does not handle.

Fig. 3 A

 3 Fig.3A. Results of the numerical simulation calculated over the initial seabed (gray) using the XBeach morphodynamic module (blue) and the Opti-Morph model (red). These are compared with the experimental data acquired during the COPTER project (green). The mean water level is denoted MWL and is set at 0.56 m. B. Zoomed in view of the sandbar, located between 6 m and 16 m. C. Zoomed in view of the plateau, located between 16 m and 24 m. D. Zoomed in view at the shoreline, located between 24 m and 32 m. E. Robustness analysis of the mobility parameter Υ. The reference profile is depicted in black. The orange (resp. light blue) profile is the result of a 50% increase (resp. decrease) in mobility, with all other parameters remaining the same. F. Robustness analysis of the maximal sand slope parameter M slope . The reference profile is depicted in black. The orange (resp. light blue) profile is the result of a 50% increase (resp. decrease) of M slope , with all other parameters remaining the same.

Fig. 4 a

 4 Fig. 4 a) Errors E L 2 (green) obtained by simulations of 10 different time steps compared to the reference simulation corresponding to a time step of 0.05 s. First order convergence (yellow). b)l Errors E L 2 (red) obtained by simulations of 10 different spatial steps compared to the reference simulation corresponding to a spatial step of 0.0002 m. First order convergence (yellow).

  A5) with ∂ ψ X = h∂ ψ k + k∂ ψ h = h∂ ψ k -k. Moreover, differentiating both sides of the dispersion equation (1) by ψ gives ∂ ψ k = k 2 cosh(kh) sinh(kh) + kh . (A6)Combining (A4),(A5), and (A6), we obtain ∂ ψ K S , and therefore ∂ ψ H.

drawn regarding the 6-day simulations, where the cumulative energy density of all three is equal to 0.177 J.m -2 .

Conclusions

Opti-Morph shows potential as a fast, robust, and low complexity morphodynamic model involving only two hyper-parameters. Despite using a basic hydrodynamic model for the description of the complex coupling of hydrodynamic and morphodynamic processes, we can nevertheless observe that a numerical model based on an optimization theory works effectively, with comparable results to a state of the art hydro-morphodynamic model requiring the tuning of dozens of hyper-parameters. Long-term simulations also show typical morphodynamic behavior, with the tendency of the seabed to evolve towards an equilibrium state. These results demonstrate the tremendous potential of Opti-Morph, a constrained energy minimization morphodynamic model.

Declarations

Availability of data and material

All data, models, and code generated or used during the study appear in the submitted article.

Conflict of interest

The authors declare that they have no conflict of interest.

Acknowledgements

Appendix A Mathematical Developments

In this section, we detail some of the mathematical results needed in the implementation of the Opti-Morph model, specifically the calculation of the gradient of the cost function J (Eq. ( 8)) with regard to the bathymetry ψ, which in turn requires the gradient of the wave height function (Eq. ( 7)) with regard to ψ. With the current choice of hydrodynamic model, this can be achieved analytically. With more sophisticated hydrodynamic models this is not always possible. In these cases, if the source code of the model is available, the calculation of the gradient can be performed using automatic differentiation of programs [START_REF] Hascoet | Tapenade user's guide[END_REF][START_REF] Griewank | Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation[END_REF] directly providing a computer program for the gradient.

A.1 Gradient of the Cost Function with respect to the Bathymetry

Opti-Morph requires the evaluation of gradient of the functional J with respect to the bathymetry ψ, denoted ∇ ψ J. For a general functional of the form J(ψ(x), H(ψ(x))) involving dependencies with respect to the bathymetry and