Ganesh Jawahar

Benjamin Muller

Amal Fethi

Louis Martin

Éric De

La Clergerie

Benoît Sagot

Djamé Seddah

Joakim Nivre

Mitchell Abrams

Željko Agić

Lars Ahrenberg

Lene Antonsen

Maria Jesus Aranzabe

Gashaw Arutie

Masayuki Asahara

Luma Ateyah

Mohammed Attia

Aitziber Atutxa

Liesbeth Augustinus

Elena Badmaeva

Miguel Ballesteros

Esha Banerjee

Sebastian Bank

Verginica Barbu Mititelu

John Bauer

Sandra Bellato

Kepa Bengoetxea

Riyaz Ahmad Bhat

Erica Biagetti

Eckhard Bick

Rogier Blokland

Victoria Bobicev

Carl Börstell

Cristina Bosco

Gosse Bouma

Sam Bowman

Adriane Boyd

Aljoscha Burchardt

Marie Candito

Bernard Caron

Gülşen Gauthier Caron

Cebiroğlu Eryiğit

Giuseppe G A Celano

Savas Cetin

Fabricio Chalub

Jinho Choi

Yongseok Cho

Jayeol Chun

Silvie Cinková

Aurélie Collomb

Çağrı Çöl- Tekin

Miriam Connor

Marine Courtin

Elizabeth Davidson

Marie-Catherine De Marneffe

Valeria De Paiva

Arantza Diaz De Ilarraza

Carly Dick- Erson

Peter Dirix

Kaja Dobrovoljc

Timothy Dozat

Kira Droganova

Puneet Dwivedi

Marhaba Eli

Ali Elkahky

Binyam Ephrem

Er- Javec Tomaž

Aline Etienne

Richárd Farkas

Hector Fernandez Alcalde

Jennifer Foster

Cláudia Freitas

Katarína Gajdošová

Daniel Galbraith

Marcos Garcia

Moa Gärdenfors

Kim Gerdes

Filip Ginter

Iakes Goenaga

Koldo Gojenola

Memduh Gökırmak

Yoav Goldberg

Xavier Gómez Guinovart

Berta Gonzáles Saavedra

Ma- Tias Grioni

Normunds Grūzītis

Bruno Guillaume

Céline Guillot-Barbance

Nizar Habash

Jan Hajič

Linh Hà Mỹ

Na-Rae Han

Kim Harris

Dag Haug

Barbora Hladká

Jaroslava Hlaváčová

Florinel Hociung

Petter Hohle

Jena Hwang

Radu Ion

Elena Irimia

Tomáš Jelínek

Anders Johannsen

Fredrik Jørgensen

Hüner Kaşıkara

Sylvain Kahane

Hiroshi Kanayama

Jenna Kanerva

Tolga Kayadelen

Václava Ket- Tnerová

Jesse Kirchner

Natalia Kotsyba

Simon Krek

Sookyoung Kwak

Veronika Laippala

Lorenzo Lambertino

Tatiana Lando

Septina Dian Larasati

Alexei Lavrentiev

Phương John Lee

Lê Hồng

Alessandro Lenci

Saran Lertpradit

Cheuk Herman Leung

Ying Li

Josie Li

Keying Li

Kyungtae Lim

Nikola Ljubešić

Olga Logi- Nova

Olga Lyashevskaya

Teresa Lynn

Vivien Macketanz

Aibek Makazhanov

Michael Mandl

Christopher Manning

Ruli Manurung

Cătălina Mărănduc

David Mareček

Héctor Katrin Marheinecke

Martínez Alonso

André Martins

Jan Mašek

Yuji Matsumoto

Ryan Mcdonald

Gus- Tavo Mendonça

Niko Miekka

Anna Missilä

Cătălin Mititelu

Yusuke Miyao

Simonetta Mon- Temagni

Amir More

Laura Moreno Romero

Shinsuke Mori

Bjartur Mortensen

Bohdan Moskalevskyi

Kadri Muischnek

Yugo Murawaki

Kaili Müürisep

Pinkey Nainwani

Juan Ignacio

Navarro Horñiacek

Anna Nedoluzhko

Gunta Nešpore-Bērzkalne

Nguyễn Lương

Huyền Thị

Thị Nguyễn

Vitaly Minh

Rattima Nikolaev

Hanna Niti- Saroj

Stina Nurmi

Adédayọstling Robert Ojala

Lilja Olúòkun

Niko Øvrelid

Elena Partanen

Marco Pascual

Agnieszka Passarotti

Siyao Patejuk

Cenel-Augusto Peng

Guy Perez

Slav Perrier

Jussi Petrov

Emily Piitulainen

Barbara Pitler

Thierry Plank

Mar- Tin Poibeau

Lauma Popel

Sophie Pretkalniņa

Prokopis Prévost

Adam Prokopidis

Tiina Przepiórkowski

Sampo Puolakainen

Andriela Pyysalo

Alexandre Rääbis

Loganathan Rademaker

Taraka Ramasamy

Carlos Rama

Vinit Ramisch

Livy Ravishankar

Siva Real

Georg Reddy

Michael Rehm

Larissa Rießler

Laura Rinaldi

Luisa Rituma

Mykhailo Rocha

Rudolf Romanenko

Da- Vide Rosa

Valentin Rovati

Olga Roca

Shoval Rudina

Shadi Sadde

Tanja Saleh

Stephanie Samardžić

Manuela Samson

Baiba Sanguinetti

Yanin Saulīte

Nathan Sawanakunanon

Sebas- Tian Schneider

Djamé Schuster

Wolfgang Seddah

Mojgan Seeker

Mo Seraji

Atsuko Shen

Muh Shimada

Dmitry Shohibussirri

Natalia Sichinava

Maria Silveira

Radu Simi

Katalin Simionescu

Mária Simkó

Kiril Šimková

Aaron Simov

Isabela Smith

Antonio Soares-Bastos

Milan Stella

Jana Straka

Alane Strnadová

Umut Suhr

Zsolt Sulubacak

Dima Szántó

Yuta Taji

Takaaki Takahashi

Isabelle Tanaka

Trond Tellier

Anna Trosterud

Reut Trukhina

Francis Tsarfaty

Sumire Tyers

Zdeňka Uematsu

Larraitz Urešová

Hans Uria

Sowmya Uszkoreit

Daniel Vajjala

Van Niekerk

ELMoLex: Connecting ELMo and Lexicon features for Dependency Parsing

 which provide contextdependent word representations; we utilize disambiguated, embedded, morphosyntactic features from lexicons (Sagot, 2018), which complements the existing feature set. Henceforth, we call our system 'ELMoLex'. In addition to incorporating character embeddings, ELMoLex leverage pre-trained word vectors, ELMo and morphosyntactic features (whenever available) to correctly handle rare or unknown words which are prevalent in languages with complex morphology. EL-MoLex 1 ranked 11 th by Labeled Attachment Score metric (70.64%), Morphology-aware LAS metric (55.74%) and ranked 9 th by Bilexical dependency metric (60.70%). In an extrinsic evaluation setup, ELMoLex ranked 7 th for Event Extraction, Negation Resolution tasks and 11 th for Opinion Analysis task by F1 score.

BiAF y (arc) y (rel) x (GP) . . .

Introduction

The goal of this paper is to describe ELMoLex, the parsing system submitted by our team 'ParisNLP' to the CoNLL 2018 Shared Task on parsing from raw 1 Code to reproduce our tagging and parsing experiments is publicly accessible at https://github.com/ BenjaminMullerGit/NeuroTaggerLex and https: //github.com/ganeshjawahar/ELMoLex respectively. text to Universal Dependencies (Zeman et al., 2018). The backbone of ELMoLex is the BiAF parser [START_REF] Dozat | Deep Biaffine Attention for Neural Dependency Parsing[END_REF] consisting of a large, well-tuned network that generates word representations, which are then fed to an effective, biaffine classifier to predict the head of each modifier token and the class of the edge connecting these tokens. In their follow-up work [START_REF] Dozat | Stanford's Graph-based Neural Dependency Parser at the CoNLL 2017 Shared Task[END_REF], the authors further enrich the parser by utilizing character embeddings for generating word representations which could help in generalizing to rare and unknown words (also called Out Of Vocabulary (OOV) words). They also train their own taggers using a similar architecture and use the resulting Part of Speech (PoS) tags for training the parser in an effort to leverage the potential benefits in PoS quality over off-the-shelf taggers.

We identify two potential shortcomings of the BiAF parser. The first problem is the context independence of the word embedding layer of the parser: the meaning of a word varies across linguistic contexts, which could be hard to infer automatically for smaller treebanks (especially) due to lack of data. To handle this bottleneck, we propose to use Embeddings from Language Model (ELMo) features [START_REF] Peters | Deep contextualized word representations[END_REF] which are context dependent (function of the entire input sentence) and obtained from the linear combination of several layers of a pre-trained BiLSTM-LM2 . The second problem is the linguistic naivety3 of the character embeddings: they can generalize over relevant sub-parts of each word such as prefixes or suffixes, which can be problematic for unknown words which do not always follow such generalizations [START_REF] Sagot | Improving neural tagging with lexical information[END_REF]. We attempt to lift this burden by resorting to external lexicons4 , which provides information for both word with an irregular morphology and word not present in the training data, without any quantitative distinction between relevant and less relevant information. To tap the information from the morphological features (such as gender, tense, mood, etc.) for each word present in the lexicon efficiently, we propose to embed the features and disambiguate them contextually with the help of attention [START_REF] Bahdanau | Neural machine translation by jointly learning to align and translate[END_REF], before combining them for the focal word.

We showcase the potential of ELMoLex in parsing 82 treebanks provided by the shared task. ELMoLex ranked 11 th by Labeled Attachment Score (LAS) metric (70.64%), Morphology-aware LAS (MLAS) metric (55.74%) and ranked 9 th by BiLEXical dependency (BLEX) metric (60.70%). We perform ablation and training time studies to have a deeper understanding of ELMoLex. In an extrinsic evaluation setup [START_REF] Fares | The 2018 Shared Task on Extrinsic Parser Evaluation. On the downstream utility of English Universal Dependency parsers[END_REF], ELMoLex ranked 7 th for Event Extraction, Negation Resolution tasks and 11 th for Opinion Analysis task by F1 score. On an average, ELMoLex ranked 8 th with a F1 score of 55.48%.

ELMoLex

The model architecture of ELMoLex, which uses BiAF parser [START_REF] Dozat | Deep Biaffine Attention for Neural Dependency Parsing[END_REF] (which in turn is based on [START_REF] Kiperwasser | Simple and accurate dependency parsing using bidirectional LSTM feature representations[END_REF]) as its backbone, is displayed in Figure 1. For our shared task submission, we assume tokenization and segmentation is already done5 ; we henceforth train ELMoLex on gold tokens and PoS tags provided by UDPipe [START_REF] Straka | UDPipe: trainable pipeline for processing CoNLL-U files performing tokenization, morphological analysis, POS tagging and parsing[END_REF]. We evaluate our model using the segmentation and PoS tags provided by UD-Pipe, except for certain languages where we use the tokens and PoS tag predicted by our own tokenizer and taggers (as respectively explained in Section 2.6 and 2.7)6 respectively.

Backbone parser

ELMoLex uses the BiAF parser [START_REF] Dozat | Deep Biaffine Attention for Neural Dependency Parsing[END_REF], a state-of-the-art graph-based parser, as its backbone. BiAF parser consumes a sequence of tokens and their PoS tags, which is fed through a multilayer BiLSTM network. The output state of the final LSTM layer is then fed through four separate ReLU layers, producing four specialized vector representations: first for the word as a modifier seeking its head; second for the word as a head seeking all its modifiers; third for the word as a modifier deciding on its label; and lastly for the word as head deciding on the labels of its modifiers. These vectors become the input to two biaffine classifiers: one computes a score for each token pair, with the highest score for a given token indicating that token's most probable head; the other computes a score for each label for a given token/head pair, with the highest score representing the most probable label for the arc from the head to the Figure 2: Architecture of embedding model used by ELMoLex for the word 'admonish'. v GP (ELM o) and v GP (lex) (red ellipses) are our major contributions. Arrows indicate structural dependence, but not necessarily trainable parameters. modifier. We refer the readers to [START_REF] Dozat | Deep Biaffine Attention for Neural Dependency Parsing[END_REF] for further details.

Formally, the BiAF parser consumes a sequence of n word embeddings (v

GP (word) 1 , . . . ,v GP (word) n
) and n tag embeddings (v

GP (tag) 1 , . . . ,v GP (tag) n
) as input, which can be written as:

x GP i = v GP (word) i ⊕ v GP (tag) i , v GP (word) i = v GP (token) i + v GP (w2v) i + v GP (char) i , v GP (tag) i = v GP (U P oS) i + v GP (XP oS) i .
(1)

In the equation 1, v GP (token) i , v GP (w2v) i , v GP (char) i
represent the holistic frequent word embedding, pretrained word embedding (fixed) and LSTM based character-level embeddings respectively, whereas the Universal PoS (UPoS) and language-specific (XPoS) tag embeddings are represented by v GP (U P oS) i and v GP (XP oS) i respectively. Note that '+' denotes element-wise addition operator, while '⊕' denotes concatenation operator. ) which provide context-dependent word representation (as discussed in Section 2.2); we utilize disambiguated, embedded, morphosyntactic features from lexicons (v

GP (lex) i
), which provide information that is especially relevant for word with an irregular morphology [START_REF] Sagot | Improving neural tagging with lexical information[END_REF], thereby complementing the existing feature set (as discussed in Section 2.3). Incorporating them, equation 1 now becomes:

x GP i = v GP (word) i ⊕ v GP (morph) i , v GP (word) i = v GP (f air) i ⊕ v GP (char) i ⊕ v GP (ELM o) i , v GP (morph) i = v GP (P oS) i ⊕ v GP (lex) i , v GP (P oS) i = v GP (U P oS) i + v GP (XP oS) i .
(2)

In the equation 2, v GP (f air) i and v GP (char) i represent the learnable embeddings that are associated with frequent words in the vocabulary (pre-initialized from FAIR word vectors [START_REF] Bojanowski | Enriching word vectors with subword information[END_REF]) and convolution-based character-level embeddings [START_REF] Ma | Stackpointer networks for dependency parsing[END_REF] 7 respectively. Apart from these changes in the embedding layer, we replace the decoding strategy (tree construction from the predicted graph) of our parser from greedy decoding (used by BiAF parser) to Chu-Liu-Edmonds algorithm [START_REF] Chu | On the shortest arborescence of a directed graph[END_REF], which further improves performance during evaluation.

ELMo features

In natural language, the meaning of a word changes when the underlying linguistic context changes. This fact is not captured by static word embeddings due to their context independence. Employing a deep, contextualized word representation, ELMo [START_REF] Peters | Deep contextualized word representations[END_REF], which is a function of the entire sentence, yields promising result for several downstream tasks such as Question Answering, Textual Entailment and Sentiment Analysis. We attempt to test whether this hypothesis holds for dependency parsing. This is an interesting experiment as the authors of ELMo obtain larger improvements for tasks with small train set (sample efficient), indicating that smaller treebanks deprived of useful information could potentially enjoy good improvements. 8 The backbone of ELMo is a BiLSTM-based neural Language Model (BiLSTM-LM), which is trained on a large raw corpus. We attempt to explore in this work whether we can train an indomain version of a BiLSTM-LM effectively using the available training data. The main challenge to accomplish this task is to learn transferable features in the absence of abundant raw data. Inspired by the authors of BiAF who use a large, well-tuned network to create a high performing graph parser, we implement a large BiLSTM-LM network (independent of the ELMoLex parser) which is highly regularized to prevent data overfitting and able to learn useful features. Our BiLSTM-LM consumes both the word and tag embedding as input, which can be formally written as:

x LM i = v LM (word) i ⊕ v LM (U P oS) i , v LM (word) i = v LM (f air) i ⊕ v LM (char) i . (3) In equation 3, the notations v LM (f air) i , v LM (char) i and v LM (U P oS) i are the counterparts of v GP (f air) i , v GP (char) i and v GP (U P oS) i
respectively. Note that ELMo, as proposed in [START_REF] Peters | Deep contextualized word representations[END_REF], builds only on character embeddings, automatically inferring the PoS information in the lower layers of the LSTM network. Since we have less training data to work with, we feed the PoS information explicitly which helps in easening the optimization process of our BiLSTM-LM network. Given a sequence of n words, x LM 1 , . . . , x LM n , BiLSTM-LM learns by maximizing the log likelihood of forward LSTM and backward LSTM directions, which can be defined as:

n ∑ i=1 (log Pr(x LM i |x LM 1 , . . . , x LM i-1 ; Θx; -→ Θ LST M , -→ Θ s)) + (log Pr(x LM i |x LM i+1 , . . . , x LM n ; Θx; ← - Θ LST M , ← - Θ s)) . (4)
We share the word embedding layer (Θ x ) of both LSTMs and learn the rest of the parameters independently. Unlike [START_REF] Peters | Deep contextualized word representations[END_REF], we do not tie the Softmax layer (Θ s ) in both the LSTM directions. Essentially, ELMo features are computed by a taskspecific linear combination of the BiLSTM-LM's intermediate layer representations. If L represents the number of layers in BiLSTM-LM, ELMo computes a 8 ELMoLex rank 8th overall for small treebanks by LAS metric.

set of 2L + 1 representations:

R k = {x LM k , ← - h LM k,j , -→ h LM k,j |j = 1, . . . , L} = {h LM k,j |j = 1, . . . , L}, (5)
where h LM k,0 is the word embedding layer (Equation 3) and

h LM k,j = ← - h LM k,j ⊕ -→ h LM k,j
, for each BiLSTM layer. The authors of ELMo [START_REF] Peters | Deep contextualized word representations[END_REF] show that different layers of BiLSTM-LM carry different types of information: lower-level LSTM states capture syntactic aspects (e.g., they can be used for PoS tagging); higher-level LSTM states model context-dependent aspects of word meaning (e.g., they can be used for word sense disambiguation); This observation is exploited by ELMoLex which can smartly select among all of these signals the useful information for dependency parsing. Thus, ELMo features for a word are computed by attending (softly) to the informative layers in R, as follows:

v GP (ELM o) i = E(R k ; Θ elmo ) = γ elmo L ∑ j=0 s elmo j h LM k,j . (6)
In equation 6, s elmo corresponds to the softmaxnormalized weights, while γ elmo lets ELMoLex to scale the entire ELMo vector.

Lexicon features

Character-level models depend on the internal character-level make-up of a word. They exploit the relevant sub-parts of a word such as suffixes or prefixes to generate word representations. They can generalize to unknown words if these unknown words follow such generalizations. Otherwise, they fail to add any improvement [START_REF] Sagot | Improving neural tagging with lexical information[END_REF] and we may need to look for other sources to complement the information provided by characterlevel embeddings. We term this problem as linguistic naivety.

ELMoLex taps into the large inventory of morphological features (gender, number, case, tense, mood, person, etc.) provided by external resources, namely the UDLexicons [START_REF] Sagot | A multilingual collection of conll-u-compatible morphological lexicons[END_REF] lexicon collection, which cover words with an irregular morphology as well as words not present in the training data. Essentially, these lexicons consist of ⟨word, UPoS, morphological features⟩ triplets, which we query using ⟨word, UPoS⟩ pair resulting in one or more hits. When we attempt to integrate the information from these hits, we face the challenge of disambiguation as not all the morphological features returned by the query are relevant to the focal ⟨word, UPoS⟩ pair. ELMoLex relies on attention mechanism [START_REF] Bahdanau | Neural machine translation by jointly learning to align and translate[END_REF] to select the relevant morphological features, thereby having the capability to handle noisy or irrelevant features by paying no attention. Put formally, given a sequence of m morphological feature embeddings (v

GP (mf ) mf 1 , …, v GP (mf ) mfm
) for a word i, the lexicon-based embedding for the word (v

GP (lex) i
) can be computed as follows:

9 v GP (lex) i = m ∑ j=1 s lex mf j v GP (mf ) mf j . ( 7
)
In equation 7, s lex mf j corresponds to the softmaxnormalized weight which is a learnable parameter for each available morphological feature (in this case, it is mf j ). The general idea to perform a weighted sum to extract relevant features has been previously studied in the context of sequence labeling [START_REF] Rei | Attending to characters in neural sequence labeling models[END_REF] for integrating word and character level features. Combining the distributional knowledge of words along with the semantic lexicons has been extensively studied for estimating high quality word vectors, also referred to as 'retrofitting' in literature [START_REF] Faruqui | Retrofitting word vectors to semantic lexicons[END_REF].

Delexicalized Parsing

We perform delexicalized "language family" parsing for treebanks with less than 50 or no train sentences (as shown in Table 1). The delexicalized version of ELMoLex throws away word-level information such as v GP (word) and v GP (char) and works with the rest.

The source treebanks are concatenated to form one large treebank, which is then used to train the delexicalized parser for the corresponding target treebank. In case of "mixed model", we concatenate at most 300 sentences from each treebank to create the training data.

Handling OOV words

Out Of Vocabulary (OOV) word problem is prevalent in languages with rich morphology and an accurate parser should come up with smart techniques to perform better than substituting a learned unknown vocabulary token ('UNK') during evaluation. To circumvent this problem, ELMoLex relies on four signals from the proposed embedding layer:

• v GP (f air) : If an OOV word is present in the FAIR word vectors, ELMoLex directly substitute the word embedding without any transformation. 10 If OOV word is absent, we resort to using 'UNK' token.

• v GP (ELM o) : For an OOV word, the ELMo layer of ELMoLex computes the context-dependent word representation based on the other vocabulary words present in the focal sentence.

• v GP (char) : Character-level embedding layer of ELMoLex computes the representation based on the known characters extracted from the OOV word naturally.

• v GP (lex) : If an OOV word is present in the external lexicon, ELMoLex queries with the ⟨word, PoS⟩ pair and computes the representation based on the known set of morphological features.

Neural Tagger with embedded lexicon

As described in [START_REF] Dozat | Stanford's Graph-based Neural Dependency Parser at the CoNLL 2017 Shared Task[END_REF], the BiAF model benefits from Part-of-Speech (PoS) inputs only if their accuracy is high enough. Our idea was therefore to design an accurate tagger in order to improve the performance of the parser. Moreover, the shared task allows the use of external resources such as lexicons. A lexicon is simply a collection of possibilities in terms of PoS and morphological features usually provided for a large amount of words. In the context of neural taggers, an external lexicon can be seen as an external memory that can be useful in two ways:

• For making training faster. At initialization, for a given token, all possible PoS tags are equiprobable of being predicted by the network. The model only learns from the example it sees. By providing the model with a constrained set of possible tags as input features we can expect the training process to be faster.

• For helping the model with OOV tokens. Indeed, the lexicon provides information -potentially complementary to the character based representation -on OOV tokens that could be useful at inference.

Generally speaking, this experience is interesting because it challenges the idea that neural models, if deep enough and trained on enough data, don't require external resources and can learn everything in an endto-end manner. As we will see for tagging, external pre-computed resources such as lexicons are of a great help.

The tagger we design is based on the neural tagging model with lexicon described in Sagot and Martínez Alonso (2017) and adapted using architectural insights from [START_REF] Dozat | Deep Biaffine Attention for Neural Dependency Parsing[END_REF]. In short, words are represented in three ways. The first part is a trainable word vector (initialized with the FAIR vectors described in [START_REF] Bojanowski | Enriching word vectors with subword information[END_REF]). The second part is a character-based representation either computed using 1-dimensional convolution or a recurrent LSTM cell. The third component is an n-hot encoded vector of the tags that appear in an external lexicon, possibly embedded in a continuous space. These three components are summed, providing a morphologically and lexically-enriched word representation. This vector is then fed to a two layer BiLSTM that encodes the sentence level context, followed by two heads, one for predicting UPoS and the other for predicting morphological features. Each head is composed of a dense layer followed by a softmax layer.

Specific Tokenization post-processing for Arabic

To improve the Arabic tokenizer, we noticed that tokenization is very error-prone wherein most of the errors come from wrong analysis of the letter .'و' Indeed, in Arabic, this letter (which is a coordinate conjunction, "and") is usually concatenated to the next word (e.g ' ,'وقطر "and Qata") but is sometimes just a part of the word (e.g ,'وافق' "he agrees"). This ambiguity confuses the UDPipe tokenizer. Our fix consists in splitting that letter from its word whenever UDPipe was unable to provide a proper UPoS tag. This simple fix led to a 0.7% improvement in word segmentation compared to the UDPipe baseline and led us to rank 4 th on Arabic in the final LAS metric.

Results

The implementation of ELMoLex as well as the neural tagger are based on the publicly available BiAF parser code provided by CMU [START_REF] Ma | Stackpointer networks for dependency parsing[END_REF]. Similar to [START_REF] Dozat | Stanford's Graph-based Neural Dependency Parser at the CoNLL 2017 Shared Task[END_REF], we use mostly the same set of hyper-parameters (as displayed in Appendix A), which makes ELMoLex robust across a wide variety of treebanks present in the shared task (Zeman et al., 2018). For treebanks with no development data, we perform a 5-fold cross validation to identify the average number of epochs taken to train each fold. By setting the maximum number of epochs to this average number, we then train ELMoLex on 90% of the training data and use the rest of the training data for selecting our best model. When we do not find external lexicon in UDLexicons [START_REF] Sagot | A multilingual collection of conll-u-compatible morphological lexicons[END_REF] for a given language, we skip the lexicon based features (v GP (lex) ) and work with the rest. ELMoLex ran for ∼26 hours on the TIRA virtual machine [START_REF] Potthast | Improving the reproducibility of PAN's shared tasks: Plagiarism detection, author identification, and author profiling[END_REF] sequentially, which can be trivially parallelized to run within two hours. Our shared task results are displayed in Appendix B.

Performance Analysis of the Tagger

Given the general architecture we presented in Section 2.6, we are able to test a few key questions: Is recurrent cell better suited at encoding word morphology compared to 1-D convolution layer ? Is embedding the lexical information into a continuous space useful for improving the performance ? And finally, is using an external lexicon always useful for better UPoS tagging ? We summarize our results as follows:

• Convolution layer works better than recurrent Table 3: Ablation study of ELMoLex. LAS dev. score along with training time (with 90% of the training data with the rest used for selecting the best model) in minutes is reported for selected treebanks. For NLM Init. and ELMoLex models, we report the time taken to train the parser (excluding the time taken to train the underlying BiLSTM-LM). All the reported models uses Chu-Liu-Edmonds algorithm [START_REF] Chu | On the shortest arborescence of a directed graph[END_REF] for constructing the final parse tree. • Leveraging an external lexicon helps the tagger for most of the languages, specifically for languages such as French (tested on f r_sequoia and f r_spoken), Greek, Danish, Hungarian and Swedish. The only language for which the lexicon did not help is Turkish.

• Using a continuous embedding layer for lexicon features always leads to better performance compare to straight n-hot encoding.

We now present in more details the performance of our model regarding these three dimensions. The results are reported on development datasets treated as a strict test set.

As we notice in Figure 3, using a convolution layer as a morphological embedding technique provides poorer results compared to a recurrent cell except for two cases: Chinese and Vietnamese. This suggests that the different morphology and tokenization complexity that we find for Europeans languages compared to Chinese and Vietnamese might well require different kind of embedding architectures. Intuitively, we could say that the character-wise sequential structure of the Europeans languages is better modeled by a recurrent cell, while a language like Chinese with overlaying phenomenons is better modeled by a convolution layer.

We now describe the impact of an external lexicon for UPoS tagging (Figure. 4). We present the results only for the datasets for which the RNN cell was providing the best results. We compare two architectures: the neural tagger using a recurrent cell for morphology with an external lexicon (embedded in a continuous space) and the same architecture without external lexicons. For all the treebanks (except Turkish), the lexicon helps the UPoS tagging performance.

The last component of the neural tagger we analyze is the input technique of the lexical information. We compare two techniques. The first one is the architecture described in [START_REF] Sagot | Improving neural tagging with lexical information[END_REF] for which the lexical information is feeded using a nhot encoded vector which is then concatenated with the other word representation vectors. The second one embeds in a continuous space the lexicon tags before averaging them providing a single vector that summarizes the lexical information of a given word. As we see in Figure 5, in all languages we experi- For most of the treebanks, we performed significantly above the UDPipe baseline for UPoS tagging. Our results are summarized in Table 2. Unfortunately, we reached these results too late before the deadline, and we did not get the time to retrain our parser on our own, predicted PoS tags. Therefore, at test time, we only used the predicted tags for treebanks for which we were confident that our predicted tags would help the parser. It resulted in using our system for four treebanks: el_gdt, hu_szeged, sv_lines and tr_imst. In Appendix C,you will find the performance of our Neural tagger and our ELMoLex parser trained on our predicted tags for which we were able to retrain the models after the system submission deadline.

Ablation Study of the Parser

To unpack the benefits obtained from each of our contributions, we perform ablation studies with the following variants of ELMoLex: ELMoLex without ELMo and lexicon features, which is effectively the vanilla BiAF model [START_REF] Ma | Stackpointer networks for dependency parsing[END_REF]; initializing the BiLSTM layer of the vanilla model with that of a pre-trained BiLSTM-LM (NLM Init.) 11 ; ELMoLex with the ELMo features only (ELMo); ELMoLex with the lexicon features only (Lex); ELMoLex with both ELMo and Lexicon features (full version); The models used in the ablation study are trained on 90% of the training data, tuned on the remaining 10% and tested on the development set provided by the organizers.

The results are displayed in Table 3 12 . We make the following important observations: (1) Utilizing ei-11 Our final submission to the shared task did not have the NLM pre-initialization feature. 12 The training time reported in the table is an over-estimate, as it is captured when several parsers (at most four of them) are running together in a single GTX 1070 GPU. In summary, in contrast with our participation to the shared task last year (de La Clergerie et al. ( 2017) 13 , we decided to focus on neural models wherein we explored many architectures and ideas both for tagging and parsing. As a result we reached superior performance in the final LAS score compared to our last year submission. To illustrate this, we compare the results of our current submission with that of the last year for four treebanks (Table 4) and observe significant improvements in the LAS score.

Extrinsic Evaluation of the Parser

A "good" parser should not only perform well in the intrinsic metrics such as LAS, BLAS and BLEX, but should strengthen a real world NLP system by providing relevant syntactic features. To understand the impact of ELMoLex in a downstream NLP application, we participiated in the shared task on Extrinsic Parser Evaluation [START_REF] Fares | The 2018 Shared Task on Extrinsic Parser Evaluation. On the downstream utility of English Universal Dependency parsers[END_REF]. The goal of this task is to evaluate the parse trees predicted by ELMoLex on three downstream applications: biological event extraction, fine-grained opinion analysis, and negation resolution, for its usefulness. Since all the tasks are based on English language, we train ELMoLex on en_ewt treebank (which is the largest English treebank provided by the organizers (Zeman et al., 2018)) without changing the hyper-parameters (as disclosed in Appendix A). We refer the readers to [START_REF] Fares | The 2018 Shared Task on Extrinsic Parser Evaluation. On the downstream utility of English Universal Dependency parsers[END_REF] for details about each of the downstream task and the accompanying system (which takes the fea- Table 5: Results on the downstream tasks for our ELMoLex system trained on the en_ewt treebank with the corresponding difference from the best system enclosed in ellipses.

tures dervied from ELMoLex) used to solve the task. Our extrinsic evaluation results14 are displayed in Table 5. ELMoLex ranked 7 th for Event Extraction, Negation Resolution tasks and 11 th for Opinion Analysis task by F1 score. On an average, ELMoLex ranked 8 th with a F1 score of 55.48%.

Conclusion

We presented our parsing system, ELMoLex, which successfully integrates context-dependent ELMo and lexicon-based representations to overcome the context independency and linguistic naivety problem in the embedding layer of the BiAF model respectively.

We showed the analysis of our neural tagger, whose competitive performance in PoS estimation is capitalized by ELMoLex to achieve strong gains in parsing quality for four treebanks. We also performed an ablation study to understand the source of gains brought by ELMoLex. We evaluated ELMoLex on three downstream applications to understand its usefulness.

In the next step of our work, we plan to: (1) compare the performance in utilizing recurrent layer over the convolution layer for character embeddings (similar to our neural tagger experiment) which underlies our parser; (2) pursue the NLM initialization feature further to inspect if using it can enrich ELMoLex; (3) observe the performance when we augment the indomain train data for our BiLSTM-LM with massive raw data (such as Wikipedia); (4) train our parser and tagger jointly with the gold PoS tags; and (5) exploit lattice information [START_REF] Amir More | Conll-ul: Universal morphological lattices for universal dependency parsing[END_REF][START_REF] Buckman | Neural lattice language models[END_REF] which captures rich linguistic knowledge.

TIRA evaluation platform [START_REF] Potthast | Improving the reproducibility of PAN's shared tasks: Plagiarism detection, author identification, and author profiling[END_REF] to whom we owe a lot. We also thank José Carlos Rosales Núñez for useful discussions. This work was funded by the ANR projects ParSiTi (ANR-16-CE33-0021) and SoSweet (ANR15-CE38-0011-01).

Figure 1 :

 1 Figure 1: Architecture of ELMoLex which uses BiAF parser as its backbone. Arrows indicate structural dependence, but not necessarily trainable parameters.

  ELMoLex reformulates the input embedding layer of BiAF parser in a few ways (as illustrated in Figure 2): we utilize an indomain version of ELMo features (v GP (ELM o) i

Figure 3 :

 3 Figure 3: Comparing morphological embedding technique: RNN vs CNN

Figure 4 :

 4 Figure 4: Impact of external lexicons

Figure 5 :

 5 Figure 5: Impact of lexicon embedding technique

Table 1 :

 1 Treebanks to source the training data for Delexicalized Parsing of a given target treebank.

	Target	Source(s)
	hy_armtdp grc_perseus, grc_proiel
	kk_ktb	tr_imst, ug_udt
	hsb_ufal	mixed
	kmr_mg	fa_seraji
	bxr_bdt	tr_imst, ug_udt, ko_gsd, ko_kaist, ja_gsd
	pcm_nsc	mixed
	en_pud	en_ewt
	th_pud	mixed
	ja_modern ja_gsd
	br_keb	mixed
	fo_oft	da_ddt,	sv_talbanken,	sv_lines,
		no_nynorsklia, no_bokmaal, no_nynorsk
	fi_pud	fi_tdt
	sv_pud	sv_talbanken
	cs_pud	cs_pdt

Table 2 :

 2 UPoS F1 on Dev. datasets (used as test)

	Treebank	Neural Tagger UDPipe
	af_afribooms	95.33	95.53
	da_dadt	95.54	95.18
	el_gdt	95.48	94.80
	fr_sequoia	96.13	95.78
	fr_spoken	95.71	93.70
	hu_szeged	93.07	92.56
	sv_lines	95.27	94.37
	tr_imst	91.08	91.02
	vi_vtb	77.10	77.80
	zh_gsd	84.26	83.24

Table 4 :

 4 Comparing LAS scores for 2017 and 2018 participation of ParisNLP. The increase in the absolute LAS points are enclosed in the ellipses.

	Treebank ParisNLP-2017 ParisNLP-2018
	el_edt	82.25	86.83 (+4.58)
	hu_szeged	66.82	74.36 (+7.54)
	sv_lines	76.61	79.92 (+3.31)
	tr_imst	56.03	61.27 (+5.24)
	ther ELMo or Lexicon or both always outperform the
	BiAF model; (2) External lexicons brings in valuable
	information about OOV words and words with irregu-
	lar morphology, thereby outperforming BiAF (which
	relies for those cases on character embeddings only)
	by a large margin; (3) ELMo entails high train time
	due to the additional LSTM operation over the entire
	sentence, but exhibits strong performance over BiAF
	model which naturally leads us to combine it along
	with the lexicon information to create ELMoLex (our
	final submission system).

Due to lack of time, we could train BiLSTM-LMs on the treebank data only (indomain version). We leave it for future work to train the model on large raw corpora from each language, which we believe could further strengthen our parser.

 3 The term linguistic naivety was introduced by[START_REF] Matthews | Using Morphological Knowledge in Open-Vocabulary Neural Language Models[END_REF] to refer to the fact that character-based embeddings for a sentence must discover that words exist and are delimited by spaces (basic linguistic facts that are built in to the structure of word-based models). In our context, we use a different meaning of this term as the term corresponds to the word-level characterbased embeddings.

We use lexicon information for treebanks from 43 languages provided by UDLexicons[START_REF] Sagot | A multilingual collection of conll-u-compatible morphological lexicons[END_REF].

We explored a wide variety of tokenization and segmentation techniques in our last year submission (de La Clergerie et al., 2017). Our primary focus for this year is to explore novel neural network layers for both tagging and parsing.

 6 Due to lack of time, we could not train the taggers effectively for all the languages and use the predicted PoS from UDPipe for training our parser.

For obtaining character emebddings, we prefer convolution operation (introduced in dos Santos and Zadrozny (2014)) over LSTM (used by BiAF and introduced in Ballesteros et al. (2015)) as the former is parallelizable and efficient especially for large treebanks.

We experienced inferior results with other lexicon-based representations such as n-hot vector (having active value corresponding to the each of the morphological feature), unweighted average of morphological feature embeddings and morphological feature group based embedded attention.

Inspired by[START_REF] Mikolov | Exploiting similarities among languages for machine translation[END_REF], we experimented with a linear transformation of the FAIR word vectors to the trained word embedding space, which resulted in poor performance. This confirms the intuition that the learned word embedding space is a non-linear transformation of the pre-trained word vectors.

In our last year joint submission with Stanford[START_REF] Schuster | Paris and Stanford at EPE 2017: Downstream evaluation of graph-based dependency representations[END_REF], we evaluated different semantic dependencies representations and also compared different parsing strategies.

Acknowledgments

We thank the organizers and the data providers who made this shared task possible within the core Universal Dependencies framework (Nivre et al., 2016), namely the authors of the UD version 2.0 datasets (Nivre et al., 2018), the baseline UDPipe models (Straka et al., 2016), and the team behind the


Thierry Goudon
email: thierry.goudon@inria.fr

Laurent Monasse
email: laurent.monasse@inria.fr

Fokker-Planck approach of Ostwald ripening: simulation of a modified Lifschitz-Slyozov-Wagner system with a diffusive correction

Keywords: Coagulation-fragmentation, Ostwald ripening, Lifschitz-Slyozov equations, Fokker-Planck equation, Convergence to equilibrium, Well-balanced scheme, Math, Subject Classification, 82C26 82C22 82D60 65M08 65M08

HAL is a

Introduction

Ostwald ripening [START_REF] Ostwald | Studien über die Bildung und Umwandlung fester Körper[END_REF] is a physical mechanism that arises in many industrial, physical or biological processes, like alloys formations [START_REF] Lifschitz | Cinétique Physique[END_REF][START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF][START_REF] Sagalovich | Diffusive decomposition of solid solutions[END_REF], synthesis of quantum dots, emulsion dynamics (it is at the origin of the so-called "Ouzo effect" [START_REF] Sitnikova | Spontaneously formed transanethol/water/alcohol emulsions: Mechanism of formation and stability[END_REF]), protein polymerization [START_REF] Hingant | Contributions à la modélisation mathématique et numérique de problèmes issus de la biologie : applications aux prions et à la maladie d'Alzheimer[END_REF], etc. The mechanism can be described as an interaction between free particles, or monomers, and polymers, which can be seen as aggregates of monomers.

Roughly speaking, the dynamics is governed by attachment to or detachment from polymers of monomers.

Having an accurate model for such phenomena is particularly important in material sciences. The models involve particle size distributions, describing the dynamics of mass exchanges between particles. A major advance is due to Lifschitz-Slyozov [START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF][START_REF] Lifschitz | Cinétique Physique[END_REF] and Wagner [START_REF] Wagner | Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung)[END_REF], and their theory is widely considered as classical. An important prediction of the LSW theory is the emergence of a universal profile, toward which all initial distributions evolve with time (up to an appropriate total-mass-rescaling). Such a conclusion is particularly relevant since the large time behavior usually corresponds to the observable states.

However, the conclusions of the LSW theory are subject to controversy. We refer the reader to [START_REF] Baldan | Review progress in Ostwald ripening theories and their applications to nickel-base superalloys part I: Ostwald ripening theories[END_REF][START_REF] Barry | Trouble with Lifshitz, Slyozov and Wagner[END_REF] for various aspects of the debate. The discussion is motivated by the combined improvements of the experiments [START_REF] Barry | Trouble with Lifshitz, Slyozov and Wagner[END_REF] and of the numerical approaches [START_REF] Chen | The dynamics of transient Ostwald ripening[END_REF][START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifschitz-Slyozov system[END_REF][START_REF] Tiné | Simulations of the Lifshitz-Slyozov equations: the role of coagulation terms in the asymptotic behavior[END_REF]. Indeed, the LSW equations, that have the misleadingly simple form of a transport equation coupled to an integral constraint, present some hidden stiffness that makes the numerical problem singularly challenging: it is particularly difficult to capture the correct large time behavior, and one definitely needs dedicated schemes. Moreover, the mathematical analysis has also established the limitations of the LSW predictions [START_REF] Niethammer | Non-self-similar behavior in the LSW theory of Ostwald ripening[END_REF][START_REF] Niethammer | The LSW model for domain coarsening: Asymptotic behavior for conserved total mass[END_REF]. In particular, the large time behavior is highly sensitive to properties of the initial conditions.

There are several options to address these issues and to modify the original LSW model in order to clarify the asymptotic behavior of the solutions. The derivation of the LSW equations assumes that the distance between clusters remains large so that they do not interact directly; however, since the dynamics produces larger and larger clusters, this assumption becomes questionable as time becomes large. Therefore, the model can be completed by adding a coagulation term into the transport equation for macroscopic grains. This coagulation term is intended to restore a selection mechanism of the LSW profile [START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF]Section 3], an intuition that has been confirmed on numerical grounds [START_REF] Tiné | Simulations of the Lifshitz-Slyozov equations: the role of coagulation terms in the asymptotic behavior[END_REF]. Another approach goes back to a discrete version of the clusters population, where clusters are just seen as aggregates of a certain number of monomers. One is led to an infinite set of ODEs, the Becker-Döring system [START_REF] Becker | Kinetische Behandlung der Keimbildung in übersättigten Dämpfen[END_REF], which has a more standard asymptotic behavior than the LSW system: under a certain critical mass condition, an equilibrium state can be identified, which indeed attracts the solutions of the Cauchy problem [START_REF] Ball | Asymptotic behaviour of solutions to the Becker-Döring equations for arbitrary initial data[END_REF][START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF][START_REF] Carr | Asymptotic behavior of solutions to the coagulationfragmentation equations. II. Weak fragmentation[END_REF][START_REF] Cañizo | Trend to equilibrium for the Becker-Döring equations: An analogue of Cercignani's conjecture[END_REF][START_REF] Cañizo | Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations[END_REF][START_REF] Jabin | On the rate of convergence to equilibrium in the Becker-Döring equations[END_REF]. By rescaling appropriately the equations, the LSW system can be interpreted as the limit of the BD equations [START_REF] Collet | The Becker-Döring system and its Lifschitz-Slyozov limit[END_REF][START_REF] Penrose | The Becker-Döring equations at large times and their connection with the LSW theory of coarsening[END_REF]. Moreover, keeping a higher-order correction term in the equation, we obtain a modified LSW system, which involves a size-diffusion term. The remarkable fact, pointed out in [START_REF] Hariz | Une version modifiée du modele de Lifshitz-Slyozov : existence et unicité de la solution, simulation numérique[END_REF][START_REF] Hariz | A modified version of the Lifschitz-Slyozov model[END_REF], is that this correction restores formally the equilibrium properties of the discrete model.

In this paper, we address the question of the numerical treatment of the modified LSW system. We identify a structure which is common to that of the Fokker-Planck equation that arises in gas dynamics and plasma physics. Inspired by [START_REF] Jin | An AP scheme for the Fokker-Planck-Landau equation[END_REF], we design a numerical strategy of Finite Volume type which has the Well-Balance property: equilibria are automatically preserved, a crucial property for large-time simulations. The scheme allows us to numerically check the conjecture of the asymptotic trend to equilibrium.

The paper is organized as follows. In Section 2 we review the basic facts about the Becker-Döring and Lifshitz-Slyozov-Wagner equations. In Section 3, we detail the construction of the scheme, which relies on a time-splitting and the resolution of a symmetric linear diffusion system. We pay attention to discuss the stability issues that govern the choice of the time step. Section 4 is devoted to numerical experiments. In particular we compare the scheme with a numerical approach recently designed in [START_REF] Jourdan | An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach[END_REF], based on an implicit-explicit strategy, coupled with a high-order method for the discretization of the transport term of the equation. Our findings can be summarized as follows:

• On the numerical side, the new scheme finds and preserves the expected equilibria, it is less constrained by stability conditions and therefore it reaches large time simulations for a reduced numerical cost.

• On the modeling side, the simulation confirms the trend to equilibrium, with an exponential rate. These indications will be a motivation for further analytical investigations.

2 From Becker-Döring to Lifshitz-Slyozov-Wagner

Discrete viewpoint: the Becker-Döring system

We start by considering that polymers are simply aggregates of i monomers, with i ranging over N\{0, 1}. Let t → c i (t) stand for the concentration at time t of i-mers and t → c 1 (t) be the monomers concentration. Mass-action kinetics apply to the reactions

(i) + (1) (i + 1),
with coagulation rate a i and fragmentation rate b i+1 , respectively. We are thus led to the Becker-Döring equations [START_REF] Becker | Kinetische Behandlung der Keimbildung in übersättigten Dämpfen[END_REF]

d dt c i = J i-1 -J i for i ≥ 2, (1)
with, for i ≥ 1,

J i = a i c i c 1 -b i+1 c i+1 . ( 2
)
Monomers are involved in all the reactions, thus the evolution of c 1 is driven by an equation with a different form

d dt c 1 = -2J 1 - ∞ i=2 J i . ( 3
)
At least formally, solutions of (1)-(3) satisfy

d dt ∞ i=1 ic i (t) = 0
which can be cast as a mass conservation property

c 1 + ∞ i=2 ic i = ρ is constant. ( 4
)
Natural assumptions on the coefficients and the data can be summarized as follows

a i ≤ Ci, b i ≤ Ci, ∞ i=1 i 2 c i (0) < ∞,
which allows us to establish the existence and uniqueness of globally defined solutions [START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF]. Equilibrium solutions (m i ) i∈N\{0} of the Becker-Döring system can be identified by imposing that the associated fluxes vanish: J i = 0 leads to the recursion relation

m i+1 = a i b i+1 m 1 m i .
Finally, we obtain a family of equilibrium states, parametrized only by the monomers concentration m 1

m i = Q i m i 1 , Q i = a i-1 a i-2 ....a 1 b i b i-1 ....b 2 . ( 5
)
We find the value of this parameter by going back to the mass constraint

∞ i=1 iQ i m i 1 = ρ. ( 6
)
This relation makes a threshold appear, in connection to the notion of critical mass. Indeed, let µ crit be the radius of convergence of the entire series in (6) (that is, when the limit exists,

µ crit = lim i→∞ b i+1 a i ) and set ρ crit = ∞ i=1 iQ i µ i crit ∈ [0, ∞]
, the critical mass. The asymptotic behavior depends on whether or not the total mass ρ exceeds the critical mass ρ crit : when 0 ≤ ρ < ρ crit , a monotonicity argument shows that there exists a unique m 1 ∈ (0, µ crit ) such that ∞ i=1 iQ i m i 1 = ρ. This equilibrium is therefore a natural candidate for the asymptotic behavior of the solutions of the system with mass ρ, and the convergence to the equilibrium (in a strong sense) can indeed be justified [START_REF] Ball | Asymptotic behaviour of solutions to the Becker-Döring equations for arbitrary initial data[END_REF][START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF][START_REF] Carr | Asymptotic behavior of solutions to the coagulationfragmentation equations. II. Weak fragmentation[END_REF], with rates that depend on the technical assumptions on the coefficients and the initial data [START_REF] Cañizo | Trend to equilibrium for the Becker-Döring equations: An analogue of Cercignani's conjecture[END_REF][START_REF] Cañizo | Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations[END_REF][START_REF] Jabin | On the rate of convergence to equilibrium in the Becker-Döring equations[END_REF]. When ρ exceeds the critical mass, as time becomes large the excess mass ρ -ρ crit concentrates in larger and larger clusters, a phenomenon interpreted as a phase transition [START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF].

Continuous viewpoint: the Lifschitz-Slyozov-Wagner system

In this description, which dates back to [START_REF] Lifschitz | Cinétique Physique[END_REF][START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF][START_REF] Sagalovich | Diffusive decomposition of solid solutions[END_REF] and, independently, [START_REF] Wagner | Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung)[END_REF], roughly speaking, the polymers are assumed to have a "large" size compared to the monomers. However, they are not that large, so that direct interactions between clusters can still be neglected. The unknowns of the model are the size-density of polymers (t, x) → f (t, x) and the monomers concentration t → c(t). Given ξ 2 > ξ 1 ≥ 0, the integral ξ 2 ξ 1 f (t, x) dx gives the number of polymers which have a volume x ∈ (ξ 1 , ξ 2 ). On the same token, the first order moment ∞ 0 xf (t, x) dx defines the mass of the aggregates within the considered solution. The attachment and detachment processes are governed by principles of overall reduction of the interface energy, where volume effects, which favor growth, compete with surface effects, which favor dissolution. The description of these processes is embodied into two nonnegative coefficients a, b, that depend on the variable x ≥ 0. The evolution is thus driven by

∂ t f + ∂ x J = 0, J(t, x) = a(x)c(t) -b(x) f (t, x), (7)
coupled to the mass conservation constraint

c(t) + ∞ 0 xf (t, x) dx = ρ is constant. (8)
A standard assumption requests

a(0)ρ -b(0) ≤ 0. ( 9
)
It means that the characteristics curves associated to the field (t, x) → a(x)c(t) -b(x) are always pointing outward the domain {x ≥ 0}, and, under this assumption, the equation does not need a boundary condition at x = 0. Moreover, the function x → b(x) a(x) is usually assumed to be non increasing so that, at each time t a critical size

x crit (t) = b a -1 (c(t)
) can be identified where the growth rate vanishes: it describes the fact that larger particles grow at the expense of smaller particles, which are thus assigned to become still smaller. We refer the reader to [START_REF] Collet | On solutions of the Lifschitz-Slyozov model[END_REF][START_REF] Laurençot | Weak solutions to the Lifschitz-Slyozov-Wagner equation[END_REF][START_REF] Laurençot | The Lifschitz-Slyozov-Wagner equation with conserved total volume[END_REF][START_REF] Niethammer | On the initial-value problem in the Lifschitz-Slyozov-Wagner theory of Ostwald ripening[END_REF][START_REF] Niethammer | Well-posedness for measure transport in a family of nonlocal domain coarsening models[END_REF][START_REF] Niethammer | Global well-posedness for an inhomogeneous LSW model in unbounded domains[END_REF] for the analysis of the existence-uniqueness issues for ( 7)-( 8).

As pointed out in [START_REF] Penrose | The Becker-Döring equations at large times and their connection with the LSW theory of coarsening[END_REF], adopting a suitable rescaling, the system ( 7)-( 8) can be derived from the discrete model ( 1)- [START_REF] Ball | The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions[END_REF], see also the analysis in [START_REF] Collet | The Becker-Döring system and its Lifschitz-Slyozov limit[END_REF]. We equally refer the reader to [START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF] for a discussion on the case where [START_REF] Carr | Numerical approximation of a metastable system[END_REF] does not hold and a how the connection with the discrete modeling can help in finding a relevant boundary condition for ( 7)- [START_REF] Carr | Asymptotic behavior of solutions to the coagulationfragmentation equations. II. Weak fragmentation[END_REF] in this case, a situation which is relevant for applications in biology (assemblies of amyloid fibrils). Quite surprisingly, despite this natural connection with the Becker-Döring system, the large time behavior of the solutions of ( 7)-( 8) is completely different. Let us focus on the standard case where a(x) = x 1/3 and b(x) = 1. The asymptotic behavior can be summarized as

f (t, x) ∼ t→∞ A ρ (1 + t) 2 M K x 1 + t where A ρ = ρ ∞ 0 yM K (y) dy -1
is a normalizing constant related to mass conservation, and z → M K (z) is a profile (which has an explicit expression), which depends on a certain constant K ∈ R. Lifschitz and Slyozov [START_REF] Lifschitz | Cinétique Physique[END_REF][START_REF] Lifschitz | The kinetics of precipitation from supersaturated solid solutions[END_REF][START_REF] Sagalovich | Diffusive decomposition of solid solutions[END_REF] conjectured a selection process which defines a universal profile, associated to a specific value of the constant K = K LS . However, both numerical simulations [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifschitz-Slyozov system[END_REF][START_REF] Tiné | Simulations of the Lifshitz-Slyozov equations: the role of coagulation terms in the asymptotic behavior[END_REF] and mathematical analysis [START_REF] Niethammer | Non-self-similar behavior in the LSW theory of Ostwald ripening[END_REF][START_REF] Niethammer | The LSW model for domain coarsening: Asymptotic behavior for conserved total mass[END_REF] have shown that the selection of the profile is much more complicated: considering a data with compact support, the large time behavior selects K according to the shape of the initial data at the tip of the support ! Such a phenomenon is highly unusual and it has motivated the introduction of sharp notions to describe the behavior of a function at the end of its support. Further details and references about the Becker-Döring and Lifschitz-Slyozov systems can be found in the surveys [START_REF] Collet | Some modelling issues in the theory of fragmentation-coagulation systems[END_REF][START_REF] Hingant | Deterministic and stochastic Becker-Döring equations: Past and recent mathematical developments[END_REF][START_REF] Slemrod | The Becker-Döring Equations[END_REF].

A model with diffusive correction

The derivation of a continuous model from the discrete equations can be pushed forward: keeping the next terms in the asymptotic expansion leads to the following Fokker-Planck equation

       ∂ ∂t g + ∂ ∂x G(g; t, x) = 0, G(g; t, x) = (a(x)c(t) -b(x))g -ε ∂ ∂x a(x)c(t) + b(x) 2 g(t, x) , ( 10
)
where ε > 0 is the scaling parameter. Equation ( 10) is supplemented by the mass conservation law

c(t) + ∞ 0 xg(t, x) dx = ρ. ( 11
)
This model has been proposed in [START_REF] Hariz | Une version modifiée du modele de Lifshitz-Slyozov : existence et unicité de la solution, simulation numérique[END_REF][START_REF] Hariz | A modified version of the Lifschitz-Slyozov model[END_REF], see also [START_REF] Conlon | On a diffusive version of the Lifshitz-Slyozov-Wagner equation[END_REF], as a variant of the Lifschitz-Slyozov model. The asymptotic analysis further developed in [START_REF] Collet | The Becker-Döring system and its Lifschitz-Slyozov limit[END_REF] has also permitted to identify a relevant boundary condition for [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifschitz-Slyozov system[END_REF]; it reads

a(0)c(t) + b(0) g(t, 0) = α c(t) 2 . ( 12
)
The coefficient α > 0 is reminiscient to the specific role of the aggregation between monomers to form 2-mers in the scaling adopted in [START_REF] Collet | The Becker-Döring system and its Lifschitz-Slyozov limit[END_REF]: the coupling between macroscopic clusters and (microscopic) monomers introduced by this reaction is considered to be weak, see also [START_REF] Dadyburjor | Kinetics of Ostwald ripening[END_REF] for related observations. We obtain the following equilibrium solutions

m c (x) = m c (0) a(0)c + b(0) a(x)c + b(x) exp 2 ε x 0 a(y)c -b(y) a(y)c + b(y) dy ,
where [START_REF] Collet | Some modelling issues in the theory of fragmentation-coagulation systems[END_REF] gives

m c (0) = αc 2 a(0)c + b(0)
.

Therefore, we have a family of equilibrium states parametrized only by the monomers concentration c. The function [START_REF] Hariz | Une version modifiée du modele de Lifshitz-Slyozov : existence et unicité de la solution, simulation numérique[END_REF][START_REF] Hariz | A modified version of the Lifschitz-Slyozov model[END_REF]. Let us denote ρ s = M (c s ). Observing that c → M (c) is increasing, for any ρ ∈ (0, ρ s ) we can find a unique c ∈ (0, c s ) such that M (c) = ρ, which in turn defines uniquely the equilibrium with total mass ρ. We thus recover a similar discussion as for the Becker-Döring system.

M : c → c + ∞ 0 xm c (x) dx is well defined for c ∈ [0, c s ), with c s = lim sup x→∞ b(x) a(x) , see
As a matter of fact, let us consider the simplest case where the coefficients a, b are constant. The equilibrium reads

m c (x) = αc 2 ac + b exp 2 ε ac -b ac + b x .
The critical mass is non trivial when 0 < a < b: c s = b a . Performing an expansion of the formulas for the equilibrium states of both the Becker-Döring and (10) as c approaches the critical value, we are led to the following analogous formulae, see [START_REF] Hariz | Une version modifiée du modele de Lifshitz-Slyozov : existence et unicité de la solution, simulation numérique[END_REF][START_REF] Hariz | A modified version of the Lifschitz-Slyozov model[END_REF]

m i m 1 exp i am 1 b -1 , m c (x) c α 2a exp x ε ac b -1 .
From this discussion, we can therefore expect that the diffusive model [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifschitz-Slyozov system[END_REF] restores the asymptotic properties of the Becker-Döring system.

Numerical scheme

The numerical treatment of coagulation-fragmentation equations could be surprisingly challenging: many comments and further references for Becker-Döring and Lifschitz-Slyozov-Wagner equations can be found for instance in [START_REF] Carr | Numerical approximation of a metastable system[END_REF][START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifschitz-Slyozov system[END_REF][START_REF] Hingant | Deterministic and stochastic Becker-Döring equations: Past and recent mathematical developments[END_REF][START_REF] Tiné | Simulations of the Lifshitz-Slyozov equations: the role of coagulation terms in the asymptotic behavior[END_REF]. Here, we focus on the system with diffusion [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifschitz-Slyozov system[END_REF]. In fact, the simulation of such a model is addressed in [START_REF] Jourdan | An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach[END_REF], using a coupling with a discrete model to describe interactions with the smallest clusters, instead of a boundary condition like [START_REF] Collet | Some modelling issues in the theory of fragmentation-coagulation systems[END_REF]. The numerical approach developed in [START_REF] Jourdan | An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach[END_REF] is based on a high order implicit Finite Volume method with slope limiters on advection. We point out that, in this approach, a proper high-order accurate size discretization of the advection term turns out to be critical to obtain valuable results.

Here, we adopt a different viewpoint inspired from the numerical treatment of the Fokker-Planck equation in gas dynamics [START_REF] Jin | An AP scheme for the Fokker-Planck-Landau equation[END_REF]. We split the resolution of the equation into two steps: the first step consists in solving the linear diffusion problem on g with c fixed, while the second step involves the numerical integration of the ODE in c.

Diffusion problem

With c > 0 given, let us consider the operator

L c g = ∂ x -(ac -b)g + ε 2 ∂ x (ac + b)g .
L c g = ∂ x dM c ∂ x g M c with d(x) = ε 2 a(x)c + b(x) . Setting h = g √ Mc ,
it is convenient to define the operator

L c h = 1 √ M c L c (h M c ) = 1 √ M c ∂ x dM c ∂ x h √ M c ( 13
)
which is symmetric for the usual L 2 inner product. Note that an alternative choice is h = g Mc which yields the operator

L c h = 1 Mc ∂ x dM c ∂ x h .
In that case, h converges to a constant, which simplifies the interpretation of the outflow boundary conditions for large x. However, the operator L c is not symmetric and the numerical resolution of the system is more involved in this case.

Given a mesh size ∆x, the operator L c is discretized by using the following formula

1 ∆x M c,j d j+1/2 M c,j+1 M c,j h j+1 / M c,j+1 -h j / M c,j ∆x -d j-1/2 M c,j M c,j-1 h j / M c,j -h j-1 / M c,j-1 ∆x = 1 ∆x 2 d j+1/2 h j+1 - 1 M c,j d j+1/2 M c,j+1 + d j-1/2 M c,j-1 h j + d j-1/2 h j-1 . ( 14
)
For further purposes, let us denote by S the corresponding matrix.

Having at hand a distribution function g, we wish to update it by solving the linear problem

g n+1 -∆tL c g n+1 = g n . ( 15
)
By using an implicit scheme we expect to relax the stability condition where ∆t should be dominated by ∆x 2 , imposed by the diffusion operator. This is crucial when we wish to investigate the large time behavior of the equation. Numerically, the computation of ( 14) can be difficult due to the fact that M c (x) displays extremely large or small values compared to M c (0) = 1 for large x, with the exponential possibly exceeding the floating point capacities. In practice, we make use of the following expression for ( 14):

1 ∆x 2 d j+1/2 h j+1 -d j+1/2 M c,j+1 M c,j + d j-1/2 M c,j-1 M c,j h j + d j-1/2 h j-1 ,
where the quotients of M c are expressed as:

M c,j+1 M c,j = a(x j )c + b(x j ) a(x j+1 )c + b(x j+1 ) exp 2 ε x j+1
x j a(y)c -b(y) a(y)c + b(y) dy .

Evolution of c

At the continuous level, expressing the conservation of total mass ρ, integration by parts leads to

d dt c(t) = - d dt ∞ 0 xg(t, x) dx = ∞ 0 x∂ x G(g; t, x) dx = - ∞ 0 (a(x)c(t) -b(x))g(t, x) dx - ε 2 αc(t) 2 ,
by using the boundary condition [START_REF] Collet | Some modelling issues in the theory of fragmentation-coagulation systems[END_REF]. Applying a discrete analogue of the integrations by parts, assuming that g n and g n+1 verify (15), the conservation of discrete mass ρ n = c n + i≥0 x i g n i leads to:

c n+1 -c n = - i≥0 x i (g n+1 i -g n i )∆x = ∆t i≥0 x i (G n+1 i+1/2 -G n+1 i-1/2 ) (16) = -∆t i≥0 G n+1 i+1/2 ∆x. ( 17
)
Replacing with the expression of the numerical fluxes G, we find that defining

c n+1 = c n + ∆t i≥0 d i+1/2 M i+1/2 g n+1 i+1 M i+1 - g n+1 i M i ∆x (18)
exactly conserves total mass.

Splitting strategy

At time t n , we have at hand a monomers concentration c n and a vector (g n 1 , ..., g n J ) whose components are intended to be an approximation of g(t n , j∆x) for j ∈ {1, ..., J}. The right endpoint is chosen large enough so that the homogeneous Dirichlet boundary condition can be assumed to define g n J+1 . It allows us to construct

M n j = M c n (j∆x).
We set

h j = g n j √ M n j
. Then, we solve

(I -∆tS)h = h + β, ( 19
)
where β accounts for the boundary condition [START_REF] Collet | Some modelling issues in the theory of fragmentation-coagulation systems[END_REF], namely all the components of β vanish but β 1 = ∆td(0)

α|c n | 2
∆x 2 (a(0)c n +b(0)) . By construction the matrix S is symmetric and the linear system can be solved efficiently by the conjugate gradient algorithm. We set g n+1 j = h j M n j and update the concentration c n+1 using ( 18). Eventually it is worth pointing out that the scheme is, by construction, wellbalanced: if the initial data is an equilibrium state, then the numerical solution remains at equilibrium forever.

Choice of the time-step

As noted previously, the implicit scheme on diffusion ( 15) is unconditionnally stable. The only stability criterion to be satisfied is the non-negativity of the monomer concentration c, and we wonder whether the time step is constrained by the preservation of this property. Suppose that the monomer concentration c n at time t n is nonnegative. Using ( 18) and ( 15), a sufficient condition for c n+1 to be nonnegative is

∆t|B T (I -∆tL) -1 (g n + ∆t β)| ≤ c n , where L = M 1/2 SM -1/2 , β = β
∆t and B is the vector of general term

B i =        -d 1/2 M 1/2 M 0 if i = 0, 1 M i d i+1/2 M i+1/2 -d i-1/2 M i-1/2 otherwise.
Since the eigenvalues of L are nonpositive, a sufficient condition on ∆t is

∆t|D -1 B||D(g n + ∆t β)| ≤ c n ,
where D is any diagonal matrix. Using a triangle inequality, the scheme is stable as long as ∆t ≤

|Dg n | 2 + 4 c n |D β| |D -1 B| -|Dg n | 2|D β| (20)
In practice, a good choice for D is the diagonal matrix with d i as diagonal elements. Note that with this choice, we observe in practice that the boundary condition contribution [START_REF] Hariz | A modified version of the Lifschitz-Slyozov model[END_REF] reduces to the simpler expression in the absence of boundary conditions:

c n |D β| |D -1 B||Dg n | 2 is negligible, so that
∆t ≤ c n |D -1 B||Dg n | .
Remark that whereas condition [START_REF] Hariz | A modified version of the Lifschitz-Slyozov model[END_REF] appears to be sharp for the fast initial phase of the system dynamics, it is suboptimal in the established regime: the variation of c becomes very slow and the term |B T (I + ∆tL) -1 (g n + ∆t β)| tends to zero, while the norms |D -1 B| and |Dg n | tend to a positive constant.

Numerical results

Infinite critical mass

Let us start with a few comments about the equilibrium states. On Fig. 1 we plot the equilibrium functions for the coefficients

a(x) = x 1/2 , b(x) = 0.05 + 0.1 × x 2/3 , ( 21
)
and several values of c. We have set α = 1.3 and ε = 0.05. Note that in this case c s = +∞, ρ s = +∞ since fragmentation dominates for large clusters; the equilibrium functions are always admissible. We observe that the shape of the equilibrium is conserved. Note in particular that the function is not a simple bell shape, and there is a steep slope for small sizes. We observe that both the "support" (where the function takes significantly positive values) and the amplitude of the equilibrium varies a lot as a function of c (note that c varies in a quite tiny interval). This sensitivity can be a numerical difficulty, since small errors on the monomers concentration can produce a large error on the equilibrium function. In Fig. 2 we plot the variation of the total mass as a function of c for these equilibrium states.

In order to check the asymptotic behavior of the solutions of ( 10) we need to find a reference profile for the equilibrium function with a given total mass ρ. To this end, we simply use a dichotomy algorithm, exploiting the fact that c → c + ∞ 0 xm c (x) dx is monotone. Fig. 3-6 illustrate the evolution with the coefficients given by [START_REF] Hingant | Contributions à la modélisation mathématique et numérique de problèmes issus de la biologie : applications aux prions et à la maladie d'Alzheimer[END_REF]. The initial data reads g(0, x) = 2e -40|x-0.5| 2 , c(0) = 1.6 so that ρ = 1.8802. The dichotomy procedure finds the equilibrium concentration c lim = 0.1547. In Fig. 3 we plot the evolution of the monomers concentration t → c(t) and the total number of polymers t → ∞ 0 g(t, x) dx, up to the time T = 10. At first sight, one might believe that the equilibrium state is reached since c(t) seems to go rapidly to the equilibrium value. However, the figure is a bit misleading: we have c(T ) = 0.1525, and, going back to Fig. 1, we realize that the corresponding local equilibrium is actually far from the expected final state. In fact the equilibrium profile requires considerably more time to establish: Fig. 4 shows the polymers size distributions at several times, up to T f = 5000. At T f = 5000 the solution indeed becomes close to the equilibrium profile, see Fig. 6 which shows how the solution, the local equilibrium and the expected equilibrium coincide. The convergence of the monomers concentration c to the equilibrium concentration c lim and of the distribution of polymers g to the equilibrium profile m as t → +∞ is shown in Figs. 7 and8. The system appears to display an exponential convergence in time to the equilibrium. In contrast to the diffusionless Lifschitz-Slyozov equation, the behavior is similar when we start from a less regular initial sate, say a step function, with the same mass. Figure 5 compares the solutions starting from a smooth initial distribution and a step function with the same mass. The solutions are clearly different for times less than 10, with the support of the step function remaining larger than that of the smooth initial distribution. Over long times, however, the solutions become indistinguishable and converge to the same equilibrium profile.

Comparison with an implicit-explicit scheme for advection diffusion

In order to assess the accuracy and efficiency of the present scheme, we compare the numerical results with an implicit-explicit (ImEx) scheme for advection-diffusion. Equation ( 10) is discretized as follows:

g n+1 -g n ∆t + A c g n -D c g n+1 = 0, (22)
where A c stands for the space-discrete advection operator with velocity a(x)c-b(x) and D c denotes the space-discrete diffusion operator associated to ∂ 2 x (d(x)•). We opt for the natural centered discretization for D c . The advection operator A c is obtained with the MP5 scheme, which consists in a 5th order expansion with monotonicity preserving flux limiting, as described in [START_REF] Jourdan | An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach[END_REF] (we also refer to the original article of Suresh and Huynh for further reference [START_REF] Suresh | Accurate monotonicity-preserving schemes with Runge-Kutta time stepping[END_REF]). The update of the monomer concentration is carried out in the same fashion as in Section 3. size distribution respectively. Note that the ImEx scheme error convergence saturates around t = 8000 due to the fact that it is not constructed to be well-balanced, contrary to the present scheme. This results in the ImEx scheme converging to a slightly inexact equilibrium state.

Regarding efficiency, the time-consuming tasks for each time-step consist of the

∆t ≤ min ∆x max x∈[0,L] a(x)c -b(x) , c n ∆x +∞ i=0 F i , ( 23
)
where F i denotes the advection flux which verifies (A c g n ) i = 1 ∆x (F i+1 -F i ). We choose ∆t as 10% of the maximal CFL conditions [START_REF] Hariz | A modified version of the Lifschitz-Slyozov model[END_REF] and [START_REF] Jabin | On the rate of convergence to equilibrium in the Becker-Döring equations[END_REF] for the present scheme and the ImEx scheme respectively. The time-step evolution is compared in log-log scale for both schemes on Fig. 11. We observe that in both cases, the time-step should be small in the initial part of the simulation, which can be related to the stiffness of the initial dynamics of the distribution. The time-step can then be increased around t = 10, which corresponds to c becoming close to the equilibrium value and the dynamics being dominated by the slow diffusion effects. Let us note that it is possible to take a time-step much larger, by one to two orders of magnitude, for the present scheme than for the ImEx scheme. The difference is particularly important for the long-term dynamics (after t = 100), which results in a significant difference in simulation time.

Finite critical mass

We turn to a case where the critical mass is finite. We set

a(x) = 1 + x 1/2 , b(x) = 0.1 + 0.75x 1/2 (24)
with α = 0.001 and ε = 0.05. We have c s = 0.75 but difficulties appear clearly with smaller monomers concentration. Fig. 12 shows the profiles that correspond to several We consider the evolution of the solution for the initial data g(0, x) = 20 e -10|x-0.5| 2 , c(0) = 0.6.

Fig. 14 shows the evolution of the monomer concentration which converges very slowly to the expected limit c lim = 0.4987; at the final time T = 10000, we find c(T ) = 0.4982. The evolution of the number of polymers is displayed in Fig. 15. Accordingly, the asymptotic profile needs a considerable time to establish: Fig. 16 shows several polymer distributions up to the final time and Fig. 17 compares the solution to the expected profile.

We now start with a different initial condition:

c(0) = 6, g(0, x) = 0.4320194 if x < 1 0 otherwise.
Both initial conditions share the same (subcritical) mass, ensuring the existence of a steady state. However, the initial monomer concentration c(0) is larger than the critical concentration c s = 0.75. As a consequence, the present scheme involves operations with a diverging exponential function. In order to assess the robustness of the scheme in that case, we compare the present scheme with the ImEx scheme (which does not involve the diverging exponential). The comparison is carried out until T = 10, since Fig. 18 shows that the monomer concentration c(t) decreases under c s as soon as t > 2.03. Figures 19 and20 show the comparison of the solutions at times t = 1 and t = 2 respectively for the present scheme and the ImEx scheme. The discrepancy between the two schemes is due to the larger numerical diffusion of the present scheme, which could be remedied by the use of a higher order approximation for the diffusion operator. Let us also mention the fact that the time-step used for the present scheme is much larger than the time-step used in the ImEx scheme. The size of the time-step has been chosen solely on stability considerations and accuracy issues should also be taken into account in the very stiff initial dynamics of the system. The position of the peak size concentration is still adequately captured and the present scheme is able to robustly accommodate the supercritical monomer concentrations until c(t) < c s .

Conclusion

We have proposed a well-balanced scheme for the Lifschitz-Slyozov-Wagner system with diffusion, which demonstrates its ability to capture accurately the long-time con- We have checked that the entropy techniques developed in [START_REF] Collet | Some remarks on the large-time asymptotic of the Lifschitz-Slyozov equations[END_REF] do not apply directly here.

ε x 0 a

 0 Let us introduce the local equilibrium, parametrized by c, M c (x) = a(0)c + b(0) a(x)c + b(x) exp 2 (y)c -b(y) a(y)c + b(y) dy which clearly makes the operator vanish: L c M c = 0. Then, the Fokker-Planck operator can be cast as

Figure 1 :Figure 2 :

 12 Figure 1: Equilibrium functions for the coefficients in (21) with c ranging over {0.154, 0.155, 0.156, 0.157, 0.158}

Figure 3 :

 3 Figure 3: Evolution of c(t) and ∞ 0 g(t, x) dx, up to the time T = 10, for the coefficients in (21)

Figure 4 :Figure 5 :

 45 Figure 4: Polymer distribution function at several times for the coefficients in (21) up to the final time T f = 5000

2 .Figure 6 :Figure 7 :

 267 Figure 6: Polymer distribution function at the final time T f = 5000, compared to the equilibrium profile, for the coefficients in (21)

2 tFigure 8 :Figure 9 :Figure 10 :

 28910 Figure 8: Evolution of g(t, •) -m L 2 for the coefficients in (21)

1 10 - 4 Figure 11 :Figure 12 :Figure 13 :

 14111213 Figure 11: Evolution of the time-step in log-log scale for the coefficients in (21) for the present scheme and the ImEx scheme

Figure 14 :Figure 15 :

 1415 Figure14: Evolution of c(t) up to the time T = 10000, for the coefficients in[START_REF] Jin | An AP scheme for the Fokker-Planck-Landau equation[END_REF]

Figure 16 :Figure 17 :

 1617 Figure 16: Polymer distribution function at several times for the coefficients in (24) up to the final time T f = 10000

Figure 18 :Figure 19 : 21 -Figure 20 :

 18192120 Figure 18: Evolution of the monomer concentration c(t) for the coefficients in (24) in the supercritical concentration case


Hervé Panetto
email: herve.panetto@univ-lorraine.fr

Paulo Cézar Stadzisz

Wenchao Li
email: wenchao@bu.edu

Qing-Shan Jia

Special Issue on (Industrial) Internet-of-Things for Smart & Sensing Systems: Issues, Trends and Applications

Smart and sensing systems represent a significant research theme that has motivated a number of research initiatives around the world. Smart systems typically consist of different components including sensors for signal acquisition, communication units for data transmission between components, control and management units for decision-making, and actuators to perform the appropriate actions. They have impacted applications in many fields such as manufacturing, healthcare, energy, environment, logistic, defence, monitoring, and mobility. As the complexity of such systems continue to grow, the challenge of developing integrated smart and sensing systems has surpassed the design complexity of their individual components. A smart and sensing system may include a large number of heterogeneous components, subsystems, smart and intelligent sensors, and may combine several correlated functionalities. Thus, the main issue of developing smart and sensing systems lies in the complexity to integrate and to manage these different components, technologies, and goals across a wide spectrum. In recent years, the emergence of IoT has amplified the capacity of sensing the world through a network of connected devices using the existing network infrastructure. Grouping together smart and sensing systems in an IoT setting to form large-scale distributed cyber-physical systems has tremendous potential in bringing smart systems to many application domains. This Journal's Special Issue (SI) aims to promote relevant researches related to strategic views, problems, trends, solutions, and applications of IoT for Smart and Sensing Systems. This represents a more specialized area of interest than the wide application of IoT. The issue focuses more specifically on how industrial IoT contributes to the development of smart and sensing systems.

50 submissions from around the world were received in response to the Calls for Papers on this Special Issue. During the review process, each paper was assigned to and reviewed by at least two experts in the related areas, with a rigorous review process. Thanks to the great support of the Editor-in-Chief of this Journal, Prof. Sherman Shen, guest editors were able to accept 20 selected papers covering issues, trends and applications of Internet-of-Things for Smart & Sensing Systems.

In the paper "Real-time monitoring of bone fracture recovery by using aware; sensing; smart and active orthopaedic devices" the authors explore the use of IoT technologies in the field of orthopaedics. It presents an Aware, Sensing, Smart and Active (ASSA) external fixation device for fractured bone treatment.

To enable collaboration among supply chains in all sectors, the paper "Trustworthy Industrial IoT Gateways for Interoperability Platforms and Ecosystems" presents a Device Driver security architecture used in the Virtual Factory Open Operating System (vf-OS) platform, orientated to manufacturing and logistics companies.

A major challenge in Industrial Internet-of-Things for Smart & Sensing Systems is the critical role that energy efficiency plays in widespread deployability of IoT applications. To address this challenge, the paper "cDIP: Channel-aware Dynamic Window Protocol for Energyefficient IoT Communication" proposes a new generalized channel-aware link layer communication strategy that exploits temporal variations of the communication channel to improve on energy efficiency.

The paper entitled "Ontology-based framework enabling smart Product-Service Systems: Application of sensing systems for machine health monitoring" proposes embedded sensing systems tailored to industrial Product-Service System (PSS) and the use of these systems in providing customized services. It presents a sensing system ontology as a framework to support the smart services in PSS.

In the paper "Closed loop controlled precision irrigation sensor network" the authors propose a closed loop irrigation system demonstrating that it fully automates the delivery of irrigation water and calculates in real time the water requirement from satellite images. Results demonstrate that combining closed loop automation and analytics can improve water use efficiency and increase yield on existing agricultural lands.

For monitoring the nitrate concentration in a field, the authors of the paper "An Internet-of-Things enabled Smart Sensing System for Nitrate Monitoring" designs a smart sensing node which can collect the water from lake, stream, or river, measure the instant nitrate concentration, and transfer the data through the gateway to a user-defined cloud server.

In the paper "IoT for Next-Generation Racket Sports Training" the authors present the application of IoT framework in the development of a smart badminton actions recognition system that includes a wearable sensor, a mobile app and cloud server. The results demonstrated that the proposed smart system can classify the clear distinction between different badminton strokes with an average accuracy of 97%. The system can automatically provide data statistic of badminton players, which can help coaches and athletes themselves to learn about the real condition change during a match or a training.

Considering the compact Operating Systems (OSes) required on low-end IoT devices, the authors of the paper "RIOT: an Open Source Operating System for Low-end Embedded Devices in the IoT" provide a comprehensive overview of RIOT covering the key components of interest to potential developers and users. RIOT is a free, open source software platform that can reach a good balance between protecting end-users and supporting industry, at large scale and in the long run.

In the paper "Inter-Community Detection Scheme for Social Internet of Things: A Compressive Sensing Over Graphs Approach" the authors investigate the intercommunity detection issue in the social IoT scenarios and present a Compressive Sensing (CS) over graphs based inter-community detection scheme for the social IoT.

For the efficient allocation of carrier and computing resources in smart manufacturing system, the authors of the paper "Internet of Things for Smart Manufacturing System: Trust Issues in Resource Allocation" propose a Vickrey-Clarke-Groves (VCG) auction based hierarchical trust computing algorithm. Results demonstrated experimentally that the utility is maximized when the IIoT devices and gateways make the smart manufacturing system be trustful.

The paper entitled "Building Data-Aware and Energy-Efficient Smart Spaces" discusses the current research in smart spaces and the challenges that arise in the execution of these smart spaces in real-time. Authors propose a smart space that considers and addresses the challenges and trade-offs between data requirements and energy-efficient data collection, along with hardware constraints.

To show how IoT could have a tangible impact on communities with reliance on sustaining infrastructures, the paper "Agile IoT for Critical Infrastructure Resilience: Cross-modal Sensing as Part of a Situational Awareness Approach" presents a method and a model for agile IoT for resilience.

In the paper "TriboMotion: A Self-Powered Triboelectric Motion Sensor in Wearable Internet of Things for Human Activity Recognition and Energy Harvesting" the authors present a new motion sensor system in wearable IoT for human physical activity recognition without any signal conditioning circuits. The triboelectricity-based physical model is explored in designing the motion sensor.

The paper entitled "Device-Free People Counting in IoT Environments: New Insights; Results and Open Challenges" presents a new open dataset with wireless signals captured over diverse physical IoT environments and explores different features and ML models so as to draw empirically informed conclusions about their performance and predictive potential for IoT scenarios and cases.

In the paper "Building and Scheduling a Mobile Sensing Platform for Smart City Services" the authors present the City Scanner project to capture the spatiotemporal variation of environmental parameters. City Scanner is a specific drive by sensing approach with several key benefits, such as a nonintrusive, selfcontained, and portable sensing components.

To allow the security community to better understand SCADA specific risk, the authors of the paper "Cybersecurity Risk Modelling for IIoT in Smart Cities" presents a customizable SCADA risk prioritization schema. Considering the distinct properties of SCADA systems, a data-driven prioritization schema helps researchers identify security gaps specific to this software subclass.

The paper entitled "Industrial Internet of Things: A Systematic Literature Review and Insights" presents results and insights of a systematic literature review (SLR) that systematically addresses Industrial Internet of Things (IIoT). The SLR is based on questions related to who is working on IIoT, the main standards and technologies to support communications, and the main research efforts and IIoT implementations.

In the paper "IoT based Vibration Analytics of Electrical Machines", the authors propose an IoT based model for real time condition monitoring of electrical machines, which addresses challenges of data storage and scalability. The proposed framework can enhance the machine condition monitoring functionality with methodologies of scalable and platform independent data aggregation and collaborative analysis that the realtime industrial applications demand extensively.

Considering the limitations in the capacity of the batteries of the wireless sensor networks (WSNs) which severely limit continuous monitoring based Internet-of-Things (IoT) applications, the authors of the paper "SBL-based Adaptive Sensing Framework for WSNassisted IoT Applications" presents a novel sparse Bayesian learning-based adaptive sensor selection framework to substantially increase the lifetime of a densely-deployed WSN for monitoring spatiotemporally varying signal.

Finally, in the paper "Prediction of frost events using Bayesian networks and Random Forest", the authors present a new approach to predict frost events which considers that frost prediction in one location could be improved by using the information of the most relevant neighbouring sensors. The authors propose to use machine learning algorithms such as Bayesian networks and Random Forest in which the training set includes new samples using SMOTE (Synthetic Minority Oversampling Technique).

The Editors of this Special Issue of IEEE IoT Journal would like to express their gratitude to the authors for their excellent contributions. We are also very grateful to all reviewers who have dedicated efforts in reviewing these papers, and for their valuable comments and suggestions that significantly improve the quality of them. We hope that this Special Issue will serve as a good reference for researchers, scientists, engineers, and academicians in the field of industrial IoT.

H.

  Panetto was born in Nancy, France. He is a Professor of Enterprise Information Systems at University of Lorraine, TELECOM Nancy. He teaches Information Systems modelling and development, and conducts research at CRAN (Research Centre for Automatic Control), Joint Research Unit with CNRS where he is managing a research project on the use of ontology for formalising models related to the interoperability of production systems, and mainly their enterprise information systems. He is member of the Academia Europaea since 2018. He received his PhD in production engineering in 1991 from the University Henri Poincaré, Nancy, France. His