Florence Bara

Dr Edouard Gentaz
email: gentaz@psycho.univ-paris5.fr

Pascale Colé

Liliane Sprenger-Charolles

The visuo-haptic and haptic exploration

Keywords: reading acquisition, multisensory, haptics Haptics in reading acquisition -3

This study examined the effect of incorporating a visuo-haptic and haptic (tactualkinaesthetic) exploration of letters in a training designed to develop phonemic awareness, knowledge of letters and letter/sound correspondences, on the children's (5 years old) understanding and use of the alphabetic principle. Three interventions, which differed in the work on letter identities, were evaluated. The letters were explored visually and haptically in "HVAM" training (Haptic-Visual-Auditory-Metaphonological), only visually in "VAM" training (Visual-Auditory-Metaphonological) and visually but in a sequential way in "VAMsequential" training. The three interventions made use of the same phonological exercises. The results revealed that the improvement in the pseudo-word reading task was greater after HVAM training than after VAM and VAM-sequential training. Similar improvements in the letter recognition test and in the phonological awareness tests were observed after the three interventions. Taken together, the results show that incorporating the visuo-haptic and haptic exploration of letters has positive effects on the reading skills of young children. The sequentiality of the exploration is not sufficient alone for explaining these improvements.

I. Introduction

This study examined the effects of incorporating a visuo-haptic and haptic (tactualkinaesthetic) exploration of letters in a training designed to develop phonemic awareness, knowledge of letters and letter/sound correspondences, on the understanding of the alphabetic principle among pre-reading kindergarten children. Researches on reading acquisition had primarily focused on the phonological dimension involved in the understanding of the alphabetic principle. Several studies have shown that one of the factors influencing reading success lies in the development of some skills related to spoken language and broadly referred to as metaphonological abilities (for recent reviews in English, see [START_REF] Ehri | Phonemic awareness instruction helps children learn to read: Evidence from the National Reading Panel's meta-analysis[END_REF][START_REF] Troia | Phonological awareness intervention research: A critical review of the experimental methodology[END_REF] for recent reviews in French, see [START_REF] Gombert | Activités métalinguistiques, lecture et illettrisme[END_REF][START_REF] Sprenger-Charolles | Phonological mediation and orthographic factors in reading and spelling[END_REF]. These are defined as the abilities which allow children to identify the phonological components of linguistic units and to manipulate them intentionally (Liberman, Shankweiker, Fisher, & Carter, 1974). It appears that the metaphonemic abilities (the ability to manipulate the phonemes in spoken words) are the best predictors of reading in alphabetic systems [START_REF] Ehri | Phonemic awareness instruction helps children learn to read: Evidence from the National Reading Panel's meta-analysis[END_REF].

Furthermore, if we very broadly believe the acquisition of reading to consist, on the one hand, in the development of phonological and orthographic representations and, on the other, in the establishment of connections between, or indeed the fusion of, these two types of representation [START_REF] Metsala | Word recognition in beginning literacy[END_REF], then the body of research devoted to the way in which these connections or this fusion come about is very slight and tends to consider that this is an "implicit" process which is triggered by the learning of letter/sound correspondences (e.g. [START_REF] Plaut | Understanding normal and impaired word reading: Computational principles in quasi regular domains[END_REF]. Reading training programs adhere strictly to this conception and make use of metaphonological and letter discrimination exercises which are associated with the learning of the letter/sound correspondences (Bus & van Haptics in reading acquisition -4 Ijzendoorn, 1999). For example, [START_REF] Byrne | Evaluation of a program to teach phonemic awareness to young children[END_REF] evaluated the efficiency of this kind of training program on 5-year-old pre-readers. The performances in phonemic awareness and word reading tasks improved more after this "experimental" training than after the "control" training which only made use of semantic activities. However, although this type of intervention really has a positive effect on reading preparation, reading acquisition generally remains slow and difficult and several months of formal instruction are necessary before young children grasp the logic of the alphabetic principle and apply it [START_REF] Colé | Syllable-sized units in visual word recognition: Evidence from skilled and beginning readers of French[END_REF]O.N.L, 1998;[START_REF] Sprenger-Charolles | Phonological mediation and orthographic factors in reading and spelling[END_REF][START_REF] Sprenger-Charolles | Development of Phonological and Orthographic Processing in Reading Aloud, in Silent Reading and in Spelling: A Four Year Longitudinal Study[END_REF].

In the light of Bryant and Bradley's work (1985), we might well imagine that one of the difficulties involved in learning to read relates in part to the task of establishing connections between the orthographic representations and the corresponding phonological representations.

More precisely, it lies in a difficulty in establishing a connection between the visual image of a word and its auditory image. In an attempt to overcome this difficulty, these authors recommended the use of a "multisensory" learning method calling not only on the visual and auditory modes as is traditionally the case, but also on the manual haptic mode. Indeed, our hands do not simply possess the motor function of moving or transforming the objects in our environment, but also a highly efficient perceptual function (e.g. Gisbon, 1962; for recent reviews, see [START_REF] Hatwell | Toucher pour connaître. Psychologie cognitive de la perception tactile manuelle[END_REF], 2003;[START_REF] Heller | Touch, representation and blindness[END_REF][START_REF] Millar | Understanding and representing space[END_REF].

This multisensory approach was first proposed by [START_REF] Montessori | The Montessori method[END_REF][START_REF] Montessori | Pédagogie scientifique, la maison des enfants[END_REF] and is still used today in a number of schools. This method had been successfully used to help children with reading difficulties and a number of older studies point out the benefits of using the haptic mode in addition to the visual and auditory modes.

Haptics in reading acquisition -5 Thus, [START_REF] Fernald | Remedial techniques in basic school subjects[END_REF] employed a "multisensory" technique (largely based on Montessori's principles) with children exhibiting reading difficulties. This technique, known as the "multisensory trace" involves the children in tracing a written word with their index finger while pronouncing it and looking at it. In line with this, [START_REF] Ofman | The kinesthetic method in remedial reading[END_REF] compared the effectiveness of the "multisensory trace" with that of the "visual trace" and "reading only" in a task involving the learning of new words. The "visual trace" task consists of asking the children to track with their eyes a word which is gradually written in front of them. The results revealed that poor readers (aged 13 years) were significantly better able to learn new words using the "multisensory trace" and the "visual trace" techniques (which are identical) than with the "reading only" technique. The introduction of an exploratory movement (visuo-haptic or visual) in the apprehension of a written word seems to facilitate the learning in this type of readers.

In a similar vein, [START_REF] Hulme | The interaction of visual and motor memory for graphic forms following tracing[END_REF] examined in younger children (8-9 years) the effect of the haptic exploration of abstract graphical figures on their memorization. The figures were explored either by looking at each of them ("Visual" condition), or by looking while simultaneously tracing them with the finger ("Visuo-Haptic" condition). The results indicated that the children in the "Visuo-Haptic" condition achieved significantly better performances in a subsequent recognition task than those in the "Visual" condition. Similar results were obtained with alphabet letters, in children with reading difficulties and normal-reading children (aged 9 years) [START_REF] Hulme | Reading retardation and multisensory teaching[END_REF].

Recently, [START_REF] Gentaz | Evaluation d'entraînements multisensoriels de préparation à la lecture pour les enfants en grande section de maternelle : une étude sur la contribution du système haptique manuel[END_REF] This positive effect of the haptic mode was explained in terms of the functional specificities of the various sensory modes in question [START_REF] Gentaz | Is haptic perception continuous with cognition?[END_REF][START_REF] Hatwell | Toucher pour connaître. Psychologie cognitive de la perception tactile manuelle[END_REF], 2003;[START_REF] Lederman | Hand movements: A window into haptic object recognition[END_REF]. Indeed, vision is characterized by its quasi-simultaneity and is therefore more suitable for processing and representing spatial stimuli such as letters. On the other hand, listening is sequential in nature and is more suitable for processing temporal stimuli such as the sounds of speech. This functional difference could explain why young children have some difficulties to establish the association between letters, which are processed visually, and sounds, which are processed auditorily. In contrast, the haptic mode shares characteristics with both the auditory and the visual modes. Even though its functioning is highly sequential in nature, the haptic mode is also a spatial mode since the exploration in this mode is not linear and subject to a fixed order.

Haptics in reading acquisition -7 Within this perspective, research into the development of the visual and haptic apprehension of objects has made it possible to better identify the characteristics of haptic exploration. Since objects are multidimensional, they possess values along a number of different dimensions such as texture, location, size, shape, etc. In the case of sight, all the dimensions are perceived practically simultaneously (separated by only a few milliseconds).

The same is not true in the haptic mode because, in this case, the exploration involved makes the perception very sequential. This is the reason why the haptic perception appears to be less "global" and more "analytical" than the visual perception. Thus, as far as our study is concerned, the incorporation of the haptic mode should require children to process letters in a more sequential and therefore a more analytical way, something which they do not do naturally when letters are presented in the visual mode only.

The first goal of the present research was to confirm that incorporating the visuo-haptic and haptic exploration of letters in phonological and letter knowledge exercises, increases their positive effects on the understanding and use of the alphabetic principle in young children and thus on their reading skills. The second goal was to investigate whether these positive effects observed after HVAM training could be explained by the sequential exploration of letter induced by the haptic perception or by the haptic perception of letter per se, in evaluating a third training.

To study these issues, we examined in kindergarten children the effect of three (insert Table 1)

II. 2. Material and procedure

The selection of the phonemes and the order in which they were studied were based on their frequency of appearance in the French oral language [START_REF] Rondal | L'évaluation du langage[END_REF]. Thus the sessions started with the study of the sounds /a/, /i/, (amongst the most frequent), continued with sounds /r/, /l/ and ended with the sounds /t/, /p/ and /b/ (less frequent). The typography of each letter resembled lowercase handwriting (Figure 1) and was used in all the exercises which required the use of the letters. The understanding and use of the alphabetic principle were measured using a pseudoword reading test and a test requiring the recognition of the alphabet letters. In the pseudoword reading test, the experimenter told the children that words were made-up and meant nothing in their language. Three of these pseudo-words consisted of two letters, six of three The children's metaphonological abilities were measured using three tests: a rhyme identification test, and two phoneme identification tests (in initial or final position in the words). In these three tests, pictures corresponding to familiar, easily depicted words were presented and the children were asked to perform a specific task. In our tests, the children were asked to look for the seven phonemes which had been included in the training sessions.

Each test consisted of three training trials, during which we made sure that the children had correctly understood the task, and six or seven trials on the basis of which a score out of six (for the rhyme test) or seven (for the two phoneme identification tests pb voir apres) was obtained. In the rhyme test, the child saw three pictures and had to indicate those corresponding to the two words "that sound the same at the end" (for example: bulle [bubble],

pull [pull over], four [furnace]). In the test requiring the identification of the phonemes in initial position, four pictures were presented to the child in each trial. The experimenter chose a first picture and said the word. This first word was then used by the child to find, among the other three pictures, the one corresponding to the word "that starts with the same sound" (for example: bague [ring] was the first word and the child had to choose between bouée [buoy],

chemise [shirt], stylo [pen]). In the test requiring the identification of phonemes in final position, we used the same principle with the difference that the child had to find the word Haptics in reading acquisition -11 "that ends with the same sound" (for example: lit, /li/ [bed]) was the first word and the child had to choose between toupie [spinning top, /tupi/], pomme [apple], clé [key]; in the French oral language, the letter /t/ of /lit/ and the letter /e/ of /toupie/ are silent). In each test, the order of presentation of the pictures was controlled. Thus, in each trial, we changed the position of the target word with reference to the word that acted as the starting point for the child in order to prevent the establishment of retrieval strategies based on the order of picture presentation.

II.2.2. Training sessions

Three equivalent groups of 20 children each were formed from eight criteria: age, vocabulary level (EVIP), Khos blocks, three metaphonological tests, recognition of letters and pseudo-word reading (Tables 1, 3 and5 the sensory modes that were requested and in the manner of exploring the letters (simultaneous or sequential).

II.2.2.1.The HVAM training sessions

a) The identification exercise At the start of each session, the experimenter gave a small, movable letter (made of foam with a thickness of 5 mm) to each child and showed the correct orientation of the letter.

The experimenter asked them to guess its identity. These small letters -a-, -i-and -r-were 2.5 cm high and the letters -t-, -l-, -p-and -bwere 4.8 cm high. The children held the letter between their hands during the two following activities (the nursery rhyme and the posters) and were free to touch it during this period.

b) The nursery rhyme

After the first identification exercise, the experimenter recited a nursery rhyme. The rhyme contained many examples of the sound on which the children were required to concentrate during the session. Its aim was to sensitise the children to the sound by means of short, playful stories which they found easy to remember and fun to repeat. During this rhyme, the children attempted to detect the sound that corresponded to the target letter. The children then repeated the rhyme sentence by sentence. Repeating the rhyme enabled them to pronounce the target sound a large number of times and to familiarize themselves with it. We expected this familiarization to encourage the learning of the letter/sound correspondences.

c) The two poster exercises

The children then started working on two posters (40 cm high by 60 cm wide). The first of these contained pictures corresponding to words starting with the learned sound and distractor words. Among the six words presented (three target words and three distractors), Haptics in reading acquisition -13 each child had to find a word starting with the sound learned during the session and whisper the answer to the experimenter. Each child's answer was then revealed to the group and discussed in order to determine whether or not it was correct. The experimenter then moved on to the poster containing pictures corresponding to words which ended with the learned sound and repeated the above exercise accordingly.

d) The visuo-haptic and haptic exercises

The large letters (fixed to a 20 cm x 25 cm board) made with foam were then handed out. The letters -a-, -i-and -r-were 5 cm high and the letters -t-, -l-, -p-and -bwere 10 cm high. The children were told to explore the relief letter with their fingers and run their index finger along its outline in a fixed exploratory order corresponding to its writing (Figure 1).

The experimenter observed and checked the way each child explored the letter. Once this exercise was finished, the haptic exploration of the letter continued under a cover which was placed above the board. The children slid their hands below the cover and were told to think of the letter while exploring it haptically. The same exercise was then used with the small letters, which were fixed to a 10 x 13 cm board (a-, -i-and -r-were 2.5 cm high and the letters -t-, -l-, -p-and -bwere 4.8 cm). It should be noted that we used two letter sizes in order to facilitate the learning and exploration of the letters. Indeed, we know that medium amplitude movements are easier to control with precision than low amplitude movements [START_REF] Hatwell | Touching for knowing[END_REF]. Thus the large letters, which induced medium amplitude movements, helped the children succeed in this visuo-haptic tracking task and give them the way of exploring the letters. This therefore facilitated the exploration of the small letters which induce low amplitude movements.

Once the haptic exploration was finished, the children performed a recognition task using the small fixed letters. They had to distinguish between the letter learned during the Haptics in reading acquisition -14 session and a "distractor letter" which physically resembled it. Thus we associated the training letter (TL) -a-with the distractor letter (DL) -e-, the TL -i-with the DL -u-, the TL -r-with the DL -n-, the TL -l-with the DL -tand the TL -t-with the DL -b-, the TL -pwith the DL -qand the TL -bwith the DL -l-. The two foam letters were arranged under the cover where the children again had to slide their hands in order to handle them. Their task was to explore both letters and to identify the target. If the children had correctly identified the letter, they were allowed to remove the cover and check for themselves. However, if they failed to identify the correct letter, the experimenter advised them to take their time and explore the letters again.

e) The two card game exercises

The final stage of the session took the form of two card games. Pictures representing the target and distractor words were spread out on the table. In the first game, the children had to take turns to choose a picture which corresponded to a word starting with the target sound. In the second game, the chosen picture had to represent a word which ended with the learned sound. This game ended when no further target pictures remained.

II.2.2.2 The VAM training sessions a) The identification exercise

The visual exploration boards were distributed to each child. Each letter, printed on a sheet of paper, was glued to a small board. The experimenter showed the correct orientation of the letter and asked them to guess its identity. The size of each letter was the same as the size of the foam letters used in the identification exercise in the HVAM training. These boards displayed the handwritten letter to be learned and remained visible throughout the two following activities (the nursery rhyme and the posters).

b) The nursery rhyme

Haptics in reading acquisition -15

The nursery rhyme was the same as in the HVAM training sessions.

c) The two poster exercises

The two poster exercises were the same as in the HVAM training sessions.

d) The visual exercises

The letter was then explored visually. The experimenter asked the children to follow the drawing of the letter with their eyes and drew their attention to its shape and the lines and curves it contained. This information allowed the children to subdivide the letter into organized elements and track its contour visually in a fixed exploratory order (Figure 1).

When the children had explored the letter for long enough, the experimenter distributed the sheets containing a visual recognition test. The children were asked to cross out the target letter presented together with a number of distractors (letters sharing physical characteristics with the target). This test was presented on a sheet of paper (A4 format) containing four lines of twelve letters each (with a small size). Each line of letters contained two types of distractor letters and a variable number of instances of the target letter in order to prevent the children from using a strategy based on the number of items to be crossed out. It should be noted that we selected two distractor letters (instead of one in the haptic task) in order to increase the difficulty of the exercise (the use of a single letter makes the visual task too easy and to fast).

The training letter (TL) -a-was associated with the distractor letters (DL) -o-and -e-; the TLi-with the DL -j-and -u-, the TL -r-with the DL -n-and -s-; the TL -t-with the DL -b-and -dand the TL -l-with the DL -t-and -h-, the TL -pwith the DL -qand -g-and the TL -bwith the DL -land -k-.

In order to attempt to equalise the global duration and the number of exercises between the interventions, we proposed an additional card game. In this game, the cards were spread on the table and each child had to take one of it. Each game consisted of 12 cards amongst Haptics in reading acquisition -16 which there were four cards corresponding to the target letter, and four cards of each of the two distractors letters. The distractor letters used for each target letter were the same as those used in the crossing-out task. The children had to judge whether the letter they had taken corresponded to the target letter or to a distractor, in placing it in one of the two boxes placed in front of them (one for the target letter and one for the distractor letters as specified by the experimenter).

e) The two card game exercises

The two card game exercises were the same as in the HVAM training sessions.

II.2.2.3 The VAM-sequential training sessions

The progress of VAM-sequential training sessions was exactly the same as the VAM training sessions, except for the presentation of the letters (visually but sequentially). Indeed, the letter presented visually took shape gradually on the screen of a computer. The letter was traced sequentially on the screen by a black spot which moved while following its outlines.

Once the letter was entirely drawn, it remained motionless and visible on the screen during 2 seconds, then disappeared during 10 milliseconds and started again to take shape. The experimenter asked the children to follow the drawing of the letter with their eyes and drew their attention to its shape and the lines and curves it contained. This sequential drawing allowed the children to subdivide the letter into organized elements and track its contours visually in a fixed exploratory order. Then, the children performed an observation work of the letter which takes shape on the screen. They had to follow with their eyes the layout of the letter and try to memorize it.

I.2.2.3. The revision session

Haptics in reading acquisition -17 A revision session was planned for each group at the end of the seven training sessions.

First of all, we summarized the work done during the sessions and reminded the children of the seven letters they had studied together with the corresponding sounds. In the HVAM training, the children were told to explore each relief letter with their index finger. In the VAM training, the children were asked to explore visually each letter. In the VAM-sequential training children explored the letters seeing them gradually take shape on the computer screen.

The purpose of this session was to reactivate the knowledge acquired by the children during the training sessions. Then they worked with the dominos, which formed part of the metaphonological training. The dominos consisted of two pictures placed side by side. All these pictures represented words which started (or ended) with one of the seven sounds learned during the training sessions. The children's task consisted of matching the dominos in such a way that the different pictures, when placed together, corresponded to words which started or finished with the same sound. We constructed two sets of dominos. The first game contained pictures corresponding to words which started with one of the sounds learned during the training sessions, while the second contained pictures corresponding to words which finished with one of these sounds. The dominos also made it possible to check the children's knowledge of the different sounds they had learned.

III. Results

III.1 Pseudo-word reading

The mean number (and standard deviations) of correctly read pseudo-words (maximum 12) before and after each of the three interventions are presented in Table 3.

(insert Table 3)

Haptics in reading acquisition -18 A 3 (training) x 3 (period) ANCOVA, with the pretest scores in vocabulary, Khos blocks, phonological tests and letters recognition as covariates, was performed on the number of correctly read pseudo-words. It revealed that the effect of period was significant

IV. Discussion

This study attempted to show that the addition of the haptic mode in a preparatory reading training program with kindergarten children can assist their understanding of the alphabetic principle and induce a major improvement in their reading skills. Moreover, the role of the sequential exploration of letters induced by the haptic perception in these positive effects was investigated. We measured the children's abilities before and after the HVAM, VAM and VAM-sequential interventions using two tests evaluating the understanding and use of the alphabetic principle (pseudo-word reading and letter recognition tests) and three tests involving phonological awareness (rhyme, initial and final identification tests).

First we observed an improvement of performances after each measures. These results were in line with the idea that any work in the field of phonemic awareness, associated with a work which develops both letters and letter/sound correspondences knowledge, assist children to understand the alphabetic principle and use it to decode written words (cf. [START_REF] Bus | Phonological awareness and early reading: A meta-analysis of experimental training studies[END_REF].

It is clear that the pseudo-word reading test is a good indicator of children's understanding of the alphabetic principle. Indeed, success in this test requires the application of the letter/sound correspondences with which the children were familiarized during the training sessions. This can only be effective if the children have understood the principle by which the sounds are represented by the letters. The results observed were consistent with those reported previously by [START_REF] Gentaz | Evaluation d'entraînements multisensoriels de préparation à la lecture pour les enfants en grande section de maternelle : une étude sur la contribution du système haptique manuel[END_REF]. The mean number of correctly read pseudowords (close to 0 before the interventions) increased significantly after each of the intervention.

The results confirmed our hypothesis since the improvement in observed performances was greater after HVAM training than after VAM training (see also Appendix), given that the Haptics in reading acquisition -21 addition of the haptic mode in the type of training inspired by [START_REF] Byrne | Evaluation of a program to teach phonemic awareness to young children[END_REF] almost makes it possible to improve the beneficial effects in pseudo-word reading. The performances observed after the HVAM training were also greater than those observed after the VAM-sequential training (which did not differ from those after the VAM training). The sequential exploration of the letters seems not to be sufficient alone. A visual sequential perception of letters did not permit to benefit more of the training than a simultaneous way of perceiving letters. The sequentiality induced by the haptic exploration of letters seems not to be the most important component of haptic perception, which permit to make easier the connection between the letter and the sound. The fact that the haptic perception is necessarily sequential did not explain, on his own the beneficial effects on reading. These results about the beneficial role of a sequential exploration were not exactly in line with those observed by [START_REF] Ofman | The kinesthetic method in remedial reading[END_REF] in poor readers (aged 13 years). Indeed, these authors found that a word was easily learned when it was explored sequentially (by sight or by touch). However, in this study, the children were older, and haptic perception is maybe less attractive than for young children.

The results in the test of recognizing the learned letters partially confirmed our hypothesis. The three interventions improved (in a similar way) the mean number of correctly identified target letters. However, as in the previous study (Gentaz & al, 2003), the improvement of the number of recognised letters was not greater after the haptic exploration than after the visual exploration.

The rhyme, initial and final phoneme identification tests make it possible to evaluate the effectiveness of the metaphonological exercises presented in the three training programs. The results were compatible with our hypotheses since the three interventions improved performances in each of the three tests in a similar way. This result could be easily explained Haptics in reading acquisition -22 by the fact that the metaphonological exercises were the same in the three interventions.

Taken together, the results showed that incorporating the haptic mode in exercises involving letter and letter/sound correspondences knowledge, combined with exercises involving phonemic awareness, increase the positive effects of this type of training on the understanding and use of the alphabetic principle in young children and thus on their reading skills. Since this specific improvement was only observed after the HVAM training on pseudo-word reading (whereas the performances were similar after the three interventions on letters recognition), we can supposed that the haptic mode facilitates not exactly the knowledge of the letters and the sounds, but the connections between the two. The haptic exploration help to establish the links between the orthographic representation of the letters and the phonological representation of the corresponding sounds. This beneficial effect of incorporating the haptic mode could be due to various functional specificities of the sensory modes. Incorporating the haptic mode would force children to process the letters in a more analytical way, something which they do not do implicitly when the letters are presented in a visual form only (simultaneously or sequentially). This would favour the association with the sound of the letter, which is processed auditorily. This hypothesis could also partially explain the beneficial effects of writing activities on the understanding of the alphabetic principle observed in some studies (cf. [START_REF] Zesiger | Ecrire: Approches cognitive, neuropsychologique et développementale[END_REF].

Haptics in reading acquisition -23

  investigated, in kindergarten children, the effects of adding a visuo-haptic and haptic exploration of letters in a reading training program, designed to develop phonemic awareness and knowledge of letters and letter/sound correspondences. Two interventions, which differed in the perceptual mode they address, were Haptics in reading acquisition -6 compared. The HVAM training (Haptic-Visual-Auditory-Metaphonological) involved the haptic, visual and auditory mode whereas the VAM training (Visual-Auditory-Metaphonological) only exploited the visual and auditory modes. Both interventions made use of the same phonological exercises. However, whereas the work on letter identity was based on a visuo-haptic and haptic exploration (the shape of the letters was traced with the finger) in the HVAM training, only the visual exploration was exploited in the VAM training. Performances were assessed before and after interventions by means of pseudo-word reading, letter recognition and phonological tasks (rhyme and phoneme identification). The results revealed that the improvement in the pseudo-word reading was greater after HVAM training than after VAM training. Similar improvements between HVAM and VAM were observed in letter recognition and in the phonological tests. These results showed that incorporating the visuo-haptic and haptic exploration of letters increases the positive effects of the interventions on the understanding and use of the alphabetic principle in young children and thus on their reading level.

  interventions that were different by the sensory mode requested(visual, auditory and haptic) and by the way of exploring the letters (simultaneous or sequential). Thus, we presented children with the two interventions proposed by[START_REF] Gentaz | Evaluation d'entraînements multisensoriels de préparation à la lecture pour les enfants en grande section de maternelle : une étude sur la contribution du système haptique manuel[END_REF] named HVAM and VAM respectively and with a third training. In the third training, labelled "VAM-sequential", the letters were explored visually and sequentially (they take shape gradually on a computer Haptics in reading acquisition -8 screen). The three interventions made use of exercises involving phonemic awareness and exercises bearing on letter recognition and learning of letter/sound correspondences. Regarding the first goal (the investigation of the haptic exploration effects), we should observe an improvement in performances on both the tasks designed to evaluate the understanding and use of the alphabetic principle (pseudo-word reading and letter recognition tests) following HVAM and VAM training, but the amplitude of this improvement should be greater after HVAM training than VAM training. Regarding the second goal (the investigation of the role of the sequentiality of exploration), if the sequential exploration of letters explained the improvements in reading observed after the HVAM training, we should observe: 1) similar improvements (in pseudo-word reading and letter recognition tests) in the HVAM and VAM-sequential training and 2) lower improvements in the VAM training. By contrast, if haptic exploration per se was totally responsible for these improvements, we should observe greater performances after the HVAM training than after both the VAMsequential and the VAM training. Finally, we also measured children's metaphonological abilities and we expected to observe similar improvements after the three interventions. II. Method II.1. Participants Sixty monolingual French children (25 girls and 35 boys) with a mean age of five years seven months (5 years 2 months to 6 years 1 month) took part in this study. These children were attending three different kindergarten classes at schools in Grenoble (France). They all belonged to average socio-economic status. The children were pre-readers and had never been trained to phonological tasks. Only the children who were not at all the training sessions were removed from the group and not taken into account in the results analyses. In each classroom, Haptics in reading acquisition -9 children in the three training groups were matched on the following criteria: age, vocabulary level (EVIP), non-verbal performance level (Kohs block from WIPSI) and metaphonological ability, knowledge of the alphabet letters and pseudo-word reading (see Tables 1, 3 and 5; all p>.25). Globally, there were 20 children in each training group.

(

  insert Figure 1) II. 2.1. Pre-and post-tests Each child's understanding and use of the alphabetic principle and her or his metaphonological abilities were individually evaluated between one and two weeks before and after the interventions. These evaluations were carried out by the same experimenter who remained blind to training assignments of each child.

  Haptics in reading acquisition -10 letters and three of four letters (score out of 12). These pseudo-words consisted only of the letters studied during the training sessions.In the letter recognition test, the experimenter said the name of a letter and the child had to indicate on a presentation card (composed of six letters) the letter he had heard. Each correct response was scored 1. Before the interventions, all the 26 letters were tested (score out of 26) whereas in the post-test, only the letters involved in the training sessions were tested (score out of 7).

  ). A specific training (namely: HVAM, VAM or VAM-sequential) was administered to each of the groups of selected children by the same experimenter. Each intervention consisted of seven sessions, conducted in the same way (the same exercises were performed in the same order) and one revision session. A different sound (and the corresponding letter) was learned in each session. By the end of the interventions, the children were familiar with the sounds/letters -a-, -i-, -r-, -l-, -t-, -p-and -b-. One session was conducted each week. As a consequence, each intervention took 8 weeks (7 training sessions and 1 revision session). The interventions were conducted from February to May. The context in which the different groups were asked to work was identical. Each training session lasted for approximately 25 minutes and took place in an acoustically insulated room in order to optimise the children's attention. The children sat in groups of 5 or 6 around a table in order to encourage their interactions. The experimenter involved each of them during the different exercises. The three interventions used the same metaphonological exercises which included the nursery rhyme, the posters and the card games. The basic difference between the three lies in Haptics in reading acquisition -12

  [F(1,57)=62.88, p<.001]: the performances were better after the interventions (M=3.86) than before (M=0.5). The training effect was not significant [F(2,50)=1.53]. The period x training interaction approached significance [F(2,57)=2.97, p=.059]. The pre-planed comparisons showed a significant difference between HVAM and VAM training [F(1,50)=9.98, p<.01]: the number of read pseudo-words was better after HVAM training (M=5.2) than after VAM training (M=2.75). The Newmans-Keuls comparisons showed that there were no differences between the three interventions in pre-test whereas, in post-test, the number of read pseudoword was significantly higher after HVAM training than after VAM-sequential training (M=3.65). No difference was found between VAM and VAM-sequential.III.2 Recognition of the seven target lettersThe mean number (and standard deviations) of correctly recognized target letters (maximum 7) before and after each of the three interventions are presented in Table4.(insert Table4) A 3 (training) x 3 (period) ANCOVA, with the pretest scores in vocabulary, Khos blocks and phonological tests as covariates, was performed on the number of correctly recognized target letters. It revealed a significant effect of period [F(1,57)=54.41, p<.001]: the performances were better after the interventions (M=4.98) than before (M=3.78). The training effect and the interaction were not significant.Haptics in reading acquisition -19II.3. Metaphonological abilitiesThe mean scores (and standard deviations) obtained in the three metaphonological tests before and after each of the three interventions are presented in Table5.(insertTable 5)a. The rhyme test (score out of 6) A 3 (training) x 3 (period) ANCOVA, with the pre-test scores in vocabulary, khos blocks and letters recognition as covariates, was performed on the scores in the rhyme test. It showed a significant effect of period [F(1,57)=18.21, p<.001]: the performances were better after the interventions (M=5.65) than before (M=5.08). The training effect and the interaction were not significant.b. The initial phoneme test (score out of 7)A 3 (training) x 3 (period) ANCOVA, with the pre-test scores in vocabulary, khos blocks and letters recognition as covariates, was performed on the scores in the initial phoneme identification test. It showed a significant effect of period [F(1,57)=51.67, p<.001]: the performances were better after the interventions (M=5.25) than before (M=3.56). The training effect and the interaction were not significant.c. The final phoneme test (score out of 7)A 3 (training) x 3 (period) ANCOVA, with the pre-test scores in vocabulary, khos blocks and letters recognition as covariates, was performed on the scores in the final phoneme identification test. It revealed a significant effect of period [F(1,57)=73.66, p<.001]: the performances were better after the interventions (M=5.36) than before (M=3.7). The training effect and the interaction were not significant.Haptics in reading acquisition -20

Figure 1 :

 1 Figure 1: The typography of each letter

Table 1 :

 1 Characteristics of the children in each group before the interventions.

		EVIP	Kohs Block	Alphabet letters Mean age
		M ()	M ()	M()	M
	HVAM	68.35 (16.06)	25.75 (4.63)	15.2 (6.3)	5 years and 6 months
	VAM-sequential 66.9 (15.03)	23.7 (4.45)	16.5 (6.9)	5 years and 7 months
	VAM	71.8 (16.77)	26.25 (5.32)	13.2 (5.45)	5 years and 7 months

Table 3 :

 3 Mean number (and standard deviations) of read pseudo-words (maximum 12) before and after each of the three interventions.

	Period

Table 4 :

 4 Mean number (and standard deviations) of recognized letters (maximum 7) before and after each of the three interventions.

	Period

Table 5 :

 5 Mean scores (and standard deviations) obtained in the three metaphonological tests before and after each intervention.

		Period
		Pre-test	Post-test
	Test x Training	M (SD)	M (SD)
	Rhyme (max. 6)
	HVAM	5.15 (0.98)	5.55 (1.14)
	VAM-sequential	4.8 (1.32)	5.75 (0.64)
	VAM	5.3 (1.17)	5.65 (0.74)
	Initial phoneme (max. 7)
	HVAM	3.7 (1.89)	5.5 (1.5)
	VAM-sequential	3.35 (1.78)	5.25 (1.44)
	VAM	3.65 (1.79)	5 (1.86)
	Final phoneme (max. 7)
	HVAM	3.4 (1.4)	5.5 (1.6)
	VAM-sequential	3.8 (1.19)	5.6 (1.18)
	VAM	3.9 (1.74)	5 (1.77)

Acknowledgments

This work was supported by the Centre National de la Recherche Scientifique, the University of René Descartes (Paris V), the University of Savoie and the Research Ministry's Cognitique program. We should like to thank Malaurie Onno for helping in the control experiment presented in Appendix. We also should like to thank the three schools that participated to this study and more particularly the teacher Maryline Aubry.

Haptics in reading acquisition -24

APPENDIX: CONTROL STUDY

The main purpose of this control study was to reproduce the results observed both in the main experiment and in [START_REF] Gentaz | Evaluation d'entraînements multisensoriels de préparation à la lecture pour les enfants en grande section de maternelle : une étude sur la contribution du système haptique manuel[END_REF] experiment. The HVAM and VAM interventions were evaluated in forty-two monolingual French children with a mean age of five years seven months (5 years 3 months to 6 years 1 month). Two equivalent groups of 21 children were formed (see Table A).

(insert Table A)

The method (pre-and post-test, the HVAM and VAM interventions) was the same as in the main experiment, except for the proposed phonemes (/a/, /i/, /r/, /l/ /t/). The results in the pseudo-word reading test confirmed the findings observed in the main experiment and by [START_REF] Gentaz | Evaluation d'entraînements multisensoriels de préparation à la lecture pour les enfants en grande section de maternelle : une étude sur la contribution du système haptique manuel[END_REF]


Jürgen Geiser
email: geiser@mathematik.hu-berlin.de

An Iterative Splitting Method via Waveform Relaxation

Keywords: Iterative operator-splitting method, waveform relaxation method, convergence analysis, Pade approximations, fixed-point schemes Iterative splitting method, Waveform-relaxation method, Pade approximations AMS subject classifications. 65M15, 65L05, 65M71

HAL is

Introduction

Historically, iterative splitting methods are related to iterative solver methods. The underlying ideas are based on waveform relaxation methods, which are fixpoint iterations or successive approximation schemes, see [START_REF] Showalter | Hilbert Space Methods for Partial Differential Equations[END_REF]. Their convergence analysis and application to second-order differential equations are studied in [START_REF] Jiang | Convergence analysis of waveform relaxation for nonlinear differential-algebraic equations of index one[END_REF][START_REF] Jiang | Waveform relaxation of nonlinear secondorder differential equations[END_REF][START_REF] Jiang | Periodic waveform relaxation solutions of nonlinear dynamic equations[END_REF][START_REF] Jiang | Periodic waveform relaxation of nonlinear dynamic systems by quasi-linearization[END_REF][START_REF] Jiang | A general approach to waveform relaxation solutions of differentialalgebraic equations: the continuous-time and discrete-time cases[END_REF] They are developed to simplify the solver process, while saving time when computing, e.g. simple diagonal matrices, see [START_REF] Vandewalle | Parallel Multigrid Waveform Relaxation for Parabolic Problems[END_REF]. On the other hand, they can be used to accelerate the iterative process of solving partial differential equations, see [START_REF] Kelley | Solving Nonlinear Equations with Newton's Method[END_REF]. In the next step, the generalization of iterative splitting schemes to unbounded operators allows them be applied to partial differential equations, see [START_REF] Geiser | Consistency of Iterative Operator-Splitting Method: Theory and Applications[END_REF], [START_REF] Geiser | Stability of Iterative Operator-Splitting Methods[END_REF]. In this paper, we deal with a general scheme, a so called multi-stage scheme, which gives a significant improvement in terms of accuracy, numerical stability and reduction of local and global errors.

We concentrate on an approximation to the solution of the linear evolution equation where L, A and B are linear operators. 1.) For the numerical method, we employ an 1-stage iterative splitting scheme, see [START_REF] Vandewalle | Parallel Multigrid Waveform Relaxation for Parabolic Problems[END_REF], also called the Waveform-Relaxation method:

∂u ∂t = Lu = (A + B)u, u(0) = u 0 , (1)
u i (t) = exp(At)u 0 + t 0 exp(A(t -s))Bu i-1 (s) ds, (2)
where i = 1, 2, 3, . . . and u 0 (t) = u(0) is the initial condition.

2.) As a second numerical method, we employ a 2-stage iterative splitting scheme, see [START_REF] Farago | Iterative Operator-Splitting Methods for Linear Problems[END_REF], [START_REF] Geiser | Iterative operator-splitting methods for nonlinear differential equations and applications of deposition processes[END_REF]:

u i (t) = exp(At)u 0 + t 0 exp(A(t -s))Bu i-1 (s) ds, (3)
u i+1 (t) = exp(Bt)u 0 + t 0 exp(B(t -s))Au i (s) ds, (4)
where i = 1, 3, 5, . . . and u 0 (t) = u(0) is the initial condition.

3.) As a third numerical method, the 2-stage scheme is improved by combining it with equations ( 3) and ( 4) in a multiple-stage scheme and can be written generally as an embedded iterative scheme with inner and outer layers.

u i k (t) = exp(At)u 0 + t 0 exp(A(t -s))Bu i k +J k-1 -1 (s) ds, (5)
u j k +I k (t) = exp(Bt)u 0 + t 0 exp(B(t -s))Au j k +I k -1 (s) ds, (6)
where i k = 1, 2, 3, . . . , I k , j k = 1, 2, 3, . . . , J k , k = 1, . . . , K, I 1 , . . . , I K are the number of iterations done with the A-operator, where J 1 , . . . , J K are the number of iterations done with the B-operator. The initialization is given as u 0 (t) = u(0) and J 0 = 0. Here, we can control the iterative steps on each operator A and B.

Remark 1. The motivation for expanding the 1-stage scheme to a 2-stage scheme comes from the reduction of local and global errors, as illustrated in the following example:

Example 1. Let A, B be constant matrices, not dependent of t, not commuting ([A, B] = O ) and no zero matrices (A, B = O). We obtain the following schemes: For an 1-stage scheme (2), for i = 1:

u 1,one (t) = exp(At)u 0 + t 0 exp(A(t -s))Bu 0 ds, (7)
further for an 1-stage scheme (2), for i = 2: and for a 2-stage scheme for i = 2:

u 2,one (t) = exp(At)u 0 + t 0 exp(A(t -s))Bu 1 (s) ds, (8)
u 2,two (t) = exp(Bt)u 0 + t 0 exp(B(t -s))Au 1,one (s) ds. (9)
So we find that equation ( 7) is locally a second order approximation of exp((A + B)t) for t > 0, meaning that:

|| exp((A + B)t)u 0 -u 1,one (t)|| ≤ Ct 2 ||u 0 || + O(t 3 ), (10)
where

C = ||BA|| 2 + ||B 2 || 2
and || • || is the L 1 -norm. Further equation ( 8) is locally a third order approximation of exp((A + B)t) for t > 0, meaning that:

|| exp((A + B)t)u 0 -u 2,one (t)|| ≤ Ĉt 3 ||u 0 || + O(t 4 ), (11)
where Ĉ = ||BAB|| 6

+ ||B 3 || 6
and || • || is the L 1 -norm. and equation ( 9) is locally a third order approximation of exp((A + B)t) for t > 0, meaning that:

|| exp((A + B)t)u 0 -u 2,two (t)|| ≤ Ct 3 ||u 0 || + O(t 4 ), (12)
where

C = ||ABA|| 6 + ||AB 2 || 6 and || • || is the L 1 -norm.
Remark 2. The motivation for expanding the 2-stage scheme to a multi-stage scheme comes from the improvement in numerical stability of the scheme. In numerical examples, the operators are not equal and often one operator can be bounded by the other operator. So, in a multi-stage scheme, we can control the iterative stage over each operator.

Example 2. Let A, B be bounded operators, but ||A|| >> ||B||, while || • || is a matrix norm. We assume 0 < ||B|| ≤ 1 otherwise we have trivially to solve only with operator A.

We apply a 2-stage method until i = 2 and obtain the error bound of equation ( 9) for bounded operators given as:

C = ||ABA|| 6 + ||AB 2 || 6 ≤ ||A 2 || 6 . (13)
We can reduce the error bound and therefore the stability of the scheme, while applying an 1-stage method to operator A, see equation [START_REF] Blanes | Splitting Methods for Non-autonomous separable dynamical systems[END_REF], and obtain the error bound of equation [START_REF] Geiser | Consistency of Iterative Operator-Splitting Method: Theory and Applications[END_REF] for bounded operators as:

Ĉ = ||BAB|| 6 + ||B 3 || 6 ≤ ||A|| 6 ( 14
)
where we have three cases: Here we have some benefits in applying different stages with respect to the underlying operators. Such benefits are discussed later in the error analysis.

F
The outline of the paper is as follows. The iterative splitting methods and their error analysis are presented in Section 2. In Section 3, we discuss an efficient computation of the iterative splitting method with a closed formulation. In Section 4, we discuss the numerical experiments and the benefits of the iterative splitting schemes. Finally, we discuss future works in the area of iterative methods.

Splitting Method and Error Analysis

In the following, the iterative splitting scheme is presented as a multi-stage scheme.

General Iterative Splitting Method (multiple stage scheme)

A general setting of an iterative splitting method is given with inner and outer iterative schemes, so that each part of a 2-stage scheme can be chosen independently to control the stages over each operator. The method is given in the following:

∂u i k +J k-1 (t) ∂t = Au i k +J k-1 (t) + Bu i k +J k-1 -1 (t), (15)
with

u i k +J k-1 (t n ) = u n , ∂u j k +I k (t) ∂t = Au j k +I k -1 (t) + Bu j k +I k (t), (16)
with

u j k +I k (t n ) = u n , where i k = J k-1 + 1, . . . , I k , j k = I k + 1, . . . , J k , k = 1, . . . , K, I k -J k-1
are the number of iterations done with the A-operator, where J k -I k is the number of iterations done with the B-operator. The initialization is given as u 0 (t) = u(0) and I 0 = J 0 = 0.

Error Analysis for the General Scheme

In this section, we analyze the convergence of the general scheme ( 15) and [START_REF] Jiang | Periodic waveform relaxation solutions of nonlinear dynamic equations[END_REF] in which Waveform Relaxation and the iterative splitting method are embedded. We consider the error of the iterative splitting method on a Banach space X with norm and induced operator norm denoted by || • ||.

Further we have the following assumptions: for some ω ≥ 0 and all t ∈ IR.

In the next we present the convergence of the iterative splitting scheme.

Theorem 1. For the numerical solution of ( 3), consider an iterative operator splitting scheme on operator A with i-th iterative steps.

If the assumption 21 is valid, then

||S n i -exp((A + B)nτ )|| ≤ C||B i ||τ i-1 , nτ ≤ T, ( 18
)
where the constant C can be chosen uniformly on bounded time intervals and in particular, independent of n and τ .

Proof. By applying the telescopic identity we obtain

(S n i -exp((A + B)nτ )u 0 = n-1 ν=0 S n-ν-1 i (S -exp((A + B)τ )) exp(ντ (A + B))u 0 , (19)
if we assume the stability bound:

||S i || ≤ exp(cωτ ), (20)
with a constant c only depends on the estimation of the method. Furthermore, if we assume the consistency bound:

||S n i -exp((A + B)nτ )|| ≤ exp(cωT ) n-1 ν=0 ||(S -exp(τ (A + B))) exp(ντ (A + B))|| (21)
≤ Cτ i , nτ ≤ T, (22)
where C is a constant and independent of τ . The desired consistency and stability bound is given in the next subsections.

Consistency and Stability:

Proof. Let us consider the iteration ( 15)-( 16) on the sub-interval [t n , t n+1 ].

For the first iteration of (15), we have:

∂u1(t) ∂t = Au 1 (t), t ∈ (t n , t n+1 ], (23)
and for the second iteration, we have: In general we have: for m = 1, 2, . . . ,

∂u2(t) ∂t = Au 2 (t) + Bu 1 (t), t ∈ (t n , t n+1 ], (24)
∂ui(t) ∂t = Au i (t) + Bu i-1 (t), t ∈ (t n , t n+1 ], (25)
where u 0 (t) ≡ 0.

We have the following solutions for the iterative scheme:

The solutions for the first two equations are given by the variation of constants:

u 1 (t) = exp(A(t n+1 -t))u(t n ), t ∈ (t n , t n+1 ], (26)
u 2 (t) = exp(At)u(t n ) + t n+1 t n exp(A(t n+1 -s))Bu 1 (s)ds, t ∈ (t n , t n+1 ], ( 27
) For m = 0, 1, 2, . . . u i (t) = exp(A(t -t n ))u(t n ) + t t n exp(sA)Bu i-1 (t n+1 -s) ds, t ∈ (t n , t n+1 ]. ( 28
)
The stability is given as:

||S i || ≤ exp(cωτ ), (29)
where c only depends on the coefficients of the method, τ = tt n and ω ≥ 0. and we assume the bound of the operators as:

|| exp(Aτ )|| ≤ exp(ω 1 τ ), ( 30
) || exp(Bτ )|| ≤ exp(ω 2 τ ), (31)
where ω 1 , ω 2 ≥ 0.

The consistency is given as:

For e 1 we have:

u 1 (τ ) = exp(A)τ )u(t n ), (32)
u(τ ) = exp((A + B)τ )u(t n ) = exp(Aτ )u(t n ) (33) + t n+1 t n exp(As)B exp((t n+1 -s)(A + B))u(t n ) ds.
We obtain:

||e 1 || = ||c -u 1 || ≤ || exp((A + B)τ )u(t n ) -exp(Aτ )u(t n )|| (34) ≤ C 1 ||B||τ ||u(t n )||.
where C 1 is a constant and independent of τ .

For e 2 we have:

u 2 (τ ) = exp(Aτ )u(t n ) + t n+1 t n exp(As)B exp((t n+1 -s)A)u(t n ) ds, (35)
u(τ ) = exp(Aτ )u(t n ) + t n+1 t n exp(As)B exp((t n+1 -s)A)u(t n ) ds + t n+1 t n exp(As)B ( 36
) t n+1 -s t n exp(Aρ)B exp((t n+1 -s -ρ)(A + B))u(t n ) dρ ds.
We obtain:

||e 2 || ≤ || exp((A + B)τ )u(t n ) -u 2 || (37) ≤ C 2 ||B 2 ||τ 2 ||u(t n )||.
For the iterations, a recursive proof is given as follows: For m = 0, 1, 2, . . ., for e i , we have :

u i (τ ) = exp(A)τ )u(t n ) (38) + t n+1 t n exp(As)B exp((t n+1 -s)A)u(t n ) ds + t n+1 t n exp(As 1 )B t n+1 -s1 t n exp(s 2 A)B exp((τ -s 1 -s 2 )A)u(t n ) ds 2 ds 1 + . . . + + t n+1 t n exp(As 1 )B t n+1 -s1 t n exp(s 2 A)B exp((τ -s 1 -s 2 )A)u(t n ) ds 2 ds 1 + . . . + + t n+1 t n exp(As 1 )B t n+1 - P i-1 j=1 s1 t n exp(s 2 A)A exp((τ -s 1 -s 2 )A)u(t n ) ds 2 ds 1 . . . ds i , u(τ ) = exp(Aτ ) + t n+1 t n exp(As)B exp((t n+1 -s)A)u(t n ) ds (39)
+ . . . + + t n+1 t n exp(As 1 )B t n+1 -s1 t n exp(s 2 A)B exp((τ -s 1 -s 2 )A)u(t n ) ds 2 ds 1 + . . . + + t n+1 t n exp(As 1 )B t n+1 - P i-1 j=1 s1 t n exp(s 2 A)B exp((τ -s 1 -s 2 )A)u(t n ) ds 2 ds 1 . . . t n+1 - P i j=1 s2 t n exp(s 2 A)B exp((τ -s 1 -s 2 )(A + B))u(t n )ds i ,
We obtain:

||e i || ≤ || exp((A + B)τ )u(t n ) -u i || (40) ≤ C||B i ||τ i ||u(t n )||,
where i is the iterative step, C is a constant independent of t.

The same idea can be applied to the operator B. Remark 3. At least we obtain for the general iterative splitting method with I iterative steps to operator A and J iterative steps to operator B the following consistency error:

||e I+J || ≤ || exp((A + B)τ )u(t n ) -u I+J || (41) ≤ C||A I || ||B J ||τ I+J ||u(t n )||,
where C is a constant and independent of τ .

Embedding the result to the convergence results we obtain at least an error of O(τ I+J-1 ).

In the next section, we describe the computation of the integral formulation with exponential functions.

Computation of the Iterative Splitting Method: Closed Formulation

In the last few years, the computational effort to compute integrals with expfunctions has increased. We present a closed form and re-substitute the integral with closed functions. Such benefits accelerate the computation and allow the ideas to be parallelized.

Here, we present a closed form for the iterative splitting method for the first 4 splitting iterations.

For i = 1, we have:

u 1 (t) = exp(At) exp(Bt)u(t n ). ( 42
)
where we have a first order method, also known as an AB splitting method, see [START_REF] Farago | Iterative Operator-Splitting Methods for Linear Problems[END_REF]. For i = 2, we have:

u 2 (t) = 1 2 (exp(At) exp(Bt) + exp(Bt) exp(At))u(t n ), (43)
where we have a second order method, also known as a parallel AB splitting method, see [START_REF] Farago | Iterative Operator-Splitting Methods for Linear Problems[END_REF]. For i = 3, we have: Here an optimization is possible by assuming that commutators are equal or at least zero, see [START_REF] Blanes | Splitting Methods for Non-autonomous separable dynamical systems[END_REF] and [START_REF] Geiser | Consistency of Iterative Operator-Splitting Method: Theory and Applications[END_REF].

u 3 (t) = 1

Exp-Approximations with Pade Approximations

In applications, we have to extend differential equations to systems of differential equations. Therefore, we have to apply matrix functions to our analytical tools.

To approximate matrix functions in the following section, we apply Pade approximations.

For the matrix exponential we apply:

I + 1 2 At 1 -1 2 At = exp(At) + O((At) 3 ) , (45)
I + 2 3 (At) + 1 6 (At) 2 I -1 3 At = exp(At) + O((At) 4 ) , (46)
where A ∈ IR n×n is the matrix.

Remark 5. The general formulation for different Pade approximations applied to exponential functions exp(At) is given in [START_REF] Baker | Pade approximants[END_REF].

In the next experiments, we apply the Pade approximations for m = n = 1, m = n = 2 and m = n = 3.

Numerical Experiments

In the following experiment, we start by presenting the enhancement of the multi-stage procedure to a standard Waveform relaxation method. Further, we present the improvement of the iterative splitting scheme to a classical splitting method.

First Example: Benchmark with 2 × 2 Matrix

In the first experiment, we deal with an ODE and separate the complex operator into two simpler operators.

We deal with the following equation :

∂u 1 ∂t = -λ 1 u 1 + λ 2 u 2 , ( 47
)
∂u 2 ∂t = λ 1 u 1 -λ 2 u 2 , (48)
u 1 (0) = u 10 , u 2 (0) = u 20 (initial conditions) , (49)
where λ 1 , λ 2 ∈ IR + are the decay factors and u 10 , u 20 ∈ IR + . We have the time interval t ∈ [0, T ]. We rewrite equation (47) in operator notation, we concentrate on the following equations :

∂u ∂t = A(t)u + B(t)u , (50) (51)
where u 1 (0) = u 10 = 1.0 , u 2 (0) = u 20 = 1.0 are the initial conditions, where we have λ 1 (t) = t and λ 2 (t) = t 2 . Our split operators are:

A = -λ 1 λ 2 0 0 , B = 0 0 λ 1 -λ 2 . ( 52
)
The actual parameters for the experiments are given as:

λ 1 = 0.05 λ 2 = 0.01 T = 1.0 u 0 = (1, 1) t
We apply AB, Strang and 3rd order splitting and compare with the unsplit solutions:

1.) Unsplit :

u exact (t) = exp((A + B)t)u(t n ). ( 53
) 2.) A-B splitting u 1 (t) = exp(Aτ ) exp(Bτ )u(t n ), (54)
where we have a first order method, also known as an AB splitting method, see where the solution is derived from the iterative splitting methods. The L 1 -error is computed as:

err num = N k=1 |u exact (t k ) -u num (t k )|, ( 57
)
where t k = k∆t, where t 0 , t 1 , . . . and ∆t = 0.1. 1. Numerical results for the first example with the iterative splitting method and 2nd-and 3rd-order method. Remark 6. Our numerical results are based on higher order iterative schemes in closed formulations. Table 1 presents the results in which the 3rd order methods can achieve more accurate results. The numerical results show that the splitting error decreases as long as the Pade approximations employed allow it. Therefore, we can say that more iterations are only sufficient when a higher order method is used. One can also see that the iterative operator-splitting method is of order i as long as the Pade approximation is also of order i.

F

Second Experiment

In the second experiment, we deal with a time-dependent ODE and separate the complex operator into two simpler operators.

We deal with the following equation :

∂u 1 ∂t = -λ 1 (t)u 1 + λ 2 (t)u 2 , ( 58
)
∂u 2 ∂t = λ 1 (t)u 1 -λ 2 (t)u 2 , ( 59
)
u 1 (0) = u 10 , u 2 (0) = u 20 (initial conditions) , (60)
where λ 1 (t) ∈ IR + and λ 2 (t) ∈ IR + are the decay factors and u 10 , u 20 ∈ IR + . We have the time-interval t ∈ [0, T ]. We rewrite the equation (58) in operator notation and concentrate on the following equations :

∂u ∂t = A(t)u + B(t)u , (61) (62)
where u 1 (0) = u 10 = 1.0 , u 2 (0) = u 20 = 1.0 are the initial conditions, where we have λ 1 (t) = t and λ 2 (t) = t 2 . and our split operators are For equation (58), we could apply a higher order Pade approximation, e.g. 3rd order.

A(t) = -λ 1 (t) λ 2 (t) 0 0 , B(t) = 0 0 λ 1 (t) -λ 2 (t) . ( 63
To start, we apply sequential splitting and the iterative operator-splitting, further we combine them by using pre-step methods to see the improvement in results.

For time-steps ∆t, we have ∆t = 1 for 1 time-partition and ∆t = 0.1 for 10 time-partitions. number of err1 (2nd order) err2 (2nd order) err1 (3rd order) err2 (3rd order) time partitions 1 4.5321e-002 3.6077e-003 4.5321e-002 3.6077e-003 10 4.6126e-004 3.6077e-003 4.6126e-004 3.6077e-003 Table 2. Numerical results for the second example with the iterative splitting method and 2nd and 3rd order method.

Third Experiment

We tackle the 2-dimensional advection-diffusion equation with periodic boundary conditions

∂ t u = -v∇u + D∆u, (64)
= -v x ∂u ∂x -v y ∂u ∂y + D ∂ 2 u ∂x 2 + D ∂ 2 u ∂y 2 , (65) u
(x, t 0 ) = u 0 (x), (66)
with parameters

v x = v y = 1, (67)
D = 0.01, (68)
t 0 = 0.25. (69)
The advection-diffusion problem has an analytical solution

u a (x, t) = 1 t exp -(x -vt) 2 4Dt , (70)
which we will use as a convenient initial function:

u(x, t 0 ) = u a (x, t 0 ). (71)
We apply dimensional splitting to our problem: where:

∂u ∂t = A x u + A y u, (72)
A x = -v x ∂u ∂x + D ∂ 2 u ∂x 2 (73)
A y = -v y ∂u ∂y + D ∂ 2 u ∂y 2 . ( 74
)
We use a 1st order upwind scheme for ∂ ∂x and a 2nd order central difference scheme for ∂ 2 ∂x 2 . By introducing the artificial diffusion constant

D x = D -vx∆x 2
we achieve a 2nd order finite difference scheme

L x u(x) = -v x u(x) -u(x -∆x) ∆x (75)
+ D x u(x + ∆x) + u(x) + u(x -∆x) ∆x 2 ,
because the new diffusion constant eliminates the first order error (i.e. numerical viscosity) of the Taylor expansion of the upwind scheme. L y u is derived in the same way.

We apply a BDF5 method to gain 5th order accuracy in time:

1 L t u(t) = 1 ∆t 137 60 u(t + ∆t) -5u(t) + 5u(t -∆t) - 10 3 u(t -2∆t) + 5 4 u(t -3∆t) - 1 5 u(t -4∆t) . (76)
Our aim is to compare the iterative splitting method with AB splitting (Lie-Trotter splitting, see [START_REF] Farago | Iterative Operator-Splitting Methods for Linear Problems[END_REF]). Since [A x , A y ] = 0 there is no splitting error for AB splitting and therefore we cannot expect to achieve better results with the iterative splitting in terms of general numerical accuracy. Instead, we will show that iterative splitting out-competes AB splitting in terms of computational effort and round-off errors. But first some remarks need to be made about the special behaviour of both methods when combined with high-order Runge-Kutta and BDF methods.

Splitting and Schemes of High Order in Time, Concerning AB-Splitting:

The principle of AB-splitting is well known and simple. The equation du dt = Au + Bu is broken up into:

du n+1/2 dt = Au n+1/2 , du n+1 dt = Bu n+1 ,
which are connected via u n+1 (t) = u n+1/2 (t + ∆t). This is pointed out in figure [START_REF] Baker | Pade approximants[END_REF]. AB splitting works very well for any given one-step method like the Crank- Nicholson-Scheme. Not taking into account the splitting error (which is an error in time) it is also compatible with high order schemes such as explicit/implicit Runge-Kutta-schemes.

Things look different if one tries to use a multi-step method like the implicit BDF or the explicit Adams method with AB splitting, as these cannot be properly applied as shown by the following example: Choose for instance a BDF2 method which, in the case of du/dt = f (u), has the scheme

3 2 u(t + ∆t) -2u(t) + 1 2 u(t -∆t) = ∆tf (u(t + ∆t)).
So, the first step of AB splitting looks like:

3 2 u n+1/2 (t + ∆t) -2u n+1/2 (t) + 1 2 u n+1/2 (t -∆t) = ∆tAu(t + ∆t)
Clearly, u n+1/2 (t) = u n (t) but what is u n+1/2 (t -∆t)? This is also shown in figure [START_REF] Baker | Pade approximants[END_REF] and it is obvious that we won't have knowledge about u n+1/2 (t -∆t) unless we compute it separately, which means additional computational effort. This overhead increases dramatically when we move to a multi-step method of higher order. The mentioned problems with AB splitting will not occur with a higher order Runge-Kutta method since only knowledge of u n (t) is needed. Remarks about iterative splitting: The BDF methods apply very well to iterative splitting. Let us recall at this point that this method, although being a real splitting scheme, always remains a combination of the operators A and B, so no steps have to be done in one direction only 2 .

In particular, we make a subdivision of our existing time-discretization t j = t 0 + j∆t into I parts. So we have sub-intervals t j,i = t j + i∆t/I, 0 ≤ i ≤ I on which we solve the following equations iteratively:

du i/I dt = Au i/I + Bu (i-1)/I (77) du (i+1)/I dt = Au i/I + Bu (i+1)/I (78) (79)
u -1/I is either 0 or a reasonable approximation 3 while u 0 = u(t j ) and u 1 = u(t j + ∆t). The crucial point here is that we only know our approximations at given times which happen not to be the times at which a Runge-Kutta (RK) method needs to know them. Therefore, in the case of a RK method, the values of the approximations have to be interpolated with at least the accuracy one wishes to attain with the splitting and this means a lot of additional computational effort. We can now summarize our results in table 4.3 which shows which methods are practicable for each kind of splitting scheme.

Numerical results

After resolving the technical aspects of this issue, we can now proceed to the actual computations. A question which arises is which of the 2 As we will see there is an exception to this. 3 In fact the order of the approximation is not of much importance if we fulfill a sufficient number of iterations. In the case of u -1/I = 0, we have the exception that a step in the A-direction is done while B is left out. The error of this step certainly vanishes after a few iterations, but mostly after only one iteration 4 Something in favour of the iterative splitting scheme is that it also takes into the account the fact that AB splitting may be used alongside the high order methods alluded to but cannot maintain the order if [A, B] = 0, while the iterative splitting scheme re-establishes the maximum order of the scheme after a sufficient number of iterations have been done. splitting methods requires the least computational effort since we can expect them to solve the problem with more or less the same accuracy if we use practicable methods with equal order, as [A x , B x ] = 0. We tested the dimensional splitting of the 2-dimensional advection-diffusion equation with AB splitting combined with a 5th order RK method after Dormand and Prince, and with iterative splitting in conjunction with a BDF5 scheme. We used 40x40 and 80x80 grids and completed n t time steps each subdivided into 10 smaller steps until we reached time t end = 0.6 which is sufficient to see the main effects. Iterative splitting was performed with 2 iterations which was already enough to attain the desired order. In tables 4 and 5, the errors at time t end and the computation times are shown. Remark 7. As we can see, the error in the iterative splitting scheme reaches the same value as the AB splitting error after a certain number of time steps and stays below it for all additional steps we accomplish. Of course, the error cannot drop below a certain value which is governed by the spatial discretization increments. It is to be noticed that, while the computation time used for iterative splitting is always about 20%-40% less than that of AB splitting 5 the accuracy is, with a sufficient number of time steps, slightly better than that of AB splitting. This is due to the roundoff error which is higher for the Runge-Kutta method A future task will be to introduce non-commuting operators in order to show the superiority of iterative splitting over AB splitting when the order in time is reduced due to the splitting error.

Conclusions and Discussions

We have presented an iterative operator-splitting method and analyzed the error bounds for linear operators. Under weak assumptions, we could prove the higher order error bounds. The benefit of higher accuracy and more computational efficiency with respect to the multi-stage scheme shows the importance of iterative splitting schemes. Numerical examples confirm the potential of applying our new scheme to differential equations. In the future, we will focus on the development of improved operator-splitting methods which are better adapted to applications involving nonlinear differential equations.

Assumption 21

 21 The linear operators A + B, A, B generate C 0 semigroups on X, and the operators A, B satisfy in addition the bounds:|| exp(Aτ )|| ≤ exp(ω|t|) and || exp(Bτ )|| ≤ exp(ω|t|)[START_REF] Jiang | Periodic waveform relaxation of nonlinear dynamic systems by quasi-linearization[END_REF]

6 (Remark 4 .

 64 exp(At) exp(Bt) exp(At) + exp(Bt) exp(At) exp(At) (44)+ exp(Bt) exp(Bt) exp(At) + exp(At) exp(At) exp(Bt) + exp(At) exp(Bt) exp(Bt) + exp(Bt) exp(At) exp(Bt))u(t n ),where we can reduce the operators by making assumptions about the commutators, e.g.[A, [A, B]] = [B, [A, A]]. Higher order iterative splitting methods involve at the very least the derivation of the remaining forms for all commutations between operators A and B.

Fig. 1 .

 1 Fig. 1. Principle of the AB-Splitting.

  greater amount of basic operations needed to compute the RK steps.

Table 3 .

 3 Practicability

	AB-splitting	X	X	-
	Iterative splitting	X	-	X

4

low order s.s.m. high order s.s.m. m.s.m. of single-and multi-step methods (s.s.m: single-step methods, m.s.m. multi-step methods).

Table 4 .

 4 Number of steps Error AB Error It.spl. AB computation time It. spl. Errors and computation times of AB splitting and iterative splitting for a 40x40 grid. Number of steps Error AB Error It.spl. AB computation time It. spl.

	computation time

Table 5 .

 5 Errors and computation times of AB splitting and iterative splitting for a 80x80 grid.