FRSA Luca F Ticini
email: luca.ticini@manchester.ac.uk

Florian Waszak

Simone Schütz-Bosbach

From goals to muscles: motor familiarity shapes the representation of actionrelated sounds in the human motor system

Keywords: action, mirror neurons, transcranial magnetic stimulation, somatotopic, sound, motor familiarity

Numerous studies corroborated the idea that the sound of familiar motor acts triggers a muscle-specific replica of the perceived actions in the listener's brain. We recently contradicted this conclusion by demonstrating that the representation of newly-learned action-related sounds is not somatotopically organised but rather it corresponds to the goal a particular action aims to achieve. In the present study, we reconciled these contrasting results by showing that a higher degree of motor familiarity obtained via long-term practice with auditorymotor associations shapes a somatotopically organised representation of action-related sounds. We measured motor-evoked potentials (MEPs) to TMS as an index of the functional correspondence between the sensory stimulation and the activity in the listener's motor cortex. Participants heard two tones of different pitch, void of previous motor meaning, before and after an acquisition phase in which they generated these tones by performing 400 free-choice button presses. After that, we disentangled the representation of the action goal (button-tone association) from the somatotopic (muscle-tone) association by reversing the muscle-button contingencies. Our result supports the hypothesis that the neuronal representations of action-related sounds depends on motor familiarity: perceptuomotor representations of newly-learned actions are muscleindependent and corresponded to the button-tone contingencies; whilst longer-term practice results in representations that relied on lower-level intrinsic parameters associated with the kinematics of specific movements.

INTRODUCTION

Despite the most apparent function of the cortical motor areas being to generate voluntary movements, their activity is also measured in the absence of overt motor acts whilst observing actions executed by others. Such a functional equivalence between action execution and action perception was originally proposed by William James [START_REF] James | The principles of psychology[END_REF]. Later on, it was experimentally substantiated by numerous studies in nonhuman and human primates (reviewed in [START_REF] Rizzolatti | The mirror-neuron system[END_REF]. Nowadays, it is generally accepted that the motor modulation (or motor resonance) associated with action perception reflects a neuronal process coding for the goal of other individuals' actions, as demonstrated in neurophysiological studies in non-human primates (Umilta et al., 2001;[START_REF] Kohler | Hearing sounds, understanding actions: action representation in mirror neurons[END_REF][START_REF] Keysers | Audiovisual mirror neurons and action recognition[END_REF] and in transcranial magnetic stimulation and fMRI experiments in humans [START_REF] Cattaneo | Representation of goal and movements without overt motor behavior in the human motor cortex: a transcranial magnetic stimulation study[END_REF][START_REF] Jastorff | Coding observed motor acts: different organizational principles in the parietal and premotor cortex of humans[END_REF].

Nonetheless, the evidence gathered (reviewed in Aglioti and Pazzaglia, 2010) when the visual information is absent and actions are perceived only through their sound (such as when hearing someone breaking a nut) has generally failed to show goal-related modulations in the listener's brain (e.g., [START_REF] Ticini | Mirror and (absence of) counter-mirror responses to action sounds measured with transcranial magnetic stimulation[END_REF]. Instead, measuring the neuronal discharge in response to action-related sounds usually revealed a faithful somatotopic (i.e., body part-specific) replica of the heard movements in a left fronto-parietal network [START_REF] Pazzaglia | The sound of actions in apraxia[END_REF][START_REF] D'ausilio | The motor somatotopy of speech perception[END_REF][START_REF] Möttönen | Motor representations of articulators contribute to categorical perception of speech sounds[END_REF][START_REF] Bangert | Mapping perception to action in piano practice: a longitudinal DC-EEG study[END_REF][START_REF] Tettamanti | Listening to action-related sentences activates fronto-parietal motor circuits[END_REF][START_REF] Gazzola | Empathy and the somatotopic auditory mirror system in humans[END_REF][START_REF] Lahav | Action representation of sound: Audiomotor recognition network while listening to newly acquired actions[END_REF]. A classical example of such muscle-specific motor pattern is the activity of the functionally defined hand motor area triggered by the sound of finger-clicks obtained in a rare intracranial EEG recording investigation conducted in an epileptic 12-year-old girl (Lepage et al., 2010a). Similarly, a magnetoencephalography study showed that the oscillatory activity associated with movement execution (cf. [START_REF] Salmelin | Spatiotemporal characteristics of sensorimotor neuromagnetic rhythms related to thumb movement[END_REF][START_REF] Murthy | Coherent 25-to 35-Hz oscillations in the sensorimotor cortex of awake behaving monkeys[END_REF] was also present after visual and auditory presentation of actions [START_REF] Caetano | Actor's and observer's primary motor cortices stabilize similarly after seen or heard motor actions[END_REF]. Other examples include hand clapping or finger tapping [START_REF] Caetano | Actor's and observer's primary motor cortices stabilize similarly after seen or heard motor actions[END_REF][START_REF] Aziz-Zadeh | Congruent embodied representations for visually presented actions and linguistic phrases describing actions[END_REF][START_REF] Gazzola | Empathy and the somatotopic auditory mirror system in humans[END_REF]Hauk, Shtyrov, & Pulvermuller, 2006;[START_REF] Pizzamiglio | Separate neural systems for processing action-or non-action-related sounds[END_REF][START_REF] Aziz-Zadeh | Left hemisphere motor facilitation in response to manual action sounds[END_REF]), speech listening (Buccino et al., 2005;[START_REF] Tettamanti | Listening to action-related sentences activates fronto-parietal motor circuits[END_REF][START_REF] Wilson | Listening to speech activates motor areas involved in speech production[END_REF][START_REF] Watkins | Seeing and hearing speech excites the motor system involved in speech production[END_REF][START_REF] Fadiga | Speech listening specifically modulates the excitability of tongue muscles: a TMS study[END_REF] and even piano playing [START_REF] Lahav | Action representation of sound: Audiomotor recognition network while listening to newly acquired actions[END_REF]D'Ausilio, Altenmüller, Olivetti Belardinelli, & Lotze, 2006;[START_REF] Bangert | Shared networks for auditory and motor processing in professional pianists: Evidence from fMRI conjunction[END_REF][START_REF] Bangert | Mapping perception to action in piano practice: a longitudinal DC-EEG study[END_REF][START_REF] Lotze | The musician's brain: functional imaging of amateurs and professionals during performance and imagery[END_REF][START_REF] Haueisen | Involuntary motor activity in pianists evoked by music perception[END_REF]. This functional correspondence between the auditory and motor aspects of actions has been further substantiated in patients with cortical injuries, and in healthy subjects by means of cortical inactivation. In both instances, impairing the execution of movements interfered with the recognition and categorisation of motor acts [START_REF] Pazzaglia | The sound of actions in apraxia[END_REF]D'Ausilio et al., 2009;[START_REF] Möttönen | Motor representations of articulators contribute to categorical perception of speech sounds[END_REF].

That is the case, for instance, in brain-damaged patients suffering from limb or buccofacial apraxia (i.e., the inability to perform specific gestures after left fronto-parietal damage), as they are unable to match representations of mouth or face-related actions with their corresponding sounds.

In an earlier transcranial magnetic stimulation (TMS) work, we tested the degree of somatotopicity in the listeners' motor cortex [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF] by measuring sound-related modulations during passive tone perception, before and after a short acquisition phase in which participants voluntarily triggered two tones by index and little finger button presses, respectively (Figure 1 in [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF]; see also, [START_REF] Elsner | Effect anticipation and action control[END_REF]. Importantly, in a subsequent test phase we disentangled the action goal (generating a tone by pressing a button) from the movements used to achieve it by changing the position of the participants with respect to the buttons. As a consequence, the representation of the finger-button mapping (visible to the participant) was reversed relative to the trained one (Experiment 2 -reverse mapping, in [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF]. The results demonstrated, to our knowledge for the first time, that a functional correspondence between the sounds and their associated actions is established very rapidly (after only 200 trials or 3 to 5 minutes) and, most interestingly, that in the reversed finger-button mapping condition the perception of each tone facilitated the action needed to produce it rather than the movement that was associated with the tone during the training phase. This was a clear indication that the cortical activity reflected a motor representation of the goal of the action (pressing a button to generate a sound in the current context) rather than the movements learned during the training.

In the present study, we aimed at assessing whether we could reconcile these results with the evidence arguing for a somatotopic motor activation during passive listening to action-related sounds. We hypothesised that familiarisation (i.e., increasing the number of trials during training to 400 trials or 6 to 10 minutes) with the experimental setup and its goals is key in determining the outcome of our experiment, as it is known that the correspondence between action execution and action perception is sensitive to previous perceptuomotor experience (e.g., [START_REF] Aglioti | Action anticipation and motor resonance in elite basketball players[END_REF][START_REF] Calvo-Merino | Seeing or doing? Influence of visual and motor familiarity in action observation[END_REF][START_REF] Cross | Building a motor simulation de novo: observation of dance by dancers[END_REF], including semantic differences (e.g., between ripping paper and vocalisation) and anticipation effects (e.g., predictability of a sequence of notes in a rehearsed piano melody; e.g., [START_REF] Lahav | Action representation of sound: Audiomotor recognition network while listening to newly acquired actions[END_REF]D'Ausilio et al., 2006).

In particular, also drawing from the motor learning literature (reviewed in [START_REF] Hikosaka | Parallel neural networks for learning sequential procedures[END_REF], we expected that newly-trained sensorimotor associations are coded in an abstract goal-directed way (mainly relying on extrinsic coordinate frames, such as the position of the buttons and the association between each button and its corresponding tone). Indeed, as executing an action in the environment often results in kinematically different motor acts producing the same sensory effects (or goals), the corresponding motor representation must be independent of the muscle that might be used to do so. Instead, we expected that longterm practice would result in effector dependency (relying on intrinsic coordinate frames that refer to the specific motor patterns associated with the sounds during training). As a matter of fact, if the association is over-learned, sensorimotor associations may tend to become more stable and consolidated [START_REF] Walker | Dissociable stages of human memory consolidation and reconsolidation[END_REF][START_REF] Krakauer | Consolidation of motor memory[END_REF] in a somatotopically organised manner.

Relying on our previous results [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF], we predicted that each training associating selfgenerated actions with arbitrary sounds would generate action-sound representations that would become apparent as a new pattern of finger-specific motor facilitation (e.g., [START_REF] Aziz-Zadeh | Left hemisphere motor facilitation in response to manual action sounds[END_REF][START_REF] Watkins | Seeing and hearing speech excites the motor system involved in speech production[END_REF][START_REF] Fadiga | Speech listening specifically modulates the excitability of tongue muscles: a TMS study[END_REF]. In particular, we expected to demonstrate that a longer training in the acquisition phase can reverse the pattern of results associated with newly-trained associations: that is, each tone would facilitate the motor program that was associated with it in the training.

To demonstrate that, we recorded motor-evoked potentials (MEPs) to single magnetic TMS pulse applied over the left primary motor cortex of healthy volunteers (see Methods). We recorded MEPs from the hand muscles specifically involved in the acquisition phase: the abductor digiti minimi (ADM; little finger muscle) and first dorsal interosseous (FDI; index finger muscle). This experimental approach allowed to measure the correspondence between auditory and motor codes in the listener's motor system at the time the TMS impulse was applied.

METHODS

Participants and Experimental Protocol

The experiment was carried out on ten right-handed healthy undergraduate students (required size determined through the G* power software as described in the Supplementary Methods; [START_REF] Faul | Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses[END_REF] who were naive to the purpose of the experiment, with ethical-committee approval and informed written consent.

We employed the same experimental equipment and setting described in [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF]. Briefly, participants sat comfortably with their forearms on a table. They were asked to observe their right hand throughout the experiment, which was positioned between two buttons (starting position): one located on the right side of the little finger and the other on the left side of the index finger. The experiment started with an Experimental Baseline (EB; Figure 1), in which we tested pre-existing motor modulations in response to action-related sounds by recording motor evoked potentials (MEPs) whilst participants passively listened to two tones of different pitch (see Supplementary Methods). Then, in an Acquisition Phase (AP), they trained to contingently associate two button presses with two corresponding tones (those already heard in EB) by performing 400 free-choice button presses by abducting (from the starting position) the index finger (to press the left button) or the little finger (to press the right button). These movements maximised the activity of the first-dorsal interosseous (FDI) and the abductor digiti minimi (ADM) muscles, respectively. The assignment of tones (high vs. low) to each button (left vs. right) of the AP was randomised and counterbalanced across participants. In other words, for half of the participants, the high tone was associated with the left button and the low tone with the right one. In the other half, the high tone was associated with the right button and the low tone with the left one. A Test Phase (TP) followed, in which MEPs were acquired whilst participants passively listened to the same tones as in EB.

In both EB and TP, during each trial we simultaneously recorded two MEPs (one from FDI and one from ADM) in response to single transcranial magnetic stimulation (TMS) pulses applied over each participant's right hand representation in the left primary motor cortex (see Supplementary Methods). In TP, each MEP was labeled as either "Congruent" or "Incongruent" (CONGRUENCY factor) in the following way.

If in a trial the tone heard was previously associated (in the AP) to the left button then the MEPs recorded from the FDI were labeled as "Congruent". Indeed, in AP this tone was associated to left button-presses and required an index finger movement (i.e. activity of the FDI muscle). On the contrary, the MEPs recorded in this trial from the ADM were labeled as "Incongruent" (given that that the little finger's movement was not associated to that sound during AP). The opposite labelling was assigned for trials with tones that were elicited by right button presses during AP (i.e., tones that were generated by little finger movement or activity of the ADM muscle): MEPs recorded from the ADM were labeled as "Congruent" while those recorded from the FDI were labeled as "Incongruent". Thus, in each TMS trial in TP, one of the (simultaneously measured) muscles was associated with the presented tone, and the other was not (see also [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF].

Importantly, in TP, in order to dissociate the representation of the button-tone relationship (each button generates a specific tone) from that of the muscle-tone relationship (each tone is triggered by the movement of a particular muscle), the participants moved to the other side of the table (i.e., they rotated 180° from the AP position, as depicted in Figure 1). In this new position, their hands were again placed between the two buttons, but this time the finger-button mapping was opposite to that trained in AP. Participants underwent no further training according to their new seating position. Instead, they passively listened to the tones generated during the previous AP, presented in random order. The buttons were differently coloured (blue and yellow) to make more explicit the fact that the finger-button mapping had changed.

Overall, in TP the CONGRUENCY factor encoded the somatotopic mapping effect (trials congruent with the finger-button mapping trained in AP). To ensure that the effect was not driven by a particular muscle (FDI or ADM), the factor MUSCLE encoding the muscle from which the MEP was recorded was also used for data analysis.

--Figure 1 about here--

Data analysis

MEP amplitudes were normalised (z-scores) to the average MEP amplitude recorded for each participant, muscle and session (see Supplementary Methods and Table S1). Statistical analysis was performed separately for each session using mean normalised MEP as dependent variable in repeated measures ANOVAs.

In EB, TONE (Tone 1 or Tone 2) and MUSCLE (FDI or ADM) were within-participants factors. In TP, CONGRUENCY (Congruent or Incongruent tone; in EB, these factors refer to the future muscle-tone combinations) and MUSCLE (FDI or ADM) were within-participants factors. We used the partial eta squared (ηp 2 ) to determine the effect size and set the alpha level at 0.05 for all statistical tests. We also computed Bayes Factors (BF; Rouder, Morey, Speckman, & Province, 2012) in JASP (Love et al., 2015) to estimate the likelihood of the null hypothesis relative to the alternative hypothesis [START_REF] Rouder | Bayesian t tests for accepting and rejecting the null hypothesis[END_REF].

RESULTS

In EB, no significant results were found (all F statistics had p-values > 0.6). This excluded that the auditory stimuli were associated with an action before the beginning of the experiment. Instead, in TP the main effect of CONGRUENCY was statistically significant [F (1,9) = 5.18, p < 0.05, ηp 2 = 0.37], whilst the main effect of MUSCLE and the interaction between CONGRUENCY and MUSCLE were not (all F statistics had p-values > 0.5). A bayesian repeated measures ANOVA (Love et al., 2015) with default prior scales revealed that the main effect model of CONGRUENCY was preferred to the interaction model by a BF of 4.6. This provided further evidence in favour of the hypothesis of a main effect of CONGRUENCY by a ratio of 4.6:1. This outcome indicated that the pattern of MEP reflected the associations trained in AP in a muscle-specific manner (Figure 2). In particular, the mean normalised MEPs measured in the Congruent condition were larger when compared to the Incongruent condition (i.e., the pattern was the same to that trained in AP, instead of being goal-directed as that in [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF].

--Figure 2 about here--

DISCUSSION

In the present study, we investigated whether the degree of sensorimotor familiarity shapes the representation of action-related sounds in the adult human motor system. To achieve this, we employed a twoactions / two-muscle paradigm in which in an Acquisition Phase (AP in Figure 1) participants learned to generate two different tones (without previous motor, verbal, or semantic meaning, as demonstrated by the null results in EB) by using two muscles executing two actions (Figure 1). Then, in a Test Phase (TP), we recorded the sound-related covert activation of the listeners' motor cortex after having reversed the fingerbutton contingencies.

The result demonstrated that intensive practice or familiarisation with sensory-motor associations (i.e., from our previous work [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF], in which newly-trained sensory-motor associations were coded in an abstract and goal-specific manner. In other words, in our earlier experiment, passive listening to tones specifically facilitated the motor program that should have generated them in the reversed context. For instance, MEPs recorded from each muscle, instead of being larger for the tone associated to it during AP (Congruent condition), they were larger for the other tone (Incongruent condition). Here we observed the opposite pattern of MEPs: each tone elicited larger MEPs in the same muscle that previously triggered it in AP (Congruent condition) when compared to the MEPs recorded from the other muscle (Incongruent condition). What does this outcome represent?

When we repeat a motor act that is causally followed by a sensory outcome, synaptic potentiation can establish a stable correspondence [START_REF] Hebb | The organisation of behaviour[END_REF][START_REF] Keysers | Demystifying social cognition: a Hebbian perspective[END_REF][START_REF] Keysers | Hebbian learning is about contingency, not contiguity, and explains the emergence of predictive mirror neurons[END_REF] between action and perception. In turn, this association can be used in the reverse direction when a movement is induced by anticipating or perceiving its sensory effects [START_REF] Herwig | Two modes of sensorimotor integration in intention-based and stimulus-based actions[END_REF][START_REF] Schütz-Bosbach | Perceptual resonance: action-induced modulation of perception[END_REF][START_REF] Hommel | The Theory of Event Coding (TEC): a framework for perception and action planning[END_REF][START_REF] James | The principles of psychology[END_REF]. We have previously demonstrated [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF] that even a short, minute-based training of voluntary self-executed actions suffices to the brain to obtain an accurate knowledge about the permanent features of the experimental setup that we employed: that is, the association between a tone and the corresponding button. The tests executed clearly argued that the motor system holds a muscle-independent representation of the actions participants trained, which takes into account the extrinsic coordinate frames for the accomplishment of the goal of the action (i.e., for the act of generating a particular tone by pressing a specific button). For this reason, in the reversed Test Phase of our earlier work, we found that the tones elicited larger MEPs in the Incongruent condition when compared to the Congruent one. Here, we went a step forward and showed that this is only true when a limited number of associative training trials is rehearsed. Indeed, when participants obviously over-familiarised with the setup, the MEPs recorded covertly replicated the kinematics of movements trained in AP.

The importance of these results lies in the reconciliation of a role of the motor cortex in goal coding with the high degree of somatotopicity observed during the perception of action-sounds (see Introduction). This to say that just changing the length of associative learning paradigms (in our case from 3 to 5 min to 6 to 10 min) may have different outcome on the representation of perceived actions in the human brain. When we use to perform goal-directed actions with multiple effectors (for instance, switching the light on with the index finger, a second time with the elbow if we are holding something in our hands, and so on) it is likely that the representation obtained is higher-order and independent from the muscles used. When, instead, we over-learn the association of a specific muscle movement with its perceivable consequences (as it may happen in sports; see also [START_REF] Wolpert | Principles of sensorimotor learning[END_REF], the representation in the brain may become effector specific. Such a phenomenon has been observed also in the brain of non-human primates, albeit in the domain of learning of movement sequences, where newly-learned sensorimotor sequences, after a long-term practice, become largely effector-dependent [START_REF] Hikosaka | Central mechanisms of motor skill learning[END_REF]. This idea is also in agreement with theories of motor learning [START_REF] Hikosaka | Parallel neural networks for learning sequential procedures[END_REF][START_REF] Hikosaka | Central mechanisms of motor skill learning[END_REF] that propose that newly-trained sensory-motor memories are coded in an abstract way in which visual-spatial coordinates play a major role. Instead, long-term practice results in effector dependency: in other words, for over-learned movements the specific activation patterns of the agonist and antagonist muscles take over. This theoretical scheme involves a fast and slower developing component that, during practice, would transform motor memories that were relying on spatial coordinates into memories that depends on body part-specific motor coordinates. One possible justification for this alteration is that relying on motor coordinates would enable a faster and more accurate performance [START_REF] Hikosaka | Parallel neural networks for learning sequential procedures[END_REF], which does not require spatial transformations and therefore demands a reduced amount of neural computation. Our results are also consistent with training-induced motor plasticity as observed in other studies (D'Ausilio et al., 2006;[START_REF] Classen | Rapid plasticity of human cortical movement representation induced by practice[END_REF]. For instance, Classen and colleagues [START_REF] Classen | Rapid plasticity of human cortical movement representation induced by practice[END_REF] demonstrated that 5 or 10 min of continuous training may establish changes that encode the kinematic details of a practiced movement. According to these authors this phenomenon "may be regarded as a short-term memory for movement and be the first step of skill acquisition".

It is also likely that these two types of sensorimotor representations (goal directed or kinematically oriented) are acquired independently, for instance by two neural system working in parallel. However, to our knowledge, no data in this regard are available. Neuroimaging studies in humans have shown that the sensory outcome of an action trigger activity in a temporal-parietal-ventral network of premotor and somatosensory areas, and the cerebellum [START_REF] Rizzolatti | Two different streams form the dorsal visual system: anatomy and functions[END_REF]; see also [START_REF] Waszak | Intention-based and stimulus-based mechanisms in action selection[END_REF][START_REF] Herwig | Two modes of sensorimotor integration in intention-based and stimulus-based actions[END_REF][START_REF] Melcher | The neural substrate of the ideomotor principle: An event-related fMRI analysis[END_REF]. Some of these areas were identified as responsive to action-effects in neurophysiological studies in the animal brain, as well [START_REF] Umilta | When pliers become fingers in the monkey motor system[END_REF][START_REF] Kakei | Muscle and movement representations in the primary motor cortex[END_REF][START_REF] Kakei | Direction of action is represented in the ventral premotor cortex[END_REF]. Whether these areas are also involved during the perception of the effects of over-learned motor acts is unclear as these experiments didn't vary the amount of training required to establish these sensorimotor associations.

Nonetheless, some work in the field of motor skills learning (reviewed in [START_REF] Hikosaka | Central mechanisms of motor skill learning[END_REF] suggested that dynamic interactions of networks composed of fronto-parietal cortices, basal ganglia and cerebellum would be able to acquire the same motor skill in different coordinates. As a matter of fact, TMS studies have demonstrated that the activity recorded from the primary motor cortex is modulated by the goal (e.g., [START_REF] Fadiga | Motor facilitation during action observation: A magnetic stimulation study[END_REF] as well as the movements of perceived actions [START_REF] Cattaneo | Representation of goal and movements without overt motor behavior in the human motor cortex: a transcranial magnetic stimulation study[END_REF]. But these variations in corticospinal excitability may be driven by premotor cortices to which the primary motor cortex has strong reciprocal connections [START_REF] Dum | Frontal lobe inputs to the digit representations of the motor areas on the lateral surface of the hemisphere[END_REF][START_REF] Shimazu | Macaque ventral premotor cortex exerts powerful facilitation of motor cortex outputs to upper limb motoneurons[END_REF][START_REF] Matelli | Afferent and efferent projections of the inferior area 6 in the macaque monkey[END_REF].

For instance, experiments (reviewed in [START_REF] Fernandino | Are cortical motor maps based on body parts or coordinated actions? Implications for embodied semantics[END_REF][START_REF] Gentilucci | Spoken language and arm gestures are controlled by the same motor control system[END_REF] have reported that the pars opercularis of the inferior frontal gyrus as well as the inferior parietal lobule encode extrinsic features (e.g., the relative position of the target and the hand in space) related to action goals, whilst the ventral premotor cortex encodes the lower-level intrinsic parameters (e.g., specific muscle, joint, and digit movements) related to specific motor acts. For instance, in an fMRI study in which participants were presented with video clips of an agent executing different motor acts with different effectors (foot, hand, and mouth), the clusters of activation in the premotor cortex were grouped according to the effectors performing the actions observed, whilst in the inferior parietal lobule the perceived actions were coded in therms of the relationship between the agent and object (e.g., bringing the object toward the agent or moving it away), regardless of the effectors used [START_REF] Jastorff | Coding observed motor acts: different organizational principles in the parietal and premotor cortex of humans[END_REF]. This reliance on extrinsic reference frames was also demonstrated in the motor cortex neurons of the monkey brain [START_REF] Graziano | Mapping behavioral repertoire onto the cortex[END_REF][START_REF] Graziano | The organization of behavioral repertoire in motor cortex[END_REF][START_REF] Graziano | Complex movements evoked by microstimulation of precentral cortex[END_REF]Umilta et al., 2001[START_REF] Umilta | When pliers become fingers in the monkey motor system[END_REF][START_REF] Kakei | Direction of action is represented in the ventral premotor cortex[END_REF], 2003;[START_REF] Kakei | Muscle and movement representations in the primary motor cortex[END_REF]Alexander & Crutcher, 1990a, 1990b;Rizzolatti et al., 1988).

Although the finding of our work may be relevant in defining that the nature of sensory-motor representations (i.e. goal-oriented or somatotopic) depends on training duration, thus potentially providing a unifying framework for previous apparently contradictory findings, we should highlight that a stronger conclusion could have been obtained by running a longitudinal study in which the same group of participants was tested at different points in time during the training.

To conclude, our result suggests that the concept of somatotopicity alone is inadequate to explain how perceived actions are mapped onto the motor system (e.g., [START_REF] Graziano | Mapping behavioral repertoire onto the cortex[END_REF][START_REF] Graziano | The organization of behavioral repertoire in motor cortex[END_REF][START_REF] Schieber | Constraints on somatotopic organization in the primary motor cortex[END_REF][START_REF] Fadiga | Visuomotor neurons: Ambiguity of the discharge or 'motor' perception?[END_REF]; see also [START_REF] Fernandino | Are cortical motor maps based on body parts or coordinated actions? Implications for embodied semantics[END_REF]. Furthermore, it indicates that the degree of somatotopicity measured in previous work may depend on the length of training of an association between an action and its perceivable consequences. As our data cannot establish the source of the modulation observed, we advocate for more research investigating the early and late learning stages of associative learning to identify how they are implemented in the human brain by, for instance, targeting with brain stimulation techniques the ventral and parietal premotor cortices as two possible sources of these different associative phenomena.

FIGURES

Figure 1 .

 1 Figure 1. Illustration of the Experiment. In an Experimental Baseline (EB), we tested pre-existing motor

Figure 2 .

 2 Figure 2. Modulation of mean normalised MEP amplitudes (z-scores: mean ± S.E.) recorded in EB and TP

  Figure 1

Supplementary material for:

From goals to muscles: motor familiarity shapes the representation of action-related sounds in the human motor system. Luca F. Ticini, Simone Schütz-Bosbach, Florian Waszak

SUPPLEMENTARY METHODS

Participants and Experimental Protocol

We determined the required size through the G* power software [START_REF] Faul | Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses[END_REF] by setting the expected effect size at 0.66 (estimated from [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF], the significance level at 0.05, and the desired power at 0.96.

The two MIDI tones (either 400 or 800 Hz, lasting 200 msec; SOA of 0 msec; instrument marimba) were randomly presented to the participants binaurally, through headphones.

To assess the attention of the participant, we introduced four catch trials (eight in EB) at a random time, in which the experimenter asked to the participant "Which tone have you just heard?" (answer: low/high).

Each TMS pulse (Magstim 200, Whitland, UK) was randomly delivered at three different time intervals (50, 150 and 300 ms) from the onset of the tone, for a total of 36 MEPs per tone in both EB and TP.

The inter-stimulation interval was of 10 to 12 seconds. We used a 70 mm figure-of-eight stimulation coil positioned tangentially over the optimal scalp position (in the left hemisphere) from which MEPs with maximal amplitude were elicited (at constant intensity) in both resting ADM and FDI finger muscles. The handle was oriented backward and laterally 45° away from the midline approximately perpendicular to the central sulcus.

The TMS intensity was set at 120% of each subject's resting motor threshold (rMT), defined as the lowest stimulator output sufficient to elicit five out of ten successive MEPs of ≥ 50µV in the relaxed ADM. Electromyographic (EMG) signal was recorded with Ag-AgCl surface electrodes fixed on the skin with a belly-tendon montage, then amplified (gain: ✕1000), filtered (10-1000 Hz bandpass), digitised at 5 kHz (with a main hum notch filter at 50 Hz), displayed on the computer screen and finally stored for off-line analysis.

No particular discomfort or negative side-effects were reported.

Data analysis

In all participants, individual peak-to-peak MEP amplitudes were calculated as the absolute distance between the minimum and maximum values observed within a search window starting at 10 msec and ending at 80 msec after the TMS pulse. We discarded trials with extremes values (SPSS Inc.) in peak-to-peak amplitudes or in the background EMG activity preceding the TMS pulse, or with exceeding SD the mean of each participant, muscle and session (EB or TP; total percentage omitted: 7.7%). For each participant, mean values were obtained from 33.2 ± 3.3 (mean ± SD) MEP per condition. [START_REF] Ticini | When sounds become actions: Higher-order representation of newly learned action sounds in the human motor system[END_REF]. When sounds become actions:

Higher-order representation of newly learned action sounds in the human motor system. Journal of cognitive neuroscience, 24(2), 464-474.

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Maya De Buhan
email: maya.de-buhan@parisdescartes.fr

Marion Darbas
email: marion.darbas@u-picardie.fr

Numerical resolution of an electromagnetic inverse medium problem at fixed frequency

Keywords: Inverse medium problem, Maxwell's equations, cost functional, minimization iterative process, Adaptive Eigenspace Method, numerical reconstruction

The aim of this paper is to solve numerically the inverse problem of determining the complex refractive index of an electromagnetic medium from partial boundary field measurements at a fixed frequency. The governing equations are the time-harmonic Maxwell equations formulated in electric field in a two-dimensional bounded domain. We express the inverse problem as the minimization of a cost function representing the difference between the measured and predicted fields. Our numerical reconstruction algorithm combines the BFGS method and an iterative process, called the Adaptive Eigenspace Inversion. The unknown complex coefficient is expanded in terms of eigenfunctions of an elliptic operator. Both the eigenspace and the mesh are iteratively adapted during the minimization procedure. Numerical experiments illustrate the performance of the reconstruction for various configurations.

Introduction

The present paper deals with the numerical resolution of an electromagnetic inverse medium problem. More precisely, we consider the problem of determining the complex refractive index of a medium, namely the dielectric permittivity (real part) and the electric conductivity (imaginary part), from a finite number of boundary field measurements at a fixed frequency. The governing equations are the time-harmonic Maxwell equations formulated in electric field in a two-dimensional bounded domain. Such an electromagnetic inverse problem arises in various areas of science and engineering with many applications, e.g. in medical imaging, geophysical exploration or nondestructive testing. For instance, microwave imaging (electromagnetic high frequencies) is under investigation for cancer screening or brain stroke detection (see Tournier et al [START_REF] Tournier | Microwave Imaging of Cerebrovascular Accidents by Using High-Performance Computing[END_REF][START_REF] Tournier | Numerical Modeling and High Speed Parallel Computing: New Perspectives for Tomographic Microwave Imaging for Brain Stroke Detection and Monitoring[END_REF]). Numerical methods that are able to highlight dielectric contrast between normal and possibly abnormal tissue are of interest.

From a mathematical point of view, the considered inverse medium problem is severely ill-posed and we refer the reader to the book [START_REF] Romanov | Inverse problems for Maxwell's equations[END_REF] by Romanov and Kabanikhin. Indeed, coefficients of elliptic problems (like the time-harmonic Maxwell problem) in a bounded domain are uniquely determined by the entire Dirichlet-to-Neumann map on the whole boundary of the domain (e.g. Ola, Païvärinta and Somersalo [START_REF] Ola | An inverse boundary value problem in electrodynamics[END_REF], Caro, Ola and Salo [START_REF] Caro | Inverse boundary value problem for Maxwell equations with local data[END_REF], Kenig, Salo and Uhlman [START_REF] Kenig | Inverse problems for the anisotropic Maxwell equations[END_REF] and references therein). A typical problem of this type is Calderón's inverse conductivity problem [START_REF] Calderón | On an inverse boundary value problem[END_REF]. Nevertheless, it is legitimate to search for reconstruction methods using partial information on the Dirichlet-to-Neumann map, which is often the case in practice. Several analytical and numerical studies have been devoted to the detection of inhomogeneities in the electromagnetic parameters of a body. Ammari et al (e.g. [START_REF] Ammari | Reconstruction of Small Inhomogeneities from Boundary Measurements[END_REF][START_REF] Ammari | Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. II. The full Maxwell equations[END_REF]) have introduced asymptotic methods to reconstruct small amplitude perturbations in coefficients from measurements on a part of the boundary. This yields constructive numerical methods for the localization of electromagnetic defects (e.g. Ammari et al [START_REF] Ammari | MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions[END_REF], Asch and Mefire [START_REF] Asch | Numerical localizations of 3D imperfections from an asymptotic formula for perturbations in the electric fields[END_REF], Darbas and Lohrengel [START_REF] Darbas | Numerical reconstruction of small perturbations in the electromagnetic coefficients of a dielectric material[END_REF]). Concerning minimization approaches, Beilina et al (e.g. [START_REF] Beilina | Adaptive finite element method for a coefficient inverse problem for Maxwell's system[END_REF][START_REF] Beilina | An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations[END_REF]) have developed an adaptive finite element method based on a posteriori estimates. Other successful methods have been proposed in the literature for the numerical solution of the electromagnetic scattering medium problem. Data are in this case measurements of the farfield pattern of the scattered field. Without being exhaustive, we can mention among them the linear sampling method of Haddar and Monk [START_REF] Haddar | The linear sampling method for solving the electromagnetic inverse medium problem[END_REF], a preconditioned Newton method initiated by Hohage [START_REF] Hohage | Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem[END_REF], or a regularized recursive linearization method used by Bao and Li [START_REF] Bao | Numerical solution of an inverse medium scattering problem for Maxwell's Equations at fixed frequency[END_REF].

Here, we propose to formulate the inverse medium problem as the minimization of a cost function representing the difference between the measured and predicted fields. To solve the minimization problem, we use a gradient-based quasi-Newton algorithm. The main goal of this paper is to present a reconstruction method for the unknown complex refractive index of the medium from boundary measurements. The idea is to consider the space spanned by some eigenvectors of the Laplacian operator as the approximation space for the unknown coefficient. Then, the method uses an iterative process to adapt the mesh and the basis of eigenfunctions to the previous approximation during the minimization procedure. This method is called Adaptive Eigenspace Inversion. We compare it with a more standard choice given by a linear piecewise approximation of the coefficient. The Adaptive Eigenspace Inversion (AEI or referred sometimes as AI) method has been initially proposed for the viscoelastic system by de Buhan and Osses [START_REF] De Buhan | Logarithmic stability in determination of a 3D viscoelastic coefficient and a numerical example[END_REF]. It has been successfully applied to an inverse scattering problem for the wave equation in a paper of de Buhan and Kray [START_REF] De Buhan | A new approach to solve the inverse scattering problem for waves: combining the TRAC and the Adaptive Inversion methods[END_REF]. In both cases, time evolution problems of hyperbolic type are treated. The geometric optics condition of Bardos-Lebeau-Rauch [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF], which allows that the associated inverse problems are uniquely solved, is satisfied. More precisely, the part of the boundary where the measurements (namely the normal derivative of the solution) are recorded, and the final observation time control geometrically the domain in the sense of [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF]. The application of the AEI method to time-harmonic problems is a new area of research. This is the aim of the present work in electromagnetics, and also the one of Grote, Kray and Nahum which study the resolution of an inverse problem for the Helmholtz equation. In [START_REF] Grote | Adaptive Eigenspace Inversion for the Helmholtz Equation[END_REF][START_REF] Grote | Adaptive Eigenspace Inversion for the Helmholtz Equation[END_REF], they proposed to combine a new AEI method and a frequency stepping process where the frequency of the incident field is iteratively increasing, with successful results. In the inverse problem we consider in this paper, we restrict ourselves to a fixed frequency. This is motivated by biomedical applications that we have in mind [START_REF] Tournier | Microwave Imaging of Cerebrovascular Accidents by Using High-Performance Computing[END_REF][START_REF] Tournier | Numerical Modeling and High Speed Parallel Computing: New Perspectives for Tomographic Microwave Imaging for Brain Stroke Detection and Monitoring[END_REF]. Biological tissues are dispersive [START_REF] Gabriel | The dielectric properties of biological tissues: I. Literature survey[END_REF], that is to say their dielectric properties are frequency-dependent. We are not interested in finding this dependency law but only in discriminating between healthy and abnormal tissues. This can be achieved with a single frequency and changing the frequency does not provide more information.

The remainder of the paper is organized as follows. In Section 2, we present the forward problem under consideration. The formulation of the inverse problem is addressed in Section 3. It is formulated as a nonlinear optimization problem. The key-point is the evaluation of the gradient of the cost function. We propose to use the adjoint method. In Section 4, we describe the reconstruction method based on the AEI method from a methodological point of view. Then, in Section 5, various numerical results are reported to discuss the advantages and limits of the method, with a particular interest in discontinuous coefficients. Finally, we give some concluding remarks.

The forward problem

Let Ω be a bounded domain in R 2 with a smooth boundary Γ := ∂Ω. We denote by µ 0 and ε 0 the permeability and the permittivity of the vacuum. We assume that Ω is filled with a non-magnetic (i.e. constant permeability µ = µ 0 ) and isotropic medium of dielectric permittivity ε = ε(x) and electrical conductivity σ = σ(x), x ∈ Ω. We consider the system of 2D Maxwell equations

∇ × E = -∂ t B, ∇ × H = ∂ t D + J , in Ω, (2.1)
where (E, H) are the electric and magnetic fields, (B, D) are the magnetic and electric flux densities and J represents the electrical current density. Notice that in two dimensions, the vector rotational operator is defined for a scalar function ϕ by ∇ × ϕ = (∂ 2 ϕ, -∂ 1 ϕ) t , whereas the scalar rotational operator acting on a vector field

v = (v 1 , v 2 ) is given by ∇ × v = ∂ 1 v 2 -∂ 2 v 1 .
We assume linear and isotropic constitutive relations

B = µ 0 H, D = εE, and J = σE. (2.
2)

The wave equation for the electric field with no source term can be derived from (2.1) and (2.2) by eliminating the magnetic field as

∇ × (∇ × E) + µ 0 (ε∂ 2 t E + σ∂ t E) = 0.
Considering the harmonic dependence in time of the form E(t, x) = (e -iωt E(x)), the electric field E satisfies the following equation in the frequency domain

∇ × (∇ × E) -k 2 κE = 0, in Ω, (2.3)
where k = ω √ ε 0 µ 0 is the wavenumber and the function

κ(x) = 1 ε 0 ε(x) + i σ(x) ω , x ∈ Ω, (2.4)
is the refractive index of the medium. We assume ε, σ ∈ L ∞ (Ω) and that there are constants ε, ε > 0, σ, σ > 0 such that ε ≤ ε(x) ≤ ε and σ ≤ σ(x) ≤ σ a.e. in Ω. Let n = (n 1 , n 2 ) t denote the outward unit normal to Γ. We impose the boundary condition

(∇ × E) × n = g, on Γ. (2.5)
For a scalar function ϕ, we have ϕ × n = (-ϕn 2 , ϕn 1 ) t . We denote by E[κ] the solution of (2.3)-(2.5) associated with the refractive index κ. Existence and uniqueness results on the boundary-value problem (2.3)-(2.5) can be found in [START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF][START_REF] Monk | Finite Element Methods for Maxwell's Equations[END_REF]. We introduce the following functional space

H(curl; Ω) = {E ∈ L 2 (Ω) 2 |∇ × E ∈ L 2 (Ω)}.
Assume that g ∈ L 2 (Γ) 2 , the variational formulation of the problem (2.3)-(2.5) is given by

   Find E ∈ H(curl; Ω) such that Ω (∇ × E)(∇ × ψ) -k 2 κE • ψ dx = Γ g • ψds, ∀ψ ∈ H(curl; Ω). (2.6)
Discretization and numerical solution of the forward problem will be mentioned in Section 5.

Formulation of the inverse problem

The inverse medium problem that interests us reads

(P)   
Given a frequency ω > 0, reconstruct the exact coefficient κ ex (x), x ∈ Ω, defined by (2.4), from the boundary measurement

E obs × n := E[κ ex ] × n on Γ 0 ,
where Γ 0 is a part of the boundary Γ including the case Γ 0 = Γ. Consequently, we recover the dielectric permittivity ε(x) and the electrical conductivity σ(x), x ∈ Ω, of the medium. We formulate the inverse problem as a minimization problem. We solve it using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) iterative algorithm [START_REF] Mordecai | Nonlinear Programming : Analysis and Methods[END_REF] (see also (5.3) for the sketch of the algorithm). This method of quasi-Newton type requires the computation of the gradient of the cost function with respect to the parameter at each iteration. The gradient is efficiently evaluated using the adjoint method.

A minimization problem

Problem (P) is written as an optimization problem, namely it is replaced by the minimization of the following functional

J(κ) = 1 2 Γ0 |E[κ] × n -E obs × n| 2 ds, (3.1)
where E[κ]| Γ0 ×n is computed by solving the forward problem (2.3)-(2.5) at a fixed frequency ω > 0 for a given refractive index κ and a boundary data g, and E obs | Γ0 × n is the measured electric field. The functional J represents the error between the observed electric field and that predicted by Maxwell equations. The minimization problem

min κ∈L ∞ (Ω) J(κ) (3.2)
is solved by using the BFGS algorithm. The Hessian of the cost function J is approximated by means of the gradient of J. The functional and its gradient have to be computed within each iteration step.

Computation of the gradient using the adjoint method

In this section, we derive an expression of the derivative of the cost function J with respect to the coefficient κ in a given arbitrary direction δκ. The directional derivative of J is defined by

D κ J(κ)δκ = lim t→0 J(κ + tδκ) -J(κ) t .
We introduce δE := δE[κ, δκ] the solution of the following linearized problem

∇ × (∇ × δE) -k 2 κδE = k 2 δκE[κ], in Ω, (∇ × δE) × n = 0, on Γ. (3.3)
We have

E[κ + tδκ] = E[κ] + tδE + o(t 2
). We obtain

D κ J(κ)δκ = lim t→0 1 2t Γ0 (|(E[κ + tδκ] -E obs ) × n| 2 -|(E[κ] -E obs ) × n| 2 )ds = lim t→0 1 t Γ0 t ((E[κ] -E obs ) × n)(δE × n) ds + 1 2 Γ0 t 2 |δE × n| 2 ds = Γ0 ((E[κ] -E obs ) × n)(δE × n)ds . (3.4)
The adjoint state method allows to simplify this expression. Let F be a test function. The variational formulation of the linearized equation (3.3) is given by

k 2 Ω δκE[κ] • Fdx = Ω ∇ × (∇ × δE) -k 2 κδE • F dx.
By integrating by parts the right-hand side of the previous formulation and using the boundary conditions satisfied by δE, we obtain

k 2 Ω δκE[κ] • Fdx = Ω ∇ × (∇ × F) -k 2 κF • δEdx - Γ ((∇ × δE) × n) • F ds + Γ ((∇ × F) × n) • δE ds = Ω ∇ × (∇ × F) -k 2 κF • δEdx + Γ ((∇ × F) × n) • δE ds.
Next, we choose

F := F[κ] the adjoint variable of δE satisfying ∇ × (∇ × F) -k 2 κF = 0, in Ω, (3.5)
with the boundary condition

(∇ × F) × n = ((E[κ] -E obs ) × n) × n, on Γ 0 , 0, on Γ \ Γ 0 . (3.6)
We get

k 2 Ω δκE[κ] • F[κ]dx = - Γ0 ((E[κ] -E obs ) × n)(δE × n)dx.
Consequently, the differential (3.4) of the functional J is given by

D κ J(κ)δκ = -k 2 Ω δκE[κ] • F[κ]dx . (3.7)
At each step of the algorithm, the computation of the gradient needs to solve the forward problem (2.3)-(2.5) and the associated adjoint problem (3.5)-(3.6) for the coefficient κ in order to obtain the variables E and F respectively.

In practice, we perform multiple observations E j obs , 1 ≤ j ≤ M , of the electric field on Γ 0 at a fixed frequency ω. Each measurement E j obs is associated with the boundary data g j . Then, we can simply define the functional

J M (κ) = 1 2 M j=1 Γ0 |(E j [κ] -E j obs ) × n| 2 ds, (3.8)
where the electric field E j is solution to the problem

∇ × (∇ × E j ) -k 2 κE j = 0, in Ω, (∇ × E j ) × n = g j , on Γ. (3.9)
We have the following expression of the gradient in the direction δκ

D κ J M (κ)δκ = -k 2 M j=1 Ω δκE j [κ] • F j [κ]dx , (3.10)
with F j satisfying the adjoint problem

     ∇ × (∇ × F j ) -k 2 κF j = 0, in Ω, (∇ × F j ) × n = ((E j [κ] -E j obs ) × n) × n, on Γ 0 , (∇ × F j ) × n = 0, on Γ \ Γ 0 .
(3.11)

In the sequel, we consider synthetic data E j obs , 1 ≤ j ≤ M (which are obtained by the numerical resolution of the forward problem and are created to simulate the real data). This will be precised in Section 5.

The Adaptive Eigenspace Inversion method

In this section, we describe the reconstruction procedure used to solve (P). It is based on the AEI method. The originality of the method comes from the parametrization space. Instead of looking for the value of the unknown coefficient κ at each node of the mesh, it is projected first in the basis of the eigenvectors of the Laplacian operator. Then, an iterative process is applied to adapt both the mesh and the basis.

Choice of the coefficient parametrization

The unknown coefficient κ ex is a function of x in Ω. In order to numerically recover κ ex in Ω, we have first to choose a discretization space. A natural choice would be to use the P 1 Lagrange basis functions (ψ ) 1≤ ≤N with N the number of internal nodes of the mesh. In the minimization method, the approximate coefficient κ N is decomposed under the form

κ N (x) = κ b (x) + N =1 d ψ (x), x ∈ Ω, (4.1)
where the function κ b is a lifting of the boundary value of κ ex , assumed to be known (and not necessarily constant), obtained as the solution of the following problem:

-

∆κ b = 0, in Ω, κ b = κ ex , on Γ.
The unknowns in (4.1) are the complex values d for = 1 to N . At each node of the computational domain, there are two degrees of freedom: the real and the imaginary parts of d . The dimension of the minimization space is thus equal to 2N .

In our method, we propose another basis. We look for an approximation κ L of the exact coefficient κ ex of the form

κ L (x) = κ b (x) + L =1 κ φ (x), x ∈ Ω, (4.2)
where φ , 1 ≤ ≤ L, are the L first eigenfunctions of the Laplacian operator. The function φ , 1 ≤ ≤ L, is solution to the following eigenvalue problem

-∆φ = λ φ , in Ω, φ = 0, on Γ, (4.3)
with λ the corresponding eigenvalue. One advantage of this approach is to decouple the mesh size N and the dimension of the minimization space (size 2L with L N ). We will give in Section 5.1 a method to find the dimension L of the basis. In Section 5.2, a numerical comparison is drawn between these two parametrizations.

Adaptation of the basis

We propose an adaptive method to improve the accuracy of the reconstruction. This iterative process works for both continuous (varying) coefficients (see Figure 9) and discontinuous coefficients. In the latter case, it allows to efficiently capture the discontinuity lines. In the sequel, we focus particularly on such coefficients. The adaptive method consists in four steps.

Step 1: We choose a parametrization for the coefficient, either (4.1) or (4.2). We solve the minimization problem (3.2) using the BFGS algorithm. We then compute a first approximate coefficient, denoted by κ

(1)
L1 , in the initial mesh.

Step 2: The information contained in the approximation κ

L1 is used both to adapt the mesh and to construct another basis that better represents the coefficient. The principle of the basis adaptation is the following: we look for the coefficient κ

(2)
L2 -κ b in the space spanned by the L 2 first eigenfunctions (φ ) 1≤ ≤L2 of an elliptic operator, that is

κ (2) L2 (x) = κ b (x) + L2 =1 κ (2), φ (x), with -∇ • (A (2) ∇φ ) = λ φ , in Ω, φ = 0, on Γ, (4.4)
where the matrix function is computed from the knowledge of the first iterate κ

L1 such that

A (2) (x) = 1 max{|∇κ (1) L1 (x)|, η} Id, x ∈ Ω.
The matrix Id is the identity matrix. The parameter η > 0 is small such as η = 10 -3 and allows the denominator of A (2) (x) not to vanish. By considering this new elliptic operator, the variations of the eigenfunctions are concentrated in the regions where the coefficient κ

(1)
L1 varies, in particular close to the discontinuities of κ ex (see Figure 2). Results on the localization and exponential decay of eigenfunctions of elliptic operators car be found for instance in [START_REF] Egorov | On Spectral Theory of Elliptic Operators[END_REF][START_REF] Nguyen | Localization of Laplacian Eigenfunctions in Circular, Spherical, and Elliptical Domains[END_REF]. We solve the minimization problem (3.2) using the BFGS algorithm to compute κ

(2) L2 .

Step 3: We search the coefficient κ

(3) L3 such that κ (3) L3 (x) = κ b (x) + L3 =1 κ (3), φ (x), where -∇ • (A (3) ∇φ ) = λ φ , in Ω, φ = 0, on Γ, with the choice A (3) (x) = 1 max{|∇κ (2) L2 (x)| 2 , η} Id, x ∈ Ω.
Here again, the BFGS method is applied to solve the minimization problem.

Step 4: In the last step, we search the coefficient κ

(4) L4 such that κ (4) L4 (x) = κ b (x) + L4 =1 κ (4), φ (x), where -∇ • (A (4) ∇φ ) = λ φ , in Ω, φ = 0, on Γ,
and A (4) is chosen following an anisotropic criterion. The idea is to take into account the orientation of the discontinuity lines of κ ex and to accord a preference to variations of the basis functions in the direction of its gradient. To do that, as presented in Figure 1, we take a new orthonormed system (x, X 1 , X 2 ) whose first axis is locally oriented by the gradient of κ

(3) L3

and we define the rotation matrix P as follows:

P (x) = 1 |∇κ (3) L3 (x)|      ∂κ (3) L3 ∂x 1 (x) - ∂κ (3) L3 ∂x 2 (x) ∂κ (3) L3 ∂x 2 (x) ∂κ (3) L3 ∂x 1 (x)      .
In the new system, we choose to give more weight to the direction of the gradient, by setting:

A (4) (x) = 1 max{|∇κ (3) L3 (x)| 2 , η} P (x)C(x)P -1 (x), with
C(x) =   1 max{|∇κ (3) L3 (x)| 2 , η} 0 0 1   . 0 x 2 x 1 X 1 X 2
x discontinuity line of κ Notice that at each step s, we increase the power q of the norm of the gradient in the definition of the matrix A (s) : we start from q = 0 at Step 1 (in that case, A (1) = Id and the associated elliptic operator is the Laplacian) to q = 4 in the direction of the gradient at Step 4. This process allows to refine increasingly the reconstrution of the coefficient κ ex . Furthermore, at each step, we use the solution obtained at the previous step to adapt the mesh. We consider a classical mesh adaptation algorithm. It is based on the Hessian of the solution κ (s) Ls , s ∈ 1, 3 . It concentrates the nodes of the mesh where the solution varies to decrease the approximation error without increasing the computational time [START_REF] Frey | Mesh Generation. Application to finite elements[END_REF]. The accuracy of the adaptative eigenspace basis is illustrated in Figure 2. The real part of the exact coefficient is presented in Figure 13(a). For s ∈ 1, 4 , we project the exact coefficient in the space spanned by the L s = 49 first eigenfunctions of the corresponding elliptic operator. At this stage, the choice L s = 49, s ∈ 1, 4 , is an arbitrary one. The aim is to provide an illustration of the efficiency of the AEI method. We give in Section 5.1 (see formula (5.4)) a method to automatically fix L s and the value can be different from a step to another.

In Figure 2, we report the mesh obtained at each step, the behavior of the first and the 26th eigenfunctions, and the projection of the exact coefficient in the successive eigenspaces. The projection error in L 2 -norm is efficiently decreased during the process.

Step 1

Step 2

Step 3

Step 4

Numerical results

In this section, we discuss the numerical solution of the inverse medium problem (P) by using the AEI method. We reconstruct the complex refractive index κ ex of the medium from synthetic noisy data. First, we describe the data set and the discretized version of the AEI method. Secondly, we perform only the Step 1 of the method to compare the coefficient parametrizations (4.1) and (4.2). Finally, we present the reconstruction of the exact coefficient κ ex via the complete iterative process for different configurations.

Description of data driven simulations and implementation

The domain Ω is the unit circle. The partition of the boundary is illustrated in Figure 3(a). The observations are collected on the boundary Γ 0 := {(cos(t), sin(t)), t ∈ [γπ, 2π]} with the parameter γ ∈ [0, 2[. We consider full data in the case γ = 0 and limited-view ones otherwise. We look for different types of coefficients κ ex :

• Continuous functions of x.

• Piecewise constant coefficients:

κ ex (x) = κ i if x ∈ ω i , i ∈ I a

Synthetic noisy data

We work with synthetic data. Let 1 ≤ j ≤ M , M ∈ N * . We take incident plane waves

E j inc (x) = η ⊥ j e ik √ κ b η j •x , (5.1)
of wave vector η j = (cos(2(j-1)π/M ), sin(2(j-1)π/M )) t . The vector η ⊥ j is a unit vector orthogonal to η j and κ b a complex constant that characterizes the reference background medium. The squareroot √ κ b stands for the classical complex square-root with branch-cut along the negative real axis.

At fixed frequency ω, recall that the wavenumber k is equal to k = ω √ ε 0 µ 0 with the permeability and the permittivity of the vacuum µ 0 = 4π 10 -7 H.m -1 and ε 0 = 8.854187 10 -12 F.m -1 . Examples of such waves (5.1) are illustrated in Figure 4. Incident waves are attenuated by the dissipative medium properties. The synthetic data E j are obtained by solving the forward problem (2.3)-(2.5) with the exact coefficient κ ex and the boundary data g j := (∇ × E j inc ) × n. We consider a triangular mesh of the computational domain Ω.

We adopt Nédélec edge elements of order 1 [START_REF] Nédélec | Mixed finite elements in R 3[END_REF] which give natural approximation spaces of H(curl; Ω). The basis functions are associated with the edges of the mesh. To avoid the inverse crime [START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF], we use Nédélec finite elements of different orders for the numerical solution of the direct and the inverse problems. Furthermore, to model possible experimental errors, we can add a Gaussian noise as

E j obs (x i ) := (1 + α rand(x i )) E j (x i ), (5.2)
where x i denotes the vertex i of a given discretization of the boundary Γ 0 , rand gives uniformly distributed random number in [-1, 1], and α is the level of noise. Note that the model of noise is a multiplicative noise and not an additive one. Our choice is motivated by Equation (3.7) page 134 in [START_REF] Colton | Recent developments in inverse acoustic scattering theory[END_REF]. The input data for the inverse problem are thus the M sets (E j obs , g j ), j = 1, . . . , M .

Broyden-Fletcher-Goldfarb-Shanno Algorithm

The BFGS algorithm is an iterative method for solving the unconstrained nonlinear optimization problem of a functional J(κ).

From an initial guess κ 0 and an approximate Hessian matrix H 0 = Id, the following steps are repeated until κ k converges to the solution. For k ≥ 0:

• Find a descent direction d k by solving H k d k = -D κ J(κ k ).
• Perform a line search to find an acceptable stepsize α k in the direction found in the first step, then update

κ k+1 = κ k + α k d k . • Set y k = D κ J(κ k+1 ) -D κ J(κ k ) and z k = α k d k .
• Update

H k+1 = H k + y k y t k y t k z k - H k z k z t k H k z t k H k z k . (5.3)
Convergence can be checked by observing the norm of the gradient.

Implementation of the reconstruction method

Let us now describe the implementation of the reconstruction method. At each step s ∈ 1, 4 of the adaptive process, we solve the minimization problem (3.2) iteratively by the BFGS algorithm with a tolerance of . The initial guess at Step s = 1 for the inverse problem is the coefficient κ := κ b that represents an a priori medium. Next, the initial guess at Step s ∈ 2, 4 is the approximate coefficient κ (s-1) computed at the previous step. The eigenvalue problems are solved by using the toolbox included in FreeFem++ [START_REF] Hecht | New development in FreeFem++[END_REF]. These tools are based on the library Arpack++ which implements the "Implicit Restarted Arnoldi Method" (IRAM) combining Arnoldi factorizations with an implicitly shifted QR method [START_REF] Lehoucq | ARPACK Users'Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods[END_REF]. For the mesh adaptation, we pay a special attention to have the more refined mesh but without increasing the total number of vertices.

The number L of eigenfunctions used in the expansion (4.2) is automatically fixed in the following way. At each step s of the AEI method, we start with L = 1 and we increase it from 10 to 10. We stop when adding 10 eigenfunctions to the basis does not allow to decrease significantly the value of the functional J M . Thus, we set L s equal to the first value L such that the following stopping criterion is met:

J M (κ (s) L ) -J M (κ (s) L+10 ) J M (κ (s) L ) < δ, (5.4)
where δ > 0 could depend on the level of noise α if it is known a priori.

At each iteration of BFGS, the computation of the gradient (3.10) needs the solution of the forward problem (2.3)-(2.5) and the adjoint problem (3.5)- (3.6). To this end, we choose Nédélec edge elements of order 0, and not of order 1 used to generate the data. We solve the 2M problems in parallel (thanks to MPI). Note that the unknown coefficient is not defined on the same mesh used to solve the state and adjoint problems. Table 1 gathers the numerical values used for all the following examples, unless specified otherwise where appropriate. Table 1 -Fixed values in the remaining of the Article (unless specified otherwise).

The complete numerical procedure for solving the inverse medium problem is the following:

AEI algorithm

• At fixed frequency ω, generate the synthetic noisy data E j obs (5.2), 1 ≤ j ≤ M , on the boundary Γ 0 := {(cos(t), sin(t)), t ∈ [γπ, 2π]}.

• AEI steps: for s ∈ 1, 4 (Step s, see Section 4.2)

-If s > 1 then adapt the mesh with respect to κ (s-1)

Ls-1 .
-Fix L = 1. While the stopping criterion (5.4) is not satisfied, * Apply the BFGS algorithm with a tolerance of .

Initialization:

κ (s) L = κ b if s = 1, κ (s-1) Ls-1
otherwise. Expand the coefficient in the space spanned by the L first eigenfunctions of the elliptic operator (associated with Step s).

At each iteration n:

Solve M forward problems in parallel. Solve M adjoint problems in parallel. Compute the gradient (3.10) of the cost functional J M . We apply the Step 1 of the method for different configurations. We study the influence of the parameter set on the relative error between the exact and reconstructed coefficients. Results are reported in Table 2. Recall that the fixed values are given in Table 1. First, we vary the length of the boundary Γ 0 on which the observations are recorded: full-data (γ = 0, i.e. Γ 0 = Γ) to limitedview data (γ = 1.75, i.e. Γ 0 := {(cos(t), sin(t)), t ∈ [1.75π, 2π]}). As expected, errors increase with partial boundary measurements and remain under 3.29%. Then, we change the number of observations. Performance of Step 1 is not very sensitive to this parameter. The value M = 16 seems to be quasi-optimal for the eigenspace approach and taking M superior to 16 does not really improve accuracy. The P 1 basis is slightly unstable with respect to this parameter. Finally, we add a Gaussian noise of level α, where α is varying between 0.005 and 0.05 (i.e. 0.5% to 5%). The approximation of the coefficient κ ex is less precise when the level of noise increases. But here again, the inclusion ω 0 is localized in each case. Now, we test the effect of the frequency and also of the dimension of the minimization space. In the latter comparison, we fix L 1 = N . Errors are reported in Table 3. Whatever the frequency and the dimension are, errors are similar. They are under 2%.

γ P 1 Eigenspace M P 1 Eigenspace α P 1 Eigenspace 0 1.58% 1.39% 4
We can conclude that the Step 1 of the AEI method provides a first reasonable approximation of the exact coefficient κ ex independently of the data set and with a slight advantage for the eigenspace approach. In the considered example, it allows the localization of the zone where the electromagnetic coefficients vary. These approximate coefficients are very good initial guesses for Table 3 -Relative errors in L 2 norm between the exact and reconstructed coefficients with respect to the frequency and to the dimension L 1 (L 1 = N ) of the minimization space.

the iterative process. We will see in the next section that the further steps of the AEI method perform an accurate reconstruction in both form and values.

Numerical reconstruction by the Adaptive Eigenspace Method

In this section, we present the numerical reconstruction for various configurations. The inhomogeneous medium is first described as a space continuous function. Then, we consider functions having some discontinuities, namely the case of piecewise constant functions. In all the figures except Figure 16, the color scale is the same: the color blue stands for value 1 and the color red represents value 2.

Continuous functions

We first try to recover the following exact coefficient, given as a continuous function of x = (x 1 , x 2 ):

κ ex (x) = 6 + cos(3(x 1 + x 2 )) + cos(5x 1 ) 4 . (5.6)
Remember that in our approach we assume that the exact coefficient is known on the boundary of the domain. We can then easily compute the lifting function κ b as the solution of the following problem:

-

∆κ b = 0, in Ω, κ b = κ ex , on Γ. (5.7)
Both functions κ ex and κ b are plotted in Figure 9, on the left and in the center respectively. As already mentioned in Section 4.2, the AEI method allows to better capture some discontinuity lines in the coefficient and is particularly efficient in that case. Here, the first step already gives satisfactory results (see Figure 9 on the right) and the following steps do not add further information.

Piecewise constant coefficients

A reference test-case: one ellipse.

We adopt the adaptive process to retrieve the exact coefficient κ ex (5.5) defined in Section 5.2, namely

κ ex (x) = 2 + 2i if x ∈ ω 0 , 1 + i otherwise,
with the ellipse ω 0 := {(x 1 , x 2 ) ∈ R 2 |(x 1 -0.2) 2 + 0.7(x 2 + 0.1) 2 ≤ 0.09}. We report on Figure 10 the successive reconstructed coefficients κ (s)

Ls , s ∈ 1, 4 . The relative L 2 -error decreases at each step of the process to reach a final value equal to 0.39%. This example illustrates how each step uses information on the coefficient obtained in the previous step to improve the reconstruction. The first step of the AEI method allows to localize the perturbations in the electromagnetic coefficients, and the others to retrieve their shapes and values. For Step 1 to 4, the dimension L s of the eigenspace is 71, 41, 51 and 51, and the corresponding number of BFGS iterations is 100, 34, 13 and 7. The total computational time is 5592, 1840, 831 and 578 seconds, including the iterative research of L s . The number of vertices of each adapted mesh is around 30000. In Figure 11, we represent the results obtained at Step 4 from noisy observation data with different levels of noise according to the formula (5.2). As expected, the reconstruction error increases with the noise level but remains stable. The results are satisfactory even with 10% noise.

Multiple inhomogeneities.

We consider the following coefficient

κ ex (x) = 2 + 2i if x ∈ ω 1 ∪ ω 2 , 1 + i otherwise.
(5.8)

The inhomogeneities of the medium are represented by two ellipses ω 1 := ω 0 and ω number of BFGS iterations is 113, 40, 16 and 17 respectively. The relative reconstruction error at the last step is 0.93%. The AEI method performs very well the separation and the reconstruction of multiple inhomogeneities, even when their sizes and the distance between them are smaller than the wavelength. Indeed, in that case for example, the incident wavelength is λ = k/2π = 3 whereas the diameter of the domain Ω is only 2.

2 := {(x 1 , x 2 ) ∈ R 2 |(x 1 +0.

Square and star.

We further test the reconstruction of inhomogeneities with irregular contour, namely

κ ex (x) = 2 + 2i if x ∈ ω, 1 + i otherwise, (5.9)
where the subdomain ω is a square centered at the origin of side 0.4, or a star delimited by the curve t → (0.1 + c(t) cos(t), 0.1 + c(t) sin(t), with c(t) = (20 + 3 sin(5t) -2 sin(15t) + sin(25t))/50. The AEI method still gives accurate results. The relative reconstruction error on the coefficient at the last step is 0.48% in the case of the square, and 1.03% for the star. As a comparison to highlight the efficiency of the AEI process, we have reported the reconstructed coefficients at the Steps 1 and 4 (cf. Figures 13 and14).

L 4 )

Figure 12 -Reconstruction of two ellipses by using the AEI method. No noise data.

Different dielectric permittivity and electrical conductivity.

The exact coefficient is given by

κ ex (x) = 21 ω1 (x) + 2i1 ω2 (x) if x ∈ ω 1 ∪ ω 2 , 1 + i otherwise.
(5.10)

The inhomogeneity in the dielectric permittivity ε (resp. in the conductivity σ) is represented by the ellipse ω 1 := ω 0 (resp. the ellipse ω

2 := {(x 1 , x 2 ) ∈ R 2 |(x 1 +0.5) 2 +0.4(x 2 -0.2) 2 ≤ (0.15) 2 }).
Notice that the inhomogeneity ω 2 is small and close to the boundary. The final reconstructed coefficient is given in Figure 15. For steps s ∈ 1, 4 , the dimension L s of the eigenspace is 71, 41, 61 and 11, and the number of BFGS iterations is 97, 34, 21 and 8 respectively. The relative reconstruction error on the permittivity and the conductivity at the last step are 1.05% and 0.81% respectively.

Small constrast.

We are interested finally in determining an inhomogeneity with a small contrast, modeled by the coefficient The method gives a very accurate result. This enables us to envisage applications in real biomedical situations where the discrepancy between the properties of healthy and sick tissues is small. We also show the results for the square and star defined previously (cf. Figure 17).

κ ex (x) = 1.1 + 1.1i if x ∈ ω 0 , 1 + i otherwise, (5.11)

Concluding remarks and prospects

We have considered the problem of determining some of the electromagnetic properties (dielectric permittivity and electrical conductivity) of a medium in a two-dimensional bounded domain from boundary measurements at a fixed frequency. We have formulated this inverse problem as a minimization problem where the functional to be minimized represents the difference between the We have presented a reconstruction procedure, called Adaptive Eigenspace Inversion (AEI) method, to solve this problem efficiently. A standard gradient-based quasi-Newton algorithm is applied to deal with the minimization problem. The originality and specificity of our approach are based on the discrete representation of the unknown complex coefficient (i.e. the refractive index of the medium). The basis composed by eigenvectors of the Laplacian operator is preferred to the more classical choice of P 1 functions. Then, during the minimization process, both the mesh and the basis are iteratively adapted. A method is proposed to compute the dimension of the successive eigenspaces automatically, which is generally very small compared to the mesh dimension. The AEI method is able to characterize simultaneously the dielectric permittivity (real part of the coefficient) and the electrical conductivity (imaginary part of the coefficient) of a medium from partial boundary measurements of electric fields. Its performance is illustrated in several examples, and in particular in the case of discontinuous functions. An attractive feature of the method is to yield accurate reconstructions both qualitatively and quantitatively even from noisy data.

This work has been motivated by biomedical applications, and in particular by the microwave imaging of cerebrovascular accidents [START_REF] Tournier | Microwave Imaging of Cerebrovascular Accidents by Using High-Performance Computing[END_REF][START_REF] Tournier | Numerical Modeling and High Speed Parallel Computing: New Perspectives for Tomographic Microwave Imaging for Brain Stroke Detection and Monitoring[END_REF]. Strokes are characterized by dielectric properties which are slightly altered (±10%) from that of a healthy brain. The AEI algorithm allows to capture small perturbations in the electromagnetic coefficients (see Figure 16 and 17) and should be a promising method to detect strokes. It has to be extended to the three-dimensional case to deal with this application.

Figure 1 -

 1 Figure 1 -Illustration of the change of coordinate system for the anisotropic case (Step 4 of the AEI method).

Figure 2 -

 2 Figure 2 -At each step: the adapted mesh, the eigenfunctions φ 1 and φ 26 , and the projection of the exact coefficient (κ ex ) (cf. Figure 13(a)) in the corresponding basis.

0 Γ

 0 finite subset of N, κ b otherwise, where κ i , i ∈ I, and κ b are complex constants. The inhomogeneities of the medium are supported in the subdomains ω i Ω satisfying ω i ∩ ω j = ∅, ∀i = j. Ω Γ \ Γ 0 (a) Observation boundary Γ 0 . (b) Triangular mesh.

Figure 3 -

 3 Figure 3 -Computational domain.

(a) j = 7 ,Figure 4 -

 74 Figure 4 -Incident waves used to illuminate the domain.

8

 8 Hz 0.1 16 0 10 -3 10 -3

Figure 5 -

 5 Figure 5 -Comparison of two parametrizations to represent the unknown coefficient κ ex . Projection error in L 2 -norm with respect to the dimension.

  (a) Real part of the exact coefficient κex (5.5), (center) 2D view, (right) 3D view. (b) Parametrization using P 1 Lagrange finite elements: (left) a mesh with 107 interior nodes, (center) the projection of (κex) in 2D view, (right) corresponding 3D view. Relative projection error in L 2 -norm = 1.0%. (c) Parametrization using the 107 first eigenfunctions of the Laplacian operator: (center) projection of (κex) in 2D view, (right) corresponding 3D view. Relative projection error in L 2 -norm = 0.6%.

Figure 6 -

 6 Figure 6 -Comparison of the two parametrizations to represent the coefficient κ ex .

Figure 7 -

 7 Figure 7 -Convergence of the BFGS minimization algorithm (Step 1 of the AEI method): decrease of the gradient norm.

Figure 8 -

 8 Figure 8 -Color scale for all the figures.

Figure 9 -

 9 Figure 9 -Exact coefficient (κ ex ) (left). Lifting κ b from the known boundary value (center). Reconstructed coefficient (κ (1) L1 ) at Step 1 with L 1 = 51. No noise data (right).

Figure 10 -

 10 Figure 10 -The AEI method for the recovery of piecewise constant functions. No noise data.

Figure 11 -

 11 Figure 11 -Reconstruction of an ellipse by using the AEI method: (a) Exact coefficient (κ ex ). (b) Reconstructed permittivity (κ (4) L4 ) with 2%-noisy data and δ = 5.10 -3 . (c) Reconstructed permittivity (κ (4) L4 ) with 5%-noisy data and δ = 1.10 -2 . (d) Reconstructed permittivity (κ (4) L4 ) with 10%-noisy data and δ = 5.10 -2 .

  (a) Exact coefficient (κex) (b) Reconstructed permittivity (κ

Figure 13 -

 13 Figure 13 -Reconstruction of a square by using the AEI method. No noise data. (a) Exact coefficient (κ ex ). (b) Reconstructed permittivity (κ (1) L1 ) at Step 1. (c) Reconstructed permittivity (κ (4) L4 ) at Step 4.

Figure 14 - 1 )

 141 Figure 14 -Reconstruction of a star by using the AEI method. No noise data. (a) Exact coefficient (κ ex ). (b) Reconstructed permittivity (κ (1) L1 ) at Step 1. (c) Reconstructed permittivity (κ (4) L4 ) at Step 4.

4 ) 4 ) L 4 )Figure 15 -

 44415 Figure 15 -Different dielectric permittivity and electrical conductivity: numerical reconstruction using the AEI method. No noise data.

Figure 16 -

 16 Figure 16 -Reconstruction of a flat ellipse by using the AEI method. No noise data.

Figure 17 - 4 )

 174 Figure 17 -Reconstructed permittivity (κ (4) L4 ) in the case of a small contrast. No noise data.

Table 2 -

 2 Relative errors in L 2 norm between the exact and reconstructed coefficients with respect to different parameters: full or limited-view data (parameter γ), the number of observations M , and the level of noise (parameter α). N = L 1 = 49.

	1.37%	1.55%	0.005 1.47%	1.45%

  5) 2 +0.4(x 2 -0.2) 2 ≤ (0.15) 2 } of different size. The final reconstructed coefficient is given in Figure 12. For steps s ∈ 1, 4 , the dimension L s of the eigenspace is 71, 51, 61 and 11, and the

			Step 1	Step 2	Step 3	Step 4
	Rel. L 2 error	1.61%	1.15%	0.5%	0.39%
		Mesh
	(κ	(s) Ls ) in 2D
	(κ

(s)

Ls ) in 3D

Acknowledgements

The authors would like to thank Frédéric Nataf and Pierre-Henri Tournier for fruitful conversations. This work has been supported by the french Agence Nationale de la Recherche through project MEDIMAX, ANR-13-MONU-0012. The numerical results presented here were obtained using the ressources of the MeCS platform of the Université de Picardie Jules Verne and the authors thank Serge Van Criekingen for his help.

* Increase L = L + 10.

-Set L s = L.

• Output: κ (4) L4 .

Coefficient parametrization: P 1 versus eigenspace method

In this section, we numerically compare the parametrizations (4.1) and (4.2) in the configuration where the exact coefficient is defined by

with the ellipse

To carry out the comparison, we perform only the Step 1 of the method by varying different parameters (such as the number of observations, the boundary Γ 0 of observation, etc). The relative error in L 2 -norm on the exact coefficient κ ex is defined by

L1 is the numerical reconstructed coefficient at the end of Step 1.

Projection error

First, we compare the projection error on the coefficient κ ex (5.5) using the parametrizations (4.1) and (4.2) with respect to the dimension, setting L = N . Results are reported on Figures 5 and6. The two methods allow a good representation of the exact coefficient and the projection error decreases with the dimension. We obtain errors equal to 0.22% and 0.07% taking L = N = 1688 respectively for the P 1 and the eigenspace basis (cf. Figure 5). Thus, the eigenspace approach provides more accurate contours of the inclusion than the P 1 basis (cf. Figure 6).

Convergence of the BFGS algorithm

We study the convergence of the BFGS algorithm for the minimization of the functional J M (cf. (3.8)). We consider the same number of degrees of freedom in both parametrizations. We fix N internal mesh nodes for the classical P1 basis and L 1 eigenvectors for the eigenspace method, with L 1 = N = 49. The BFGS tolerance is fixed equal to = 10 -4 . We report on Figure 7 the history of the relative norm r := D κ J M (κ) l2 / D κ J M (κ b ) l2 in logarithm scale with respect to the number n of iterations. Here, κ is the approximate coefficient at iteration n. The convergence is faster for the eigenspace approach than for the P 1 basis (135 iterations against 260).