Clément Jailin

Stéphane Roux

Modal decomposition from partial measurements

Keywords: Modal analysis, Proper Generalized Decomposition, Dynamic stereo-vision, Dynamic tomography, Field recovery, Gappy Proper Orthogonal Decomposition

A data set over space and time is assumed to have a low rank representation in separated spatial and temporal modes. The problem of evaluating these modes from a temporal series of partial measurements is considered. Each elementary instantaneous measurement captures only a window (in space) of the observed data set, but the window position varies in time so as to cover the entire region of interest and would allow for a complete measurement would the scene be static.

A novel procedure, alternative to the Gappy Proper Orthogonal Decomposition (GPOD) methodology, is introduced. It is a xed point iterative procedure where modes are evaluated sequentially. Tested upon very sparse acquisition (1% of measurements being available) and very noisy synthetic data sets (10% noise), the proposed algorithm is shown to outperform two variants of the GPOD algorithm, with much faster convergence, and better reconstruction of the entire data set.

Introduction

In experimental mechanics, full eld measurements are an essential ingredient for the identication and validation of mechanical laws. Initially developed for 2D plane measurements with Digital Image Correlation, (DIC) [START_REF] Ma Sutton | Determination of displacements using an improved digital correlation method[END_REF], and extended to stereo-correlation [START_REF] Ma Sutton | Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications[END_REF] and 3D analyses (Digital Volume Correlation, Preprint submitted to Comptes-Rendus Mécanique August 19, 2019 DVC [START_REF] Bk Bay | Digital volume correlation: three-dimensional strain mapping using x-ray tomography[END_REF]), they provide rich data to challenge complex models or to feed datadriven approaches. Performed in-situ (i.e., during a mechanical test), the recent evolution of those approaches to space-time measurements (both for 2D [START_REF] Besnard | A spacetime approach in digital image correlation: Movie-dic[END_REF] and 3D [START_REF] Buljac | Digital volume correlation: review of progress and challenges[END_REF]) provided an enhanced sensitivity fostering the identication of strongly non-linear behaviors. Therefore, acquiring a complete set of in-situ space and time measurements is a grail in experimental mechanics.

However, the acquisition of a complete stack of data is generally inaccessible.

Incomplete acquisition, in a mechanical context, can be due to many reasons and to mention just of few of them:

Intrinsically incomplete measurement devices (e.g., mobile camera imaging a large or curved surface [START_REF] Newcombe | Live dense reconstruction with a single moving camera[END_REF], partial space-time measurements in a Continuous Scanning Laser Vibrometer (CSLV) [START_REF] Sj Rothberg | An international review of laser doppler vibrometry: Making light work of vibration measurement[END_REF], etc.)

Partially masked eld of view, either unwanted (e.g., due to the experimental setup [START_REF] Dufour | Shape, displacement and mechanical properties from isogeometric multiview stereocorrelation[END_REF]) or voluntary (e.g., because of the detector saturation of a part of the sample due to specular reection in [START_REF] Berny | On the analysis of heat haze eects with spacetime dic[END_REF] or due to the degradation of the imaged texture and speckles [START_REF] Ma Iadicola | Uncertainties of digital image correlation due to pattern degradation at large strain[END_REF]).

Fast rate phenomena that can be studied with an acquisition rate that is too low. This problem can be circumvented if the exploited image is processed from the fast acquisition of a collection of raw data (e.g., in computed-tomography, MRI, diraction patterns in EBSD, etc.). In tomography, while 3D measurements require the (very) long acquisition time of volumes, the development of projection-based measurement methods [START_REF] Leclerc | Projection savings in ct-based digital volume correlation[END_REF][START_REF] Jailin | Fast 4d tensile test monitored via X-CT: Single projection based digital volume correlation dedicated to slender samples[END_REF] made it possible to image fast phenomena. As they are based on projection and not on volumes, these methods exploit, at each time/loading step, only a partial (projected) spatial information.

An intentionally sparse acquisition aiming at a reduction of data storage and processing.

Although incomplete, other modalities / sensors are often coupled to full eld measurements as they provide complementary information (generally continu-ous in time and sparse in space) such as load measurements, extensometers, impedance measurements, etc.

In mechanical tests, the measurement of the kinematics is generally based on highly redundant (yet noisy) data [START_REF] Neggers | Big data in experimental mechanics and model order reduction: today's challenges and tomorrow's opportunities[END_REF] (e.g., refer to the 6050 (2048 × 2048 pix) images acquired in [START_REF] Berny | On the analysis of heat haze eects with spacetime dic[END_REF]). Moreover, those space-time measurements are themselves also redundant when dealing with model identication. Hence a natural trend to perform fast experiments and to tolerate noisy data, is to investigate the possibility to deal with as little information as possible. Namely, having a way to deal eciently with a sparse support in space and or time for the measurement would be highly desirable.

Field recovery from measurements with missing data or data corruption, is addressed in a very abundant and diverse literature relevant to various elds.

Dierent categories can be highlighted such as 1. Assumption on the low rank of the recovered eld which leads to the determination of few modes composed of space and time functions. Such is the case e.g., in robust-Principal Component Analysis (PCA) [START_REF] Netrapalli | Nonconvex robust pca[END_REF][START_REF] Wen | A survey on nonconvex regularizationbased sparse and low-rank recovery in signal processing, statistics, and machine learning[END_REF] where a low-rank dynamic background corrupted by sparse defects may be cleaned assuming few modes.

2. Methods based on a-priori knowledge. In model-based measurements (called Integrated-DIC [START_REF] Mathieu | Estimation of elastoplastic parameters via weighted femu and integrated-dic[END_REF][START_REF] Buljac | Experimental-numerical validation framework for micromechanical simulations[END_REF][START_REF] Buljac | Digital volume correlation: review of progress and challenges[END_REF]), the displacement eld is expressed from a model driven by a (partially-known) behavior. In previously introduced projection-based methods [START_REF] Jailin | Fast 4d tensile test monitored via X-CT: Single projection based digital volume correlation dedicated to slender samples[END_REF][START_REF] Jailin | Fast four-dimensional tensile test monitored via x-ray computed tomography: Elastoplastic identication from radiographs[END_REF], an incomplete data set is supplemented by assumptions regarding the smooth temporal evolution.

When the knowledge is on the variable statistics, optimal interpolation methods as Kriging, can be carried out.

3. Approaches based on regularization penalties (l 1 -norm and nuclear norm being the most popular in Compressed-Sensing [START_REF] Baraniuk | Compressive sensing[END_REF][START_REF] Wen | A survey on nonconvex regularizationbased sparse and low-rank recovery in signal processing, statistics, and machine learning[END_REF], e.g., applied for the image reconstruction of single-pixel cameras [START_REF] Mf Duarte | Single-pixel imaging via compressive sampling[END_REF]).

4. Other methods, based on prior full analyses (e.g., Gappy-POD, Gappy-PCA, (GPCA) for the recovery of images [START_REF] Everson | Karhunenloève procedure for gappy data[END_REF] and elds [START_REF] Willcox | Unsteady ow sensing and estimation via the gappy proper orthogonal decomposition[END_REF][START_REF] Ne | An application of gappy pod[END_REF]) allow recovering complete eld from learned dictionaries.

In the rst category (robust-PCA), the data corruption is assumed to be sparse, whereas we are motivated by the opposite limit where reliable measurement is sparse.

In the second approach, regularity assumptions issued from a physical model, or from an a priori assumption (as for Kriging) allows to swiftly interpolate from known data to ll missing patches. This category can be seen as the use of an appropriate lter. Although ecient and physically sound, such approaches are discarded from the present analysis to address the very question of incompleteness, but they can be easily merged with the proposed methodology to supplement it with a specic lter.

In the third approach, (Compressed Sensing), the sought information (after possibly an appropriate transformation ...) is assumed to have a sparse support, and it is this sparsity that is exploited to retrieve the information. For instance, when an image contains few phases, boundaries, being the support of non-zero gradients of the image, are sparse, and hence a total variation may be used to enforce a limited number of phases.

In the fourth approach, Gappy-PCA (as PCA itself ) does not necessitate any regularity assumption, although it is easy (and benecial when relevant) to supplement them with a specic lter. This is so common, that it is dicult to apprehend what contributes to the success of GPCA when both the PCA methodology and a regularizing lter are both present. In the present study, the choice is made to exclude all lters, or any recourse to continuity, to dierentiability or even to low power of high frequencies.

Thus, even if space and time are referred to, they may be thought of as disconnected positions in space and instant in time, that can be arbitrarily reshued either in space or in time. Hence, time is to be seen as a mere discrete labeling of measurement instant, and space is equally a labeling of discrete position. Hence, the focus is made only on the sparsity of the measurement. Nevertheless, it is obviously needed that the number of measurements is somewhat larger than the information content of the exploited signal, the low information content (compatible with the sparse measurement) lies in the low rank of separated mode representation of the signal. The latter is indeed robust with respect to any permutation in space or in time. In such a context, GPCA is the reference methodology.

In the present study, a variant algorithm is introduced that provides similar or better results, with fewer iterations, in the limit of very sparse and noisy data.

Dierent measurement methods in mechanics have been developed to exploit this low rank of separated mode representation. Such is the case for vibration measurements [START_REF] Passieux | High resolution digital image correlation using proper generalized decomposition: Pgd-dic[END_REF] and also for quasi-static behaviors for the measurement [START_REF] Jailin | Fast 4d tensile test monitored via X-CT: Single projection based digital volume correlation dedicated to slender samples[END_REF] and identication [START_REF] Jailin | Fast four-dimensional tensile test monitored via x-ray computed tomography: Elastoplastic identication from radiographs[END_REF] and ultra-fast dynamics with a crack propagation [START_REF] Berny | Mode-enhanced spacetime dic: applications to ultra-high-speed imaging[END_REF].

Inspired from the Proper Generalized Decomposition (PGD) technique [START_REF] Ladevèze | Nonlinear computational structural mechanics: new approaches and non-incremental methods of calculation[END_REF],

PGD-DIC consists of a progressive enrichment of the space-time modes for displacement corrections. As a side remark, those modal full eld measurement methods were initially applied for the decomposition of the dierent direction of the space displacements [START_REF] Passieux | High resolution digital image correlation using proper generalized decomposition: Pgd-dic[END_REF][START_REF] La Gomes Perini | A multigrid pgd-based algorithm for volumetric displacement elds measurements[END_REF]. As such, PGD algorithms can be seen as a complete Gappy-POD method with the progressive modal identication.

After a presentation of the treated problem in section 2, the proposed method is described and discussed in section 3. Then, various applications are carried out to challenge the approach considering dierent measurement protocol and noise. The comparison with Gappy-POD methods is nally shown and highlights the eciency of the proposed procedure.

Addressed problem

Notations

A space-time phenomenon is characterized by a physical quantity (which can be of any nature) whose space x and time t expression is denoted A(x, t) which is discretized in space and time and represented as a matrix A aα = A(x a , t α ).

Conventionally, in the following Latin (resp. Greek) indices will refer to space (resp. time). This eld is assumed to be well represented by an expansion over space-time modes

A aα = n i=1 f i ϕ i α Φ i a (1)
where, conventionally, it is chosen to normalize all space Φ i and time ϕ i modes to unity, Φ i a = 1 and ϕ i α = 1, and hence f i is the amplitude of mode i.

The eld A aα is unknown, but it is studied through dierent measurement devices. The measurement device acting at time t α , is dened as a linear operator, [M α ], that extracts from the eld A aα , a set of measurements gathered in a vector B aα = [M α ] ab A bα . Although the problem could easily be generalized, measurements are assumed instantaneous. A measurement is said to be partial if [M α ] is rank decient and cannot be inverted (the entire eld A •α cannot be obtained from the snapshot B •α ). Additionally, in the following, the measurements B are polluted by a random Gaussian noise, characterized by its variance σ. Arbitrary spatial covariance matrix could be considered, using its inverse as the optimal metric tensor. For the sake of simplicity, noise will be considered as white (or spatially uncorrelated).

An example of such partial measurement is provided by a 3D phenomenon captured from a 2D detector. Inasmuch as one individual measurement device is incomplete, a full experiment consists in combining such partial measurements in time from dierent points of view so that the 3D consistency can be captured. For instance, using stereo-vision, one can access 3D shape measurements making the combination of two (or several) partial measurement devices (2D cameras) a complete system. In the similar case of radiography, the combination of a (potentially large) number of projection directions allows the entire 3D microstructure to be reconstructed provided the studied object remains still during the scan rotation, a process known as tomography.

Another trivial example of such partial measurements is such that, at each instant of time t α , [M α ] is simply the identity restricted to a limited set of points Ω(t α ). In this case, [M α ] = [I] Ω(tα) is a diagonal matrix (projection over Ω(t α )). In the following, this elementary case will be considered as a toy problem to investigate the retrieval of A from B, where the collection of data along time will supplement the partial measurement at each elementary instant, and exploit the property that only few space-time modes are sucient.

This example is chosen so as to focus on the key point of compensating for lacking data, and not on the specic particularities of the measurement operator

[M α ].
In space and time, Figure 1 shows several examples of partial measurements in space and time, for a rather low coverage. Often, measurements occur over a compact support (window) which appears here as intervals. It is chosen here to consider intervals of the same length at each instant of time, however the position of this set of measurement may move over time in discrete steps, in a regular fashion, and randomly. Those examples are representative of dierent acquisition strategies encountered in practice. The rst case is encountered in the Scanning Laser Vibrometry technique [START_REF] Ab Stanbridge | Modal testing using a scanning laser doppler vibrometer[END_REF]), or for the modal analysis of a vibrating object from radiographs acquired at few directions [START_REF] Jailin | Measurement of 1-10 Hz 3D vibration modes with a CT-scanner[END_REF]. The second case illustrates the projection-based measurement methods [START_REF] Jailin | Fast 4d tensile test monitored via X-CT: Single projection based digital volume correlation dedicated to slender samples[END_REF] in which the projection of the sample is imaged during rotation and loading. Another application for the same case is the Continuous Scanning Laser Vibrometry techniques [START_REF] Sj Rothberg | An international review of laser doppler vibrometry: Making light work of vibration measurement[END_REF]. The third case is met when the window motion is erratic, as for instance for random acquisition patterns of the single-pixel camera [START_REF] Mf Duarte | Single-pixel imaging via compressive sampling[END_REF].

In the limit of a very sparse acquisition, the question of invertibility will be discussed in the following. Eventually an additional information, coming from additional sensors (generally continuous in time and local in space) such as load measurements, can be included to make the problem well-posed.

Static problem

Before addressing the modal analysis, let us specify our notations and recall how a static problem would be characterized optimally with respect to the set of noisy measurements. Here, for the sake of illustration, a large fraction of possible measurements has been chosen, η = 35%, while in the test cases chosen in section 4, η = 1%

The following cost function is introduced

T = (1/2) α [M α ]{A α } -{B α } 2 (2)
where the sought eld and measurement in space at each instant t α are gathered into a vector {A α }, and the norm is the one resulting from a maximum likelihood based on the noise covariance, Cov,

{X} 2 ≡ {X} • Cov -1 • {X} . (3)
For the white noise considered herein, this norm reduces to the Euclidian (L2) one.

Minimizing the cost function with respect to {A} leads to

α [M α ] ([M α ]{A α } -{B α }) = 0 (4) or α [M α ] [M α ] • {A α } = α [M α ] {B α } (5) Let us dene [H α ] = [M α ] [M α ] and [H] [H] ≡ α [H α ] (6)
It is this matrix, discussed in the introduction of section 2, that has to be inverted, whereas [H α ] cannot be as it is rank decient. The sum over α allows all complementary measurements to participate to the identication. The minimizer of T then reads

{A α } = [H] -1 • α [M α ] {B α } (7)

Proposed algorithm

Let us rst recall the basic principle of the GPOD algorithm [START_REF] Everson | Karhunenloève procedure for gappy data[END_REF], and a variant of it [START_REF] Beckers | Eof calculations and data lling from incomplete oceanographic datasets[END_REF].

GPOD relies on a plain POD algorithm, applied to a complete set of data.

However, because some data are lacking, fake values are used over this missing part. To initiate the process, an average value of the known data is used to ll in those parts, where averages may be computed over time or space or both [START_REF] Everson | Karhunenloève procedure for gappy data[END_REF].

POD is run over the resulting data set, with a specied number of modes. Then, the same procedure is repeated after data completion with the identied modes, and keeping the actual measurements unchanged. Upon iterations, the missing part are progressively lled with data that are more and more consistent with the measured ones. The procedure is stopped when a stationary solution has been reached. This algorithm will be refered to as GPOD1. There exists variants of this algorithm that are faster but approximate and they are not discussed herein.

Let us however mention a variant hereafter referred to as GPOD2 such that the number of modes is not xed for the entire procedure. On may run the same GPOD1 procedure, searching for one single mode to start with. After convergence, the number of sought modes is incremented up to the desired value (or after the incremental benet of using an additional mode is not considered sucient). The resulting modes obtained with this variant is similar to the previous (when the problem admits a unique solution) because the last iteration loop is similar but convergence is faster [START_REF] Beckers | Eof calculations and data lling from incomplete oceanographic datasets[END_REF].

Let us stress that the beauty of the GPOD algorithms is to turn an absence of measurement into a fake one, but where the alleged measured value is tuned to be the least inconsistent with the actual ones. This substitutes to each Hessian [H α ], a completed one which restores invertibility and makes the identication from each measurement instant a well-posed problem.

Proposed algorithm

As for GPOD2 procedure, the initial step is to identify a single mode. However, in contrast to the above two algorithms, missing data are not fudged even temporarily. When no measurement is performed, no extraneous equation is introduced.

From a set of measurements, {B α }, one looks for the spatial mode {Φ}, the temporal function ϕ α and amplitude γ such that the following cost function is minimized

T ({Φ}, {ϕ}) = (1/2) α γϕ α [M α ]{Φ} -{B α } 2 (8)
under two additional normalization conditions ϕ 2 α α = 1, and Φ 2 a a = 1. Thus, one has to solve the following two equations Eqs. ( 9) and [START_REF] Ma Iadicola | Uncertainties of digital image correlation due to pattern degradation at large strain[END_REF].

{Φ} = Argmin

{Φ * } T ({Φ * }, {ϕ}) (9)
ϕ(t) = Argmin {ϕ * } T ({Φ}, {ϕ * }) (10)
On the one hand, the minimization of T with respect to {Φ * }, leads to

γ α ϕ 2 α [M α ] [M α ] • {Φ} = α ϕ α [M α ] {B α } (11)
On the other hand, the determination of the associated temporal amplitudes is obtained by a minimization of T with respect to ϕ α giving

γϕ α = {Φ} [M α ] {B α } {Φ} [H α ]{Φ} (12)
It is proposed to use a xed point algorithm, where ϕ α is rst initialized to 1, and from Eq. ( 11), γ{Φ} is computed. The spatial mode and its amplitude are then obtained from their product by using the normalization condition Φ 2 a a = 1. Then from the determined spatial mode, the temporal evolution, γ{ϕ}, is computed from Eq. ( 12), and again, γ is determined by the normalization condition ϕ 2 α α = 1. The staggered determination of the spatial and temporal modes is iterated until a xed point is reached. This concludes the determination of the rst (dominant) POD mode.

As for PGD methods, the measurement is performed by adding modes progressively, one at a time. Once a rst mode is measured, a residual, expressed only on the measured areas, is computed as

{R α } = {B α } -γϕ α [M α ]{Φ} (13)
This residual {R α } is substituted to the measurement data, {B α }, in the above single-mode procedure, providing the second mode, which is subtracted from the starting data to give a second residual. The latter substitution of successive residuals is iterated to generate as many modes as needed until the residual is comparable to noise.

It is to be underlined that the solution of Eqs. ( 9) never involves fake measurements. As compared to GPOD, where the latter are included to restore invertibility at each instant of time, they also endow the algorithm with a kind of inertia, that slows down convergence. Thus the proposed algorithm is expected to show a much faster convergence. Yet, in the absence of noise, all those algorithms should converge toward the same (exact) solution. For noisy data, the proposed scheme that always gives the proper weight to each measurement is expected to be more robust.

It is to be noted that such a greedy approach for the case of complete measurements is strictly equivalent to the plain POD approach where all modes are sought simultaneously. In the following, because of our choice of considering a toy problem for which the metric is just trivial, PCA, POD or PGD are three dierent names for an equivalent result. However, it is to be observed that, when the norm used in the functional to be minimized results from an arbitrary variational formulation, (a PGD type problem), the very same procedure can be duplicated providing the corresponding generalization to sparse data acquisition. Just to mention a simple example, when noise is spatially correlated, the introduction of a metric based on the inverse covariance matrix, (see Eq. ( 3)) is trivially taken into account in the above cost function, and it would prevent a straightforward use of GPOD.

Uniqueness for one single mode

Let us now consider the particular case of the toy model of direct measure-

ments, [M α ] = [I] Ω(tα) . Thus, [H] = α [M α ] 2 = α [M α ] is also diagonal
and the ath element along the diagonal is d a = α I α (a), or in other words, the number of times position a has been measured. Hence, the only condition for being able to invert the Hessian [H] is that all sites should be visited at least once. Obviously, this condition guaranties the uniqueness of the solution for the static case (where ϕ α = 1). More generally, when only one spatio-temporal mode is present, but the time evolution is not constant, then it is needed that all positions in space are visited at least once, but similarly along the time axis, at all considered instants, at least one measurement has been performed.

However, assessing uniqueness of the mode determination requires a more complete discussion. Let us consider the case of two batches of measurements, with for instance, a rst half of sites being measured during a rst period of time, after what the second half is measured over the same amount of time. The rst domain is labeled (X) and the second (Y ). These two domains corresponds to a natural partition in two sub-problems each of which being solved by a standard POD. Considered globally, this problem is within the class of partial measurements. Let us suppose that this problem has been solved and that a mode has been computed

A aα = γ (1) ϕ (1) α Φ (1) a ( 14
)
We can now construct another triplet (γ (2) , ϕ (2) , Φ (2) ) such that

ϕ (2) α =    ω X ϕ (1) α when α ∈ (X) ω Y ϕ (1) α when α ∈ (Y ) Φ (2) a =    (λ/ω X )Φ (1) a when a ∈ (X) (λ/ω Y )Φ (1) a when a ∈ (Y ) (15)
Let us dene ξ 2 = ϕ (1) 2 (X) and η 2 = Φ (1) 2 (X) . In order to fulll the nor- malization conditions of (ϕ (2) , Φ (2) ), the following two conditions are to be met

ω 2 X ξ 2 + ω 2 Y (1 -ξ 2 ) = 1 (λ/ω X ) 2 η 2 + (λ/ω Y ) 2 (1 -η 2 ) = 1 (16)
Provided these two conditions are satised, (f (2) , ϕ (2) , Φ (2) ) is just equivalent to (f (1) , ϕ (1) , Φ (1) ), however two conditions to determine three degrees of freedom λ/ω X is lacking a constraint. Thus, the solution cannot be unique. Hence, although all observables B may be accounted for as a single space-time mode (one separated representation), the solution is not unique if no overlap between the measurement domains exist. This solution can be further extended to a collection of blocks with no overlap. Note however that overlap is to be considered after an arbitrary permutation of measurement of sites or instants. To evaluate the degeneracy of the solution in terms of modes, it suces to count the number of connected components using simple graph-cut algorithms.

Furthermore, let us note that connectedness of the components is not yet the ultimate criterion: if connectedness of a single component is only due to a unique measurement at a particular location ã and time α, its value may be seen as providing the additional equation needed to complement Eq. [START_REF] Mathieu | Estimation of elastoplastic parameters via weighted femu and integrated-dic[END_REF] enforcing ω X = ω Y = λ = 1. However, this necessitates the value of the mode at this specic site and time, ϕ

(1) ã , be non-zero. This implies that the uniqueness of the solution is not only due to the measurement set-up (the choice of measurement positions and instants), but also to the modes which are measured. In particular, the locations where the mode vanishes (or simply assumes a low absolute value as compared to the surrounding) are potential sources of fragility for the solution. Those situations are typically those that are easily cured by requiring a smooth behavior for both spatial mode and/or time function.

As an alternative to having an overlap, let us note that adding an additional information may allow to recover a well posed problem. One natural assumption is that stationary modes are present, e.g., because a vibrating system is subjected to a random but steady excitation [START_REF] Jailin | Full eld modal measurement with a single standard camera[END_REF][START_REF] Jailin | Measurement of 1-10 Hz 3D vibration modes with a CT-scanner[END_REF]. In such a situation, possibly a wealth of information may be accessible (exploiting a statistical distribution that may be Gaussian), but the most robust ones rely on low order statistical moments of the distribution per block, i.e., having an equal total power per block of measurements, ϕ 2 α block = 1 (similar to the global normalization). This implies only that the number of measurements is large per block as the decay of uctuations (law of large numbers) is slow.

Uniqueness in the case of multiple modes

When multiple, N mode , space-time modes are present, one may generalize the previously mentioned graph theoretic argument based on connected components. However, at least N mode common sites should be shared between blocks so as to be considered connected.

The above discussion of uniqueness may be further extended to the analysis of uncertainty. Thus rather than just a determination of well-posedness of the problem, an assessment of the uncertainties may reveal ghosts of the above undeterminations. The least stable eigenmodes may display such block-like multiplicative features as those above analyzed. Such were indeed observed in test cases, especially when spatial and temporal modes crossed a 0 value. This point emphasizes the property that stability is not only a matter of experimental design, but also involves properties of the observed modes. This observation was the motivation for introducing the previous discussion.

Test cases

The procedure is applied for the recovery of dierent synthetic test cases.

In order to test the proposed method, we chose extreme cases of sparse acquisition representing η = 1% of the entire (x, t) domain, where the latter is chosen to be of size N x = 1024 locations in space and N t = 1024 instants of time for measurements. Therefore the number of measurements is ηN x N t .

Moreover, an additional Gaussian white noise is added to the measured data, with a standard deviation that represents 10% of the standard deviation of the reference data. (The signal to noise ratio is thus 20 dB).

The reference space-time eld was synthesized by the Fourier transform of random amplitudes and phases. High frequencies were cut out to produce smooth elds. However, as earlier mentioned no regularity assumption was exploited to retrieve the modes. Thus, this regularity is more relevant to pinpoint unsatisfactory recovery from visual inspection.

For the approach to make sense, some redundancy is necessary, and hence the reference data should require less data than the measured ones. Hence, it was chosen to consider only N mode = 4 separated modes. Each mode requires (N x + N t ) components. Therefore, (when the measurements are homogeneous in the space-time domain), the redundancy is

β = Measurements Information content = ηN x N t N mode (N x + N t ) = 1.28 (17)
so that no more than 28% of the data is redundant and is to be used to reduce noise. The three cases of distribution of measurements in space and time that are shown in Figure 1 were tested together with a completely random distribution of measurements, but the subsampling was kept to a constant value η = 1%.

In order to evaluate the proposed algorithm a comparison is proposed with the two reference algorithms discussed earlier, GPOD1 and GPOD2. All algorithms are iterative, and hence the same criterion was chosen to terminate the iterative procedure. When the increment in relative error between two successive iteration became smaller than 10 -4 , iterations were stopped. Hence the number of iterations varies from one case to the other. Moreover, the maximum number of iterations was set to a maximum value of 400 if the previous criterion was not reached. However, because both GPOD2 and the proposed algorithm involve many loops, their total number of iterations (summed over all loops) may exceed 400.

Case Algorithm Iterations Final Error

Step scan GPOD1 1, the results of the four dierent cases illustrated in Fig. 1 for all three algorithms are reported. The nal error is computed as the norm (over space and time) of the dierence between the recovered eld, and the original reference eld without noise. This norm is scaled to the standard deviation of the original reference eld. The presence of a high level of noise and of the very sparse acquisition η = 1% is such that the computation cannot recover exactly the original data. The sole presence of noise would be responsible for an error of about 4 to 5%. The exploitation of the presence of only few modes, allows all algorithms to reduce this error to a lower level in all cases. However, all algorithms do not reach the same level of nal error. Both variants of Gappy-POD provide results of comparable quality in terms of errors, but GPOD2 requires often more iterations than GPOD1, so that the claimed of a progressive enrichment of modes does not appear so obvious in the studied case (which is clearly an extreme case). The proposed method shows generally a markedly lower residual error (down to 50% smaller), and yet with a much lower number of iterations.

Two cases are chosen for a more detailed illustration, the random acquisition case in Figure 2 and the continuous scan in Figure 3. In both gures, the reference eld (without noise) is shown on the top line together with the support of the measurement. The elds reconstructed from the four identied modes are displayed for the three algorithms. Finally, the evolution of the error with iterations is shown for the three algorithms. The fact that both GPOD2 and the proposed method consists of N mode = 4 loops can be seen from the error evolution. In this latter set of graph, the error that would correspond only to the presence of noise (but no missing data, and no mode recovery treatment) is indicated as a dotted line. It is to be noted that the range of variation of both axes dier from one algorithm to the next.

In Figure 2, GPOD1 shows a marked dissymmetry between time where the mode are well captured and space where amplitudes are depressed. It is also to be noted that convergence is quite slow, and the criterion for interrupting the mode recovery stopped the computation well before its asymptotic result. GPOD2 performs much better for the reconstructed eld, but required more than 200 additional iterations. The proposed algorithm has a residual error level better than that of noise after only 2 iterations (to be compared to 450 for GPOD1, and 420 for GPOD2).

In Figure 3, relative to the continuous scan case, the temporal evolution is well captured, but the spatial modes seem to have been erased, for both GPOD1 and GPOD2. (This dissymmetry between space and time originates from the way missing data are lled in with spatial averages at each instant of time.) This is to be contrasted with the present method which gives a much more balanced picture of the space-time eld, and reaches an error level which is cut down by a factor of 2 as compared to Gappy-POD results. For all three algorithms, convergence is faster than for the previous random case, and the convergence criterion is reached much earlier in terms of iterations. The very formulation of the introduced algorithm makes its extension to Proper Generalized Decomposition (PGD) straightforward. Namely, the cost function that was used herein, Eq. ( 8), involves a simple L2 norm, but can simply be tailored to any other problems where a variational formulation is accessible.

Figure 1 :

 1 Figure 1: An illustration of dierent partial measurements in space x and time t in the simple case of an indicator (shown in grey), for stepwise scan (a), continuous scan (b), randomly moving window (c), and random acquisitions (d).Here, for the sake of illustration, a large fraction of possible measurements has been chosen, η = 35%, while in the test cases chosen in section 4, η = 1%

Figure 2 :

 2 Figure 2: Random acquisition case. (a) Reference eld; (b) measurement support with random acquisitions; (c) reconstructed eld using GPOD1; (d) reconstructed eld using GPOD2; (e) reconstructed eld using the proposed method; (f) relative error between reconstruction and reference as a function of iteration number for GPOD1; (g) same relative error for GPOD2; (h) same relative error for the proposed method.

Figure 3 :

 3 Figure 3: Continuous scan case. (a) Reference eld; (b) measurement support; (c) reconstructed eld using GPOD1; (d) reconstructed eld using GPOD2; (e) reconstructed eld using the proposed method; (f) relative error between reconstruction and reference as a function of iteration number for GPOD1; (g) same relative error for GPOD2; (h) same relative error for the proposed method.

Table 1 :

 1 Comparison for dierent test cases (Step scan, Continuous scan, Randomly moving scan, and Random acquisition) of the three algorithms (GPOD1, GPOD1, Present Method). For each combination of test case and algorithm, the number of iterations at convergence is given, together with the relative error at the end of the computation In Table

	400 *	0.024

Acknowledgements

The authors acknowledge the support of Safran Tech for a project where the above procedure extended to Digital Volume Correlation was applied successfully. This work benetted from useful discussions with N. Belkhir, D. Duclos, E. Parra and J. Schneider.


A tour of my soft matter gardenfrom shining globules, soap bubbles to cell aggregates Françoise Brochard-Wyart Sorbonne Université, Professor Emeritus, UMR 168, Institut Curie, 26 rue d'Ulm Institut Universitaire de France CNRS, UMR 168, Institut Curie, 26 rue d'Ulm, 75248 Paris Cedex 05, France Like The Magic Flute, my career has been paved by wonderful and unexpected stories played by enthusiastic and talented students, in close contact with experiments and industry. I participated to the birth of soft matter physics under the impulse of Pierre-Gilles de Gennes: polymer, liquid crystals, colloids and wetting, which I have later applied to the study of living matter. By teaching in the early days at the Institute Universitaire de Technologies d'Orsay, I got into contact with industry, which give me the chance to establish collaboration with several companies: Rhône Poulenc, Dior, St-Gobain, Rhodia and Michelin. These partners have not only largely financed my research activity in physical chemistry, but they also offered us a wealth of innovative research topics. In 1996, when Professor Jacques Prost became of the director of the PCC Curie laboratory, in the Pavillon Curie built for Marie Curie, I turned to biophysics. Throughout this scientific adventure, I had a great guide, Pierre-Gilles de Gennes.

I) Childhood in Annecy

I need to begin this autobiography by describing my childhood and my parents, who have been a model of life teaching to their five children to be strong and simple and to get to the bottom of things without fanfare. In March 2016, I was made "Officier de la Légion d'Honneur" by the Professor Yves Quéré, from Ecole Polytechnique. He summarized beautifully my origins: « What parents you had! Pierre Wyart, your father, whose own father was driving locomotives to the North Railroad Company, and whose brother, Jean Wyart, was the famous mineralogist and crystallographer, some of us were able to attend his classes at the Sorbonne. Your father started the Ecole Polytechnique at 17 years old in 1928, magnificently illustrating how one could climb the French social ladder of the timenot so blocked! His undergraduate studies were soon after interrupted by tuberculosis, which penetrated the vertebrae. He therefore entered in a sanatorium in Briançon where he stayed for 5 years. Dura calamitas for this young man full of ardor, vitality and desire to undertake, but also Felix calamitas since it is there, in the sanatorium of Briançon, that he met his future wife, Liliane Weber, your mother. Finally, he recovered and entered a School of Application in Saint Etienne where he learns hydraulics before being named in Annecy. This is during that time that you are born, on Sept 11 th , 1944. The family now lives in the small town of Verrier du Lac, where you were a pupil in a single-class school. During these years, you have a lot of fun with the children of the neighboring farms. So much fun that when the family has to leave to go to Annecy where you were already enrolled in primary school, you burst into tears and repeat a class: bad luck for Francoise that worries her parents until she passes the 6th grade exam, her first intellectual high. Back to her father: in 1946, Pierre Wyart participated in the creation of nothing less than the company Electricity of France (EDF), which he quickly climbed the ladder, a company that he will even be proposed to become the general director.

To the highest functions this man of intelligence, action and dialogue, a sort of Julien Sorel of modern times, will never forget that he is the son of a worker. Always active and full of energy, when he retires, he creates and leads an agency dedicated to the resolution of international conflicts on energy. Your mother was an amazing woman, full of energy and vision, who wished to be a doctor but she was born a little too early: studies are for men, so she became a nurse. Your mother, who became a nurse traveling to Northern African countries such as Algeria, from where, anticonformist as she was, she brought back a monkey and an antelope and moved to Briancon's hospital, where she met a charming young man suffering from Tuberculosis... Raising her 6 children, your mother practiced throughout her life the taste for culture (she managed libraries in old people home) and openness to others,

From the roofs of Building 210 to Building 510

After a brief work on superconductivity under the supervision of Fyl Pincus, I joined as a theorist the Liquid Crystal Group to work in close collaboration with several experimental teams: Etienne Guyon, my former professor at ENS Cachan, who was key to build my confidence, Georges Durand, Madeleine Veyssié, Robert B. Meyer, Francis Rondelez and Pawel Pieranski. My colleague Liliane Leger initiated me to her beautiful experiments as we performed twin Ph.D., hers in experimental physics and mine in theory. Nematic liquid crystals are very sensitive to external fields. At the time of my Ph.D., it was interesting and important for applications in display to understand their transient responses to magnetic or electrical fields suddenly applied. I therefore studied the dynamics of the Frederick's transition of a nematic liquid crystal. The nematic is aligned by the surfaces of the sample and distorted by a magnetic field perpendicular to the nematic orientation. A distortion is observed above a threshold field H c 1,2 . I studied the hydrodynamic "back flows" 3 induced by the rotation of the molecules. For a given field H > H c , the system may choose between two different distortions patterns and the nematic slab breaks into domains. I analyzed the structure of the walls separating two domains and showed that the thickness of the wall diverges at H c . I modeled the motion of the domain and the slowing down of the dynamics near the critical magnetic field [START_REF] Brochard | Mouvements de parois dans une lame mince nématique[END_REF] . This project was a first step to study the dynamics of liquid crystals phase transitions. The different types of liquid crystals are indeed connected by remarkable transitions: at these transitions appear critical fluctuations much subtler than those of the usual systems (liquid -gas, magnetic...). My background in superconductivity became very useful in this area, as certain transitions, like smectic A / nematic or smectic C / A, resemble from the point of view of their symmetry superfluid transitions. Therefore, I built dynamic scaling laws in analogy with the models of Halperin and Hohenberg [START_REF] Brochard | Dynamique des fluctuations près d'une transition smectique A-nématique du 2e ordre[END_REF][START_REF] Jähnig | Critical elastic constants and viscosities above a nematicsmectic a transition of second order[END_REF][START_REF] Brochard | Damping of second sound for smectic a liquid crystals[END_REF] . This work, which made up most of my Ph.D. thesis, was done in collaboration with several groups in Orsay. At the end of my Ph.D., we extended these works to lipid-water mixtures, referred to as lyotropic liquid crystal. We calculated the collective modes of lamellar phases and found six classical modes of thermotropic smectics, and an additional mode, the slippage mode, due to the addition of one variable, the lipid concentration 8 . This 7 th mode was later observed at Harvard University by forced Rayleigh diffusion (Pershan, 1978). I defended my thesis in 1974 with the title "Dynamic properties of mesomorphs". At that time, PGG put me in contact with J.F. Lennon, a researcher from the Laboratory of Hematology from Pasteur Institute, who has carried out the first quantitative studies on the scintillation of red blood cells, a strange phenomenon observed for a long time in phase contrast microscopy, but totally unexplained at the time. The flicker of the red blood cell was seen as an active movement of the cell, at a physiological frequency related to the life of the cell, as the heartbeats. There a nice collaboration started to understand this phenomenon. Suddenly, the walls of my office in Orsay became decorated by the experimental data of Jean-François Lennon. He measured experimentally the correlations functions for the flicker intensities at two different points on the cell surface, at various filtering frequencies. We interpreted theoretically these results in term of the thermal fluctuations of the thickness of the biconcave cell shown in figure1, which modulate the transmitted light. In physiological conditions, the membrane surface tension vanishes and the resistance to deformation is mainly due to curvature energy. In this approximation, the fluctuations are of very large amplitude (a fraction of a micron) as required by the observations. The detailed shape of the correlation functions is in rather good agreement with the theory [START_REF] Brochard | Frequency spectrum of the flicker phenomenon in erythrocytes[END_REF] . We conclude that a purely physical model is sufficient to interpret the flicker effect, but that rather stringent physiological conditions are required to maintain the zero-surface tension, which is critical to the effect. This early work will be crucial for my transition to biophysics many years later! Figure3) Une bulle dégonflée : le globule rouge (La Recherche 1977)

A deflated bubble: the red blood cell

The end of the Ph.D. after Mai 68 was a period of doubt for scientists of my generation in search for meaning. Some became physicians or moved to farms raising sheep in remote areas such as the plateau of Larzac in the south of France. On my side, in order to experience the life of a scientific editor, I wrote an article for the public journal La Recherche (Figure 3)

Dynamics of polymers

Polymers confined in a pore An interesting problem, important for enhanced oil recovery, is the dynamic behavior of polymeric chains trapped in very fine pores. My first calculation in polymers was to transpose the physics of polymer solutions in 3d to 1d, where polymers are confined to capillaries. I constructed dynamic scaling laws for these problems [START_REF] Brochard | Dynamics of confined polymer chains[END_REF] , which disproved a previous Debye analysis as PGG was proud to claim. Subsequently, I worked closely with S. Daoudi on the forced passage of one single polymer chain through the entry of a pore and on the behavior of polymer solutions in convergent / divergent pores [START_REF] Daoudi | Flows of Flexible Polymer Solutions in Pores[END_REF] . The two mechanisms shown in the Figure 4 Later on, I studied the properties of melted polymer mixtures in confined geometry that was part of E. Raphael's thesis [START_REF] Brochard-Wyart | Scaling theory of molten polymers in small pores[END_REF] . More recently, with C. Gay and T. Sakaue, we studied the passage of branched polymers, and showed that the forced penetration into a pore does not select the chains according to their length or their topology [START_REF] Gay | Injection Threshold for a Statistically Branched Polymer inside a Nanopore[END_REF][START_REF] Sakaue | Flow injection of branched polymers inside nanopores[END_REF] .

Polymers in solution Polymers in bad solvents

With PGG, I studied the movements of a chain in bad solvent where it is strongly folded on itself, and where it forms many self-nodes [START_REF] Brochard | Dynamical Scaling for Polymers in Theta Solvents[END_REF] . This aspect, ignored in classical literature, leads to curious effects observed by neutron and light scattering (G. Jannink-M.Adam CEA), which do not follow simple dynamical scaling laws because there are two characteristic lengths, the monomer-monomer correlation length and the distance between the nodes [START_REF] Brochard | Gel-like modes. of polymer solutions in « θ » solvents[END_REF] . We then studied the dynamics of swelling of a polymer droplet immersed in a good solvent, where we have introduced the "Magic length". Polymer dissolution is a significant problem for many industrial processes, with a lot of empirical data available, but without an overall view. We have been able to analyze different steps in the process, and have shown on a (non-physical) case where n = 0. However, we know, many examples in theoretical physics, that the extension of some results to non-physical values of the parameters can be fruitful (as well as complex angular moments in collision physics). The "n = 0 theorem", established by us in 1972, has made it possible to transpose the enormous theoretical arsenal accumulated with regard to phase transitions to polymer problems Thanks to the n = 0 theorem, the problems related to self avoiding random walks have been fundamentally related to the physics of phase transitions, all of them having a very great universality, that is, laws independent of the local structure. practical level that there is an optimum size for the concentrated polymer drops that must be prepared (by mechanical stirring) before starting the true dissolution: the time of dissolution decreases with the size of the droplets, but reaches a plateau value, because the chains have to disentangle. We also studied later with C. Williams the dynamics of collapse of one single chain [START_REF] Williams | Polymer Collapse[END_REF] and the swelling of a gel with free polymer chains [START_REF] Brochard | Polymer networks swollen by a homopolymer solution[END_REF] . Polymers in liquid crystal solvents I followed the experimental work of M. Veyssié, J. Dubault, on the usual polymer -liquid crystals mixtures. I modified my first description, which was limited to weak coupling situations. Quite often, the nematic order is completely destroyed inside the chain and one has a strong coupling situation: isotropic solvent droplet in the region of the cloud of monomers. For short chains (smaller than the coherence length  of the nematic), the unexpected result is an "ideal" chain behavior. For long chains (R> ), there is a "collapsed" regime. We also calculate the phase diagram of nematic polymer mixtures with P.Levinson [START_REF] Brochard | Phase diagrams of mesomorphic mixtures[END_REF] .

Dynamics of polymer melt

Couplings of melt-compatible polymers are rare, but of great importance for industrial applications (polymer metallurgy, self-adhesion). For such couples, I had predicted unusual interdiffusion laws [START_REF] Brochard | Polymer diffusion in blends: effects of mutual friction[END_REF] . Following a collaboration with experimenters Prof. Lauwrence (Amherst), J. Klein (Cambridge) and E. Kramer (Cornell), it turned out that my results explained well the interdiffusion in the case of comparable lengths, but did not apply to the case of asymmetrical junctions. My first theoretical interpretation, based on the model of reptation of polymer chains crawling in fixed "tubes", was incorrect for the case where one of the species is much shorter than the other. It took me five years to arrive at a satisfactory theoretical description, in which the tubes themselves move with a velocity unambiguously determined by a balance of forces on the tube [START_REF] Brochard-Wyart | Interdiffusion de polymères flexibles très enchevêtrés[END_REF] : the small chains swell the transient network of the long chains. I was very proud of this result, but I published it in French in the Comptes Rendus de l'Académie des sciences (Paris): Masao Doi who was first skeptical to the motion of tubes, recognized his importance, used it and asked me to put my name in his article!!! because my own article will be lost.

Polymers at interfaces: Role of Connectors

The graft polymers are used as connectors (example: metal / polymer adhesion), as colloid stabilizers (for example: Indian ink, "hairy" vesicles formed of partially water-soluble lipids and copolymers, which stand two days in blood, while simple vesicles are instantly destroyed), as wetting control agents (example: silanized silicon wafers). These long graft chains also play an important role in the rheology of polymers at interfaces, which was studied at the College by Léger and H.Hervet. A polymer melt slides on a smooth surface. On a surface onto which a few chains have been grafted, the slip is destroyed at low shear rates because of the strong friction but restored at high shear rate because the chains undergo a coil-stretch transition and disentangle with the melt. With PGG, we introduced the concept of a "marginal state" for a wide range of sliding velocities (four decades), where friction becomes solid type [START_REF] Brochard | Shear-dependent slippage at a polymer/solid interface[END_REF] . We extended this study to the slip of a rubber on a grafted surface mainly with L. Leibler, A. Ajdari and C. Gay. It's amazing that a marginal state has been observed recently in the shearing of entangled fire ants glued to the wall of the rheometer with Velcro! I also participated in the development of a theoretical model of brush-rubber adhesion, extended later to cellular adhesion induced by specific binders [START_REF] Brochard-Wyart | Adhesion induced by mobile binders: Dynamics[END_REF] and its measurement by enforced cell's detachment.

Wetting

The physics of wetting was initiated at the College de France thanks to the Petroleum French Institute IFP, concerned in the wettability of rocks and the role of polymers used in oil recovery. The different regimes of wetting of a droplet deposited on a substrate (partial versus complete wetting) depend upon the sign of the spreading parameter S. For the case of complete wetting (S positive) the structure and the dynamics of the precursor film, named van der Waals pancake, was studied by J.F. Joanny, at this time a young student of PGG. I decided to extend their work to the wetting of fibers with two Ph.D. students D.Quéré and J.M.di Méglio [START_REF] Brochard | Spreading of liquid drops on thin cylinders: The '"manchon/droplet"' transition[END_REF][START_REF] Quere | Spreading of Liquids on Highly Curved Surfaces[END_REF] . If the spreading on a solid plane was relatively well understood, the spreading on fibers (textiles, fiberglass, carbon, hair) remained mysterious. Why does a drop spreading on a plane no longer spread on a fiber? We have shown that in this situation, the excess L / G surface linked to the cylindrical geometry leads to a new situation, which we named "pseudo-partial wetting": the drop does not spread even if S is positive! The equilibrium state is a droplet in co-existence with a microscopic film. We predicted "Droplet-Manchon" transitions and studied the dynamical properties of the spreading: a drop placed on a fiber spreads by emitting two films of microscopic thickness. We also studied the stability of films deposited on a fiber, the threshold thickness beyond which the liquid sheath splits into droplets and the dynamics of this Rayleigh instability (Fig. 5). This work is crucial for the coating of textile fibers: during the extrusion of the polymer wire at gigantic speeds over the earth moon distance, the fiber must be covered with a lubricating liquid film. Any fault causes a break in the fiber and the coil becomes entangled. Visiting l'Oreal, I discovered that they used our work for the formulation of mascara: the drying must be faster than the rise time of the instability, to get a regular coating of the cilia. We have also investigated the wetting phase diagram of solid substrates coated with a molecular film [START_REF] Brochard-Wyart | Spreading of nonvolatile liquids in a continuum picture[END_REF] . Comparing the polarizability of the film versus the polarizability of the solid, we predicted complete, partial and pseudo-partial wetting, for the case of a low polarizability solid (plastic) covered with a polarizable layer (gold). A sessile droplet spreads on gold but does not spread on golden plastic: at equilibrium a macroscopic droplet with a finite contact angle coexists with a nanoscopic precursor film. At the end of my stay at the College de France, I attended a PGG's lecture on soap films. He described the growth of Newton black films observed by K. Mysels, which nucleate during the drainage of a vertical soap film. For fun, I transposed the theory that PGG described for suspended films to the case of films supported by a solid substrate. A black film becomes a dry patch, that opens up on a wet substrate. The dewetting was born! and it was going to keep me busy for almost 10 years! I can see dewetting every day in my shower when dry region open on my wet skin. In 1984, my son Olivier was born.

IV) Sorbonne University 1986-1991

In 1986, I was appointed Professor at Paris VI. With J.P. Badiali, M.P. Pileni and F. Rondelez, we create a team centered on the problems of structure and reactivity at interfaces in the laboratory of Physical Chemistry, in the Jean Perrin building. In this new environment, I tried to keep the same style of work, dealing with various areas of physical chemistry (surfactants, polymers, hydrodynamics), and following fairly closely the industrial needs (textile, petroleum, cosmetic industries, chemical engineering). We started to work on wetting and dewetting, Ph.D. C.Redon, financed by Dior: we want to produce a shampoo to keep our hair dry when going to the pool! We did not succeed in achieving what nature has been able to do for the feathers of ducks, that remain dry thanks to nanoparticles which make them super hydrophobic, but we started the first fundamental studies of dewetting by nucleation and growth of a dry patch [START_REF] Redon | Dynamics of dewetting[END_REF] (Fig. 10A) and spinodal decomposition [START_REF] Wyart | Drying of solids wetted by thin liquid films[END_REF] . We monitored and modeled the dynamics in both viscous and inertial regimes, where capillary waves are induced by the motion of the rim collecting the liquid of the dry region [START_REF] Buguin | Shocks in Inertial Dewetting[END_REF] . Wetting is a science that seduced me because it intervenes in our daily life. On the aesthetic level, the experiences are very beautiful but not as simple as it seems! It also leads to many industrial processes, as illustrated in our book Capillary and wetting phenomena with PGG and D. Quéré (fig 16). This action gave me the opportunity to work with Dior, Atochem, Rhône-Poulenc, Michelin, Valéo and Airbus. Thanks to Francis Rondelez, I became interested in phase transition in Langmuir monolayers. With D.Andelman and J.F.Joanny, we have shown that the interactions between permanent dipoles modify the phase diagram [START_REF] Andelman | Phase transitions in Langmuir monolayers of polar molecules[END_REF] : at the liquid / gas or liquid-liquid / condensed liquid transitions, long-range repulsions between dipoles give rise to periodic "super crystal" arrangements visualized directly by epifluorescence . When the laboratory SRI moved to Jussieu in 1991, we decided with Francis Rondelez to join the Curie Institute, in the same building! in the section of Physics directed by Monique Pagès, which was changing orientation towards soft matter. The same year, my son Marc was born.

V) Institut Curie 1991-present

In 1992, I entered the Physics and Chemistry section of the Institute Curie where we founded, with F. Rondelez and D. Chatenay, the team "Physics of Surfaces and Interfaces". In 1994, J. Prost brought our team and a team of chemists together to create the PCC, Physico Chimie Curie Unit. It lies at the interface between physical chemistry and cell biology. I started the team Surface Douces "Soft Surfaces" focused on hydrodynamics and polymers at interfaces with an outstanding experimentalist Axel Buguin and a great time started for me! In 2000s, I began a gradual transition from soft matter to biology by studying biomimetic systems. Then we validated our approaches on cells, which raised new questions that I developed with my team, and in collaboration with groups in France and abroad. My experience with polymers was useful because the cells, from a mechanical point of view, can be considered as viscoelastic jelly beads, surrounded by a liquid membrane which gives them a surface tension, such as droplets.

In 2003, I was nominated member of the Institute of France IUF.

Soft matter 1992-2008

During these years, I was able to pursue an activity in soft matter and develop new projects in biology.

Dewetting at soft interfaces: aquaplaning and cellular adhesion

On the problem of adhesion, we started with Michelin a long-term program to investigate rubber / wet surface contacts. We first studied theoretically [START_REF] Brochard-Wyart | Dewetting of a water film between a solid and a rubber[END_REF] the dynamics of dewetting of a liquid film intercalated between a soft rubber and a rigid substrate. Using Reflection Interference Contrast Microscopy (RICM) P. Martin was able to follow the dynamics of growth of dry adhesive contacts . With a macro AFM built by Axel Buguin where the tip is replaced by a millimetric elastic ball, we studied the adhesion and the friction of rubber beads pressed on a rigid plate in air or immersed in a liquid. When the plate slides at a velocity U larger a threshold velocity, the contact is lubricated, corresponding to the aquaplaning that happens when a car looses control on a wet road (Fig. 6 A). However, if the sliding surface carries an asperity, a "dry wake" can be induced ( Fig. 6 B) in analogy with Čerenkov radiation, with a Mach angle derived from a competition between dewetting velocity v and invasion velocity U [START_REF] Martin | Čerenkov" dewetting at soft interfaces[END_REF] , which restores adhesion. We used the same set up to study cellular adhesion [START_REF] Puech | Adhesion between Giant Vesicles and Supported Bilayers Decorated with Chelated E-Cadherin Fragments[END_REF][START_REF] De Gennes | Adhesion Induced by Mobile Stickers: A List of Scenarios[END_REF] which is also done by the elimination of a liquid film of comparable thickness. After having worked on model systems (elastomer ball immersed in a viscous liquid and sliding on smooth glass), adhesion was studied on systems closer to reality: I) microstructured surfaces (pillars) (Fig. 6 C) to simulate the asperities of the soil; ii) high speed friction. This project required microfabrications of surfaces and the realization of a rotating disk force machine to explore very high speed ranges. In addition, we built biomimetic models of cells at all scales, ranging from mm to micron: viscoelastic beads decorated with proteins to simulate the properties of the cytoskeleton and specific adhesion, soft shells to represent the cortex, suction cups for simulating the adhesion of amoebae in the intestine. These giant systems allowed us to more easily study the dynamics of adhesion and detachment on smooth (Fig. 6 D) and rough surfaces [START_REF] Brochard-Wyart | Adhesion of soft objects on wet substrates[END_REF] .

Bursting of viscous bubbles

We are in 1991, when PGG receives the Nobel Prize. Rhône Poulenc offers us a box in plastic containing a magic paste with the inscription "la matière molle: vous connaissez?" This paste has kept all its properties: rolled in small ball, it bounces at short times and it spreads like a liquid. Playing with this paste to make a thin film, I saw the formation of holes. If for a hundred years, physicists (Dupré, Rayleigh,...) have been passionate about understanding the bursting of soap films, G. Debrégeas is the first to have made films of viscous polymers without surfactants, and studied their bursting (Fig. 7) which gives new laws: the inertia is negligible and the viscoelastic properties of the film control its rupture which is as fast as the bursting of soap bubbles, because of the radial plug flows that gives rise to a very small viscous dissipation [START_REF] Debrégeas | Viscous Bursting of Suspended Films[END_REF][START_REF] Debrégeas | The life and death of 'Bare' viscous bubbles[END_REF] .

Hydrodynamic extrusion of membranes tubes

A hydrodynamic probe was developed by O. Rossier based on the extrusion of membrane tubes from giant vesicles [START_REF] Rossier | Giant Vesicles under Flows: Extrusion and Retraction of Tubes[END_REF] .A vesicle is attached to a microstick and submitted to a flow (Fig. 8A). A tube is extruded above a threshold force f 0 , at which we have coexistence between a spherical vesicle and a membrane tether. We modeled this transition in analogy to the unbinding of a polymer chain in a bad solvent. Nicolas Borghi applied the tube hydrodynamic extrusion to test the properties of cell membranes (Fig. 8B) and the adhesion of the plasma membrane to the cytoskeleton [START_REF] Borghi | Hydrodynamic extrusion of tubes from giant vesicles[END_REF][START_REF] Tabdanov | Role of E-Cadherin in Membrane-Cortex Interaction Probed by Nanotube Extrusion[END_REF] . We modeled the non-linear relation between the force and the velocity of extrusion by a thinning of the tube when the viscous stress increases [START_REF] Brochard-Wyart | Hydrodynamic narrowing of tubes extruded from cells[END_REF] . Manufacturing microstructured substrates to attach dozens of cells, it will be possible to establish statistical laws characterizing the extrusion of nanotubes of different cell lines, and to determine the effect of drugs or genetic mutations. The membrane nanotubes are used in vitro and in vivo to transport active molecules between vesicles or cells. We have shown that the transport is achieved by a Marangoni flow from the tense vesicle towards the soft vesicle [START_REF] Dommersnes | Marangoni transport in lipid nanotubes[END_REF] .

Antagonist wetting: "fried eggs droplet"

Rhone Poulenc was interested in the wetting of mixtures where the components have antagonist wetting properties. They came to me with a bottle filled with sticky complex liquids. I simplified the system, using a polymer in solution in a good solvent: the monomer. With R. Fondecave [START_REF] Fondecave | Wetting laws for polymer solutions[END_REF][START_REF] Fondecave | Polymers as Dewetting Agents[END_REF][START_REF] Fondecave | Application of statistical mechanics to the wetting of complex liquids[END_REF] , we explored the laws of wetting of polymer solutions on model surfaces (salinized wafer) in antagonistic situation: the solvent totally wets the surface, but the molten polymer wets it only partially (finite contact angle). By diluting the polymer, it was thought naively to encounter a wetting transition. In fact, we observed a transition from a new type of "leak out" where the wetting solvent escapes from the drop and wets the substrate by forming an aureole around the drop, looking like a fried egg. Below this composition, the contact angle remains finite. A phase separation is coupled to the wetting.

Floating contact lines (X.Noblin) [START_REF] Noblin | Triplon Modes of Puddles[END_REF][START_REF] Noblin | Cascade of Shocks in Inertial Liquid-Liquid Dewetting[END_REF] . We have explored the dewetting of liquid films on a liquid substrate and observed a cascade of shocks waves(capillary waves in the front, gravity waves in the back) and the fluctuations of the contact lines, named Triplons, studied in parallel at ENS using superfluid helium [START_REF] Brochard-Wyart | Triple line dynamics in superfluid helium[END_REF] .

DNA under strong flow: trumpet and stem-flower In labelling DNA with chromophores, Steve Chu (Standford) was able to observe single macromolecule and manipulate it with optical or magnetic tweezers: he visualized the reptation of one chain in a polymer solution. In parallel, I described the deformation regimes of a DNA molecule attached at one end and subjected to a uniform flow (velocity V) [START_REF] Brochard-Wyart | Deformations of One Tethered Chain in Strong Flows[END_REF] . I predicted for the configuration of the DNA a shape of a "trumpet" of size L (Fig. 9a) obeying to simple scaling laws and I send my article to Steve Chu. He told me "Thanks to you, we did the experiments but your result is wrong!". I realized that the regime of the trumpet is valid only for a narrow range of velocities, and above a threshold velocity, we predicted a "stem and flowers" regime (Figure 9b) [START_REF] Brochard-Wyart | Polymer Chains Under Strong Flows: Stems and Flowers[END_REF] . The total length L do not obey simple scaling laws, but the length of the flowers does. With A. Buguin, we became interested in the unfolding and relaxation of the globular DNA in "bad" solvent. This work could have an impact on the denaturation of proteins. It also led to a new theoretical model of the collapse of a polymer chain suddenly placed in bad solvent. We then described the relaxation of a chain initially stretched, which reforms a coil from the free end, in collaboration with P. Cluzel and J.L. Viovy, who watched the relaxation of highly stretched DNA molecules. This model can also describe the helix-coil transition of synthetic polypeptides. With Fyl Pincus, we studied the transient regimes: the unfolding of the chain when, at time t = 0, a force or a flow is applied. I studied the eigen modes of flexible chains stretched by a flow, in the shape of trumpets (thesis Y. Marciano) [START_REF] Marciano | Normal modes of stretched polymer chains[END_REF] .

Adhesion of vesicles

The non-specific adhesion of giant and fluctuating vesicles to substrates was first studied by fluorescence microscopy by Olivier Sandre, who saw that strong adhesion can lead to the bursting of the vesicle. To slow down the dynamics, I told to Olivier to use a viscous solvent and a great discovery occurred! He saw the opening and the closure of a succession of transient pores. It was in June. Before the summer vacation, it was published in PNAS [START_REF] Sandre | Dynamics of transient pores in stretched vesicles[END_REF] .

Opening of transient pores (Fig. 10): When a floppy vesicle (Fig. 10A) is stretched by an external action (Fig. 10B), pores open to relax the tension of the membrane. If the vesicle is immersed in a very viscous liquid, macroscopic pores open and close slowly because the leakage of the internal liquid is slowed down. In water, the leak is so fast that the pores close again before reaching an observable size. We performed a complete theoretical description of the dynamics of these transient pores.

Line tensiometer: N. Borghi and E. Karatekin [START_REF] Karatekin | Cascades of Transient Pores in Giant Vesicles: Line Tension and Transport[END_REF] studied the role of surfactants on the mechanical properties of the membrane, and the line energy associated with holes in the membrane. Some surfactants are able to drop by a factor of 100! this line energy and make the membrane permeable. We tested several families of surfactants based on their hydrophilic / hydrophobic balance, and their affinity for insertion into the membrane.

These large membrane permeabilities, induced by tension and favored by certain surfactants, may have applications in the transfer of drugs or to understand certain biological processes (endocytosis, exocytosis) [START_REF] Berthaud | Spreading of porous vesicles subjected to osmotic shocks: the role of aquaporins[END_REF] . They are now a model for the pores observed in the nuclear membrane, when cells migrate in small capillaries or through the extracellular matrix. They can also be harmful to the life of our cells. Bursting induced by curling: Later, we extended these studies to the bursting of asymmetric light sensitive polymersomes [START_REF] Mabrouk | Bursting of sensitive polymersomes induced by curling[END_REF] observed by E. Mabrouk and M.Li (Fig. 10D), which represents a new phenomenon because the light induces an increase of surface area of one of the two leaflets, which should lead to a decrease in membrane tension , and not an increase causing usually the bursting. I proposed a mechanism inspired by a very simple and beautiful experiment shown to me by E. Reyssat when visiting Mahadevan at Harvard : the curling of tracing paper gently deposited at the surface of water, caused by the swelling of one side of the paper. We suggested that the growth of a pore in polymersome membranes is driven by the same curling instability. The opening of the pore relaxes the curvature energy stored in the membrane as shown by the membrane curling of the rim (Fig. 10D).

Living matter 2008-2018

Over the past ten years, our Physico Chimie Curie laboratory has been moving towards cellular organization under the impetus of the new Department of Developmental Biology. Our culture in soft matter, critical phenomena, liquid crystals and self-organized liquids, wetting, brings new points of view to biologists. Thus, in the early stages of Drosophila development, a cellular organization close to a two dimensional foam, exhibits a symmetry breaking with an isotropic → anisotropic organization that is similar to that of crystals liquids. A bold idea of J. Prost was the analogy between cancer and 1st order phase transition, that is, discontinuous. A tumor would be indicative of growth instability, and could only grow beyond a critical radius of nucleation, an idea that we tried to test and which was the starting point for my work on cell aggregates these last ten years, and that has jostled the field anyway.

We have introduced and worked on the broad field of entangled active matter, a novel class of non-equilibrium materials composed of many interacting units that individually consume energy and collectively generate motions or mechanical stresses. Unlike swarms of fish and flocks of birds, both ants and cells can support static loads. This is because both cells and ants are also entangled, so that the individual units (cell, ant) are bound by transient links. With David Hu, we have explored and establish analogies between aggregates of both ants and cells [START_REF] Hu | Entangled active matter: From cells to ants[END_REF] . For experimental purposes, assemblies of cells are among the simplest examples of active matter made of live entities. Increasingly, biologists, pharmacologists, and toxicologists are using cell aggregates as substitutes to animal studies. They are reliable and cost-effective systems for drug screening, bridging the gap between 2D cell-based assays and animal studies. Our team, in collaboration with S. Dufour, has been working on applying soft matter physics [START_REF] Gonzalez-Rodriguez | Soft Matter Models of Developing Tissues and Tumors[END_REF] , to study the biophysics of model tissues (tissue rheology, aspiration, spreading, mechanosensitivity, wetting and dewetting, adhesion and fracture). Our practice of phase transitions in liquid crystals and instabilities of wetting films has been useful for understanding the spreading of cell aggregates and wetting transitions by playing on adhesion with substrate and intercellular adhesion. The results obtained from such analogies have suggested important implications to both tissue development and cancer. In September 2013, I was appointed Professor Emeritus, my group Surface Douce was dissolved and I joined the Theoretical group of Jean-François Joanny Since 2014, I collaborate with Françoise Winnik's team in Japan on inert matter-living matter hybrid systems with two main questions: 1) Nanoparticles can stick the cells together and substitute for cell adhesion molecules and 2) Cellular noise can give movement to the inert macroparticles "Active Granular Matter " I remain concrete theoretician, working closely with experiments, and in collaboration with biologists who present us with complex situations, where we have to unmask a simple idea and develop it.

Cellular dewetting Macroapertures in epithelial cells (Collaboration E. Lemichez Nice)

Pathogenic bacteria migrate from blood and lymphatic vessels to host tissues by opening transient macro apertures, MAs, in endothelium cells. To accomplish this, staphylococcus aureus infect cells with EDIN, which produces a sudden disruption of the contractile cytoskeleton network. Cell membrane tension is no longer resisted by contractile acto-myosin fibers, leading to the opening of MAs (Fig. 11B). The opening is opposed by the line tension at the rim of the hole, which is induced by specific lipid sorting in the curved membrane. This induced line tension limits MA maximal size and eventually closes the hole, limiting endothelium permeability and preventing cell death. We modeled the opening and closure of MAs by analogy with the dewetting of a liquid film [START_REF] Redon | Dynamics of dewetting[END_REF] (Fig. 11A) and transient pores in vesicles [START_REF] Sandre | Dynamics of transient pores in stretched vesicles[END_REF][START_REF] Karatekin | Cascades of Transient Pores in Giant Vesicles: Line Tension and Transport[END_REF] . We calculate the minimal radius for hole nucleation, as well as the maximal MA size as a function of the initial membrane tension, assuming a constant line tension T. We show that the radius of a transient MA obeys simple scaling laws if T is small, leading to large MAs.

Cellular monolayer dewetting

We have shown that cohesive cellular monolayers deposited on non-adhesive substrates are metastable and "dewet" by nucleation and growth of dry patches [START_REF] Douezan | Dewetting of cellular monolayers[END_REF] . The dewetting can be induced either chemically by a non-adhesive surface treatment or physically by a decrease in the substrate rigidity. We interpret the dynamics of growth of holes in the monolayer by an analogy with the dewetting of viscous films. This analogy can be used to estimate parameters characterizing the mechanical properties of cellular sheets.

Spreading and aspiration of living drops

Since 2010, we have started a new research project on the characterization of the mechanical properties of biological tissues, the role of cell-cell adhesion on the tissue mechanics and eventually on the conditions for the growth of cancer tumors and formation of metastasis. Our studies are performed on spherical cellular aggregates, which are useful systems to study the mechanical properties of tissues since the adhesion energy between the cells can be controlled. We have used murine sarcoma (S180) cell lines transfected to express various levels of E-Cadherin molecules at the surface of the cells (coll. S. Dufour UMR 144), thereby controlling the intercellular adhesion energy and CT26 murine carcinoma cells in collaboration with D. Vignevic interested in the role of fibroblast in cancer extravasation.

Wetting transitions of cellular aggregates

We have studied the spreading of spheroidal aggregates of cells (S180), expressing a tunable level of cadherins (adhesion energy W cc ), on glass substrates decorated with fibronectin PEG mixtures (adhesion energy W cs ) [START_REF] Douezan | Spreading dynamics and wetting transition of cellular aggregates[END_REF] . As shown on the phase diagram (Fig. 11C), we observed both partial and complete wetting depending upon the sign of the spreading parameter S= W cs -W cc .We monitored the contact area by optical interferometry and the profile by side-view microscopy. At short times, the aggregate flattens and the contact area increases as t 2/3 . We interpret these results by modeling the aggregate as a viscoelastic droplet. At long times, we observe a precursor film with two possible states: in strongly cohesive aggregates this film is in liquid state, while in weakly cohesive aggregates the constitutive cells escape from the aggregate forming a 2D gas. The progression of a non-invasive tumor into a metastatic malignant carcinoma, known as the epithelial-mesenchymal transition, can be interpreted as a wetting (liquid-gas) transition. On soft gels decorated with fibronectin and strongly cohesive aggregates, we have observed a wetting transition induced by the substrate rigidity [START_REF] Douezan | Wetting transitions of cellular aggregates induced by substrate rigidity[END_REF] . We model the spreading dynamics of the monolayer expanding from the aggregate by balancing driving forces at the film periphery and viscous forces associated to the penetration of the cells from the (3D) aggregate into the (2D) film. By confocal microscopy, we observe this new mechanism named permeation by tracking single cell [START_REF] Beaune | How cells flow in the spreading of cellular aggregates[END_REF] . Using Particle Imaging Velocimetry, we characterize the flow field versus substrate rigidity. If cells spread like a viscous liquid on stiff substrate, the flows become irregular, with formation of holes as the rigidity decreases. This work will shed light on the dynamics of tissue spreading occurring during cancer progression and embryonic development. We have developed a new method based on micropipette aspiration technique to measure the viscoelasticity and the surface tension [START_REF] Guevorkian | Aspiration of biological viscoelastic drops[END_REF][START_REF] Douezan | Curling instability induced by swelling[END_REF][START_REF] Guevorkian | Mechanosensitive shivering of model tissues under controlled aspiration[END_REF] of cellular aggregates. As shown in figure 12, the aspiration curve has two distinct regimes; a rapid deformation followed by a slow flow. This creep behavior is a signature of viscoelastic materials. At short time scales the material deforms like an elastic rubber, whereas at long time scales it flows like a liquid. Surface tension, viscosity, and elastic modulus of these aggregates have been deduced by analogy between cellular aggregates and viscoelastic liquids. We found that the surface tension of the aggregate is stress dependent, suggesting that upon the application of an external stress, tissue cohesion is reinforced.

Aspiration of cellular aggregates

We have observed that the aggregates behave like "living" viscoelastic liquids (Guevorkian K, 2010). However, unlike an inert fluid, we observe aggregate reinforcement with pressure, which for a narrow range of pressures results in pulsed contractions or "shivering" (Guevorkian K, Gonzales-Rodriguez D, 2011). We interpret this reinforcement as a mechanosensitive active response of the acto-myosin cortex.

Spontaneous Migration: "Giant Keratocytes"

Figure 13) Spontaneous migration: Cell aggregates on soft gels move spontaneously. They crawl with a stick-slip motion and adopt the fan shape of giant keratocytes or a bipedal motion: penguins.

Single cell migration is studied extensively, but processes such as embryonic development or tumor metastasis often require the collective motion of a group of cells. We have observed a spontaneous collective migration of cellular aggregates on soft gels coated with fibronectin.

During the spreading, the cell monolayer expanding outwards becomes unstable and dewets [START_REF] Douezan | Wetting transitions of cellular aggregates induced by substrate rigidity[END_REF] . It leads to a symmetry breaking of cell polarities causing the entire aggregate to move. Decreasing the substrate rigidity, we observe stick-slip motions and different shapes (Fig: 13): "Giant Keratocytes", where the lamellipodium is a multicellular monolayer expending in the front and retracting in the back and "Penguins" characterized by bipedal locomotion and "Running Spheroids". We characterize the flow field and the stresses that drive the migration.

Our study provides a powerful in vitro model of the mechanisms of multicellular cell migration. These results underscore the biological predisposition for living material to migrate, independent of length-scale.

Cellular aggregatenanoparticles hybrid systems.

In January 2015, I got with Françoise Winnik a fellowship from WPI International Center for Material Nanoarchitectonics (MANA-NIMS) for "Developing new theories based on soft matter physics to describe and understand the mechanical properties of cell as they interact with nanoparticules". Grégory Beaune, my postdoc in Curie accepted to move to Japan.

We have studied how cells and particles play together. We name this project "eatingdancing": nanoparticles and microparticles are eaten by the cells or adsorbed on the cell's membrane. On the other hand, macroparticles, which are too big to be digested, are put into motion by the cells, they dance: this is our project for the following year.

Figure 14) nanoparticles sticking cells together and multicellular uptake of microparticles.

I) Nano stickers (size 20 nm):

We show that nanoparticles can be used as a glue "Nano stickers" to enable the formation of self-assembled aggregates by promoting cell-cell interactions [START_REF] Brunel | Nanostickers for cells: a model study using cell-nanoparticle hybrid aggregates[END_REF] (Fig. 14). We find that carboxylated polystyrene NPs are more efficient than the silica NPs of the same size, which were reported to induce fast wound healing and to glue soft tissue by L. Leibler. Nano stickers by increasing the cohesion of tissues and tumors may have important applications for cellular therapy and cancer treatment.

II) Microparticles (size 1micron): gluttonous cells: We study the spreading of cell aggregates deposited on adhesive substrates decorated with microparticles. A cell monolayer expands around the aggregate. The cells on the periphery uptake the microparticles by phagocytosis, clearing the substrate and forming an aureole of cells full of particles (Fig. 14). We study the dynamics of spreading, the width of the aureole, and the level cell internalization. It leads to an easy, fast, and inexpensive cellparticle internalization measurement [START_REF] Beaune | How gluttonous cell aggregates clear substrates coated with microparticles[END_REF] . We want to study the role of the rigidity on phagocytosis by using microfluidic fabrication of soft jelly beads.

Project: Hybrid dead-living matter

After Japan, I continue to collaborate with G. Beaune who joined the Active Matter Group of J. Timonen in Finland. Our aim is to study granular active matter by using hybrid aggregates of microparticles and cells. The hybrid aggregates will be prepared by a flocculation of a solution of micrometric passive beads (soft or rigid) and active cells. For micrometric particles, we do not expect Brownian motion: aggregates of beads are frozen. On the other hand, the aggregates of cells are liquids because of the activity of the cells produces a large non-thermal noise: they form spheroids to minimize their surface energy. Our project is to investigate the conformation and the physical state of cell-particles hybrid aggregates versus the particle volume fraction for which we predict a jamming transition and a phase separation. It's a fundamental question, with applications for all types of mixtures of active matter, as for example cell's nucleus or cell mixtures 1) The hybrid aggregate is a solid or a liquid? By the pipette aspiration technique, we shall measure the elastic modulus and the viscosity of the aggregate. We expect a liquid-solid transition as the volume fraction of the cells decreases. We will compare our data to simulations performed by David Gonzalez Rodriguez. Moreover, we expect to observe active Brownian motion of the particles in the dilute regime, which will allow to measure an effective temperature of the cells.

2) Do cells and beads form an aggregate of uniform composition? As in the case of the genome of eukaryotes that is partitioned into domains of functionally distinct chromatin states, euchromatin containing active genes and heterochromatin containing silent genes, we anticipate that hybrid aggregates will contain domains enriched/depleted in particles. Phase separation is predicted also by simulations and analytically in a mixture of particles with different level of activity, where one phase is enriched in active particles (cells) and the other in passive particles. This hypothesis will be tested by imaging with phase contrast microscopy or confocal fluorescence microscopy (for fluorescently labeled particles or/and cells) the distribution of passive particles within hybrid aggregates versus the composition of the suspension used to prepare the aggregates. The 'hallmark' of your scientific output is very strong: most of your articles are those of a theoretician (you) who are joined by excellent experimenters, in small groups that you form, inspire and animate. You're not the only one working in this way but you do it in an exemplary way. You, dear Françoise, gave birth to a lot of new knowledge, but also to many beings, the hundreds of your university students, the dozens of researchers with whom you worked who all love you, and also of course to six wonderful children whom I am glad they are all present today at your side.

Figure 1 )

 1 Figure 1) FBW and PGG à Sanary en 1991.

Figure 2 )

 2 Figure 2) Liliane and Pierre Wyart at the EDF dam in La Girotte View of Mont Blanc from the early childhood of FBW.

  are an affine deformation under convergent flows at the entry of the pore, and a suction of the polymer by the flow of solvent inside the pore. They both give the same threshold flow for the passage of the polymer. The experiments were carried out in the following years by a collaborative effort of researchers from the French Institute of Petroleum (G. Chauveteau) and the College of France (F. Rondelez and Léger).

Figure 4 )

 4 Figure 4) Forced passage of a flexible polymer chain under strong flows: a) affine deformation b) suction c) Free Energy of the chain versus the number of monomers sucked in the capillary

Figure 5 )

 5 Figure 5) Instability of liquid films on fibers: An enlarged view of beaded thread (D.Quéré)

Figure 6 )

 6 Figure 6) A: Aquaplaning: sliding of an adhesive rubber-glass contact immersed in a liquid versus the sliding velocity U: dry, semi lubricated and lubricated regimes (Pascal and Arnaud Martin). B: Cerenkov dewetting a) Mach angle and b) profile of the rim collecting the liquid. C: from left to right: « suspended» contact of a soft bead on a microstructured substrate (E. Echalier), pillars; Instability of an antifrost polymer film exposed to a moist atmosphere (H. Gerardin). D: Detachment of a soft bead adhering on smooth glass (J. Clain).

Figure 7 )

 7 Figure 7) Bursting of suspended polymer films and viscous bubbles (G. Debrégeas)

Figure 8 )

 8 Figure 8) Extrusion of membrane tube from vesicles (A) and red blood cells (B).

Figure 9 )

 9 Figure 9) Deformation of a tethered chain under uniform flow: a) trumpet b) stem and flower.

Figure 10 )

 10 Figure 10) A: Floppy vesicle. B: Tense vesicle. C: Opening and closure of transient pore (O. Sandre). And D: Light sensitive polymersome bursting under UV illumination.

Figure 11 )

 11 Figure 11) A) Snapshots of the opening of a dry patch in a liquid film (C. Redon), B) Opening of MA in endothelial cells, C) wetting phase diagram of cellular aggregates 58 .

Figure 12 )

 12 Figure 12) Aspiration of a cellular aggregate. (A)-(C) snapshots of aspiration of an aggregate inside a pipette, scale bar 50 µm, ∆P = 1200 Pa. From the length of the tongue L(t), one can measure the surface tension, the elastic modulus and the viscosity of the tissue 63 .

Figure 15 )

 15 Figure 15) Books coauthored by FBW Finally, I will quote the conclusion of Yves Quéré's speech, which expresses my style of work and the meaning of my life.

Figure 16 )

 16 Figure16) Poster for a meeting on Oct. 1-3, 2018 to celebrate the scientific career of FBW and her contribution to soft and biological matter.

  receiving at home young refugees and isolated elderly. She died in the summer of year 2000, with your father following her less than two months after, crushed with grief. The young Françoise was a brilliant student who was born at the end of the Second World War and had the opportunity, that her mother did not have, to complete her education. After high school, Françoise went to the Ecole Normale Supérieure (ENS) Cachan in 1964 and pass her aggregation of physics in 1968, an unusual year to prepare such a hard exam! In the same year, she started her Ph.D. in the Solid State Physics laboratory in Orsay."II) Orsay 1968 -1977Two years after my admission in ENS Cachan, I enrolled in 1966 the master of Solid Physics where classes were instructed by notorious professors such as J. Friedel, A. Guinier, P. Nozières and P.G. de Gennes. Together, these great minds conveyed a novel, personal and exciting vision of physics. After passing the Aggregation of Physics, I joined the Laboratory of Solid Physics (Orsay) to prepare a Ph.D. under the supervision of P.G. de Gennes, referred

to as "PGG". After the events of May 1968, one could not enroll in a Ph.D. without a permanent position. Consequently, we were very few Ph.D. students! On my side, I got in the fall of 1968 a permanent assistant professor position at University Paris 11 so I started to teach and to do research while soon taking care of my first child, Pascale, born in October 68 and soon to be followed by my first son, Virgile.

  [START_REF] Brochard-Wyart | Une bulle dégonflée: le Globule Rouge[END_REF] , but I do not like the experience, so I decided to stay in academic research.

	developed for critical phenomena replaced mean field approaches. The scaling laws were
	visualized by PGG, who introduced the picture of "blobs", a statistical object to characterize
	the monomer-monomer correlations. My previous training in hydrodynamics and phase
	transitions became extremely useful.
	III) College de France 1977-1985
	Pierre Gilles de Gennes had been nominated in 1971 as Professor for the Chair of Physics of
	Condensed Matter in College de France.
	In 1977, I joined his laboratory where I found some of my "old friends" from Orsay,
	Madeleine Veyssié, Liliane Léger, Françis Rondelez. The same year my daughter Claire was
	born, and in 1978, my son Matthieu. I took part in the fast evolution of polymer physics,
	initiated by de Gennes with Jannink and des Cloiseaux. The famous theorem "n = 0" of PGG
	in 1972 1 was a revolution in polymer physics, where suddenly all the theoretical technics

This theorem is presented by PGG in his activity report "On the theoretical level, it has been possible to establish a rigorous relation between the statistics of the chains and the transitions of phase. This relation is quite abstract: it is to study a magnetic system whose magnetization possesses independent elements and to pass to the

Acknowledgements: I thank all my colleagues and collaborators in France and abroad for their friendship, my students and my post-docs who achieved all this beautiful experiment, Gregory Beaune and F.Winnik for their wonderful collaboration at the Mana Institute in Tsukuba, Basil Gurchenkov for his drawing "Be simple", Claire Wyart, Madeleine Veyssié , Thomas Risler, Uzay Girit and Nada Khalifat for their precious advice in the production of this text, and last but not least Gisèle Vergand who took care of my children when I was working and travelling like a second mother.

Publications