Duong Nguyen

Oliver S Kirsebom

Fábio Frazão

Ronan Fablet

Stan Matwin

RECURRENT NEURAL NETWORKS WITH STOCHASTIC LAYERS FOR ACOUSTIC NOVELTY DETECTION

Keywords: acoustic modeling, novelty detection, variational recurrent neural network, stochastic recurrent neural network

In this paper, we adapt Recurrent Neural Networks with Stochastic Layers, which are the state-of-the-art for generating text, music and speech, to the problem of acoustic novelty detection. By integrating uncertainty into the hidden states, this type of network is able to learn the distribution of complex sequences. Because the learned distribution can be calculated explicitly in terms of probability, we can evaluate how likely an observation is then detect low-probability events as novel. The model is robust, highly unsupervised, end-toend and requires minimum preprocessing, feature engineering or hyperparameter tuning. An experiment on a benchmark dataset shows that our model outperforms the state-of-the-art acoustic novelty detectors.

INTRODUCTION

Audio processing in general, and acoustic novelty detection in particular has attracted significant attention recently. A number of studies have used acoustic data to detect abnormal events, mostly for surveillance purposes, such as human fall detection [START_REF] Zhuang | Acoustic fall detection using Gaussian mixture models and GMM supervectors[END_REF], [START_REF] Salman Khan | An unsupervised acoustic fall detection system using source separation for sound interference suppression[END_REF], abnormal jet engine vibration detection [START_REF] Clifton | Novelty detection in jet engine vibration spectra[END_REF], hazardous events detection [START_REF] Ntalampiras | Probabilistic Novelty Detection for Acoustic Surveillance Under Real-World Conditions[END_REF].

The main challenge of novelty detection is we do not have a large amount of novel events to learn their characteristics, while the normal set is usually very big and contains a large amount of uncertainty. The common approach is to use unsupervised methods to learn the normality model, then consider events that do not fit this model as abnormal (novel). Most of these systems use Gaussian Mixture Model (GMM) or Hidden Markov Model (HMM) [START_REF] Kumar | A Multimodal Audio Visible and Infrared Surveillance System (MAVISS)[END_REF], [START_REF] Ntalampiras | Probabilistic Novelty Detection for Acoustic Surveillance Under Real-World Conditions[END_REF], [START_REF] Atrey | Audio Based Event Detection for Multimedia Surveillance[END_REF]. Bayesian Networks have also been explored [START_REF] Zajdel | CAS-SANDRA: audio-video sensor fusion for aggression detection[END_REF], [START_REF] Giannakopoulos | Audio-Visual Fusion for Detecting Violent Scenes in Videos[END_REF]. Recently, advances in deep learning [START_REF] Lecun | Deep learning[END_REF], especially in Recurrent Neural Networks (RNNs) and their extensions (Long Short-Term Memory -LSTM [START_REF] Sak | Long Short-Term Memory Recurrent Neural Network Architectures for Large Scale Acoustic Modeling[END_REF], Gated Recurrent Unit -GRU [START_REF] Chung | Gated Feedback Recurrent Neural Networks[END_REF]) have opened new venues for acoustic modeling. In [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF], the authors employed LSTMs to create an AutoEncoder (AE) to model normal sounds and detect abnormal sounds using the reconstruction errors. This idea has been extended in [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF] by applying an adversarial training protocol.

However, acoustic signals are stochastic. RNN-based networks, whose hidden states are deterministic, can hardly capture all the variations in the data. Recent efforts to improve the modeling capacity of RNNs by including stochastic factors in their hidden states have shown impressive results, especially for generating text, music and speech [START_REF] Bayer | Learning Stochastic Recurrent Networks[END_REF], [START_REF] Boulanger-Lewandowski | Modeling Temporal Dependencies in High-Dimensional Sequences: Application to Polyphonic Music Generation and Transcription[END_REF], [START_REF] Chung | A Recurrent Latent Variable Model for Sequential Data[END_REF], [START_REF] Fraccaro | Sequential Neural Models with Stochastic Layers[END_REF].

In this paper, we adapt these models to create an unsupervised acoustic novelty detector. Our approach performs an end-to-end learning of a probabilistic representation of acoustic signals. Given this representation, we can evaluate how likely an observation and state the detection of novel events as the detection of observations with a low probability. We argue that this model is robust, highly unsupervised, end-to-end and requires minimum preprocessing, feature engineering or hyperparameter tuning. Our empirical evaluation on a dataset for novel event detection in audio data shows that the proposed model outperforms the state-of-the-art.

The paper is organized as follows: in Section 2, we present the details of the proposed approach; we compare the model with state-of-the-art methods to point out its advantages in Section 3; the experiment and results are shown in Section 4; finally in Section 5 we give conclusions and some perspectives for future work.

THE PROPOSED APPROACH

Recurrent Neural Networks with Stochastic Layers (RNNSLs)

For time series modeling, the two most common approaches are State Space Models (SSMs) and Recurrent Neural Networks (RNNs). SSMs such as Kalman filters [START_REF] Brown | Introduction to Random Signals and Applied Kalman Filtering[END_REF] and particle filters [START_REF] Doucet | A Tutorial on Particle Filtering and Smoothing: Fifteen years later[END_REF] have been explored for a long time and are the state-of-the-art model-driven schemes thanks to their ability to model stochasticity. However, these models are limited by their mathematical assumptions (for example, Kalman filters assume the data generating process is Gaussian). RNNs, on the other hand, have attracted a lot of attentions recently by their capacity to represent long-term dependencies in time series [START_REF] Lecun | Deep learning[END_REF]. The main drawback of RNNs is that their hidden states are deterministic, making them unable to capture all the stochastic components of the data. A number of efforts have been made to bring together the power of SSMs and RNNs [START_REF] Bayer | Learning Stochastic Recurrent Networks[END_REF], [START_REF] Chung | A Recurrent Latent Variable Model for Sequential Data[END_REF], [START_REF] Fraccaro | Sequential Neural Models with Stochastic Layers[END_REF], [START_REF] Krishnan | Deep Kalman Filters[END_REF]: Recurrent Neural Networks with Stochastic Layers (RNNSLs).

RNNSLs aim to learn the distribution p, which can be factored through time, over a sequence of T observed random variables {x t } ,t=1..T :

p(x 1:T ) = T t=1 p t (x t |x <t ), (1)
where x <t denotes x 1:t-1 . Following a SSM formulation, we assume that the data generation process of x 1:T relies on a sequence of T latent random variables {z t } ,t=1..T . At each time step t, the joint distribution p t (x t , z t |x <t z <t ) can be factored into:

p t (x t , z t |x <t z <t ) = p t (x t |x <t , z ≤t )p t (z t |x <t , z <t ), (2)
where z ≤t denotes z 1:t . In other words, each time step of the network is an autoencoder, conditionally to the historical information.

Depending on the stochastic nature of the considered data, the emission distribution p t (x t |x <t , z ≤t ) may be highly nonlinear. However, this nonlinearity usually leads to the intractability of the inference distribution p t (z t |x ≤t , z <t ). The most common solution to overcome this obstacle is the variational approach [START_REF] Boulanger-Lewandowski | Modeling Temporal Dependencies in High-Dimensional Sequences: Application to Polyphonic Music Generation and Transcription[END_REF], [START_REF] Chung | A Recurrent Latent Variable Model for Sequential Data[END_REF], [START_REF] Fraccaro | Sequential Neural Models with Stochastic Layers[END_REF], which introduces an approximation q t (z t |x ≤t , z <t ) of the posterior distribution p t (z t |x ≤t , z <t ) then estimates p t (x t |x <t ) by the Evidence Lower BOund (ELBO) L(x, p t , q t ):

log p t (x t |x <t ) ≥ L(x, p t , q t ) = E zt∼qt log p t (x t |x <t , z ≤t ) -KL q t (z t |x ≤t , z <t )||p t (z t |x <t , z <t ) (3)
where KL q t ||p t is the Kullback-Leibler divergence between two distributions q t and p t .

There are several types of RNNSLs, differing in the way that they model the structure of the latent space. The most common types are Variational Recurrent Neural Networks (VRNNs) [START_REF] Chung | A Recurrent Latent Variable Model for Sequential Data[END_REF], Stochastic Recurrent Neural Networks (SRNNs) [START_REF] Fraccaro | Sequential Neural Models with Stochastic Layers[END_REF] and Deep Kalman Filters (DKFs) [START_REF] Krishnan | Deep Kalman Filters[END_REF]. We experimented most of these types, however, in this paper, for simplicity purposes, we only report the VRNNs, introduced by Chung et al. [START_REF] Chung | A Recurrent Latent Variable Model for Sequential Data[END_REF]. In VRNNs, the historical information (x <t , z <t ) is encoded by the dynamics of the hidden states of their RNN (LSTM) h t = h(x t-1 , z t-1 , h t-1 ). More precisely, it involves the parameterization of the following distributions, namely the emission distribution p t (x t |x <t , z ≤t ) = p(x t |z t , h t ), the prior distribution p t (z t |x <t , z <t ) = p(z t |h t ) and the variational posterior distribution q t (z t |x ≤t , z <t ) = p(z t |x t , h t ) as neural networks. Here, we consider fully connected networks with Gaussian formulation of these three distributions. For more details of VRNNs, please refer to [START_REF] Chung | A Recurrent Latent Variable Model for Sequential Data[END_REF].

RNNSLs for Acoustic Novelty Detection

RNNSLs were initially designed for generating text, music, speech. They are currently the state-of-the-art in these domains [START_REF] Boulanger-Lewandowski | Modeling Temporal Dependencies in High-Dimensional Sequences: Application to Polyphonic Music Generation and Transcription[END_REF], [START_REF] Chung | A Recurrent Latent Variable Model for Sequential Data[END_REF], [START_REF] Fraccaro | Sequential Neural Models with Stochastic Layers[END_REF], [START_REF] Maddison | Filtering Variational Objectives[END_REF]. The interesting point of this type of models in comparison to other state-of-the-art methods like Wavenet [START_REF] Oord | WaveNet: A Generative Model for Raw Audio[END_REF] is that these models calculate the distribution p(x 1:T ) explicitly, so that after learning this distribution from the training set, we can evaluate the probability for each new sequence. The idea of using RNNSLs for novelty detection was first introduced in [START_REF] Nguyen | A Multi-task Deep Learning Architecture for Maritime Surveillance using AIS Data Streams[END_REF] for the detection of abnormal behaviors of vessels, we adapt this model to novelty detection in acoustic data.

Here, an acoustic signal is modeled as a time series {x t } ,t=1..T where x t can be a chunk of n samples of the waveform, or n frequency bins in a spectrogram at a given time t. A RNNSL first learns the distribution over x 1:T in the training set, which may or may not contain some abnormal sequences. Then, for any new acoustic signal, we can evaluate its log-probability. If this log-probability is smaller than a threshold, the sequence will be considered as abnormal (or novel), as illustrated in Fig. 1.

To choose the threshold, we create a validation set, which again may or may not contain some abnormal sequences and compute the mean µ valid and the standard deviation σ valid of the log-probability of the sequences in this set. The value of the threshold is then chosen as: θ = µ valid -α * σ valid . α is usually chosen as 3.

The training set and the validation set may contain some abnormal sequences. However, since RNNSLs are probabilistic models, they will eventually ignore these "outliers" (this conjecture is confirmed experimentally). This property helps to reduce data cleaning efforts. x t is the original signal at the given time step t, h t is the hidden state of the RNN (LTSM), z t is the latent stochastic state, x t is the reconstructed output of the AE. The solid arrows denote the calculation processes, while the dashed arrows show how the cost function is calculated. We use the same notation as [START_REF] Fraccaro | Sequential Neural Models with Stochastic Layers[END_REF], circles for stochastic factors, diamonds for deterministic factors.

RELATED WORK

A number of researches have explored deep neural networks to detect novelty in acoustic surveillance. We point out here the advantages of our model over those used in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF], which are currently the state-of-the-art methods.

Both [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF] used RNNs (LSTMs in particular) as an AutoEncoder (AE) which can reconstruct the original signal from a compressed representation (Compression AutoEncoders -CAEs) or from a corrupted version of it (Denoising AutoEncoders -DAEs). However, as discussed in [START_REF] Chung | A Recurrent Latent Variable Model for Sequential Data[END_REF], [START_REF] Fraccaro | Sequential Neural Models with Stochastic Layers[END_REF] and [START_REF] Krishnan | Deep Kalman Filters[END_REF], the fact that the hidden states of RNNs are deterministic reduces their capacity to capture all data variabilities, especially for data that contain high levels of randomness.

Moreover, the detection criterion used in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] is the Euclidean distance between the original input and the reconstructed output of the autoencoder. This criterion is very sensitive to noise. [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF] addressed this drawback by using an adversarial strategy, however, the ultimate idea is also to compare the original input and the reconstructed output from the autoencoder. By contrast, our method detects novel events by directly evaluating the probability of the received signal. Besides the improved detection criterion, the architecture of our model is also more robust to noise [START_REF] Nguyen | A Multi-task Deep Learning Architecture for Maritime Surveillance using AIS Data Streams[END_REF].

These differences are sketched in Fig. 2. The hidden space of our model has stochastic factors, which help to increase modeling capacity. The decision rule of our model is a function of the distribution learned by the network, making the model more robust to noise.

The selection of the thresholding value for novelty detection is another important difference compared to previous works. The approach in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] is not fully unsupervised, because it needs some information about the proportion of abnormal events in the data. Our method, in contrast, only uses the information from the training set and the validation set to chose the threshold, without any prior knowledge of the annotations, based on a statistically-sound criterion, i.e. the false alarm rate.

EXPERIMENT AND RESULT

Dataset

We tested our model1 on the same dataset used in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF], which is part of the PASCAL CHiME speech separation and recognition challenge dataset [START_REF] Barker | The PASCAL CHiME speech separation and recognition challenge[END_REF]. The original dataset contains 7 hours of in-home environment recordings with two children and two adults performing common activities, such as talking, eating, playing and watching television. The author of [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] took a part of those recordings and created a dataset for acoustic novelty detection (100 minutes for the training set and 70 minutes for the test set). In the new dataset, the sounds of the PASCAL CHiME are considered as background, the test set was generated by digitally adding abnormal sounds like alarms, falls, fractures (breakages of objects), screams. The details of the dataset were presented in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF].

Experimental Setup

In order to use the models in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF] as baselines, we set up our model to have the same evaluation metric that was used in those papers. However, instead of transforming the data to mel spectrograms like in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF], we worked directly with the waveform (end-to-end model). The dataset was recorded by a binaural microphone at a sample rate of 16kHz. We converted each audio to 1 channel and then split it into sequences of 160-dimensional frames, each frame corresponds to 0.01s, as in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF]. [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF] evaluated the detection at each frame instead of at the whole sequence, so we also applied the thresholding step to each log p(x t |x <t ), instead of log p(x 1:T ).

We tested different topologies of VRNN, with the latent size of 64, 80, 160 and 200. The models were trained using Adam optimizer [START_REF] Kingma | Adam: A Method for Stochastic Optimization[END_REF], with a learning rate of 3e -5.

Results

Different configurations gave different log-likelihoods on the dataset, however the final detection results were quite similar. We report here only one of the topologies, which gave the best result: VRNN with 160 latent units (the models with 80 hidden units also gave similar results). We compare the performance of our model with the result of GMM, HMM, those in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] (LSTM-based CAE, LSTM-based DAE) and in [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF] (Adversarial AE). The result is shown in Table 1 2, we also used the same technique as in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] to chose the optimal threshold value, denoted as VRNN* .

Our method not only outperformed the state-of-the-art methods, but also has the ability to work online, which is highly beneficial for real-time surveillance. Models that use bidirectional LSTM (BLSTM-CAEs, BLSTM-DAEs) can not reach online processing the because a look-ahead buffer is required. The online processing ability of Adversarial AEs depends on the structure that they use (LSTM or BLSTM).

When investigating the cases where the proposed model misdetected the novelty, we found that actually the model could detect all the novel events, however, the way the detection was evaluated reduced the accuracy. As in [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF], the detection was evaluated at each time step of 0.01s. Our model has a memory effect (the memory of its LSTM cells), so it tends to merge the abnormal events that are very close to each other, as shown in Fig. 3. In other cases, the model missed a part of the sound, especially for the tail of the fractures, as shown in Fig. 4. These sounds have a long tail which is gradually submerged in the background. These misdetections are not detrimental in real life applications, because we are more interested in whether or not there is a novel event than on how long the event is.

We also conducted a robustness test where we added Gaussian noise to the test set. The additive noise is unknown by the model. This is a common scenario in audio surveillance, when the background environment changes (e.g. because of winds) or when noise appears in the electronic system. Table 2 3 shows the performance of the proposed approach (with optimal threshold) on the corrupted test sets with different level of Signal to Noise Ratio (SNR). Thanks to the nature of VRNNs and the improved detection criterion, our model is robust to noise.

CONCLUSIONS AND PERSPECTIVES

We have presented a novel unsupervised end-to-end approach for acoustic novelty detection. This approach exploits RNNs with stochastic layers, which are the state-of-the-art frameworks for time series modeling. Given the learned probabilistic representations, novelty detection can be stated as a classic statistical test, which fully accounts for the stochasticity of the considered acoustic datasets. Reported experiments on a benchmarked dataset showed that the model outperforms the state-of-the-art detectors [START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF], [START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF].

The dataset used in this paper is quite simple, the novel events in it are quite easy to be detected. Future work could involve applying this model to more complex signals, e.g. underwater acoustic signals which depict even greater variabilities. The impact of the threshold is also being studied to obtain better threshold selection rule.

Fig. 1 .

 1 Fig. 1. Architecture of the proposed RNNSL-based novelty detector.

Fig. 2 .

 2 Fig.2. Architecture and decision rule of the proposed model (VRNN) in compared to previously proposed AE-based models. x t is the original signal at the given time step t, h t is the hidden state of the RNN (LTSM), z t is the latent stochastic state, x t is the reconstructed output of the AE. The solid arrows denote the calculation processes, while the dashed arrows show how the cost function is calculated. We use the same notation as[START_REF] Fraccaro | Sequential Neural Models with Stochastic Layers[END_REF], circles for stochastic factors, diamonds for deterministic factors.

Fig. 4 .

 4 Fig. 4. An example where the model missed a part of the novelty event. This figure shows the waveform of the sound of a fracture of a dish. The tail of the sound is very mall and gradually becomes submerged in the background.

Table 1 .

 1 2 . Besides choosing the threshold automatically as discussed in Section Detection result, in comparison with state-of-the-art methods.

	Method	Online Processing	Precision	Recall	F1 score
	GMM	Yes	99.1	87.8	89.4
	HMM	Yes	94.1	88.9	91.1
	LSTM-CAE	Yes	91.7	86.6	89.1
	BLSTM-CAE	No	93.6	89.2	91.3
	LSTM-DAE	Yes	94.2	90.6	92.4
	BLSTM-DAE	No	94.7	92.0	93.4
	Adversarial AE	?	?	?	93.3
	VRNN	Yes	95.4	91.8	93.6
	VRNN*	Yes	95.4	92.8	94.1

Table 2 .

 2 Robustness test.

	SNR	Precision	Recall	F1 score
	5dB	96.0	91.2	93.6
	10dB	96.1	91.9	94.0
	15dB	96.1	92.1	94.0

The code is available at https://github.com/dnguyengithub/AudioNovelty

The values in Table1are from[START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and[START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF],[START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF] did not show the precision and recall of their model

[START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] and[START_REF] Principi | Acoustic novelty detection with adversarial autoencoders[END_REF] did not provide sufficient detail to replicate their codes for this test. Fig.3. An example where the novelty events were merged. This figure shows the waveform of two alarms, each alarm consists of there "beeps", our model considered this "beep beep beep" as one event, while the annotation made by the authors of[START_REF] Marchi | A novel approach for automatic acoustic novelty detection using a denoising autoencoder with bidirectional LSTM neural networks[END_REF] separates these "beeps".

supported by the UBL Mobility Fund and the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would like to thank A3Lab for the dataset.


Nicolas Renard
email: nico-las.renard@onera.fr.

Sébastien Deck

Pierre-Élie Weiss

On the Wall-Modeled LES use of ZDES in compressible flows

The present study is devoted to the use of the Zonal Detached Eddy Simulation (mode 3) hybrid RANS/LES method as a Wall-Modeled Large Eddy Simulation technique in compressible flows. The first section is devoted to a supersonic turbulent boundary layer test case, showing that the performance of ZDES mode 3 is satisfying in supersonic conditions, validating this WMLES method previously analyzed at low Mach numbers. Then the interaction of this boundary layer with an oblique shock wave is considered, with a promising qualitative agreement with experimental data and enhanced predictions compared with RANS. Finally, a new strategy by Deck et al.

[1] for a low-noise resolved turbulence generation is presented, which is expected to be instrumental for the next use of ZDES mode 3 in aeroacoustic studies.

I. Introduction

The increasing demand for high-fidelity computational aerodynamics motivates more and more frequently unsteady turbulence modelling. Among the possible approaches, the hybrid RANS/LES paradigm is suited for the industrial needs since it combines a high degree of resolution in the zones of interest, which are treated in LES, with a cost-efficient RANS treatment of the other regions of the flow. The Zonal Detached Eddy Simulation technique (ZDES) [START_REF] Deck | Recent improvements in the Zonal Detached Eddy Simulation (ZDES) formulation[END_REF] developed since 2002 enables a flexible definition of the zones of interest and has been widely validated and used for aerospace engineering applications (see e.g. [START_REF] Deck | High-fidelity simulations of unsteady civil aircraft aerodynamics: stakes and perspectives. Application of Zonal Detached Eddy Simulation (ZDES)[END_REF]) as well as for academic research. The ZDES approach features two modes for massively separated flows and a third mode for the resolution of wall-bounded turbulence. Resolving turbulent fluctuations in attached boundary layers may be required to reach a specified level of description of the unsteady flow (e.g. for unsteady loading or aeroacoustic predictions), to enhance universality compared with RANS models (e.g. with respect to adverse pressure gradient effects) or to include upstream history effects associated with the boundary layer unsteady dynamics (e.g. mild flow separation, shock wave / boundary layer interaction). The cost of resolving the inner layer dynamics of attached boundary layers is however prohibitive for industrial applications, so that wall modelling is mandatory. In ZDES mode 3, a near-wall RANS zone is defined and performs as a wall model for the LES of the outer layer, as detailed in [START_REF] Deck | Zonal Detached Eddy Simulation of a spatially developing flat plate turbulent boundary layer[END_REF][START_REF] Deck | Zonal Detached Eddy Simulation (ZDES) of a spatially developing flat plate turbulent boundary layer over the Reynolds number range 3 150 ≤ Re θ ≤ 14 000[END_REF][START_REF] Renard | Improvements in Zonal Detached Eddy Simulation for Wall Modeled Large Eddy Simulation[END_REF]. This WMLES strategy relies on the hybrid RANS/LES framework, like other methods such as the WMLES branch of IDDES [START_REF] Shur | A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities[END_REF] or the LES/RANS blending by Choi, Edwards & Baurle [START_REF] Choi | Compressible Boundary-Layer Predictions at High Reynolds Number Using Hybrid LES/RANS Methods[END_REF].

More specifically, in ZDES mode 3 the near-wall RANS / outer LES interface is treated in a passive way by simply switching the length scale used in the Spalart-Allmaras turbulence model [START_REF] Spalart | A One-Equation Turbulence Model for Aerodynamic Flows[END_REF] from the wall distance d w in the RANS zone to the cell size estimate (based on the cell volume) in the LES zone (i.e. d = d w if d w < d interface w , min(d w , C DES ∆ vol ) otherwise). This simple treatment is robust and does not introduce empiricism or spurious noise, but the height of the RANS/LES interface must be carefully set, independent from the mesh resolution [START_REF] Deck | Zonal Detached Eddy Simulation of a spatially developing flat plate turbulent boundary layer[END_REF] (which means there is no meshing constraint, unlike other methods) and at an outer-scaled position, for instance d interface w /δ = 0.1 [START_REF] Deck | Zonal Detached Eddy Simulation (ZDES) of a spatially developing flat plate turbulent boundary layer over the Reynolds number range 3 150 ≤ Re θ ≤ 14 000[END_REF] (which requires preprocessing). Further refinement is proposed in [START_REF] Renard | Improvements in Zonal Detached Eddy Simulation for Wall Modeled Large Eddy Simulation[END_REF] but not crucial here.

The investigation and validation of ZDES mode 3 [START_REF] Deck | Zonal Detached Eddy Simulation of a spatially developing flat plate turbulent boundary layer[END_REF][START_REF] Deck | Zonal Detached Eddy Simulation (ZDES) of a spatially developing flat plate turbulent boundary layer over the Reynolds number range 3 150 ≤ Re θ ≤ 14 000[END_REF][START_REF] Renard | Improvements in Zonal Detached Eddy Simulation for Wall Modeled Large Eddy Simulation[END_REF][START_REF] Deck | Large scale contribution to mean wall shear stress in high Reynolds number flat plate boundary layers up to Re θ = 13 650[END_REF] and of the turbulent inflow conditions required for such WMLES [START_REF] Deck | A rapid and low noise switch from RANS to WMLES on curvilinear grids with compressible flow solvers[END_REF][START_REF] Laraufie | A dynamic forcing method for unsteady turbulent inflow conditions[END_REF][START_REF] Laraufie | Assessment of Reynolds stresses tensor reconstruction methods for synthetic turbulent inflow conditions. Application to hybrid RANS/LES methods[END_REF] has been mainly focused on subsonic boundary layers at high Reynolds numbers. Compressibility effects that might affect ZDES mode 3, for instance in applications at higher Mach numbers or in aeroacoustic studies, have been addressed in a limited number of studies so far. The present paper investigates compressibility effects in ZDES mode 3 in three steps. First, simulations of a supersonic turbulent boundary layer are presented. The interaction of the resolved turbulent boundary layer with a shock wave is then considered. Finally, the new strategy by Deck et al. [START_REF] Deck | A rapid and low noise switch from RANS to WMLES on curvilinear grids with compressible flow solvers[END_REF] to reduce the spurious acoustic footprint of the resolved turbulence generation needed for a WMLES is discussed.

II. WMLES of a supersonic turbulent boundary layer

The first test case is a flat-plate zero-pressure-gradient turbulent boundary layer at an outer Mach number M ∞ = 2. A wide range of Reynolds numbers is covered by the ZDES simulations, as illustrated by fig. 1. The mesh counts 7.7 • 10 6 grid points, much less than what would be needed for a wall-resolved LES (approximately 100 • 10 6 points). In a compressible context, the following definitions of the Reynolds numbers are adopted:

θ = ∫ ∞ 0 ρ u ρ ∞ U ∞ 1 - u U ∞ dy (1)
Re θ = θU ∞ ρ ∞ µ ∞ ( 2
)
Re δ 2 = θU ∞ ρ ∞ µ w = F θ Re θ = µ ∞ µ w Re θ (3)
τ w = µ ∂ y u (y = 0) ( 4
)
τ w = ρ w u 2 τ ( 5
)
Re τ = δu τ ρ w µ w (6) √ Re τ , i.e. close to the geometric center of the logarithmic layer. The results are post-processed in order to enable a comparison with incompressible turbulent boundary layers following the findings of Hopkins & Inouye [START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF]. Regarding the mean skin friction coefficient, it is first defined by:

y + = y u τ ρ w µ w (7)
C f = τ w 1/2ρ ∞ U 2 ∞ ( 8
)
Then the mean skin friction C f is reduced to the incompressible case C f i for comparison using the Van Driest II approach [START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF]:

C f i = F c C f (9)
F c = T w T ∞ -1 arcsin 1 - T ∞ T w 1/2 -2 (10)
The relation C f i (Re δ 2 ) is almost independent of the Mach number ( [START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF]). In addition, two boundary layers at different Mach numbers have relatively close Re τ if they have the same Re δ 2 ([14]). For this reason, C f i and the turbulent profiles are compared with the incompressible case at the same Re δ 2 (or at the same Re τ for quantities other than C f i ).

The prediction of mean skin friction illustrated in fig. 2 and compared with incompressible data after rescaling is quite satisfying. After the relaxation following the inflow condition, the ZDES mode 3 WMLES technique provides an accuracy better than 5% with respect to the Coles-Ferhnolz correlation calibrated by [START_REF] Nagib | Approach to an asymptotic state for zero pressure gradient turbulent boundary layers[END_REF] for both RANS/LES interface height settings. This result is consistent with the previously reported behavior of ZDES mode 3 at low Mach numbers, indicating that the transposition of the method into supersonic conditions seems successful. This is further confirmed by the turbulent profiles compared with experimental / DNS data at close values of Re δ 2 , with the mean velocity profile in fig. 3 and the Reynolds stresses in fig. 4 and5. Here the mean velocity has been Van Driest transformed according to:

u * = ∫ u 0 ρ ρ w du (11)
as well as the Reynolds stresses:

R i j = u i u j (12)
R * i j = ρ ρ w R i j (13)
Note that δ is still defined by u (y = δ) = 0.99 U ∞ . The results confirm that the behaviour of the ZDES mode 3 technique at M ∞ = 2 is very similar to its low Mach-number performance. It should be emphasized that most of the outer layer dynamics is resolved rather than modelled, as shown in fig. 4, making it a true WMLES approach. Similarly to the low Mach number results, the mean velocity profile somewhat deviates from the logarithmic law near the RANS/LES interface. The streamwise turbulent intensity is slightly underpredicted in the middle of the outer layer while it is overpredicted just below the RANS/LES interface when the latter is set at d interface + w = 3.9 √ Re τ . These results confirm that the outer-scaled interface setting (d interface w /δ = 0.125 here) is a good compromise for ZDES mode 3.

III. WMLES of a shock-wave boundary layer interaction

The second test case is designed to investigate the capacity of the ZDES mode 3 technique to simulate the interaction of a supersonic M ∞ = 2 turbulent boundary layer with an oblique shock wave. The previous case of a flat-plate zero pressure gradient turbulent boundary layer is retained but an incident shock wave producing a deviation angle of 8 • is added. The interaction features a mild flow separation near the shock foot, as illustrated by a RANS SA [START_REF] Spalart | A One-Equation Turbulence Model for Aerodynamic Flows[END_REF] simulation (fig. 6). It occurs at a Reynolds number Re δ 2 = 5 860 which is higher than what could be easily achieved by DNS or WRLES (the latter would have required an estimated 100 • 10 6 grid points), demonstrating the interest of WMLES in this hybrid RANS/LES context where the mesh counts only 7.7 • 10 6 points. The interface setting d interface w /δ = 0.125 is The interaction region is further described in figure 10. The mean wall pressure evolution reveals the smoothing of the average pressure jump at the foot of the shock wave by the unsteadiness resolved in ZDES compared with the steady RANS simulation. This gradual pressure increase is a feature of the experimental data in a slightly different case (IUSTI experiment [START_REF] Piponniau | A simple model for low-frequency unsteadiness in shock-induced separation[END_REF] with an upstream flow at M ∞ = 2.3 and a deviation of 8 • ). It should be noted that the interaction in the IUSTI experiment is at a lower Reynolds number (Re δ 2 of the order of 3 000). The non-dimensionalised pressure jump ∆P/(2τ w ) is however very close in both cases (around 48 in the IUSTI case as well as in the present case). Remarkably, the resolved streamwise turbulent intensity obtained using ZDES in the present test case is qualitatively similar to the experimental results of the IUSTI case (fig. 10), especially emphasizing the proper resolution of turbulent fluctuations in the shear layer associated to the separation bubble. Overall, the ZDES technique seems mature enough for the prediction of shallow flow separation at the shock foot, even though a more thorough analysis of the results with comparison with reference data in the exact same conditions would be needed to gain even more confidence in the technique. It should be emphasized that the essential unsteady phenomena are obviously captured at a computational cost more than ten times as small as for a WRLES, and this saving ratio would be even larger at higher Reynolds numbers.

IV. Low-noise turbulent inflow for compressible WMLES

For WMLES of fully turbulent boundary layers, resolved turbulence must be injected into the flow. This means that the ZDES mode 3 technique is used in conjunction with a turbulent injection method. However, in a compressible framework, the acoustic footprint of this injection may be problematic, especially for aeroacoustic studies. The last test case is consequently designed for the study of the acoustic footprint of the resolved turbulence generation needed for WMLES. A low Mach number (M ∞ = 0.21) flat-plate zero-pressure-gradient turbulent boundary layer case is simulated by the ZDES mode 3 technique in WMLES mode with the RANS/LES interface set at d interface w /δ = 0.125. The new approach of resolved turbulence generation with a low-noise signature recently introduced by Deck et al. [START_REF] Deck | A rapid and low noise switch from RANS to WMLES on curvilinear grids with compressible flow solvers[END_REF] is compared with more classical approaches, namely based on the Synthetic Eddy Method [START_REF] Pamiès | Contrôle d'une chouche limite turbulente au moyen d'un micro-système distribué[END_REF] and on White Noise injection [START_REF] Laraufie | A dynamic forcing method for unsteady turbulent inflow conditions[END_REF][START_REF] Laraufie | A Rapid switch from RANS to WMLES for spatially developing boundary layers[END_REF][START_REF] Deck | Numerical investigation of the flow dynamics past a three-element aerofoil[END_REF]. The new approach relies on roughness elements described as immersed boundaries (ZIBC, Zonal Immersed Boundary Conditions as described in [START_REF] Weiss | On the coupling of a zonal body-fitted/immersed boundary method with ZDES: application to the interactions on a realistic space launcher afterbody flow[END_REF]) whose length scales are selected in order to disturb the boundary layer profile and generate large scale vorticity. The fluctuations are then enhanced by a dynamic forcing approach (inspired by Spille-Kohoff & Kaltenbach and adapted to ZDES by [START_REF] Laraufie | A dynamic forcing method for unsteady turbulent inflow conditions[END_REF][START_REF] Deck | Numerical investigation of the flow dynamics past a three-element aerofoil[END_REF]), enabling a quick transition from the vorticity generated by the roughness to fully developed realistic turbulence as illustrated in fig. 11. The parameters of the different approaches are summarized in fig. 12. Using the new approach, the mean skin friction relaxation distance is as short as with more common turbulent inflow conditions (fig. 13) as it can be as low as 7δ 0 . Besides, satisfying velocity profiles are obtained (fig. 14), the only visible shortcomings being related to ZDES mode 3 as a WMLES technique as already described earlier in this paper, but not to the injection approach.

In comparison with the strong spurious acoustic footprint of turbulence injection by the classical synthetic eddy method [START_REF] Pamiès | Generation of synthetic turbulent inflow data for large eddy simulation of spatially evolving wall-bounded flows[END_REF] illustrated in fig. 15, the new approach significantly reduces the spurious noise (fig. 16). This is quantified by a comparison of the pressure levels in fig. 17 and18. Comparing the profile of rms pressure with DNS data especially suggests that only the computations made with the new injection approach are not excessively polluted by spurious noise. Of interest, it can be seen that the pressure fluctuations do penetrate through the inner RANS layer down to the wall, which is coherent with the penetration of Reynolds stresses already observed in previous studies, e.g. [START_REF] Deck | Zonal Detached Eddy Simulation (ZDES) of a spatially developing flat plate turbulent boundary layer over the Reynolds number range 3 150 ≤ Re θ ≤ 14 000[END_REF][START_REF] Renard | Improvements in Zonal Detached Eddy Simulation for Wall Modeled Large Eddy Simulation[END_REF]. For this reason, the wall pressure signals are further investigated in the following.

The spurious nature of the wall pressure field when classical turbulent inflow conditions are used is confirmed by spectral analysis, whereas the power spectral density obtained with the new approach shows no spurious peaks (fig. 19). The present results and their extrapolation to higher Reynolds numbers based on predictions by [START_REF] Aupoix | Extension of Lysak's Approach to Evaluate the Wall Pressure Spectrum for Boundary Layer Flows[END_REF] suggest that at high Reynolds number, approximately 50% of the wall pressure variance may be resolved by a technique such as ZDES mode 3, corresponding to the lower-half of the spectral frequency range. Given the significant reduction of computational cost compared with WRLES, this makes ZDES mode 3 a technique of interest for the prediction of low-frequency wall pressure fluctuations. The physical relevance of the wall pressure signals obtained with ZDES and the new injection approach is further confirmed in fig. 20, where the intermittency of the wall signal, related to the intermittent nature of turbulence, is revealed by the higher tails of its probability density function compared with a Gaussian behavior (which corresponds to the spurious noise cases where the intermittent physical signal is masked by the spurious signal).

From a theoretical point of view, using the new turbulence injection approach in a supersonic boundary layer does not require any a priori modification. This has to be verified in a new test case which is out of the scope of the present study.

V. Conclusion

The present study has demonstrated the use of the ZDES technique for WMLES studies using its mode 3 in a compressible context. Firstly, the performance in supersonic boundary layers has been shown to be equivalent to the previous results obtained at low Mach numbers. Secondly, this paves the way towards the simulation of complex shock wave / boundary layer interactions, as illustrated by an example at Mach 2 where the most salient features of the unsteadiness appear to be properly captured at a relatively low computational cost. Finally, a new resolved turbulence injection approach introduced by [1] is compared with more classical approaches, revealing in a low Mach number test case that the new approach strongly mitigates the spurious acoustic footprint usually induced by classical methods. The analysis of the wall pressure signals suggests that ZDES mode 3 may be used together with the new injection strategy for the prediction of the low-frequency range of the spectrum corresponding to 50% of the wall pressure variance at high Reynolds number. Future work should involve the use of the new injection approach in supersonic flows. Overall, this study suggests the maturity of the ZDES mode 3 technique for compressible flows with both supersonic and aeroacoustic problems being within reach of the method.

Fig. 1

 1 Fig. 1 Reynolds number range covered by the FP ZPG TBL case at M ∞ = 2. For the ZDES mode 3 computations, two different RANS/LES interface settings are considered, as introduced in [6]. The first interface is set at d interface w /δ = 0.125 based on the local boundary layer thickness. The second interface is located at d interface + w = 3.9 √ Re τ , i.e. close to the geometric center of the logarithmic layer. The results are post-processed in order to enable a comparison with incompressible turbulent boundary layers following the findings of Hopkins & Inouye [13]. Regarding the mean skin friction coefficient, it is first defined by:

Fig. 2

 2 Fig. 2 Mean skin friction predicted by the ZDES simulations of the flat plate zero-pressure gradient turbulent boundary layer at M ∞ = 2 rescaled for comparison with incompressible flows following [13].

Fig. 3 Fig. 4

 34 Fig. 3 Mean velocity profile predicted by the ZDES simulations of the FP ZPG TBL at M ∞ = 2 rescaled for comparison with incompressible flows following [13].

Fig. 5

 5 Fig. 5 Normal Reynolds stress profiles predicted by the ZDES simulations of the FP ZPG TBL at M ∞ = 2 rescaled for comparison with incompressible flows following [13].

Fig. 6 RANSFig. 7

 67 Fig. 6 RANS SA simulation of the shock-wave boundary layer interaction case.

Fig. 8

 8 Fig. 8 Eddy viscosity field in the ZDES simulation of the shock-wave boundary layer interaction, together with pressure iso-surfaces.

Fig. 9 Fig. 10

 910 Fig. 9 Isosurface of the Q criterion in the ZDES simulation of the shock-wave boundary layer interaction, together with pressure iso-surfaces.

Fig. 11

 11 Fig. 11 Iso-surface of the Q criterion coloured by the streamwise vorticity, near the inflow of the FP ZPG TBL case with the new resolved turbulence generation approach.

Fig. 13

 13 Fig. 13 Mean skin friction evolution near the inlet with different resolved turbulence generation methods: A, synthetic eddy method; B, white noise; C, new approach.

Fig. 15

 15 Fig. 15 Iso-surface of the Q criterion coloured by the streamwise vorticity and instantaneous field of -∂ ρ/∂t (gray scale) with a Synthetic Eddy Method inflow.

Fig. 16

 16 Fig. 16 Iso-surface of the Q criterion coloured by the streamwise vorticity and instantaneous field of -∂ ρ/∂t (gray scale) with the new resolved turbulence generation approach.

Fig. 18

 18 Fig. 18 Wall-normal distribution of pressure rms at Re θ = 5 200 with different resolved turbulence generation methods: A, synthetic eddy method; B, white noise; C, new approach.

Fig. 19 Power

 19 Fig. 19 Power Spectral Density of wall pressure fluctuations at Re θ = 5 200 with different resolved turbulence generation methods: A, synthetic eddy method; B, white noise; C, new approach.

Fig. 20 PDF

 20 Fig. 20 PDF of wall pressure fluctuations normalized by their root mean squared value at Re θ = 5200.

Acknowledgments

The authors wish to thank all the people involved in the past and present evolution of the FLU3M code. The ZIBC approach has been developed in the framework of the research project ALLIGATOR funded by ONERA. Parts of this work are also related to the EU collaborative research project Go4Hybrid, funded by the European Community in the 7th Framework Programme, under Contract No. 605361, and to the research project OPOSSUM funded by ONERA.


Elin Käck, "Swarming European Consciousness". Europe and Tradition in the Work of William Carlos Williams. Studies in language and culture 27, Department of Culture and Communication, Linköping University. Linköping 2015.

In the lines of his preface to Paterson, William Carlos Williams expresses his disapproval of those poets that had opted for expatriation, deserting the American territory for the more polished, but less fertile grounds of Europe. Whereas he sees himself as the old "dog, " who is also an old "god, " he considers that "The rest have run out --/After the rabbits" (Paterson 1958, 3). Williams's commitment to the production of a truly American poetics, one countering the imperial hold of Europe over the arts and literature, is a career-long undertaking. The demands and conditions of this commitment are central to all of his texts, and those of many, if not all, other American experimental authors. Elin Käck's book "Swarming European Consciousness": Europe and Tradition in the Work of William Carlos Williams makes the reader return to texts that are fundamental and fascinating and that retain their difficulty despite the numerous exegeses of many brilliant critics. The texts elude definite interpretation, and acquire new significance over time, so that it is not sufficient to take for granted that interpretation necessarily is plural and contradictory rather than univocal and homogeneous: it is also mutable, and adaptable, and consequently requires the recovery of already covered ground. Williams's texts are heterogeneous and hybrid and as such generate heterogeneous and hybrid analyses that need constant reassessment and revaluation. Over an impressive volume of 252 pages of text, complete with a bibliography of more than twenty pages and an index, Elin Käck demonstrates the necessity of these re-readings with the finesse of one that continuously resists the temptation to resolve dilemmas and aporias.

Hence the dynamics of Käck's three chapters is crucial to understanding the spiraling quality of her reflection, her reluctance to bring closure to debate and her attention to the minutiae of textual analysis. Place, power, and language are the notions that constitute the focal points of the book, since they are so intertwined in Williams's work and in his approach to Europe and European culture that they fold onto themselves and cross-pollinate one another in ways that transcend the apparent neatness of these distinctions. By choosing such transparent terms as titles for her chapters, Käck questions these categories and unsettles rather than she confirms this clarity: the corpus will prove recalcitrant to the notions, and the very perception of Europe through Williams's works will thus be complexified and transformed.

Logically, in her introduction, Elin Käck begins with a redefinition of Europe, not to be understood as the geopolitical entity one might at first envision, but as both a place and a concept. She sees Williamsian Europe as "a construct" or a "trope." Quite systematically, the introduction in fact reviews the various terms of the book's title, activating each of its words as the intellectual trajectory moves from the issue of Europe to that of tradition. If defining what tradition represents to Williams proves complex in the long run, a primary assessment of Williams as part of the American poetic tradition today is made: he has been long excluded from anthologies in the U.S., but not anymore. He has met great interest in South America, and in Europe: in France, for instance, Williams features as the main subject of a whole issue of the ground-breaking poetics journal edited by Jacques Darras in Amiens (In'hu i 1980). This special issue gathers original translations and critical essays, and gives a podium to Williams's poems in France and in French as early as 1980. The very same year, some of Williams's works were published in Darras's own translation by the major publisher Christian Bourgois: In the American Grain, and The Great American Novel (Au Grain d' Amérique, suivi de Le grand roman américain 1980). Williams's status is thus, as Käck underlines it, problematic and unstable, since one cannot fully argue that he is an outsider to the poetic canon, nor can one argue that he is part of it. He has been integrated as a foil to the non-democratic, dogmatic and at times elitist proclamations by authors such as Ezra Pound or T.S. Eliot, but this image of Williams as the "unlearned" poet ( 20) is a distortion. According to Käck, Williams posits Europe and European culture at the center of his work in the 1920s (but also later, one could argue) as a trope generating specific literary strategies (rather than as a theme).

In her assessment of the state of the art in Williams studies, Käck lays the emphasis on the idea that "America and Europe" may be contrasted as the sites for different epistemological conditions, something which is powerfully taken up by Wendell Berry in his 2011 assessment of the usefulness of Williams's poetry: in The Poetry of William Carlos Williams of Rutherford (2011), one reads an account of the conceptualization of the poetry as one of place and "local adaptation" (11). The insistence, besides the comments about poetics, measure, and rhythm and the mysteries of making a poem, is laid on Williams's commitment to Rutherford as the place from which to look at the world. The portrait of the poet that thus emerges is extremely powerful, but also in keeping with the self-made image of Williams as the poet of re-contextualization and locality. To this extent, Käck's review of previous research outlines a blind spot that rests in the obliteration of the European dimension of the poetry, notwithstanding the rootedness of the poet and of his poetic objects. It is in the textual consequences of the patterns of influence that one can recognize this more cosmopolitan dimension of Williams. What such studies might obliterate is the "discrepancy between what Williams explicitly states in his work, in his so-called rhetoric, and the operations the works themselves perform" (25). The summary of these different approaches in Williams criticism leads Käck to the statement of her own hypotheses, one that considers the text as embodying the process rather than as the resolution of the complex negotiation between the Europe (rather than European) trope and the American impetus.

The theoretical framework integrates the reflections around hegemonic power, and the counterhegemonic strategies at work in social discourse, including in literature and the arts. Williams's text is to be read through a reinvestment of Foucauldian and Bakhtinian conceptions of discourses as sites of resistance and of form as the embodiment of intention. Käck suggests that Williams's texts are at the same time the examples and the theoretical statements of a poetics, thus simultaneously accounting for and altering their context. To this extent, the poems come to embody the Williamsian work of the imagination, fully creative and performative as it transforms the world it inhabits.

So, the methodology used by Elin Käck is far from being solely "thematic" (35), as demonstrated in her analysis of the poem "The Prelude" from the 1917 collection Al Que Quiere: the reading is of topoi in the poem's textual space and the interpretation relies on a rigorous examination of intertextual echoes to calculate the intent and portent of the poem. Going beyond the critical tensions between close reading and historicism, the method makes a very convincing argument for a more general process of analysis combining the two strategies and allowing for a stronger conceptualization, and the composition of a "socio-poesis." The reference to the poetic and critical practice of Rachel Blau Du-Plessis ( 35) is symptomatic of the innovative qualities of Käck's approach, and opens onto the notion of a "poethics" in the term coined by, among others, Joan Retallack (The Poethical Wager 2003).

Thus, the discussion of appropriation is important, since it raises the issue of counter-hegemonic strategies that would aim at asserting new authority over previously hegemonic material. One might contend, however, that Williams does not perform appropriation in the sense of assuming authorial control over someone else's text. As Käck under-lines it, his modes of appropriation bear more on the allusive and evocative potentialities of words or expressions which are rather integrated or incorporated rather than imperialistically appropriated.

The three lines of reflection chosen to articulate the critical discourse offer a synchronic, rather than diachronic, vision of texts that were published between 1917 and 1928: this is the most openly experimental decade in Williams's writing, the period in which he tests a wide gamut of modes and forms. The later texts in this respect are more "recognizable" to use a Williamsian term, more specific of a poetics that has found its major tenets. This definition of the corpus evidences interest in formative texts whose very heterogeneity works against the homogeneity of an established canon, and contribute to a recovery of the poet from the classics of twentieth-century American poetry. The first chapter about "Place, " the second about "Power, " and the third focused on "Language" outline the different alleys of Williams's anti-dogmatic push: place to define localism and the possibility for a text to be informed by the very geomorphological "delineaments" of the land; power to account for the poetic ideal of marriage and a mystical "contact" with the radically "other"; language to reveal in the very structures of syntax the ideological "machine" that crushes the wild and enforces the predatory principles of empire.

The first chapter, "Place, " starts by emphasizing the intricate links between the birth of an American poetics and its difficult separation from the body of Western-European culture. The issue of autonomy is at the heart of the questioning, as well as the patterns of a progressive decentering that has a lot to do with Jacques Derrida's advocacy in "Structure, Sign and Play in the Discourse of the Human Sciences" (Of Grammatology 1997). The example of Williams's letter to Pound given on page 57 is telling in this respect: Dear Ezra, Fer the luv of God snap out of it! I'm no more sentimental about "murika" than Li Po was about China or Shakespeare about Yingland or any damned Frog about Paris. I know as well as you do that there's nothing sacred about any land. But I also Know (as you do also) that there's no taboo effective against any land, and where I live is no more a "province" that I make it. To hell with youse I ain't tryin' to be an international figure. All I care about is to write. (Williams to Pound, 4 January 1921, Pound/Williams: Selected Letters of Ezra Pound and William Carlos Williams 1996, 52).

By questioning the very definition of center and "province, " Williams adumbrates the necessary distancing from the dichotomy opposing self-centeredness to self-marginalization. Neither at the margins, nor at the center, the poet performs a more discrete reorganization of hierarchies from the vantage point of no vantage point: this important notion is inscribed between the lines of Käck's argument at this point, before she moves on to better-known comments on the position of Williams's suburb to the hyperactivity of Manhattan Dada. Now even the New York effervescence still sounds constraining when compared with the freedom of Paris as expounded on page 60: in this early twentieth century, as Käck points out very astutely, "America is in fact a colony." What this implies is the need to recover a language, poetic forms and poetic objects in a radical manner, which might have been the opportunity to put to good use the tools of postcolonial studies, particularly those honed by Gayatri Chakravorty Spivak in the eponymous interview of The Post-Colonial Critic: Interviews, Strategies, Dialogues (1990). Instead, maybe to avoid the debate over who colonized whom in America, Käck transitions from this paradoxical metamorphosis of the colonizer into the colonized and the notion of American literature as bearing at least some of the characteristics of postcolonial literature (notably because it is written in the language of a reconstructed colonizer) to the more widely investigated relations between Williams and the New York avant-garde of the 1910s and 1920s.

Amidst a number of biographical elements that help explain Williams's powerful links to European culture and languages, Käck underlines a specificity of Williams's writing as it is inscribed inside the hegemonic paradigms, exposing them and evidencing the limits they unavoidably impose on the poet (69). There ensues an overview of Williams's readings, from his library, which contributes to debunking the myth of Williams as a relatively uncultured poet. Williams's reading includes all the classics, and many more authors, French and Spanish, that were not so currently read. Through the analysis of the poem "March, " one understands that in fact the reading material, as well as the material gathered from the museum or the avant-garde, are considered as raw material both for the creations of their time, and for the creations of Williams's own time. The capability of the artist to transmute this material into a work of art guarantees, in Williams's eyes, the possibility of an actual American culture. But above all Käck transcends the more conventional work on Williams's Europe, which is based on biographical investigations and the reading of Voyage to Pagany: condensed in one chapter of Herbert Leibowitz's recent biography ("Something Urgent I Have to Say to You": The Life and Works of William Carlos Williams 2011), Williams's Europe could be seen as made of multiple encounters with varied individuals, with a special note for Surrealist poet Philippe Soupault or Romanian sculptor Contantin Brancusi. Yet, with Käck's revisitation of a more conceptual Europe in texts other than those about Europe, there emerges a different understanding of a phenomenon of European influence and inspiration that proves pervasive throughout the work.

A longer development is indeed devoted to Williams's claim of Shakespeare in the American lineage (and rather than focusing on a rejection of Shakespeare as the emblematic classic of British literature): by reclaiming Shakespeare, Williams integrates a lineage of dissent and linguistic inventiveness that is in keeping with his interest in and fantasized link to Emily Dickinson. The quotation on page 97 of Williams's description of Shakespeare is in this sense evocative: not only does he, as Käck states, define himself as an heir of Shakespeare, but he redefines Shakespeare as he sees himself -in his own image. To some extent, this self-made reading of tradition constructs every piece of interpretation, reinvention, or appropriation as an avatar of the self-portrait.

In chapter two, the interplay of power relationships in the achievement of poetic recognition is shown to be connected with the issue of violence and the use of war semantics in Williams's poems. A first step in this chapter is to specifically consider Kora in Hell: Improvisations: the approach is not so much centered on the possible formal or thematic common ground with the French prose poem, or with the Surrealist products of automatic writing, since those have been widely explored elsewhere. Rather it attempts to grapple with "a different kind of European presence" (115). Summoning and qualifying Mikhail Bakhtin's assertions on the monolingual and monological quality of poetry, Käck shows that poetic language cannot be radically divorced from context: in Williams's case, its very dynamics is based on constant decontextualization and recontextualization, notably when transferring the words of Europe to the American scene.

The comments on the use of myth in Kora in Hell as threatening the modernist project and as "conservative" (121) may fall a bit short of the actual status of myth in Williams. As a cosmogonic discourse, the aim of myth is to provide an explanation for the world, its organization, or the chaotic lack thereof. Beyond the intertextual reference to Kora, there is an ethnographic dimension to the myth, and a reference to its Greek etymology, which implies a poetics based on the breath of life, and the spirit. What Williams does then could also be seen as an attempt to revivify discourse in 1917 and 1918, at a time of severe crisis in the Western world. To successfully return from Hell, from the realm of the dead, was a serious and very real stake at the time of Williams's improvisations. Laying more emphasis on the inscription of war in the poems may have served to better assess this value of myth as seminal, rather than merely referential, in Williams's texts.

Similarly, the point about polemics in Williams's 1920s writings singularly bypasses the actuality of war to focus on literary disputes: to the sentence "antagonism as a literary strategy is vital to Williams' construction of himself as a literary writer" (134-135), one might be tempted to respond that, in keeping with the urge to contextualize that pervades his work, Williams sees antagonism as ingrained in human relations and on many occasions, dramatizes it to denounce it. His disgust with Eliot also has to do with Eliot's detachment from the struggles of his contemporaries in real life, whereas his disapproval of Pound has a lot to do with Pound's political choices and his subsequent diatribes against the common man. Käck shows great insight in her reading of Spring and All and its method of including and excluding the reader from the text however: on page 146, for instance, she describes Williams's project of persuasion in extremely precise terms. The subsequent merging of criticism and poetry might have been more explicitly connected to the project that underpins Williams's work: his attempt at suppressing the notion of a poetic language that would be essentially different from the prosaic. In Williams's work since Kora in Hell, the poetic status of the poem lies somewhere beyond semantics or a conventional, metrically identifiable diction.

And indeed, this is what Käck goes on to explain as she comments on the role of the imagination as the power of actualization, and later on the performative qualities of The Great American Novel. The novel deals with the "complex network of relations that have a purchase on [Williams's] own lit-erary efforts" (166). But it is probably in the analysis of A Voyage to Pagany that the Europe trope and its validity come across most clearly: there, Europe is seen "as the locus of the plot, as a powerful construct of past glory and as matter for textual rearrangement and subversion" (168). This threefold enumeration is crucial to the general argument of the book as it naturally translates into the definition of the function of the Europe trope: it is the locus of the poet's struggle, a powerful construct of aesthetic tradition, and the matter for rearrangement and subversion into an American trope. The comparison between the beginning of the novel and the beginning of the Cantos is very fertile and inspiring: it adds a lot to the investigation of the Pound-Williams tradition that M.L. Rosenthal first defined in his groundbreaking 1984 article, "William Carlos Williams: A Memoir" (William Carlos Williams Review 10, 1).

In the third and last chapter of the book, Elin Käck considers the linguistic strategies at work in Williams's texts to carry out what she calls his counter-hegemonic project. The battlefield is displaced a third time, from the primary issues of place and position, then power and agonistics, to anchor itself in text and textuality. She outlines the "de-potentiation of language, " after Richard Sheppard in "The Crisis of Language" (Modernism: A Guide to European Literature 1991). Remarkably, one reads there a synthetic paragraph that outlines, in my opinion, the horizon of the considerations about Williams and Europe:

Through meta-language, on the one hand, Williams' work actively destabilizes representations and description of reality, and in so doing they enact a "re-framing of the 'real' " to borrow Rancière's phrasing. Through code-switching, a linguistic term denoting a speaker's shift from the use of one language to another in conversation, furthermore, Williams destabilizes both the concept of time and of place in his work. In his use of polyphony and multiple and competing discourses within a single work, Williams enacts dissent, enacts the power-structure against which he writes, and enacts his proposition of the poem as a field of action […]. (193) In Williams, language becomes "a transformative, performative and subversive medium" (193). Through a thorough analysis of the various interpretations of the first poem of Section VI of Spring and All, Käck consequently distinguishes the dif-ferent levels on which the poem can situate itself, as well as a whole gamut of possible reinterpretations. The poem crystallizes around the materiality of language and the arbitrariness of the sign unless it is made to radiate and establish relationships with other signs in a political and social network. Bakhtin's contention that in poetry words are suspended "in timeless air, " separate from their objects and their history is turned into a theoretical foil, since Williams emerges as one of the counter-examples to this contention. What the poem does indeed is to demonstrate the vanity of metalinguistic comment when not connected to a specific context: to problematize agency on the purely grammatical level, as Williams does it, is, as Williams puts it, to "have done nothing." To have done nothing is tantamount to having enforced nothingness, to having brought to the fore an emptiness that is depressing, entropic, a waste of energy leading to inertia. This entropic "nothing" necessarily calls forth "new linguistic configurations for social change" (203), that find their source in the re-energizing of the void. This is further demonstrated in fact in a masterful analysis of the poem "The rose is obsolete, " where Käck explicates or extricates the different levels on which the poem functions, particularly the way it provides a criticism of the rose as poetic signifier, the objective correlative for an entire tradition of romance and sentimentality, and a devalued topos of lyricism (201)(202)(203)(204)(205)(206)(207)(208)(209)(210)(211). This leads to considerations about the form of the poem as one realizes that such form as the sonnet is, to Williams, "a superstructure that orders and almost scripts the words." Here one wants to add that there is another worse crime of the sonnet which is that it also orders and scripts the world: in Williams's vision, the constraints to the poem are at the same time the symptoms and the causes of worldly constraints that need critiquing. This is another way to account for the Williamsian notion of structure as the contact zone between language and reality. As Käck astutely formulates it (216), it is "this power of language to structure and categorize that is Williams's chief concern."

Deriving from further considerations about foreign languages and polyphonic voices, the discussion of the unrecognizability of the American language is original and much to the point. Why would Williams seek to elaborate "a recognizable image" if he had not recognized its very unrecognizability? As the book moves towards a close, it is the expression of this paradox in Williams's poetics that gets refined: Williams seeks to resolve the aporias of writing the American poem through their very reenactment, and undermining. Whether this succeeds or not is more a matter of opinion than of actual analysis, since the work is characterized by the iterations of the attempts, the composition of "new" processes rather than by the assessment of their results, thus remaining resistant to stabilization and categorization.

In the conclusion, the characterization of the part played by the Golden Age of Spanish literature as less markedly colonial than that played by English literature is probable, but it downplays the linguistic dimension of this literature that to a large extent must have addressed Williams's preoccupations with the vernacular, the instability of voice, and the precariousness of authorship and agency, that are also part and parcel of the American dilemma.

Elin Käck's work brings new concepts and innovative arguments to the field of Williamsian studies at the same time as it revisits the Euro-American relationship that is inaugurated by the poet's texts. One reads in Williams's work a dramatization of this complex relationship. The method of moving from close reading to socio-poetics suggests the possibility of a reading that will reveal the negotiation of this relationship, how it became reformulated and transformed through its questioning, precisely at a historical moment when the hegemonic status of Europe had become fragile and a reversal in hegemonic empire was taking place. The study of the Europe trope in Williams's works helps us understand the critical moment in history when America and the America trope become prevalent. The theoretical choice of Bakhtin over Rancière and his The Politics of Aesthetics: The Distribution of the Sensible (2013), or Derrida and his notion of decentering and différance, or Deleuze and his notion of the minor mode in his study of Kafka (Kafka: Towards a Minor Literature 1986), may at times seem disconcerting, but it does show us Williams as the "inventor" that he was, in Ezra Pound's terms, beautifully breaking out of the frame of the norm, out of the structure, and away from the normative.

Where The Cambridge Companion to William Carlos Williams (2016) provides important syntheses about the key aspects of the poet's life and works, addressing in turn his relations to his contemporaries, his links to the visual arts, his grounding in the American material and cultural land-scapes, as well as his legacy, Elin Käck provides important insight to understand a more nuanced and less antagonistic relationship of his work to Europe and to grapple with the formal experiments that pervade the poems and to a considerable extent stem from the reading of more or less marginal Europeans. To this extent, it converges with Alexander Leicht's volume entitled The Search for a Democratic Aesthetics (Heidelberg, 2012) to reach out for original tools to return to the texts in themselves, their shifting status, and the new tradition they seek to empower. Leicht parallels Williams's work with that of Robert Rauschenberg and Walker Evans, in that they all explore their medium to produce "democratic theory in the language of art." Similarly Käck investigates Williams's poems implementing Europe as a crucial trope that helps theorize their polyglot, polyphonic, and polymorphous intermediality, and thus makes [them] relevant to Ezra Pound's poetic paradigm as it is defined in ABC of Reading (London, 1934): they are "news that stays news."

Hélène Aji