Loriane Weber

Max Langer

Peter Cloetens

Francoise Peyrin

come

In-line phase contrast tomography Regularized phase retrieval • X-ray phase contrast imaging offers several advantages versus absorption X-ray imaging • With coherent X-rays, a simple experimental setup based on propagation can be used [1] • Phase contrast is achieved by moving the detector downstream • The object is modelled by the complex refractive index n=1-δ+iβ, where δ is the refractive index decrement and β the absorption index • It depends on the material and the energy of the incoming beam. For soft tissue, at low energies (as used in radiology), δ is three order of magnitude larger than β • Propagation modelled by Fresnel diffraction • In holotomography [1], images are acquired at:

-several source-to-sample distances -under different angles of rotations • Phase projections are obtained by combining these multidistance projections for each angle  using phase retrieval.

• No prior (i.e. ψ θ,0 (x)=0) induces low-frequency noise.

• Possible introduction of prior information to alleviate this problem.

• Homogeneous object: prior is designed assuming that phase is proportional to the attenuation (δ/β-ratio is constant) and applied in the projection domain [3].

ψ θ,0 (x) = f(x) × δ 2β ln[I θ,0 (x)]
• For heterogeneous object of known material, the prior can be designed using several δ/β-ratios:  by thresholding the attenuation scan [4]  by assuming that the δ/β-ratio is a (continuous) function of the attenuation coefficient [5] • • Method C seems more quantitative, and image quality is improved with regards to the other methods.

• Introduce spatial constraints in prior, i.e. reconstruction while getting rid of the glass capillary • Further quantitative studies using a calibrated phantom for example.

• Heterogeneous object approaches can clearly improve image quality

1 -

 1 Creatis, Université de Lyon, CNRS UMR5220, Inserm U1044, INSA-Lyon, Université Lyon 1, Villeurbanne, France. 2-European Synchrotron Radiation Facility, Grenoble, France. Image reconstruction Setup: propagation-based parallel beam phase micro-CT

•

  Modeling of the Direct Problem • Attenuation B and phase shift ψ induced by the object described as projections perpendicular to the propagation direction z , with x=(x,y) B(x)= 2π λ  β dz and ψ(x)= 2π λ  δ dz • At each angle θ, the interaction between the objet and the X-ray can be described as a transmittance function T θ (x)= exp[-B θ (x) + i ψ θ (x)] • The free space propagation over a distance D can be modeled by the Fresnel transform involving the propagator P D T θ,D (x)=(T θ * P D )(x) • The intensity recorded by the detector at a distance D is I θ , D (x)=|T θ,D (x)|² • Inverse problem • Can be split in two steps • Phase retrieval: calculate the phase shift ψ θ (x) from the phase contrast images at different propagation distances {I θ,D (x), D=D 1 …D n }  Ill-posed inverse problem, which requires regularization. ψ θ (x)= arg min ψ σ D [ | ሚ I θ,D,ψ (x) -I θ,D (x)|² + α|ψ θ (x) -ψ θ,0 (x)|² ] • Tomographic reconstruction: reconstruct the 3D refractive index image from the phase shifts ψ θ (x) at different projection angles • Usually solved by filtered back-projection. • Phase retrieval using homogeneous [A] and heterogeneous [B] prior. • The sample is a phantom composed of multiple wires (Al, PET and PP and Al 2 O 3 ), mounted inside a glass capillary. • Experimental conditions o 1500 projections o 22.5 keV o 4 distances o Pixel size = 0.7 µm o Field of view = 1.4 mm Application on 3D cell cultures • Biomaterial (Skelite®) that can be used for 3D cell culture and artificial bone scaffolds in surgery (coll. R Cancedda, University of Genova) • Composed of Si-TCP(67%) and β-TCP/HA (33%) • 3 scaffolds imaged: discs of 9 mm diameter, seeded with osteoblasts Regularization • Reconstruction using the homogeneous approach (method A) allows quantification of scaffolds but causes artifacts in soft tissue [7] • We applied 3 heterogeneous objects approaches to improve soft tissue imaging • Method B (resp. C): using 1 st threshold (resp. 2 nd threshold) • Method D: using a functional relationship between the attenuation coefficient and δ/β values • Experimental conditions o 2000 projections o 30 keV o 4 sample-to-detector distances o Pixel size=5 µm o Field of view = 10.2 mm Example of a filtered slice of the attenuation scan, and the corresponding histogram; the histogram is clearly tri-modal, so that 2 different thresholds are possible. They were used in method B and method C. Functional relationship between the attenuation and the δ/β values, used in method D.ResultsTheoretical vol. mass densities:o ρ(air)=1.3 x 10 -3 g/cm 3 o ρ(Collagen)=1 g/cm 3 o ρ(Skelite®)=1.8 g/cm 3Design of priorApplication on phantom[5] Crop of a reconstructed slice, using homogeneous approach [A], heterogeneous approach with 1 st threshold [B] or 2 nd threshold [C], and heterogeneous approach with a functional relationship[D]. Only 3 modes are visible using methods A and B, while we can distinguish 4 modes with methods C and D.• Method A presents fringes around soft tissue • Phase contrast artefacts are visible using methods B and D.

Acknowledgements

This work was supported by the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


Widad Es-Soufi
email: widad.es-soufi@ensam.eu

Esma Yahia
email: esma.yahia@ensam.eu

Lionel Roucoules
email: lionel.roucoules@ensam.eu

A Dynamic Contextual Change Management Application for Real Time Decision-Making Support

Keywords: Change management, Dynamic change propagation, Decision making, Process patterns, Business process

Decision making is a fundamental process within organizations for many reasons. It is indeed involved at all levels (new product decisions, management and marketing decisions, etc.) and has a direct impact on companies' efficiency and effectiveness. Many researches are conducted to enhance the decision-making process by proposing decision support systems where the most frequent challenge is the change management. Indeed, all businesses operate within an environment that is subject to constant changes (like new customers' needs and requirements, organisational and technological changes, changes in key information used to derive decisions, etc.). These changes have a major impact on the quality and accuracy of the proposed decision if they are not detected and propagated, at the right time, during the decision-making process. The present work attempts to resolve this challenge by proposing a dynamic change management technique that allows three tasks to be automatically performed. First, continuously detect changes and note them. Second, retrieve from the detected changes those that are related to the decision rules. Finally, propagate them by computing the new value of the decision rule. The proposal has been fully implemented and tested in the supervision process of gas network exploitation.

Introduction

The 21 st century is characterized by rapid changes that are due to the technological progress, as well as the economic, political and social changes. In this context, it becomes critical that organizations perform two tasks. The first one is to develop capabilities to implement changes in order to improve their competitiveness and sustain their business growth. The second task consists in informing the decisionmaking process of these changes. Indeed, the latter is considered as one of the most complex and critical processes in an organization for three reasons among others. First, it is the key function to a company's success where any decision made, no matter how simple it may be, has a significant impact on the other processes and thus on the business performance [START_REF] Blenko | Decision Insights: The five steps to better decisions[END_REF]. Second, it is highly constrained by the decision context, that is often subject to several changes which need to be considered to make the right decision. Third, it sometimes involves several teams (subject matter experts, business analysts, engineers, etc.) that collectively analyse problems, evaluate alternatives and decide. Many researches focus on supporting engineers by developing decision supports systems (DSS), which are computational tools that enable decision makers to select, at each time, the best alternative among a set of alternatives that are sometimes contradictory. After the appearance of artificial intelligence and learning techniques, the concept of intelligent decision support system (IDSS) arose. It is a DSS that provides direct access to expertise by supporting learning in the decision-making process [START_REF] Arnott | Eight key issues for the decision support systems discipline[END_REF]. In [START_REF] Es-Soufi | A Process Mining Based Approach to Support Decision Making[END_REF], authors propose a process mining based IDSS that performs two tasks. First, it analyses the past process executions, using process mining techniques, to identify and generate the different process patterns as well as the decision points and the rules that control them (cf. Fig. 1). Second, it filters the detected patterns and propose a ranking of the ones that fit the decision context.

Fig. 1. The identified process patterns, decision points and decision rules

In the present paper, authors focus on developing an executable application whose objective is to inform process mining based IDSSs of the changes that dynamically occur during the decision-making process. It updates the decision context whenever a change is detected and automatically filters the patterns that are generated from the IDSS based on the decision rules, also generated from the IDSS. Authors assume that, no matter how many changes occur, the developed application helps to make the right decision given a context at time ti, where i goes from 0 (start of the decision-making process) to n (end of the process). Furthermore, authors assume that the application enables innovation by discovering new patterns from the already existing ones. This paper is organized as follows. In Section 2, a literature review on change management during the decision-making process is presented and discussed. In Section 3, the proposed application is presented, then tested and verified in an industrial case study in Section 4. Finally, the future work is discussed before concluding the paper.

Contextual Change Management During Decision Making

Change management is the process in which a change is identified, analysed, evaluated, planned and propagated [START_REF] Crnkovic | Implementing and Integrating Product Data Management and Software Configuration Management[END_REF]. Three major types of change are identified in Process patterns: P1: A1A2A3A4A5A6A7 P2: A1A2A3A5A4A6A7 P3: A1A2A3A5A6A4A7 P4: A1A2A5A6A3A4A7 P5: A1A2A5A3A6A4A7 P6: A1A2A5A3A4A6A7 P7: A1A8A9A11 P8: A1A8A10A11 P9: A1A8A9A10A11 10-A1A8A10A9A11 literature. First, the change that comes before making a decision such as an organisational change, technological change, etc [START_REF] Helfat | Dynamic Capabilities: Understanding Strategic Change in Organizations[END_REF]. This type of change helps companies to remain relevant by keeping up with technological advances and changes in the marketplace and is generally planned and evaluated before deciding to successfully implement it. The second type is the product change management also called engineering design change [START_REF] Ullah | Engineering Product and Process Design Changes: A Literature Overview[END_REF]. This type concerns the modifications that are applied to products to meet the clients' needs among others and is, thus, linked to the first change type. The product change management consists in propagating the changed (i.e. new) product data throughout the product lifecycle. The third type is the contextual change management [START_REF] Burstein | Supporting real-time decision making : The role of context in decision support on the move[END_REF]. It consists in predicting, identifying or managing the context changes for successfully addressing dynamic (also called real time) decision making support. Indeed, in dynamic environments, only real time contextual data can provide decision makers with the accurate knowledge that is necessary to make decisions [START_REF] Delic | Towards an architecture for real-time decision support systems: challenges and solutions[END_REF]. In the present paper, authors are interested in the contextual change management since it responds to their objective, which is to inform the IDSS of the changes that affect the decision-making context. Therefore, only the last type of change is explored and discussed in the following sub-sections.

Literature review

A decision is defined by [START_REF] Resulaj | Changes of mind in decisionmaking[END_REF] as an irreversible commitment to a proposition or a plan of action based on evidence, the expected metrics associated with the outcome and the context in which the decision is being made. The context is an essential and powerful concept for effective real time decision-making. The context definition that is adopted by most researchers is that it is "any information that can be used to characterise the situation of an entity" [START_REF] Dey | Understanding and Using Context[END_REF]. This context information can be imported from sensors, derived from other information, reasoned, computed or explicitly entered by users [START_REF] Burstein | Supporting real-time decision making : The role of context in decision support on the move[END_REF].

There are two categories of context [START_REF] Haghighi | Reasoning about Context in Uncertain Pervasive Computing Environments[END_REF]. The dynamic context that consists of dynamic information such as heart rate, pressure data streams and the static context where information are static such as decision makers names, etc [START_REF] Burstein | Supporting real-time decision making : The role of context in decision support on the move[END_REF].

There are many approaches in literature that focus in understanding the behaviour of the decision-making context. Some approaches intend to model the context and are based on Bayesian reasoning techniques [START_REF] Fox | Bayesian filtering for location estimation[END_REF], fuzzy logic [START_REF] Haghighi | Reasoning about Context in Uncertain Pervasive Computing Environments[END_REF], graphical modelling [START_REF] Henricksen | Modeling Context Information in Pervasive Computing Systems[END_REF] and some other ontology-based modelling approaches that allow a formal representation of concepts and their relationships that is used for knowledge reasoning about context [START_REF] Truong | Modeling uncertainty in context-aware computing[END_REF]. In [START_REF] Ziebart | Navigate Like a Cabbie: Probabilistic Reasoning from Observed Context-aware Behavior[END_REF], authors propose an overview of the context using the user behavior modelling. In addition to the modelling based approaches, there are other approaches that intend to predict the future context, in order to improve the contextual change management, by using data mining techniques such as classification, neural networks, and statistics-based techniques [START_REF] Burstein | Supporting real-time decision making : The role of context in decision support on the move[END_REF]. Other approaches use expert systems and rule engines for context prediction [START_REF] Burstein | Supporting real-time decision making : The role of context in decision support on the move[END_REF].

Discussion

Most of the approaches explored in the precedent sub section are related to the context modelling or prediction. these techniques face some challenges that concern, among others, the prediction of human actions that may be part of the real time decision context [START_REF] Nurmi | Enabling proactiveness through context prediction[END_REF]. The latter, as defined in [START_REF] Burstein | Supporting real-time decision making : The role of context in decision support on the move[END_REF], is a representation of the environment as a set of contextual data which provide the decision maker with the necessary knowledge to decide as soon as these data are obtained. In authors' context, these contextual data are both static and dynamic and are not always pre-defined or inferred since some of them change constantly, will be known only in real time or interfere with other contexts (e.g. A station's pressure may depend on the other surrounding stations behaviour, as well as on its lifetime and/or the climate change).

In addition, authors need to express the context in a deterministic way, since it is used to filter the end-to-end patterns based on some decision rules in the form "if (pressure>25) then send an emergency technician", here, a precise value of the pressure (i.e. 25) is needed rather than the distribution or prediction of pressure values. To achieve their objective, authors need continuous insights on the environmental changes and explicit knowledge of the current context. Therefore, they assume that the explored approaches cannot be used to achieve their objective and propose in the next section an application that, based on the contextual data that are obtained on the move, aims at supporting real time decision making in an informed way.

Proposal: The Dynamic Change Management Application

The workflow presented in Fig. 2 is the overview of the proposal that enables the current objective to be achieved. The first step consists in executing the static IDSS to generate the "decision rules" and the "ranked patterns" files from the process trace base as explained in [START_REF] Es-Soufi | A Process Mining Based Approach to Support Decision Making[END_REF]. This base contains the past process executions, also called traces, that were generated with respect to the trace metamodel identified in [START_REF] Es-Soufi | Collaborative Design and Supervision Processes Meta-Model for Rationale Capitalization[END_REF]. The "decision rules" file contains the rules that control how the process flows and is expressed in PNML (Petri Net Markup Language) format. The latter is an XML based standardized interchange format for Petri nets that allows the decision rules to be expressed as guards, this means that the transition from a place (i.e. activity) to another, can fire only if the guard, and thus the rule, evaluates to true. For instance, rule 1 in Fig. 1 is the decision rule that enables the transition from A1 to A2. The second document, generated by the static IDSS, is the "ranked patterns file". It contains the distinct end-to-end process patterns (e.g. patterns P1 to P9 in Fig. 1), that were identified in the process trace base and sorted by frequency.

Once the two files are generated by the IDSS, and if it is its first execution for a given decision problem, the static filtering module is invoked. This latter requests the current context from the IDSS (step 2) which transmits the request to the connected systems via the communication layer (step3). This layer is beyond the scope of the present paper, authors aspire to present it in detail in another article. Among the systems that are connected to the IDSS, authors cite the data acquisition server that collects data from the different field devices (e.g. pressure sensors), the CMMS (Computerized Maintenance Management System) that delivers maintenance data such as the date of failure/repair, the ERP (Enterprise Resource Planning) that delivers the operational processes information (e.g. delivery date) and PLM software (Product Lifecycle Management) that contains all the product/process information throughout the product lifecycle (e.g. design data).

Fig. 2. Workflow of the proposal (Expressed in BPMN 1 )

Once the IDSS receives the context from the connected systems, it copies it in a "contextual data" file which is sent to the static filtering module (step 8). The execution of this module generates the "statically filtered ranked patterns" file (step5) that associates to each pattern, of the "ranked patterns" file, a decision rule flag whose value is either 1 or 0. 1 means that the pattern satisfies the decision rules in the current context's conditions and is, thus, ready to be proposed to the user (step 6) via the IDSS graphical user interface presented in [START_REF] Roucoules | Engineering design memory for design rationale and change management toward innovation[END_REF]. The user chooses one of the proposed patterns and starts to execute its first activity (step 7). If the executed activity does not refer to the end of the process (i.e. "activityEvent ≠ end" as explained in [START_REF] Es-Soufi | Collaborative Design and Supervision Processes Meta-Model for Rationale Capitalization[END_REF]), then both the activity and the pattern to which it belongs are sent to the dynamic filtering module (step 9). The latter requests the context at time t (execution of the steps 2, 3, 4, 8), and takes as inputs the "contextual data" and the "statically filtered ranked patterns" files as well as the last executed activity and its pattern in order to update the statically filtered ranked patterns to a new file called "Dynamically filtered ranked patterns", where the decision rule flag is updated according to whether the decision rules are satisfied by the new current context. If the latter satisfies the rule 1 (cf. Fig. 1) for example, the activity A2, and all the patterns that contain it (P1 to P6) are updated to 1 while the patterns (P7 to P9) are updated to 0. The patterns whose decision rule flag equals to 1 are proposed to the user based on their frequency order (step 6). Then the steps 7, 9, 2, 3, 4, 8 are repeated in a loop until the last executed activity refers to the end of the process (i.e. "activityEvent = end" as explained in [START_REF] Es-Soufi | Collaborative Design and Supervision Processes Meta-Model for Rationale Capitalization[END_REF]). In this case, the new pattern that contains all the executed activities is sent to the process trace base to enrich it (step 11). Note that the new executed pattern is sent to the base even if it is one of the existing patterns, since the component that generates the "ranked patterns" selects only the distinct patterns and associates to each one the number of its occurrence (frequency).

The proposed change management application enables the IDSS, developed in [START_REF] Es-Soufi | A Process Mining Based Approach to Support Decision Making[END_REF], to be dynamically aware of context changes. It is fully implemented using the Java programming language.

Case Study: Supervision of Gas Network Exploitation

As discussed in [START_REF] Es-Soufi | A Process Mining Based Approach to Support Decision Making[END_REF], systems supervision is a business process that consists of complex decision based activities. During this process, supervisors have to take actions (i.e. decisions) depending on both the supervision result and the set-point (i.e. target value for the supervised system) in order to maintain the well-functioning of an industrial process, in case an alarm is received or when the performance target is seeming far from being met.The authors prefer to reuse the same supervision case study, presented in [START_REF] Es-Soufi | A Process Mining Based Approach to Support Decision Making[END_REF], where the supervisor of an industrial process should take, in the shortest time, the right decision in case an alarm is received. The IDSS is intended to support this supervisor during his problem-solving process by proposing the right decision even if the context (that can be described by the field sensors values, season, supervisor's name, etc.), changes considerably and quickly (e.g. the pressure of a gas station at time ti may be very different from the pressure at time ti+1). The problem-solving process starts by receiving the malfunction alarm. The IDSS identifies the decision points and rules (c.f. Fig. 3) as well as the different end-to-end patterns and export them to two files: The decision points/rules file in pnml format and the ranked patterns file in csv format [START_REF] Es-Soufi | A Process Mining Based Approach to Support Decision Making[END_REF].

After encountering the first gateway, the IDSS asks the connected systems for the context at time t1 (step 2,3 in Fig. 2) based on which it statically filters the ranked patterns by filling the decision rule column to 1 if the pattern satisfies the rule, 0 otherwise (step 5 in Fig. 2). All the patterns are then displayed in the IDSS interface, based on their frequency, where those whose decision rules flags equal to 1 are printed in dark black and can be selectable while the other patterns are displayed in grey but can also be selectable as the IDSS aims at supporting the user, not restricting his choices. Therefore, as depicted in Fig. 4(a), the pattern "P4T1John" is the only one that satisfies the context at time t1 (cf. Table 1 and rules in Fig. 3) and is selected by the user. The latter starts to execute the next proposed patterns activities, namely, "Send an emergency technician" and "Measure the flow rate". Before proposing these activities, the IDSS does not ask for any context information since it does not encounter any gateway, where one choice among others must be made. After the activity "Measure the flow rate" is executed, the IDSS asks for the context at time t2, according to which it filters the patterns. In this case, the pattern that satisfies the new context is still the same (P4T1John) and the IDSS continues proposing its next activity "Simulate the pressure setpoint decrease". Once this latter is executed, the IDSS asks for the context at time t3, based on which it filters the patterns. The new pattern that satisfies the rules is henceforth "P3T1John" as depicted in Fig. 4(b). The IDSS proposes the activity "inform clients" and repeats all the steps until the activity "Problem solved", that refers to the end of the process (activityEvent=end), is met. Note that the user may refuse to execute the proposed pattern and start to execute new activities. In this case the IDSS listens and traces the full executed pattern in order to send it to its process trace base to enrich it.

Conclusion and Future Work

The objective of this paper is to inform process mining based IDSSs (Intelligent Decision Support Systems) of the changes that dynamically occur when supporting engineers during their decision-making processes. Through the proposed technique, the IDSS starts by statically filtering the identified end-to-end patterns, given the context at the beginning of the decision-making process, to fit the identified decision rules. Then the filtered patterns are proposed to the user who chooses one and starts to execute its activities. The IDSS requests the new context whenever a decision point is encountered to make sure that the patterns filtering takes account of the correct context, and thus propose the effective decision. The proposed technique is generic and can be used by every IDSS that uses process mining techniques to learn from data to infer decisions. Authors assume that the use of the process mining based IDSS along with the proposed change management application, promote innovation within organizations, since it helps to create new patterns (way of thinking or solving a problem) from several existing ones (Fig. 4(c)). Future work consists in testing this assumption and evaluate the IDSS using a real-world design problem. It also consists in continuing the development of the IDSS' interface to make it more interactive and to enable the user to add his own activities to the proposed pattern and/or trace his new executions.

Fig. 3 .

 3 Fig. 3. Part of the supervision process with the identified decision points and rules

Fig. 4 .

 4 Fig. 4. Real time switching from one pattern to another while supporting decision making

Table 1 .

 1 Contextual data at times t1, t2 and t3

		t1	t2	t3
	Supervisor	John	John	John
	Season	Winter	Winter	Winter
	Pressure	25	28	28
	Flow rate	Abnormal	Normal	Normal
	Network status	Unfavourable	Unfavourable	Favourable

http://www.bpmn.org/

Acknowledgments. This research takes part of a national collaborative project (Gontrand) that aims at supervising a smart gas grid. Authors would like to thank the companies REGAZ, GDS and GRDF for their collaboration.


Loriane Weber

Max Langer

Peter Cloetens

Francoise Peyrin

Regularized phase retrieval algorithms for X-ray phase tomography of 3D bone cell culture analysis

In-line phase contrast tomography Application to 3D cell cultures

Conclusions & Perspectives

• Attenuation B and phase shift ϕ induced by the object described as projections perpendicular to the propagation direction Ԧ 𝑧 (note that x =(𝒙,𝒚) represents the coordinates in the real domain and f =(𝒇 𝒙 , 𝒇 𝒚 ) the coordinates in the Fourier domain). Material and methods

B(x)=

B C D

Crop of a reconstructed slice, using homogeneous approach [A], heterogeneous approach with 1 st threshold [B] or 2 nd threshold [C], and heterogeneous approach with a functional relationship [D].

A

Histograms of volumetric mass densities of the reconstructed slice with the 4 methods

Thresholded slice reconstructed with method C

• Only 3 modes are visible using methods A and B, while we can distinguish 4 modes with methods C and D.

• Method A presents fringes around soft tissue, and phase contrast artefacts are visible using methods B and D.

• Method C seems more quantitative, and image quality is improved with regards to the other methods.

Statistical analysis

1 -•

 1 Creatis, Université de Lyon, CNRS UMR5220, Inserm U1044, INSA-Lyon, Université Lyon 1, Villeurbanne, France. 2-European Synchrotron Radiation Facility, Grenoble, France. The object is modelled by the complex refractive index n(x,z)=1-δ(x,z)+iβ(x,z), where δ is the refractive index decrement and β the absorption index • For soft tissue, at low energies (as used in radiology), δ is three order of magnitude larger than β [1] • Phase contrast is achieved by moving the detector downstream • A two-step process: phase retrieval and tomographic reconstruction • Phase retrieval: calculate the phase shift φ θ (x) from the phase contrast images at different propagation distances {I θ,D (x), D=D 1 …D n }  Ill-posed inverse problem, which requires regularization  Usually solved by least-square method • Tomographic reconstruction: reconstruct the 3D refractive index image from the estimated phase shifts ෝ φ θ (x) at different projection angles θ  Usually solved by filtered back-projection (FBP). • Least-square method (LSQ), based on the Contrast Transfer Function φ 𝒇 = σ D sin(πλD 𝒇 2 ) ሚ 𝐼 D 𝒇 -δ 𝑫𝒊𝒓𝒂𝒄 (𝒇) σ D 2sin²(πλ𝐷 𝒇 2 ) + α • Advanced regularized method ("quasi-iterative methods") • ෝ φ θ (x)= arg min ψ σ D [ | ሚ I θ,D,ψ (x) -I θ,D (x)|² + α|ψ θ (x) -ψ θ,0 (x)|² ] • Designing the prior Regularized phase retrieval algorithms for X-ray phase tomography of 3D bone cell culture analysis L. Weber 1,2 , M. Langer 1,2 , P. Cloetens 2 , F. Peyrin 1,2 .

2πλ

   β(x,z)d𝑧 and φ(x)= 2π λ  δ(x,z),d𝑧 • At each angle θ, the interaction between the objet and the X-ray can be described as a transmittance function T θ (x)= exp[-B θ (x) + i φ θ (x)] • The free space propagation over a distance D can be modelled by the Fresnel propagator P D (Fresnel diffraction) T θ,D (x)=(T θ * P D )(x) • The intensity recorded by the detector at a distance D is I θ , D (x)=|T θ,D (x)|²• In holotomography[2], images are acquired at:-several source-to-sample distances -under different angles of rotations • Phase projections are obtained by combining these multi-distance projections for each angle  using phase retrieval.Heterogeneous objects• Must be introduced in object domain • Langer[5]: Thresholding of the attenuation scan  Bi-material object (methods B & C) • Langer[6]: Functional relationship between the attenuation and the phase (method D) 0 Phase retrieval methods • Work to be submitted to Phys. Med. Biol. • Heterogeneous object approaches can clearly improve image quality • Choice of good parameters (threshold, function) is of crucial importance • Introduce spatial constraints in prior, i.e. reconstruction while getting rid of the glass capillary • Further quantitative studies using a calibrated phantom for example. • Biomaterial (Skelite®) used for 3D cell culture and artificial bone scaffolds in surgery (coll. R Cancedda, University of Genova) [7] • Composed of Si-TCP(67%) and β-TCP/HA (33%) • 9 scaffolds imaged: discs of 9 mm diameter, seeded with bone cells • Experimental conditions: 2000 projections, 30 keV, 4 sample-todetector distances, pixel size=5 µm, Field of view = 10.2 mm • Reconstruction methods : see above Theoretical vol. mass densities: o ρ(air)=1.3 x 10 -3 g/cm 3 o ρ(Collagen)=1 g/cm 3 o ρ(Skelite®)=1.8 g/cm 3

  Can be introduced in projections • Paganin[3]: proportionality between phase and attenuation (method P) : 3 samples / type of seeded cells (OBs, pre-OCs, OBs+pre-OCs)• Quantification of volumes and corresponding thicknesses: Mineralized Volume (MV), Immature bone Volume (IBV), volume of cells (C) • ANOVA : whether there is a difference between the groups or not • Tukey's HSD (honest significant difference): which groups are significantly different from each other. • Statistical analysis showed no influence on seeded cells, except on the Immature Bone Volume. • /!\ BUT: a very few samples imaged and analyzed.

Acknowledgements

This work was supported by the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


Cédric Méaux

Giampiero Mancinelli

Justine Serrano

Aoife Bharucha

Diogo Boito

Boito Diogo

Rare

Rare heavy mesons decays to leptons

la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

For B decay, predictions not done when resonances stands in the q 2 -phase space For D decay, we cannot do this because it's equivalent to throw away most of the phase space ⇒ We need to model these resonances.

c u C k q ρ 0 , ω 0 , φ ... q γ q 2 = m 2
Predictions done in the framework of Effective Field Theory : In blue, effective operators with as "coupling" the Wilson coefficient C k . (all which is not relevant at the scale mc)

[Grinstein] The main difference between the Boer-Hiller reference and our work are :

• the treatement of resonances (inspired from arXiv 1406.0566).

• the implementation of non factorizable (n.f.) correction via the QCD factorization approach (inspired from arXiv 1705.05891).

Phenomenology project on D → π : Framework

The effective hamiltonian :

H eff = -4G F √ 2 q=b,d V * cq Vuq i •C i • O i
In our case O 1 , O 2 and O 9 are the dominant operators.

The amplitude can schematically be written :

π + | H eff | D + = C • ff + Φ D + ⊗ T ⊗ Φ π +
where ff stands for form factors, the Φ are the light-cone distribution amplitudes.

The C and T factors are caculables in renormalization-group improved perturbation theory and can be further decomposed into :

C = C (0) + as • C (1) and T = T (0) + as • T (1) with as = C F αs 4π
Phenomenology project on D → π : Leading Order (LO) and Next-to-Leading Order (NLO) contributions Our strategy : improve resonances modelization by using the Shifman parametrisation and by extracting these parameters from a fit to e + e -experimental data R by virtue of the optical theorem : Experimental measurements :

π 3 R(q 2 ) = Im[L](q 2 ) dispersion relation → L(q 2 ) (
• B(D + → π + µ + µ -) < 7.3 • 10 -8 at 90% C.L. LHCB-PAPER-2012-051 • B(D 0 → µ + µ -) < 6.2 • 10 -9 at 90% C.L. LHCB-PAPER-2013-013
Take-home messsage :

The work is still ongoing ! Thanks to a more accurate estimation of the hadronic contribution as well as the non factorizable contribution, we aim at giving more accurate bounds on NP Wilson coefficients.

This will help us to estimate the room left for other observables like the angular observables (A FB , F H ), the CP assymetries, lepton flavor universality ratio...

Eye candy example :

If C T and C S real and the other NP C i null, we get : Preliminary plot (doesn't include either n.f. corrections or hadronic contributions) Experimental project on B 0 (s) → τ + τ -: Introduction

• B 0 (s) → are golden channels, theoretically clean and with clear experimental signature.

• B(B 0 s → µµ) = (3.0 ± 0.6± +0.3 -0.2 ) × 10 -9 in agreement with SM LHCB-PAPER-2017-001

• Nevertheless, several models predict BR higher than SM arXiv 1505.05164 , 1609.09078

• But the τ channel is more challenging :

B 0 s → τ + τ - B 0 → τ + τ - SM 7.7 × 10 -7 2.2 × 10 -8 Limit at 95% C.L. < 6.8 × 10 -3 < 2.1 × 10 -3
Best world limit set with LHCb Run 1 data (Lumi 3fb -1 ) LHCB-PAPER-2017-003

2 axes to improve these limits : 1 Explore another channel with Run 1 data :

B 0 (s) → τ (3π ± )τ (µ) 2 Run 2 data (Lumi 7fb -1 , energy × 2)
Experimental project on B 0 (s) → τ + τ -: Axe 1 : (3π, µ) Run 1, Motivations and regions Motivations for the (3π, µ) channel :

Higher effective branching ratio : 17.4% for τ → µνν vs 9.3% for τ → 3πν Only 4 tracks required in the detector acceptance. µ trigger more efficient than the hadronic trigger. Plus one ν / only 1 τ vertex, less handle to discriminate signal and background. Substantial semi-leptonic B and D decays as background.

Regions : 3 ν in the final state ⇒ no narrow mass peak to fit ! Idea : Exploit the ρ 0 resonances of the τ → 3π decay

τ -→a1(1260) -ντ → π - 1 ρ(770) 0 → π + 2 π 3 - Region (3π, µ) (3π, 3π)
Signal τ in 5 τ + and τ -in 5 Bkg.

τ in 1/3/7/9 τ + (or τ -) in 1/3/7/9 Control τ in 4/8 τ + in 4/5/8 and τ -in 4/8 Experimental project on B 0 (s) → τ + τ -: Axe 1 : (3π, µ) Run1, Workflow Workflow (inspired from the (3π, 3π) Run 1 analysis) :

1 Selection : Loose cut-based + BDT-based Based on kinematical, geometric variables and custom-made variables, mainly isolation variables.

2 1D binned Maximum Likelihood fit :

• Fit a new BDT output for data in the signal region.

• Fit error for zero signal events estimated via a toys study.

3 Normalisation : channel :

B 0 → D + (K -π + π + )π - B(B 0 (s) → τ + τ -) = N obs τ τ tot τ τ N B 0 ≡ α • N obs

Back-Up

Rare heavy mesons decays to leptons Cédric Méaux

Introduction•• 5 •

 5 Flavor Changing Neutral Current (FCNC) meson decays are only at loop level in Standard Model (SM) ⇒ Rare but very sensitive to new physics (NP) ! Many deviations from SM seen in FCNC (b → s ) : R K 0( * ) = B(B 0 →K 0( * ) µ + µ -) B(B 0 →K 0( * ) e + e -) , P But also at tree level R D ( * ) = B(B 0 →D +( * ) τ -ν) B(B 0 →D +( * ) µ -ν) , NP scenario with lepton flavor dependence ? Need to pursue all possible ways to constrain NP. In this light, working on 2 projects : Experimental • CPPM, LHCb • Purely leptonic b → s transition Pursue the search for B 0 (s) → τ + τ - Phenomenology • CPT and D. Boito (São Paulo University) • Semi-leptonic c → u transition Improve predictions for D → π Phenomenology project on D → π : Introduction GOAL : Improve predictions around D → π • FCNC decay of type c → u • Should be sensitive to the same NP as FCNC B decay like b → s • So why it's not a golden channel yet ? Because of resonances entering the quark loop in one of the leading order diagram.

  LO : (a) and (b) included in C = C (0) , weak annihilation diagram (c) included in T (0) (not taken into account in the Boer-Hiller reference.) NLO form factor correction : inlcuded in C = C (1) , partly taken into account in the Boer-Hiller reference. NLO hard spectator scattering correction : inlcuded in C = T (1) , not taken into account in the Boer-Hiller reference Phenomenology project on D → π : Treatment of resonances In the Boer-Hiller reference, resonances are treated by simple Breit-Wigner peaks.

  function describing the resonances entering into C 9 ) Shifman parametrisation : hep-0009131 Resonances modelled by an infinite tower of equidistant vector resonances Rare heavy mesons decays to leptons Cédric Méaux Phenomenology project on D → π : Conclusion B(D + → π + µ + µ -) depends on C 7 , C 9 and NP : C 10 , C S , C P , C T , C T 5

Figure

  Figure -MC simulation of B 0 s → τ + τ -

  → τ + τ -: Axe 1 : (3π, µ) Toy estimate obtained by : U L = 2 • α • signal-yield-error (no systematics, no efficiency correction, no template correction) Axe 1 explored : U L(3π,µ) U L(3π,3π) for B 0 (s) : 3.7(4.5) • Appeared to be difficult to fight the B and D substantial background. • Internal note delivered on June 2018 LHCb-INT-2018-021 Experimental project on B 0 (s) → τ + τ -: Conclusion [Courtesy of Kristof De Bruyn] • If not for LHC, B 0 (s) → τ + τ -will be a golden mode for the future collider (like FCC). Comparison with sister processes with e or µ → test of LFU. Richer phenomenelogy (angular observables) • Definitively worthwhile to work on it and to prepare the future of this channel ! ! Back-Up

Rare heavy mesons decays to leptonsCédric Méaux

Experimental project on B 0 (s) → τ + τ -: Axe 2 : (3π, 3π) analysis workflow and results

The signal yield obtained for Run 1 data s = -15 +67

-56 (stat) +44 -42 (syst) Uncertainty mainly statistical, hence adding Run 2 data is highly motivated.

Axe 2 started 2 month ago, two strategy explored in parallel :

• Apply the exact same selection than for Run 1.

• Re-optimize the selection, the used regions, etc...

What is done :

• Normalization channel (except systematics) • A possible re-selection has been designed (performance estimation ongoing).

• Apply the Run 1 selection (ongoing).

Prospects :

• Deliver soon an estimate of the limit we could reach for both strategy Rare heavy mesons decays to leptons Cédric Méaux

Phenomenology project on D → π : Angular observables

The decay rate can be expressed like :

3 with θ angle between the D and -direction of motion in the dipleton center of mass frame.

Two clean null-tests can be defined (more deltails in arXiv:0709.4174) • Forward-backward asymmetry :

• Flat term :


Guillaume Briffoteaux
email: guillaume.briffoteaux@umons.ac.be

Nouredine Melab
email: nouredine.melab@univ-lille1.fr

Mohand Mezmaz
email: mohand.mezmaz@umons.ac.be

Daniel Tuyttens
email: daniel.tuyttens@umons.ac.be

An adaptive evolution control based on confident regions for surrogate-assisted optimization

Keywords: Surrogate-modeling, multi-objective optimization, evolution control, direct fitness replacement, machine learning

In simulation-based optimization the objective function is often computationally expensive for many optimization problems. Surrogate-assisted optimization is therefore a major approach to efficiently solve them. One of the major issues of this approach is how to integrate the approximate models (surrogates or metamodels) in the optimization process. The challenge is to find the best trade-off between the quality (in terms of precision) of the provided solutions and the efficiency (in terms of execution time) of the resolution. In this paper, we investigate the evolution control that alternates between the simulator and the surrogate within the optimization process. We propose an adaptive evolution control mechanism based on the distance-based concept of confident regions. The approach has been integrated into an ANN-assisted NSGA-2 and experimented using the ZDT4 multi-modal benchmark function. The reported results show that the proposed approach outperforms two other existing ones.

I. INTRODUCTION

In simulation-based optimization, the evaluation of the fitness function relies on complex simulations often characterized by a high computational cost [START_REF] Duchaine | Computationalfluid-dynamics-based kriging optimization tool for aeronautical combustion chambers[END_REF]. For many engineering problems, the dimensionality is high, the cost function is non-convex and no gradient information is available. For these problems, Evolutionary Algorithms (EAs) have proven to be efficient optimization techniques [START_REF] Talbi | Metaheuristics: From Design to Implementation[END_REF]. The idea behind these algorithms is to simulate the natural evolution of species. An initial population of solutions, generated by sampling the search space, is iteratively evolved by selecting parent individuals, mating them and replacing them by the generated offsprings if they improve the considered objectives. The process usually leads to a high number of fitness assessments. Improving the design of a manufactured product, for instance, could require to evaluate a great number of different models implying a great number of simulator calls and making the procedure unrealizable given certain computational time budgets. It is worth noticing that the computational time budget can be expressed in terms of number of simulator calls since other operations are negligible in a lot of engineering cases such as studies involving Computational Fluid Dynamic (CFD) [START_REF] Duvigneau | Hybrid genetic algorithms and artificial neural networks for complex design optimization in cfd[END_REF].

A way to circumvent such computational constraints is to resort to Meta-Models (MMs). Meta-models, also called surrogate-models, aim at emulating and simplifying an actual model making its evaluation cheaper but at the cost of a coarser accuracy. The trade-off between the accuracy of the MM and the computational costs entailed to build it has to be controlled to respond to the final purpose. The term model management is usually used to refer to this task. Indeed, when addressing meta-modeling, several possibilities can be examined [START_REF] Yaochu | Surrogate-assisted evolutionary computation: Recent advances and future challenges[END_REF] as the number of MMs, the way they collaborate with each other, their fidelity, their type, the region of the decision space they act upon and the objective they emulate but also and more importantly the way they cooperate with the real fitness function and the EA. In this sense, a taxonomy has been proposed by A.Díaz-Manríquez et al. [START_REF] Hugo | A review of surrogate assisted multiobjective evolutionary algorithms[END_REF] to categorize the studies of this field, with an emphasis on fitness replacement. Since the surrogate-model is constructed based on the real fitness function and aims at substituting this last, a strategy has to be devised to determine whether calling the real fitness function or the MM at a given step of the evolution, and when building or updating the surrogate. Two fitness replacement modes can be envisioned: Direct Fitness Replacement (DFR) and Indirect Fitness Replacement (IFR). In the former, it is assumed that a surrogate-evaluated solution carries the same degree of accuracy that a real-evaluated one and the EA is allowed to embed it into the population at the replacement stage of the evolution. In the latter, the EA appeals to the approximated model so as to give a lift to exploitation, letting it only proposing new offsprings then re-evaluated with the original fitness function if they are selected. The drawback of the first approach is to converge toward a false optimum while the second approach might converge prematurely.

In DFR, the MM is used alternately with the simulator according to an alternation criterion, or evolution control. For low dimensional problems, the EA may exclusively call the surrogate-model previously built thanks to simulations performed in the past. This concept is called No Evolution Control (NEC) since it is decided a priori that only the MM serves the evolution. A particular attention has to be paid to the accuracy of the approximated model. For problems of higher dimension, it could be better to alternate between the fitness functions during the evolution. A control parameter sets the percentage of original and surrogate evaluations performed at a generational or individual level. This strategy is referred to as Fixed Evolution Control (FEC). The advantage is the opportunity of updating the surrogate through the search and thus refining its precision on new regions of interest. Nevertheless, the disadvantage is the adding of a user-defined parameter for the control. To avoid this drawback, the Adaptive Evolution Control (AEC) suggests to test the accuracy of the MM as an alternation criterion. The issue that arises is how to evaluate the surrogate precision.

In this paper, an heuristic is proposed to define an alternation criterion for AEC between the MM and the original fitness function based on the known sub-regions of the decision space. In Section 2, a review of some studies illustrating the different model management approaches is presented. Section 3 is dedicated to introduce the Non-dominated Sorting Genetic Algorithm (NSGA-II) for Multi-objective Optimization (MO) and Artificial Neural Network (ANN) for surrogate modeling. In Section 4, the confidence regions AEC is proposed along with a detailed presentation of two evolution controls from past studies. Preliminary experiments and results are reported in Section 5, and conclusions and some directions about further works are made in Section 6.

II. RELATED WORKS

A cooperation strategy between elitist NSGA-II and an ANN is proposed in [START_REF] Deb | An Evolutionary Multi-objective Adaptive Meta-modeling Procedure Using Artificial Neural Networks[END_REF] and validated in a cantilever plate optimal shape problem according to two objectives. The percentage of real fitness evaluations and surrogate fitness evaluations is fixed a priori. For a predefined number of Q generations, the n first are produced based on the real fitness function and the remaining ones based on the surrogate fitness function. The model management considered is DFR with a FEC.

The drawbacks of the previous work is that MM precision is not taken into account to determine the values of the parameters Q and n. From this observation, A.Gaspar-Cunha and A.Vieira offer to take advantage of the ANN approximation error to switch from one fitness function to the other [START_REF] Gaspar | A multi-objective evolutionary algorithm using neural networks to approximate fitness evaluations[END_REF]. The evolution control thus becomes adaptive. The real optimization problem considered in this study consists in maximizing the mass and the mixing degree of a screw produced by a single-screw polymer extruder.

In [START_REF] Müller | Influence of ensemble surrogate models and sampling strategy on the solution quality of algorithms for computationally expensive black-box global optimization problems[END_REF], a combination of different surrogate-models is presented, in an IFR flavor, to evaluate candidate solutions chosen randomly in the decision space. The best candidate solution is then re-evaluated with the real fitness function and compared to the best solution found so far. This single-objective nonevolutionary optimization technique is validated solving a 12dimensional ground water bioremediation problem where the objective is to decontaminate ground water at minimum cost. The fitness function evaluation requires a system of nonlinear partial differential equations reflecting both water flow and chemical reactions to be treated and takes up to 4 minutes.

In [START_REF] Duchaine | Computationalfluid-dynamics-based kriging optimization tool for aeronautical combustion chambers[END_REF], a MO strategy based on an iteratively enhanced Kriging meta-model coupled to a fully parallel 3D CFD solver is proposed to improve the design of gas turbine engines. Kriging, also named Gaussian Processes, is a regression method appreciated for the uncertainty measure it provides on the predictions. Predictions and uncertainties are combined into a new function on which local search is performed to propose new candidate solutions. The model management applied here is IFR since candidate solutions are all evaluated with the solver. The total number of CFD evaluations is one of the stopping criteria of the whole search.

C.Poloni et al. propose an approach based on the collaboration between a two-objective Genetic Algorithm (GA), a Conjugate Gradient (CG) algorithm, an ANN as surrogatemodel and a 3D Navier-Stokes CFD solver to enhance the hydro-dynamism of a sailing yacht [START_REF] Poloni | Hybridization of a multi-objective genetic algorithm, a neural network and a classical optimizer for a complex design problem in fluid dynamics[END_REF]. The GA is first run to provide both an approximation of the Pareto front and an initial database to construct the ANN. The second step of the search is carried by CG, initialized thanks to the Pareto approximation, in cooperation with the ANN which is updated after each step. The model management applied here is IFR since, at each CG step, the solver is called to evaluate the proposed solution.

Scheduling problems are also tackled by MM-assisted optimization as shown in [START_REF] Syberfeldt | A parallel surrogateassisted multi-objective evolutionary algorithm for computationally expensive optimization problems[END_REF] where ANN approximations and uncertainties are used to optimize a factory production planning maximizing devices utilization and minimizing tardiness. In a parallel EA, offsprings are evaluated and ranked thanks to the meta-model, the ones marked with the better rank are re-evaluated with the original cost function before to be inserted into the population. Given that the population is solely constituted of solutions evaluated with the real fitness function, the model management is qualified as IFR.

To improve the aerodynamism of an airfoil, an ANN surrogate-model assists a GA based on a 3D flow simulator in [START_REF] Duvigneau | Hybrid genetic algorithms and artificial neural networks for complex design optimization in cfd[END_REF]. The first generations are obtained with the simulator and evaluations are kept to built the ANN. Then, GA relies on the ANN and only the best solutions obtained at the end of a generation are re-evaluated with the simulator. The model management is then a mix between DFR and IFR.

M.Mlakar et al. combine Gaussian Processes with a multi-objective differential evolutionary algorithm [START_REF] Mlakar | Gp-demo: Differential evolution for multiobjective optimization based on gaussian process models[END_REF]. The alternation between the surrogate and the real fitness function happens at the replacement step of the EA and depends on the uncertainty upon the prediction of the offspring objective values. If the original version of the MM is good enough, the model management is a DFR, otherwise it can be defined as IFR or as a mix of both fitness replacements. Optimization of the quality of cast steel, is chosen to validate the strategy. Given a model of the steel casting process, the goal is to determine the best input values of the simulator to obtain the desired outputs. It is indicated than one simulation lasts approximately 4 minutes on a single computing core.

Inspired by [START_REF] Deb | An Evolutionary Multi-objective Adaptive Meta-modeling Procedure Using Artificial Neural Networks[END_REF] and [START_REF] Gaspar | A multi-objective evolutionary algorithm using neural networks to approximate fitness evaluations[END_REF], an heuristic-based adaptive evolution control is presented in this paper and compared to the previous studies. NSGA-II and ANN are chosen as MO algorithm and surrogate model respectively.

III. BACKGROUND ON NSGA-II AND ANN A. Comparison in multi-objective optimization

In multi-objective optimization the comparison between two solutions is performed according several criteria. A set of dedicated comparison operators are available [START_REF] Huband | A review of multiobjective test problems and a scalable test problem toolkit[END_REF] in order to assist the decision maker in his/her final choice.

Given a, b two decision vectors, from the decision space, and X a set of decision vectors containing a, it is stated that:

• a dominates b iff a is at least better than b regarding one criterion and as good as b regarding the remaining criteria.

• a and b are incomparable iff no solution dominates the other and there exists at least one criterion whose value differs from a to b. • a is a non-dominated solution of X iff there is no solution in X that dominates a. These definitions still stand for objective vectors picked up within the objective space.

Given a population of solutions, the subset composed of non-dominated individuals, such that one non-dominated individual is incomparable with all other non-dominated individuals, is called the Non-Dominated Front (NDF). To confront NDFs with each other, the accuracy of the solutions just as their spread and distribution over the objective space can be analyzed [START_REF] Okabe | A critical survey of performance indices for multi-objective optimisation[END_REF]. The best NDF for a given problem is called the Pareto front of this problem.

B. Non-dominated Sorting Genetic Algorithm (NSGA-II)

The idea behind population-based evolutionary algorithms for optimization is to simulate the natural evolution of species. Algorithm 1 gives the general skeleton of EAs emphasizing the main operations performed over the population: initialization, selection, reproduction (mating) and replacement. The actual choice of operators and their parameters depends on the desire trade-off between diversification, understood as the exploration of the entire search space, and intensification, the exploitation of solution characteristics encountered in a promising region of the search space.

The particularity of NSGA-II is the use of the crowdedcomparison operator both at the selection and replacement step [START_REF] Deb | A fast and elitist multiobjective genetic algorithm: Nsga-ii[END_REF]. When comparing two solutions, both the accuracy, Algorithm 1 Basic sketch of population-based evolutionary algorithms.

Initialize a population P while the stopping criterion is not met do Select a set of parents from P Mate the parents to father a set of offsprings Evaluate the offsprings if the replacement criterion is met then Replace the parents by the offsprings to form the new population end if end while return Non-dominated front from the last population represented by the rank of the NDF the solutions belong to, and the distribution, shown by the crowded distance are taken into account. To the NDF of the whole population is assigned rank 1 (the best), then after removing solutions of rank 1, the NDF of the remaining population is ranked 2. Repeating the procedure allows the individuals to be classified by group depending on their accuracy. To distinguish solutions of the same rank, the crowded distance indicating the average distance from one solutions to its closest neighbors along each objective is computed. A higher value of the crowded distance exhibits a solution with a wider empty neighborhood, which is attractive to enhance the distribution of the population.

In NSGA-II, crossover and mutation are the reproduction operators employed to generate offsprings from parents. Crossover combines good features extracted from each parent to produce a child thus increasing intensification while mutation disrupts the child by modifying the features inherited from its parents thus reinforcing diversity. A lot of versions of these operators have been proposed in order to reach different shade of exploitation and exploration under the constraint of producing valid candidate solutions [START_REF] Talbi | Metaheuristics: From Design to Implementation[END_REF].

C. Artificial Neural Networks (ANN)

Artificial Neural Networks are often selected as surrogate models thanks to their approximation universality [START_REF] Hornik | Multilayer feedforward networks are universal approximators[END_REF] and ease of update. An ANN architecture is composed of units organized in 1-D fully connected layers: input layer, hidden layers and output layer. One unit of a given layer is linked with all units from the previous and next layers by a weighted connection. Each unit is also equipped with an activation function f supposed to be the same for all hidden units in this application. Training a network requires a set of sample data called training set and composed of several inputs with corresponding outputs. Inputs flow through the layers such that x (i) , the input of layer l i , is given by:

x (i) = W (i-1) .f (x (i-1) ) + b (i-1) (1)
with W (i-1) and b (i-1) respectively the matrix of weights and the vector of biases from the l i-1 → l i connection. The output layer gives finally an estimation of the expected output. A gradient descent (GD) algorithm is used iteratively to improve the estimation accuracy by updating weights and biases thanks to backpropagation [START_REF] Goodfellow | Deep Learning[END_REF].

The way data is fed into the network during the training phase is of important relevance for the training quality. Two training procedures have been proposed. The first, called batch training, consists in gathering training data in batches and update the ANN's weights once for the whole batch. The second, called iterative learning, considers one sample at a time. On the one hand, all the batch samples contribute together to modify the ANN weights and biases whereas on the other hand, only one sample leads to the update.

Hyperparameters (such as the number of neurones, number of layers and learning rate) and methods (such as GD algorithm, sample normalization and early stopping) specializing the ANN are difficult to be tuned because of the number of possible configurations and the problem dependence.

IV. A NEW CONFIDENCE-REGIONS ADAPTIVE EVOLUTION CONTROL

The general purpose of the adaptive evolution control proposed hereafter is to reduce the number of real fitness assessments while keeping a good convergence toward the Pareto front as well as preserving a good spread of solutions. Strategies proposed in [START_REF] Deb | An Evolutionary Multi-objective Adaptive Meta-modeling Procedure Using Artificial Neural Networks[END_REF] and [START_REF] Gaspar | A multi-objective evolutionary algorithm using neural networks to approximate fitness evaluations[END_REF] are reproduced with NSGA-II as EA and ANN as MM for comparison purposes.

In the DFR-FEC strategy, called B Q n in [START_REF] Deb | An Evolutionary Multi-objective Adaptive Meta-modeling Procedure Using Artificial Neural Networks[END_REF] and reproduced in [START_REF] Gaspar | A multi-objective evolutionary algorithm using neural networks to approximate fitness evaluations[END_REF], cycles (shown in Figure 1), each producing (p + q) generations, are run until the real evaluations budget is exhausted. The first p populations are fathered using the real fitness function only, then the surrogate is constructed or updated and the last q generations are created relying on the surrogate only. The determination of the parameters p and q such as the training mode are let to the user. Except for the first p generations of the first cycle, the population can be composed of individuals evaluated with the surrogate function, consequently a low value of p with regard to q provokes a low accuracy on the surrogate predictions and may drive the search in a wrong direction. Conversely, a high value of p, particularly at the beginning of the EA, leads to restricting the search to a narrow location and low diversity.

The model management suggested in [START_REF] Gaspar | A multi-objective evolutionary algorithm using neural networks to approximate fitness evaluations[END_REF], named e N N -EC in this paper, offers to automatically adapt the number q of MM-based generations depending on the approximation error commited by the ANN. For a given cycle i, p generations are first built with the real fitness function, the surrogate is then built or updated and the very next generation is obtained with the surrogate. During this latter a few number M of solutions are assessed with both fitness functions to compute the approximation error e N N . If e N N > e 0 , with e 0 a given threshold, the surrogate is not trusted any more and the cycle ends with q i = 1, otherwise the same procedure is repeated. In order to ensure the end of a cycle, a limit q max is imposed on the number of surrogate-based generations. This trust criterion alleviates the probable damages sustained by an inaccurate surrogate with the drawback of defining the parameters p, q max , e 0 and M . Moreover, the selection of individuals to be assessed with the real fitness function to compute e N N is carried out randomly so that these evaluations might not benefit the next surrogate update.

In covariance-based MM such as Kriging [START_REF] Mlakar | Gp-demo: Differential evolution for multiobjective optimization based on gaussian process models[END_REF], the expected error on predictions is low at points used to construct the surrogate and increases when moving away from these known points. Even if ANN is not an interpolation method, a similar behavior could be expected assuming an acceptable training error. Based on this observation and on the two methods depicted previously, it is proposed to alternate between the fitness functions at an individual level depending on the distance between the candidate individual and the known solutions. In other terms, given S * the set of solutions evaluated with the original cost function that have been used to construct the surrogate and s a candidate solution to be evaluated. If there exists s * ∈ S * such that d(s, s * ) < then the candidate solution resides within the confidence region where the surrogate is trusted and s is thus assessed with the surrogate, otherwise, the simulator is called. The union of the hyperspheres H with center s * and radius in the decision space is defined as the confidence region CR:

CR = ∅ if S * = ∅ ∪ s * ∈S * H(s * , ) otherwise
The fact that the alternation criterion is tested at an individuallevel and not at a population-level reduces greatly the probable damages sustained by an inaccurate surrogate that is to converge to a false Pareto front. Another advantage of this method is the saving in real evaluations provided by removing the need to compute the approximation error e N N . In [START_REF] Gaspar | A multi-objective evolutionary algorithm using neural networks to approximate fitness evaluations[END_REF], training is realized in batch mode with a training set composed of 50% of all available real evaluations. The underlying hypothesis is to build a strong surrogate on promising regions as the search moves forward. Besides, in order to grant the same importance to each training sample, training targets (outputs) are preprocessed by log (x + 1) and These assumptions are respected in this study to maintain comparison fairness. The number of real evaluations between two surrogate updates has to be analyzed too. NSGA-II, completing a generation implies pop size real with pop size being the size of the population. So in B Q n -EC, p * pop size real evaluations are performed between two updates. To that amount has to be added the small number of real evalutions needed to compute e N N in e N N -EC. In the confidence region AEC (CR -EC), it is decided to trigger the update step after p * pop size real evaluations.

V. PRELIMINARY EXPERIMENTS

A preliminary experiment is set up so as to reproduce design engineering problem conditions. It is assumed that the real evaluation budget is small in comparison to the number of real evaluations needed, theoretically, to converge to the Pareto front. Benchmark problem ZDT4 [START_REF] Zitzler | Comparison of multiobjective evolutionary algorithms: Empirical results[END_REF] is chosen for its multi-modality aspect, a feature recognized to cause difficulty in MO. ZDT4 real fitness function, presented in Equation [START_REF] Talbi | Metaheuristics: From Design to Implementation[END_REF], is a continuous bi-objective function with 10 decision variables and admitting 21 local Pareto-optimal fronts. It has been shown that 30000 real evaluations are needed to converge to the Pareto front with NSGA-II [START_REF] Deb | An Evolutionary Multi-objective Adaptive Meta-modeling Procedure Using Artificial Neural Networks[END_REF], so the budget is fixed to 5000 here to respect the previous assumption.

F 1 (x) = x 1 F 2 (x) = g(x) 1 - x 1 g(x) g(x) = 91 + 10 i=2 (x 2 i -10 cos (4πx i )) x 1 ∈ [0, 1] and x i ∈ [-5, 5]∀i ∈ {2, . . . , 10} (2)
The Hypervolume indicator H is the performance index used to compare NDFs in this study. Given a reference point O , H is the area generated by the set of solutions composing the NDF and O . The higher the value of H, the better is the NDF as suggested in Figure 3 where H is displayed for a set of 3 solutions {s 1 , s 2 , s 3 }. In this application, O is set to [START_REF] Duchaine | Computationalfluid-dynamics-based kriging optimization tool for aeronautical combustion chambers[END_REF]406) by roughly computing an upper bound for both objective functions. The interest of this index is to reflect the accuracy, the distribution and the spread of the solutions at once. Optimization with EAs is a stochastic process because of the choice of the initial population and the frequency of applying reproduction operators. To get the trend of efficiency of the different evolution controls, several searches are launched, each with a fixed initial population for all the strategies. For each search, the initial population is constituted randomly over the decision space and, at the end, a point is given to the strategy that produces the best H value. The EA without MM and the three evolution control strategies (B Q n -EC, e N N -EC and CR -EC) are finally ranked.

The Pagmo [START_REF] Biscani | esa/pagmo2: pagmo 2[END_REF] implementation of NSGA-II is parameterized as follows:

• Population size: 100 • Crossover probability: 0.9 • Distribution index for crossover: 10 • Mutation probability: 0.1 • Distribution index for mutation: 50 For B Q n -EC, p is set to 3 and q to 7 in accordance with the guidelines of [START_REF] Deb | An Evolutionary Multi-objective Adaptive Meta-modeling Procedure Using Artificial Neural Networks[END_REF]. To ensure comparison fairness in terms of number of samples in the training sets and to follow guidelines of [START_REF] Gaspar | A multi-objective evolutionary algorithm using neural networks to approximate fitness evaluations[END_REF], p = 3 and q max = 10, e 0 = 0.03 and M = 5 in e N N -EC. In CR -EC, is fixed to 1% of the maximum distance between two decision vectors. Specifications of the Scikit-Learn [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] implementation of ANN are as follows:

• Number of hidden layer: 1

The first tournament between the different methods is composed of 50 searches. ANN training is stopped when the coefficient of determination doesn't improve by at least 10 -4 during 4 iterations. Results, shown on the left of Table I, indicate the superiority of NSGA-II without MM in this tournament. The explanation resides in the poor training quality of the ANN as shown in Figure 4. This same figure suggests that training is more complex in the case of CR -EC than in other cases. Even if CR -EC admits the worst training quality, it is the best method out of the three evolution controls as shown on the right of Table I. From Figure 5, it is worth noticing that e N N -EC limits the use of the inaccurate surrogate, with q i = 1 on average, as expected.

To ensure fairness regarding the training quality and to keep a low training time, it is suggested to stop training as soon as R 2 reaches 0.1 for the second tournament. The results, presented in Table II, demonstrate the superiority of e N N -EC for this tournament. Since the training quality is maintained to a low level, the limitation of the inaccurate surrogate implied by e N N -EC gets the drop on the heuristic proposed in CR -EC as it can be remarkably seen from cycle 14 in Figure 6.

Since the computational costs to attain a high R 2 score are prohibitive with ANN trained sequentially (more than an hour for CR-EC with 2500 training samples), Random Forests are considered as MM [START_REF] Breiman | Random forests[END_REF]. A forest of 10 regression trees each based on a sub-sampling created with replacement is used, given a training time of approximately 0.1 second. The quality of the MM is high on the training set in all the cases as shown in Figure 7. eN N -EC trusts more the surrogate in this tournament as shown in Figure 8. Table III gives CR -EC Fig. 6. Percentage of real evaluations along the cycles for the second tournament. as the best method for the third tournament.

The curves of mean percentage of real evaluations over the cycles are the same in all three tournaments for CR-EC. This observation indicates a convergence toward regions containing the best known solutions.

VI. CONCLUSION

The new confidence regions adaptive evolution control (CR -EC) proposed in this paper relies on a distance-based heuristic evolution control at an individual level during the evolution. If a candidate solution resides nearby a solution already evaluated and that has been valued to develop the surrogate then this latter is called, otherwise a complete simulation has to be launch. The preliminary experiments led in this study are set up in a way that imitates the processing of a complex engineering problem with a small real evaluations budget.

CR -EC appears to be a promising surrogate-assisted MO method according to the preliminary results. However, further experiments have to be carried out, in particular, to analyze the dependence between the training quality and the value of the hyperspheres radius. It is also planed to use GPU-based parallel computing to improve training quality with ANN in a reasonable time. An engineering problem as well as other benchmark problems have to be considered and other EAs and MMs will be investigated to comparison.

Fig. 1 .

 1 Fig. 1. Cycle of generations. B Q n -EC on the left and e N N -EC on the right.

Fig. 2 .

 2 Fig. 2. Cycle of evaluations in CR -EC. The first cycle, on the left, is realized with real fitness function only. Once the surrogate is built, cycles are realized with both real fitness and surrogate function depending on the confidence region criterion.

Fig. 3 .

 3 Fig. 3. Hypervolume indicator.

Fig. 5 .

 5 Fig. 5. Mean percentage of real evaluations along the cycles for the first tournament.

Fig. 8 .

 8 Fig. 8. Mean percentage of real evaluations along the cycles for the third tournament.

TABLE I RESULTS

 I OF THE FIRST TOURNAMENT WITH NSGA-II ALONE (ON THE LEFT) AND WITHOUT NSGA-II ALONE (ON THE RIGHT).

	Strategy NSGA-II NSGA-II/B Q n-EC NSGA-II/eNN-EC NSGA-II/CR-EC	Grade 46/50 0/50 1/50 3/50	Strategy NSGA-II/B Q n-EC NSGA-II/eNN-EC NSGA-II/CR-EC	Grade 1/50 22/50 27/50
	Fig. 4. Average training quality for the first tournament.

nation R 2 [21] computed over the training set: R 2 (y, y pred ) = 1 -(y -y pred ) 2 (y -E(y)) 2

TABLE II RESULTS

 II OF THE SECOND TOURNAMENT WITH NSGA-II ALONE (ON THE LEFT) AND WITHOUT NSGA-II ALONE (ON THE RIGHT).

	Strategy NSGA-II NSGA-II/B Q n-EC NSGA-II/eNN-EC NSGA-II/CR-EC	Grade 13/50 7/50 19/50 11/50	Strategy NSGA-II/B Q n-EC NSGA-II/eNN-EC NSGA-II/CR-EC	Grade 11/50 21/50 18/50

TABLE III RESULTS

 III OF THE THIRD TOURNAMENT WITH NSGA-II ALONE (ON THE LEFT) AND WITHOUT NSGA-II ALONE (ON THE RIGHT).

	Strategy NSGA-II NSGA-II/B Q n-EC NSGA-II/eNN-EC NSGA-II/CR-EC	Grade 14/50 14/50 6/50 16/50	Strategy NSGA-II/B Q n-EC NSGA-II/eNN-EC NSGA-II/CR-EC	Grade 19/50 7/50 24/50

Fig. 7. Average training quality for the third tournament.


Lieven Meerpoel

Maria-Miranda Vrahami

Brigitte Deguin

Pierre Vogel

ene Derivatives

Face Selectivity of the Diels-Alder Additions and

Cheletropic Additions of Sulfur Dioxide to 2-Vinyl-7-oxabicyclo[2.2.1]hept-2-


Delphine Lejri
email: delphine.lejri@entpe.fr

Cécile Becarie
email: cecile.becarie@entpe.fr

Xavier Olny
email: xavier.olny@developpement-durable.gouv.fr

Cécile Bécarie

Sound exposure of a pedestrian going through an urban boulevard

Keywords: noise exposure, pedestrian, annoyance, health risks, dynamic noise assessment, urban environment, traffic dynamics exposition au bruit, piéton, gêne, évaluation dynamique, milieu urbain

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Résumé

L'exposition au bruit a une incidence sur la santé des individus, telles que le démontrent les études épidémiologistes. En milieu urbain, la situation est d'autant plus critique que la dynamique du trafic routier y impose de fortes variations des niveaux sonores auxquelles nous sommes sensibles et des niveaux sonores élevés pour lesquels nous cumulons les effets négatifs au cours de la journée. D'autre part, les études d'impact réglementaires cherchent à estimer les niveaux sonores en façade des bâtiments afin de limiter l'exposition de leurs occupants, mais ne se préoccupent pas de l'exposition des piétons. La méthode décrite dans cet article permet de déterminer la dose de bruit perçu par un piéton parcourant un boulevard urbain à partir d'une simulation dynamique couplée trafic -bruit, en s'appuyant sur un maillage de récepteurs prédéfini. Les résultats analysés nous permettent de caractériser l'exposition des piétons en fonction de leur trajet à l'aide d'indicateurs traditionnels et dynamiques.

Introduction

Noise exposure

Noise exposure may constitute a health risk as for instance hypertension or heart diseases. These health effects have been examined in epidemiologic studies [START_REF] Passchier | Noise exposure and public health[END_REF]. The exposure settings are quite diverse, but a common environmental noise source is traffic. This is the reason why annoyance due to traffic noise is one of the major concerns of French people when it comes to their quality of life (INSEE, 2002). Moreover the health risk depends not only on sound pressure levels but also on the duration of the exposure. Indeed, we do not get used to noise but cumulate its effects. Thus annoyance is related to sound pressure levels, but also to the cumulated noise exposure. Furthermore, noise with highly variable levels increases the negative impacts. In urban areas, such situations are common due to traffic dynamics.

Proof of concept

The relation between traffic dynamics and dynamical noise levels was studied in [START_REF] Leclercq | Dynamic prediction of urban traffic noise: a real case study[END_REF]. The modelling chain coupling (i) a dynamic traffic simulator Symuvia 2 , (ii) noise emission laws [START_REF] Lelong | Dynamic evaluation of the noise emitted by a bimodal (passenger cars/buses) traffic flow[END_REF] and (iii) the geometrical propagation model NMPB-routes-96 implemented in (Mithra, 2002), has been validated by a measure campaign in an urban boulevard (Lafayette, Lyon, France). Noise levels in buildings' frontage, as required by noise regulations for day and night periods, were dynamically assessed at each time step (1s). Nevertheless, these indicators are describing the situation at one place. The focus of the study was to know if the same simulation could also describe the exposure of a mobile receptor. Indeed, as we said before, pedestrians in urban areas are quite exposed to annoyance due to noise. We will present the method proposed to assess the whole exposure of the pedestrian during his route, from a predefined set of receivers and the calculation of the dynamic noise levels at such locations (the time evolution of the short-term sound level L Aeq,1s ). One can assume that the sound exposure of the pedestrian can be assessed using the sound level of the nearest receptor. For this, at each time step t i (usually 1s), the following steps have to be achieved: (i) Searching the nearest receiver from pedestrian's position X i using a nearest neighbour algorithm: R Xi, (ii) Extracting the sound level L Aeq,1s (R Xi , t i ), (iii)Assigning this sound level to the pedestrian exposure. Then we will show the results in the urban boulevard we studied.

Method

Nearest neighbor and sound level associated

As exposed in the introduction, in order to assess the pedestrian exposure, we have to lean on the information we have, which is the dynamic noise level (L Aeq,1s at each second) in buildings' frontage. In operational conditions, these receivers are placed 2 meters in front of each floor of the buildings, as imposed by French acoustic regulation. Consequently we make the following approximation: at every time step, the pedestrian is exposed to the sound level of the nearest receiver. The first step of the method consists in looking for the nearest neighbor of the pedestrian's position. For this search, we used a nearest neighbor algorithm developed by [START_REF] Cao | Practical method for determining the minimum embedding dimension of a scalar time series[END_REF], which gives good results in very short time.

We have then to pick up the appropriate sound level in the simulation results, in parsing the xml file, which contains at each second the sound level at each receiver.

Pedestrian exposure

Pedestrian exposure is defined as the sum of the levels to which the pedestrians are exposed through their route. Noise dose D was then computed using the following formula:

D(T) =10 log [(1/T) T n 10 (Ln/10) ]

(1) with T : total duration of the route (s)

T n : duration of the period n (in our case each period is 1s long) L n : L Aeq of the period n (dB(A))

Case study

Dynamic noise simulation of the urban boulevard

The study concerns a section of an urban boulevard with six traffic lights and some bus lanes in both directions.

The coupled simulation was produced in (Can and al., 2008a). This site has been described by 335 acoustic sources and 731 receivers (ground floor and in front of each building), as shown in fig. 1(b).

Pedestrian trajectory

In this study, the pedestrian trajectory was quite simply simulated with the traffic model, as if it were a vehicle. However we had some constraints: (i) the mean speed is 1m/s (ii) the pedestrian walks on the pavement observing the traffic lights. As for the vehicles, the pedestrian position is estimated each second. His route duration is approximately 10 minutes from one side of the zone to another. Different trips were simulated (same route; different beginning times). However, the sources do not change much, therefore pedestrian exposure during a mean time route is quite the same. This would probably not be the case in a larger zone, where various circulation conditions could appear.

Results

The results we obtained when we exploited the simulation on the urban boulevard allow us (i) to analyze the results, and (ii) to see the feasibility and the limits of this method.

Fixed vs. mobile receiver

We first analyzed the results comparing the exposure of a pedestrian walking through the simulated zone and the exposure at one fixed point. We compared this exposure to two different receivers: (i) receiver n°495 situated near a bus station and (ii) receiver n°700 in a quieter zone, during the same period T. In each case, the noise levels are contained between 55 and 85dB(A), with numerous emergences due to the important traffic and the geometry of the site. However the distributions of L Aeq,1s already announce that the nature of the exposure is different.

The other indicators used to compare these three sound exposures are explicitly given in table 1. The traditional indicators (equivalent noise level and statistical indicators) seem to show that the mobile receiver is a kind of mean situation between the two contrasted situations (R495 and R700). The interpretation is somewhat different if we look at the dynamical noise indicators that were defined in [START_REF] Can | Capturing urban traffic noise dynamics through relevant descriptors[END_REF]. The quietness and loudness are especially adapted in urban noise characterized by the rhythm of the traffic lights cycles. These criteria conduct us to qualify differently the pedestrian exposure D(T): during 4% of the route time the pedestrian is confronted to "calm" situations, which is not the case in a fixed point in the site.

Limits

After having analyzed the results obtained for various pedestrian's routes, we can conclude that the main sources can be identified. For instance, the emergences that result from the buses passing near the pedestrian or across the street are well reproduced.

However we can admit that the simulation could be improved. The number of receivers can probably be augmented without affecting the time calculation. Indeed in our case study, the distance between the pedestrian's position and the nearest receiver can be at most 20 meters. This is particularly due to the fact that there is (i) only one receiver for each building, no matter how big it is and (ii) no receiver in the crossroads. This lack of receivers can produce some anomaly: the presence of two emergences for only one bus passing near the pedestrian crossing the street.

Conclusion

The method described in this article gives the opportunity to assess the pedestrian exposure from a coupled dynamic simulation « traffic-noise » determined in only some locations in the site (positioned in regard to the French acoustic regulation).

We showed that it is worth looking at the pedestrian exposure because it has other characteristics than exposure in a fixed point. We also formulated some recommandations on the distance between the receivers (10m seems to be a good distance), which supposes that some receiver has to be added especially in crossroads. Another parameter could be tested: we could change the height of the receiver to see if a child is exposed to the same levels as an adult. This method will be further applied to a larger zone, an urban quarter, with more contrasted traffic conditions and a measure campaign.

Fig. 1

 1 Fig. 1 Dynamic coupled simulation: (a) the traffic simulation in Symuvia (b) sources and receivers in Mithra

Fig. 2

 2 Fig. 2 Comparison between a pedestrian, a fixed point near a bus station and a quieter place (a) distribution of noise levels (b) noise levels time series

Fig. 3

 3 Fig. 3 (a) Position of the receivers (b) Example of selected receivers

Table 1 .

 1 Comparison of the pedestrian exposure PE and the receivers sound levels

		L Aeq,T	L1	L90	Quietness 3 (%)	Loudness 4 (%)
	D(T)	75	84	61	4	51
	R495	77	86	68	0	71
	R700	73	80	62	1	35

The quietness indicator is defined as the percentage of period of

seconds where the noise level is less than 60dB(A)4 The loudness indicator is defined as the percentage of period of 4 seconds where the noise level is more than 70dB(A)