J Eleutério

D Martinez

A Rozan

Developing a GIS tool to assess potential damage of future floods

Keywords: flood damage evaluation, GIS model, risk analysis, vulnerability assessment, natural hazard, decision support tool

The evaluation of potential damage of future floods is an essential part of flood management project appraisals. Analysis results reliability is an important issue when comparing flood risk reduction project scenarios. Geographic Information System (GIS) technology plays a crucial role on flood risk analyses. On one hand, the evaluation process requires data on flood hazard and on vulnerability of assets at risk, both spatial data. On the other hand, this data must be combined in order to evaluate flood risk. Even though the role of GIS is central in the evaluation process, GIS software does not offer specific tools for achieving flood damage analysis. The use of standard methods for assessing and combining different data influences accuracy and comparability of evaluation results. Few countries in the world have developed national standard methods to assess flood damage potential. The construction of GIS-based methodologies and GIS models can be the first step toward standardisation of the whole evaluation process. In this purpose, this article presents the development of a GIS tool for evaluating future flood damage potential: F.R.A. GIS tool extension, for use with ArcGIS (ESRI). It first explains how GIS technology is used in flood risk analysis procedures. Then, the tool pre-and post-functions as well as structure of the model are detailed. Finally, a brief case study is presented in order to illustrate the functionalities of the model developed in this article. The methodology described here can be used to guide analysts on the realisation or the automation of flood risk analysis using other GIS software.

Introduction

Flood risk brings different aspects together, e.g. human, social, economic and environmental (Messner and Meyer [START_REF] Messner | Flood damage, vulnerability and risk perception -Challenges for flood damage research[END_REF]). The general purpose of different risk analyses is to understand and/or measure the possible consequences associated with the occurrence of flooding in areas occupied by vulnerable systems. This article focuses on the economic aspect of the risk. The evaluation of potential damage of future floods is an essential part of flood management project appraisals, getting more importance over time (Penning-Rowsell and Green [START_REF] Penning-Rowsell | New Insights into the Appraisal of Flood-Alleviation Benefits: (1) Flood Damage and Flood Loss Information[END_REF]), e.g. the European Water Framework Directive (EWFD) 2007/60/EC determines that flood management projects must taken into consideration cost-benefit analysis principles. The damage evaluation results reliability is an important issue when comparing flood risk reduction project scenarios, potentially influencing decision making process. Several studies have been developed to improve damage evaluation (Messner et al [START_REF] Messner | Evaluating flood damages: guidance and recommendations on principles and methods[END_REF], Penning-Rowsell et al [START_REF] Penning-Rowsell | The benefits of flood and coastal risk management: a manual of assessment techniques[END_REF]).

Even though the role of GIS is central in the evaluation process, GIS software does not offer specific tools for achieving flood damage analysis e.g. ArcGIS system produced by Environmental Systems Research Institute (ESRI). The use of standard methods for assessing and combining different data contributes to the improvement of the methodology and influences accuracy and comparability of evaluation results. Few countries in the world have developed national standard methods to assess flood damage potential (Dutta et al [START_REF] Dutta | Direct Flood Damage Modeling Towards Urban Flood Risk Management[END_REF]). The objectives of the EWFD 2007/60/EC incite the development of national and/or international standard methods. The construction of GIS-based methodologies and GIS models is an important step toward the standardisation of the whole evaluation process (Yi et al [START_REF] Yi | GIS-based distributed technique for assessing economic loss from flood damage: pre-feasibility study for the Anyang Stream Basin in Korea[END_REF]). In this purpose, this article presents the development of a GIS tool for evaluating future flood damage potential: F.R.A. GIS tool extension, for use with ArcGIS (ESRI).

GIS and flood risk analyses

Geographic Information System (GIS) technology plays a crucial role on flood damage evaluation process. Flood damage assessment is based on different data: flood hazard data i.e. flood parameters spatial distribution, exceedence probability; and assets vulnerability data i.e. assets exposure, susceptibility to suffer damage, and damaging potential (Penning-Rowsell et al [START_REF] Penning-Rowsell | The benefits of flood and coastal risk management: a manual of assessment techniques[END_REF]).

In a first moment, we have to assess these different datasets. Typical GIS applications include analyses of Digital Elevation Models (DEM) over which several hydrological and hydraulic analyses can be realised (Moglen and Maidment [7]). Specific tools have been developed to explore GIS spatial data processing functionalities for use as data pre-and post-processing for hydrodynamic models like Hec-RAS, Mike Flood and Hydrarive e.g. Hec-GeoRas (Ackerman et al [START_REF] Ackerman | Hec-GeoRAS Linking GIS to Hydraulic Analysis Using Arc-Info and Hec-RAS[END_REF]) and Mike11 GIS, both developed for use within ArcGIS software; and HydraGIS for use within MapInfo GIS software. These different software items have similar functionalities. They are generally used to: prepare input data inside the GIS structure; export data according to requirements of hydrodynamic numerical models; and to process results issues from hydrodynamic models for generating flood hazard distribution maps. The use of these tools provides more efficiency for the analyst, reducing technical efforts and optimising the analyses in terms of analysis possibilities. Several hydrodynamic models can be used in practice to simulate floods (Stelling and Verwey [START_REF] Stelling | Numerical flood simulation[END_REF]), but by the end of the process, GIS software or integrated interfaces are generally used to represent and analyse the results of these models. Different studies have been carried out in order to link the hydraulic/hydrologic models to GIS e.g. Renyi and Nan [START_REF] Renyi | Flood area and damage estimation in Zhejiang, China[END_REF] developed linkage between HecRAS model and GIS equivalent to Hec-GeoRAS. Other studies have been carried out in order to simulate different flood characteristics through numerical schemes inside GIS software (Tsanis and Boyle [START_REF] Tsanis | A 2D hydrodynamic/pollutant transport GIS model[END_REF], Dutta et al [START_REF] Dutta | Direct Flood Damage Modeling Towards Urban Flood Risk Management[END_REF]).

GIS technology is also largely used to store and analyse stake data. Several land-use datasets have been developed for different purposes and by different agents e.g. CORINE Land Cover, BD Topo from the French National Institute of Geography (IGN) and local GIS datasets. Vulnerability data refers specifically to information concerning the assets and their susceptibility to flood water. Generally, stake vulnerability data is explored in basis of land-use datasets with different levels of detail, varying from the description of units (Gilard [12], Oliveri and Santoro [START_REF] Oliveri | Estimation of urban structural flood damages: the case study of Palermo[END_REF], Edlenbruch et al [START_REF] Edlenbruch | Etude socio-économique des inondations sur le bassin versant de l'Orb[END_REF]) to large homogeneous areas (Simpson and Human [START_REF] Simpson | Large-scale vulnerability assessments for natural hazards[END_REF], Van der Veen and Logtmeijer [START_REF] Van Der Veen | Economic Hotspots: Visualizing Vulnerability to Flooding[END_REF], Dutta et al [START_REF] Dutta | A mathematical model for flood loss estimation[END_REF]).

In a second part of the evaluation process, data on hazard and vulnerability must be combined in order to calculate damage potential and support flood management. GIS technology is crucial in this step once spatial data should be combined. Basic GIS functions provide the user the possibility to realise this step. However, high levels of professional skills are required to achieve them. Generally, the evaluation of flood damage relays on the use of damage-functions to represent damaging potential for assets at risk (Penning-Rowsell et al [START_REF] Penning-Rowsell | The benefits of flood and coastal risk management: a manual of assessment techniques[END_REF]). These damage-functions relate monetary damage potential with vulnerability and hazard characteristics, which are in the evaluation process assessed by combining previous datasets. Depending on the types of damage-function, they are used to represent damage potential for homogeneous land-uses or individual elements at risk. Few models have been developed for achieving combination of spatial data and calculation of damage potential in GIS software (Yi et al [START_REF] Yi | GIS-based distributed technique for assessing economic loss from flood damage: pre-feasibility study for the Anyang Stream Basin in Korea[END_REF], Dutta et al [START_REF] Gilard | Les bases techniques de la méthode inondabilité[END_REF], Betts [START_REF] Betts | Flood damage analysis using GIS at Gold Coast City Council[END_REF]). One of the challenges in research and practical fields is the development of methods and tools for achieving flood risk evaluations and generate flood risk maps, in order to support decision-making process (De Moel et al [START_REF] Zerger | Beyond Modelling: Linking Models with GIS for Flood Risk Management[END_REF], Zerger and Wealands [START_REF] Zerger | Beyond Modelling: Linking Models with GIS for Flood Risk Management[END_REF]).

Flood Risk Analysis GIS extension

The F.R.A. GIS tool extension is a combination of GIS functions and tools developed using Visual Basic computational language and Visual Basic Applications (VBA) for use with ArcGIS system (ESRI). The tool enhances GIS classic applications with the purpose of realising flood risk analyses through the combination of data related to flood hazard and assets vulnerability. Input data includes flood maps and vulnerability maps in GIS layer format, and damage-functions. The method developed to analyse flood risk places the elements or assets at risk in the centre of the evaluation so that the assets are analysed individually, following the principles of the "unit damage evaluation" method (Penning-Rowsell et al [START_REF] Penning-Rowsell | The benefits of flood and coastal risk management: a manual of assessment techniques[END_REF]).

Method background

The general equations used in the model are: ).

) ( , ) ( ) , ( i F A f i A y x P E FP = ( 1)
where A FP is the intensity of flood parameters inside/over each asset for specific flood annual exceedence probability (i); A E is the exposure of the asset represented by its location (x, y, z) during the flood event; F P(x,y) is the maximum intensity of flood parameter(s) for a given flood event with specific (i).

(

) .

, ), ( ) (

DP SSD FP DP A A i A f i A = ( 2)
where A DP (i) is the asset damage potential related to a specific flood annual exceedence probability (i); A SSD is the asset susceptibility to suffer damage; A DP is the asset damaging potential represented by damage-functions. . ) (

1 0 ∫ × = i i A A DP EAD ( 3)
where A EAD is the asset expected annual damage (or average annual cost) caused by floods. The EAD value is the assets damage potential average considering flood events with annual exceedence probability varying from 0 to 1.

General structure and interface

The algorithm of the tool is composed of independent functions and modules: pre-processing functions; flooding potential module; damage potential module; expected annual damage module; and post-processing functions (Fig. 1).

The model general equations; eqns. ( 1), ( 2) and ( 3) are individually solved using the different tool modules. The resolution of these equations depends of the datasets and parameters defined by the user. The pre-processing functions facilitate the import of input data in the model. Other equation variables as well as solution hypotheses can be entered by the user through the tool interface in ArcMAP platteform (Fig. 1).

Figure 1:

Tool structure.

The interface of the model consists of a toolbar composed of different menus and shortcuts that allow the user to access windows with the pre-and post-processing functions, the model parameters and the model execution setup (Fig. 2). This organisation provides flexibility in the analysis procedure. Input datasets are entered in the model individually and the different calculation modules can be executed together or individually.

Input data and pre-processing functions

Data input requirements vary for the different evaluation modules. Basic data input requirement includes: -For assets flooding potential module: assets vulnerability maps (vector data) including at least spatial localisation of assets (x, y, z coordinates, where z is the relative position in relation to the digital elevation model used to construct flood maps); flood hazard maps (raster or vectors formats) including at least water-depth spatial distribution (x, y coordinates); -For assets potential damage module: previous datasets; assets vulnerability Additional data such as other flood parameters, e.g. flow velocity, duration of submersion, or vulnerability parameters influencing damages, e.g. assets value, assets surface, can be also incorporated in the analysis.

The tool pre-processing functions allow the user to easily process input data in order to respond to format requirements of the model. Another functionality of the pre-processing functions is that unit transformations and layer conversions can be realised without modifying the original data (new layers are generated for data processing). One example of pre-processing function interface is displayed in figure 3. This interface is used to add damage-functions in the model project. A table containing all the variables for the different damage-functions entered is generated and stored in the project output data folder.

Model parameters and execution options

Before running the model, the user must specify the characteristics of the scenario(s) of evaluation. The model parameters interface (Fig. 4, left) provides the user the possibility to enter the number of scenarios as well as the characteristics of the assets layers and flood layers which will be used in the model.

When pressing the button ADD the architecture of the analysis is generated and the different input layers are added to the model project. Finally the model execution setup must be realised in order to specify which modules will be executed and to define all hypotheses for solving the model basic equations eqns. ( 1), ( 2) and (3) related to the different modules (Fig. 1). The model execution interface is used for this purpose (Fig. 4, right). The asset layers are individually analysed. All scenarios can be analysed simultaneously or individually as well as the different modules, according to user specifications. The users can also choose to change the standard module parameters in order to adapt the model to their input data and study site characteristics. When choosing to calculate expected annual damages, the user must enter the return period of the first event for which damage occurs in the study site. The other parameters are explained in next session.

Running the model and solving the basic equations

The scheme (Fig. 5) displays how input datasets are processed to solve the model basic equations eqns. ( 1), ( 2) and (3) according to the 3 different modules respectively: 1) Assets Flooding Potential; 2) Assets Damage Potential; and 3) Expected Annual Damages.

Figure 5:

Tool resolution scheme.

The first module solves eqn. ( 1) through the combination of flood parameters with assets exposure characteristics (Fig. 5). The model routine combines GIS functions (layers over-position and table jointures) in order to import information from flood hazard layers to the assets layer, taking into account the spatial localisation of the elements at risk. The coordinates (x and y coordinates) of both are combined and water-depth is subtracted by the elevation (z coordinate) of the asset in order to determine the water depth inside/over the elements at risk. The model interface (Fig. 4, right) provides to the user the possibility to choose between different approaches to combine data.

The second module solves eqn. (2) through the application of damage-functions (representing assets damaging potential). The asset susceptibility to suffer damages is represented by a vulnerability index, which must correspond to the identification of an existing damage-function. The model routine links the assets to the damage-functions using table jointures. In the damage-functions, damage can be dependent or independent of characteristics of the asset. In the case damage is linked to one characteristic of the asset, e.g. proportional to asset price, the correlation must be indicated in the parameter definition, and the characteristic of the asset must be described in the assets layer. Assets damage is finally calculated using the flooding parameters inside/over the elements at risk provided by the previous module and vulnerability input data (Fig. 5). The model interface (Fig. 4, right) provides to the user the possibility to choose between different methods to calculate damages.

For solving eqn (3), at least 3 flood return periods must be used to achieve the previous calculations (Fig. 5). The module 3 solves this equation using data concerning the flood events return periods and the assets damage potential previously calculated. The equation 3 is solved by means of linear regression of the known values. Two values must be entered by the user in order to do this calculation (Fig. 4, right): first damaging event return period and damage value for the null exceedence probability flood event i.e. theoretical infinity return period event. This last information must be entered in a factor format, representing how many times the null exceedence probability damage is superior to the more elevated flood return period calculated damage.

Output data and post-processing functions

By running the GIS tool with appropriate input datasets, the analyst is able to: estimate intensity of flood parameters inside/over assets at risk for different flood events; estimate assets damage potential related to different flood events; calculate expected annual damages for the assets; realise and compare several scenarios of evaluation; produce flood risk maps; and analyse damage and expected annual damage spatial distribution All data calculation is stored in the assets layers, which allows the user to generate risk maps by means of simple GIS functions. Displaying options can also be changed by using post-processing functions. These functions were elaborated to provide quick analysis of overall damage. The computational routines also create text files containing a resume of total damage by summing up unit damage for different return period events and summing up unit expected annual damages. The global results of different scenarios can be easily compared in the context of uncertainty or cost-benefit analysis.

Case study

The case study presented here illustrates the functionality of the model. We apply the F.R.A. model to calculate buildings potential damage and estimate expected annual damages in an urban community in eastern France. Assets vulnerability map was generated using aerial photography observation and site survey results. This map represented: building exposure by their geo-referenced contours and their ground-floor height in relation to the natural terrain; building susceptibility to suffer damage by vulnerability indexes created in function of building occupation type and construction characteristics; and building damaging potential by 17 damagefunctions -2 for residential buildings (Torterotot [START_REF] Torterotot | Le coût des dommages dus aux inondations: Estimation et analyse des incertitudes[END_REF]) and 15 for public and commercial buildings (DNRM [START_REF] Dnrm | Guidance on the assessment of tangible flood damages[END_REF]).

Flooding water depth maps have been generated through simulations using the 1D/2D hydrodynamic Mike Flood software. We simulated floods for 10, 30 and 100 years return period floods. The maps were generated in raster formats with pixel resolution of 20x20 meters. Figure 6 provides an overview of raw data for flood and vulnerability layers. Case study raw data.

The 17 damage-functions were entered in the tool by means of the specific pre-processing function (Fig. 3). The ID for each damage-function corresponded to one specific asset vulnerability index. We used hazard preprocessing functions to transform the flood water depth raster layers into vector layers and to enter the information about the return period of the different flood events. Vulnerability pre-processing functions were used to create a new vulnerability layer (copying the information contained in the raw layer, changing the name of fields and the measure units, responding to the model input requirements).

Through the model parameters interface, the number of scenarios, number and type of vulnerability layers, and the number and parameters for the flood events were defined (Fig. 4, left). Closed surfaces option represents polygonal vector layers, which correspond to the building contours input data. In the model execution interface (Fig. 4, right), all modules were selected and the different required parameters entered: the maximum combination approach was used to estimate assets flooding potential; simple calculation method using only flood water depth parameter was used for calculating damages (the damage-functions used are dependent of only water depth parameter); 5 years return period flood was considered as first damaging flood event and asset damage for the null probability flood event was estimated 1.5 times flood damage caused by the 100 years return period flood event.

Then, the three general equations eqns. ( 1), ( 2) and (3) were solved when we executed the model, determining respectively the assets flooding potential, damage potential and expected annual damage. Figure 7 displays the graphic results for assets expected annual damage. Total expected annual damage were calculated at 350,000 € per year for the community evaluated. The bigger the damage is for a specific asset, the darker the colour of red in the figure above (Fig. 7). The expected average annual damage for individual assets vary from 0 (non exposed assets) to 20,000 € per year (for high vulnerable buildings in the area). Several results could be explored e.g. number of buildings impacted, distribution of damage per event return period, level of flooding water inside buildings. These simple examples of results represent some types of analysis we can easily realise using post-processing functions.

Conclusions

This article presented the development of the first version of a GIS tool for evaluating future flood damage potential: F.R.A. GIS tool extension, for use with ArcGIS 9.2. (ESRI). It explained in detail the structure of the model as well as its main functionalities including: input data pre-processing functions; assets flooding potential estimation module; assets damage potential estimation module; expected annual damage calculation module; and results analysis and interpretation functions. We also explained how GIS technology is used in flood risk analysis procedures. The brief case study presented here illustrated the functionalities of the model developed. The main gains enhanced by this model are: the standardisation of a general method to evaluate future floods potential damage; the generation of a friendly interface which allow the user to easily realise the analysis without depending on great background on GIS; the possibility of comparing different scenarios of evaluation in project appraisal or uncertainty analysis contexts. The utilisation of this model by stakeholders should bring great improvement on the evaluation process, and it could also provide feedback in order to improve the actual existing functionalities. The methodology described here can be used to guide analysts on the realisation or the automation of flood risk analysis using other GIS software.

The authors thank to Maif Foundation for financial support of this research.

(

Figure 2 :

 2 Figure 2: Model interface toolbar.

F

  least vulnerability index (expressing asset susceptibility to suffer damages); damagefunctions for the different vulnerability indexes; -For expected annual damage module: previous datasets including at least three hazard maps for events with different return periods; information concerning return periods and first damaging flood return periods.

Figure 3 :

 3 Figure 3:Pre-processing interface for damage-functions.

Figure 4 :

 4 Figure 4: Model parameters interface (left) and model execution interface (right).

Figure 6 :

 6 Figure 6:Case study raw data.

Figure 7 :

 7 Figure 7:

Risk Analysis 2010, 7th International Conference on Computer Simulation of Risk Analysis and Hazard Mitigation, Algarve Portugal, 13 -15 September 2010


C Werey
email: caty.werey@engees.unistra.fr

Z Larabi

A Rozan

DIME Workshop :Environmental innovation in Infrastructure sectors

Introduction

Rehabilitation of sewer networks has to serve various objectives, both in terms of present general performance, and in terms of long term sustainability. Part of these stakes are external to the sewerage utility operation, such as social impacts of sewer failures, or social impacts of works. These external impacts are to some extend considered in real life decisions, and should therefore also be taken into account in decision support systems. CARE-S project (2003-05 EU 5 th Framework Program), "Computer Aided Rehabilitation of Sewer systems", has developed a decision support environment and software prototype, including data management procedures, a set of performance indicators, models describing the condition and evolution of sewers, models describing the risks and consequences of failures, a database describing rehabilitation technologies with their characteristics, an assessment of socio-economic criteria, a long term planning tool, multi-criteria decision tools. The objective is to help utility managers to rehabilitate the right pipe, at the right time, with the right technology. Criteria concern on one hand impacts during rehabilitation works, criteria used for the selection of the best technique for rehabilitating the concerned pipe in a multi-criteria decision making procedure, on the other hand impacts during a failure, criteria used for the selection of pipes candidates for rehabilitation. We worked out criteria such as time loss due to traffic deviation, economic loss for trades due to accessibility problems during works but also criteria concerning ground water or surface water pollution damages risk during overflow or due to exfiltrations from the pipes. Concerning floods damages in cellars during overflow, valuation has taking into account methods used for river flooding damages concerning material goods [START_REF] Werey | Rehabilitation impacts on socio-economic costs[END_REF] A second project INDIGAU " Performance Indicators for urban sewer networks asset management" (financed by the ANR, French Research Agency, 2007-10) goes further on with the valuation of the impact criteria using methods coming from the environmental economics such as "Willingness to pay "or contingent ranking recently used for giving a value to intangible damages or impacts such as odours problems or loss of "sentimental affect" goods such as photographs during floods due to overflows of the networks [START_REF] Larabi | Evaluation des dommages intangibles liés au dysfonctionnements des réseaux d'assainissement mémoire de Master2 Expertise Economique[END_REF].

In our paper we will first present the impacts of failures or works identified in sewers management, the we will present the economic valuation methods and finally present two examples of criteria valuation: on one hand the valuation of loss of trade due to works using compensation data, belonging to indirect valuation methods, and on the other hand the valuation of dysfunction impacts on intangible goods using a "stated choice preferences" direct method.

-sewer networks management: how taking into account environmental and socio-economic impacts

1.1 Impacts definition CARE-S project has developed a decision support environment and software prototype, including data management procedures, a set of performance indicators, models describing the condition and evolution of sewers, models describing the risks and consequences of failures, a database describing rehabilitation technologies with their characteristics, an assessment of socio-economic criteria, a long term planning tool, multi-criteria decision tools. The motto of the project was " how to rehabilitate the right pipe, at the right time, with the right technology".

So socio-economic criteria were developed [START_REF] Werey | Rehabilitation impacts on socio-economic costs[END_REF] to take into account the impacts due to failures or to works on the network, this("indirect", or "external") costs have then be evaluated to feed the multi-criteria tools relying on both technical and socio-economic criteria: -decision criteria for comparing a limited set of technologies when considering a given single pipe (impacts of rehabilitation works), -decision criteria for comparing various rehabilitation projects, defined each at pipe level, given the related failures or failures hazards (impacts of failures).

So two set of impacts are to be taken into account:

Impacts of failures

10 criteria have been worked out concerning failures: 1: flooding damage to buildings and their contents, and business losses 2: flooding intangible damage 3: traffic disruption and trouble due to flooding 4: degradation of surface water quality and consequences on present or future water uses 5: degradation of groundwater quality and consequences on present or future water uses 6: waste water service interruption 7: traffic disruption and trouble, annoyances to life quality, due to soil depression 8: annoyances and damage from dry weather flooding in basements 9: annoyances and damage from dry weather flooding on the street 10: annoyances due to odours, insects, rodents Our definition of failure is inspired by the French AFNOR standard on maintenance terminology NF EN 13306: termination of the ability of a pipe or of a network to perform a required function; a failure is a defect or a performance deficiency and is defined in reference to a required level of performance.

Impacts during rehabilitation works

In this case the following 5 criteria have been worked out: 1: noise 2: dust 3: service interruption 4: traffic disturbance and business losses

Link between dysfunctions and impacts

The failure impacts are linked with one or more different defects and dysfunctions and presented in table 1 Table 1 failures impacts considered in defining decision making criteria for selecting priority pipes for rehabilitation (Werey & al,2005) The impacts have been evaluated taking into account the different probabilities of failures or dysfunction occurrences and the effect of the natural or socio-economic environment of the pipe. The criteria taken into account represent impacts on the natural environment ( groundwater, surface water..) but also to the urban environment (damage to buildings, traffic disruptions…) and to the users disturbance by service interruption or flood damages as to economic activity.

Typologies considered for describing the pipe environment

For describing and possibly quantifying various components of vulnerability, we may consider an endless detailed description of population, buildings, river quality and morphology… Not only is there a limitation in available and manageable information, but moreover the decision maker has to be proposed a "reasonable" trade-off between detailed description and quantification on one hand, time and resources for processing on the other hand. This is true for hydraulic modelling as well as for socio-economic parameters. The paragraphs below show the way we chose to describe various dimensions of the pipe and network environment.

land uses

Dealing with vulnerability leads to taking into account different land use types, as they include for instance different population densities, different commercial activities… A CARE-S common typology corresponding to the typology addressed by standard EN752 for recommending levels of flood protection :

• rural • urban housing • city centre • industrial area • shopping area.
The decision-maker must also be able to define "special areas or buildings", in a case-by-case way, for especially sensitive or special places (hospitals…)..

roads and traffic

Roads and the corresponding traffic are described by several parameters: daily traffic flow: number of vehicles per day number of lanes: absolute number for a street section public transport through 5 classes

2-How making an economic valuation of these impacts

In literature, several papers present works using economic valuation methods coming from environmental economics to evaluate externalities liked with a given environmental problem. Such methods were developed within cost-benefit approaches. In care-s and INDIGAU projects we use multi criteria decision tools, so "monetarising" criteria is not compulsory but we consider that it is a good way to give a value to a criteria, less subjective that a notation method. These methods are generally based on the measurement of the variation of the individuals welfare variation due to a qualitative modification of the environment.

There are two methods developed in environmental economics: indirect methods based on revealed preferences and direct methods dealing with expressed preferences. Works exist making economic valuation of the impact of a pollution but very few concern sewerage networks or equipments. In a general way, effects are similar (water pollutions, air pollution, noise, mortality, morbidity) and approaches used in other contexts ( waste, noise pollution) can be used as example for a monetary valuation of sewer defects impacts.

Indirect methods

Revealed preferences methods suppose "Willingness to pay " of agents can be established by observing their behaviour transactions concerning trade possessions trade goods. The objective is to reconstitute the characteristics of "'unobservable " behaviours using " empiric proofs". They are revealed approaches.

the most common methods are: the travel cost method, the hedonist prices method, the protection cost method and the accounted costs method.

Travel cost method

This method has been developed for giving a value to leisure demand by economic agents (Clarwson& Knecht, 1967).the idea was that the spendings The spendings granted by the households to go to natural, tourist or leisure sites and the entry cost establish a revelation of the value they give to the existence of the site. So this method can give a value of the impact in case of pollution and concerning sewer dysfunction such as pollution discharge to surface receiving water, when it concern surface water having recreation uses.

Hedonist prices methods

The objective of this method was to appreciate the value that individuals grant to the change of a component of the environment, given the localisation of their habitation considering that they would accept to pay their habitation more to avoid environment problems. This method was used to evaluate air pollution and noise effect near airports, and also concerning nuisances due to solid waste treatment plants [START_REF] Chèze | Une méta-analyse des études d'évaluation monétaire par la méthode des prix hedinistes du coût externe des installations de traitement des déchets, économix, working paper[END_REF]. It needs many data so that the other characteristics of habitations are sufficiently represented. Concerning sewer's dysfunctions this method could be a way to evaluate the impact of odours.

Protection costs method

This method bases on the consumer choice theory, and more specifically on the observation of the production function of households. We can calculate, indirectly, the cost of the pollution impact by taking into account the spendings made by households to protect themselves from an environment degradation . This methodology has been used for noises impacts due to airports by evaluating the costs of Double glazing, it has also been applied for the valuation of the cost of the contamination of ground water in Pennsylvania [START_REF] Abdalla | valuing environmental quality changes using averting .expenditures: an application to ground water contamination[END_REF]. Concerning sewer dysfunctions, protection costs due to double glazing could be imagined during works with a long duration.

Accounted costs method

This method relies on the data analysis of compensations that have been given similar cases. For instance concerning the realisation a new tramway lines, prealable compensation procedures exists to take into account the loss of trade for the activities in the street concerned by the new line. We used such data to make a typology of the different activities, we will present his valuation in the third part.

Direct methods

The idea is here to create an hypothetical market to observe directly the value of goods for which there exists no market. The specificity is that the value will be given by the individuals during interviews. We will focus on the two principal methods, "contingent valuation" and "choice modelling" approaches.

Contingent valuation

This method bases on the agents expression of their "Willingness to pay " , using a contingent market, that is to say an hypothetical market were environmental goods can be sold and bought. It is widely used for environmental impacts valuation. It has been used by the American Water Resources Council and is promoted by the water framework directive (2000). The methodology uses a questionnaire making the agent give the value he is ready to pay to avoid or to reduce a given impact. It can give an answer ex-ante to the impact of a environmental policy This methodology has been applied to value benefits of river floods protection ( Deronzier & Chegrani, 2005), also in 2003 concerning annoyances due to a waste water treatment plant ( CMV2, 2003).

Choice modelling

The Choice modelling is one of the stated preferences. The difference comes from the fact that contingent evaluation gives a value at a global level, were as choice modelling decomposed the good to be evaluated in several attributes, an one of this attribute is a cost to pay to reduce the impact. So the agent doesn't have to give the value of his willingness to pay, given the discribed situation as in the precedent method, but to classify different scenario reducing more or less the impact and costing more or less.

This approach has been used by [START_REF] Adamowicz | Combining revealed and stated preference methods for valuing environmental amenities[END_REF] for valuing recreative preferences on two Canadian rivers Alberta. Dachary-Bernard (1994) used the same approach to analyse landscape economical value in the Monts d'Arrée (France).

This methodology has been applied to value intangible damages due to floods or odours coming from dysfunctions on the network, his work is presented in part 4..

-Valuation of the impact of works on loss of trade

Loss of trade means that because of the works, trades suffer from economic losses. This is linked essentially to the duration of the works, the location of the street, its accessibility, the technique put into place…People can decide to delay their purchases another day because the shop is in a works area and there are difficulties to access it by car or by bus for traffic jam reasons, or because the pavement is uneven and it is difficult to walk on it… People can also decide to go to another shop and come back to the "works concerned shop" once the works are finished, or maybe never come back… Goods delivery may also be disturbed. In specific cases, if the goods are not in place at a given time of the day, they will not be sold: newspaper, bread…. Parking possibilities and public transportation facilities will of course have be significant in the consumers' attitude, we will focus here on the question of accessibility of the shops at the street level. In literature, loss of trade is studied as a social cost mainly for situations of service interruptions (essentially concerning water delivery for sensitive activities) or during big public works such as new tramway laying.

methodology

One example of monetary valuation is given by Angot (2001) when using a technique with trench opening on the pavement. takes into account the fact that parking and access to shops can be disturbed. One possibility for valuation is to consider the tax that shops are paying to the municipality to occupy the public area of the pavement in front of their shop or trade. Another way is to take into account the loss for the parking manager, in this case, the cost of losing parking space can be calculated as follows: Other valuation made by [START_REF] Awwarf | Costs of infrastructure failure[END_REF] or CSIRO [START_REF] Speers | Determining customer service levels -Development of a methodology[END_REF] used for water supply disruption take into account the decrease in turnover concerning the economic activity and propose results within a typology of economic sectors. We identified also a method used in France in relation with big urban projects such as tramway works; because of the duration of the works and the area of the working site, works like laying a new tramway line often lead to compensation requests. The loss of trade during works is due essentially to accessibility problems of the shop itself or of the works area. In France, several towns have decided to construct new tramway lines, and have put into place a conciliatory compensation procedure. The compensation risk is taken into account in the project cost valuation. So more and more information is provided to shopkeepers, craftsmen and freelancers concerning their possibility of asking for compensation, which is possible if they are directly concerned by the works. The calculation of the compensation is given in equation 2 and used by chartered accountants:

w w w w - CM ) RTO - (TTO on Compensati SC ∆ ⋅ =
(2) with w period concerned by the works TTO average theoretical turnover during the 2 years before the works for the corresponding period w RTO real turn over during the period w CM gross income ratio for the victim or the national average for the economic sector concerned ∆SC variation of staff costs on period w

The first part represents the loss of contribution margin and the second one the saving in fixed costs concerning the staff. The gross income ratio is calculated on the 2-years-period before works and compared to the value of the national gross income ratio in the same activity sector. Depending on the activity, seasonal factors can be introduced.

In regard to this valuation method,, a survey has been made in Strasbourg for works corresponding to one tramway line. The average duration of these works was 15 months. 93 requests have been examined on which we had the information of the compensation value. We classified them into a 14 activities typology presented in table 2.. In our analysis, we focused on the valuation of 28 of them on which we got detailed information of the calculation steps. With this reduced sample we defined for each activity sector the ratio (average loss of trade/theoretical turn over) on the works period with the loss of trade given by the difference (loss of contribution margin -economy on staff costs), that gives an idea of the relative loss of each activity sector and in this way of the sensitivity of the economic activities. These results are presented in table 3. We can see that the most sensitive activities concern food products, equipment for the persons, restaurant and service station.

On the whole sample we looked at the relative average compensation cost per sector activity, that gives an idea of the scale of the cost for the utility, showing for instance that for a supermarket the ratio previously presented can be low but that the compensation amount will be high because of the size of the activity. The analysis of the global sample is presented on table 4 Here we can see that the activities culture-leisure, equipment for the house, supermarket represent the highest compensations. These results have been integrated in the proposed valuation as following:

LossOfTradeW criteria calculation

For dig techniques In this case the working area is whole length of intervention LossOfTradeW is given in equation 3

dig dig E p r d L eW LossOfTrad ⋅ ⋅ ⋅ ⋅ = (3)
with L length of the works (m) d duration of the works (days) r if "need to later reopening of laterals" or "digging for reconnecting laterals" r=1,2 if no "need to later reopening of laterals" or "digging for reconnecting" laterals r=1 p in case of "digging requirements: pit damage", p=1,1 if no pit damage p = 1 E dig l oss of trade factor presented in table .5

Table 5: E dig loss of trade factor for dig technique [START_REF] Werey | Rehabilitation impacts on socio-economic costs[END_REF] Noise Concerning this impact in the case of failure, the valuation is included in the criteria 1: "flooding damage to buildings and their contents, and business losses" and relies of results coming from river flooding impacts valuations linked with the characteristics of the buildings.

-valuation of the impact of dysfunction on intangible goods

The function of a sewer network is to collect , waste water and/or rain water and to forward them to the treatment plant. Networks are generally dimensioned for collecting rain water up to events of 10 years frequency. So the risk of flooding exist when rain events are more important. Our purpose are flooding events due to dysfunction of he sewer system.

Causes and consequences of flooding

Damages can concern different goods: tangible that is to say material goods on one hand, theses are goods whose monetary value is established by the insurance companies, intangible goods on the other hand which are hard to value. Intangible goods gather for instance:

degradation of personal effects with high affective value such as photographs stress due to the fair of a new flood to come, much more existing in relation with river floods odours coming from the sewers… ...

Objectives

Intangible goods lead to subjective costs for individuals. In order to take into account the welfare of individuals, we wanted to evaluate the amounts people would be ready to pay to avoid these damages.

The method we used is choice modelling because essentially of the decomposition in attributes which is interesting here, because we have several intangibles. The method relies on the fact that by presenting several combination of these attributes, individuals, here sewer consumers, are invited to reveal there preferences.

methodology

We defined our attributes and their levels and built so fictive amelioration programs. Attributes and levels are given in the following table 7.

analysis of the survey data

The analysis was made with a Random Utility Model [START_REF] Hanley | Choice Modelling Approaches: A Superior Alternative for Environmental Valuation?[END_REF]. the utility is an index giving the importance of each attribute. Random utility models assume, as neoclassical economic theory, that the decision-maker has a perfect discrimination capability. In this context, however, the analyst is supposed to have incomplete information and, therefore, uncertainty must be taken into account. Manski (1997) identifies four different sources of uncertainty: unobserved alternative attributes, unobserved individual attributes (called ``unobserved taste variations'' by Manski, 1997), measurement errors and proxy, or instrumental, variables.

The utility is modeled as a random variable in order to reflect this uncertainty. Similarly to the neoclassical economic theory, the alternative with the highest utility is supposed to be chosen.

results: willingness to pay

The Multinomial Logit Parameter Estimates showed that only 2 attributes were significant: Reduction of flooding risk and reduction of odours. The willingness to pay value correspond to the marginal rate of substitution between the level of the considered attribute and the monetary attribute, the results are given in the following table 9 t a b l e 9 willingness to pay results (Larabi,2009) Attributes :

Willingness to pay (€/an)

RI 50% 60€

RI 80% 99€

RO 50% -16€

Thus people are ready to pay 60 euros per year to reduce of 50% the risk of flooding, 99 euros to reduce this same risk of 80 %.

Concerning the odours the negative results means that people are ready to support the impact but to be paid 16 euros per year

Conclusion

Our purpose in this paper was to show how environmental impacts can be taken in decision tools for sewer rehabilitation work programs. Environment is considered on the natural point of view but also on the human activity point of view. Environment economics methods seem to be well adapted to give trends and values but they are heavy to put in place because needing many data coming from different services or company of the network territory or needing he realisation of enquiry. Many evaluation still exist but not concerning sewer networks, the most nearly works concern evaluation of damages by rivers [START_REF] Euleterio | Sensibilité de l'évaluation des dommages potentiels à la caractérisation des enjeux : Application aux inondations dans la basse vallée de la Bruche[END_REF].

  hours of a ticket machine per day.

Table 2

 2 activities typology in the sample[START_REF] Werey | Rehabilitation impacts on socio-economic costs[END_REF]

	categories	Detail
	FOOD PRODUCTS	Butchery, Bakery, coffee, tea houses, chocolate shops, health food shops
	HEALTH CARE	Doctor, Dentist
	PETROL SUPPLY	wholesaler to service station
	CULTURE & LEISURE	Travel agency, Music and instruments shop, Gymnastic centre
		Cinema
	EQUIPMENT FOR THE HOUSE	Furniture shop
	EQUIPMENT FOR THE PERSON	Florist, Sport wear shop, Clothing shop, Shoes shop, Jewellery
	MEDICAL EQUIPMENT	Pharmacy, Optician, Biological products
	BOOKSHOP
	RESTAURANT	Restaurant,, Fast food, Cake-shop, Tea-room
	HEALTH, BEAUTY and HYGIENE	Hairdresser, Beauty centre
	SERVICES TO PERSONS	School of motoring, Insurance office, Dry cleaning
	SERVICE STATION
	SUPERMARKET
	TOBACCONIST

Table 3 :

 3 reduced sample -loss trade/turn over[START_REF] Werey | Rehabilitation impacts on socio-economic costs[END_REF]

	Activity categories	Number of requests	%(Loss of trade)/theoretical turn over
	ALIMENTARY PRODUCTS	4	14
	HEALTH CARE	1	4
	FUEL DELIVERY	1	2
	CULTURE & LEISURE	1	2
	EQUIPMENT FOR THE HOUSE	1	4
	EQUIPMENT FOR THE PERSON	3	26
	MEDICAL EQUIPMENT	3	1
	BOOKSHOP	0	/
	RESTAURANT	6	22
	HEALTH, BEAUTY and HYGIENE	1	19
	SERVICES TO PERSONS	0	/
	SERVICE STATION	1	28
	SUPERMARKET	2	4
	TOBACCONIST	4	7
	total	28

Table 4 :

 4 whole sample -compensation scale[START_REF] Werey | Rehabilitation impacts on socio-economic costs[END_REF]

	Activity categories	Number of requests	Loss of trade scale
	FOOD PRODUCTS	6	30
	CONSULTING	5	13
	FUEL DELIVERY	1	37
	CULTURE & LEISURE	4	71
	EQUIPMENT FOR THE HOUSE	2	64
	EQUIPMENT FOR THE PERSON	20	40
	MEDICAL EQUIPMENT	5	47
	BOOKSHOP	4	55
	RESTAURANT	24	31
	HEALTH, BEAUTY and HYGIENE	6	53
	SERVICES TO PERSONS	4	39
	SERVICE STATION	1	31
	SUPERMARKET	3	100
	TOBACCONIST	8	30
	total	93

  These scenario were compared with the statue-quo (paying nothing and doing nothing) situation by using choice cards. People had to class the different programs, from the best to the worst.We have 4 attributes and 3 levels for each, that means 81 possible programs. Using the orthogonal design function within he SAS software we could reduce the number of cards, keeping a significant result. So 18 programs in 6 cards were proposed during to enquiry to each person. It is necessary that the number of cards is few to keep a good attention of people on all the cards. So people had to classify 6 choice cards like the following one in table 8:The survey has been done on 200 habitants from 5 municipalities of 28 of the Urban Community of Strasbourg by a consultancy specialised in inquiries linked with environment impact valuation, within the INDIGAU project during may 2009. The people concerned by the survey are or have been concerned by .odours or flooding events

	table 8 one of the six choice cards (Larabi, 2009)
	1	Program A	Program B	Program C	Program D
						Statue quo
	Reduction of
	flooding risk of	80%		0%	50%	0%
	Reduction of
	the water level	15cm		10cm	0cm	0
	of..
	Reduction of
	odours of	0%		50%	80%	0%
	Contribution
		50€		25€	10€	0€
	Classement
	4.3 sample
	t a b l e 7 attributes and levels (Larabi,2009)
	Attribute		level
	Reduction of flooding risk by		1-0 %
				2-50%
				3-80%
	Reduction of the water level by..	1-0 cm
				2-10 cm
				3-15 cm
	Reduction of odours by		1-0 %
				2-50%
				3-80%
	Contribution		1-10 euros
				2-25 euros
				3-50 euros


R Rischette
email: romain.rischette@insa-lyon.fr

T N Baranger

N Debit

Numerical analysis of an energy-like minimization method to solve a parabolic Cauchy problem with noisy data

This paper is concerned with solving Cauchy problem for parabolic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to an adapted stopping criterion depending on noise rate for the minimization process. Numerical experiments are performed and confirm theoretical convergence order and the good behavior of the minimization process.

Introduction

The Cauchy problem considered here consists of solving a parabolic partial differential equation on a domain for which over-specified boundary conditions are given on a part of its boundary. It entails solving a data completion problem and identifying the missing boundary conditions on the remaining part of the boundary. This kind of problem is encountered in many industrial, engineering and biomedical applications.

Since J.Hadamard's works [START_REF] Hadamard | Lectures on Cauchy's problem in Linear Partial Differential Equation[END_REF], the Cauchy problem is known to be ill-posed and considerable numerical instability may occur during the resolution process. It provides researchers with an interesting challenge for carrying out numerical procedures to approximate the solution of the Cauchy problem in the specific case of noisy data. Many theoretical and applied works have been dedicated to this subject, using iterative methods [START_REF] Baumeister | On iterative methods for solving ill-posed problems modeled by partial differential equations[END_REF], regularization methods [START_REF] Tikhonov | Méthode de résolution de problèmes mal posés[END_REF][START_REF] Cimetière | Solution of the Cauchy problem using iterated tikhonov regularization[END_REF], quasireversibility methods [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for laplace's equation[END_REF] and minimal error methods [START_REF] Marin | Numerical solution of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials[END_REF][START_REF] Marin | The minimal error method for the Cauchy next term problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity[END_REF][START_REF] Chakib | Convergence analysis for finite element approximation to an inverse Cauchy problem[END_REF].

In this paper, we focus on a method introduced in [START_REF] Andrieux | Solving Cauchy problems by minimizing an energy-like functional[END_REF][START_REF] Baranger | An optimization approach for the Cauchy problem in linear elasticity[END_REF][START_REF] Andrieux | An energy error-based method for the resolution of the Cauchy problem in 3d linear elasticity[END_REF][START_REF] Andrieux | Energy methods for Cauchy problems of evolutions equations[END_REF][START_REF] Baranger | Constitutive law gap functionals to solve Cauchy problem for a linear elliptic pde: a review[END_REF] based on the minimization of an energy-like functional. More precisely, we introduce two distinct fields, each of which fulfills one of the over specified boundary conditions. They are therefore solutions of two well-posed problems. Next, an energy-like error functional is introduced to measure the gap between these two fields. If the Cauchy problem solution exists and is unique, it is obtained when the functional reaches its minimum. Then, the resolution of the ill-posed Cauchy problem is achieved by successive resolutions of well-posed problems. This method provides promising results. Nevertheless, like many other methods, it becomes unstable in the case of noisy data. To overcome this numerical instability, we propose an adequate stopping criterion parametrized by the noise rate deduced by numerical convergence analysis. This analysis has already been performed for elliptic Cauchy problems in [START_REF] Rischette | Numerical analysis of an energy-like minimization method to solve Cauchy problem with noisy data[END_REF].

The outline of the paper is as follows. In section 2, we give the Cauchy problem and report classical theoretical results. In section 3, we formulate the Cauchy problem as a data completion problem and introduce the related minimization problem. In sections 4 and 5, we present finite element and time discretization, convergence analysis and the study of noise effects for the minimization problem. An a priori error estimate is then given, taking into account data noise, and a stopping criterion is proposed to control the instability of the minimization process. Finally, the numerical procedure and results are presented.

Statement of problem

We consider a Lipschitz bounded domain Ω in Ê d , d = 2, 3 with n being the outward unit normal to the boundary Γ = ∂Ω. Let us assume that Γ is partitioned into two parts, Γ u (for unknowns) and Γ m (for measurements), of the non-vanishing measurement, such that Γ u ∩ Γ m = ∅.

Q = Ω×]0, D[, Σ u = Γ u ×]0, D[, Σ m = Γ m ×]0, D[.
Given an initial temperature u 0 in Ω, a source term f , a conductivity field k, a density ρ and a heat capacity c in Q, a flux φ and the corresponding temperature T on Σ m , the aim is to identify the corresponding flux and temperature on Σ u . The nondimensionalized Cauchy problem is then written as

         ∂u ∂t -∇ • k(x)∇u = f in Q k(x)∇u • n = φ on Σ m u = T on Σ m u(•, 0) = u 0 in Ω. (1)
where

k(x) = k(x)/ρc, f = f /ρc and φ = φ/ρc.
A problem is well-posed according to Hadamard (see [START_REF] Hadamard | Lectures on Cauchy's problem in Linear Partial Differential Equation[END_REF][START_REF] Isakov | Inverse problems for partial differential equations[END_REF][START_REF] Tikhonov | Méthode de résolution de problèmes mal posés[END_REF]) if it fulfills the following properties : the uniqueness, existence and stability of the solution. The extended Holmgren theorem relating to Sobolev spaces (see [START_REF] Isakov | Inverse problems for partial differential equations[END_REF]) guarantees uniqueness under regularity assumptions for the solution of the Cauchy problem. Since the well known Cauchy-Kowalevsky theorem (see [START_REF] Lions | -Functional and Variational Methods[END_REF]) is applicable only in the case of analytical data, the existence of this solution threefore depends on the verification of a compatibility condition difficult to formulate explicitly. In addition to the fact that for one fixed datum, the set of compatible data is dense within the full set of data (see [START_REF] Fursikov | Optimal control of distributed systems[END_REF]), this compatibility condition implies that the stability assumption is not satisfied in the sense that the dependence of solution u of (1) on data (φ, T ) is not continuous. Hereafter, we assume that data (φ, T ) in (1) are compatible.

A few notations : Let x be a generic point of Ω. The space of squared integrable functions L 2 (Ω) is endowed with a natural inner product written as (•, •) 0,Ω . The associated norm is written as • 0,Ω . We note H p (Ω) the Sobolev space of functions of L 2 (Ω) for which their p-th order and lower derivatives are also in L 2 (Ω). Its norm and semi norm are written as • p,Ω and | • | p,Ω respectively. Moreover, let u = (u 1 , u 2 ) ∈ (H p (Ω))

2 , the semi-norm of this space is written

u p,Ω = |u 1 | 2 p,Ω + |u 2 | 2 p,Ω 1/2 . Let γ ⊂ Γ, we define the space H 1 0,γ (Ω) = {v ∈ H 1 (Ω); v |γ = 0} and H 1/2 00 (γ)
is the space of restrictions to γ of the functions of H 1/2 (Ω) = tr H 1 (Ω) . Its topological dual is written as

H -1/2 00 (γ) = H 1/2 00 (γ)
′ . The associated norms are written as • 1/2,00,γ and • -1/2,00,γ respectively and •, • 1/2,00,γ states for the duality inner product. Now, let t be the time variable. We denote by L 2 (0, D; F ) the space of squared integrable functions in [0, D] with values in F , where F is a normed functional space. In the same way, C n (0, D; F ) defines the space of n times continuously derivable functions in [0, D] with values in F . The space of distributions in ]0, D[ is written as D ′ (]0, D[). In the sequel, C indicates a positive generic constant.

3 Energy-like minimization method

Let f ∈ L 2 0, D; L 2 (Ω) , k(x) ∈ L ∞ (Ω) positive, φ ∈ L 2 0, D; H -1/2 00 (Γ m ) and T ∈ L 2 0, D; H 1/2 00 (Γ m ) .
The Cauchy problem can be written as a data completion problem:

Find (ϕ, ξ) ∈ L 2 0, D; H -1/2 00 (Γ u ) × H 1/2 00 (Γ u ) such that u ∈ L 2 0, D; H 1 (Ω) is the solution of          ∂u ∂t -∇ • k(x)∇u = f in Q u = T, k(x)∇u • n = φ on Σ m u = ξ, k(x)∇u • n = ϕ on Σ u u(•, 0) = u 0 in Ω. (2)
Remark 1 : We note that in the case Γu ∩ Γm = ∅, as given in figure 2 illustrating the ring numerical tests of section 6.2, spaces H -1/2 (Γ u ) × H 1/2 (Γ u ) and H -1/2 (Γ m ) × H 1/2 (Γ m ) for the unknowns and the data respectively, would be more appropriate. Nevertheless, the general functional framework is not restrictive because spaces H s 00 (Γ u ) and H s 00 (Γ m ) are dense in H s (Γ u ) and H s (Γ m ), respectively, for s = ±1/2.

Following [START_REF] Andrieux | Energy methods for Cauchy problems of evolutions equations[END_REF], we now introduce two distinct fields u 1 and u 2 which are the solutions of well posed problems differentiated by their boundary conditions. We attribute to each of them one datum on Σ m and one unknown on Σ u . Then, we obtain

         ∂u 1 ∂t -∇ • k(x)∇u 1 = f in Q u 1 = T on Σ m k(x)∇u 1 • n = η on Σ u u 1 (•, 0) = u 0 in Ω, (3)
         ∂u 2 ∂t -∇ • k(x)∇u 2 = f in Q u 2 = τ on Σ u k(x)∇u 2 • n = φ on Σ m u 2 (•, 0) = u 0 in Ω. (4)
We denote a i (•, •) and l i (•), i = 1, 2 the bilinear and linear forms associated to the weak forms of the problems (3) and ( 4) respectively. They are given by

a i (ũ i (t), v) = Ω k(x)∇ũ i (t)∇v dx, for i = 1, 2, (5)
l 1 (v; t) = Ω f (t)v dx - d dt (ū 1 (t), v 1 ) H -a 1 (ū 1 (t), v) + η(t), v 1/2,00,Γu , (6)
l 2 (v; t) = Ω f (t)v dx - d dt (ū 2 (t), v 1 ) H -a 2 (ū 2 (t), v) + φ(t), v 1/2,00,Γm , (7)
where ū1 (t) and ū2 (t) are the lifting of the extended Dirichlet conditions T (t) and τ (t) respectively and ũi = u iūi , i = 1, 2. We have by summation the following weak problem:

Find u = (ũ 1 , ũ2 ) ∈ L 2 (0, D; V ) ∩ C 0 (0, D; H) such that d dt (u(t), v) H + a(u(t), v) = L(v; t), ∀ v = (v 1 , v 2 ) ∈ V in D ′ (]0, D[) (8)
u(., 0) = u 00 = (ũ 10 , ũ20 ), with a(u(t), v) = a 1 (ũ 1 (t), v 1 ) + a 2 (ũ 2 (t), v 2 ), and L(v; t) = l 1 (v 1 ; t) + l 2 (v 2 ; t), where H = L 2 (Ω)

2 endowed with the scalar product (u, v)

H = (u 1 , v 1 ) 2 0,Ω + (u 2 , v 2 ) 2 0,Ω 1/2 , V = H 1 0,Γm (Ω) × H 1 0,Γu (Ω) and v V = ( v 1 2 1,Ω + v 2 2 1,Ω ) 1/2
is the norm associated with space V . It is easy to show that the linear form L(•) is continuous and that the bilinear form a(•, •) is continuous and V -elliptic. Then, by using a theorem formulated by J.L. Lions (see [START_REF] Lions | Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF]), the weak problem (8) admits a unique solution.

We now consider the following energy-like functional:

E(η, τ ) = D 0 Ω k(x) (∇u 1 (η, t) -∇u 2 (τ, t)) 2 dx dt + 1 2 Ω (u 1 (η, D) -u 2 (τ, D)) 2 dx, (9)
and the following minimization problem:

   (η * , τ * ) = argmin (η,τ )∈U E(η, τ ), U = L 2 0, D; H -1/2 00 (Γ u ) × H 1/2 00 (Γ u ) ,
with u 1 and u 2 solutions of ( 3) and ( 4) respectively.

(

) 10
By using the convexity of space U , the existence and uniqueness of the solution to the Cauchy problem in the case of compatible data allows proving that the solution (η * (t), τ * (t)) of the minimization problem [START_REF] Baranger | An optimization approach for the Cauchy problem in linear elasticity[END_REF], if it exists and is unique, is the solution of the data completion problem (i.e.(η * , τ * ) = (ϕ, ξ)).

Discretization method and error estimation

Let X h be the finite element space for which the following classical assumptions are verified:

(i) Ω is a polyhedral domain in Ê d , d = 2, 3. (ii) T h is a regular triangulation of Ω i.e. h = max K∈T h h K → 0 and max K∈T h h K ρ K ≤ c being a constant independent of h, h K is the element, K the diameter and ρ K the inscribed circle diameter of K.
(iii) Γ u and Γ m can be written exactly as the merger of the faces of several finite elements K ∈ T h .

(iv) The family (K, P K , Σ K ), K ∈ T h for all h is affine-equivalent to a reference finite element ( K, P , Σ) of C 0 regularity.

(v) The following inclusion is satisfied:

P l ( K) ⊂ P ⊂ H 1 ( K) for l ≥ 1.
These assumptions imply that X h ⊂ H 1 (Ω). We define the following spaces:

X uh = {v h ∈ X h ; v h|Γu = 0}, X mh = {v h ∈ X h ; v h|Γm = 0},
and

V h = X mh × X uh ⊂ V the finite dimensional approximation space.
Given u 0h ∈ V h the V h -interpolation of u 00 , the semi-discrete problem associated with ( 8) is written as

Find u h (t) = u 1h (t), u 2h (t) ∈ L 2 (0, D; V h ) such that d dt u h (t), v h H + a u h (t), v h = L(v h ; t), ∀ v h = (v 1h , v 2h ) ∈ V h in D ′ (]0, D[) (11)
u h (0) = u 0h .
We now turn to time discretization. We introduce time step ∆t and time t n = n∆t, 0 ≤ n ≤ N , and denote the approximation of u(•, t n ) by u n h ∈ V h . This gives the discrete problem based on the backward Euler scheme:

Find {u n h ∈ V h ; 0 ≤ n ≤ N } such that 1 ∆t (u n+1 h -u n h , v h ) H + a(u n+1 h , v h ) = L(v h ; t n+1 ), 0 ≤ n ≤ N -1, ∀ v h ∈ V h ( 12
)
u 0 h = u 0h .
The same argument as for the weak problem (8) provides the existence and uniqueness of the solutions of ( 11) and [START_REF] Andrieux | Energy methods for Cauchy problems of evolutions equations[END_REF].

Using the standard procedure described in [START_REF] Thomée | Galerkin Finite Element Methods for Parabolic Problems[END_REF], we report the following error estimate:

Proposition 4.1 In addition to the assumptions stated above, let us assume that the integer l ≥ 1 exists such that the following inclusion is satisfied:

H l+1 ( K) ⊂ C s ( K) with continuous injection ( 13
)
where s is the maximal order of the partial derivatives occurring in the definition of the set Σ.

Then, if the solution u of the variational problem (8) also verifies u(t) ∈ H l+1 (Ω) 2 for all t ∈ [0, D], there is a constant C independent on h and ∆t such that

u(t n ) -u n h V ≤ C h l u 00 l+1,Ω + tn 0 du dt (s) 2 l+1,Ω ds 1/2 + +∆t tn 0 d 2 u dt 2 (s) 2 l+1,Ω ds 1/2 , (14)
where

{u n h ∈ V h , 0 ≤ n ≤ N } is the discrete solution.
5 Noisy data, error estimates and stopping criterion

Error estimates and data noise effects

In the case of given perturbed data, say (φ δ , T δ ), problem ( 12) is written as:

Find {u n hδ ∈ V h ; 0 ≤ n ≤ N } such that 1 ∆t (u n+1 hδ -u n hδ , v h ) H + a(u n+1 hδ , v h ) = L δ (v h ; t n+1 ), 0 ≤ n ≤ N -1, ∀ v h ∈ V h ( 15
)
u 0 hδ = u 0h ,
where L δ (•) is given by the expression of L(•) with (φ, T ) being replaced by (φ δ , T δ ).

Proposition 5.1 Under assumptions of proposition 4.1, if the solution u of the variational problem (8) also verifies u(t) ∈ H l+1 (Ω) 2 for all t ∈ [0, D], then there is an independent constant C on h, ∆t and data such that

u(t n ) -u n hδ V ≤ C h l u 00 l+1,Ω + tn 0 du dt (s) 2 l+1,Ω ds 1/2 + +∆t tn 0 d 2 u dt 2 (s) 2 l+1,Ω ds 1/2 + √ ∆t   n j=1 d(T -T δ ) dt (t j ) 1/2,00,Γm + 1 h δ(t j )      , (16)
where

δ(t) = T (t) -T δ (t) 2 1/2,00,Γm + φ(t) -φ δ (t) 2 -1/2,00,Γm 1/2 (17)
is the norm of the data noise and {u n hδ ∈ V h ; 0 ≤ n ≤ N } is the solution of the discrete problem (15) associated with the noisy Cauchy problem.

Proof We write u(t n ) -u n hδ = u(t n ) -u n h + u n h -u n hδ = ρ n + θ n δ .
An estimation for ρ n V is given immediately by the proposition 4.1.

From ( 12) and ( 15), we obtain

1 ∆t (θ n+1 δ -θ n δ , v h ) H + a(θ n+1 δ , v h ) = L δ (v h ; t n+1 ) -L(v h ; t n+1 ), ∀ v h ∈ V h . ( 18
)
ω n δ ∈ V h is defined by (ω n δ , v h ) H = L δ (v h ; t n+1 ) -L(v h ; t n+1 ) = d(ū 1 -ūδ 1 ) dt (t n ), v h H + a ū1 (t n ) -ūδ 1 (t n ), v h + φ(t n ) -φ δ (t n ), v 2h -1/2,00,Γm , (19)
where ūδ 1 (t) is the lifting of the extended Dirichlet conditions with noise T δ (t). By using trace and lifting operator properties and inverse inequalities, we prove that

ω n δ H ≤ C d(T -T δ ) dt (t n ) 1/2,00,Γm + 1 h δ(t n ) . (20)
Choosing [START_REF] Lions | Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF], it gives

v h = θ n+1 δ -θ n δ ∆t in
θ n δ 2 V ≤ C∆t n j=1 ω j δ 2 H . (21)
and then, with [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF],

θ n δ 2 V ≤ C∆t n j=1 d(T -T δ ) dt (t n ) 1/2,00,Γm + 1 h δ(t n ) 2 . ( 22
)
Therefore, by using ( 20) and ( 22) we obtain an estimation of θ n+1 δ V that leads to (16).

Stopping criterion for the minimization process

When noise is introduced in the Cauchy data, we observe during the optimization process that the error reaches a minimum before increasing very fast, leading to a numerical explosion. At the same time, the energy-like functional asymptotically attains a minimal threshold, which is a strictly positive constant dependent on the noise. It is noteworthy that this constant vanishes in the case of compatible Cauchy data. The aim now is to theoretically determine this threshold in order to propose a stopping criterion dependent on the noise rate and which allows stopping the minimization process just before the numerical explosion.

We introduce a general quadrature formula where nodes and weights are denoted by (t j , α j ) to approximate the integral of a continuous function f on the time interval,

I N (f ) = N j=0 α j f (t j ) ∼ D 0 f (t) dt. ( 23
)
The noisy discrete functional is then given by

E δ h (η, τ ) = N j=0 α j Ω k(x) ∇u j 1hδ (η) -∇u j 2hδ (τ ) 2 dx + 1 2 Ω u N 1hδ (η) -u N 2hδ (τ ) 2 dx. ( 24
)
Proposition 5.2 Under the assumptions of proposition 4.1, if the solution u of the variational problem (8) also verifies u(t) ∈ H l+1 (Ω) 2 for all t ∈ [0, D] and if (η * , τ * ) is the solution of the minimization problem [START_REF] Baranger | An optimization approach for the Cauchy problem in linear elasticity[END_REF], then there is an independent constant C on h and the data such that

E δ h (η * , τ * ) ≤ C    h 2l N n=0 α n u 0 l+1,Ω + tn 0 du dt (s) 2 l+1,Ω ds 1/2 2 + +∆t 2 N n=0 α n tn 0 d 2 u dt 2 (s) 2 l+1,Ω ds + ∆t N n=0 α n   n j=1 d(T -T δ ) dt (t j ) 1/2,00,Γm + 1 h δ(t j )   2      . (25)
Proof Let (η * , τ * ) be the solution of the minimization problem [START_REF] Baranger | An optimization approach for the Cauchy problem in linear elasticity[END_REF] with compatible Cauchy data.

After several algebraic operations and taking into account the fact that u 1 (η * ; t) = u 2 (τ * ; t), ∀t ∈ [0, D], we can write

E δ h (η * , τ * ) = N j=0 α j Ω k(x) ∇u j 1hδ (η * ) -∇u 1 (η * ; t j ) -∇u j 2hδ (τ * ) -∇u 2 (τ * ; t j ) 2 dx+ + 1 2 Ω u N 1hδ (η * ) -u 1 (η * , t N ) -u N 2hδ (τ * ) -u 2 (τ * ; t N ) 2 dx. ( 26
)
It follows that

E δ h (η * , τ * ) ≤ 2 k L ∞ (Ω) N j=0 α j |u j 1hδ (η * ) -u 1 (η * ; t j )| 2 1,Ω + |u j 2hδ (τ * ) -u 2 (τ * ; t j )| 2 1,Ω + + (u N 1hδ (η * ) -u 1 (η * ; t N ) 2 0,Ω + u N 2hδ (τ * ) -u 2 (τ * ; t N ) 2 0,Ω , (27)
and then, ther exists a costant C that may depend on ∆t, such that

E δ h (η * , τ * ) ≤ C N j=0 α j u j hδ (η * , τ * ) -u(η * , τ * ; t j ) 2 V . (28)
Therefore, using proposition 5.1, we derive (25).

Nevertheless, at each time step except the last, the error semi-norm in V is majorized by the error norm in V . Then, this overestimation must be taken into acocunt to be more precise. We prove that

u n hδ (η * , τ * ) -u(t n ; η * , τ * ) 2 H ≥ 1 2 u n 1hδ (η * ) -u n 2hδ (τ * ) 2 0,Ω (29)
Hence, when the noisy discrete functional (24) reaches its minimum and, when h and ∆t are sufficiently small, we obtain through (25) and ( 29),

E δ h (η * , τ * ) ∼ O (S δ (η * , τ * ))), (30)
where

S δ (η * , τ * ) = ∆t N n=0 α n      n j=1 d(T -T δ ) dt (t j ) 1/2,00,Γm + 1 h δ(t j )   2 - 1 2 u n 1hδ (η * ) -u n 2hδ (τ * ) 2 0,Ω . (31)
In order to propose a stopping criterion based on these theoretical estimates, let us use (η j , τ j ) to denote the unknowns and E j the value of the discrete noisy functional at the j-th iteration.

At first, the stopping criterion relies on verifying that the noisy discrete functional has reached S δ . Taking into account the asymptotical behavior of the functional, we want to stop the optmization algorithm when the functional variations become lower than the functional itself, and then lower than S δ . Moreover, the ratio E j /E j-1 < 1 tends to 1. Then, multiplying S δ by this ratio, the threshold S δ is weakened before the asymptote is reached by the functional. Thus a consistent stopping criterion based on the description of the behavior of E δ h (•, •) and the estimation (25), could be max

{E j , |E j -E j-1 |} ≤ E j E j-1 S δ (η j , τ j ). ( 32
)
6 Numerical issues

Numerical procedure

Let us describe the calculation method of the elements required for the optimization procedure, more specifically the gradient of the functional. We assume that the triangulation T h of Ω is characterized by n nodes. Let p and q denote the number of nodes on the boundaries Γ u and Γ m respectively and (ω i ) 1≤i≤n = (ω 1i , ω 2i ) 1≤i≤n the which is the canonical basis of V h . We write the unknowns η(t n ) and τ (t n ) as X n η and X n τ respectively. Vectors U n 1 and U n 2 correspond to fields u 1 (t n ) and u 2 (t n ) respectively, vectors T n and Φ n correspond to the Dirichlet and Neumann data T (t n ) and φ(t n ) respectively. We introduce the following notations, (K 1 ) kl = a 1 (ω 1k , ω 1l ), (K 2 ) kl = a 2 (ω 2k , ω 2l ), (F n 1 ) k = l 1 (ω 1k ; t n ) and (F n 2 ) k = l 2 (ω 2k ; t n ) depending on the Neumann data φ(t n ). As the bi-linear forms are similar, we note

K = K 1 = K 2 .
The linear systems associated with (3) and ( 4) respectively are given by:

       M ∆t + K U n+1 1 = F n+1 1 (X n+1 η ) + M ∆t U n 1 L m U n+1 1 = T n+1 U 0 1 = U 0 , (33)
       M ∆t + K U n+1 2 = F n+1 2 (Φ n+1 ) + M ∆t U n 2 L u U n+1 2 = X n+1 τ U 0 2 = U 0 .
(34) The functional can be written as follows:

E(X η , X τ ) = 1 2 N n=0 α n (U n 1 -U n 2 ) t K(U n 1 -U n 2 ) + (U N 1 -U N 2 ) t M (U N 1 -U N 2 ) (35)
We want to calculate the functional derivatives with respect to each component with index i and at each time step k of the two unknowns X η and X τ written as X i,k η and X i,k τ respectively.

First, we derive the functional with respect to the unknown X η .

∂E ∂X

i,k η (X η , X τ ) = 2 N n=k α n ∂U n 1 ∂X i,k η t K(U n 1 -U n 2 ) + δ N k ∂U N 1 ∂X i,N η t M (U N 1 -U N 2 ), k = 1, ..., N ; i = 1, ..., p. (36)
In the sequel, we denote by U n 1,i,k the derivative of U n 1 with respect to X i,k η . By deriving the linear system (33), U n+1 1,i,k is the solution of:

M ∆t + K U n+1 1,i,k = ∂F n+1 1 ∂X i,k η + M ∆t U n 1,i,k , (37)
As T n and U 0 are independent on X n,k η , their derivatives vanish. Moreover,

U n 1,i,k = 0 if k > n and ∂F n 1 ∂X i,k η = 0 if k = n. We note F n 1,i = ∂F n 1 ∂X i,n η = (δ j i ) 1≤j≤m . We have then        M ∆t + K U n+1 1,i,k = δ k n+1 F n+1 1,i + ½ {k>n+1} M ∆t U n 1,i,k L m U n+1 1,i,k = 0 U 0 1,i,k = 0, 1 ≤ i ≤ m, 0 ≤ n ≤ N -1, 1 ≤ k ≤ N. (38)
Now, we derive the functional with respect to the unknown X τ .

∂E ∂X i,k τ (X η , X τ ) = - N n=k α n ∂U n 2 ∂X i,k τ t K(U n 1 -U n 2 ) -δ N k ∂U N 2 ∂X i,N τ t M (U N 1 -U N 2 ), k = 1, ..., N ; i = 1, ..., p. (39)
In the sequel, we denote by U n 2,i,k the derivative of U n 2 with respect to X i,k τ . By deriving the linear system (34), U n+1 2,i,k is the solution of

M ∆t + K U n+1 2,i,k = ∂F n+1 2 ∂X i,k τ + M ∆t U n 2,i,k , (40)
As F n 2 (Φ n ) and U 0 are independent on X n,k τ , their derivatives vanish. Moreover,

U n 2,i,k = 0 if k > n and ∂X n τ ∂X i,k τ = 0 if k = n. We note Xn τ,i = ∂X n τ ∂X i,n τ = (δ j i ) 1≤j≤m . We then have        M ∆t + K U n+1 2,i,k = ½ {k>n+1} M ∆t U n 2,i,k L u U n+1 2,i,k = Xn+1 τ,i U 0 2,i,k = 0, 1 ≤ i ≤ m, 0 ≤ n ≤ N -1, 1 ≤ k ≤ N. (41)
Here, we consider the case of real applications where we have only measured and noisy data (T δ , φ δ ) given with a noise rate 0 < a < 1. We are therefore not able to calculate exactly the norm of the difference between the exact and noisy data involved in the stopping criterion (32). We must therefore estimate these norms. This is done as follows:

T (x, t) -aT (x, t) ≤ T δ (x, t) ≤ T (x, t) + aT (x, t), ∀ x ∈ Γ m (42) ⇐⇒ -a 1 -a T δ (x, t) ≤ T (x, t) -T δ (x, t) ≤ a 1 + a T δ (x, t) (43)
and then

T (t) -T δ (t) 2 1/2,00,Γm ≤ max a 1 -a , a 1 + a T δ (t) 1/2,00,Γm (44)
Proceeding in the same way for the Neumann data and the time derivative of the Dirichlet data, we have:

δ(t) ≤ a 1 -a T δ (t) 2 1/2,00,Γm + φ δ (t) 2 -1/2,00,Γm 1/2 (45) and d(T -T δ ) dt (t) 1/2,00,Γm ≤ a 1 -a dT δ dt (t) 1/2,00,Γm . ( 46
)
The stopping criterion (32) can then be written as follows:

max{E j , |E j -E j-1 |} ≤ E j a 2 ∆t E j-1 (1 -a) 2 N n=1 α n n k=1 dT δ dt (t k ) 1/2,00,Γm + 1 h T δ (t k ) 2 1/2,00,Γm + φ δ (t k ) 2 -1/2,00,Γm 2 - 1 2 N n=1 α n (U n 1 -U n 2 ) t M (U n 1 -U n 2 ). (47)

Numerical results

We consider the following Cauchy problem on the domain Ω given by figure [START_REF] Baumeister | On iterative methods for solving ill-posed problems modeled by partial differential equations[END_REF]:

         ∂u ∂t -△u = 0 in Ω×]0, 1[ u = g on Γ m ×]0, 1[ ∇u • n = h on Γ m ×]0, 1[ u(•, 0) = u 0 in Ω, ( 48
)
where g, h and u 0 are the Cauchy data extracted from the exact solution that we intend to approximate.

Γu Γm Ω r 2 = 1 r 1 = 0.5

Axisymmetric example

The Cauchy problem is written in polar coordinates (ρ, θ) and we assume that its solution does not depend on the angular coordinate. The state equation of ( 48) is therefore

∂u(ρ, t) ∂t - ∂ 2 ∂ρ 2 + 1 ρ ∂ ∂ρ u(ρ, t) = 0. ( 49
)
An analytical solution of this equation is given by u(ρ, t) = e -t J 0 (ρ) where J 0 (•) is the Bessel function of the first kind of order 0.

Figure 3 represents the finite element discretization error with respect to the maximum edge size of the mesh. We choose a sufficiently small ∆t such that time discretization is negligible. Similarly, figure 4 shows the time discretization error with respect to the time step, by considering h sufficiently small. These results are in agreement with the theoretical error estimate [START_REF] Rischette | Numerical analysis of an energy-like minimization method to solve Cauchy problem with noisy data[END_REF].

Two dimensional example

We now consider the resolution of the Cauchy problem (48) in two dimensions. An analytical solution of this problem is given by u(x, y; t) = e -2t cos(x + y) which provides Cauchy data on Γ m .

The figure 5 represents the discrete solution of the Cauchy problem along with the selected points p 1 , p 2 and p 3 used to represent the time evolution of the solution to the data completion problem. Figures 6 and7 represent the solution and the discrete solution of the data completion problem obtained by using energy-like method. We can see that the temperature and heat flux recovered are close to the exact ones. Figure 9 represents the finite element discretization error with respect to the maximum edge size of the mesh with a sufficiently small ∆t. Similarly, figure 8 shows the time discretization error with respect to the time step with h sufficiently small. These results are in agreement with the theoretical error estimate [START_REF] Rischette | Numerical analysis of an energy-like minimization method to solve Cauchy problem with noisy data[END_REF]. Nevertheless, the optimization process is significantly perturbed when the discretization steps h and ∆t are not of the same order. Indeed, since the error associated with the largest discretization step behaves like numerical noise, the energy-like method does not provide a discrete solution with the required accuracy. Moreover, since the results obtained in the axisymmetric case unambiguously confirm the theoretical estimate, this numerical noise could also be related to a mesh effect.

We introduce a Gaussian random noise on data with an amplitude depending on a rate a. Figures 10 and 11 represent the error and the energy-like functional at each iteration of the optimization process for different noise rates. These behaviors make it necessary to introduce a criterion to stop the optimization process before numerical explosion. Next, we choose h and ∆t such that the discretization error is negligible in comparison to the error due to noise and we observe error and functional behaviors with respect to the satisfactory noise measurements. These noise measurements correspond to terms dependent on the data in the estimates ( 16) and (25). They are denoted by

m n δ = √ ∆t   n j=1 d(T -T δ ) dt (t j ) + 1 h δ(t j )   and M δ = N n=0 α n (m n δ ) 2 . ( 50
)
These results, shown in figure 13, are in agreement with the error estimates ( 16) and (25). As illustrated by figures 14, 15 and 16, the proposed stopping criterion allows identifying a consistent solution.

Stratified inner fluid problem

We now explore the efficiency of the proposed stopping criterion on the stratified inner fluid problem already studied in [START_REF] Andrieux | Solving Cauchy problems by minimizing an energy-like functional[END_REF][START_REF] Rischette | Numerical analysis of an energy-like minimization method to solve Cauchy problem with noisy data[END_REF]. We therefore consider the reconstruction of temperature and flux in a pipeline of infinite length. This application is used in several industrial processes. Indeed, knowledge of the temperature on the internal wall of a pipeline is necessary for controlling material safety: stratified inner fluid generates mechanical stresses which may cause damage such as cracks. We assume that the temperature does not depend on the longitudinal coordinate and then consider the following problem on the geometry defined by figure 17 :

         ∂u ∂t -∇ • (k∇u) = 0 in Ω k∇u • n + αu = 20 on Γ m k∇u(x, t) • n + αu(x, t) = 250. ½ {Γu,up(t)} (x) + 50. ½ {Γ u,lo (t)} (x) on Γ u u(., 0) = u 0 (51)
where k = k/ρc, α = α/ρc with k = 17 W.m -1 .K -1 is the constant thermal conductivity, α = 12 on Γ m and 1000 on Γ u is the Fourier coefficient, ρ and c are the density and the heat capacity such that ρc = 1. The radius of the inner and outer circles in figure 17 are the same as in figure 2. The boundary Γ u is partitioned into two parts, the lower arc Γ u,lo (t) = {(x, y) ∈ Γ u ; y < y s (t)} and the upper arc Γ u,up (t) = {(x, y) ∈ Γ u ; y ≥ y s (t)}. Angle θ(t) evolves linearly from 0 to π with time. Therefore, the upper and lower parts of Γ u , and then temperature on Γ u , depend on t. The initial condition u 0 is the stationary solution of the problem (51) with θ = 0.

The Cauchy data are generated by solving the forward problem defined by (51). Then, a random noise with a rate of a = 5% is introduced in the Dirichlet data while we assume that the flux is known exactly on Γ m . The results presented here are obtained by using the proposed stopping criterion. Figure 18 represents the temperature field that has to be identified along with the selected points p 1 , p 2 and p 3 used to represent the time evolution of the solution of the data completion problem. On the one hand, figure 19 shows the temperature and heat flux recovered in comparison to the solution of the data completion problem given by the numerical resolution of (51) at times t 2 = 0.2, t 6 = 0.6 and t 10 = 1. On the other hand, figure 20 shows the time evolution of the temperature and heat flux recovered in comparison to the time evolution of the data completion problem solution on points p 1 , p 2 , p 3 . Figure 21 represents the temperature field identified relating to figure 18. It should be noted that the reconstructed field is close to the field to be recovered. Finally, figure 22 represents the solution of the generic optimization algorithm. A numerical explosion without the proposed stopping criteria can be clearly observed in this case.

Conclusion

In this work, we stated the Cauchy problem as being the minimization of an energy-like functional and presented classical theoretical results. Then, we gave the finite element discretization and performed numerical convergence analysis. A priori error estimates were derived by taking into account the effects of noisy data. A stopping criterion was then proposed, dependent on the noise rate in order to control the numerical instability of the minimization process due to noisy data. A numerical procedure was proposed and numerical experiments performed to confirm the theoretical error estimates. Finally, we illustrated the robustness and efficiency of the proposed stopping criterion, especially in the case of singular data. It would now be interesting to couple this approach with a regularization procedure. [20]

Figure 1 :

 1 Figure 1: An example of geometry The most common problem consists in solving the heat transfer equation in a given domain Ω and a time interval [0, D], assuming temperature distribution and heat flux are given over the accessible region of the boundary. We denote for D > 0

Figure 2 :

 2 Figure 2: Ring

Figure 3 :Figure 4 :

 34 Figure 3: Evolution of u(t N )u N h V with respect to h for different ∆t and for the axisymmetric example.

Figure 5 :

 5 Figure 5: Exact temperature and selected points p 1 , p 2 and p 3 , h = 0.1, ∆t = 0.1.

Figure 6 :

 6 Figure 6: Exact (filled markers) and identified (empty markers) temperature and heat flux on Γ u at times t 2 = 0.2, t 6 = 0.6 and t 10 = 1 for the 2D example, h = 0.1, ∆t = 0.1.

Figure 7 :

 7 Figure 7: Time evolution of exact (filled markers) and identified (empty markers) temperature and heat flux on selected points p 1 , p 2 and p 3 on Γ u for the 2D example, h = 0.1, ∆t = 0.1.

Figure 8 :

 8 Figure 8: Evolution of u(t N )u N h V with respect to ∆t for different h for the 2D example.

Figure 9 :

 9 Figure 9: Evolution of u(t N )u N h V with respect to h for different ∆t for the 2D example.

Figure 10 :

 10 Figure 10: Evolution of u(t N )u N hδ V during the optimization process for different noise rates and for the 2D example.

Figure 11 :

 11 Figure 11: Evolution of E δ h (η, τ ) during the optimization process for different noise rates and for the 2D example.

Figure 12 :

 12 Figure 12: Evolution of u(t N )u N hδ V with respect to m N δ for different noise rates and for the 2D example, h = 0.09, ∆t = 1/12.

Figure 13 :

 13 Figure 13: Evolution of E δ h (η * , τ * ) with respect to M δ for different noise rates and for the 2D example, h = 0.09, ∆t = 1/12.

Figure 14 :

 14 Figure14: Exact (filled markers) and identified (empty markers) temperature and heat flux on Γ u at times t 2 = 0.2, t 6 = 0.6 and t 10 = 1 for the 2D example, h = 0.1, ∆t = 0.1, a = 5%.

Figure 15 :

 15 Figure 15: Time evolution of exact (filled markers) and identified (empty markers) temperature and heat flux at selected points p 1 , p 2 and p 3 on Γ u for the 2D example, h = 0.1, ∆t = 0.1, a = 5%.

Figure 16 :Figure 17 :

 1617 Figure 16: Identified solution for the 2D example, a = 5%, h = 0.1, ∆t = 0.1.

Figure 18 :

 18 Figure 18: Exact temperature and selected points p 1 , p 2 and p 3 for the stratified inner fluid example at time t = 1, h = 0.1, ∆t = 0.1.

Figure 19 :

 19 Figure19: Exact (filled markers) and identified (empty markers) temperature and heat flux on Γ u at times t 2 , t 6 and t 10 for the stratified inner fluid example, h = 0.1, ∆t = 0.1, a = 5%.

Figure 20 :

 20 Figure 20: Time evolution of exact (filled markers) and identified (empty markers) temperature and heat flux on selected points p 1 , p 2 and p 3 on Γ u (cf. figure 18) for the stratified inner fluid example, h = 0.1, ∆t = 0.1, a = 5%.

Figure 21 :

 21 Figure 21: Identified solution for the stratified inner fluid example using the proposed stopping criterion at time t = 1, h = 0.1, ∆t = 0.1, a = 5%.

Figure 22 :

 22 Figure 22: Identified solution for the stratified inner fluid example without using the proposed stopping criterion at time t = 1, h = 0.1, ∆t = 0.1, a = 5%.