SYSTEM APPROACH-BASED BAYESIAN NETWORK TO AID MAINTENANCE OF MANUFACTURING PROCESS

Keywords: maintenance, Bayesian Network, system approach, diagnosis strategy, prognosis strategy, FMECA

The prospective work reported here explores a new methodology to develop Bayesian Network-based diagnosis and prognosis aids for manufacturing processes. This work is justified with the complex systems by the need of controlling and maintaining in dynamical way the global system performances in order to optimise the enterprise strategies. The added value of our methodology is to formalise the maintenance aid models from a priori knowledge both on the system functioning and malfunctioning by means of Bayesian Networks. The networks are built on adaptability principles and integrate uncertainties on the relationships between causes and effects. The feasibility of this methodology is tested in a manufacturing context with the maintenance aids on a lathe machine.

INTRODUCTION

One of the main challenges of the Extended Enterprise is to maintain and optimise, in dynamics, the quality of the services delivered by industrial objects all along their life cycle. These objects are increasingly complex and constrained today by societal, economical and technological environments having more and more requirements. During the object-manufacturing phase, the improvement of the global performances of the manufacturing system is an issue to be solved to ensure the optimisation of the Enterprise needs. In that way, to control the direct and indirect costs of the system compared with the necessary availability, is a first challenge raised. The objective is thus to have maintenance processes mainly in terms of decision-making aids in order to guarantee maximum components availability keeping the system in operation. These processes do not have the aim to replace the operator but to assist him in its final decision-making to avoid consequences of an error in judgement. Today, most of current automated systems do not provide the means for intelligent interpretation of information copying with large process disturbances (diagnosis) and for predicting the consequences of a future action (prognosis). However tools issued from Artificial Intelligence [START_REF] Khoo | A fuzzy based genetic approach to the diagnosis of manufacturing systems[END_REF], [START_REF] Hu | A systematic approach to integrated fault diagnosis of flexible manufacturing systems[END_REF] are used now to bring a first decision-making aid for manufacturing systems. In addition, recent works on system safety and Bayesian Networks (BNs) are developed [START_REF] Kang | A Bayesian belief network-based advisory system for operational availability focused diagnosis of complex nuclear power systems[END_REF]. For example, [START_REF] Bouissou | Assessment of a Safety-Critical System Including Software: A Bayesian Belief Network for Evidence Sources[END_REF] propose, within the SERENE project, a hierarchical decomposition of the decision-making model for system safety analysis. As for them, [START_REF] Bobbio | Improving the analysis of dependable systems by mapping fault trees into Bayesian networks[END_REF] explain how the Fault Tree can be achieved using BNs.

In relation to these works, the proposed methodology has originality on formalising, by means of BNs (section 2), the diagnosis and prognosis aid models (section 3) from a priori knowledge on the system. So, issued from design, the networks are built: -by inheriting the adaptability principles as advocated by the IMS (Intelligent Manufacturing Systems) initiative; it is a first answer to knowledge structuring and reusability as looked for in the expert systems, -by incorporating uncertainties on the relationships between causes and effects to be more consistent with system reality. To show the feasibility and the added value of this prospective methodology, an application is developed in section 4 on the case of maintenance aid for lathe machine. Finally, in section 5, conclusions and prospects are presented.

BAYESIAN NETWORK THEORY

The BNs are Directed Acyclic Graph (DAG) and used to represent uncertain knowledge in Artificial Intelligence [START_REF] Jensen | An Introduction to Bayesian Networks[END_REF]. A BN is defined as a pair: G=((n,e),CPT), where (n,e) represents the DAG; "n" is a set of nodes, defined by different states; "e" is a set of directed edges describing the probabilistic dependencies between nodes. In this work, the nodes represent discrete random variables of the process. In pair G, each node is associated to a Conditional Probability Table (CPT) to quantify the dependencies between random variables as conditional probabilities. A node without parent is called a root node.

A B

Fig. 1. Basic example of a BN.

A probability is allowed to each state of the node. This probability is defined, a priori, for a root node and computed by inference for the others. The computation is based on the probabilities of the parents states and the CPT. For instance, two nodes A and B with each of them two states (S *1 and S *2 ) compose the BN in Fig. 1. The a priori probabilities of the node A are defined as:

S A1 P(A=S A1 ) A S A2 P(A=S A2 )
The probabilities of the states allowed to B are computed using a CPT. This CPT is defined by the probability of each B state knowing the state of A.

A S A1 S A2 S B1 P(B=S B1 |A=S A1 ) P(B=S B1 |A=S A2 ) B S B2 P(B=S B2 |A=S A1 ) P(B=S B2 |A=S A2 ) Thus to compute P(B=S B1 ), the following eq. 1 is used:

) S A ).P( S |A S B P( ) S A ).P( S |A S B P( ) S B P( 2 A 2 A 1 B 1 A 1 A 1 B 1 B = = = + = = = = = (1)
The added value of a BN is linked to the computation of the probabilities allowed to a node state, knowing the state of one or several variables. If the knowledge on the process modelled by the BN is unavailable then the computation is based on a priori probabilities. However if the knowledge is increasing, the computation takes it into account and the results are adapted to the process state. It is then possible to estimate the impacts of a random variable on the process.

MODELLING APPROACH

The proposed modelling approach consists, from functioning systemic analysis based on SADT1 graphical representation, (a) in representing the abnormal operation (malfunctioning) based on FMECA2 and then (b) in formalising and unifying these two results in a unique model by means of BNs.

From system complementary functioning-malfunctioning representation…

The functioning and malfunctioning of the system are dual and must be studied together to control each system variable [START_REF] Léger | An innovative approach for new Distributed Maintenance System: application to Hydro Power Plants of the REMAFEX project[END_REF]. It leads, first, to focus on the system functioning in relation to its environment and its global internal and external resources. This action can be made by means of a functioning modelling using SADT graphical representation. This modelling is based on the principle of activity and sub-activities until elementary activities, supported by components, are emerging. Each activity (Fig. 2) fulfils a finality, which is to modify a "product" carried out by the manufacturing system. For example, the output flow WD is a report (RWD) that represents the informational result of the Input HD product flow transformed by the activity. From this functioning, the malfunctioning is induced [START_REF] Léger | Methodological approach to modelling of degradation detection and failure diagnosis in complex production system[END_REF] by considering that the relationship between these two modes is directly linked to the relationship between the normal and abnormal states of the system. The malfunctioning modelling can be made by means of FMEA study in order to identify the failure or degradation modes of each activity, the elements which are at the failure origin (causes) and the possible consequences of these failures (effects). For example, the RWD flow can take the value "fulfilled or realised (R)" corresponding to the nominal state of the activity or the values "No true to the nominal or Not conform (NC)"; "Not fulfilled or Not Realised (NR)" requiring to identify the causes and the effects related to these two abnormal states. The failure causes are either external with the activity (Input flows) or internal ones because linked to the AD activity support flow (components). A whole of states can be thus associated with each component. These states correspond to: nominal operation (OK), breakdown 1, breakdown 2… In the same way, the consequences are observable either on activity output flows or on the influence of the component degradation development on itself (to go towards a breakdown state). To sum up, a failure cause leads to a failure mode (e.g. the modification of the function state reported in RWD), which has as effect for the function not to produce nominal flow HD any more. This FMEA study is completed by a FMECA study where each failure mode is characterised by a criticality (risk priority number) resulting from the product of the three criteria, which are the Frequency (F), the Severity (S) and the No Detection (ND) [START_REF] Suhner | Use of FMECA for fault diagnosis of complex automotive systems[END_REF].

… to unified Bayesian Network representation

The BN is directly built from the dual functioning/malfunctioning analysis presented above leading to a unified representation. This representation is structured as a BNs tree. Its root is a BN representing the highest abstraction level. The elementary activities represent the lowest functional levels modelled by BNs. To keep the generic function concepts (Fig. 2), its inputs are modelled, in the BN, by input nodes defining the random variables associated to the flows AD, HD, KHD and WD. From this step, simplifying assumptions are made to carry out our prospective study until a feasibility phase. Therefore just the RWD flow is taken into account as main output. This flow is an informational view of the function finality, so it is assumed to be the same as the added value on product flow represented in the Fig. 2 by the HD flow transformed by the activity. It is thus transferred as informational view of physical result through the input flow HD of another function. The generic function represented in BN formalism is given in Fig. 3. On this basis, the BNs are built in different ways according to the abstraction level and the design project advancement. The high functional levels. To model high functional levels, the BNs are composed of generic sub-functions with the structure defined in Fig. 3. The number of nodes states is adapted and it is possible to duplicate several inputs or outputs nodes (AD, HD…) if the function carries out several missions. Moreover, it is possible to model subfunctions in parallel or in series (Fig. 4). In Fig. 4, as the generic sub-functions F1 and F2 are in series, the report RWD1 is transferred to F2 through the input flow HD. As the functions F2 and F3 are in parallel, a node OR is linked to RWD2 and RWD3 to compute the RWD of the global function. In this level, only the connections between functions in parallel are defined as CPT. The CPT of a OR node, for two functions whose RWD2 and RWD3 are composed of three states (R, NC, NR cf. section 3.1), is defined as follows:

AD

RWD2 R NC NR RWD3 R NC NR R NC NR R NC NR R 1 1 1 1 0 0 1 0 0 NC 0 0 0 0 1 1 0 1 0 NR 0 0 0 0 0 0 0 0 1
The low functional levels. The last BN of a branch is based on the FMECA analysis. The nodes constituting this BN are:

• the component nodes (CMP) or a set of components nodes according to the model explanation,

• and elementary function nodes (EF).

The CMP are defined only in the low BN levels. They are directly linked to the EF nodes representing their functionality. The CMPs states are defined by the causes analysed by means of FMECA. The causes are either internal of the low BN level i.e. linked to CMP, or external i.e. linked to the input nodes. The common causes are defined in higher hierarchical levels and the information forwards by heritage between the levels through the input and output nodes. There is thus duplication either for the CMPs or for the external causes. The EF nodes are linked to the CMPs and the input nodes leading to compute the RWD states probabilities (Fig. 5). If all the EFs are realised then the RWD is realised. Evaluation of the probabilities. The FMECA analysis assigns a criterion F to the causes of failure. A probability of occurrence is associated with F. For sake of simplicity, only this criterion is presented. Nevertheless information such as the Severity (S) and the No Detection (ND) can also be considered. On this basis, the parameter F is used to determine the a priori probability that a CMP is in a state among a list of causes (considered as exhaustive). An example of F and causes is described below (with F = 2 when one failure occurred / month, and 3 for one / week):

Failure Modes Causes Effects F CMP failure P1 Effect 1 2 Elementary function in mode 1 CMP failure P2 Effect 2 3

The a priori probabilities associated to the CMP node states in the BN are estimated for a given operating time (T), for example T=2600 h / year: 0.005

) 1 P CMP P( = = h -1 (2) 0.020 ) 2 P CMP P( = = h -1 (3)
The a priori probability of correct operation of CMP is then deduced by:

)) 2 P CMP P( ) 1 P CMP (P( -1

) OK CMP P( = + = = = (4)
So, the a priori probabilities for each cause are:

• correct operation (OK):

0.975 h -1 • failure 1 (P1): 0.005 h -1 • failure 2 (P2): 0.020 h -1
The EF states are defined by the failure modes. For instance, the EF nodes in the BN take the following states:

• Realised (R),

• Not Conform or partially realised (NC),

• Not Realised (NR). No a priori probability is associated to these states because they are calculated according to the states of their parents i.e. the internal causes (CMP node states) and the external causes (input node states). The CPT of the EF is defined thanks to the columns of the causes and the modes of the FMECA analysis.

AD

R NC NR CMP OK P1 P2 OK P1 P2 * R 1 0.2 0 0 0 0 0 NC 0 0.8 0 1 0.2 0 0 EF NR 0 0 1 0 0.8 1 1
The probabilities link the causes to the EF modes. Nevertheless the power of the BNs representation is that a combination of causes (for instance AD=R and CMP=P1) can lead to several failure modes with different probabilities allowing then to model the uncertainty.

A prognosis model. The BN model defined above allows the analysis of the degradation influences on the flows produced by the activity. This analysis is based on the simulation of a component failure, a common cause or an unconformity of a sub-function. The objective is to forecast the impacts of failures on the functions. It is then possible to analyse the upstream and downstream consequences on the whole system. For example, if a component failure is supposed, an evidence is defined as for instance P(CMP=P1) = 1. Then the probabilities associated to sub-functions are updated after the BN's inference. The RWD of each function relates the failure impacts on each functional level. Thus in a design phase, the BN model allows to aid the designer in his decision-making by the estimation of the various impacts of the solutions [START_REF] Weber | System architecture design based on a Bayesian Networks method[END_REF].

Diagnosis

The diagnosis process tries to determine the origin of the degradations or the failures. Based on the observed situation, the causality represented as a relation between the activities leads to determine internal cause or common cause.

Principle. After an observation of a failure mode on the process, the evidences are defined in the BN. These modes correspond in states R, NC or NR of the observed variables. The BN's inference computes the probabilities of the unobserved variables states in order to determine the most probable causes. The cause with the most significant probability is suspected. This cause indicates the component, which is, a priori and according to the knowledge, the origin of the observed failure mode. It is then necessary to check its state. If the component is at the origin of the failure, then the diagnosis is finished. Otherwise, the knowledge is considered as not enough sufficient. In this case, the component state is taken into account as new evidence and another BN's inference determines a new cause.

Diagnosis aid with BN modelling. To facilitate the investigation into the most probable cause, the diagnosis is fulfilled using the hierarchical decomposition of the BNs model. So, a solution consists in inserting diagnosis nodes for each functional level to specify the cause location as: internal, upstream or downstream from the level.

The "Input Diagnosis" node is connected to the input nodes of the function and defines if they are conforms to carry out the mission. The "Internal Diagnosis" node is connected to the RWD flows to evaluate if the mission is carried out (RWD) (Fig. 6). The states of these diagnosis nodes are: R, NC and NR. The rules allowing to define the CPT of the "Function Diagnosis" node are the following:

• The cause location is internal if the function is not correctly realised (Internal Diagnosis=NC or NR) and the input nodes are conform to carry out the mission (Input Diagnosis=R). • The cause location is upstream if the input nodes are not conform to carry out the mission (Input Diagnosis=NC or NR), independently to the internal diagnostic. • The cause location is downstream if the mission is carried out (Internal Diagnosis=R) and the input nodes are conform to carry out the mission (Input Diagnosis=R).

APPLICATION

The application chosen to test the feasibility of the approach is a lathe. Its functional analysis (activities and flows) of which the level A0 is presented in Fig. 7, is composed of three main functions (to grip/to loosen the part, to put in rotation the part, to machine the part). Each function is broken down into one level at least allowing to make the link by a "physical support mechanism" with the real components of the lathe system. For example, the function "to grip/to loosen" is broken down into elementary generic functions such as "pre-actuate", "actuate". The function "pre-actuate" is supported by a distributor and the function "actuate" by a jack.

BN's Model

The software used to build BNs model is Serene: (web link: http://www.hugin.dk/serene/).

Level 0: Function "Lathe" (Fig. 8). The BN represents high level of the functional decomposition of the lathe. Input nodes are grey: AD (electric and pneumatic power), WD (request to transform), KHD (programs), and HD (the raw part).

Table 1. Part of FMECA analysis on the function "actuate" supported by a jack. The sub-functions are represented as sub BN (framed nodes), input nodes are linked to power, programs, requests, raw part and the output represents the RWD flow of the sub-function. The function "to grip/to loosen the part" is considered as composed of two sub-functions. The inputs and the outputs of this function correspond to gripping or loosening according to the state of the part. There is duplication of the input nodes HD and output nodes RWD, nevertheless any cycle is made in the networks thanks to the tree decomposition adopted.

Level 012: Function "Actuate" (Fig. 9). The BN presents the low functional level given by the FMECA analysis of the sub-function "actuate". The Fig. 9 represents the component "JACK" and its different EF: "going in" and "going out". These EF are linked to the "pneumatic power" and the "JACK". The "Internal and Input diagnosis nodes" are linked to the "Function Diagnosis" node. First inference. The a priori probabilities associated to the states of the variables representing the parts are calculated with an inference of the BN. The HD raw parts and RWD of each sub-function are presented in Table 2. Thus according to the characteristics of reliability used for the components, and as output of this lathe, 65.5% of the parts are machined correctly, 22.7% of the parts are machined but with no conformities, and 11.7% are not machined. This calculation is carried out according to the quality of the raw parts. Then the raw parts machined in conformity knowing the raw parts in state R before machining, are 72.8%.

BN inference Diagnosis

The diagnosis starts when the "RWD producing" is not in conformity or not realised. Let us develop the diagnosis whose scenario is the following: on the level 0 of the general function, the RWD producing is not in conformity NC.

Initially, input flows (power electric, pneumatic, and raw parts) are checked. The checking leads to consider these flows as all in conformity (state R).

CONCLUSION AND FURTHER WORK

The proposed model based on the functioning and malfunctioning studies carried out when design makes it possible to build easily the structure of the BN. This structure is based on several levels of abstraction carried out from a generic formalism.

The phase of quantification of the BN is based on the occurrence probability criterion of the FMECA. The translation of the FMECA criterion F makes it possible to obtain prior probabilities. This assessment will have to be updated by the data of operating feedback collected during the exploitation of the system, for example by using Bayesian approach [START_REF] Suhner | Utilisation de l'AMDEC pour formaliser le retour d'expérience en utilisant une approche bayésienne[END_REF]. Thus, the operating feedback will enable to refine the probabilities contained in the CPT by learning.

It is also necessary to associate with all probabilities an uncertainty in order to undertake studies of sensitivity.

The use of the network in order to simulate the impact of a failure on the total performance of the system is easy.

The networks can also be used in design to validate the effectiveness of the corrective actions suggested following the FMECA analysis.

During the exploitation of the system, it is effective to use BN for the diagnosis. The strategy described here is only based on the probabilities of occurrence. It is necessary to improve it by taking account other parameters: aptitude of detection, costs of the components…

The taking into account of the economic aspect is also possible in the BN formalism while adding to the network, nodes of costs (e.g. costs related to the detection of a failure) and nodes of decision (e.g. strategy of maintenance).

BNs constitute a powerful tool for decision-making aid in maintenance. The model combines a priori approach in design and a posteriori approach in operation.

It remains to improve the automatic generation of BNs starting from model SADT and FMECA analysis. It is also necessary to validate the model by applying it to a real system in order to show industrial feasibility and to confirm its added value compared to the traditional computerised decision-making systems.

Fig. 2 .

 2 Fig. 2. Flows and Activity Representation.

Fig

  Fig. 3. Generic BN input and output nodes structure.

Fig. 4 .

 4 Fig. 4. High level of the functional decomposition.

Fig. 5 .

 5 Fig. 5. Low level of the functional decomposition.

Fig. 6 .

 6 Fig. 6. Diagnosis aid nodes.

Fig. 7 .

 7 Fig. 7. Level 0 of the lathe function analysis.

Fig. 8 .

 8 Fig. 8. Level 0: High BN.

Fig. 9 .

 9 Fig. 9. Level 012: Actuate BN.

  The Function Diagnosis nodes of each sub-function of the level 0 are examined after the BN inference:It is a priori "TOOL" abnormal wear which is the cause of the no conformity of the parts. The diagnosis is coherent; the LATHE is not incremented because a failure of the LATHE leads to parts in the state NR. Let us suppose that the checking of the tool shows that it is OK. Then the evidence is added to the knowledge:The states of BN are updated and a new inference proposes a new diagnosis taking account the new information. The procedure thus consists in going at the hierarchical level higher (i.e. the level 0). In this case, the failure mode is caused only from the sub-function "To grip/To loosen":

			Diagnosis	To grip/	To put in	To
				To loosen	rotation	machine
			Internal	0.6438	0	0.7122
			Upstream	0.3562	0	0
			Downstream	0	1	0.2878
	The function suspected is a priori "To machine the parts" because the probability of "Internal diagnosis" is the
	strongest. Thus the low level of this function is checked. The states of the variables representing the elements
	implied are observed and taken the following states:
			LATHE		TOOL
			OK	1		OK	0.2878
			HS	0		Wear	0.7122
						Broken	0
					TOOL
				OK		1
				Wear	0
				Broken	0
			Diagnosis	To grip/	To put in	To
				To loosen	rotation	machine
			Internal	1	0	0
			Upstream	0	1	1
			Downstream	0	0	0
	On the level "To grip/To loosen", the diagnosis nodes of sub-functions indicate that the failure is due to the sub-
	function "To actuate". Within this sub-function, the component "JACK" is the cause:
					JACK
				OK		0
				Jammed	1
				Blocked	0
				Table 2. First inference.
		HD				RWD
		Raw Part	Gripping	Putting in rotation	Machining	Loosening	Producing
	R	0.900	0.854	0.845	0.746	0.655	0.655
	NC	0.075	0.056	0.060	0.142	0.227	0.227
	NR	0.025	0.088	0.093	0.111	0.117	0.117

Structured Analysis and Design Technique

Failure Mode, Effects and Criticality Analysis


Persi Diaconis

Gilles Lebeau
email: lebeau@unice.fr

Laurent Michel

Gibbs/Metropolis algorithms on a convex polytope

This paper gives sharp rates of convergence for natural versions of the Metropolis algorithm for sampling from the uniform distribution on a convex polytope. The singular proposal distribution, based on a walk moving locally in one of a fixed, finite set of directions, needs some new tools. We get useful bounds on the spectrum and eigenfunctions using Nash and Weyltype inequalities. The top eigenvalues of the Markov chain are closely related to the Neuman eigenvalues of the polytope for a novel Laplacian.

Introduction 1.Overview

The Metropolis algorithm and the Gibbs sampler (also known as Glauber dynamics) are often used together as one of the basic tools of scientific computation. We treat the following example: let Ω be a polyhedral convex set in d dimensions. To sample from the uniform distribution on Ω, from a point x in Ω, pick a direction e from a fixed finite collection. Set y = x + ue where u is chosen uniformly in [-h, h]. If y ∈ Ω, move to y. Else, stay at x. Under a mild generality condition on the set of directions in relation to Ω, this Markov chain converges to the uniform distribution on Ω. Our main result gives a sharp determination of the exponential rate of convergence of this algorithm. It is ce -ng(h) with g(h) asymptotic to h 2 ν for ν the first non zero eigenvalue of a novel Laplacian defined on Ω with Neumann condition on the boundary.

Sampling from a convex set is a practical problem. For example, choosing a uniformly distributed 100 × 100 doubly stochastic matrix [START_REF] Chatterjee | Properties of uniform doubly stochastic matrices[END_REF] or a uniformly distributed 100 × 100 tri-diagonal doubly stochastic matrix [START_REF] Diaconis | On random, doubly stochastic, tri-diagonal matrices[END_REF]. It is also a basic problem of study in theoretical computer science [START_REF] Lovász | Random walks in a convex body and an improved volume algorithm[END_REF][START_REF] Lovász | Hit-and-run is fast and fun[END_REF]. Many algorithms have been proposed and studied. A readable textbook description of the Gibbs sampler is in Liu [START_REF] Liu | Monte Carlo Strategies in Scientific Computing[END_REF]. See [START_REF] Diaconis | What do we know about the Metropolis algorithm?[END_REF] for a review of rigorous results for the Metropolis algorithm in finite spaces. The popular hit and run algorithm [START_REF] Diaconis | Hit and run as a unifying device[END_REF][START_REF] Lovász | The geometry of logconcave functions and sampling algorithms[END_REF][START_REF] Smith | Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions[END_REF] was introduced for this purpose.

Hit and run makes long moves and will probably be preferred in practice to the local algorithms studied here.

Spectral techniques for analysis of the Metropolis algorithm on continuous spaces are developed in [START_REF] Diaconis | Micro-local analysis for the Metropolis algorithm[END_REF][START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF][START_REF] Lebeau | Semiclassical analysis of a random walk on a manifold[END_REF]. The proposal distributions there are "ball walks" choosing from the uniform distribution on the interior of a ball. The discrete set of directions studied here is widely used in practice and necessitates new ideas. Present problems can also be studied by Harris recurrence techniques [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF][START_REF] Rosenthal | Minorization conditions and convergence rates for Markov chain Monte Carlo[END_REF] and by the path techniques of Yuen [START_REF] Yuen | Applications of geometric bounds to the convergence rate of Markov chains on R n[END_REF]. These give useful results but do not get the sharp rates on the exponents derived here.

The remainder of this section gives a careful description of the Markov chain and the geometric connection between the underlying directions and the convex set Ω required for ergodicity. Section 2 gives bounds on the spectrum and eigenvectors using Nash inequalities and Weyl-type inequalities. Section 3 uses this spectral information to get rates of convergence. Section 4 proves that our operator (suitably rescaled) converges, in the strong resolvent sense, to a novel Laplace operator on Ω with Neumann boundary conditions. A similar convergence of the ball walk Metropolis operator to the usual Neumann Laplacian is a key ingredient of [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF][START_REF] Lebeau | Semiclassical analysis of a random walk on a manifold[END_REF]. The final section shows how to modify the argument to handle a continuous choice of direction.

Basic definitions

Let Ω be an open convex polytope in R d , d ≥ 2. Thus there exists linear forms j : R d → R, j = 1, . . . , m and real numbers b j such that

Ω = x ∈ R d , ∀j = 1, . . . , m, j (x) > b j (1.1)
Assume also that Ω is bounded and non empty. Consider E = {e 1 , . . . , e p } a family of vectors in R d . For any j ∈ {1, . . . , p} we introduce the operator acting on continuous functions M j,h f (x) = m j,h (x)f (x) + K j,h f (x), where

K j,h (f )(x) = 1 2 t∈[-1,1]
1 Ω (x + hte j )f (x + hte j ) dt (1.2) and m j,h (x) = 1 -K j,h (1)(x).

The local Metropolis operator associated to the family E is

M h (f )(x) = 1 p p j=1 (K j,h f (x) + m j,h (x)f (x)) . (1.3)
In the sequel, denote m h = 1 p p j=1 m j,h and K h = 1 p p j=1 K j,h . Let M h (x, dy) be the Markov kernel associated to this operator. This defines a bounded self-adjoint operator on L 2 (Ω). Moreover, since M h (1) = 1, M h L 2 →L 2 = 1. Thus the probability measure dx vol(Ω) on Ω is stationary. For n ≥ 1, denote by M n h (x, dy) the kernel of the iterated operator (M h ) n . For any x ∈ Ω, M n h (x, dy) is a probability measure on Ω, and our main goal is to get some estimates on the rate of convergence, when n → +∞, of the probability M n h (x, dy) toward the stationary probability dy vol(Ω) . A good example to keep in mind is the case where Ω = A N is the set of N × N doubly stochastic matrices. In other words,

A N = (a i,j ) 1≤i,j≤N , ∀i, j, a i,j > 0, k a ik = k a kj = 1 .
(1.4)

The set A N can be viewed as convex open polytope in A 0 N = {(a i,j ) 1≤i,j≤N , k a ik = k a kj = 1}. A good way to sample from A N is to use the Metropolis strategy in the following manner. Starting from a matrix A ∈ A N choose two distinct rows R i 1 , R i 2 and two distinct columns C j 1 , C j 2 at random. Denote i = (i 1 , i 2 , j 1 , j 2 ) and F = F ( i) the matrix such that

F i,j = δ i 1 j 1 -δ i 1 ,j 2 -δ i 2 j 1 +δ i 2 j 2 .
For h > 0 given, build the family of matrices ( Ã(t) = A + tF ( i)) t∈ [-h,h] . For any t ∈ R the matrix Ã(t) belongs to the set A 0 N . Taking t ∈ [-h, h] at random and keeping the move A → Ã(t) only if it results in an element of A N , we are exactly in the above situation with E = {F ( i)}. This algorithm is used in [START_REF] Chatterjee | Properties of uniform doubly stochastic matrices[END_REF] to study things like the distribution of typical entries or the eigenvalues of random doubly stochastic matrices.

Let us go back to the general problem. From the definition of Ω, a point x ∈ R d belongs to ∂Ω iff there exists a partition I ∪ J = {1, . . . , m} such that I = ∅ and ∀i ∈ I, i (x) = b i and ∀j ∈ J, j (x) > b j .

(1.5)

Define the following function c :

R d → N ∪ {+∞} by c(x) = 0 if x ∈ Ω = +∞ if x ∈ R d \ Ω (1.6) = card(I) if x ∈ ∂Ω.
To proceed, the following geometric condition is needed; it shows how the generating set E must be related to the convex set Ω. Proposition 1.5 shows the condition is equivalent to M h having a spectral gap.

Definition 1.1. The family E is weakly incoming to the set Ω if for any point x 0 ∈ ∂Ω there exists > 0, θ ∈ {±1} and e ∈ E such that, for c defined in (1.6),

c(x 0 + θte) < c(x 0 ) ∀t ∈]0; ]. (1.7)
The following observation is simple and fundamental. Suppose that E is weakly incoming, then span(E) = R d . Indeed, otherwise there is a hyperplane H = (Rν) ⊥ of R d such that span(E) ⊂ H. Since Ω is compact, the function x ∈ Ω → x, ν would have a global minimum in some x 0 ∈ ∂Ω.

Since Ω is open, Ω ⊂ x 0 + H + , where H + = {y ∈ R d , y, ν > 0}. As E is weakly incoming, there is u ∈ span(E) such that c(x 0 + u) = 0. In other words,

x 0 + u ∈ Ω ∩ (x 0 + H). This contradicts Ω ⊂ x 0 + H + . Example 1.1. Consider Ω the convex hull of on equilateral triangle (ABC) in R 2 and E = {e 1 , e 2 } like on Figure 1. For α ∈]0, π/3], E is weakly incoming to Ω whereas for α ∈]π/3, π[, condition (1.7) is satisfied in every point x 0 of the boundary excepted in point A.
Remark 1.2. In the above case of doubly stochastic matrices, the set E = {F ( i)} is weakly incoming. Indeed, if A is in the boundary of A N , there exists i 1 and j 1 such that A i 1 j 1 = 0. Since A is doubly stochastic, there exists i 2 , j 2 such that A i 1 j 2 > 0 and

A i 2 j 1 > 0. Let = min(A i 1 j 2 , A i 2 j 1 )/2, then for all t ∈]0, ], c(A + tF (i 1 , i 2 , j 1 , j 2 )) < c(A).
Denote H k = ker( k ) and let ν k be the unit vector such that k (ν k ) > 0 and

H + k = {y ∈ R d , k (y) > b k }.
H k if u, ν k ≥ 0. Further, u is strictly incoming to H k if u, ν k > 0; u is strictly outgoing to H k if u, ν k < 0; u is parallel to H k if u ∈ H k .
Lemma 1.3. Suppose that E is weakly incoming to Ω and let x 0 ∈ Ω and k = c(x 0 ). There exists r > 0 and I ⊂ {1, . . . , m} such that I = k and for B(x 0 , r) the open ball of radius r about x 0 ,

Ω ∩ B(x 0 , r) = B(x 0 , r) ∩ ∩ i∈I H + i . (1.8)
Further, there exists β 1 , . . . , β k , ∈ {1, . . . , p}, a family (θ n ) n=1,...,k of numbers in {±1} and a bijection {1, . . . , k} n → i n ∈ I such that for all n ∈ {1, . . . , k},

θ n e βn is strictly incoming to H in ;

θ n e βn is incoming to H im , ∀m > n.

(1.9)

Proof. Proceed by induction on k = c(x 0 ). When k = 0, there is nothing to prove. Suppose that the property holds true at rank k ≤ k -1 and let x 0 ∈ ∂Ω be such that c(x 0 ) = k. By definition of Ω, there exists r > 0 and I ⊂ {1, . . . , m} with I = k such that

Ω ∩ B(x 0 , r) = B(x 0 , r) ∩ (∩ i∈I H + i ) (1.10)
Since E is weakly incoming to Ω, there exists q ∈ {1, . . . , k}, θ 1 = ±1, β 1 ∈ {1, . . . , p} and i 1 , . . . , i q ∈ I such that θ 1 e β 1 is strictly incoming to H in for n = 1, . . . , q and θ 1 e β 1 is parallel to H i for i ∈ I := I \ {i 1 , . . . , i q }. By definition of Ω there exists x 0 close to x 0 and r > 0 such that B(x 0 , r

) ∩ Ω = B(x 0 , r ) ∩ (∩ i∈I H + i ).
From the induction hypothesis, there exists β q+1 , . . . , β k ∈ {1, . . . , p}, θ q+1 , . . . , θ k = ±1 and a bijection {q + 1, . . . , k} n → i n ∈ I such that for all n ≥ q + 1,

θ n e βn is strictly incoming to H in ;

θ n e βn is incoming to H im ∀m > n.

(1.11) and the proof is complete.

Corollary 1.4. Suppose that E is weakly incoming to Ω. There exist r > 0 and ∈]0, 1], such that for all x 0 ∈ Ω, there exists q ∈ {1, . . . , p} and θ q = ±1 such that

x + tθ q e q ∈ Ω ∀x ∈ B(x 0 , r) ∩ Ω, ∀t ∈ [0, ].

(1.12)

Proof. The fact that r, > 0 can be chosen uniformly with respect to x 0 follows easily from compactness of Ω. The statement is trivial when x 0 ∈ Ω. Suppose that x 0 ∈ ∂Ω. From Lemma 1.3, there exists r > 0 and I = {i 1 , . . . , i k } ⊂ {1, . . . , p} such that

Ω ∩ B(x 0 , 2r) = B(x 0 , 2r) ∩ ∩ i∈I H + i (1.13)
and θ 1 = ±1, β 1 ∈ {1, . . . , p} such that θ 1 e β 1 is strictly incoming to H i 1 ;

θ 1 e β 1 is incoming to H iq , ∀q > 1.

(1.14)

Let x ∈ B(x 0 , r) ∩ Ω and ∈]0, r[. Then

ν i , θ 1 β 1 ≥ 0, ∀i ∈ I =⇒ x + tθ 1 e β 1 ∈ Ω. (1.15)
Thanks to (1.14), the left hand side of the above property is satisfied and the proof is complete.

Proposition 1.5. The family E is weakly incoming to Ω iff 1 is not in the essential spectrum of M h .

Proof. If E is weakly incoming to Ω, 1 is not in the essential spectrum of M h thanks to Proposition 2.2 of this paper and Theorem 1.1 in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF]. Suppose now that E is not weakly incoming to Ω. This means that there exists x 0 ∈ ∂Ω such that (1.7) does not hold. Let k = c(x 0 ). There exists a neighborhood V of x 0 and I ⊂ {1, . . . , m} with I = k such that V ∩ Ω = V ∩ (∩ i∈I H i ). Then, for any θ = ±1 and any j ∈ {1, . . . , p}, the following holds true: If θe j is strictly incoming to one of the (H i ) i∈I , then θe j is strictly outgoing to one of the (H i ) i∈I .

Otherwise, there is j ∈ {1, . . . , p} and θ = ±1 such that θe j is strictly incoming to one of the (H i ) i∈I and incoming to the other. Then for t > 0 small enough, c(x 0 + θte j ) < c(x 0 ).

Hence, assume that there exists r ≥ 1 such that

• for any j ∈ {1, . . . , r}, e j and -e j are strictly outgoing to some of the (H i ) i∈I ;

• for any j ∈ {r + 1, . . . , p}, e j is parallel to the (H i ) i∈I .

Recall that ν i denotes the unit incoming orthogonal vector to

H i . Let W = span(ν i , i ∈ I) and near x 0 use the variable x = x 0 + (x , x ) with x ∈ W and x ∈ W ⊥ . Let χ(x ) = 1l 1 2 <|x |<1
and for

λ, h > 0 denote f λh (x) = (λh) -dim(W )/2 χ( x λh ). Since any v ∈ W ⊥ is parallel to the (H i ) i∈I , there exists λ 0 , c 0 > 0 such that for all h ∈]0, 1] and λ ∈]0, λ 0 ], f λh L 2 (Ω) ≥ c 0 .
For any j ∈ {r + 1, . . . , p}, e j is parallel to the (H i ) i∈I . Hence, the function t → f λh (x + hte j ) is constant and (M j,h -1)f λh (x) = 0.

On the other hand, for any j ∈ {1, . . . , r} there exists i j , i j ∈ I such that e j is strictly outgoing to H i j and -e j is strictly outgoing to H i j . Consequently, there exists γ j , δ j > 0 such that for t > 0,

x ∈ Ω and x -te j ∈ Ω =⇒ dist(x -te j , H i j ) ≤ dist(x, H i j ) -γ j t x ∈ Ω and x + te j ∈ Ω =⇒ dist(x + te j , H i j ) ≤ dist(x, H i j ) -δ j t. (1.16)
Let us compute the potential m j,h on the support of f λh . For x ∈ supp(f λh ), |x j | ≤ λh for all j = 1, . . . , r. In particular dist(x, H i j ) ≤ λh and dist(x, H i j ) ≤ λh and thanks to (1.16),

1 -m j,h (x) = 1 0 1 Ω (x + hte j ) + 1 Ω (x -hte j ) dt ≤ 0≤t≤dist(x,H i j )/(δ j h) dt + 0≤t≤dist(x,H i j )/(γ j h) dt ≤ λ 1 γ j + 1 δ j .
(1.17)

Finally,

(

-M h )f λh , f λh L 2 (Ω) = 1 p r j=1 (1 -M j,h )f λh , f λh L 2 (Ω) ≤ 1 p r j=1 Ω (1 -m j,h (x)) |f λh (x)| 2 dx ≤ Cλ f λh 2 L 2 (Ω) . 1
(1.18)

Here we used the fact that for any non-negative fonction f , one has K j,h f, f ≥ 0. Finally, we conclude by taking λ = 2 -n → 0 as n → ∞. Indeed, the functions f 2 -n h are mutually orthogonal. Their norm is bounded uniformly from below and they satisfy 0

≤ (1 -M h )f 2 -n h , f 2 -n h ≤ C2 -n .

Spectral Analysis of the Metropolis Operator

This section is devoted to the analysis of the spectral theory of the Metropolis operator. For this purpose, we introduce a Laplace operator associated to the family E to be used as a model. For any e ∈ R d \ {0} and any smooth function u, define ∂ e u(x) = d dt (u(x + te)) |t=0 . Then, consider the operator ∆ E , defined by

∆ E u = 1 6p p j=1 ∂ 2 e j u D(∆ E ) = u ∈ H 1 (Ω), ∆ E u ∈ L 2 , ∂ n,E u |∂Ω = 0 (2.1) with ∂ n,E u(x) = 1 p p j=1 n(x
), e j ∂ e j u(x), n(x) denoting the outgoing normal vector to the boundary at point x. If the domain Ω has smooth boundary, the normal derivative is well defined. In the case where it is Lipschitz, it can be defined by duality in the following way.

Define first the gradient and divergence associated to the family E, by div

E u = 1 p p j=1 ∂ e j u j for any u = (u 1 , . . . , u p ) and ∇ E u = (∂ e 1 u, . . . , ∂ ep u). Then, define a trace operator γ E by γ E : u ∈ L 2 (Ω) p , div E (u) ∈ L 2 (Ω) → H -1/2 (∂Ω) (2.2)
and and for v ∈ H 1 (Ω),

Ω div E (u)(x)v(x) dx = - 1 p Ω u(x), ∇ E v(x) C p dx + ∂Ω γ E (u)v| ∂Ω dσ(x). (2.3) In particular, for u ∈ H 1 (Ω) satisfying ∆ E u = 1 6 div E ∇ E u ∈ L 2 (Ω) define ∂ n,E u| ∂Ω = γ E (∇ E u) ∈ H -1/2 (∂Ω) and the set D(-∆ E ) is well defined. The Dirichlet form associated with -∆ E is E E (u) = 1 6p p j=1 Ω |∂ e j u(x)| 2 dx. (2.4)
Let E 0 be the canonical basis in R d . Then, ∆ E 0 = 1 6d ∆ where ∆ is the usual Laplace operator and E E 0 (f ) = 1 6d Ω |∇f | 2 dx is the usual Dirichlet form. Since E = {e 1 , . . . , e p } spans R d , a simple calculation shows that there exists a constants C > 0 such that

C -1 E E 0 (f ) ≤ E E (f ) ≤ CE E 0 (f ). (2.5)
Then, it is standard to show that -∆ E is the self-adjoint realization of the Dirichlet form E E . A standard argument using Sobolev embedding shows that -∆ E has compact resolvant. Denote its spectrum by ν 0 = 0 < ν 1 < ν 2 < . . . and by m j the associated multiplicities. Observe that m 0 = 1. Section 4 shows that h -2 (1 -M h ) converges to -∆ E in the strong resolvent sense so that eigenvalues and eigenvectors converge; see [START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of Operators[END_REF].

The main theorem of this section follows.

Theorem 2.1. Suppose that E is weakly incoming to Ω, then the following hold true.

i) There exists h 0 > 0, δ 0 ∈]0, 1 2 [ and a positive constant C such that for any h ∈]0,

h 0 ], the spectrum of M h is a subset of [-1 + δ 0 , 1], 1 is a simple eigenvalue and Spec(M h ) ∩ [1 -δ 0 , 1] is discrete. ii) For any h ∈]0, h 0 ] and 0 ≤ λ ≤ δ 0 h -2 , the number of eigenvalues of M h in [1 -h 2 λ, 1] (with multiplicity) is bounded by C(1 + λ) d/2 .
iii) For any R > 0 and ε > 0 such that ν j+1 -ν j > 2ε for ν j+2 < R, there exists h 1 > 0 such that one has for all h ∈]0,

h 1 ], Spec 1 -M h h 2 ∩]0, R] ⊂ ∪ j≥1 [ν j -ε, ν j + ε], (2.6)
and the number of eigenvalues of

1-M h h 2 in the interval [ν j -ε, ν j + ε] is equal to m j .
A consequence of this theorem is that M h has a spectral gap g(h) = 1-sup(Spec(M h )\{1}) > 0 and that lim h→0 + h -2 g(h) = ν 1 . This will be used in the proof of total variation estimates.

The strategy used to prove the first part of Theorem 2.1 is very close to the one given in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF]. First, show that some iterate of the Markov kernel "controls" the random walk on a ball. Next, this ball walk on the polytope is compared to the same walk on a large torus containing Ω. Finally the information on the torus is transferred back to the original problem.

The proof of the last part of Theorem 2.1 is slightly different from the proof in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF]. Indeed, the starting point of the analysis in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF] is that for regular function ϕ with normal derivatives vanishing on the boundary, h -2 (1 -T h )ϕ is close to -∆ϕ up to the boundary, where T h is the Metropolis operator associated to the kernel vol(B(0, 1)) -1 h -d 1 |x-y|<h . Here, this property fails to be true. Suppose for instance that Ω ⊂ R 2 and that its boundary is given near (0, 0) by x 1 ≥ 0. Suppose that e 1 = (a, b) and e 2 = (b, -a) for some a, b > 0. Then

h -2 (1 -M 1,h )f (x) = 1 2h 3 |t|<h,x+te 1 ∈Ω (f (x) -f (x + te 1 )) dt = - 1 2h 3 ∂ e 1 f (x) |t|<h,x+te 1 ∈Ω t dt + O(1) = 1 4h 3 ∂ e 1 f (x)1 ]0,ah] (x 1 ) h 2 - x 2 1 a 2 + O(1).
(2.7)

A similar expression holds for M 2,h and summing these equalities gives

h -2 (1 -M h )f (x) = 1 4h 3 ∂ e 1 f (0, x 2 )1 ]0,ah] (x 1 ) h 2 - x 2 1 a 2 + ∂ e 2 f (0, x 2 )1 ]0,bh] (x 1 ) h 2 - x 2 1 b 2 + O(1) (2.8) If a = b, ∂ e 1 f + ∂ e 2 f
is proportional to the normal derivative of f and hence, the above quantity is bounded. Suppose now that a < b. Then the above quantity is bounded on x 1 ∈ [ah, bh] provided ∂ e 2 f (0, x 2 ) = 0. Then the same argument on [0, ah] shows that ∂ e 1 f (0, x 2 ) = 0 also.

In order to avoid these difficulties, we work directly on the quadratic form and show that the Dirichlet form associated to the Metropolis operator converges to the Dirichlet form of the Laplace operator with Neuman boundary conditions. The end of this section is devoted to the proof of Theorem 2.1.

Proposition 2.2. There exists N ∈ N and constants c 1 , c 2 > 0 such that for all h ∈]0, 1]

M N h (x, dy) = µ h (x, dy) + c 1 h -d 1 |x-y|<c 2 h dy (2.9)
where for all x ∈ Ω, µ h (x, dy) is a positive Borel measure.

Proof. The proof follows the lines of [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF].

Denote K h = 1 p p j=1 K j,h . Since for any h 2 > h 1 > 0 and any non-negative function f , h 2 K h 2 f ≥ h 1 K h 1 f , it
is sufficient to prove the following: there exists h 0 > 0, c 1 , c 2 > 0 and N ∈ N * such that for all h ∈]0, h 0 ], one has, for all non-negative continuous functions f ,

K N h (f )(x) ≥ c 1 h -d y∈Ω,|x-y|≤c 2 h
f (y) dy.

(2.10)

First note that it is sufficient to prove the weaker version: for all x 0 ∈ Ω, there exist N (x 0 ), α = α(x 0 ) > 0, c 1 = c 1 (x 0 ) > 0, c 2 = c 2 (x 0 ) > 0, h 0 = h 0 (x 0 ) > 0 such that for all h ∈]0, h 0 ], all x ∈ Ω and all non-negative functions

f |x -x 0 | ≤ 2α =⇒ K N (x 0 ) h (f )(x) ≥ c 1 h -d y∈Ω,|x-y|≤c 2 h f (y) dy. (2.11)
Let us verify that (2.11) implies (2.10). Decreasing α(x 0 ) if necessary, it may be assumed that 2α(x 0 ) < r(x 0 ), where r(x 0 ) is given by Lemma 1.3. Since Ω is compact, there exists a finite set

F such that Ω ⊂ ∪ x 0 ∈F {|x -x 0 | < α(x 0 )}. Let N = sup{N (x 0 ), x 0 ∈ F }, c i = min x 0 ∈F c i (x 0 )
and h 0 = min x 0 ∈F h 0 (x 0 ). One has to check that for any x 0 ∈ F and any x with |x -x 0 | ≤ α(x 0 ), the right inequality in (2.11) holds true with N = N (x 0 ) + n in place of N (x 0 ) for some constants c 1 , c 2 , h 0 . Moreover, one may assume that h 0 max |e j | ≤ min x 0 ∈F α(x 0 )/N . Let > 0, q ∈ {1, . . . , p} and θ q = ±1 be given by Corollary 1.4. Then for |x-

x 0 | < (2-1 N )α(x 0 ), one has K N (x 0 )+1 h f (x) ≥ 1 p K h,βq K N (x 0 ) h f (x) ≥ 1 p 1 0 K N (x 0 ) h f (x + htθ q e βq ) dt ≥ c 1 h -d p min( , c 2 2 max j |e j | ) 0 y∈Ω, |y-x-htθqe βq |<c 2 h f (y) dydt ≥ c 0 h -d y∈Ω, |y-x|<c 2 h/2
f (y) dy It remains to prove (2.11). If x 0 ∈ Ω, the proof is obvious. Indeed, since E spans R d , it is easy to see that for any δ > 0, there exists c 3 , c 4 > 0 such that for any non-negative function f , dist(y, ∂Ω)

≥ δh =⇒ K d h (f )(y) ≥ c 3 h -d z∈Ω, |y-z|<c 4 h f (z) dz ∀y ∈ Ω. (2.13)
Suppose that x 0 ∈ ∂Ω and denote k = c(x 0 ). Let (i j ) 1≤j≤k , (β j ) 1≤j≤k , (θ j ) 1≤j≤k be as in Lemma

1.3. Let 1 = γ 1 > γ 2 > • • • > γ k > 0 and δ 1 , .
. . , δ k > 0 be such that for all j, γ j -δ j > γ j+1 . Let G j = [γ j -δ j , γ j ] and G = Π k j=1 G j . In the following computation, c denotes a positive constant independant of f and h that may change from line to line. Since f is non-negative,

K k h (f )(x) ≥ p -k K β 1 ,h . . . K β k ,h f (x) ≥ c t∈A h (x) f (x + h k j=1 θ j t j e β j ) dt (2.

14)

where A h (x) = {t = (t 1 , . . . , t k ) ∈ G, ∀l = 1, . . . , k, x + h l j=1 θ j t j e β j ∈ Ω}. Since θ 1 e β 1 is strictly incoming to H i 1 , there exists some constant c 5 , c 6 > 0 such that for any

t ∈ I, dist   x + h k j=1 θ j t j e β j , H i 1   ≥ c 5 ht 1 -c 6 h(t 2 + • • • + t k ) ≥ c 5 h(γ 1 -δ 1 ) -c 6 h(γ 2 + • • • + γ k ) ≥ c 5 h(γ 1 -δ 1 )/2
(2.15) by taking γ 2 , . . . , γ k small with respect to γ 1 . Similarly, by taking γ j very small with respect to γ j+1 for j = 2, . . . , k, there is c 7 > 0 such that for any j = 1, . . . , k,

∀(t 1 , . . . , t j ) ∈ G 1 × • • • × G j , dist x + h j i=1 θ i t i e β i , R d \ Ω ≥ c 7 h.
(2.16)

Hence,

K k h f (x) ≥ c t∈G f   x + h k j=1 θ j t j e β j   dt (2.17)
and for any N ≥ 0

K k+N h f (x) ≥ c t∈G K N h (f )   x + h k j=1 θ j t j e β j   dt. (2.18)
Combining (2.13), (2.16) and (2.18), there is c 8 > 0 small enough such that any y ∈ R d such that |x + h k j=1 t j e β j -y| < c 8 h belongs to Ω and hence

K d+k h f (x) ≥ ch -d t∈G |x+h P k j=1 t j e β j -y|<c 8 h f (y) dydt. (2.19) Since, K k h f (y) ≥ p -k K h,β k . . . K h,β 1 f (y), then K d+2k h f (x) ≥ ch -d (t,s,y)∈B h (x) f (y -h k j=1
s j e β j ) dtdsdy (2.20)

where

B h (x) =    (t, s, y) ∈ G × G × R d , |x + h k j=1 θ j t j e β j -y| < c 8 h and ∀l = 1, . . . , k, y -h k j=l θ j s j e β j ∈ Ω    . (2.21)
Using the new variable z = y -h k j=1 θ j s j e β j ,

K d+2k h f (x) ≥ ch -d (t,s,z)∈D h (x) f (z) dtdsdz (2.22) with D h (x) =    (t, s, z) ∈ G × G × Ω, |x + h k j=1 (t j -s j )θ j e β j -z| < c 8 h and ∀l = 1, . . . , k, z + h l-1 j=1 θ j s j e β j ∈ Ω    . (2.23)
Since in the above integral, |t j -s j | < δ j , taking the δ j 's small enough gives

D h (x) ⊃    (t, s, z) ∈ G × G × Ω, |x -z| < c 8 h/2, ∀l = 1, . . . , k, z + h l-1 j=1 θ j s j e β j ∈ Ω    . (2.24)
Now using (2.16), it follows that

D h (x) ⊃ {(t, s, z) ∈ G × G × Ω, |x -z| < c 8 h/2} . (2.25)
Combined with (2.22), this yields the announced result.

Following the strategy of [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF], introduce the Dirichlet form associated to the iterated kernel M k h :

E h,k (u) = 1 -M k h u, u L 2 (Ω)
.

(2.26) Also, put Ω in a large box B =] -A/2, A/2[ d and define an extension map E : L 2 (Ω) → L 2 (B) which is continuous from H 1 (Ω) into H 1 (B) and vanishes far from Ω. This is possible since ∂Ω has Lipschitz regularity. Finally, introduce the Dirichlet form on B:

Ẽh (u) = h -d B×B,|x-y|<h |u(x) -u(y)| 2 dxdy. (2.27)
Then Proposition 2.2 easily yields the following (see [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF] for details).

Lemma 2.3. There exists C 0 , h 0 > 0 such that for any h ∈]0, h 0 ] and any u ∈ L 2 (Ω),

Ẽh (E(u)) ≤ C 0 E h,N (u) + h 2 u 2 L 2 (Ω) .
(2.28)

Moreover, any function u ∈ L 2 (Ω) such that u 2 L 2 (Ω) + h -2 (1 -M h )u, u L 2 (Ω) ≤ 1 admits a decomposition u = u L + u H with u L ∈ H 1 (Ω), u L H 1 ≤ C 1 , and u H L 2 ≤ C 1 h.
We are now in position to prove the first part of Theorem 2.1. First, assume that M h u = u. Then, it follows from Proposition 2.2, that

c 1 h -d Ω×Ω,|x-y|<c 2 h (u(x) -u(y)) 2 dxdy ≤ Ω×Ω (u(x) -u(y)) 2 M N h (x, dy) dx. (2.29)
On the other hand, the right hand side in the above inequality is equal to E h,N (u) which is actually equal to zero. Hence, u is constant and 1 is a simple eigenvalue.

Using the Markov property of M N h , positivity of µ h and the fact that ∂Ω has Lipschitz regularity, easily yields

µ h L ∞ →L ∞ = µ h (Ω) ≤ 1 -c 1 h -d min x∈Ω Ω 1 |x-y|<c 2 h dy < 1 -δ 0 (2.30)
for some δ 0 > 0 independent of h. Working as in the proof of Theorem 1 in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF] shows that there exists δ 0 ∈]0, 1 2 [ such that for any u ∈ L 2 (Ω) and any n ≥ N ,

M n h u, u L 2 (Ω) ≥ (-1 + δ 0 ) u 2 L 2 (Ω) . (2.31)
Hence, the same holds true for n = 1 with a possibly different δ 0 .

To show that there is δ 0 > 0 sufficiently small so that the spectrum of M h is discrete in [1-δ 0 , 1] it suffices to work as in the proof of Theorem 4.6 in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF], using again Proposition 2.2.

Similarly, the Weyl bound on the number of eigenvalues follows from Lemma 2.3 as in Lemma 4.8 in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF]. This proves Part i.

To prove the last part of the theorem, work on the Dirichlet form is needed. In the following, denote E h = E h,1 . Introduce the bilinear form associated with E h :

B h (u, v) = (1 -M h )u, v L 2 (Ω) , ∀u, v ∈ L 2 (Ω).
(2.32)

A standard computation shows that B h (u, v) = 1 p p j=1 B j,h (u, v) with B j,h (u, v) = 1 4h x∈Ω,x+te j ∈Ω,|t|<h (u(x) -u(x + te j )) (v(x) -v(x + te j )) dxdt (2.33) Lemma 2.4. Let θ ∈ C ∞ (Ω) be fixed and let (ϕ h , r h ) ∈ H 1 (Ω) × L 2 (Ω) be such that r h L 2 (Ω) = O(h) and ϕ h converges weakly in H 1 (Ω) to some ϕ. Then lim h→0 + h -2 B h (r h , θ) = 0 (2.34)
and lim

h→0 + h -2 B h (ϕ h , θ) = 1 6p Ω ∇ E ϕ(x), ∇ E θ(x) C p dx. ( 2

.35)

Proof. To prove (2.34), observe that since θ is smooth,

(1 -M j,h )θ(x) = h -1 2 |t|<h,x+te j ∈Ω (θ(x) -θ(x + te j )) dt = ∂ e j θ(x) 2h |t|<h,x+te j ∈Ω t dt + O(h 2 ).
(2.36) Denoting

ρ h (x) = ∂ e j θ(x) 2h |t|<h,x+te j ∈Ω t dt observe that supp(ρ h ) ⊂ {x ∈ Ω, d(x, ∂Ω) < h} and ρ h L ∞ = O(h). Hence ρ h L 2 = O(h 3/2 ) and since r h L 2 = O(h), it follows that h -2 B j,h (r h , θ) = h -2 r h , (1 -M j,h )θ L 2 = h -1 r h , h -1 ρ h L 2 + O(h) = O(h 1/2 ) (2.37)
which goes to zero as h goes to zero.

To prove (2.35) observe that

θ(x + te j ) -θ(x) = tψ(t, x) ϕ h (x + te j ) -ϕ h (x) = t 1 0 ∂ e j ϕ h (x + tze j ) dz (2.38)
with ψ(t, x) smooth and ψ(0, x) = ∂ e j θ(x). Hence

h -2 B j,h (ϕ h , θ) = 1 4h
u 2 ∂ e j ϕ h (x)ψ(hu, x -huze j ) dudzdx.
Taylor expansion of ψ shows that ψ(hu, x -huze j ) = ∂ e j θ(x) + O(h). Hence, for any δ > 0 and any h ∈]0, 1],

h -2 B j,h (ϕ h , θ) = 1 4 x-huze j ∈Ω,x+hu(1-z)e j ∈Ω,|u|<1,z∈[0,1] u 2 ∂ e j ϕ h (x)∂ e j θ(x) dudzdx + O(h) = I δ (h) + J δ (h) + O(h) (2.40)
with I δ (h) equal to the above integral over d(x, ∂Ω) ≥ δ and J δ (h) the integral over d(x, ∂Ω) < δ.

Then, by Cauchy-Schwartz, |J δ (h)| ≤ C(θ)δ 1/2 ϕ h H 1 . On the other hand, for any h ∈]0, δ[,

I δ (h) = 1 6 x∈Ω,d(x,∂Ω)>δ ∂ e j ϕ h (x)∂ e j θ(x) dx = 1 6 x∈Ω ∂ e j ϕ h (x)∂ e j θ(x) dx + O δ 1/2 ϕ h H 1 .
(2.41)

Given > 0, it is easy to find δ > 0 small enough such that for any h ∈]0, δ[, |J δ (h)| < and |I δ (h) - 1 6 x∈Ω ∂ e j ϕ h (x)∂ e j θ(x)dx| < . Now make h → 0 + , δ being fixed, and use the fact that ϕ h converges weakly in H 1 to get lim

h→0 + h -2 B j,h (ϕ h , θ) = 1 6 Ω ∂ e j ϕ(x)∂ e j θ(x) dx (2.42)
and the proof is complete.

To complete the proof of Theorem 2.1, denote

|∆ h | = h -2 (1 -M h ). Let R > 0 be fixed and observe that if ν h ∈ [0, R] and f h ∈ L 2 (Ω) satisfy |∆ h |f h = ν h f h and f h L 2 = 1, then, thanks to Lemma 2.3, f h can be decomposed as f h = ϕ h + r h with r h L 2 (Ω) = O(h) and ϕ h bounded in H 1 .
Hence (extracting a subsequence if necessary) it may be assumed that ϕ h weakly converges in H 1 to a limit ϕ and that ν h converges to a limit ν. It now follows from Lemma 2.4 that for any

θ ∈ C ∞ (Ω), 1 6p Ω ∇ E f (x), ∇ E θ(x) C p dx = ν ϕ, θ L 2 . (2.43)
Since θ is arbitrary, it follows that (-∆ E -ν)ϕ = 0 and ∂ n,E ϕ |∂Ω = 0. In fact, this also proves that for any > 0 small, there exists h > 0 such that for h ∈]0, h ], one has

Spec(|∆ h |) ∩ [0, R] ⊂ ∪ j [ν j -, ν j + ] (2.44) and Spec(|∆ h |) ∩ [ν j -, ν j + ] ≤ m j (2.45)
In fact, there is equality in (2.45). The following proof is a simplification of the one in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF]. Proceed by induction on j: let > 0, small, be given such that for 0 ≤ ν j ≤ M +1, the intervals I j = [ν j -, ν j + ] are disjoint. Let (µ h j ) j≥0 be the increasing sequence of eigenvalues of |∆ h |, σ N = N j=1 m j and (e k ) k≥0 an othonormal basis of eigenfunctions of -∆ E such that for all k ∈ {1 + σ N , . . . , σ N +1 }, one has (-∆ E -ν N +1 )e k = 0. As 0 is a simple eigenvalue of both -∆ E and |∆ h |, clearly ν 0 = µ 0 = 0 and

m 0 = 1 = Spec(|∆ h |) ∩ [ν 0 -, ν 0 + ].
Suppose that for all n ≤ N ,

m n = Spec(|∆ h |) ∩ [ν n -, ν n + ]. Then by (2.44), for h ≤ h ε , µ h 1+σ N ≥ ν N +1 -. (2.46)
By the min-max principle, if G is a finite dimensional subspace of

H 1 with dim(G) = 1 + σ N +1 , µ h σ N +1 ≤ sup ψ∈G, ψ =1 |∆ h |ψ, ψ L 2 (Ω) . (2.47)
Let G be the vector space spanned by the e k , 0

≤ k ≤ σ N +1 . Then, dim(G) = 1 + σ N +1 and it follows from Lemma 2.4, for any k, k ≤ 1 + σ N +1 , lim h→0 + h -2 B h (e k , e k ) = 1 6p Ω ∇ E e k (x), ∇ E e k (x) C p dx. (2.48)
Hence lim

h→0 + h -2 B h (ψ, ψ) = 1 6p Ω |∇ E ψ(x| 2 dx ≤ ν N +1 (2.49)
for any ψ ∈ G with ψ L 2 = 1. Since G has finite dimension, a standard compactness argument shows that there exists h > 0 such that for any h ∈]0, h ] and any ψ ∈ G with ψ L 2 ≤ 1,

h -2 B h (ψ, ψ) ≤ ν N +1 + . ( 2
N +1 = Spec(|∆ h |) ∩ [ν N +1 -, ν N +1 + ].
The proof of Theorem 2.1 is complete.

Total Variation Estimates

This section gives estimates on the convergence speed of the iterated kernel M n h (x, dy) towards its stationary measure dy V ol(Ω) . Recall that the total variation µ -ν T V between two probability measures µ and ν on Ω is defined by

µ -ν T V = sup |µ(A) -ν(A)| (3.1)
where the sup is taken over all measurable sets A. Equivalently,

µ -ν T V = 1 2 sup f ∈L ∞ , f L ∞ =1 |µ(f ) -ν(f )|. (3.2) Theorem 3.1.
Assume that E is weakly incoming. Then there exists C > 0 and h 0 > 0 such that for all h ∈]0, h 0 ] and all n ∈ N, the following estimate holds true, with g(h) the spectral gap studied in Section 2:

sup x∈Ω M n h (x, dy) - dy vol(Ω) T V ≤ Ce -ng(h) . (3.3)
Proof. The proof is very close to the proof of Theorem 4.6 in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF] and is just sketched for the reader's convenience. Observe first that n ≥ h -2 can be assumed, since otherwise the estimate is trivial thanks to the lower bound on the spectral gap. Let Π 0 be the othogonal projector in L 2 (Ω) on the constant functions. Observe that

2 sup x∈Ω M n h (x, dy) - dy vol(Ω) T V = M n h -Π 0 L ∞ →L ∞ . (3.4)
Using the spectral decomposition of M h , let 0

< λ 1,h ≤ • • • ≤ λ j,h ≤ • • • ≤ h -2 δ 0 be such that the eigenvalues of M h in the interval [1 -δ 0 , 1] are the 1 -h 2 λ j,h with associated orthonormalized eigenfunctions M h (e j,h ) = (1 -h 2 λ j,h )e j,h . Then write M h -Π 0 = M h,1 + M h,2 + M h,3
, so that the operators M h,1 , M h,2 have kernels

M h,1 (x, y) = λ 1,h ≤λ j,h ≤h -α (1 -h 2 λ j,h )e j,h (x)e j,h (y) (3.5)
M h,2 (x, y) = h -α ≤λ j,h ≤h -2 δ 0 (1 -h 2 λ j,h )e j,h (x)e j,h (y) (3.6)
where α ∈]0, 2] is a small constant that will be chosen later. Then

2 sup x∈Ω M n h (x, dy) - dy vol(Ω) T V ≤ 3 j=1 M n h,j L ∞ →L ∞ (3.7)
and terms on the right hand side must be estimated. From (2.30), it is easy to prove that any eigenfunction M h (u) = λu with λ ∈]1 -δ 0 , 1] satisfies

u L ∞ ≤ Ch -d/2 u L 2 . (3.8)
As in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF], using in particular the bound on the number of eigenvalues, we show that for n ∈ N,

M n h,2 L ∞ →L ∞ + M n h,3 L ∞ →L ∞ ≤ C (1 -h 2-α ) n + (1 -δ 0 ) n h -3d/2 (3.9)
For n ≥ h -2 , this implies that . Let E α denote the space spanned by the eigenvectors e j,h such that λ j,h ≤ h -α . Then, thanks to Part ii of Theorem 2.1, dim(E α ) ≤ h -dα/2 . As in [START_REF] Diaconis | Geometric analysis for the Metropolis algorithm on Lipshitz domains[END_REF], Lemma 2.3 shows that there exists α > 0 and p > 2 such that for any u ∈ E α , .11) This gives the following Nash estimate, with 1 D = 2 -4 p > 0:

M n h,2 L ∞ →L ∞ + M n h,3 L ∞ →L ∞ ≤ C α e -nh
u 2 L p ≤ Ch -2 E h,N (u) + h 2 u 2 L 2 . ( 3
u 2+ 1 D L 2 ≤ Ch -2 E h,N (u) + h 2 u 2 L 2 u 1 D L 1 ∀u ∈ E α .
(3.12)

This inequality allows an estimate of M h,1 from L 1 into L 2 and this leads to

M kN h,1 L ∞ →L ∞ ≤ Ce -kN g(h) for k ≥ h -2 .
As M h is bounded by 1 on L ∞ it follows that kN can be replaced by n ≥ h -2 in this estimate, and the proof of Theorem 3.1 is complete.

Convergence of the Resolvants

Let us denote |∆

h | = h -2 (1 -M h ).
Recall ∆ E from (2.1). This section proves strong resolvent convergence of |∆ h | to ∆ E . For background and consequences, see [START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of Operators[END_REF].

Theorem 4.1. Let z ∈ C \ [0, +∞[ and g ∈ L 2 (Ω). Then lim h→0 + (|∆ h | -z) -1 g -(-∆ E -z) -1 g L 2 (Ω) = 0. ( 4.1)
Proof. Let z ∈ C \ [0, +∞[ and g ∈ L 2 (Ω) be fixed. For any h > 0 let f h ∈ L 2 (Ω) be the solution of (|∆ h | -z)f h = g. Hence

-z f h , f h L 2 + 1 -M h h 2 f h , f h L 2 = g, f h L 2 . (4.2) Since z / ∈ [0, ∞[ and |∆ h | is a positive operator, it follows that f h L 2 ≤ dist(z, [0, ∞[) -1 g L 2 is
bounded uniformly with respect to h. It follows from the above equation that there exists C 0 > 0 such that

f h 2 L 2 + h -2 E h (f h ) ≤ C 0 g 2 L 2 . (4.3)
It now follows from Lemma 2.4 that there exists C > 0 depending on z and g L 2 such that for any h ∈]0, 1], we can write

f h = ϕ h + r h with ϕ h H 1 ≤ C and r h L 2 ≤ Ch. Let f ∈ H 1 (Ω)
and (h k ) k∈N be a sequence of positive numbers such that (ϕ

h k ) k converges weakly to f in H 1 . Let θ ∈ C ∞ (Ω) be fixed. Then -z f h k , θ L 2 + h -2 k B h k (f h k , θ) = g, θ L 2 (4.4)
and taking the limit k → ∞ it follows from Lemma 2.4 that

-z f, θ L 2 + 1 6p Ω ∇ E f (x)∇ E θ(x) dx = Ω g(x)θ(x) dx. (4.5)
Since θ is arbitrary, this implies (-∆ E -z)f = g and ∂ n,E f |∂Ω = 0. Since, this is true for any subsequence (h k ), this shows that f h -f L 2 → 0 when h → 0, which is exactly (4.1).

Some Generalizations

Here we present a possible generalization of the previous results. It is still assumed that Ω is a convex polytope in R d . Suppose that E ⊂ R d is endowed with a Borel probability measure µ. For any e ∈ E, define

K e,h f (x) = 1 2 t∈[-1,1],x+hte∈Ω f (x + hte) dt (5.1)
and

K h f (x) = e∈E K e,h f (x) dµ(e).
(5.

2)

The associated Metropolis operator is defined by

M h f (x) = m h (x)f (x) + K h f (x) with m h (x) = 1 -K h (1).
Definition 5.1. Say that (E, µ) is weakly incoming to Ω if for any x 0 ∈ ∂Ω there exists > 0, θ ∈ {±1} and a measurable subset F ⊂ E such that µ(F ) > 0 and

c(x 0 + θte) < c(x 0 ) ∀t ∈]0, ], ∀e ∈ F. (5.3)
Lemma 5.1. There exists some measurable subsets F 1 , . . . , F d ⊂ E such that µ(F j ) > 0 for all j and any (f 1 , . . . , f d ) ∈ Π d j=1 F j spans R d . Moreover the sets F j can be chosen with arbitrary small diameters.

Proof. From the same argument as in remark following Definition 1.1, we can easily see that µ can not be supported in an hyperplane of R d . Let us prove by induction that for k = 1, . . . , d, there exists F 1 , . . . F k ⊂ E such that µ(F j ) > 0 for all j and for any (f 1 , . . . , f k ) ∈ Π k j=1 F j , rank(f 1 , . . . , f k ) = k. If k = 1, it suffices to take F 1 ⊂ F \ {0} with µ(F 1 ) > 0, which is possible thanks to the fact that F is weakly incoming to Ω.

Assume that the property holds true at rank k -1 < d. There exists F 1 , . . . F k-1 ⊂ E such that µ(F j ) > 0 for all j and any (f 1 , . . . , f k-1 ) ∈ Π k-1 j=1 F j , H = span(f 1 , . . . , f k-1 ) has dimension k -1. Since supp(µ) is not contained in H, there exists F k ⊂ F \ H with µ(F k ) > 0. Then F 1 , . . . , F k satisfy the property at rank k.

The fact that we can take diam(F j ) arbitrary small can be shown as follows. Let > 0 and assume by contradiction that there exists j 0 such that for any f ∈ F j 0 , µ(B(f, ) ∩ F j 0 ) = 0. Then any compact subset of F j 0 would have measure zero, which is impossible since µ(F j 0 ) > 0.

Introduce the following differential operators associated to the set E:

∇ E : H 1 (Ω) → L ∞ E, L 2 (Ω) (5.4) defined by ∇ E u(e, x) = ∇u(x), e C d for any (e, x) ∈ E × Ω; div E : L ∞ E, H 1 (Ω) → L 2 (Ω) (5.5) defined by div E f (x) = E ∇ x f (e, x
), e C d dµ(e) for any x ∈ Ω; and

∆ E : H 2 (Ω) → L 2 (Ω) (5.6)
given by ∆ E = 1 6 div E ∇ E .

Define also the following trace operator:

γ 0 E : f ∈ L ∞ E, H 1 (Ω) , div E f ∈ L 2 (Ω) → H -1 2 (∂Ω) (5.7) by ∂Ω γ 0 E f (x)v |∂Ω (x) dσ(x) = Ω div E f (x)v(x) dx + E Ω f (e, x)∇ E v(e, x) dxdµ(e) (5.8) for any v ∈ H 1 (Ω). Observe that if f ∈ L ∞ (E, C 1 (Ω)), then γ 0 E f (x) = E e, n(x) C d f (x, e) dµ(e) (5.9)
where n(x) denotes the unit outgoing normal vector to the boundary ∂Ω at point x.

For u ∈ H 1 (Ω) such that ∆ E u ∈ L 2 (Ω), the function f = ∇ E u satisfies div E f ∈ L 2 (Ω).
Hence, the operator

γ 1 E : u ∈ H 1 (Ω), ∆ E u ∈ L 2 (Ω) → H -1 2 (∂Ω) (5.10) defined by γ 1 E u(x) = γ 0 E ∇ E u(x) is continuous. Finally, introduce the following quadratic form on H 1 (Ω): E E (u) = 1 6 E Ω |∇ E u(e, x)| 2 dxdµ(e) ∀u ∈ H 1 (Ω). (5.11)
From Lemma 5.1 it follows that, since E is weakly incoming to Ω, there exists some subsets F 1 , . . . , F d with arbitrary small diameters and µ(F j ) > 0 such that any (f

1 , . . . , f d ) ∈ F 1 × • • • × F d spans R d .
Taking the diameter of the F j sufficiently small, it is easy to show that there exists C > 0 such that for any u ∈ H 1 (Ω),

1 C ∇u 2 L 2 (Ω) ≤ E E (u) ≤ C ∇u 2 L 2 (Ω) .
(5.12)

Then, the operator -∆

E = -1 6 div E ∇ E with domain D(-∆ E ) = {u ∈ H 1 (Ω), ∆ E u ∈ L 2 (Ω), γ 1 E u = 0}
is the self-adjoint realization of the Dirichlet form E E . Moreover, it follows from (5.12) that -∆ E has compact resolvant. Denote its spectrum by ν 0 = 0 < ν 1 < ν 2 < . . . and by m j the multiplicity associated to ν j . Observe that m 0 = 1. Theorem 5.2. Suppose that (E, µ) is weakly incoming to Ω, then the following hold true. i) There exists h 0 > 0, δ 0 ∈]0, 1 2 [ and a positive constant C such that for any h ∈]0, h 0 ], the spectrum of M h is a subset of [-1 + δ 0 , 1], 1 is a simple eigenvalue and Spec(M h ) ∩ [1 -δ 0 , 1] is discrete.

ii) For any h ∈]0, h 0 ] and 0 ≤ λ ≤ δ 0 h -2 , the number of eigenvalues of M h in [1 -h 2 λ, 1] (with multiplicity) is bounded by C(1 + λ) d/2 .

iii) For any R > 0 and ε > 0 such that ν j+1 -ν j > 2ε for ν j+2 < R, there exists h 1 > 0 such that one has for all h ∈]0, h 1 ], Spec 1 -M h h 2 ∩]0, R] ⊂ ∪ j≥1 [ν j -ε, ν j + ε] (5.13)

and the number of eigenvalues of 1-M h h 2 in the interval [ν j -ε, ν j + ε] is equal to m j .

Here are two examples of (E, µ) which are weakly incoming to Ω. The first is the case where E = {e 1 , . . . , e p } is discrete and µ is simply the measure 1 p p j=1 δ e=e j . Then it suffices to assume that E is weakly incoming to Ω in the sense of Definition 1.1. Moreover, in that case the conclusion of Theorem 5.2 are exactly those of Theorem 2.1.

A second example is the following. Let E be equal to the sphere S d-1 and µ = dσ d be the surface measure. Assume that ρ : S d-1 → R + is a continuous function such that S d-1 ρ(ω)dσ d (ω) = 1 and let µ = ρ(ω)dσ d (ω). Then (E, µ) will be weakly incoming to Ω iff there exists a family of vectors e 1 , . . . , e p ∈ supp(ρ) such that (e 1 , . . . , e p ) is weakly incoming in the sense of Definition 1.1. For instance, if ρ is strictly positive on S d-1 then these assumptions are automatically satisfied.

The proof of Theorem 5.2 is very close to that of Theorem 2.1 and only the main steps are given. The following proposition is a version of Lemma 1.3 adapted to the present setting.

Proposition 5.3. Assume that (E, µ) is weakly incoming to Ω, let x 0 ∈ Ω and denote k = c(x 0 ). There exists > 0 and some subsets F 1 , . . . , F k ⊂ E such that µ(F i ) > 0 for all i = 1, . . . , k and • there exists r 0 > 0 and I ⊂ {1, . . . , m} with I = k such that Ω ∩ B(x 0 , r 0 ) = ∩ k i∈I H + i ∩ B(x 0 , r 0 );

(5.14)

• there exists θ 1 , . . . , θ k ∈ {±1} and a bijection {1, . . . , k} n → i n ∈ I such that for any n = 1, . . . , k and any f n ∈ F n , θ n f n is strictly incoming to H in (5.15) and θ n f n is incoming to H im ∀m > n.

(5.16)

Moreover the sets F 1 , . . . , F k can be chosen with arbitrary small diameter.

Proof. First, it is clear that (5.14) holds true. We prove (5.15) and (5.16) by induction on k = c(x 0 ). For k = 0 there is nothing to prove. Assume now that the property holds true for all x 0 such that c(x 0 ) ≤ k -1 and suppose that c(x 0 ) = k. Since (E, µ) is weakly incoming to Ω, there exists F ⊂ E, θ 1 ∈ {±1} and > 0 such that c(x 0 + tθ 1 f ) < c(x 0 ) for all t ∈]0, ]. Assume without loss of generality that θ 1 = 1. Since µ(F ) > 0, there exists f 0 ∈ F such that for all ρ > 0, µ(B(f 0 , ρ) ∩ F ) > 0 and c(x 0 + tf ) < c(x 0 ) ∀f ∈ B(f 0 , ρ) ∩ F, ∀t ∈]0, ].

(5.17)

In particular, there exists q 1 ∈ {1, . . . , k} and i 1 , . . . , i q 1 ∈ I such that f 0 is strictly incoming to H iq ∀q = 1, . . . , q 1 (5.18) and f 0 is parallel to H iq ∀q ≥ q 1 + 1.

(5.19)

Let F q = B(f 0 , ρ) ∩ F with ρ > 0 for q = 1, . . . , q 1 . Then µ(F q ) > 0 and it follows from (5.18) that for ρ small enough, any f ∈ F q is strictly incoming to H iq . Moreover, thanks to (5.17), any f ∈ F q is incoming to H i for i ∈ I \ {i 1 , . . . , i q 1 }. Then we can use the induction hypothesis with x 0 = x 0 + f 0 close to x 0 such that c(x 0 ) = k -q 1 < k to build F q 1 +1 , . . . , F k . The statement concerning the diameter of the F j is a trivial consequence of the construction.

Figure 1 :

 1 Figure 1: Weakly incoming condition in the case of an equilateral triangle.

(2. 12 ) 2 2

 122 since for any t ∈ [0, min( , c max j |e j | )], {|y -x| < c 2 h/2} ⊂ {|y -x -htθ q e βq | < c 2 h}. Iterating this computation n ≤ N times gives (2.10).

* Supported in part by NSF grant 0804324 † Supported in part by ANR-06-BLAN-0250-03 § Supported in part by ANR-06-BLAN-0250-03 and ANR-09-JCJC-0099-01

 Corollary 5.4. Assume that (E, µ) is weakly incoming to Ω and let x 0 ∈ Ω. Then there exists r 0 > 0, > 0, F ⊂ E with µ(F ) > 0 and θ ∈ {±1} such that

Using these results and working as in Section 2 easily proves the following.

Proposition 5.5. There exists N ∈ N and c 1 , c 2 > 0 such that

where for all x ∈ Ω, µ h (x, dy) is a positive Borel measure.

Proof. The starting point of the proof is to observe that for any k ∈ N and any non-negative function f ,

for any F 1 , . . . , F k ⊂ E. Then the proof is the same as the proof of Proposition 2.2. In fact, (2.13) remains valid thanks to Lemma 5.1. Then we can mimick the end of the proof, using the fact that in Proposition 5.3 the set F j can be chosen with arbitrary small diameter. Details are left to the reader.

Proposition 5.5 implies a lemma analogous to Lemma 2.3 for the operator M h considered in this section. In particular, any function u ∈ L 2 (Ω) satisfying Part iii is implied by the following lemma (where B h still denotes the Dirichet form associated to M h ). Lemma 5.6. Let θ ∈ C ∞ (Ω) be fixed and let (ϕ h , r h ) ∈ H 1 (Ω) × L 2 (Ω) be such that r h L 2 (Ω) = O(h) and ϕ h converges weakly in H 1 (Ω) to some ϕ. Then

and lim

)∇ E θ(e, x) dxdµ(e).

(5.25)

Proof. The proof is the same as that of Lemma 2.4.

Total variation estimates for rates of convergence now follow as in Section 3.


Rui Xue
email: rxue@laas.fr

Claude Baron
email: claude.baron@laas.fr

Philippe Esteban
email: philippe.esteban@laas.fr

Li Zheng
email: li.zheng@laas.fr

Alignment of practices for an efficient management of Systems Engineering processes during the development of systems of systems

Keywords: systems engineering, project management, standards or guides, alignment I

Nowadays, because of the increasing complexity of systems and projects, more and more people focus on the systems of systems. At the same time, Systems Engineering and Project Management are considered as two independent activities during the development of systems of systems, but many inconsistencies appear between Systems Engineering and Project Management teams during engineering projects although their relationships is needed to be integrated as they get involved in large scale systems of systems. For the most part, these inconsistencies are due to a not close enough coordination and sometimes a lack of communication. As a result, there is a great need for international companies to understand the standards or guides from both domains to better integrate them and enhance their chances of success. Although numerous Systems Engineering and Project Management standards or guides have been published to help people get better organized and improve the quality of products or services, a lot of their projects still fail due to the poor integration of both domains. At the same time, the number of small and medium enterprises has quickly increased because the social division of labor is becoming more and more meticulous with the rapid development of the science and technology. The ability for small and medium enterprises to analyze and apply the Systems Engineering and Project Management standards or guides is not the same as for international companies, due to their limited resources. So the aim of this paper is to select two references from the most frequently used standards or guides in Systems Engineering and Project Management domains, the ISO/IEC 15288 and the PMBoK, and to illustrate how to help the systems engineers and project managers to compare and align the references from both domains quickly and effectively.

INTRODUCTION

At present, with the increasing complexity of projects, it is obvious that Systems Engineering (SE) and Project Management (PM) teams need to intensify their relationships as they get involved in large scale projects or systems [START_REF] Clemens | Systems Engineering (SE) & Project Management[END_REF]. As a result, there is a need for systems engineers and project managers to understand the standards or guides from both domains to better integrate them and enhance their chances of success. Although numerous SE and PM standards or guides have been published to help people get better organized and improve the quality of products or services, a lot of projects still failed. The Standish Group pointed out that only 16.2% of software projects are completed successfully, on time and on budget. However not all of successful projects achieve features and functions originally proposed. It is also worth noting that only 9% of projects in big companies were successful while 31.1% of projects will be cancelled before they ever get completed [START_REF] Standish | The Standish Group Report[END_REF]! According to the report drawn up by INCOSE UK, the effective use of systems engineering can save not only 10 to 20% of the project budget, but can also prevent half of all failures [START_REF] Incoseuk | Why do Systems Engineering? Manage Complexity[END_REF]. It is well-known that some failures are due to inconsistencies and a lack of consultation between the different teams involved in the same project from Systems Engineering and Project Management [START_REF] Weingart | Intra-Team Conflict Through Collaboration[END_REF]. The technical team will follow the project from a technical perspective, making use of engineering knowledge in an attempt to solve the technical problems, whereas the managerial team will follow the project globally by using key performance indicators to evaluate the success of the project. So in this paper, we consider the Systems Engineering and Project Management as two essential activities of the project. This paper aims to select one reference amongst the most commonly used ones in SE and in PM and to compare them, in order to assess whether a bridge could be built in the framework of systems of systems between the two references so as to provide a view shared by systems engineers and project managers enabling them to carry out the project successfully.

II. INTEGRATING SE AND PM: A RECENT ISSUE!

In a highly competitive environment, companies have to improve their performance and practices to successfully implement their projects. The current issues deal with how to simplify and speed up processes to better coordinate, control and manage these projects. Thus, there is a need not only to integrate systems engineering processes and project management processes, but also to help engineers and managers supervise and conduct the project by implementing methods and support tools allowing them to make decisions jointly. So far, international organizations have devised many SE and PM standards, guides or references (e.g., ANSI/EIA 632, IEEE 1220, INCOSE HandBook and SEBoK for SE, or PMBoK and ISO 21500 for PM). However not a single standard or guide contemplates an advanced cooperation between SE and PM, despite the fact that engineers and manager have to cooperate closely throughout the whole project development. So one has to compare and analyze the China Scholarship Council (CSC) differences and similarities between SE and PM standards or guides and to supplement them during project implementation.

In order to improve the performance of SE and PM, research has recently been carried out. In 2011, INCOSE (the International Council on Systems Engineering) and PMI (Project Management Institute) recognized the importance of integrating SE with PM and agreed to tackle the question to help organizations reduce risks and improve returns on investment [START_REF] Conforto | Survey Report: Improving Integration of Program Management and Systems Engineering[END_REF]. Based on this consensus, the MIT took out a survey and pointed out the need to overcome the barriers between systems engineers and project managers. They suggested four methods to enhance cooperation based on the analysis of several cases: using standards from both domains, formalizing the definition of integration, developing integrated engineering program assessments and sharing responsibility for risk management, quality, lifecycle planning and external suppliers [START_REF] Conforto | Survey Report: Improving Integration of Program Management and Systems Engineering[END_REF]. In May 2012, a guide named "the guide to lean enablers for managing engineering programs" was published by the joint MIT-PMI-INCOSE community of practice on lean in program management after a one year research on the integration of operations research, systems engineering and project management. It is suggested to be used by the managers and engineers to improve their performances during the whole project. The authors collected and synthesized data to provide the best guidance on how to implement the project more successfully [START_REF] Oehmen | The Guide to Lean Enablers for Man-aging Engineering Programs[END_REF]. This paper addresses the first suggestion made by MIT: to have standards cooperate. To do so, we reviewed the standards and guides from both domains [START_REF] Xue | Towards the success of design projects by the alignment of processes in collaborative engineering[END_REF], compared them [7][8], and conducted a survey on a panel of industrialists to identify the most commonly used standards and guides: in SE, the ISO/IEC 15288 standard is the one most often used while in PM, the PMBoK is also the most internationally followed guide. In the next section both references are considered.

III. INTRODUCTION TO ISO/IEC 15288 STANDARD AND PMBOK GUIDE

With the increasing globalization of markets, international standards have become critical to the trading project, ensuring the product and services meet internationally recognized levels of performance and safety. So companies are encouraged to use international standards or guides. A great deal of standards and guides are used by companies as reference [START_REF]ISO: What is a standard[END_REF]. They can be subdivided into three categories: Product standards or guides, Process standards or guides and Project management standards or guides. Among them are many popular standards, such as ISO 9000 about quality management, ISO 14000 about environmental management and ISO 31000 about risk management. The PMI only focuses on drawing up the guides or standards related to management.

A. ISO 15288

The ISO/IEC 15288 is a Systems Engineering standard covering processes and life cycle stages. [START_REF]IEEE Systems and Software Eng. -System Life Cycle Processes[END_REF]. Many other standards or SE references, such as INCOSE SE Handbook and SEBoK, are based on the ISO/IEC 15288. This standard defines a set of processes that are applied to the development of product, systems or services and the associated terminology. It defines the systems life cycles as six stages: conception, development, production, utilization, support and retirement. It can be applied concurrently, iteratively and recursively to a system and its elements. The systems considered in this international standard are man-made, created and utilized to provide products and/or services for the benefit of users and other stakeholders [START_REF]IEEE Systems and Software Eng. -System Life Cycle Processes[END_REF].

ISO/IEC 15288 standard is broken down into 3 levels processes (Figure 1). The first level features four process groups; there are 25 processes in total in the four process groups at the second level. Each process is presented in terms of purpose, outcomes and activities at the third level.

B. PMBoK

The PMBoK is a guide about project management; it provides the widely recognized standard and guidelines for project managers to deal with the management of the project effectively. Not only does it contain the standard that describes processes, input and output of the processes, but also guides about the tools and methods to be used as a resource in managing projects while considering the overall approach and methodology to be followed [START_REF]PMI A Guide to the Project Management Body of Knowledge: PMBOK V5[END_REF]. The PMBoK can be used before starting a project or at any steps to formalize it without overlooking some key parameters.

The PMBoK is also broken down into 3 levels (Figure 2); there exists 10 Knowledge Areas at the first level. A Knowledge Area represents a complete set of concepts and activities that make up a professional field, project management field, or area of specialization. To these 10 areas correspond 5 process groups at the second level. These have clear dependencies and interact with one another.

A. Comparison on the First Level of Decomposition

After analyzing the processes of the 10 knowledge areas in PMBoK, we will find that almost in each Knowledge Area (KA) have two similar processes: the planning process and the controlling process, there are nine Knowledge Areas (KAs) that have the planning processes and eight KAs have the processes in the monitoring and controlling process groups. It is as shown in Figure 3 and Figure 4. After analyze parts of the two process groups, we found that all the process in the Figure 3 of PMBoK can be founded in the project planning process of ISO/IEC 15288. The processes of monitoring and controlling process group in the Figure 4 can be founded in the Project Assessment and Control Process of ISO/IEC 15288. Based on the analysis above, we give some detailed alignment of both references; the result is shown as the table I.

B. Comparison on the second level of decomposition

Let us now consider the relationships between the tasks and activities of ISO/IEC 15288 and the 5 process groups of the PMBoK. We analyze all processes of the ISO/IEC 15288, we extract and classify the verbs that are been used to describe the tasks and activities of the processes. A distinct classifications can be found, they can be classified into five groups as shown in the left column of the table II. It is worth noting that the five verb groups exactly correspond to the names of the five process groups of PMBoK. This corresponding relationship between the two references is shown as in the table II. Monitoring and Controlling Accept, Close, Improve, Maintain, Perform, Support, Finalize Closing Figure 6 shows one example about the alignment of activities and tasks of ISO/IEC 15288 and processes of PMBoK.

C. Conclusion on the comparison of the ISO/IEC 15288 and PMBoK structures

As the organizations of both references have already been presented and compared, levels of decomposition can be highlighted (see Figure 1 and Figure 2). Hence, general structures are compatible and this is the first similarity in both references. With respect to these figures, both references are broken down into three levels. However, these levels do not refer to the same thing, but a linear match between levels cannot be achieved at the same degree of decomposition. Nonetheless, by going deeper into each level, another more sophisticated match can be found. This is shown below in Figure 7.

A. Focus on different systems in both references

After analyzing the focus of each system in both references, it becomes apparent that the two references focus on a different system. Indeed, the PMBoK's system is a project whereas ISO/IEC 15288's system is a product or a service. This point occurs in each knowledge area and in each sub-process.

For example, "Risk Management Process" in ISO/IEC 15288 deals with a product or service as shown in the following definition: "The Risk Management process is a continuous process for systematically addressing risk throughout the life cycle of a system product or service". Likewise, "Project Risk Management" in the PMBoK deals with a project as shown in the following definition: "Project Risk Management includes the processes of conducting risk management planning, identification, analysis, response planning, and controlling risk on a project". Nevertheless, approaches, processes and steps are still the same. For example, every tasks and activities of the subprocess "Risk Management Process" (ISO/IEC 15288) matches every process of the knowledge area "Project Risk Management" as shown in the Table III.

B. Chronologically versus concurrently

How to execute the processes is an important point if they can be implemented simultaneously or chronologically.

For the PMBoK, the 10 knowledge areas can be executed concurrently. Indeed, each knowledge area is an important project management field, and information about cost or time must always be available. All the knowledge areas will not begin and end at the same time but they are all independent. Conversely, the five processes must be executed one after the other (see Figure 8). For ISO/IEC 15288, the four process groups: "Agreement processes", "Technical processes", "Project processes" and "Organizational-project-enabling processes" can be executed concurrently. For sub-processes, some of them can run simultaneously while the others must be executed in a chronological order. For example, the two sub-processes "Acquisition Process" and "Supply Process" (pertaining to the "Agreement Processes") can run simultaneously (see Figure 9). However, almost all sub-processes which belong to the "Technical processes" must be executed one after the other. Finally, all "Tasks and Activities" in each sub-processes have to be executed in a chronological order.

C. Conclusion on the comparison of the ISO/IEC 15288 and PMBoK content

The first general remark that can be made is that the degree of explanations differs in both references. The PMBoK is more detailed than ISO/IEC 15288. This can be easily verified by the number of pages. ISO/IEC 15288 is 84 pages long whereas PMBoK is 616 pages long. Moreover, the manner in which both documents are decomposed also accounts for these differences. On the one hand, in ISO/IEC 15288, there are just chapters for the 4 main categories of processes and one small part for each subprocess; on the other hand, in PMBoK, one chapter is devoted to each knowledge area and a small portion in each chapter is devoted to each process.

Secondly, it has been stated that in the decomposition of the 5 processes of PMBoK, there are some introductions about the tools and methods corresponding to the processes. It is really useful to know those tools or methods allowing the processes to be implemented effectively. However, the ISO/IEC 15288 standard does not mention the tools and methods that can be used during the implementation of the processes. Thirdly, one has to check whether words convey the same meaning in both references. Indeed, after reviewing the words used in both references, it appears that there are no real differences on the technical side; for example, the word "specification" or "risk" has the same meaning in both references. The only thing worth noticing is that the word "system" has a different meaning. Indeed, ISO/IEC 15288 focuses on technical systems whereas PMBoK deals with the "system" as projects. So this may account for the communication issue observed between stakeholders.

VI. CONCLUSION

To develop the systems of systems quickly and effectively, it is better to know the references from the Systems Engineering and Project Management domains since they are two initial and critical elements of the systems of systems during their developments. How to better integrate them during the development for the systems engineers and project manager is a great challenge. In this paper, we considered the SE and PM as two important aspects of the development of systems of systems, and then the comprehensive study of the PMBoK and ISO/IEC 15288 has been conducted. Firstly, we introduced the two references briefly. Then, we compared them to highlight differences and similarities at different levels. We first compared them based on the general structures of both references, we analyzed ISO/IEC 15288 and PMBoK in terms of the processes and activities, and we also analyzed those parts of the reference covered by one or the other, in this part, we give some detail example of the similarities and differences. In the second part, we analyzed both contents to study their points of view. It was concluded that the type of systems on which the ISO/IEC 15288 reference focuses is a product or service, whereas the PMBoK focuses on a project. ISO/IEC 15288 focuses essentially on technical aspects and only a few concepts of project are treated. Conversely, PMBoK focuses only on the project aspect even if there are some same processes or activities. Then, each reference standard features its own specific parts but they are consistent, complementary and follow the same approach. When implementing the project management processes of ISO/IEC 15288, the tools or methods from PMBoK can be used as reference. These two references can also be employed differently. We can use the technical processes from m ISO/IEC 15288 standard to complete the PMBoK, when the technical process is needed during implementation of the project, we use ISO /IEC 15288 standard as reference, then we use the Project Management processes of the PMBoK In order to complete the ultimate goal of integrating methods and tools from SE and PM during the development of systems of systems, one may align both references to facilitate the management of the technical project. However, how to mix these references remains an interesting area to explore.

Figure 1 .

 1 Figure 1. Structure of ISO/IEC 15288

Figure 2 .

 2 Figure 2. Structure of PMBoK

Figure 3 .

 3 Figure 3. Relationship between planning process groups and 10 Knowledge Areas

Figure 4 .

 4 Figure 4. Relationship between the monitoring and controlling process group and 10 Knowledge AreasAfter aligning the two process groups PMBoK to the two processes of ISO/IEC 15288, we compare all the processes of ISO/IEC 15288 to the 10 Knowledge Areas of PMBoK. This comparison is based on ISO/IEC 15288; so, Figure5highlights those parts of the PMBoK covered or not covered by said standard. If a knowledge area in the longer dashed rectangle, it means that this knowledge area is only partially covered by ISO/IEC 15288. If it is in the solid rectangle, it means that the whole knowledge area can be found in said standard.

Figure 5 .

 5 Figure 5. Comparison between ISO/IEC 15288 and PMBoK at the first level

Figure 6 .

 6 Figure 6. Process relationship between ISO 15288 and PMBoK

Figure 7 .

 7 Figure 7. Match between the three levels of both references

Figure 8 .

 8 Figure 8. Major time considerations for ISO/IEC 15288 and PMBoK

Figure 9 .

 9 Figure 9. Some time considerations for ISO/IEC 15288

TABLE I .

 I SOME DETAILED ALIGNMENT OF PROCESSES OF THE TWO REFERENCES

	ISO/IEC 15288	PMBoK
	1. Project Planning Process-Define the project.
	2. Stakeholder Requirements Definition Process
	3. Architectural Design Process

TABLE II .

 II RELATIONSHIP BETWEEN ISO/IEC 15288 AND THE PMBOK

ISO/IEC 15288: Tasks & activities PMBoK: 5 process groups Prepare

, Initiate, Identify, Establish, Define, Elicit Initiating Advertise, Develop, Plan, Manage Planning Execute, Evaluate, Acquire, Activate, Analyse Executing Monitor, Deliver, Assess, Provide, Control, Treat

TABLE III .

 III FOCUS ON DIFFERENT SYSTEMS IN ISO/IEC 15288 AND PMBOK

		5. Monitor risks	6.Control risks
		6. Evaluate the risk
		management
		process
	Risk Management Process	Project Risk Management
	(ISO/IEC 15288)	(PMBoK)
	Tasks and Activities:	5 processes:
	1. Plan risk	1. Plan risk management
	management
	2. Manage the risk	2. Identify risks
	profile
	3. Analyse risks	3. Perform qualitative risk
		analysis
		4. Perform quantitative risk
		analysis
	4. Treat risks	5.Plan risk responses