E Agullo

L Giraud

S Nakov

J Roman

Hierarchical hybrid sparse linear solver for multicore platforms

Keywords: High Performance Computing (HPC), multicore architecture, sparse linear solver, hybrid method, direct method, Krylov method, MPI, multi-threading, hardware locality Calcul haute performance (HPC), mulicore architecture, algèbre linéaire creuse, solveur hybride, méthode directe, méthode de Krylov, multi-threading, localité matérielle

The solution of large sparse linear systems is a critical operation for many numerical simulations. To cope with the hierarchical design of modern supercomputers, hybrid solvers based on Domain Decomposition Methods (DDM) have been been proposed. Among them, approaches consisting of solving the problem on the interior of the domains with a sparse direct method and the problem on their interface with a preconditioned iterative method applied to the related Schur Complement have shown an attractive potential as they can combine the robustness of direct methods and the low memory footprint of iterative methods. In this report, we consider an additive Schwarz preconditioner for the Schur Complement, which represents a scalable candidate but whose numerical robustness may decrease when the number of domains becomes too large. We thus propose a two-level MPI/thread parallel approach to control the number of domains and hence the numerical behaviour. We illustrate our discussion with large-scale matrices arising from real-life applications and processed on both a modern cluster and a supercomputer. We show that the resulting method can process matrices such as tdr455k for which we previously either ran out of memory on few nodes or failed to converge on a larger number of nodes. Matrices such as Nachos_4M that could not be correctly processed in the past can now be efficiently processed up to a very large number of CPU cores (24, 576 cores). The corresponding code has been incorporated into the MaPHyS package.

Introduction

Parallel sparse linear algebra solvers are often the innermost numerical kernels in scientific and engineering applications; consequently, they are one of the most time consuming parts. In order to cope with the hierarchical hardware design of modern large-scale supercomputers, the HPC solver community has proposed new sparse methods. One promising approach towards the highperformance, scalable solution of large sparse linear systems in parallel scientific computing consists of combining direct and iterative methods. To achieve a high scalability, algebraic domain decomposition methods are commonly employed to split a large size linear system into smaller size linear systems that can be efficiently and concurrently handled by a sparse direct solver while the solution along the interfaces is computed iteratively [START_REF] Yamazaki | On techniques to improve robustness and scalability of a parallel hybrid linear solver[END_REF][START_REF] Rajamanickam | ShyLU: A hybrid-hybrid solver for multicore platforms[END_REF][START_REF] Luc | Parallel algebraic hybrid solvers for large 3D convection-diffusion problems[END_REF][START_REF] Gaidamour | HIPS: a parallel hybrid direct/iterative solver based on a Schur complement approach[END_REF]. Such an hybrid approach exploits the advantages of both direct and iterative methods. The iterative component allows us to use a small amount of memory and provides a natural way for parallelization. The direct part provides its favorable numerical properties; furthermore, this combination provides opportunities to exploit several levels of parallelism.

With the need of solving ever larger sparse linear systems while maintaining numerical robustness, multiple variants for computing the preconditioner for the Schur complement of such hybrid solvers have been proposed. PDSLin [START_REF] Li | Factorization-based sparse solvers and preconditioners[END_REF], ShyLU [START_REF] Rajamanickam | ShyLU: A hybrid-hybrid solver for multicore platforms[END_REF] and Hips [START_REF] Gaidamour | A parallel direct/iterative solver based on a Schur complement approach[END_REF] first perform an exact1 factorization of the interior of each subdomain concurrently. PDSLin and ShyLU then compute the preconditioner with a two-fold approach. First, an approximation S of the (global) Schur complement S is computed. Second, this approximate Schur complement S is factorized to form the preconditioner for the Schur Complement system, which does not need to be formed explicitly. While PDSLin has multiple options for discarding values lower than some user-defined thresholds at different steps of the computation of S, ShyLU [START_REF] Rajamanickam | ShyLU: A hybrid-hybrid solver for multicore platforms[END_REF] also implements a structure-based approach for discarding values named probing and that was first proposed to approximate interfaces in DDM [START_REF] Tony | The interface probing technique in domain decomposition[END_REF]. Instead of following such a two-fold approach, Hips [START_REF] Gaidamour | A parallel direct/iterative solver based on a Schur complement approach[END_REF] forms the preconditioner by computing a global ILU factorization based on the multi-level scheme formulation from [START_REF] Hénon | A parallel multistage ILU factorization based on a hierarchical graph decomposition[END_REF]. Finally, object of this study, MaPHyS [START_REF] Luc | Parallel algebraic hybrid solvers for large 3D convection-diffusion problems[END_REF] computes an additive Schwarz preconditioner for the Schur complement.

To ensure numerical robustness while exploiting all the processors of a platform, an important effort has been devoted to propose two levels of parallelism for these solvers. Relying on the SuperLU_DIST [START_REF] Li | SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems[END_REF] distributed memory sparse direct solver, PDSLin implements a 2-level MPI (MPI+MPI) approach with finely tuned intra-and inter-subdomain load balancing [START_REF] Yamazaki | On techniques to improve robustness and scalability of a parallel hybrid linear solver[END_REF]. A similar MPI+MPI approach has been assessed for additive Schwarz preconditioning in a prototype version of MaPHyS [START_REF] Luc Giraud | Using multiple levels of parallelism to enhance the performance of domain decomposition solvers[END_REF], relying on the Mumps [START_REF] Amestoy | A fully asynchronous multifrontal solver using distributed dynamic scheduling[END_REF][START_REF] Amestoy | Hybrid scheduling for the parallel solution of linear systems[END_REF] and ScaLAPACK [START_REF] Blackford | ScaLAPACK Users' Guide[END_REF] sparse direct and dense distributed memory solvers, respectively. On the contrary, expecting a higher numerical robustness thanks to multi-level preconditioning, Hips associates multiple subdomains to a single process and distributes the subdomains to the processes in order to maintain load balancing [START_REF] Gaidamour | A parallel direct/iterative solver based on a Schur complement approach[END_REF]. Finally, especially tuned for modern multicore platforms, ShyLU implements a 2-level MPI+thread approach [START_REF] Rajamanickam | ShyLU: A hybrid-hybrid solver for multicore platforms[END_REF].

Whereas PDSLin and ShyLU can be virtually turned into to a pure direct method if no dropping is performed, and whereas Hips may expect robustness by relying on a multilevel scheme, additive Schwarz preconditioners are extremely local. As a result, their computation is potentially much more parallel (and scalable), but their application may lead to a dramatic increase of the number of iterations (or even to non convergence) if the number of subdomains becomes too large. The objective of the present study is to assess whether a 2-level parallel approach allows additive Schwarz preconditioning for Schur Complement methods to achieve an efficient trade-off between numerical robustness and performance on modern hierarchical multicore platforms. Following the parallelization scheme adopted in [START_REF] Rajamanickam | ShyLU: A hybrid-hybrid solver for multicore platforms[END_REF], we have designed an MPI+thread approach (Section 3) to cope with these hardware trends. Contrary to the MPI+MPI approach investigated in [START_REF] Luc Giraud | Using multiple levels of parallelism to enhance the performance of domain decomposition solvers[END_REF], such a parallelization allows for a better usage of the memory (e.g., symbolic data structures such as the elimination tree used within a direct solver are shared between threads of a same process) and a better exploitation of the CPU cores local to a node at each step of the parallel computation [START_REF]PLASMA Users' Guide, Parallel Linear Algebra Software for Multicore Architectures[END_REF][START_REF] Faverge | Dynamic scheduling for sparse direct solver on NUMA architectures[END_REF][START_REF] Excellent | A study of shared-memory parallelism in a multifrontal solver[END_REF], as we show in Section 4.

The rest of the paper is organized as follows. In Section 2, we present the numerical schemes used in sparse hybrid solvers; in particular, we introduce the additive Schwarz preconditioning for the Schur Complement and its MPI parallelization in MaPHyS. The 2-level MPI+thread extension (first contribution) of this baseline version is then introduced in Section 3. A detailed performance analysis (second contribution) on large test cases is then presented in Section 4 before concluding in Section 5.

Baseline flat MPI design of an hybrid solver

Let Ax = b be the linear problem and G = {V, E} the adjacency graph associated with A. In this graph, each vertex is associated with a row or column of the matrix A and it exists an edge between the vertices i and j if the entry a i,j is non zero. In the sequel, to facilitate the exposure and limit the notation we voluntarily mix a vertex of G with its index depending on the context of the description. The governing idea behind substructuring or Schur complement methods is to split the unknowns in two categories: interior and interface vertices. We assume that the vertices of the graph G are partitioned into N disconnected subgraphs I 1 , ..., I N separated by the global vertex separator Γ. We also decompose the vertex separator Γ into non-disjoint subsets Γ i , where Γ i is the set of vertices in Γ that are connected to at least one vertex of I i . Notice that this decomposition is not a partition as Γ i ∩ Γ j = ∅ when the set of vertices in this intersection defines the separator of I i and I j . By analogy with classical domain decomposition in a finite element framework, Ω i = I i ∪ Γ i will be referred to as a subdomain with internal unknowns I i and interface unknowns Γ i . If we denote I = ∪I i and order vertices in I first, we obtain the following block reordered linear system

A II A IΓ A ΓI A ΓΓ x I x Γ = b I b Γ (1)
where x Γ contains all unknowns associated with the separator and x I contains the unknowns associated with the interiors. Because the interior vertices are only connected to either interior vertices in the same subgraph or with vertices in the interface, the matrix A II has a block diagonal structure, where each diagonal block corresponds to one subgraph I i . Eliminating x I from the second block row of Equation (1) leads to the reduced system

Sx Γ = f (2)
where

S = A ΓΓ -A ΓI A -1 II A IΓ and f = b Γ -A ΓI A -1 II b I . (3)
The matrix S is referred to as the Schur complement matrix. This reformulation leads to a general strategy for solving [START_REF]PLASMA Users' Guide, Parallel Linear Algebra Software for Multicore Architectures[END_REF]. Specifically, an iterative method can be applied to solve [START_REF] Agullo | Task-based sparse hybrid linear solver for distributed memory heterogeneous architectures[END_REF]. Once x Γ is known, x I can be computed with one additional solve for the interior unknowns via

x I = A -1 II (b I -A IΓ x Γ ) .
We illustrate in Figure 1a all these notations for a decomposition into 4 subdomains. The local interiors are disjoint and form a partition of the interior I = I i (blue vertices in Figure 1b). It is not necessarily the case for the boundaries. Indeed, two subdomains Ω i and Ω j may share part of their interface (Γ i Γ j = ∅), such as Ω 1 and Ω 2 in Figure 1b which share eleven vertices. Altogether, the local boundaries form the overall interface Γ = ∪Γ i (red vertices in Figure 1b), which is not a disjoint union. While the Schur complement system is significantly smaller and better conditioned than the original matrix A, it is important to consider further preconditioning when employing a Krylov method. We refer to [START_REF] Li | Factorization-based sparse solvers and preconditioners[END_REF], [START_REF] Rajamanickam | ShyLU: A hybrid-hybrid solver for multicore platforms[END_REF] and [START_REF] Gaidamour | A parallel direct/iterative solver based on a Schur complement approach[END_REF] for details on preconditioning strategies used in PDSLin, ShyLU and Hips, respectively, and we now introduce the additive Schwarz preconditioner considered in MaPHyS. This preconditioner was originally proposed in [START_REF] Carvalho | Local preconditioners for two-level nonoverlapping domain decomposition methods[END_REF] and successfully applied to large problems in real life applications in [START_REF] Giraud | Parallel scalability study of hybrid preconditioners in three dimensions[END_REF][START_REF] Haidar | On the parallel scalability of hybrid linear solvers for large 3D problems[END_REF][START_REF] Nakov | On the design of sparse hybrid linear solvers for modern parallel architectures[END_REF]. To describe the main preconditioner in MaPHyS, we define Si = R Γi SR T Γi , that corresponds to the restriction of the Schur complement to the interface Γ i . If I i is a fully connected subgraph of G, the matrix Si is dense.

With these notations the additive Schwarz preconditioner reads

M AS = N i=1 R T Γi Si -1 R Γi . (4)
With all these components, the classical parallel implementation of MaPHyS can be decomposed into four main phases (see [START_REF] Haidar | On the parallel scalability of hybrid linear solvers for large 3D problems[END_REF] for more details):

• the partitioning step consists of partitioning the adjacency graph G of A into several subdomains and distribute the A i to different processes. For this, we are able to use two state-ofthe-art partitioners, Scotch [START_REF] Pellegrini | Sparse matrix ordering with scotch[END_REF] and METIS [START_REF] Karypis | MeTiS -Unstructured Graph Partitioning and Sparse Matrix Ordering System -Version 2[END_REF];

• the factorization of the interiors and the computation of the local Schur complement factorizes A i with the PaStiX [START_REF] Hénon | PaStiX: A High-Performance Parallel Direct Solver for Sparse Symmetric Definite Systems[END_REF] or the Mumps [START_REF] Amestoy | A fully asynchronous multifrontal solver using distributed dynamic scheduling[END_REF] sparse direct solver and furthermore provides the associated local Schur Complement S i thanks to recent developments from the development teams of those sparse direct solvers;

• the setup of the preconditioner assembles diagonal blocks of S i via a few neighbour to neighbour communications to form Si and factorize this local dense matrix with Mkl;

• the solve step is twofold: a parallel preconditioned Krylov method is performed on the reduced system (Equation 2) to compute x Γi where all BLAS operations are provided by Mkl, followed by a back solve on the interiors to compute x Ii with the sparse direct solver.

Design of a 2-level MPI+thread extension of MaPHyS

The baseline MPI model presented above presents some strong limitations on modern multicore supercomputers. Indeed, modern platforms tend to have more and more CPU cores. Yet, the baseline model assigns exactly one subdomain per core being used. As a consequennce, if one wants to exploit all available computational cores, he has to decompose the matrix in as many subdomains. For instance, if two nodes of eight cores each are being used, 16 subdomains have to be used in order to exploit all the 16 cores (see Figure 2(a)). As shown later on this study (Section 4), this approach can (1) deteriorate or even prevent the convergence of the method at scale and (2) lead to excessive memory consumption, possibly leading the method to run out-ofmemory. Solving those issues with the baseline MPI MaPHyS implementation would impose to assign fewer subdomains per node, hence fewer cores. For instance, on the same example platform, imposing four subdomains would limit us to use four cores only (see Figure 2(b)). To overcome these limitations while using all available cores, we have designed a 2-level MPI+thread method. With such a method, each process associated to a subdomain can use multiple cores thanks to multithreading. In our example, each subdomain would thus be processed by four cores using four threads per process (see Figure 2(c)).

In this section, we present the design of our MPI+thread extension of MaPHyS. Because MaPHyS is modular, we first need to select and possibly tune multithreaded versions of the external software packages MaPHyS relies on (Section 3.1). We then need to make sure that those libraries cooperate correctly in order to maximize the overall performance of our hybrid solver (Section 3.2). Among other important issues, the physical mapping of the threads on the processing units is critical. We present the three different binding strategies that have been studied in Section 3.3.

Multithreaded building block operations

We focus in this section on the description of multithreaded version of the external software packages that are used by MaPHyS. The 2-level parallel implementation will be effective for our hybrid solver if its three main computational phases can be efficiently performed in parallel: factorization of the interiors, setup of the preconditioner and solve step. We do not consider the parallel implementation of the partitioning step for two reasons. First, with the current state of MaPHyS, the domain decomposition is performed sequentially in a pre-processing phase, which computes the decomposition and sends the needed data on the corresponding processes. Furthermore, the current software version of the partitioners Scotch and METIS are not multithreaded. Let us quickly recall the main numerical kernels of the three computational steps of our solver and for each of them describe the multithreaded strategy.

For the factorization of the interiors with the current state of our package, we are able to use two sparse direct solvers, PaStiX and Mumps. These packages are able to perform Cholesky, LU or LDL T factorizations of symmetric definite, symmetric and general sparse matrices respectively. Both solvers are state of the art software, implemented with a large number of numerical and Inria technical functionalities. The major difference between the two solvers is that Mumps is based on the multifrontal approach while PaStiX is a supernodal solver. With the recent development of the Mumps group, activities towards a multithreaded version have been accomplished [START_REF] Excellent | A study of shared-memory parallelism in a multifrontal solver[END_REF], but when this study was developed, only the PaStiX package offered the possibility of using multithreading. In this study only PaStiX is considered and a more detailed description is given below.

Two different versions of the setup of the preconditioner are possible in our solver, sparse and dense. For each one, a different library is used. For the sparse version, the PaStiX solver is used, while for the dense version, the LAPACK routines of the Mkl library are used. For the iterative solver, the BLAS operations are provided by the Mkl library and the application of the preconditioner is performed by the Mkl library or by the back-solve operation of the sparse direct solver that is used. Finally the black-solve on the interiors is performed by the sparse direct solver. All in all, we use the PaStiX sparse direct solver and the Mkl library.

Multithreaded MKL

The Mkl library provides highly optimized BLAS and LAPACK routines for Intel processors. The multithreaded version of the Mkl library is implemented on top of the Intel OpenMP runtime system.

Multithreaded sparse direct solver

Sparse direct solvers usually consist of three distinct steps. First, the analysis step performs a reordering of the variables (by calling a third-party library such as Scotch or METIS) and a symbolic factorization in order to compute data structures that cope with expected fill-in. Second, the matrix is decomposed in factors in the so-called numerical factorization or factorization step for short. In MaPHyS, these first two steps occur in sequence during the factorization of the interiors (and separately during the computation of a sparse preconditioner). Third, the solve step computes the solution from the right-hand side and the factors. This third step occurs during the back-solve on the interiors (and separately during the application of the preconditioner if a spare preconditioner is used).

PaStiX is based on the supernodal technique. Once the block symbolic factorization has been performed, the block elimination tree and the supernode partition are available. Basically the supernodal method consists of regrouping several nodes (columns) of the elimination tree in a larger set called "supernode". From the matrix point of view, the supernodal approach can be viewed as grouping several contiguous columns in a single column-block, each of them corresponding to a supernode (see [START_REF] Hénon | PaStiX: A High-Performance Parallel Direct Solver for Sparse Symmetric Definite Systems[END_REF] for details). Instead of using inefficient kernels with low fetch/compute ratio, well suited BLAS-3 operations can then be used and exploit much more efficiently modern processing units. The supernodal factorization occuring in PaStiX consists of performing three operations for each column-block of the matrix, as depicted in Figure 3 for the first column-block. First the factorization of the diagonal block is computed (red block in Figure 3). Then a triangular solve is performed on the off-diagonal part of the column-block (yellow blocks in Figure 3). For each off-diagonal sub-block of the factorized column-block, an update to the facing column-block that owns the same rows is applied (green blocks in Figure 3). This is repeated for each column-block until the whole matrix is factorized. In PaStiX each operation is performed by one thread and the parallelism is expressed by performing several independent matrix operations simultaneously.

In the baseline version of PaStiX, multithreading was implemented for the factorization of the interiors A IiIi but the computation of the Schur (S i ) was still performed sequentially. For the purpose of this study, we relieved this bottleneck by enabling concurrent updates of S i as long as the updates are performed on disjoint matrix blocks (the three green blocks in the red bottom-right block in Figure 3). In that respect, we considered a 2D block decomposition of the square dense Schur complement matrix. The concurrency is protected by a simple deadlock-free algorithm consisting of performing active waiting until all the blocks accessed by an update are free for being updated. This additional software development designed to remove the final bottleneck

Coexistence of different multithreaded libraries

The main issue of interoperability between PaStiX and Mkl is related to the different ways these libraries manage the parallelism. As explained above, PaStiX expresses the parallelism through individual tasks associated with different supernodes; those tasks are handled by single threads. The supernode calculation is performed by sequential (mono-threaded) BLAS-3 routines from the Mkl library. However, the multithreading exploited in the other steps of MaPHyS such as the setup of the preconditioner and the Krylov iterations is based on calls to multithreaded Mkl routines. Consequently, both mono and multithreaded Mkl subroutines need to be used in different contexts by MaPHyS.

Because the Mkl library relies on OpenMP, there are two different ways to control multithreading: through the OpenMP functionalities or directly using functions defined in the Mkl library such as mkl_set_num_threads(nb_threads) or omp_set_num_threads(nb_threads). The Mkl functions override the OpenMP functions. We point out below the adaptation that needs to be performed for the main three steps of MaPHyS.

First during the factorization of the interiors, PaStiX is used so the mono-threaded version of Mkl needs to be used. For the setup of the preconditioner, depending on the dense or sparse version of the preconditioner that is used, either Mkl or PaStiX is used respectively. In a similar manner, during the iterative method, Mkl or PaStiX is used depending on the preconditioner that is used. For the rest of the routines within the iterative part, the multithreaded version of Mkl is used. For the back-solve on the interiors, PaStiX is used with the mono-threaded Mkl.

Binding strategies

Three binding variants have been initially explored. The first strategy simply consists of not enabling any binding and let the operating system handle the mapping of processes and threads (see Figure 4(a)). The second strategy consists of binding processes to a subset of cores and let the operating system handle the mapping of threads (see Figure 4(b)). The third strategy consists of binding both processes and threads within processes (see Figure 4(c)).

The mechanism for binding processes and threads may differ depending on the considered target operating system and architecture. To perform a consistent and portable binding, we rely on the Portable Hardware Locality (hwloc) software package [START_REF] Broquedis | hwloc: a Generic Framework for Managing Hardware Affinities in HPC Applications[END_REF]. The hwloc library indeed provides a portable interface that is an abstraction of all the low level technical details (operating system and architecture). It also offers a consistent numbering of the physical cores, ensuring that cores that share cache memory will be numbered next to each other. This is not always the case with the raw information provided by the operating system. The numbering of hwloc is thus better suited for designing portable binding algorithms.

Modern multithreaded BLAS libraries commonly perform mapping of processes and threads. It is for instance the case of the Mkl library on which we rely in this study. In order to control the binding by our own, we therefore prevent Mkl from doing it. This is achieved by setting the environment variable KM P _AF F IN IT Y to "disable".

We further discuss the impact of binding on performance in Section 4.2 where we show that the second strategy consisting of binding processes to a subset of cores and let the operating system handle the mapping of threads (see Figure 4(b)) is consistently optimum. For this reason, this strategy will be consistently used in the rest of this paper.

Performance analysis

In this section, a large set of experiments is presented to analyze the parallel and numerical performance of the proposed MPI+thread extension of MaPHyS. After selecting the most appropriate binding strategy (Section 4.2), we study the impact of multithreading on performance (Section 4.3). In that context, the number of subdomains is set to the number of nodes and we only vary the number of threads per subdomain. We then study the flexibility of our 2-level parallel implementation to achieve the best trade-off between the numerical and parallel behaviors (Section 4.4). For those experiments, we vary the number of subdomains and the number of threads per subdomain, so that the total number of cores remains constant.

Experimental setup

RR n°8960

The experiments presented in this paper were conducted on two platforms: PlaFRIM, located at Inria Bordeaux-Sud-Ouest and the Hopper machine from NERSC, located at the Lawrence Berkeley National Laboratory. The PlaFRIM platform is composed of 68 nodes. Each node is equipped with two Quad-core Nehalem Intel Xeon X5550 with a total of eight cores per node. Each node has a total of 24Gb of RAM memory. The nodes are interconnected in a fat tree fashion using an Infiniband QDR network delivering a 40Gb/s bandwidth. The Hopper platform is composed by 6384 nodes, each being two twelve-core AMD 'MagnyCours' 2.1-GHz processors. Each node has 24Gb of RAM. Hopper compute nodes are connected via a custom high-bandwidth, low-latency network provided by Cray.

We display in Table 1 the main characteristics of the test matrices considered in this section. They arise from different application domains ranging from structural mechanic analysis to electromagnetics calculation using both classical and discontinuous finite element modeling.

The normwise backward error is used for our study. As the iterative method is applied on the reduced system introduced in Equation 2, one option consists in using scaled residual norm associated to that system:

ε f = Sx Γ -f f . (5)
However, this criterion is related to the reduced system but not necessarily to the global system. Indeed, assuming the backsolve on the interior is performed in exact arithmetic, we have:

SxΓ -f f = Ax-b f
. Because f depends on the number of subdomains, this criterion may thus not be appropriate for comparisons between executions whose number of subdomains differs. We rely on the alternative stopping criterion:

ε b = Sx Γ -f b . (6)
This latter stopping criterion indeed reflects the error on the global system because, assuming the backsolve on the interior is performed in exact arithmetic, we have:

SxΓ -f b = Ax-b b
. We use this latter convergence criterion which we set to ε b = 10 -10 when not stated otherwise. The right preconditioned GMRES restart parameter is set to 500 and the maximum number of iterations is set to 7000. We use the version 4.0.1 of METIS to perform the domain decomposition and rely on the multithreaded version of Mkl and the modified version of PaStiX discussed in Section 3.

Impact of the thread binding strategies

As discussed in Section 3, three binding strategies have been designed consisting either of fully delegating the mapping management to the operating system (see Figure 4(a)), binding processes to a subset of cores and let the operating system handle the mapping of threads (see Figure 4(b)) or binding both processes and threads within processes (see Figure 4(c)), respectively. Intensive experiments (not reported here) have been conducted on all matrices from Table 1 showing that the second strategy consisting of binding processes to a subset of cores and letting the operating system handle the mapping of threads (see Figure 4(b)) is consistently optimum. Indeed, this configuration outperforms the case where threads are also binded by the application for two reasons. First, binding threads at the application level prevents Mkl from doing internal optimizations related to multithreading. Second, every time the number of threads used by the Mkl library is changed, the binding needs to be performed again. This effects turns out to be especially significant in the iterative method when the sparse preconditioner is used, because it has to be done twice at each iteration for switching between calls to the dense operations performed by Mkl and the call to the sparse direct solver for applying the preconditioner.

It also outperforms the case where the mapping is fully delegated to the operating system. We report here a detailed analysis on a particular example, considering matrix Matrix211 processed on 8 nodes of PlaFRIM with a dense preconditioner. We compare the behavior of MaPHyS when x-axis) is inversely proportional to the number of subdomains (8, 4, 2, 1).

Inria

no binding is performed (see Figure 5) to the case where processes are binded (see Figure 6). When only one process per node is used (eight threads per domain), both versions are equivalent and therefore achieve the same performance (right-most part of the plots). However, when multiple processes compete on the same node (i.e., when one, two or four threads per domain are used), the performance gap may be significant, especially because of the solve step.

Multithreading performance

In this section, we investigate the performance of the multithreaded implementation on multicore nodes. First we are interested in the performance behavior of MaPHyS for the test cases described in Table 2. For those experiments, the number of subdomains is only four and we use the dense variant of the preconditioner. Processes (hence subdomains) have dedicated nodes. When the number of threads increases, more and more cores can thus be exploited (see Figure 7). The parallel performance, measured as a speed-up with respect to the single threaded reference, is plotted in Figure 8 using bar charts with the corresponding thread number on the top of the bars.

For each matrix, we vary the number of threads from one to eight and compute the speed-ups for the three numerical steps of MaPHyS as well as the overall speed-up. Table 2: Size of the interior (A IiIi ), size of the local Schur complement ( Si ) and total size of the Schur complement (S) are given for each matrix when four subdomains are used. The number of iterations (#iter) needed with four subdomains and dense preconditioner is also given. It can be seen that significant speed-ups are achieved for all test cases. An average speed-up of 4.53 is achieved for the overall execution time (see Figure 8(a)) of our matrix collection. In this configuration, the overall execution time is consistently dominated by the factorization of the interiors. An explanation for this is the fact that only four subdomains are used. Most of the unknowns are associated with the interior resulting to relatively small Schur complements. As a consequence, the factorization of the interiors takes more time to be performed compare to the setup of the preconditioner. Using only four subdomains and a dense preconditioner results with a preconditioner with good numerical quality, thus only a small number of iterations is needed in order to achieve the target accuracy.

Since the overall time is dominated by the factorization of the interiors, it has a behavior similar to the overall speed-up. An average speed-up of 4.6 is obtained for the factorization of the interiors (see Figure 8 Figure 8: Overall speed-up with using 1, 2, 4 or 8 threads (and cores) with respect to the time when 1 thread (and core) per process, for the matrices from Table 1. The dense preconditioner is used. Four nodes are used and one process per node is assigned. The number of threads [START_REF]PLASMA Users' Guide, Parallel Linear Algebra Software for Multicore Architectures[END_REF][START_REF] Agullo | Task-based sparse hybrid linear solver for distributed memory heterogeneous architectures[END_REF][START_REF] Amestoy | Hybrid scheduling for the parallel solution of linear systems[END_REF] or 8) for each histogram is given on the top of the bars.

Inria

(see Table 2) for the multithreaded direct sparse solver. During the setup of the preconditioner with this configuration, the dense solver is used for the factorization of Si . The size of the local Schur complements is large enough (at least a few thousands unknowns). As a result, the direct dense solver is able to exploit efficiently the multicore machines (see Figure 8(c)). For the solve step, an average speed up of only 1.94 (out of 8) is achieved (see Figure 8(d)). Two main reasons explain this behavior. First, the iterative method is expensive in terms of communication, which do not scale with the multithreaded version. Additionally, the operations that are done during the iterative method have a low level of computational intensity. These operations do not exploit efficiently the multicore machines. In addition to what has been presented, for the Haltere and Tdr455k matrices the back-solve on the interior represents a large portion of the solve step. This is a consequence of the fact that only 9 and 5 iterations are needed for the Haltere and Tdr455k matrix respectively (see Table 2). When the number of iterations increases, the portion that corresponds to the back-solve decreases. When more than a few dozen of iterations are used, the back-solve becomes inconsiderable compared to the time needed for the solve step.

In the above study we have shown that MaPHyS is able to efficiently exploit the multicore platforms with a dense preconditioner and four nodes. To further explore the capabilities of the 2-level parallel implementation, we consider experiments at a scale related to the problem size and relying on a preconditioning strategy tuned for each matrix.The characteristics of this second set of experiments are reported in Table 3 while the observed speed-ups are displayed in Figure 9. Table 3: Configuration (#nodes and preconditioner) used in Section 4.3 and 4.4. The size of the interiors (I i ), the size of the local Schur ( Si ), the total size of the Schur complement (S) and the number of iterations are given for each matrix.

Matrix

Similarly to the situation described in Figure 8, we observe significant (but lower) speed-ups (see Figure 9(a)). Indeed, although the overall time is dominated by the factorization of the interiors, the number of subdomains increases with the number of nodes yielding smaller interiors (see Table 3) and resulting in a factorization of the interiors that is harder to parallelize efficiently with the use of multiple threads (see Figure 9(b)).

Concerning the preconditioner step, in the case where the sparsified preconditioner is used, the involved matrices almost do not benefit from the multithreading (see Figure 9(c)). The main reason is that sparsifing considerably decreases the amount of computation (factorization of the local assembled Schur Si ), which thus becomes dominated by the assembly step. Because the assembly step consists only of communications, it is not affected by multithreading. The second reason is that the local Schur complement become too small for a multithreaded sparse direct solver. For example, for the Audi_kw matrix, the average size of Si is 5,884. After applying the dropping with a threshold of 10 -3 , approximately 2.75 percent of the entries are kept, resulting in a sparsified Si that contains around 692K of non zero values. This is not a large enough computational load for the sparse direct solver to exploit efficiently the multicore processors. As a consequence, the overall setup of the preconditioner is hard to accelerate with multithreading when a sparse preconditioner is used. On the other hand, thanks to the sparsification, the setup of the preconditioner turns out to be only a very limited proportion of the total execution time (see Figure 9(a)). For instance, when eight threads are used on four nodes with the Audi_kw matrix, Figure 9: Overall speed-up when using 1, 2, 4 or 8 threads (and cores) with respect to the time when 1 thread (and core) per process for the matrices from Table 3. The configuration (#nodes and preconditioner) is given in Table 3 for each matrix. The number of threads (1, 2, 4 or 8) for each histogram is given on the top of each bar.

Inria the dense factorization is performed in 2.55 seconds, while the sparse factorization takes only 0.95 seconds. Regarding the solve step, it follows the same behvavior for all the matrices for the same reasons as explained above.

Numerical and parallel flexibility of the 2-level implementation

In the previous section, we mainly focused on the parallel efficiency of the 2-level implementation when the number of cores is increased while keeping constant the number of nodes. We are now investigating the capability of the code to fully exploit the computing resources of a hierarchical multicore platform. To highlight the flexibility we fix the number of cores used for the solution of a given problem and we vary the number of subdomains and the number of threads per subdomain so that nb_nodes×nb_cores = nb_threads_per_process×nb_domains (see Figure 10). Those isocomputing resource experiments show that our 2-level approach enables us to find the best trade-off between the numerical behavior of the solver (which depends on the number of subdomains) and its parallel performance (which depends on the number of threads per subdomain). For those experiments we consider the same test examples as in Table 3. With this configuration, when the number of threads per process increases, the number of subdomains decreases; this tends to reduce the number of iterations but enlarges the elapsed time of the factorization of the interiors. For instance, for the Audi_kw matrix, the first histogram presents a run with 32 subdomains with one thread per process, while the last one, when 8 threads are used, corresponds to a run with 4 subdomains. Compared to the results presented in the previous section, for each run not only the number of cores assigned per process changes, but also the numerical problem solved. With an increased number of subdomains, the size of the domain interior is decreasing (see Figure 11(a)), while the size of the total Schur complement (S) is increasing (see Figure 11(b)). With a larger Schur complement and the larger number of subdomains, the numerical difficulty of the iterative part tends be higher, increasing the number of iterations and possibly the overall solution time (see Figure 11(d)). With our 2-level parallel approach, we are able to explore the trade-off between larger subdomains and numerical efficiency of the iterative method in order to exploit the multicore machines in a most optimal way. The results are presented in Figure 12.

For the Haltere matrix, the best performance is obtained when one thread per process is used. More specifically, each step yields the best performance when one thread per process is used. Considering the solve step, increasing the number of subdomains does not strongly deteriorate the quality of the preconditioner for this matrix. For instance, in this case, when the number of subdomains varies from 4 to 32, the number of iterations increases from 11 to only 14 respectively (see Figure 11(d)). On that example, the best performance is obtained when the number of subdomains is the largest. Additionally, for this matrix with all configurations, the overall time is dominated by the factorization of the interiors. The factorization of the interiors yields best performance when the number of subdomains is the largest (see Figure 12 3 and the number of threads used per process is given on top of each histogram bar. All histograms are normalized with respect with the histogram when one thread per process is used.

this behavior is the fact that when the number of subdomains is doubled, the size of each interior is approximately divided by two (see Figure 11(a)) and the number of threads are doubled. Since the complexity in terms of calculation for 3D problems for the sparse direct solver is squared (see Figure 12(d)), larger number of domain yields better performance.

For the Nachos matrix, the best configuration is also when one thread per process is used. Contrary to the the Haltere matrix, the solve step is expensive for this matrix. Nevertheless, with the increasing of the number of subdomains, the number of iterations does not decrease drastically (see Figure 11(d)). Since the solve step does not exploit efficiently multithreading (see Figure 9(d)), similar numbers of iterations will be executed in approximately the same amount of time (see Figure 12(d)). Due to this, the overall execution time is driven by the factorization of the interiors, and, as explained above, the factorization of the interior yields the best performance when one thread per process is used. The Amande and Matrix211 matrices both exhibit similar behaviors. The optimal performance is obtained when two threads per process are used. Compared to the Nachos matrix, the number of iterations decreases more quickly. As a consequence the solve step becomes less expensive in terms of time consumption when the number of subdomains decreases (see Figure 12(d)). Moreover, both matrices are issued from difficult numerical problems, resulting in a large number of iterations during the solve step. For instance, when one thread per process is used, 223 and 468 iterations are needed for the Matrix211 and Amande matrices, respectively. Therefore, the benefits of reducing the cost of the solve step have larger consequences on the overall computational time (see Figure 12(d)).

For the two remaining matrices, Audi_kw and Tdr455k, this phenomena is even stronger. Additionally, when the number of subdomains is increased, the size of the local Schur complement is not increased, even slightly decreased (see Figure 11(c)). Since the dense preconditioner is used for the Tdr455k matrix, the setup of the preconditioner yields better performance with the multithreaded version for this matrix. For these matrices increasing the number of subdomains has the most significant influence on the number of iterations. Therefore the solve step yields the best performance when eight threads per process are used.

Furthermore, the Tdr455k matrix is the most difficult matrix to solve in our collection for our algorithm. The results presented in Figure 12 correspond to a dense preconditioner. Performing dropping on the local Schur complement strongly decreases the quality of the preconditioner (see Table 4). Additionally, decomposing the matrix in larger number of subdomains increases strongly the number of iterations for each preconditioner, failing to converge in most cases. This matrix has strong numerical barriers for our algorithm. With the multithreaded version, we are able to break the numerical barrier by assigning larger number of cores per subdomain. On the Hopper platform we are able to both efficiently exploit as much as 384 cores (see Figure 13(a)) and in the same time lower the memory peak per node (see Figure 13(b)).

In order to push the limit of our algorithm, we have furthermore performed tests on the Nachos4M matrix (see Table 1) on the Hopper platform. In these series of experiments we have increased the number of nodes up to one thousand (see Figure 14). With the increased number of nodes, the number of subdomains is also increased up to 2 13 when three threads per process are Although not illustrated up-to now, depending on the targeted accuracy, the best choice of the number of subdomains can vary for a given number of cores to be used. In Figure 15, we display the convergence history as a function of the elapsed time and iteration number for the Audi_kw matrix (see Figures 15(a) and 15(b) respectively). Lower the number of subdomains (i.e., larger number of threads per subdomain), larger the time needed for factorization of the interiors and setup of the preconditioner is, and faster is the convergence. However, for a moderated target accuracy, this setup cost is not worth to invest. It might be more effective to enlarge the number of subdomains to shrink the setup time and having slower convergence rate but overall faster solution. For instance, for a 10 -4 accuracy, the best choice is 64 subdomains (one thread each) while for a 10 -9 accuracy the best choice is 16 subdomains (i.e., 4 threads).

Conclusion

In this paper we have described a 2-level parallel implementation of the hybrid solver MaPHyS and shown how multithreaded libraries, namely Mkl and PaStiX, have been composed to design an efficient parallel implementation. The resulting MPI+thread parallel implementation does match the hierarchical structure of modern multicore parallel platforms. Not only it enables us to better exploit the computer architectures features but it also introduces an additional numerical flexibility to balance the numerical and parallel performances of the MaPHyS solver. Thanks to the new implementation we demonstrated that large computing platforms, up to a few tens of thousand cores, can be exploited to solve 3D linear systems that were not tractable before.

We have considered classical programming paradigms based on MPI and threads to design our 2-level parallel hybrid solver. An alternative approach would be to consider a more disruptive approach based on a task graph description of the algorithm that can be efficiently mapped and scheduled on multicore and manycore platforms by a modern runtime systems as discussed in [START_REF] Agullo | Task-based sparse hybrid linear solver for distributed memory heterogeneous architectures[END_REF] Because the preconditioner used in MaPHyS is local, we are also considering for symmetric positive definite problems to combine the 2-level parallel design proposed in the present article with coarsegrid correction techniques [START_REF] Spillane | Achieving robustness through coarse space enrichment in the two level Schwarz framework[END_REF][START_REF] Jolivet | Scalable domain decomposition preconditioners for heterogeneous elliptic problems[END_REF]. with sparse preconditioner when a dropping is applied to the preconditioner with threshold of 10 -2 . The detailed histogram for all computational steps is given in 14(b) and the sizes for the interior, the local Schur complement and the total size of the Schur complement are given in 14(d). The tests were performed on the Hopper platform. All the available cores per node are used and the statement is satisfied: nb_nodes × nb_cores = nb_threads_per_process × nb_subdomains. In the legend t p refers to number of threads (t) per process (p). . Dropping with a threshold of 10 -2 is applied to the preconditioner.

Inria

4 Γ

 4 (b) Graph subdomains.(c) Block reordered matrix.

Figure 1 :

 1 Figure 1: Domain decomposition into four subdomains Ω 1 , . . . , Ω 4 . The initial domain Ω may be algebraically represented with the graph G associated to the sparsity pattern of matrix A (a). The local interiors I 1 , . . . , I N form a partition of the interior I = I i (blue vertices in (b)). They interact with each others through the interface Γ (red vertices in (b)). The block reordered matrix (c) has a block diagonal structure for the variables associated with the interior A II .

  (a) Baseline MPI -16 subdomains. (b) Baseline MPI -4 subdomains. (c) 2-level MPI+thread -4 subdomains. (d) Caption.

Figure 2 :

 2 Figure 2: Baseline MPI (a, b) versus MPI+thread (c) models on a 16 cores platform composed of two nodes of eight cores each. Imposing four subdomains limits the baseline MPI model to exploit four cores (b) but not the MPI+thread model (c).

Figure 3 :

 3 Figure 3: Operations related to the factorization of the first supernode (red block): triangular solve on the off-diagonal blocks (yellow blocks) and update to the facing column-blocks that owns the same rows (green blocks).

Figure 4 :

 4 Figure 4: Binding strategies.

Inria ( a )

 a All computantional steps. (b) Factorization of the interiors. (c) Setup of the preconditioner. (d) Solve step.

Figure 5 :

 5 Figure 5: Performance of MaPHyS when binding management is fully delegated to the operating system (configuration of Figure 4(a)) for matrix Matrix211 on eight nodes of PlaFRIM with dense preconditioner. On each node, all eight cores are consistently used but the number of threads per subdomain (1, 2, 4 or 8, x-axis) is inversely proportional to the number of subdomains (8, 4, 2, 1).

( a )

 a All computantional steps. (b) Factorization of the interiors. (c) Setup of the preconditioner. (d) Solve step.

Figure 6 :

 6 Figure 6: Performance of MaPHyS when processes are binded to a subset of cores (configuration of Figure 4(b)) for matrix Matrix211 on 8 nodes of PlaFRIM with dense preconditioner. On each node, all eight cores are consistently used but the number of threads per subdomain (1, 2, 4 or 8,x-axis) is inversely proportional to the number of subdomains (8, 4, 2, 1).

Figure 7 :

 7 Figure 7: Using 1, 2, 4 or 8 threads and per process (set up in Section 4.3).

  (b)) when 8 threads are used. The size of the interiors is large enough (a) All computational steps. (b) Factorization of the interiors. (c) Setup of the preconditioner. (d) Solve step.

  (a) All computational steps. (b) Factorization of the interiors. (c) Setup of the preconditioner. (d) Solve step.

Figure 10 :

 10 Figure 10: Setup in Section 4.4 illustrating the numerical and parallel flexibility of the 2-level implementation for exploiting all cores of a node with 1, 2, 4 or 8 subdomains per node.

  (b)). The main reason for

Figure 11 :

 11 Figure 11: Average size of the interiors (11(a)), of the local Schur complement (11(c)), of the total Schur complement (11(b)) and number of iterations (11(d)) for each matrix shown when all available cores are used and the following statement is satisfied : nb_nodes × nb_cores = nb_threads_per_process × nb_subdomains. The number of nodes and the preconditioner are given in Table3and the number of threads used per process is given on top of each histogram bar. All histograms are normalized with respect with the histogram when one thread per process is used.

  (a) All computational steps. (b) Factorization of the interiors. (c) Setup of preconditioner. (d) Solve step.

Figure 12 :

 12 Figure 12: In these experiments all available cores are used and the following statement is satisfied : nb_nodes × nb_cores = nb_threads_per_process × nb_subdomains. The number of nodes and the preconditioner are given in Table3and the number of threads used per process is given on top of each histogram bar. Each histogram is normalized in comparison to the time obtained when one thread per subdomain is used.

Figure 13 :

 13 Figure 13: The maximum time 13(a) and memory peak per node 13(b) when the Tdr455k matrix is used with dense preconditioner. All the available cores per node are used and the statement is satisfied: nb_nodes × nb_cores = nb_threads_per_process × nb_subdomains. In the legend t p

  (a) Elapsed time for all computational steps. (b) All computational steps. (c) Memory peak per node. (d) Sizes.

Figure 14 :

 14 Figure 14: The maximum time 14(a) and memory peak per node 14(c) for the Nachos4M matrix with sparse preconditioner when a dropping is applied to the preconditioner with threshold of 10 -2 . The detailed histogram for all computational steps is given in 14(b) and the sizes for the interior, the local Schur complement and the total size of the Schur complement are given in 14(d). The tests were performed on the Hopper platform. All the available cores per node are used and the statement is satisfied: nb_nodes × nb_cores = nb_threads_per_process × nb_subdomains. In the legend t p refers to number of threads (t) per process (p).

Inria

Figure 15 :

 15 Figure15: Convergence history with respect with the elapsed time for the Audi_kw matrix when four nodes are used in respect with the iteration (see Figure15(a)) and in respect with the time (see Figure15(b)). Dropping with a threshold of 10 -2 is applied to the preconditioner.

Table 1 :

 1 Matrices used in this study.

	Matrix	Matrix211	Audi_kw	Nachos	Haltere	Tdr455k	Nachos4M	Amande
	n	0.8M	0.9M	1.1M	1.3M	2.7M	4.1M	7.0M
	nnz	129M	392M	40M	10M	113M	256M	58M
	Symmetry	non-symmetric	SPD	non-symmetric	symmetric	non-symmetric	non-symmetric	symmetric
	Arithmetic	real	real	complex	complex	complex	complex	complex

Table 4 :

 4 Number of iterations for different configurations (#subdomains and preconditioner) for the Tdr455k matrix with the GMRES algorithm. "-" means that the algorithm fails to achieve convergence in 7000 iterations with restart of 500.

				#subdomains
			8	16	32	64	128
	Precond.	dense 10 -4 10 -3 10 -2	9 31 168 1002 21 108 502 107 315 6323 -----6496 ----

There are also options for computing Incomplete LU (ILU) factorizations of the interiors but the related descriptions are out the scope of this paper.RR n°8960

RR n°8960

Inria

avenue de la Vielle Tour 33405 Talence Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Acknowledgement

The authors would like to thank Sherry Li (LBNL), Esmond Ng (LBNL) and Ichitaro Yamazaki (UTK) for constructive exchanges on the design of high-performance hybrid schemes. They are also grateful to the PaStiX team for their great support in the integration of the direct solver within MaPHyS. They also thank the NERSC and Plafrim support teams. All this work has been conducted in the context of the Depth Imaging Partnership (DIP) between Total and Inria and the authors would specially like to thank Henri Calandra (Total) for this support.


Marysia Ogrodnik
email: marysia.ogrodnik@univ-paris1.fr

INVESTIGATING THE DETERMINANTS OF SMOKING CESSATION: FROM THE DESIRE TO QUIT TO THE EFFECTIVE ATTEMPT

Keywords: JEL Classification: C25, D83, I12 addiction, motivation, self-control, smoking, stages of change

Smoking behavior involves complex mechanisms such as addiction (physical and psychological dependence, self-control, social smoking) and smoking related perceptions (self-exempting beliefs and smoking norms). The latter evolve through consumption career described by the stages of change. Those effects are tested on a sample of French smokers and ex-smokers who completed an onlinesurvey.

A first model tests the role of addiction on the motivation to quit. The results show a strong relationship: physical and psychological dependence are positively related to the motivation to quit, as self-control is negatively related. Moreover, as cigarette consumption becomes more anchored in everyday life and more solitary, reported motivation to quit increases. The second model, studies the relationship between smoking related perceptions and motivation to quit. Results show that smoking denormalization beliefs increase, and self-exempting beliefs decrease across the smoking career.

In order to re-establish smokers' self-control, public policies should act on physical dependence (by helping consumers to adopt a smoking cessation strategy through the funding of different kind of treatments), but also on environmental cues that trigger the desire to smoke (by limiting them). The second possible strategy is to act on the relationship between smoking-related beliefs and the motivation to quit by promoting nudges, normative change, and moderate fear-appealing campaigns associated to high levels of efficacy and self-efficacy.

INTRODUCTION

Smoking prevalence

French public authorities invested substantial resources in order to overcome the smoking issue.

During the past decade, public bans were multiplied, cigarette price has doubled, and preventive campaigns have been intensified. In response, the evolution of French cigarette consumption showed puzzling results. Between 2010 and 2014, among the population aged 15 to 75 years, the rate of smoking has weakly increased from 33.7% up to 34.1%, although daily smoking rate diminished from 29.1% to 28.2% [START_REF] Guignard | La consommation de tabac en France en 2014: caractéristiques et évolutions récentes[END_REF]. In addition, daily cigarette consumption has stagnated at 11. Abstention process involves a succession of changes including desire to quit appearance, followed by intention to quit, and effective attempt. Understand which factors enhance those evolutions and which ones hinder them is fundamental to interpret the stagnation of smoking prevalence and its tendency to increase during the last years, and to propose effective instruments that could be adopted by public authorities, in order to improve current situation.

The evolution of smoking behavior: the Transtheoretical Model

Smoking behavioral pattern has been described by the Transtheoretical Model [START_REF] Abel | Treating Addictions With EMDR Therapy and the Stages of Change[END_REF][START_REF] Connors | Substance abuse treatment and the stages of change: Selecting and planning interventions[END_REF][START_REF] Prochaska | Applying the stages of change[END_REF] from initiation to final abstention, smokers follow five stages of change. The consumer who begins to smoke is in precontemplation stage: he does not consider himself as dependent and underestimates the noxiousness of cigarette smoking compared to other smokers and non-smokers. The contemplation stage corresponds to the emergence of an ambivalence: however, he admits that smoking could have negative impact on his life; he only expresses a desire to change in the long run. Then comes the preparation stage when self-control issue appears leading the consumer to plan to quit cigarette within the six months. An effective attempt to abstain from smoking drives him into the action stage (abstention for less than six months) and then the maintenance stage (abstention for more than six months). Some factors should naturally push the smoker into the next stage. Indeed, as cigarette consumption increases and becomes anchored in the smoker's everyday life, he realizes that he has developed a dependence and has less control over his smoking behavior, which enhances successively the desire to quit, the intention to quit, and the effective attempt. However, addiction is also an impediment to a smoking cessation success that can trap the individual in preparation stage. Moreover, as described below, some cognitive mechanisms such as self-exempting beliefs Documents de travail du Centre d'Economie de la Sorbonne -2016. 52 or the perception that smoking is socially approved also intervene to slow down the stage of change succession, and can even cause a return to a previous stage.

Addiction problem

The level of addiction to cigarettes is directly linked to the motivation to quit [START_REF] Prochaska | Stages of change and decisional balance for 12 problem behaviors[END_REF]. However, this concept is complex and covers various dimensions. First, it is associated to physical dependence described by the tolerance (an increase of the amounts smoked across time in order to get a constant satisfaction), and reinforcement (an increase of the desire to smoke) effects [START_REF] Becker | An Empirical Analysis of Cigarette Addiction[END_REF]. There is also a psychological component represented by environmental cues [START_REF] Upadhyaya | Reactivity to smoking cues in adolescent cigarette smokers[END_REF]. The presence of the latter increases the subjective value of the product [START_REF] Hayashi | Dorsolateral prefrontal and orbitofrontal cortex interactions during self-control of cigarette craving[END_REF]. Those cues can be complementary good to cigarette as alcohol or coffee, or situations that were associated to cigarette consumption such as waiting in the street or boredom [START_REF] García-Rodríguez | Validation of smoking-related virtual environments for cue exposure therapy[END_REF][START_REF] Van Gucht | Smoking behavior in context: Where and when do people smoke[END_REF]. Combined together physical and psychological addiction to a product induce self-control problems that explain why quitting attempts often fail [START_REF] Hagger | Cue-Induced Smoking Urges Deplete Cigarette Smokers' Self-Control Resources[END_REF][START_REF] Kim | The effect of self-control depletion on approach tendencies toward smoking cues in relapsers[END_REF][START_REF] Vangeli | Predictors of attempts to stop smoking and their success in adult general population samples: a systematic review[END_REF]. Indeed, Cigarettes belong to a category of goods that are associated to visceral effects [START_REF] Loewenstein | Out of Control: Visceral Influences on Behavior[END_REF]. The latter arise when the amounts smoked are insufficient considering the level of addiction, and make the consumer suffer. The only way to get rid of this sufferance is to smoke more. He can therefore be "forced" to smoke even if he does not want to, losing his self-control.

In case of an addictive consumption, visceral effects are intensified by repeated consumption, as well as by environmental cues. This lack of self-control is a predictor of the motivation to quit [START_REF] De Ridder | Taking stock of self-control a meta-analysis of how trait self-control relates to a wide range of behaviors[END_REF] but also a factor of failure in quitting attempts. The last dimension of addiction to cigarette is the level of social smoking, which is firstly a social activity associated to less nicotine dependence and less intention to quit [START_REF] Moran | Social smoking among US college students[END_REF][START_REF] Song | Social smoking among young adults: investigation of intentions and attempts to quit[END_REF].

With time, it becomes more and more a solitary habit as cigarette consumption becomes anchored in everyday life.

Smoking related-perceptions

Prevention campaigns induce a cognitive dissonance among smokers [START_REF] Festinger | A theory of cognitive dissonance[END_REF] between their cognitions (the information that smoking is dangerous), and their behavior (smoking). This dissonance induces a state of tension that the individual resolves via behavioral adaptation -by Documents de travail du Centre d'Economie de la Sorbonne -2016.52 quitting smoking -or, when it is not immediately possible, by cognitive rationalization -by manipulating his beliefs [START_REF] Goethals | Fabricating and ignoring social reality: Self-serving estimates of consensus[END_REF][START_REF] Kruglanski | Lay epistemics and human knowledge: Cognitive and motivational bases[END_REF][START_REF] Kruglanski | Bias and error in human judgment[END_REF][START_REF] Shah | Heuristics made easy: an effort-reduction framework[END_REF]. The result is that "many addicts deny that they have a problem or, if they admit it, deny that they can do anything about it" [START_REF] Elster | Strong feelings: Emotion, addiction, and human behavior[END_REF]. Many smokers show unrealistic optimism [START_REF] Weinstein | Smokers' unrealistic optimism about their risk[END_REF], and self-exempting beliefs, especially in precontemplation stage [START_REF] Oakes | Bulletproof skeptics in life's jungle": which self-exempting beliefs about smoking most predict lack of progression towards quitting?[END_REF]. The problem is that dissonance reduction in the form of denial or distortion of threatening information affects the readiness to change and delay the decision to quit [START_REF] Dillard | Unrealistic optimism in smokers: Implications for smoking myth endorsement and self-protective motivation[END_REF][START_REF] Oakes | Bulletproof skeptics in life's jungle": which self-exempting beliefs about smoking most predict lack of progression towards quitting?[END_REF][START_REF] Radtke | Smoking-specific compensatory health beliefs and the readiness to stop smoking in adolescents[END_REF]. In addition, the perceived norm of smoking also appears to be related to the motivation to quit. The more the smoker is ready to change, the less he perceives that smoking is a valued behavior [START_REF] Mourre | Proposition d'un modèle intégrateur de la résistance aux messages antitabac[END_REF][START_REF] Zhang | The impact of social norm change strategies on smokers' quitting behaviours[END_REF]. Indeed, injunctive norms, more than descriptive norms (van den [START_REF] Van Den Putte | Social influences on smoking cessation: a comparison of the effect of six social influence variables[END_REF] are good predictors of individual smoking behavior [START_REF] Steele | Contending with group image: The psychology of stereotype and social identity threat[END_REF].

Smokers who attempt to quit are very supportive of anti-smoking policies [START_REF] Amador | Self-control and support for anti-smoking policies among smokers, ex smokers, and never smokers[END_REF].

Paper objectives

Few studies have distinguished the different steps involved in the motivation to quit smoking, i.e. the desire, the intention and the effective attempt to quit. In addition, no paper analyzing the link between different dimensions of addiction (dependence, environmental cues, social smoking and perceived self-control) and motivation to quit has been found. Thereby, the first study (model 1) aims to establish the impact of those factors on the different levels of motivation to quit.

Moreover, existing studies about the evolution of smoking related perceptions (regrouping selfexempting beliefs and the injunctive norm of smoking) across the smoking career are scarce [START_REF] Bursey | Attitudes, Subjective Norm, Perceived Behavioral Control, and Intentions Related to Adult Smoking Cessation After Coronary Artery Bypass Graft Surgery[END_REF]. In order to overcome this weakness, a second study (model 2) links those variables to the whole stage of change succession.

It is assumed that the power of physical dependence and environmental cues to trigger the desire to consume is increasing through the smoking stages of change, whereas reported self-control decreases and the smoking habits becomes a solitary activity through a decrease of social smoking (model 1). It is also hypothesized that, from smoking initiation to abstention, selfexempting beliefs weaken, and smoking denormalization reinforces (model 2). Documents de travail du Centre d'Economie de la Sorbonne -2016.52

MATERIAL AND METHODS

Survey procedure

The invitation to respond to the self-administered questionnaire has been spread on social networks via snowball sampling [START_REF] Goodman | Snowball sampling[END_REF] between March and May 2013. It included a brief description of the survey, instructions on how to complete it and a direct internet link. The condition of eligibility was past or present smoking experience and to be aged 15-65. Population aged 66 and more have been excluded from the analysis due to a weak access to the Internet [START_REF] Gombault | L'internet de plus en plus prisé[END_REF].

The questionnaire electronic format permits to show or to hide some questions according to the respondents' characteristics. Thereby, ex-smokers do not answer to questions concerning current smoking behavior. The online administration also avoids the duplication of cases by IP tracking, and prevents the problem of missing values by implementing forced responses (nonetheless this also causes drops out that are not measured). The online surveys also permit to avoid a part of interview biases such as social desirability [START_REF] Crowne | A new scale of social desirability independent of psychopathology[END_REF][START_REF] Kiesler | Response effects in the electronic survey[END_REF], and the recruitment process on social networks increases respondent level of confidence since the researcher shows his personal information (profile) and participates to the groups [START_REF] Baltar | Social research 2.0: virtual snowball sampling method using Facebook[END_REF]. Moreover there is no significant difference between the response of smokers who has an access to the Internet and smokers who has not [START_REF] Bigot | La diffusion des technologies de l'information et de la communication dans la société française[END_REF][START_REF] Nagelhout | Is web interviewing a good alternative to telephone interviewing? Findings from the International Tobacco Control (ITC) Netherlands Survey[END_REF].

Participant who clicked on the link, read an introduction reminding the scope of the survey, after which the questions were asked. In order to alleviate the problem of non-representativeness of the sample generated from snowball sampling, the initial sample was as much representative as possible e.g. from different age groups, geographical regions, and professions [START_REF] Morgan | Snowball sampling[END_REF]. Despite this representativeness issue, online recruitment is a suitable strategy for studying adults smokers behavior [START_REF] Bauermeister | Innovative Recruitment Using Online Networks: Lessons Learned From an Online Study of Alcohol and Other Drug Use Utilizing a Web-Based, Respondent-Driven Sampling (webRDS) Strategy[END_REF][START_REF] Klein | Self-Reported Smoking in Online Surveys: Prevalence Estimate Validity and Item Format Effects[END_REF][START_REF] Ramo | Reliability and validity of self-reported smoking in an anonymous online survey with young adults[END_REF][START_REF] Ramo | Broad reach and targeted recruitment using Facebook for an online survey of young adult substance use[END_REF]. However, the results cannot be generalized to the whole population. Documents de travail du Centre d'Economie de la Sorbonne -2016.52

Survey measures

Dependent variable : Stage of change

Respondents are first associated to a stage of change, the dependent variable [START_REF] Prochaska | Predicting change in smoking status for self-changers[END_REF]: (a) precontemplation, for those who answered "no" to the question "would you like to quit smoking?"; (b) contemplation when they answered "yes" to the previous question and "no" to the question "do you plan to quit smoking within the six months?"; (c) preparation if they answered "yes" to both questions2 ; and (d) action-preparation stages, for respondents who reported to be ex-smokers.

Independent variables

Addiction measures (model 1)

Four measures are computed to indicate respondents' level of addiction to cigarettes, and concerns only current smokers (in precontemplation, contemplation and preparation stages).

Physical dependence is measured through the heaviness smoking index -HSI [START_REF] Heatherton | Measuring the Heaviness of Smoking: using self-reported time to the first cigarette of the day and number of cigarettes smoked per day[END_REF] composed of the addition of two items: the daily cigarette consumption (0=10 or less, 1=11-20, 2=21-30, 3=31 and more) and the duration between wake up and the first cigarette (0= more than 60 min, 1=31-60 min, 2=6-30 min, 3=5 min or less). The score (from 0 to 6), is considered to be a good brief screen for high nicotine dependence [START_REF] Borland | The reliability and predictive validity of the Heaviness of Smoking Index and its two components: Findings from the International Tobacco Control Four Country study[END_REF][START_REF] Pérez-Ríos | Fagerstrom test for nicotine dependence vs heavy smoking index in a general population survey[END_REF].

Sensitivity to environmental cues is measured by asking the smokers which of the situations, proposed in García-Rodríguez et al. ( 2012) usually enhance their cigarette consumption (0=no, 1=yes): "drinking coffee", "eating", "making a pause during work", "relaxing time", "boredom".

The score (from 0 to 5) is the addition of the responses to the items and shows a satisfactory internal consistency (Cronbach 𝛼=0.72). Indeed, the amounts of contexts triggering the desire to smoke is a good indicator of the level of addiction [START_REF] Van Gucht | Smoking behavior in context: Where and when do people smoke[END_REF]. Self-control over tobacco consumption is assessed through the item "I can easily control my cigarette consumption when I am in a situation of craving" (0=never, to 10=always). The last component of addiction is the propensity to smoke with peers (0=0% of cigarettes consumed, to 10: 100% of cigarettes consumed).

Markers of denormalization (model 2)

Respondents' perception of the smoking norm is appraised by two variables. The injunctive norm is an indirect measure. It is assessed by questioning respondents on how they think that smokers are generally perceived (0=totally disagree, to 4=totally agree) through items retrieved from [START_REF] Chapman | Markers of the denormalisation of smoking and the tobacco industry[END_REF] : "people think that smokers are selfish", "people think that smokers are less productive", "people think that smokers are excessive users of public health services", "people think that smokers are litterers". The items of the environmental cues scores (from 0 to 16) show a correct internal consistency (Cronbach 𝛼=0.73). A second measure of smoking denormalization is more direct and refers to the level of disturbance caused by secondhand smoke (0=not bothered at all, to 10=very bothered).

Perceived cigarette noxiousness (model 2)

Just like markers of smoking denormalization, personal vulnerability to cigarette noxiousness is evaluated directly and indirectly. [START_REF] Oakes | Bulletproof skeptics in life's jungle": which self-exempting beliefs about smoking most predict lack of progression towards quitting?[END_REF] item measuring bulletproof smokers belief "I think I must have the sort of good health or genes that means I can smoke without getting any of the harms" is used here to assess the direct perception of the respondent vulnerability. The end of sentence is reformulated to "… that means I'm not sensitive to smoking harms" (1= totally disagree, to 10=totally agree), in order to be applicable for ex-smokers. Evaluation of the percentage of lung cancer cured (0=0% to 10=100%) is an indirect manner to measure the perceived level of vulnerability to smoking noxiousness [START_REF] Weinstein | Smokers' unrealistic optimism about their risk[END_REF].

Statistical analysis

All statistics analyses were performed of R software. Effects were considered significant at 𝛼 level of 0.05 or less. First, for each model, the relationship between the dependent variable and the independents variables was examined through bivariate analyses. Factorial ANOVAs were used for continuous variable whereas Pearson chi square tests were done for categorical variables. Afterwards, logistic regressions are run in order to test the hypotheses made in 1.6. Documents de travail du Centre d'Economie de la Sorbonne -2016.52

Descriptive statistics

Participants general features

The sample is made up of the N=269 French participants (54.3% female), with a mean age of 32.4 years (s.d.=13.5). Among those participants, 66.9% reported to be smokers (occasional and daily smokers): 28.6% of the total sample is in pre-contemplation stage (42.7% of current smokers), 20.8% is in contemplation stage (31.1%) and 17.5% in preparation (26.1%). Those proportions are similar to those found by the INPES3 [START_REF] Guignard | La consommation de tabac en France en 2014: caractéristiques et évolutions récentes[END_REF] for 2014. The remaining 33.1% reported to be ex-smokers, which is less than in the INPES study 4 , probably due to the over-representation of young respondents.

Whereas gender is not related to the smoking stage (𝜒 2 (3)=3.62, p=.302), it is not surprising to observe that age shows a significant relation (F(3,365)=35.87, p<.001). Indeed, most smokers begin young, and individuals who reach their mid-20 as non-smokers are unlikely to become smokers [START_REF] Aldrich | The role of epidemiology in evidence-based policy making: a case study of tobacco use in youth[END_REF]. Thus, age can be directly related to consumption history. Moreover, stages of change follow each other during the consumption career. As for consequences the more the smoker ages, the more he has chances to be in a more advanced stage.

Addiction measures (model 1)

As expected, a more advanced stage of change is associated with a higher HSI (F(2,177)=21.57, p<0.001)), a greater sensitivity to environmental cues (F(2,177)=16.70, p<.001), a lower self-control over cigarettes consumption (F(2,177)=28.21, p<.001), and a smaller propensity to smoke with peers (F(2,177)=17.85, p<.001).

Markers of denormalization (model 2)

Bivariate analyses showed that injunctive norm deteriorates (F(3,265)=9.39, p<.001), and that the level of discomfort caused by smoke increases (F(3,265,)=15.77, p<.001) through the stages of change.

Perceived cigarette noxiousness (model 2)

Perception of the amount of lung cancers cured (F(3,265)=12.17, p<.001) as well as the role of luckiness as a factor of protection from the danger (F(3,265)=9.40, p<.001) are significantly related to the stage of change.

Test for parallel lines assumption

Cumulative logistic models work under parallel lines assumption: each cumulative logit has its own intercept and model assumes the same effects for each logit. Thereby, effects are simple to summarize since each predictor is accompanied by one single parameter. Each cumulative odds ratio is proportional to the distance between the 𝑋 𝑗 . The same proportionality constant applies for each logit [START_REF] Agresti | Categorical data analysis[END_REF]. Here, a likelihood ratio test is used to test the parallel lines assumption. In other words, the following assumption is tested [START_REF] Erkan | Parallel lines assumption in ordinal logistic regression and analysis approaches[END_REF]:

𝐻 0 = 𝑎 𝑗,1 = … = 𝑎 𝑗,𝐾-1 = 𝑎 𝑗 , 𝑘 = 1, … , 𝐾 -1
If parallel lines assumption does not hold for at least one variable, the assumption is breached.

For both regressions, likelihood ratio tests of the proportional odds assumption permit to find that there is no evidence of non-proportional odds for any independent variable. Results are presented in tables 3 and 4, and are accompanied by indicators of goodness of fit (Akaike Information Criteria, and deviation measure e.g. LogLikelihood). This suggests that there is a linear relationship between the stage of change and the independent variables in both models.

Check for multicollinearity

The degree of multicollinearity of the independent variables is measured by the Variance Inflation Factor (VIF). For each variable k, it corresponds to: 𝑉𝐼𝐹(â 𝑘 ) 1 1-𝑅 𝑘 2 were 𝑅 𝑘 2 corresponds to the unadjusted coefficient of determination. The latter is obtained by running a linear regression using 𝑘 as the response variable [START_REF] Hendrickx | Collinearity involving ordered and unordered categorical variables[END_REF]. From each regression, VIF values are manually computed. The results are provided in tables 5 and 6. All VIF values are smaller than 10 so there is no multicollinearity between the dependent variables [START_REF] Hair | Multivariate data analysis[END_REF][START_REF] Menard | Applied logistic regression analysis: Sage university series on quantitative applications in the social sciences[END_REF].

RESULTS

Model 1: The role of addiction factors on the motivation to quit smoking

As assumed, HSI increases from precontemplation to preparation stage (OR=1.

DISCUSSION AND CONCLUSION

The results of the first study demonstrate a relationship between addiction and the motivation to quit. Motivation to quit is induced by the awareness that addiction is increasing (through an increase of the physical dependence) and that consumption is more solitary and anchored in everyday life (an increasing amount of elements from environment are associated to smoking). directions [START_REF] Thaler | Nudge. Improving Decisions About Health, Wealth, and Happiness[END_REF], retain attention. They are a very hot topic that captured attention of researchers, as well as policy makers with the creation of Behavioral Insight Team 5 in the United Kingdom, or the recent Obama 6 executive order toward an application of behavioral science in public policies. According to [START_REF] Reisch | Do Europeans Like Nudges?[END_REF], nudges aiming to discourage smoking behavior are accepted in most European countries examined, among which France, but less than other nudges since smoking is still a socially accepted behavior. Deposit programs 7 are for instance an effective strategy to quit smoking [START_REF] Giné | Put your money where your butt is: a commitment contract for smoking cessation[END_REF], although they are less attractive than an award program [START_REF] Halpern | Randomized Trial of Four Financial-Incentive Programs for Smoking Cessation[END_REF]. Other interventions permitting to reestablish individual self-control aim to diminish environmental cues. They include smoking bans [START_REF] Boes | The Impact of Smoking Bans on Smoking and Consumer Behavior: Quasi-Experimental Evidence from Switzerland[END_REF]Kan, 2007), making smaller cigarette pack available [START_REF] Marti | Smaller Cigarette Pack as a Commitment to Smoke Less? Insights from Behavioral Economics[END_REF] with a unitary price similar to bigger packages, or making substitutions treatments more available [START_REF] Cornuz | Costeffectiveness of pharmacotherapies for nicotine dependence in primary care settings: a multinational comparison[END_REF][START_REF] Hoogendoorn | Long-term effectiveness and cost-effectiveness of smoking cessation interventions in patients with COPD[END_REF].

The second study shows that the different perceptions associated to smoking activity, are worsened as the individual evolves throughout the stages of changes. Injunctive norms about smoking worsen, as the cigarette smoke becomes more and more disturbing. On the other side, perceived danger of smoking increases as the individual gets closer to the effective abstention, and self-exempting beliefs weaken as the individual reaches a stage in which he is be less concerned by the danger. However, there is no evidence about the direction of causality i.e. if the modification of the perception causes the change in stage (behavioral adaptation), or if the change in stage causes the modification in perceptions (cognitive adaptation). As for consequences, this point needs further investigations in the context of a longitudinal study.

In the first case (behavioral adaptation), policies aiming a normative change would be effective in reducing smoking behavior by accelerating the abstention desire, plan and attempt. Use mass media campaign in order to change indirectly social norms works for health behavior [START_REF] Wakefield | Use of mass media campaigns to change health behaviour[END_REF] among which smoking behavior [START_REF] Lazuras | Social cognitive predictors of smoking cessation intentions among smoker employees: The roles of anticipated regret and social norms[END_REF], but this method requires to be 5 http://www.behaviouralinsights.co.uk/ 6 https://www.whitehouse.gov/the-press-office/2015/09/15/executive-order-using-behavioral-science-insightsbetter-serve-american 7 Commitment contracts consists on registering on a platform as stick.com where the individual fixes his objective (here stop smoking), a deadline, a way to verify that he attained his objectives (for instance a cotinine blood test in a given laboratory). Then he put money in a virtual account, that he will only get back if he attains his objective. Documents de travail du Centre d'Economie de la Sorbonne -2016.52 careful about stigmatization by accompaigning population at risk. Indeed, stigma can be a barrier for help-seeking [START_REF] Brown-Johnson | Exploring Smoking Stigma, Alternative Tobacco Product Use, and Quit Attempts[END_REF][START_REF] Clement | What is the impact of mental health-related stigma on help-seeking? A systematic review of quantitative and qualitative studies[END_REF]. Social stigma can also lead to unintended consequences such as "guilt, loss of self-esteem, defensiveness and resolve to continue smoking" [START_REF] Bell | Smoking, stigma and tobacco "denormalization": Further reflections on the use of stigma as a public health tool[END_REF][START_REF] Dunlop | When Your Smoking Is Not Just About You: Antismoking Advertising, Interpersonal Pressure, and Quitting Outcomes[END_REF] that prevent the smoker from a long term cessation.

In the second case, cognitive adaptation prevents the individual from relapse to a previous stage, but it also means that in case of relapse, perception will also adapt. Thus solution would be to use fear campaigns that follow the extended parallel process model [START_REF] Witte | Putting the fear back into fear appeals: The extended parallel process model[END_REF][START_REF] Witte | A Meta-Analysis of Fear Appeals: Implications for Effective Public Health Campaigns[END_REF] recommendations, i.e. inducing a moderate level of fear and enhancing high degree of selfefficacy and response efficacy [START_REF] Gharlipour | The effect of preventive educational program in cigarette smoking: Extended Parallel Process Model[END_REF][START_REF] Peters | Threatening communication: a critical re-analysis and a revised meta-analytic test of fear appeal theory[END_REF]. Self-efficacy and response efficacy are strengthened when the fear-appealing message is accompanied by the proposition of credible solutions providing support to an abstention from addictive consumption, that consumer would feel able to put in place reinforcing his self-efficacy. Therefore, accessibility of quitting strategies (i.e. the cost of avoidance and rehabilitation strategies) plays a key role. For instance, propose quitting lines are an effective device to help motivated smokers to quit [START_REF] Inpes | Tabac Info Service, une efficacité démontrée dans l'arrêt du tabac[END_REF][START_REF] Lichtenstein | Smoking cessation quitlines: An underrecognized intervention success story[END_REF][START_REF] Neri | Use and effectiveness of quitlines versus Web-based tobacco cessation interventions among 4 state tobacco control programs[END_REF][START_REF] Nguyen-Thanh | Aide à l'arrêt du tabac par téléphone: Efficacité et organisation[END_REF], however, this device is scarcely used [START_REF] Kaufman | Awareness and Use of Tobacco Quitlines: Evidence from the Health Information National Trends Survey[END_REF].

Since the external validity from online samples in unknown [START_REF] Barratt | Hidden Populations, Online Purposive Sampling, and External Validity Taking off the Blindfold[END_REF], the present research has been conducted without losing sight of the fact that the results cannot be inferred or generalized beyond the sample itself. However, our conclusions provide a trail for further researches on larger and more representative populations. Documents de travail du Centre d'Economie de la Sorbonne -2016.52

Finally, losses of

  self-control overconsumption lead the agent to consider to stop smoking. On the other side, those factors are also an obstacle for a quitting success. Facing such a contradiction between what the consumer wants to do, and what he actually does, nudges, i.e. approaches to law and policy that maintain freedom of choice and at the same time steer people in given Documents de travail du Centre d'Economie de la Sorbonne -2016.52

Table 1 -

 1 Participant characteristics

	Variable	Precontemplation	Contemplation	Preparation	Action Maintenance
		No. (%)	No. (%)	No. (%)	No. (%)	𝜒 2	p-value
	Respondents	77 (29.6)	56 (20.8)	47 (17.5)	89 (33.1)
	Female	41 (53.2)	26 (46.4)	24 (51.1)	55 (61.8)	𝜒 2 (3)=3.62	0.302
		Mean (s.d.)	Mean (s.d.)	Mean (s.d.)	Mean (s.d.)	F	p-value
	Age	24.5 (10.8)	27.1 (10.4)	34.2 (12.2)	41.6 (21.1)	F(3,365)=35.87 <0.001

Table 2

 2 -addiction, smoking norm and self-exempting beliefs among the stages of change

	Variable	Precontemplation	Contemplation Preparation	Action Maintenance
		Mean (s.d.)	Mean (s.d.)	Mean (s.d.)	Mean (s.d.)	F-value	p-value
	Heaviness smoking index	0.78 (1.16)	1.73 (1.42)	2.43 (2.42)	-	F(2,177)=21.57	<0.001
	Environmental cues	2.09 (1.63)	2.92 (1.58)	3.76 (1.47)	-	F(2,177)=16.70	<0.001
	Self-control	6.40 (2.71)	4.48 (2.38)	3.14 (1.82)	-	F(2,177)=28.21	<0.001
	Smoking with peers	6.92 (2.48)	5.77 (2.42)	4.29 (2.14)	-	F(2,177)=17.85	<0.001
	Injunctive norm	4.97 (3.13)	5.25 (2.85)	6.23 (3.47)	7.44 (3.50)	F(3,265)=9 .39	<0.001
	Discomfort caused by smoke	4.09 (2.77)	4.09 (2.08)	5.38 (2.97)	6.56 (2.60)	F(3,265)=15.77	<0.001
	Percentage of lung cancers cured	4.54 (2.07)	3.52 (1.88)	3.66 (2.29)	2.72 (1.66)	F(3,265)=12.17	<0.001
	Luckiness as a protection factor	5.22 (2.88)	4.66 (2.94)	4.36 (2.83)	3.07 (2.57)	F(3,265)=9.40	<0.001
	Logistic regression
	A cumulative logistic regression model (using ordered variable response), has been chosen for
	two studies. Such a specification permits to compare a set of groups (stages of change) to all the
	groups that are below. If there are K groups, K-1 logits are computed by using the following
	equation: 𝐿𝑂𝐺𝐼𝑇(𝑃(𝑌 > 𝑘|𝑋)) = ln	𝑃(𝑌>𝑘/𝑋) 𝑃(𝑌≤ 𝑘/𝑋)	= 𝑎 0 + 𝑎 1 𝑋 1 + ⋯ + 𝑎 𝐽 𝑋 𝐽 ,	𝑘 = 1, … , 𝐾 -1

were coefficients 𝑎 𝑗,𝑘 can be read as follows: when a variable 𝑋 𝑗 increases by ∆, the different LOGIT increase by : ∀𝑘, 𝐿𝑂𝐺𝐼𝑇 𝑘 (𝑋 𝑗 + ∆) -𝐿𝑂𝐺𝐼𝑇 𝑘 (𝑋 𝑗 ) = â 𝑗,𝑘 × ∆. Documents de travail du Centre d'Economie de la Sorbonne -2016.52

Table 3 -

 3 Likelihood ratio tests of the proportional odds assumption -Model 1

	Variable	Df	LogLik	AIC	LRT	p-value
	<None>	-	-153.82	319.65	-	-
	HSI	1	-153.51	321.01	0.6332	0.4262
	Environmental cues	1	-153.14	320.29	1.3602	0.2435
	Smoking with peers	1	-153.79	321.58	0.0679	0.7944
	Self-control	1	-153.78	321.57	0.0778	0.7803

Table 4 -

 4 Likelihood ratio tests of the proportional odds assumption -Model 2

	Variable	Df	LogLik	AIC	LRT	p-value
	<None>	-	-316.08	646.16	-	-
	Injunctive norm	2	-315.12	648.25	1.9141	0.3840
	Discomfort caused by smoke	2	-314.04	646.07	4.0883	0.1295
	Percentage lung cancers cured	2	-314.87	647.74	2.4190	0.2983
	Luckiness as a protection factor 2	-315.08	648.75	2.0084	0.3663

Documents de travail du Centre d'Economie de la Sorbonne -2016.52

Table 5 -

 5 Check for predictors multicollinearity -Model 1

	Dependent variable	Independent variables	Coefficient of determination	VIF
	HSI	Environmental cues, Smoking with peers, Self-control	0.2444	1.3235
	Environmental cues	HSI, Smoking with peers, Self-control	0.1693	1.2038
	Smoking with peers	HSI, Environmental cues, Self-control	0.2129	1.2705
	Self-control	HSI, Environmental cues, Smoking with peers	0.2902	1.4088

Table 6 -

 6 Check for predictors multicollinearity -Model 2

	Dependent variable	Independent variables	Coefficient of determination	VIF
	Injunctive norm	Discomfort caused by smoke. Percentage of lung cancers cured. Luckiness as a protection factor	0.1194	1.1356
	Discomfort caused by smoke	Injunctive norm. Percentage of lung cancers cured. Luckiness as a protection factor	0.0949	1.1049
	Percentage lung cancers cured	Injunctive norm Discomfort caused by smoke. Luckiness as a protection factor	0.0932	1.1027
	Luckiness as a protection factor	Injunctive norm. Discomfort caused by smoke. Percentage of lung cancers cured. Luckiness as a protection factor	0.0893	1.0981

Table 7 -

 7 Cumulative logistic regression (one smoking stage vs. previous stages) -Model 1

	Variable	OR	95% CI	p-value
	HSI	1.38	(1.10, 1.72)	0.004**
	Environmental cues	1.32	(1.09, 1.60)	<0.001***
	Smoking with peers	0.84	(0.74, 0.96)	0.011*
	Self-control	0.78	(0.68, 0.90)	<0.001***
	OR= Odd-ratio, CI= confidence interval, levels of significance: * <0.05, **<0.01, *** <0.001.
	Model 2
	Smoking injunctive norm worsens (OR=1.12, p<0.001), and individuals reports to be more
	bothered by smoke as they evolve through the stages of change (OR=1.26, p<0.001). Perception
	of dangerousness of smoking, and bulletproof beliefs are less strong: the perceived percentage of
	lung cancer cured decreases (OR=0.76, p<0.001) and luckiness is less perceived as a factor of
	protection from danger (OR=0.89, p=0.005).

Table 8 :

 8 Cumulative logistic regression (smoking stages vs. previous stages) -Model 2

	Variable	OR	95% CI	p-value
	Injunctive norm	1.12	(1.03,1.19)	<0.001***
	Discomfort caused by smoke	1.26	(1.16,1.38)	<0.001***
	Percentage lung cancers cured	0.77	(0.68,0.86)	<0.001***
	Luckiness as a protection factor	0.89	(0. 81,0.96)	0.005**
	OR= Odd-ratio, CI= confidence interval, levels of significance: * <0.05, **<0.01, *** <0.001.

No respondent reported to plan quit smoking and not to desire quit smoking. Otherwise, the agent would be placed in preparation stage. This kind of situation can occur for instance when the smoker cannot afford cigarettes anymore, or when he feels too much pressure to quit from his relatives. Documents de travail du Centre d'Economie de la Sorbonne -2016.52

According to[START_REF] Guignard | La consommation de tabac en France en 2014: caractéristiques et évolutions récentes[END_REF], 59.5% of smokers report a desire to quit. Therefore

40.5% (1-0.595) are in precontemplation stage. Moreover, 27.3% plan to quit within six months, so they are 32.2% (0.595-0.273) in contemplation stage and 27.3% in preparation stage.4 Compared to[START_REF] Guignard | La consommation de tabac en France en 2014: caractéristiques et évolutions récentes[END_REF] study, ex-smokers are underrepresented in the present study (33.1%) versus 47.6% (when non-smokers are excluded from the sample). Documents de travail du Centre d'Economie de la Sorbonne -2016.52