F Ben Belgacem

V Girault
email: girault@ann.jussieu.fr

F Jelassi
email: fjelassi@utc.fr

Analysis of Lavrentiev-Finite Element Methods for Data Completion Problems

Keywords: Cauchy problem, ill posed problem, Lavrentiev regularization, finite elements, bias-variance decomposition, error analysis

The variational finite element solution of Cauchy's problem, expressed in the Steklov-Poincaré framework and regularized by the Lavrentiev method, has been introduced and computationally assessed in [Inverse Problems in Science and Engineering, 18, 1063-1086 (2011)]. The present work concentrates on the numerical analysis of the semi-discrete problem. We perform the mathematical study of the error to rigorously establish the convergence of the global bias-variance error.

Introduction

Solving data completion problems consists in implementing numerical procedures for reconstructing missing data on a portion of the boundary of the domain which is inaccessible to measurements.

The success of the (computational) reconstruction depends on the data that users are able to collect along the accessible boundary; these are Cauchy's data. As a matter of fact, 'Cauchy's problems' is an alternative terminology for these models. The amount of necessary boundary data is tightly related to the (partial) differential equation considered. When the differential equation is elliptic such as the Laplace equation, the data completion problem turns out to be severely ill-posed (see [START_REF] Hadarmard | Lectures on Cauchy's Problem in Linear Partial Differential Equation[END_REF][START_REF] Ben Belgacem | Why is the Cauchy's Problem Severely Ill-posed?[END_REF]). In particular, numerical schemes cannot be accurate and stable at the same time. These two properties should be carefully balanced to ensure satisfactory results. A wide literature exists on the subject, in both theoretical and numerical analysis, and also in computations and applications. A non-exhaustive recent bibliography is [START_REF] Andrieux | On the determination of missing boundary data for solids with nonlinear material behaviors, using displacement fields measured on a part of their boundaries[END_REF][START_REF] Feng | Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method[END_REF][START_REF] Cheng | A coupled complex boundary method for the Cauchy problem[END_REF][START_REF] Liu | The pre/post equilibrated conditioning methods to solve Cauchy problems[END_REF][START_REF] Habbal | Neumann-Dirichlet Nash Strategies for the Solution of Elliptic Cauchy Problems[END_REF][START_REF] Jin | A Bayesian inference approach to the ill-posed Cauchy problem of steadyheat conduction[END_REF][START_REF] Kadria | Identification of Contact Pressures in Two and Three-Dimensional Solid Bodies from Cauchy Data[END_REF][START_REF] Delvare | A robust data completion method for two dimensional Cauchy problems associated with the Laplace equation[END_REF][START_REF] Li | New approach for solving the inverse boundary value problem of Laplace's equation on a circle: Technique renovation of the Grad-Shafranov (GS) reconstruction[END_REF][START_REF] Duranda | Numerical solution of Cauchy problems in linear elasticity in axisymmetric situations[END_REF][START_REF] Boukari | A convergent data completion algorithm using surface integral equations[END_REF][START_REF] Langrenne | Vector intensity reconstruction using the data completion method[END_REF]. A list of earlier works is found in [START_REF] Azaïez | On Cauchy's Problem. II. Completion, Regularization and Approximation[END_REF]. Various computational methods have been designed and different frameworks, variational or not, have been introduced. Many have been experimented and evaluated with varying success. The quasi-reversibility [START_REF] Bourgeois | Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation[END_REF][START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data Inverse Problems[END_REF][START_REF] Dardé | An H div -Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems[END_REF], the complex method [START_REF] Cheng | A coupled complex boundary method for the Cauchy problem[END_REF], the boundary integral method [START_REF] Boukari | A convergent data completion algorithm using surface integral equations[END_REF], the method of the fundamental solution [START_REF] Marin | The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity[END_REF][START_REF] Shigeta | Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points[END_REF], the least-squares or optimal control formulation [START_REF] Rischette | Numerical analysis of an energy-like minimization method to solve the Cauchy problem with noisy data[END_REF] are among popular approaches.

The variational framework elaborated in [START_REF] Ben Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF], letting aside its computational relevance, brings about great facilities to the analysis and the proof of important properties related to the severe illposedness of the problem. The variational problem introduced is set on the incomplete boundary and is the result of the Kohn-Vogelius duplicating procedure associated to the Steklov-Poincaré condensation concept. We refer to [START_REF] Azaïez | On Cauchy's Problem. II. Completion, Regularization and Approximation[END_REF][START_REF] Ben Belgacem | The Lavrentiev regularization of the data completion problem[END_REF][START_REF] Du | A Lavrentiev Finite Element Model for the Cauchy Problem of Data Completion: Analysis and Numerical Assessement[END_REF][START_REF] Ben Belgacem | Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality[END_REF] for a comprehensive exposition, especially when Lavrentiev regularization is used. A finite-element discretization has also been presented in [START_REF] Azaïez | A Finite Element Model for the Data Completion Problem: Analysis and Assessment[END_REF].

This reference discusses related computational issues such as existence and uniqueness criteria for the resulting linear system, and presents a wide range of numerical experiments that show the reliability and relevance of the numerical method. The ultimate goal, namely the convergence analysis of the full approximation, turns out to be really hard, and thus was missing so far. Methodologically, we choose to explore the problem gradually( 1) and the present work is a first step in this direction. We consider the (semi-discrete) problem, where the use of finite elements is confined to the incomplete boundary. This problem inherits and still suffers from the main instability features of the continuous completion process. We focus on the complications specifically arising from the ill-posedness and attempt to handle them in a mathematical way.

The outline of the paper is as follows. Section 2 recalls the variational formulation of Cauchy's problem, where the unknown is the missing Dirichlet boundary condition, and lists useful properties of the variational solution. Section 3 mainly describes the Lavrentiev regularization and its Galerkin finite element approximation. In Section 4, we conduct the numerical analysis of the discrete solution and state a convergence result in polygonally shaped domains. This study uses sharp tools from the theory of elliptic regularity. We address afterward the effect of noisy data and discuss how they affect the selection of the regularization parameter and the mesh-size, to guarantee satisfactory computations. Finally, we investigate the issue of local convergence, away from the incomplete boundary. By means of a Carleman inequality, Hölderian convergence rates with respect to the regularizing parameter ̺ and mesh-size h are therefore derived.

Notation.-Let Ω be a bounded Lipschitzian domain in R d , d = 2, 3. The symbol x denotes the generic point of Ω. As usual, L 2 (Ω) is the Lebesgue space of square integrable functions, with inner product (•, •) L 2 (Ω) and associated norm • L 2 (Ω) . The Sobolev space H 1 (Ω) is the space of the functions that are in L 2 (Ω) as well as their first order derivatives. Let Υ ⊂ ∂Ω be a closed subset (surface or curve) of the boundary. The space H 1/2 (Υ) is the set of the traces over Υ of all the functions of H 1 (Ω) and we consider the notation H -1/2 (Υ) for the dual space of H 1/2 (Υ). These are the basic functional tools we use repeatedly here. We refer to [START_REF] Adams | Sobolev Spaces[END_REF] for further constructions of fractional Sobolev spaces.

Setting of the problem

Let Ω be a given domain, with boundary Γ = ∂Ω. To avoid unnecessary technicalities we assume that Γ is divided into two disjoint components Γ C and Γ I (where C stands for complete and I for incomplete) as indicated in Fig. 1. We have then Γ C ∩Γ I = ∅. Unless explicitly stated, both portions of the boundary Γ I and Γ C are taken smooth. We focus on the case when all measurements are recorded on the complete boundary Γ C whereas data are missing on Γ I , the incomplete boundary.

The data completion model we study is therefore designed to recover the missing data on Γ I , by Let the conductivity function a be given in L ∞ (Ω) ; it is supposed smooth enough, say a ∈ C 1 (Ω), with a(x) ≥ a * > 0 for all x ∈ Ω. The following norm, equivalent to the H 1 (Ω) norm for functions that vanish on Γ I (or on Γ C ), is well adapted to our problem:

|v| a,H 1 (Ω) = √ a∇v L 2 (Ω) , (1)
and shall be used in this work. Now, let a pair of Cauchy's boundary conditions (g, ϕ)

∈ H 1/2 (Γ C ) × H -1/2 (Γ C ) be prescribed.
The Data Completion problem we propose to solve consists in the following:

find u ∈ H 1 (Ω) solution of -div(a∇u) = 0 in Ω, u = g, a∂ n u = ϕ on Γ C . u = ? on Γ I . (2)
This Cauchy problem is ill-posed in the sense that it may not have a solution. If it has one then by Holmgren's theorem it is unique (see [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF]). Thus there is always uniqueness but not always existence. Existence depends on the data (g, ϕ), but the condition that guarantees existence can hardly be checked on the data. Furthermore, the set of data (g, ϕ) for which (2) has a solution

is dense in H 1/2 (Γ C ) × H -1/2 (Γ C
), but the complementary set of pairs (g, ϕ) that does not yield existence is also dense in

H 1/2 (Γ C ) × H -1/2 (Γ C ).
In reference [START_REF] Ben Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF], problem (2) is set in a variational formulation derived by associating the duplication argument of Kohn-Vogelius [START_REF] Kohn | Determining conductivity by boundary measurements. II. Interior results[END_REF] with the condensation Steklov-Poincaré approach (see [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF]).

This variational formulation uses a pair of functions: the solution u D (µ, g) ∈ H 1 (Ω) of a Poisson problem with pure Dirichlet boundary conditions,

-div(a∇u D (µ, g)) = 0 in Ω, u D (µ, g) = g on Γ C , u D (µ, g) = µ on Γ I , (3)
and the solution u N (µ, ϕ) ∈ H 1 (Ω) of a Poisson problem with mixed Dirichlet-Neumann boundary conditions,

-div(a∇u N (µ, ϕ)) = 0 in Ω, a∂ n u N (µ, ϕ) = ϕ on Γ C , u N (µ, ϕ) = µ on Γ I . (4)
Both functions u D (µ, g) and u N (µ, ϕ) are well defined owing to the coerciveness of these problems

(3) and ( 4). The variational formulation we propose is then based on the following result:

Proposition 2.1 Problem (2) has a solution if and only if there exists λ ∈ H 1/2 (Γ I ) for which the following fluxes identity holds:

a∂ n u D (λ, g) = a∂ n u N (λ, ϕ), on Γ I . (5)
If (5) is satisfied then we have u = u D (λ, g) = u N (λ, ϕ).

Proof: Assume that (5) holds, then the difference w = u D (λ, g)u N (λ, ϕ) satisfies the homogeneous data completion problem div(a∇w) = 0 in Ω, w = 0, a∂ n w = 0 on Γ I .

Holmgren's theorem implies that necessarily w = 0. The converse is obvious. Details can be found in [START_REF] Ben Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF].

The variational translation of the fluxes equality (5) prescribed on the incomplete boundary Γ I reads then as:

find λ ∈ H 1/2 (Γ I ) such that s(λ, µ) = ℓ(µ), ∀µ ∈ H 1/2 (Γ I ), (6)
where the bilinear and linear forms s(•, •) and ℓ(•) are respectively defined by: ∀χ, µ ∈ H 1/2 (Γ I ),

s(χ, µ) = Ω a∇u D (χ) • ∇u D (µ) dx - Ω a∇u N (χ) • ∇u N (µ) dx, ℓ(µ) = - Ω a∇ȗ D (g) • ∇u D (µ) dx -ϕ, u N (µ) 1/2,Γ C .
Here u N (µ) (respectively, u D (µ)) stands for u N (µ, 0) (respectively, u D (µ, 0)) and ȗN (ϕ) (respectively, ȗD (g)) replaces u N (0, ϕ) (respectively, u D (0, g)). The forms s(•, •) and ℓ(•) are made of two contributions each, (s D (•, •), s N (•, •)) and (ℓ D (•), ℓ N (•)) with no ambiguity about their definitions.

The equation ( 6) is derived by suitable applications of Green's formula and a substitution of (5).

Since

s D (µ, µ) = |u D (µ)| 2 a,H 1 (Ω)
for all µ ∈ H 1/2 (Γ I ) (with the norm defined in (1)), the mapping µ → s D (µ, µ) is a norm equivalent to the natural norm • H 1/2 (Γ I ) (see [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF]). For convenience, it is denoted by • s D and it will be used in the sequel as norm on H 1/2 (Γ I ),

µ s D = s D (µ, µ), ∀µ ∈ H 1/2 (Γ I ), (7)
with associated scalar product s D (•, •).

Regarding the symmetric bilinear form s(•, •), there holds that

s(λ, µ) ≤ |u D (λ)| a,H 1 (Ω) |u D (µ)| a,H 1 (Ω) + |u N (λ)| a,H 1 (Ω) |u N (µ)| a,H 1 (Ω) .
Considering the bound,

|u N (µ)| a,H 1 (Ω) ≤ |u D (µ)| a,H 1 (Ω) = µ s D ,
we finally obtain that

s(λ, µ) ≤ 2|u D (λ)| a,H 1 (Ω) |u D (µ)| a,H 1 (Ω) = 2 λ s D µ s D . (8)
Thus s(•, •) is continuous with respect to • s D . Next, owing to the obvious identity

Ω a∇u N (µ) • ∇(u D (µ) -u N (µ)) dx = 0,
it can be stated that

s(µ, µ) = Ω a∇ u D (µ) -u N (µ) • ∇ u D (µ) -u N (µ) dx.
Hence, we deduce that

s(µ, µ) = |u N (µ) -u D (µ)| 2 a,H 1 (Ω) . (9)
It follows that the symmetric form s(•, •) is non-negative. Moreover, the condition s(µ, µ) = 0 implies that u N (µ) = u D (µ)(= w). As a result, w satisfies the data completion problem with homogeneous Cauchy data on Γ C . This problem has w = 0 as the only solution. This yields that µ = 0. Therefore s(•, •) determines an inner product with associated norm denoted by • s ,

µ s = s(µ, µ), ∀µ ∈ H 1/2 (Γ I ), (10)
but we must keep in mind that it is not a Hilbertian norm on H 1/2 (Γ I ). The, with this very weak norm, the following sharp bound holds sup

µ∈H 1/2 (Γ I ) ℓ(µ) µ s = |ȗ D (g) -ȗN (ϕ)| a,H 1 (Ω) . (11)
This is useful when studying noisy data. The proof, which is not straightforward, can be found in [10, Lemma 2.1].

Lavrentiev-Finite Element Method

The inexact Cauchy conditions (g, ϕ) have inevitably dramatic effects on the quality of approximation of the problem. Indeed, even if (g, ϕ) belongs to the set where existence of a solution of problem ( 6) is ensured, this existence may typically fail when (g, ϕ) is slightly perturbed, since the set of non existence is dense in

H 1/2 (Γ C ) × H -1/2 (Γ C ).
Crude computations are therefore risky and should be avoided. Regularization is mandatory. To dampen the pollution that may damage the solution λ, because Cauchy's data suffer from noise, Lavrentiev method turns out to be well suited to the symmetric positive-definite problem ( 6), as illustrated in [START_REF] Ben Belgacem | The Lavrentiev regularization of the data completion problem[END_REF].

Lavrentiev Regularization

Let ̺ be a small positive real parameter and consider the regularized problem:

find λ ̺ ∈ H 1/2 (Γ I ) such that, ̺s D (λ ̺ , µ) + s(λ ̺ , µ) = ℓ(µ), ∀µ ∈ H 1/2 (Γ I ). ( 12
)
The term ̺s D (•, •) ensures the ellipticity of the problem, λ ̺ exists in H 1/2 (Γ I ) whatever the data ℓ(•). For compatible data (g, ϕ), the convergence proof of the Lavrentiev solution λ ̺ towards λ may be found in [START_REF] Azaïez | On Cauchy's Problem. II. Completion, Regularization and Approximation[END_REF]. We refer also to a more extensive study in [START_REF] Ben Belgacem | The Lavrentiev regularization of the data completion problem[END_REF] for the regularization strategy;

the Lavrentiev method strengthened by the Morozov Discrepancy Principle, and for the selection of the parameter ̺.

Lemma 3.1 Assume that problem (6) has a solution λ ∈ H 1/2 (Γ I ). There holds that

lim ̺→0 λ -λ ̺ s D λ s D = 0. ( 13
)
Moreover, we have

lim ̺→0 1 √ ̺ λ -λ ̺ s λ s D = 0. ( 14
)
Proof: The limit ( 13) is proved in [4, Proposition 3.2]. Let us prove [START_REF] Boukari | A convergent data completion algorithm using surface integral equations[END_REF]. By subtracting ( 6) from [START_REF] Bernardi | Optimal Finite-Element Interpolation on Curved Domains[END_REF], we obtain

̺s D (λ ̺ -λ, µ) + s(λ ̺ -λ, µ) = -̺s D (λ, µ). The choice µ = λ ̺ -λ gives ̺ λ ̺ -λ 2 s D + λ ̺ -λ 2 s = ̺ 2 [s D (λ, λ) + s D (λ ̺ -λ, λ ̺ -λ) -s D (λ ̺ , λ ̺ )] . This yields ̺ 2 λ ̺ -λ 2 s D + λ ̺ -λ 2 s = ̺ 2 ( λ 2 s D -λ ̺ 2 s D ). ( 15
)
Then the limit (14) follows by using [START_REF] Bernardi | Discrétisations variationnelles de problèmes aux limites elliptiques[END_REF].

Remark 3.1 Note that (15) implies the particular bound

∀̺ > 0, λ ̺ s D ≤ λ s D . ( 16
)
Remark 3.2 Apart from the convergence of λ ̺ towards λ in the strong norm • s D , no Hölderian estimate can be expected. Without further regularity assumption on λ, the convergence result [START_REF] Bernardi | Discrétisations variationnelles de problèmes aux limites elliptiques[END_REF] is the best we can aim at. In contrast, the gap (λ ̺λ) decays faster than √ ̺, with respect to the norm • s . The extreme weakness of the norm • s explains this result.

Finite Element semi-discretization

In practice, running computations for the numerical simulation of ( 12) prompts practitioners to select a fitting approximation method. Here, we propose a finite element method of degree one.

However, since the numerical analysis of the full discretization of u D and u N is highly technical, in this work we study a semi-discretization where only λ ̺ is discretized while u D and u N are computed by solving exactly ( 3) and ( 4) with these discrete boundary functions. Moreover, to avoid inessential complications and to focus on difficulties inherent to Cauchy's problem, especially the approximation effects on the solution λ ̺ , we assume that the boundary Γ I is polygonal when d = 2

or polyhedral when d = 3. Thus there is no error in approximating the boundary, but the price to pay is that a corner domain generally reduces the regularity of the solution of such problems as ( 3) and ( 4). And as convergence results for finite element methods are tightly connected to the regularity of the underlying functions (here u D and u N ), corners of the polygonal or polyhedral domain will be responsible for lowering these convergence rates (see [START_REF] Strang | An Analysis of the Finite Element Method[END_REF]), thus making the ultimate results possibly suboptimal. We henceforth call the approximation a semi-discretization procedure. Let h > 0 be the discretization parameter and let T h be a subdivision of Γ I made of simplicial elements with maximum mesh size (i.e., diameter h),

Γ I = κ∈T h κ.
We assume that T h is compatible with the polyhedral shape of Γ I in the sense that each κ is located in exactly one face of Γ I . Each face is the exact union of elements in a subset of T h . In addition, we suppose that T h is regular in the sense of Ciarlet [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF]; there exists a constant σ > 0, independent of h such that

h κ ρ κ := σ κ ≤ σ, ∀κ ∈ T h ,
where h κ denotes the diameter of κ and ρ κ the diameter of the ball inscribed in κ.

Let P 1 be the space of polynomials with degree ≤ 1. We introduce the following finite elements subspace on Γ I

H h = ψ h ∈ C (Γ I ) : ∀κ ∈ T h , (ψ h ) |κ ∈ P 1 .
The discretization transforms the problem (12) into a linear system, where ill-posedness is circumvented by the Lavrentiev regularization method. The (semi) discrete problem to explore reads as follows:

find λ ̺,h ∈ H h such that ̺s D (λ ̺,h , µ h ) + s(λ ̺,h , µ h ) = ℓ(µ h ), ∀µ h ∈ H h . ( 17
)
This is nothing but the Ritz-Galerkin discretization with exact integration

(1 + ̺) Ω a∇u D (λ ̺,h ) • ∇u D (µ h ) dx - Ω a∇u N (λ ̺,h ) • ∇u N (µ h ) dx = - Ω a∇ȗ D (g) • ∇u D (µ h ) dx -(ϕ, u N (µ h )) 1/2,Γ C , ∀µ h ∈ H h .
In the remainder of this work, we investigate the behavior of λ ̺,h as the parameters ̺ and h decay both to zero. Our aim is to exhibit a sufficient condition on these parameters that ensures the convergence of the Lavrentiev-finite element solution λ ̺,h towards λ, the exact solution of (6), in the case when this solution exists.

Remark 3.3 Note that Problem [START_REF] Bramble | On the stability of the L 2 projection in H 1 (Ω)[END_REF] still has a solution when ̺ = 0, whatever the Cauchy boundary conditions. The reason is that • s is a norm on the finite dimensional space H h and therefore the matrix of the linear system (17) is invertible. However, this invertibility is not uniform with respect to h, things may go wrong for small h, and the computed solution may blow up and regularization (by discretization) needs to be strengthened.

Numerical Analysis

The purpose of this section is to bound the error (λ ̺ -λ ̺,h ), called the bias in the ill-posed problems community. The variance, i.e., the error caused by noisy data is addressed later on. We thus assume for a while that the data are exact and problem (6) has a unique solution λ. The starting point of the analysis is Céa's abstract result (see [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF]).

Lemma 4.1 The following inequality holds for all µ h ∈ H h :

λ ̺ -λ ̺,h 2
s D + 1 ̺ λ ̺ -λ ̺,h 2
s ≤ λ ̺ -µ h 2 s D + 1 ̺ λ ̺ -µ h 2 s .
Remark 4.1 The right-hand side of Céa's estimate shows the critical effect of the division by the parameter ̺. Indeed, in the computations ̺ is usually small, and the numerical analysis must take into account its asymptotic decay to zero. Therefore the order of λ ̺µ h 2 s as h tends to zero must be strong enough to compensate the factor 1/̺. As in Remark 3.2, our hope is to obtain the best convergence rate of this term with respect to h.

In view of Lemma 4.1, a bound is needed for the approximation error of λ ̺ by a suitably chosen µ h . The L 2 -orthogonal projection on H h will be selected, i.e. (µ h = π h λ ̺ ). The point is hence to derive optimal approximation errors, for functions with low regularity. Let q be real-number in [0, 1], we have

∀µ ∈ H q (Γ I ), µ -π h µ L 2 (Γ I ) ≤ Ch q µ H q (Γ I ) . (18)
This requires the use of some special 'interpolation' operators, rather than the standard Lagrange interpolant. Indeed, the fact that point-wise values of µ may not be accessible because µ is not necessarily smooth suggests using instead some sort of regularized approximation operator such as the one introduced by Scott & Zhang type [START_REF] Scott | Finite element interpolation of nonsmooth functions satisfying boundary conditions[END_REF] and extended by Girault & Lions in [START_REF] Girault | Two-grid finite-element scheme for the transient Navier-Stokes problem[END_REF] to L 1 functions. The proof of the approximation result [START_REF] Cao | A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation[END_REF] in fractional Sobolev spaces follows [START_REF] Ciarlet | Analysis of the Scott-Zhang interpolation in the fractional order Sobolev spaces[END_REF].

Adapting those proofs to our non planar polyhedral geometry of Γ I is checked out in [START_REF] Ben Belgacem | Interplation in Non-Planar Polyhydral Surfaces[END_REF]. Moreover, optimal estimates hold in some dual Sobolev spaces. The proof is short and may be found in [START_REF] Bernardi | Discrétisations variationnelles de problèmes aux limites elliptiques[END_REF],

but for the sake of completeness, we provide it here.

Lemma 4.2 For any real numbers p, q ∈ [0, 1], the following estimate holds for all µ ∈ H q (Γ I )

µ -π h µ H -p (Γ I ) ≤ Ch q+p µ H q (Γ I ) . (19)
Proof: We use the duality Aubin-Nitsche argument. Let χ ∈ H p (Γ I ), we have

Γ I (µ -π h µ)χ dγ = Γ I (µ -π h µ)(χ -π h χ) dγ ≤ µ -π h µ L 2 (Γ I ) χ -π h χ L 2 (Γ I ) .
By estimate [START_REF] Cao | A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation[END_REF], we derive the bound

Γ I (µ -π h µ)χ dγ ≤ Ch q µ H q (Γ I ) h p χ H p (Γ I ) = Ch q+p µ H q (Γ I ) χ H p (Γ I ) .
Observing that

µ -π h µ H -p (Γ I ) = sup χ∈H p (Γ I ) 1 χ H p (Γ I ) Γ I (µ -π h µ)χ dγ,
completes the proof.

The convergence of (µπ h µ) in norms stronger than the L 2 norm, such as the H 1/2 norm, follows from the stability of the operator π h established in the early works of Crouzeix & Thomée in [22,1987] or more recent publications by Bramble & Pasciak and Bank & Yserentant in [START_REF] Bramble | On the stability of the L 2 projection in H 1 (Ω)[END_REF][START_REF] Bank | On the H 1 -stability of the L 2 -projection onto finite element spaces[END_REF].

Thus, at the cost of a mild assumption on the meshes, by using these results and the equivalence of the norms • s D and • H 1/2 (Γ I ) , we have the stronger convergence,

lim h→0 µ -π h µ s D µ s D = 0. ( 20
)
We do not describe accurately the criterion on the meshes, suffices it to know that the result is available for a large class of meshes such as graded and refined meshes used in adaptativity.

Let us choose µ h = π h λ ̺ in Lemma 4.1. The convergence (20) can be used for the term λ ̺µ h s D . But in contrast, convergence of λ ̺µ h s , that follows from the continuity of the bilinear form s(•, •), is not sufficient because of the division by ̺. Recalling that both ̺ and h tend to 0, a far better estimate of λ ̺µ h s is required to compensate for this division. The quality of the approximation depends in general upon the regularity of the function to approximate, λ ̺ in our case. However, due to the particular expression of the norm • s , the effective regularity is the smoothness of either u D (λ ̺ ) or u N (λ ̺ ) away from Γ I . Each of these functions enjoys more regularity than expected at the vicinity of Γ C .

In the sequel, we need some elliptic regularity results for solutions to Poisson problems in polyhedral domains. Even for boundary data that are very smooth, the regularity of the Poisson solutions may be limited by the geometry. According to [START_REF] Jerison | The Neumann problem on Lipschitz domains[END_REF][START_REF] Jerison | The Inhomogeneous Dirichlet Problem in Lipschitz Domains[END_REF], and assuming that the boundary data are smooth enough on Γ C and Γ I , the solution of the Poisson problem belongs to the Sobolev spaces H 3/2+p (Ω), for any p < p * , where the number p * ∈]0, 1/2[ depends on the geometry. Such is the case for u D (µ, g) and u N (µ, ϕ), even if g, ϕ and µ are very smooth. No better results can be expected. Then, p * is the Sobolev exponent that drives and limits the regularity of the solution of Poisson problems, because of the geometry of Γ I . This exponent will be used in several places.

Remark 4.2 Things are well understood and documented in two dimensions, where p * depends on the angles of the closed polygonal boundary Γ I (see [START_REF] Dauge | Elliptic boundary value problems in corner domains[END_REF][START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF]). Indeed, let θ i be the measure of the (internal) angle at the corners of Γ I (recall that Γ C is assumed to be smooth). There is of course a finite number i * of corners and we denote by θ the largest angle, that is θ = max 1≤i≤i * θ i (see Fig. 2). This angle lies in the interval ]π, 2π[. The angle 2π corresponds to a cusp which is not allowed, since Γ I is a Lipschitz boundary. Then, the real number p * , is given by

p * = π θ - 1 2 ∈ 0, 1 2 .

Technical lemmas

Let µ be given in H 1/2 (Γ I ), our aim is to extract more regularity than expected on the function

u D (µ) -u N (µ)
, in the vicinity of Γ I . This is related to the elliptic regularity in non smooth geometries of the following problem: for ψ given in

H 1/2 (Γ C ), find v ∈ H 1 (Ω) solution of, -div(a∇v) = 0 in Ω, v = ψ on Γ C , v = 0 on Γ I .
This problem has a unique solution and this solution satisfies

v H 1 (Ω) ≤ C ψ H 1/2 (Γ C ) . (21)
Considering that ψ has no further smoothness than being in H 1/2 (Γ C ), the function v cannot be more globally regular than being in H 1 (Ω). Nevertheless, far away from Γ C , there is no reason why v will not be smoother. This is the object of the following lemma:

Lemma 4.3 Let p < p * be any positive real number. Then, there exists a constant C, depending only on Ω, such that

a∂ n v H p (Γ I ) ≤ C ψ H 1/2 (Γ C ) . ( 22
)
Proof: Let ξ be a smooth cut-off function defined in Ω, that takes the value one in a neighborhood V I of Γ I , the value zero in a neighborhood V C of Γ C and intermediate values elsewhere in Ω, i.e.,

ξ(x) = 0, ∀x ∈ V C , ξ(x) = 1, ∀x ∈ V I , 0 ≤ ξ(x) ≤ 1, ∀x ∈ Ω.
Then, define v ξ = ξv. Given that div(a∇v) = 0, it can be checked out that

-div(a∇v ξ ) = -v div(a∇ξ) -2a∇v∇ξ in Ω, v ξ = 0 on Γ C , v ξ = 0 on Γ I .
The right-hand side of the first equation lies in L 2 (Ω). Therefore, the elliptic regularity results of [START_REF] Dauge | Elliptic boundary value problems in corner domains[END_REF] imply that v ξ belongs to H 3/2+p (Ω) with the following stability, consequence of ( 21):

v ξ H 3/2+p (Ω) ≤ C v H 1 (Ω) ≤ C ψ H 1/2 (Γ C ) .
Owing to the fact that v ξ coincides with v in V I , the estimate ( 22) is obtained by a trace theorem.

Lemma 4.6 Let p < p * be any positive real number. There holds that

λ ̺ -π h λ ̺ s ≤ Ch 1/2+p λ ̺ s D .
Proof: Let η the residual function, η = λ ̺π h λ ̺ . We start then from

η 2 s = Ω a∇u D (η) • ∇u D (η) dx - Ω a∇u N (η) • ∇u N (η) dx.
Applying Green's formula, and after setting once more v = u Du N , we obtain

η 2 s = Γ I (a∂ n v(η))η dγ = Γ I (a∂ n v(η))(λ ̺ -π h λ ̺ ) dγ.
But according to Lemma 4.3, a∂ n v(η) belongs to H p (Γ I ), therefore we derive by duality that

η 2 s ≤ a∂ n v(η) H p (Γ I ) λ ̺ -π h λ ̺ H -p (Γ I ) .
Owing to [START_REF] Cheng | A coupled complex boundary method for the Cauchy problem[END_REF] with q = 1/2, we get the following estimate:

η 2 s ≤ C a∂ n v(η) H p (Γ I ) h 1/2+p λ ̺ H 1/2 (Γ I ) = Ch 1/2+p a∂ n v(η) H p (Γ I ) λ ̺ s D .
Finally, the bound of Lemma 4.5 implies

η 2 s ≤ Ch 1/2+p η s λ ̺ s D .
The proof is complete.

Remark 4.4 At least in two dimensions, a sharp study of the geometrical singularities responsible for the limitation of the regularity allows to obtain a quasi-optimal estimate for the critical Sobolev regularity exponent p * . In fact, following [START_REF] Babuška | The hp version of the finite element method with quasiuniform meshes RAIRO[END_REF], it should be possible to state that

λ ̺ -π h λ ̺ s ≤ C h 1/2+p * | log h| b λ ̺ s D ,
for some b > 0, but that is beyond the scope of this work.

By combining Lemma 4.6 and the convergence (20), we come up with an estimate of the error between the Lavrentiev and Lavrentiev-finite elements solutions.

4.7

Let p < p * . There exist two functions ǫ ̺ and ǫ h that tend to zero for small ̺ and h such that,

λ ̺ -λ ̺,h s D ≤ C ǫ ̺ + ǫ h + h 1+2p ̺ λ s D .
Proof: First, by combining ( 16) (where the Lavrentiev solution is dominated by the exact one)

and Lemma 4.6, we infer,

1 √ ̺ λ ̺ -π h λ ̺ s ≤ C h 1+2p ̺ λ s D ,
with p is a real number such that p < p * . Next, we bound λ ̺π h λ ̺ s D as follows:

λ ̺ -π h λ ̺ s D ≤ (λ ̺ -λ) -π h (λ ̺ -λ) s D + λ -π h λ s D .
Owing to the uniform stability of π h given by ( 20), we deduce that

λ ̺ -π h λ ̺ s D ≤ C λ ̺ -λ s D + λ -π h λ s D = (ǫ ̺ + ǫ h ) λ s D .
The proof is complete after using [START_REF] Bernardi | Discrétisations variationnelles de problèmes aux limites elliptiques[END_REF], another application of [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF], and Lemma 4.1.

Remark 4.5 A sufficient condition to ensure the convergence of the finite element solution λ ̺,h towards the Lavrentiev solution λ ̺ is to fix the mesh size h = h(̺) so that lim

̺→0 h 1+2p ̺ = 0.
Now, when evaluated with respect to the weaker norm • s , the error is bounded as follows:

λ ̺ -λ ̺,h s ≤ C (ǫ ̺ + ǫ h ) √ ̺ + h 1/2+p λ s D . (24)
This error decays towards zero provided that each of ̺ and h tends to zero independently. Notice that the overall estimates derived here do not require any additional smoothness on the exact solution λ.

Remark 4.6

We have already pointed out that a polygonal or polyhedral boundary Γ I slows down substantially the convergence rate of the finite element method. Indeed, assume that Γ I is smooth and the problem is discretized by curved finite elements of degree one, as defined in [12, section 2] and [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF]Section 4.3]). On one hand, since these finite elements fit the exact shape of Γ I , they induce no approximation error of the boundary. On the other hand, the smoothness of the exact solution is not limited by the boundary. As a result, the analysis elaborated above permits to take a full advantage of the finite element approximation estimates for smooth functions. The final estimate is hence changed to

λ ̺ -λ ̺,h s D ≤ C ǫ ̺ + ǫ h + h 5 ̺ λ s D .
We refer to [START_REF] Du | A Lavrentiev Finite Element Model for the Cauchy Problem of Data Completion: Analysis and Numerical Assessement[END_REF]Chapter 5] where this particular case is discussed.

The final error bound evaluates the difference between λ, the solution of equation ( 6), and λ ̺,h , the regularized-discrete solution of problem [START_REF] Bramble | On the stability of the L 2 projection in H 1 (Ω)[END_REF].

Theorem 4.8 Let p < p * be any positive real number. There exists two functions ǫ ̺ and ǫ h that decay both towards zero, for small ̺ and h respectively, such that

λ -λ ̺,h s D ≤ C ǫ ̺ + ǫ h + h 1+2p ̺ λ s D .
Proof: The estimate is ensued after assembling results of Lemma 3.1 and Proposition 4.7.

Noisy Data

Crude computations with problem (6), or rather with its discrete version, are certainly affected by perturbations that fatally alter the data (g, ϕ), with high risk of irrelevant results. Thus, regularization is necessary for a safe solution. We focus on this issue in the sequel and try to control the effects of disturbances on the Lavrentiev-finite element approximation due to inexact Cauchy data.

Assume we are given the polluted data (g ǫ , ϕ ǫ ) = (g, ϕ) + (δg, δϕ), with a known noise level ǫ > 0.

This means that ȗD (δg) -ȗN (δϕ)

H 1 (Ω) ≤ ǫ. (25)
It is true that the noise level should be evaluated in the natural norms of (δg, δϕ) which means that

δg H 1/2 (Γ C ) + δϕ H -1/2 (Γ C ) ≤ Cǫ.
This bound implies necessarily [START_REF] Delvare | A robust data completion method for two dimensional Cauchy problems associated with the Laplace equation[END_REF]. Conversely, it may be ensued from ( 25) if a statistical independence assumption is added on the errors δg and δϕ. However, from a practical point of view, the (statistical) evaluation in ( 25) is easier to obtain. Furthermore, as will be seen later on it is well fitted to the Kohn-Vogelius approach we follow here. We refer to [10, Remark 5.1] for more clues on this issue.

We consider now the Lavrentiev regularization in the finite element context applied to the perturbed problem as follows:

find λ ǫ ̺,h ∈ H h such that ̺s D (λ ǫ ̺,h , µ h ) + s(λ ǫ ̺,h , µ h ) = ℓ ǫ (µ h ), ∀µ h ∈ H h . (26)
The linear form ℓ ǫ is given by the same formula as ℓ with (g ǫ , ϕ ǫ ) instead of (g, ϕ). Our aim here is to estimate the difference λ ̺,hλ ǫ ̺,h , called the variance.

Lemma 4.9 There holds that

λ ̺,h -λ ǫ ̺,h s D ≤ ǫ 2 √ ̺ .
Proof: For the sake of simplicity, we set η ǫ = λ ǫ ̺,hλ ̺,h . By subtracting ( 17) from ( 26), we obtain

̺s D (η ǫ , µ h ) + s(η ǫ , µ h ) = ℓ ǫ (µ h ) -ℓ(µ h ), ∀µ h ∈ H h . Let us set δℓ = ℓ ǫ -ℓ. The choice µ h = η ǫ gives ̺ η ǫ 2 s D + η ǫ 2 s = (δℓ)(η ǫ ). ( 27
)
Using the stability [START_REF] Ben Belgacem | Interplation in Non-Planar Polyhydral Surfaces[END_REF] and accounting for [START_REF] Delvare | A robust data completion method for two dimensional Cauchy problems associated with the Laplace equation[END_REF] implies that

|(δℓ)(η ǫ )| ≤ ȗD (δg) -ȗN (δϕ) H 1 (Ω) η ǫ s ≤ ǫ η ǫ s .
Then Cauchy-Schwarz inequality yields that

|(δℓ)(η ǫ )| ≤ ǫ 2 4 + η ǫ 2 s .
By substituting this inequality into identity [START_REF] Duranda | Numerical solution of Cauchy problems in linear elasticity in axisymmetric situations[END_REF], we obtain

̺ η ǫ 2 s D ≤ ǫ 2 4 ,
hence the desired result. The proof is complete.

Remark 4.7 A by product of the proof is a bound on the weak norm of η ǫ ,

η ǫ s = λ ̺,h -λ ǫ ̺,h s ≤ ǫ. (28)
We are well equipped to give an estimate of the global error and state a sufficient condition to ensure a convergence result when all the parameters involved (̺, h, ǫ) decay towards zero. We need the bias-variance decomposition

λ -λ ǫ ̺,h s D ≤ λ -λ ̺,h s D + λ ̺,h -λ ǫ ̺,h s D .
We have the following final result, a straightforward consequence of the results of Theorem 4.8 and Lemma 4.9.

Theorem 4.10 Let p < p * be any positive real number. There exist two functions ǫ ̺ and ǫ h , both tending to zero, for small ̺ and h respectively, such that

λ -λ ǫ ̺,h s D ≤ C ǫ ̺ + ǫ h + h 1+2p ̺ λ s D + ǫ 2 √ ̺ .
4.11 Let p < p * any positive real number. There holds This suggests that ̺ cannot be chosen arbitrarily small, but depends on the noise magnitude ǫ.

u -u D (λ ǫ ̺,h , g) H 1 (Ω) + u -u N (λ ǫ ̺,h , ϕ) H 1 (Ω) ≤ C ǫ ̺ + ǫ h + h 1+2p ̺ λ s D + ǫ 2 √ ̺ .
In our opinion, the condition on (̺, ǫ) is more stringent and will prevail over the condition linking (h, ̺). This means that h is in general small enough for fulfilling the last limit.

Local Super-Convergence

The study conducted above concludes to the convergence of the regularization-discretization method.

No convergence rates, of any form Hölderian or logarithmic, can be supplied in the global domain without additional assumptions on the solution of ( 6). However, super-convergence holds far away from the incomplete boundary, as observed in [START_REF] Cao | A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation[END_REF][START_REF] Ben Belgacem | Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality[END_REF]. We propose to establish super-convergence results for λ ̺,h and their incidence on the semi-discrete solution. The ideas we follow may be found in [START_REF] Cao | A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation[END_REF] or in [START_REF] Klibanov | Carleman estimates for the regularization of ill-posed Cauchy problems[END_REF]. They have been successfully extended for the Lavrentiev solution of problem [START_REF] Bernardi | Optimal Finite-Element Interpolation on Curved Domains[END_REF] in [START_REF] Ben Belgacem | Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality[END_REF].

The subsequent analysis needs slightly more regularity on the Neumann data ϕ; it must be in

L 2 (Γ C ) instead of H -1/2 (Γ C
). This is consistent with the actual measurements on ϕ, since noise affecting Neumann data is also estimated in the Lebesgue L 2 -norm. Consequently, we assume that

δg H 1/2 (Γ C ) + δϕ L 2 (Γ C ) ≤ Cǫ. (29)
The pillar tool here is a fitting Carleman inequality. Let then ψ ∈ C 2 (Ω) be a smooth function defined in Ω that satisfies the following properties:

|∇ψ(x)| > 0, ∀x ∈ Ω, ψ(x) > 1, ∀x ∈ Ω \ Γ I ψ(x) = 1, ∀x ∈ Γ I . (30)
Thus the maximum of ψ is attained on Γ C . We shall use the following Carleman estimate, valid in a bounded Lipschitz domain: for large ζ > 0,

Ω a(∇v) 2 + ζ 2 v 2 e 2ζψ dx ≤ C 1 ζ Ω -div(a∇v) + v 2 e 2ζψ dx + ∂Ω (a∂ n v) 2 + ζ 2 v 2 e 2ζψ dγ , ∀v ∈ H 2 (Ω). (31)
The constant C is independent of ζ. We refer to [START_REF] Tataru | A-priori estimates of Carleman's type in domains with boundary[END_REF] for the proof. Next, for a given small parameter τ > 0, we define

Ω τ = x ∈ Ω : ψ(x) ≥ 1 + τ .
Owing to the last condition in [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF], the closure Ω τ does not intersect Γ I . Moreover, τ may be chosen small enough so that Ω \ Ω τ determines a thin tubular neighborhood of Γ I . The right plot of Fig. 2 illustrates the geometry of Ω τ . Now, for a given pair (τ, ν) with τ > ν > 0, we shall use the smooth cut-off function ξ τ,ν defined by the following:

ξ τ,ν (x) = 1, ∀x ∈ Ω τ , ξ τ,ν (x) = 0, ∀x ∈ Ω \ Ω ν , 0 ≤ ξ τ,ν ≤ 1, ∀x ∈ Ω. ( 32
)
Notice that the support of ξ τ,ν is contained in Ω ν .

The Bias

The local super-convergence analysis begins with the bias. We mainly resume and adapt the arguments exposed and sharpened in [START_REF] Ben Belgacem | Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality[END_REF]. The following estimate holds:

Lemma 5.1 Let β > 0 be a small parameter. There exists q = q(β) ∈ [0, 1/2[ and a constant

C = C(β) such that |u N (λ ̺,h , ϕ) -u| a,H 1 (Ω β ) ≤ C̺ q 1 + h 1+2p ̺ λ s D , (33)
for all pairs (̺, h) of small enough numbers.

Proof: In addition to the notation η ̺ = λ ̺,hλ ̺ , we denote

w N,̺ = u N (λ ̺,h , ϕ) -u N (λ ̺ , ϕ) = u N (λ ̺,h -λ ̺ ) = u N (η ̺ ).
Then, we choose (τ, ν) with β > τ > ν > 0, and consider ξ τ,ν , the cut-off function defined in [START_REF] Habbal | Neumann-Dirichlet Nash Strategies for the Solution of Elliptic Cauchy Problems[END_REF].

To alleviate the presentation, we drop the indices N and τ,ν and we use ξw ̺ instead of ξ τ,ν w N,̺ .

As stated in Section 4, the function w ̺ cannot be expected to belong to H 2 (Ω) for at least one of two reasons: the trace η ̺ of w ̺ on Γ I lies in H 1/2 (Γ I ) and Γ I is polygonal or polyhedral. However these sources of singularities are both made ineffective by multiplication with the cut-off function ξ and the product ξw ̺ is indeed in H 2 (Ω). We can therefore apply Carleman's estimate [START_REF] Hào | A Variational Method and Approximations of a Cauchy Problem for Elliptic Equations[END_REF] to

ξw ̺ ∈ H 2 (Ω). It yields Ωτ a(∇(ξw ̺ )) 2 + ζ 2 (ξw ̺ ) 2 e 2ζψ dx ≤ C 1 ζ Ω -div(a∇(ξw ̺ )) + ξw ̺ 2 e 2ζψ dx + Γ C (a∂ n (ξw ̺ )) 2 + ζ 2 (ξw ̺ ) 2 e 2ζψ dγ .
By carrying out the calculations, observing that (a∂ n w ̺ ) |Γ I = (a∂ n w N,̺ ) |Γ I = 0 together with the

fact that ξ ≡ 1 in Ω τ , we derive, Ωτ a(∇(w ̺ )) 2 + ζ 2 (w ̺ ) 2 e 2ζψ dx ≤ C 1 ζ Ων \Ωτ a(∇w ̺ ) 2 + (w ̺ ) 2 e 2ζψ dx + 1 ζ Ωτ (w ̺ ) 2 e 2ζψ dx + ζ 2 Γ C (w ̺ ) 2 e 2ζψ dγ .
By choosing a large enough ζ, the integral over Ω τ in the right-hand side can be absorbed by the same in the left-hand side, and we have

a(∇(w ̺ )) 2 + ζ 2 (w ̺ ) 2 e 2ζψ dx ≤ C 1 ζ Ων \Ωτ a(∇w ̺ ) 2 + (w ̺ ) 2 e 2ζψ dx + ζ 2 Γ C (w ̺ ) 2 e 2ζψ dγ .
The integral in the left-hand side can be restricted to Ω β ; thus denoting σ = max x∈Γ C ψ(x) -1 > 0 and accounting for the specification of Ω β , Ω τ and Ω ν , we obtain

e 2ζ(1+β) Ω β a(∇w ̺ ) 2 + ζ 2 (w ̺ ) 2 dx ≤ C 1 ζ e 2ζ(1+τ ) Ων \Ωτ [a(∇w ̺ ) 2 + (w ̺ ) 2 ] dx + ζ 2 e 2ζ(1+σ) Γ C (w ̺ ) 2 dγ .
Then, obvious simplifications lead to

Ω β a(∇w ̺ ) 2 + ζ 2 (w ̺ ) 2 dx ≤ C 1 ζ e 2ζ(τ -β) Ων \Ωτ [a(∇w ̺ ) 2 + (w ̺ ) 2 ] dx + ζ 2 e 2ζ(σ-β) Γ C (w ̺ ) 2 dγ .
On one hand the stability of w ̺ (= w N,̺ ) with respect to η ̺ , and Lemma 4.4 on the other hand,

give

Ω β a(∇w ̺ ) 2 + ζ 2 (w ̺ ) 2 dx ≤ C 1 ζ e -2ζ(β-τ ) η ̺ 2 s D + ζ 2 e 2ζ(σ-β) η ̺ 2 s .
In view of ( 24), we have the following bound

Ω β a(∇w ̺ ) 2 + ζ 2 (w ̺ ) 2 dx ≤ C 1 ζ e -2ζ(β-τ ) η ̺ 2 s D + ζ 2 e 2ζ(σ-β) (̺ + h 1+2p ) λ 2 s D .
Now, let t = 1 ζ e -2ζ(β-τ ) . Since βτ is positive, this quantity decays towards zero for large ζ and the above estimate can be transformed into

|w ̺ | a,H 1 (Ω β ) ≤ C t η ̺ 2 s D + (̺ + h 1+2p ) t s λ 2 s D 1/2 , ( 34
)
where2 s = [(σβ)/(βτ )] + and C = C(β, τ ) is a positive constant.

Let us select t so as to minimize the right-hand side of [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF]. With this minimum value (34) becomes,

|w ̺ | a,H 1 (Ω β ) ≤ C(̺ + h 1+2p ) 1 2(1+s) η ̺ s 1+s s D λ 1 1+s s D .
Hence, according to Proposition 4.7, we deduce

|w ̺ | a,H 1 (Ω β ) ≤ C(̺ + h 1+2p ) 1 2(1+s) 1 + h 1+2p ̺ s 2(1+s) λ s D .
To close the proof of (33), we need the estimate established in [START_REF] Ben Belgacem | Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality[END_REF]Theorem 3.2],

|u N (λ ̺ , ϕ) -u| a,H 1 (Ω β ) ≤ C̺ q λ s D , with the same exponent q = 1 2(1+s) . The proof is complete by the triangle inequality.

v = ξw ǫ (that belongs to H 2 (Ω)) in [START_REF] Hào | A Variational Method and Approximations of a Cauchy Problem for Elliptic Equations[END_REF], and proceeding as in the proof of Lemma 5.1, we arrive In the right-hand side, the first integral is bounded by using Lemma 4.9. Owing to [START_REF] Girault | Two-grid finite-element scheme for the transient Navier-Stokes problem[END_REF], in the last integral the normal derivative (a∂ n w ǫ ) = δϕ is bounded by Cǫ and the bound on the trace w ǫ comes from Lemma 5.2. Assembling all the bounds leads to

Ω β a(∇w ǫ ) 2 + (w ǫ ) 2 dx ≤ C 1 ζ e -2ζ(β-τ ) ǫ 2 ̺ + (1 + ζ 2 )e 2ζ(σ-β) ǫ 2 .
By introducing again t = 1 ζ e -2ζ(β-τ ) , we obtain

|w ǫ | 2 a,H 1 (Ω β ) ≤ Cǫ 2 ( t ̺ + 1 t s ).
Then the choice of t that minimizes the above right-hand side yields (s+1) .

|w ǫ | a,H 1 (Ω β ) ≤ Cǫ̺ -s 2
Given that q = 1 2(1+s) , the final bound is expressed by Cǫ̺ -1 2 +q . The proof is complete.

Super-Convergence

According to the bias-variance decomposition principle, and after assembling the results of Lemmas 5.1 and 5.3, we are able to derive local convergence rates for the Lavrentiev-finite elements solution.

Theorem 5.4 Let β > 0 be a small parameter. There exists q = q(β) ∈ [0, 1/2[ and a constant C = C(β) such that the following bound holds

|u N (λ ǫ ̺,h , ϕ ǫ ) -u| a,H 1 (Ω β ) ≤ C̺ q 1 + h 1+2p ̺ λ s D + ǫ̺ -1 2 .

Conclusion

The variational formulation, proposed in [START_REF] Ben Belgacem | On Cauchy's Problem. I. A Variational Steklov-Poincaré Theory[END_REF] for the Cauchy problem, provides a suitable framework where Lavrentiev's regularization has been successfully applied and studied (see [START_REF] Ben Belgacem | The Lavrentiev regularization of the data completion problem[END_REF]). The present contribution, which yields the expected error estimates, establishes that this same framework is well adapted to the convergence analysis of a semi-discrete Galerkin Finite Elements method. This is a first step in the theoretical confirmation of the work in [START_REF] Azaïez | A Finite Element Model for the Data Completion Problem: Analysis and Assessment[END_REF] where the computational efficiency of the fully discrete scheme has been assessed. We emphasize the fact that the overall results obtained are valid in two and three dimensions. The next step will consist in extending the present analysis to the fully discrete finite element approximation where all the intermediate Poisson problems are also discretized. This work in progress, which will be the subject of a forthcoming article, is based on non-standard, sharp local convergence results such as those proved by A. Schatz and his co-workers, see for instance [START_REF] Nitsche | Interior estimates for Ritz-Galerkin methods[END_REF][START_REF] Wahlbin | Local behavior in finite element methods[END_REF].

Figure 2 :

 2 Figure 2: Polygonal incomplete boundary Γ I (left). The sub-domain Ω τ (right).

Remark 4 . 8 ̺

 48 In general, the noise level ǫ drives the convergence. The above theorem brings about a (sufficient) criterion for choosing the regularization parameters (̺, h) that ensure the convergence of the Lavrentiev-finite-element approximation towards the exact solution. A sufficient condition is to choose ̺ = ̺(ǫ) and h = h(ǫ) which satisfy lim ǫ→0

Ωτ

  ǫ ) 2 + ζ 2 (w ǫ ) 2 dx ≤ C 1 ζ e 2ζ(τ -β) Ων \Ωτ [a(∇w ǫ ) 2 + (w ǫ ) 2 ] dx + e 2ζ(σ-β) Γ C [(a∂ n w ǫ ) 2 + ζ 2 (w ǫ ) 2 ] dγ .

  using the abundant Cauchy data on Γ C .

	Γ C
	000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 111111111111111 111111111111111 Ω 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 Γ

I Figure 1: The boundary Γ C is accessible to measures and Γ I is out of reach

Actually, have we the choice?

The symbol α+ stand for any real number strictly larger than α.

Recall that u N (µ) is a lifting to the whole domain Ω of a given trace µ on Γ I . The next result provides a bound of the trace of u N (µ) on Γ C by the weak norm µ s . Lemma 4. [START_REF] Azaïez | On Cauchy's Problem. II. Completion, Regularization and Approximation[END_REF] The following holds: for all µ ∈ H 1/2 (Γ I ),

Proof: The trace theorem applied to the function (u D (µ)u N (µ)) yields that

Then, [START_REF] Dardé | An H div -Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems[END_REF] follows from ( 9) and [START_REF] Ben Belgacem | The Lavrentiev regularization of the data completion problem[END_REF].

From these two lemmas, we deduce the following estimate which will have a preponderant impact on the subsequent analysis.

Lemma 4.5 Let p < p * be any positive real number. Then, there exists a constant C such that:

The constant C depends on p.

Proof: Let µ be given in

We know from the Lemma 4.3 that (a∂ n v) lies in H p (Γ I ) and

The proof is completed by substituting the estimate of Lemma 4.4.

Convergence Results

We have at hand the tools for deriving bounds of the approximation error, λ ̺π h λ ̺ s , without any particular smoothness requirement on λ ̺ . This is the object of the next lemma.

Remark 5.1 The following bound holds true:

As a result, a sufficient condition to ensure the convergence in the sub-region Ω β is as follows:

This is a weaker condition on the mesh-size h than the one reported earlier for the whole domain Ω.

The variance

In the same spirit, we derive a sharp convergence rate for the variance error, away from the incomplete boundary. We retain the notation of Section 4.3 and set η ǫ = λ ̺,hλ ǫ ̺,h . Then, we define

We need the following preliminary result:

Proof: The inequality [START_REF] Girault | Two-grid finite-element scheme for the transient Navier-Stokes problem[END_REF] gives the straightforward bound, ȗN (δϕ)

On the other hand, owing to the stability of the trace and the fact that u D (η ǫ ) vanishes on Γ C ,

The last bound is given in [START_REF] Feng | Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method[END_REF]. The proof is obtained by the triangle inequality.

Lemma 5.3 Let β > 0 a small parameter. There exists a constant

where q = q(β) is defined as in Lemma 5.1.

Proof: We retain the notation of Lemma 5.1. The argument of the proof in inspired by [START_REF] Ben Belgacem | Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality[END_REF]Theorem 3.5]. We apply Carleman's estimate to ξ τ,ν w N,ǫ . Again, we drop the indices N and τ,ν . By taking


domain: sdv stat

text:
Sarah Berdot
email: sarah.berdot@aphp.fr

Virginie Korb-Savoldelli

Emmanuel Jaccoulet

Vincent Zaugg

Patrice Prognon

Laetitia Minh

Maï Lê

Brigitte Sabatier

A centralized automated-dispensing system in a French teaching hospital: return on investment and quality improvement

Keywords: centralized automated-dispensing system, return on investment, medication process, satisfaction

Objectives: To evaluate the return on investment (ROI) and quality improvement after implementation of a centralized automated-dispensing system after 8 years of use. Design: Prospective evaluation of ROI; before and after study to evaluate dispensing errors; user satisfaction questionnaire after 8 years of use. Setting: The study was conducted at a French teaching hospital in the pharmacy department, which is equipped with decentralized automated medication cabinets in the wards. Participants: Pharmacy staff (technicians and residents). Intervention(s): Implementation of a centralized automated-dispensing robot. Main Outcome Measure(s): The true ROI was prospectively and annually compared to estimated returns calculated after implementation and upgrade of the robot; dispensing errors determined by observation of global deliveries and the satisfaction of users based on a validated questionnaire were evaluated. Results: Following the upgrade, we found little difference for the ROI (+1.86%). The payback period increased by almost 3 years. There was a significant reduction of dispensing errors, from 2.9% to 1.7% (P < 0.001). User satisfaction of the robot by the pharmacy staff was reported (score of 5.52 ± 1.20 out of 7). Conclusions: These systems are worthwhile investments and largely contribute to improving the quality and safety of the medication process.

Introduction

Centralized and decentralized automated-dispensing systems have been developed in hospital pharmacies to secure drug dispensing and administration process [START_REF] Lehnbom | Evidence briefings on interventions to improve medication safety. Automated dispensing systems[END_REF][START_REF] Fox | ASHP national survey on informatics: assessment of the adoption and use of pharmacy informatics in U.S. hospitals-2013[END_REF]. In 2014, 37% of surveyed hospitals with more than 600 beds in the United States had a dispensing robot and all had automated-dispensing cabinets [START_REF] Fox | ASHP national survey on informatics: assessment of the adoption and use of pharmacy informatics in U.S. hospitals-2013[END_REF][START_REF] Pedersen | ASHP national survey of pharmacy practice in hospital settings: dispensing and administration-2014[END_REF]. These complementary systems are widely established in the European hospital medication process and many studies have demonstrated their contribution to decreasing iatrogenic injuries by reducing potentially preventable events [START_REF] James | The impact of automation on pharmacy staff experience of workplace stressors[END_REF][START_REF] Tsao | Decentralized automated dispensing devices: systematic review of clinical and economic impacts in hospitals[END_REF][START_REF] Chapuis | Automated drug dispensing system reduces medication errors in an intensive care setting[END_REF]. However, the high cost of these available technologies must be balanced against their efficiency.

Despite nominative unit-dose dispensing is largely promoted in European healthcare institutes, this dispensing process is less developed in France. Global dispensing still persists but presents a high risk of error in the medication process and needs to be secured. Consequently, a centralized automated-dispensing system was implemented in our hospital in July 2008, in addition to decentralized automated cabinets, before the growth of unit-dose delivery. We calculated the return on investment (ROI) and evaluated medication errors and user satisfaction to evaluate the financial impact of this system and quality improvements due to its implementation.

We aimed to compare the ROI and evaluate quality improvement after implementation of a centralized automated-dispensing robot. We compared the ROI after 8 years of use with the initial estimated ROI and assessed quality improvement based on dispensing errors and user satisfaction.

Materials and Methods

Setting

The European Georges Pompidou Hospital (HEGP) is a teaching hospital in Paris, France with 714 beds and 106 day hospital places (43 clinical wards). The mean duration of hospitalization is 6.1 days. The hospital is equipped with a patient information system, integrating an electronic health record and a computerized physician order entry system (DxCare ® , Medasys), and 50 automated medication cabinets (Omnicell ® Inc.). The total number of referenced drugs managed in the pharmacy is 1700. Before implementation of the robot, the pharmacy team for the global medication dispensing process consisted of three pharmacy technicians, two pharmacy technician aides, three pharmacy residents and seven pharmacists.

Medication process

Each electronic medication prescription by a physician is reviewed by a pharmacist. Drugs are dispensed by pharmacy technicians. Before implementation of the robot, the technicians manually dispensed drugs via decentralized automated-dispensing cabinets or urgent global deliveries. After implementation, the technicians dispensed drugs using the robot. Cabinets are refilled twice a week per unit. Since 2008, a centralized automated-dispensing robot (Rowa ® system from ARX ® ) was implemented in the pharmacy to store and deliver drugs.

Centralized automated-dispensing system

The centralized automated-dispensing system consists of tandem robots with a duplicate stock for a total approximate capacity of 22 000 boxes. In 2008, each robot included a specific refrigerated unit, a loading unit (ProLog system) to input drug boxes using barcodes, and a dispensing system. In 2013, the system was upgraded to improve the quality process. In contrast to the first robot, the new system integrated the capacity to read datamatrix available on each drug box. It was equipped with three specific scanners (ROWA VMAX). Due to the absence of a refrigerated unit, refrigerated drugs were removed.

The two systems, controlled by ARX ® software, were interfaced with the drug stock management software Pharma ® (Computer Engineering Company). Almost 95% of the drugs are stored in the robot. Some drugs are not stored in the robot, such as plasmaderived medicinal products and narcotics drugs, due to weight, large dimensions, lack of a barcode on the drug box, or classification.

Comparison of return on investment

ROI is commonly used to gauge an investment's profitability, because of its versatility and simplicity to calculate. An estimated ROI was initially calculated to justify the investment for the hospital financial department.

We assessed the ROI of robot implementation by annually comparing the true ROI to the estimated ROI, calculated after implementation in 2008 (initial estimated ROI) and its upgrade in 2013 (estimated ROI with upgrade). Since 2013, the initial estimated ROI and estimated ROI with upgrade were combined.

ROIs were calculated by deducting the costs of investment from the cost savings since implementation of the robot. The investment costs included buying and installation of the robot, annual maintenance and repairs, and upgrading of the robot. The cost savings were evaluated by the reduction of drug stocks and the decrease in pharmacy staff dedicated to global dispensing.

Quality improvement

Dispensing errors

We conducted a before and after study to evaluate the benefit of the centralized automated-dispensing robot on dispensing errors. Observations were performed by a pharmacist before robot implementation and after 2 years of use. Twenty prepared global deliveries were examined on five consecutive days. Before implementation, deliveries were manually prepared by pharmacy technicians, whereas they have been ordered from the robot by pharmacy technicians after implementation and manually completed for drugs stored outside the system. We evaluated the mean number and standard deviation (SD) of drug units per global delivery. The error rate was calculated as the number of dispensing errors divided by the total unit doses ordered, multiplied by 100.

User satisfaction

Every new system generally faces user reluctance. We thus aimed to assess acceptability of the robot. A survey questionnaire was administered to recurrent users, including pharmacy technicians and pharmacy residents working at the HEGP pharmacy. Senior pharmacists were not surveyed as they did not use the robot. We used the questionnaire developed by Palm et al. [START_REF] Palm | Determinants of user satisfaction with a Clinical Information System[END_REF][START_REF] Palm | Clinical information system postadoption evaluation at the Georges Pompidou University Hospital[END_REF]. It consists of six dimensions, evaluating compatibility, expectations, user support, ease of use, usefulness and user satisfaction. Each component is evaluated using a Likert scale, for which 1 = strongly disagree, 2 = disagree, 3 = somewhat disagree, 4 = neutral (neither disagree nor agree), 5 = somewhat agree, 6 = agree and 7 = strongly agree. We added an 'intention to continue using the robot' dimension, also evaluated by Palm et al.

Statistical analysis

The mean numbers of drug units per global delivery were compared using Student's t-test. Dispensing error rates were compared using the Chi-squared test. Descriptive statistics were used to summarize the information for the satisfaction dimension. Results were computed using aggregate variables for each dimension by the mean score and standard deviation (SD). The mean ages for pharmacy residents and pharmacy technicians were compared by Fisher's exact test. Other statistical analyses were performed using Welch's t-test. Analyses were performed using the statistical online software BiostaTGV (https://marne.u707.jussieu.fr/biostatgv).

Results

Comparison of return on investment

We estimated the yearly savings and investment costs from 2008 to 2016 (Fig. 1). The ROI was positive after 8 years of use (+$294 498). The total costs savings and investment costs were +$1 315 158 and -$1 020 660, respectively. Costs savings were divided between decreasing drug stocks (+$75 084), observed the first year after implementation of the robot, and cumulative reduction of pharmacy staff in the dispensing area (+$1 240 074), corresponding to a mean annual reduction of +$155 009. Before implementation of the robot, the pharmacy staff for medication dispensing consisted of three pharmacy technicians and two pharmacy technician aides. Since 2008, one pharmacy technician and 1.5 pharmacy technician aides were moved to other pharmaceutical activities, except in 2013 for the upgrade of the robot. Indeed, during this period, one pharmacy technician aide was necessary to compensate for the absence of the automatic dispensing system. The investment costs included purchase of the robot, technical and computer support, and the first year of maintenance (-$483 892), robot maintenance (-$226 985), robot repairs (-$30 784) and robot upgrade (-$276 000). No repairs have been necessary since the upgrade.

The comparison between the estimated and real ROI are presented in Fig. 2 to 2016) was +$528 946 based on cost savings of +$1 367 496 and investment costs of -$838 550. However, the estimated ROI was reevaluated in 2013 due to the necessity of the robot upgrade. Until 2010, there was little difference between the estimated and real ROIs. In 2011, a major repair of the robot was needed following human misuse, explaining the 26.1% increase of the real ROI in 2012. The robot upgrade in 2013 altered the estimated ROI for 2016 to + $289 129. We observed no major deviations since 2013. In 2016, a positive difference between the real and new estimated ROI (+1.86%) was reported.

Quality improvement

Dispensing errors

The mean number of drug units per global delivery was 40.0 (SD: 28.6) before implementation and 32.8 (SD: 25.2) after 2 years of implementation. There was no significant difference (P = 0.40). Over the 40 global deliveries assessed, we controlled a total of 25 719 drug units. The error rate before implementation was 2.9% (419 errors/14 526 units), which decreased to 1.7% (194 errors/11 193 units) after. The error rates were significantly different (P < 0.001).

User satisfaction

We analyzed a total of 13 and 18 questionnaires from pharmacy residents and technicians (Table 1). The average age of the respondents was 31.0 ± 6.9 years and 22 (71.0%) were women (P = 0.11).

The mean duration of experience with the robot was 2.31 ± 2.99 years = 0.009). Results of the aggregate variables for the six dimensions of the robot survey are presented in Table 2. The perception of experience with the robot did not differ significantly between pharmacy residents and technicians, except for user satisfaction (P = 0.04). No dimension scored below the average level on the scale of 1-7 points for either pharmacy residents or technicians. The lowest average perception was reported by pharmacy residents for expectations (4.7 1 ± 1.27) and by pharmacy technicians for support (4.71 ± 1.63). Overall, the highest average appreciation was obtained for the perceived ease of use (5.81 ± 0.96), followed by compatibility (5.63 ± 1.06) and user satisfaction (5.52 ± 1.20). The mean score was 4.8 ± 2.1 for the additional questions defining the 'intention to continue using the robot' dimension.

Discussion

After 8 years of using the robot (from 2008 to 2016), we recorded a ROI of +$294 498. The estimated ROI for 8 years of use was +$528 946 (+$1 367 496 of cost savings and -$838 550 of investment costs). We observed a 26.1% difference of ROI in 2012, due to major repairs. The payback period was consequently increased by almost 3 years. Following the upgrade of the robot in 2013, the estimated ROI was recalculated and there was little difference with the real ROI (+1.86%). Moreover, there was an improvement in the medication dispensing process, with a significant reduction of 41% in medication error rates following implementation of the robot. Following implementation, all pharmacy staff received training before using the robot. The perception of their experience with the robot did not differ significantly between pharmacy residents and technicians in the user satisfaction survey, despite that pharmacy residents had less experience with the robot. Indeed, due to the limited duration of the resident internship in pharmacy, the ease of robot use contributed to the increase in compliance of the staff and optimization of the quality process. This was underlined by the highest score of 5.81 (SD: 0.96) given by both types of users for this dimension. We found that this pharmacy-based robot improved workflow efficiency and provided greater storage capacity, in accordance with the literature. Franklin et al. showed that the implementation of dispensing robots in the UK hospital pharmacies resulted in a 23% increase in storage capacity over traditional storage in one site and a 123% increase in another [START_REF] Franklin | An evaluation of two automated dispensing machines in UK hospital pharmacy[END_REF]. Robots ensure that drug resources are secure, organized, tracked and ready for use, leading to improved safety of the medication process [START_REF] Pedersen | ASHP national survey of pharmacy practice in hospital settings: dispensing and administration-2014[END_REF][START_REF] Franklin | An evaluation of two automated dispensing machines in UK hospital pharmacy[END_REF]. A significant reduction in dispensing errors, from 16% to 60%, has been observed in several studies [START_REF] James | The impact of automation on pharmacy staff experience of workplace stressors[END_REF][START_REF] Franklin | An evaluation of two automated dispensing machines in UK hospital pharmacy[END_REF][START_REF] Fitzpatrick | Evaluation of an automated dispensing system in a hospital pharmacy dispensary[END_REF]. This was confirmed in our study in which the reduction of dispensing errors was 41%. We did not evaluate the impact of the robot in the management of expiry dates and batch number in this study, but a substantial quality improvement would be expected. A centralized automated-dispensing system optimizes drug traceability. Information present in datamatrix, such as quantity, expiry dates and batch number are automatically recorded by uploading unit, which is crucial for batch number recall management. In addition to identifying the batch number concerned, the robot improved time savings and patient safety by increasing pharmacy staff reactivity.

Centralized automated-dispensing systems also help pharmacy staff to reduce or eliminate labor-intensive tasks by automating the management of drugs and supplies where they are needed. We found there to be a significant reduction in drug delivery times after implementation of the robot (data not shown). Franklin et al. found a significant reduction in median picking time [START_REF] Franklin | An evaluation of two automated dispensing machines in UK hospital pharmacy[END_REF]. Finally, robots induce departmental reorganization, with a net reduction of pharmacy technicians for the global dispensing process and their reassignment to the unit-dose dispensing process [START_REF] Temple | Implementation and evaluation of carousel dispensing technology in a university medical center pharmacy[END_REF][START_REF] Bedouch | La sécurisation du circuit du médicament dans les établissements de santé: données actuelles et expérience du centre hospitalier universitaire de Grenoble (Drug supply chain safety in hospitals: Current data and experience of the Grenoble university hospital)[END_REF][START_REF] Oborn | The adoption of dispensing technology: an occasion for restructuring hospital pharmacy work[END_REF]. In our hospital, one pharmacy technician and 1.5 pharmacy technician aides were moved to more value-added activities, such as unit-dose dispensing activities and the delivery of chemotherapy drugs to units. This is the first study to compare ROI following the implementation of a centralized automated-dispensing system. However, this study has some limitations. It was monocentric, not allowing generalization of the data for all centers. The calculation of the ROI may have been underestimated, as the cost savings did not include the management of expired drugs and the effect of the reduction of drug stocks after the first year of implementation of the robot.

Despite their benefits, these technologies are not all-inclusive. Centralized automated-dispensing robots and automated-dispensing cabinets in the wards must be associated with barcode medication administration (BCMA) systems at the patient bedside [START_REF] Poon | Effect of bar-code technology on the safety of medication administration[END_REF] to complete drug traceability and ensure patient safety during the medication process.

The ROI after 8 years of use was positive, despite the initial and upgrade costs, but was close to the estimated ROI recalculated following the robot upgrade. Economic projections should be calculated with caution because of technological and regulatory changes that can lead to premature obsolescence of the system. In our experience, such systems are worthwhile investments, leading to a ROI within a few years. Over the last decades, centralized automateddispensing systems have largely contributed to improve the quality and safety of the medication process. In addition to optimizing drug stock management, such systems result in greater efficiency of the global dispensing process by significantly reducing dispensing errors and improving time savings, with good compliance of pharmacy staff and redeployment of some members to value-added activities.

Figure 1

 1 Figure1Real and estimated costs before and after upgrade divided between pharmacy staff (A) and those of the robot and maintenance (B) from 2008 to 2016.

Figure 2

 2 Figure 2 Real and estimated returns on investment (ROI) from 2008 to 2016 in a French hospital, before and after upgrade.

Table 1

 1 Respondent demographic characteristics and perceptions for the six dimensions of the survey of robot use for pharmacy residents and pharmacy technicians.

	Residents	Technicians	Total	P-value

Table 2

 2 Perceptions of robot use for the six dimensions of the survey from Palm for pharmacy residents and pharmacy technicians-mean (SD)

	Items from questionnaire developed by Palm [10, 11]	Residents	Technicians	Total	P-value
	Compatibility	5.85 (0.93)	5.48 (1.06)	5.63 (1.02)	0.26
	Support	5.37 (1.37)	4.75 (1.63)	5.01 (1.55)	0.09
	Usefulness	5.52 (1.39)	5.43 (1.22)	5.47 (1.29)	0.76
	Ease of use	6.16 (0.74)	5.61 (1.00)	5.81 (0.96)	0.12
	User satisfaction	5.90 (1.12)	5.25 (1.20)	5.52 (1.20)	0.04
	Expectations	4.71 (1.27)	4.85 (0.97)	4.79 (1.11)	0.73

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Acknowledgements

We thank Anne Pont, Béatrice Dejean and Nathalie Valin for their help during this study.