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AND NUUTTI Antti Hannukainen

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AN H div -BASED MIXED QUASI-REVERSIBILITY METHOD FOR SOLVING ELLIPTIC CAUCHY PROBLEMS

Keywords: 2010 Mathematics Subject Classification. 35A15, 35N25, 35R25, 35R30, 65M32, 65M60 quasi-reversibility method, Cauchy problem, inverse obstacle problem, mixed variational problem

This work considers the Cauchy problem for a second order elliptic operator in a bounded domain. A new quasi-reversibility approach is introduced for approximating the solution of the illposed Cauchy problem in a regularized manner. The method is based on a well-posed mixed variational problem on H 1 × H div , with the corresponding solution pair converging monotonically to the solution of the Cauchy problem and the associated flux, if they exist. It is demonstrated that the regularized problem can be discretized using Lagrange and Raviart-Thomas finite elements. The functionality of the resulting numerical algorithm is tested via three-dimensional numerical experiments based on simulated data. Both the Cauchy problem and a related inverse obstacle problem for the Laplacian are considered.

Introduction

We consider the Cauchy problem for a second order elliptic operator in a bounded domain Ω ⊂ R d , d ≥ 2. Such problems have been studied extensively (see, e.g., [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF][START_REF] Belgacem | Why is the Cauchy problem severely ill-posed?[END_REF][START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 domains[END_REF][START_REF] Hadamard | Lectures on Cauchy's Problem in Linear Partial Differential Equations[END_REF]), with the common understanding that they are 'exponentially' ill-posed. Cauchy problems for elliptic operators are encountered in many practical applications such as electrocardiography (ECG) [START_REF] Gulrajani | The forward and inverse problems of electrocardiography[END_REF] and plasma physics [START_REF] Blum | Numerical simulation and optimal control in plasma physics with application to tokamaks[END_REF], and they are also closely related to the inverse source problems arising from, e.g., electroencephalography (EEG) and magnetoencephalography (MEG) [START_REF] Hämäläinen | Magnetoenephalography -theory, instrumentation, and applications to noninvasive studies of signal processing in the human brain[END_REF]. In addition, Cauchy problems play an important role in inverse obstacle problems (cf. [START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF]), which are studied, e.g., in connection with inclusion detection by electrical impedance tomography (EIT) when only one pair of boundary current and voltage is used for probing the examined body [START_REF] Borcea | Electrical impedance tomography[END_REF][START_REF] Hanke | Convex source support and its application to electric impedance tomography[END_REF].

Several methods have been proposed to regularize the Cauchy problem; see, e.g., [START_REF] Andrieux | Solving Cauchy problems by minimizing an energy-like functional[END_REF][START_REF] Azaiez | On Cauchy's problem: II. Completion, regularization and approximation[END_REF][START_REF] Ben Belgacem | Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality[END_REF][START_REF] Ben Belgacem | Extended-domain-Lavrentiev's regularization for the Cauchy problem[END_REF][START_REF] Cao | A Carleman estimate and the balancing principle in the quasireversibility method for solving the Cauchy problem for the Laplace equation[END_REF], 27] and the references therein. Among these approaches, the variants of the quasi-reversibility (QR) method [START_REF] Lattès | Méthode de Quasi-réversibilité et Applications[END_REF] comprise a technique with some useful properties. The main idea of the original QR method [START_REF] Lattès | Méthode de Quasi-réversibilité et Applications[END_REF] is to approach the ill-posed second order Cauchy problem by a family of well-posed fourth order problems depending on a (small) regularization parameter. It is a non-iterative technique which can be applied numerically using finite elements methods (FEM), and thus it is adaptable for solving the Cauchy problem in complicated domains. However, the main drawback of the original QR method is the fourth order nature of the regularizing variational problems: one has to use C 1 finite elements, which are difficult to handle and seldom available in numerical solvers, especially in three dimensions.

The first attempt to get rid of this technical difficulty was introduced in [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation Inverse Problems[END_REF], where a QR method based on a mixed variational regularization of the Cauchy problem was proposed. The method involves two unknown H 1 (Ω)-functions, one of which approximates the solution of the Cauchy problem and the other, loosely speaking, the error in the corresponding right-hand side (cf. (3.3)). In particular, the second member of the regularized solution does not carry any explicit information on the solution of the underlying Cauchy problem. (The approach of [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation Inverse Problems[END_REF] has also other minor drawbacks as explained in Section 3 below.)

The purpose of this article is to introduce a new QR method that is based on a mixed variational formulation on H 1 (Ω) × H div (Ω). As for the previous mixed QR method in [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation Inverse Problems[END_REF], the first member of the regularized solution approximates the actual solution of the Cauchy problem, but in our case also the second member provides explicit information on the solution: it gives an estimate for the corresponding flux. As the regularization parameter tends to zero, the solution of our mixed QR problem converges monotonically to the solution of the Cauchy problem, if such exists, and diverges otherwise. According to our knowledge, monotonic convergence has not previously been shown for any QR formulation; our technique for proving this property can straightforwardly be adapted for the original QR method of [START_REF] Lattès | Méthode de Quasi-réversibilité et Applications[END_REF] as well. Moreover, we briefly consider how the regularization parameter of the mixed QR method should be chosen in the case of noisy data and demonstrate how the method can be discretized by combining Lagrange and Raviart-Thomas finite elements, which are standard tools supported by many finite element solvers.

In order to illustrate the functionality of our mixed QR method, we test it numerically in a realistic three-dimensional setting; according to our knowledge, this is the first implementation of any QR method in three dimensions. Both the Laplace operator and a constant-coefficient, anisotropic operator of the divergence form are considered. In addition, we incorporate our method as a part of the level set algorithm for solving the inverse (Dirichlet) obstacle problem for the Laplace equation introduced in [START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF]]. According to our numerical studies, the new QR formulation produces acceptable solutions both for Cauchy problems and for the inverse obstacle problem, even with considerable amounts of noise.

The paper is organized as follows. Section 2 introduces the considered Cauchy problem together with the related function spaces and terminology. In Section 3, we present the standard formulation of the QR method from [START_REF] Lattès | Méthode de Quasi-réversibilité et Applications[END_REF] and the mixed one proposed in [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation Inverse Problems[END_REF]. We highlight their main qualities and drawbacks. Section 4 is dedicated to the formulation of the new mixed QR method, its properties and discretization. Finally, Section 5 presents the numerical experiments and Section 6 lists the concluding remarks.

The setting

Let Ω ⊂ R d be a bounded domain with a Lipschitz boundary (cf. [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]). We denote by ν ∈ L ∞ (∂Ω, R d ) the exterior unit normal of ∂Ω, and suppose that ∂Ω is divided into two open subsets of positive measure Γ and Γ c such that

∂Ω = Γ ∪ Γ c . Let A = [a ij ] : Ω → R d×d be a real matrix valued function such that A ∈ W 1,∞ (Ω) d×d and A T = A, ξ T Aξ ≥ c|ξ| 2 for all ξ ∈ R d
almost everywhere in Ω. The Cauchy problem we are interested in is defined as follows:

Definition 2.1 (Cauchy problem). For (f, g D , g N ) ∈ L 2 (Ω) × H 1/2 (Γ) × H -1/2 (Γ), find u ∈ H 1 (Ω) such that        ∇ • A∇u = f in Ω, u = g D on Γ, ∂u ∂ν A = g N on Γ.
Here,

H 1/2 (Γ) = φ |Γ : φ ∈ H 1 (Ω)
and H -1/2 (Γ) is defined as the dual of (see, e.g., [START_REF] Fernandes | Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions[END_REF])

H1/2 (Γ) = φ ∈ L 2 (Γ) : ∃v ∈ H 1 (Ω) s.t. v |Γ = φ, v |Γc = 0 . Moreover, (1)
∂ ∂ν A : v → d i,j=1 a ij ∂v ∂x j ν i |Γ , H 1 (Ω, ∇ • A∇) → H -1/2 (Γ)
is the linear and bounded conormal derivative operator defined on (cf., e.g., [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF])

H 1 (Ω, ∇ • A∇) := v ∈ H 1 (Ω) : ∇ • A∇v ∈ L 2 (Ω) .
Since Hadamard, it is well-known that the Cauchy problem of Definition 2.1 is severely ill-posed: although it has at most one solution (see, e.g., [START_REF] Rousseau | Applications to unique continuation and control of parabolic equations[END_REF]), it may have none, and if a solution exists, it does not depend continuously on the data (f, g D , g N ) in any reasonable topology. Therefore, regularization is needed to stabilize the problem.

Previous quasi-reversibility methods

For simplicity, we focus in this section on the special case that A is the identity matrix, i.e., we consider the Cauchy problem for the Poisson equation. We will return to the general setting in Section 4, where the new mixed formulation of the QR method will be introduced.

Standard formulation.

For the standard formulation of the QR method, we need to make some extra smoothness assumption on ∂Ω, g D and g N . More precisely, we suppose that ∂Ω is of the class C 1,1 , g D ∈ H 3/2 (Γ) and g N ∈ H 1/2 (Γ); under this assumption on the regularity of ∂Ω, the trace operation

H 2 (Ω) ∋ v → v, ∂v ∂ν ∈ H 3/2 (Γ) × H 1/2 (Γ)
is linear, bounded and surjective [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]. The QR method, as introduced in [START_REF] Lattès | Méthode de Quasi-réversibilité et Applications[END_REF], relies on the following problem:

Definition 3.1 (Standard QR problem). For ε > 0, find u ∈ H 2 (Ω), with u |Γ = g D and (∂u)/(∂ν) |Γ = g N , such that for all v ∈ H 2 0 (Ω), (2)
Ω ∆u ∆v dx + ε Ω d i,j=1 ∂ 2 u ∂x i ∂x j ∂ 2 v ∂x i ∂x j + ∇u • ∇v + uv dx = Ω f ∆v dx.
It follows directly from the Lax-Milgram theorem that the standard QR problem has a unique solution.

Proposition 3.1. For any ε > 0, the standard QR problem of Definition 3.1 admits a unique solution

u ε ∈ H 2 (Ω).
Moreover, one can deduce the following convergence result [START_REF] Lattès | Méthode de Quasi-réversibilité et Applications[END_REF][START_REF] Dardé | The 'exterior approach': a new framework to solve inverse obstacle problems[END_REF]:

Theorem 3.2.
Suppose that the Cauchy problem admits a (unique) solution u ∈ H 2 (Ω). Then, the solution to the standard QR problem u ε converges to u in H 2 (Ω) as ε tends to zero, and it holds that

∆u ε -f L 2 (Ω) ≤ √ ε u H 2 (Ω) .
By nature, the QR method is non-iterative. Moreover, in case the data (g D , g N ) are corrupted by noise, a method to set the regularization parameter ε > 0 as a function of the amplitude of the noise, say α, based on the Morozov's discrepancy principle and the duality in optimization has been developed in [START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF]; the solution u ε(α) of the QR problem tends to the exact solution of the Cauchy problem when the amplitude of the noise goes to zero (cf. Section 4.3). Unfortunately, the standard QR formulation of Definition 3.1 has also two drawbacks, which are linked.

First of all, one needs relatively smooth data to ensure that (2) has a solution. To make matters worse, the convergence of the approximate solution u ε toward the exact solution u has only been proved if u ∈ H 2 (Ω), although it is well-known that in some cases u is only in H 1 (Ω, ∆) because the Cauchy data is a priori known to be H 3/2 × H 1/2 -smooth only on a subset of ∂Ω.

However, the main issue with the standard QR formulation is arguably related to its discretization: to obtain an approximate solution of the Cauchy problem, one has to discretize the variational formulation (2) using, e.g., some FEM. Since (2) is a fourth order problem, it cannot be discretized using standard C 0 finite elements. In order to obtain a conforming discretization of the problem, one has to use C 1 finite elements, which are difficult to handle and seldom available in numerical solvers, especially for three-dimensional problems; see [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF] for a description of such elements in the two-dimensional case. According to our knowledge, a conforming discretization of the variational equation (2) has never been performed. However, (2) has been successfully discretized using non-conforming finite elements [START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF], namely the Fraeijs de Veubeke 1 elements [START_REF] Lascaux | Some nonconforming finite elements for the plate bending problem[END_REF], which are simpler than C 1 elements, but unfortunately rarely implemented in numerical solvers as well. It should also be mentioned that the QR problem has been successfully discretized using difference schemes [START_REF] Lattès | Méthode de Quasi-réversibilité et Applications[END_REF] and splines [START_REF] Clason | The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium[END_REF], but these approaches are typically limited to simple geometries.

To overcome this technical difficulty with the discretization, we will propose a novel QR method based on a mixed variational formulation. The leading idea is to introduce an additional unknown, which deals with the second order derivatives, resulting in a lower order problem that can be discretized using standard finite elements. However, our mixed QR formulation is not the first of its kind, as indicated in the following section.

3.2.

A previous mixed formulation. According to our knowledge, the first (and thus far only) mixed QR approach has been proposed in [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation Inverse Problems[END_REF]. It relies on the following problem, for which the assumptions on ∂Ω, g D and g N are as listed in Section 2. Definition 3.3 (Mixed QR problem). For ε, δ > 0, find u ∈ H 1 (Ω), with u |Γ = g D , and λ ∈ H 1 (Ω), with λ |Γc = 0, such that

               ε Ω (∇u • ∇v + u v) dx + Ω ∇λ • ∇v dx = 0 for all v ∈ H 1 (Ω), v |Γ = 0, Ω ∇u • ∇µ dx - Ω λ µ dx -δ Ω (∇λ • ∇µ + λ µ) dx = Ω f µ dx + g N , µ H -1/2 (Γ), H1/2 (Γ) for all µ ∈ H 1 (Ω), µ |Γc = 0.
The following theorem follows directly from the material in [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation Inverse Problems[END_REF].

Theorem 3.4. For all ε, δ > 0, the mixed QR problem of Definition 3.3 has a unique solution

(u ε,δ , λ ε,δ ) in H 1 (Ω) × H 1 (Ω). Furthermore, if δ > 0 is defined as a function of ε > 0 so that lim ε→0 ε δ(ε) = 0
and if the Cauchy problem has a (unique) solution u ∈ H 1 (Ω), then

(u ε,δ(ε) , λ ε,δ(ε) ) ε→0 -----→ H 1 ×H 1 (u, 0).
Obviously, this mixed formulation of the QR method carries some properties that we were looking for. First of all, it does not require any additional smoothness assumptions on the boundary, the Cauchy data or the exact solution of the Cauchy problem. Secondly, the variational form appearing in Definition 3.3 can be discretized using standard C 0 Lagrange finite elements (cf. [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation Inverse Problems[END_REF]), as the solution pair lies in H 1 (Ω) 2 .

Unfortunately, this formulation has also some mild flaws. The first one concerns the additional unknown λ ε , which can in a way be considered known: it can be interpreted as an estimate for ∆uf , with u being the solution of the original Cauchy problem (assuming that it exists). In other words, λ ε approximates the zero function, and does not provide any additional information on the Cauchy problem in hand.

Arguably, the most important drawback of the above mixed formulation of the QR method is that there currently exists no method for choosing the regularization parameters (ε, δ) in case of noisy data. In particular, the method developed in [START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF] for the standard QR formulation cannot be used for this mixed formulation. As the Cauchy problem is severely ill-posed and its solution thus very sensitive to noise, this can be considered a major issue. Due to the above described difficulties with the mixed QR problem of Definition 3.3, we propose in the following section a new mixed QR formulation. Our aim is to circumvent the flaws of the original mixed QR approach without loosing its good qualities (no additional smoothness assumption, discretization with standard finite elements).

A new H div -based mixed quasi-reversibility method

Recall the original Cauchy problem of Definition 2.1 and assume for now that it has a solution. If we define p := A∇u, it is clear that p ∈ L 2 (Ω) d . Furthermore, as ∇ • A∇u = f belongs to L 2 (Ω), we actually have p ∈ H div (Ω), with the standard definition

H div (Ω) := q ∈ L 2 (Ω) d : ∇ • q ∈ L 2 (Ω) .
Hence, we can rewrite the Cauchy problem in the following, equivalent form:

Definition 4.1 (Reformulated Cauchy problem). For (f, g D , g N ) ∈ L 2 (Ω) × H 1/2 (Γ) × H -1/2 (Γ), find (u, p) ∈ H 1 (Ω) × H div (Ω) such that (3)        A∇u = p in Ω, ∇ • p = f in Ω, u = g D on Γ, p • ν = g N on Γ.
The leading idea behind our new mixed QR method is to regularize the reformulated Cauchy problem of Definition 4.1.

Formulation and basic properties.

To begin with, let us introduce a few auxiliary sets and spaces:

V := v ∈ H 1 (Ω) : v |Γ = g D , V 0 := v ∈ H 1 (Ω) : v |Γ = 0 , D := q ∈ H div (Ω) : (q • ν) |Γ = g N , D 0 := q ∈ H div (Ω) : (q • ν) |Γ = 0 ,
which are well defined due to trace theorems in H 1 (Ω) and H div (Ω) (cf., e.g., [START_REF] Fernandes | Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions[END_REF][START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]). We consider the following QR problem:

Definition 4.2 (New mixed QR problem). For ε > 0, find (u, p) ∈ V ×D such that for all (v, q) ∈ V 0 ×D 0 (4)        Ω (A∇u -p) • A∇v dx + ε Ω (∇u • ∇v + u v) dx = 0, - Ω (A∇u -p) • q dx + Ω (∇ • p)(∇ • q) dx + ε Ω (∇ • p)(∇ • q) + p • q dx = Ω f (∇ • q) dx.
It follows relatively straightforwardly from the Lax-Milgram theorem that the above mixed variational problem has a unique solution.

Proposition 4.1. For all ε > 0, the new mixed QR problem of Definition 4.2 has a unique solution

(u ε , p ε ) ∈ V × D. Proof. Since the mappings v ∈ H 1 (Ω) → v |Γ ∈ H 1/2 (Γ) and q ∈ H div (Ω) → (q • ν) |Γ ∈ H -1/2 (Γ)
are linear, continuous and surjective [START_REF] Fernandes | Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions[END_REF][START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], they have continuous right inverses. Hence, for all (g D , g

N ) ∈ H 1/2 (Γ) × H -1/2 (Γ), there exists (u D , p N ) ∈ V × D such that (5) u D H 1 (Ω) ≤ c g D H 1/2 (Γ) , p N H div (Ω) ≤ c g N H -1/2 (Γ) ,
where c > 0 does not depend on the data (g D , g N ). In consequence, by adding the two equations in (4), we can rewrite the new mixed QR problem in the following equivalent form: find (ũ, p)

∈ V 0 × D 0 that satisfies (6) a ε (ũ, p), (v, q) = L ε (v, q) for all (v, q) ∈ V 0 × D 0 , with a ε (ũ, p), (v, q) := Ω (A∇ũ -p)(A∇v -q) dx + Ω (∇ • p)(∇ • q) dx + ε Ω (∇ũ • ∇v + ũ v) dx + ε Ω (∇ • p)(∇ • q) + p • q dx and L ε (v, q) := Ω f (∇ • q) dx -a ε (u D , p N ), (v, q) .
The product space V 0 × D 0 , endowed with the scalar product

(u, p), (v, q) H 1 ×H div := Ω (∇u • ∇v + u v) dx + Ω (∇ • p)(∇ • q) + p • q dx,
is a Hilbert space. Furthermore, a ε and L are obviously bilinear and linear functionals on (V 0 × D 0 ) 2 and V 0 × D 0 , respectively, and for all (u, p)

∈ V 0 × D 0 and (v, q) ∈ V 0 × D 0 it holds that a ε (v, q), (v, q) ≥ ε (v, q) 2 H 1 ×H div , a ε (u, p), (v, q) ≤ 1 + 2 A ∞ + A 2 ∞ + ε (u, p) H 1 ×H div (v, q) H 1 ×H div , |L ε (v, q)| ≤ C (v, q) H 1 ×H div , where C = f L 2 (Ω) + c 1 + 2 A ∞ + A 2 ∞ + ε g D H 1/2 (Γ) + g N H -1/2 (Γ) .
Hence, an application of the Lax-Milgram theorem [START_REF] Brézis | Analyse Fonctionnelle, Théorie et Applications[END_REF] proves the unique existence of a solution (ũ ε , pε ) to [START_REF] Ben Belgacem | Extended-domain-Lavrentiev's regularization for the Cauchy problem[END_REF], leading in turn to the existence of a unique solution [START_REF] Blum | Numerical simulation and optimal control in plasma physics with application to tokamaks[END_REF] (

u ε , p ε ) = (ũ ε + u D , pε + p N ) ∈ V × D to (4).
Our main theorem states that (u ε , p ε ) converges to the unique solution of the original Cauchy problem if such exists, and diverges otherwise.

u ∈ H 1 (Ω), then (u ε , p ε ) converges to (u, A∇u) in H 1 (Ω) × H div (Ω) as ε > 0 tends to 0. If the Cauchy problem has no solution, then (u ε , p ε ) H 1 ×H div → ∞ as ε > 0 goes to 0.
The mixed formulation (4) can thus be considered as a regularized version of the ill-posed Cauchy problem. Therefore, by solving the mixed QR problem, we obtain an approximation of the solution to the Cauchy problem, if such exists. Note that the additional unknown p ε is an estimate of A∇u, and hence it provides additional information on the solution u.

Remark 4.4. The new mixed QR method can also be viewed as a compatibility test for the data (f, g D , g N ). Indeed, if one wants to know if there exists a solution to the Cauchy problem, one can solve (4) for various ε > 0 and consider the behavior of the function R + ∋ ε → (u ε , p ε ) H 1 ×H div when ε tends to zero. If this function blows up, the Cauchy problem has no solution. However, it is probably difficult to use this criterion in practice -especially with noisy data -, as the blow-up may be very slow.

To prove Theorem 4.3, we divide it into two propositions, because the proofs of the two cases require slightly different arguments. Proposition 4.2. Suppose that the Cauchy problem of Definition 2.1 has a (unique) solution u ∈ H 1 (Ω). Then,

lim ε→0 u ε -u H 1 (Ω) = 0, lim ε→0 p ε -A∇u H div (Ω) = 0,
and the estimates

A∇u ε -p ε 2 L 2 (Ω) + ∇ • p ε -f 2 L 2 (Ω) 1/2 ≤ √ ε (u, A∇u) H 1 ×H div hold.
Proof. Because u is the solution of the Cauchy problem, we know that

u |Γ = g D , ν • A∇u |Γ = g N and A∇u ∈ H div (Ω). Hence, choosing v = u ε -u ∈ V 0 and q = p ε -A∇u ∈ D 0 in (4) leads to                  Ω (A∇u ε -p ε ) • (A∇u ε -A∇u) dx + ε Ω ∇u ε • ∇(u ε -u) + u ε (u ε -u) dx = 0, - Ω (A∇u ε -p ε ) • (p ε -A∇u) dx + Ω (∇ • p ε ) ∇ • (p ε -A∇u) dx + ε Ω (∇ • p ε )(∇ • (p ε -A∇u)) + p ε • (p ε -A∇u) dx = Ω f ∇ • (p ε -A∇u) dx.
Summing these equalities and using the fact that ∇

• A∇u = f , it follows that (8) A∇u ε -p ε 2 L 2 (Ω) + ∇ • p ε -f 2 L 2 (Ω) + ε (u ε , p ε ), (u ε -u, p ε -A∇u) H 1 ×H div = 0.
Therefore, we have

(9) (u ε , p ε ), (u ε -u, p ε -A∇u) H 1 ×H div ≤ 0, which, in particular, means that (10) (u ε , p ε ) H 1 ×H div ≤ (u, A∇u) H 1 ×H div .
Equation ( 9) also leads to

(11) (u ε -u, p ε -A∇u) 2 H 1 ×H div ≤ -(u, A∇u), (u ε -u, p ε -A∇u) H 1 ×H div , which in turn implies that (12) (u ε -u, p ε -A∇u) H 1 ×H div ≤ (u, A∇u) H 1 ×H div .
Finally, by applying [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 domains[END_REF] and ( 12) to (8), we deduce the needed convergence estimates:

A∇u ε -p ε 2 L 2 (Ω) + ∇ • p ε -f 2 L 2 (Ω) ≤ ε (u ε , p ε ), (u ε -u, p ε -A∇u) H 1 ×H div ≤ ε (u ε , p ε ) H 1 ×H div (u ε -u, p ε -A∇u) H 1 ×H div ≤ ε (u, A∇u) 2 H 1 ×H div . ( 13
)
According to [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 domains[END_REF], the family of solutions (u ε , p ε ) is bounded in H 1 (Ω)×H div (Ω), and thus there exists a sequence of real positive numbers (ε n ) n∈N converging to zero such that the corresponding solutions of (4), i.e., (u n , p n ) := (u εn , p εn ), weakly converge to some (w, r) in H 1 (Ω) × H div (Ω). Since the operations

H 1 (Ω) × H div (Ω) ∋ (v, q) → A∇v -q ∈ L 2 (Ω) d , H div (Ω) ∋ q → ∇ • q ∈ L 2 (Ω), H 1 (Ω) ∋ v → v |Γ ∈ H 1/2 (Γ), H div (Ω) ∋ q → (q • ν) |Γ ∈ H -1/2 (Γ)
are linear and bounded, they are also weakly continuous. In consequence, by taking the weak limit of (u n , p n ) and employing [START_REF] Brézis | Analyse Fonctionnelle, Théorie et Applications[END_REF] with ε = ε n , we deduce that (w, r) satisfies

         A∇w = r in Ω, ∇ • r = f in Ω, w = g D on Γ, q • ν = g N on Γ,
which means that (w, r) is the solution of (3), i.e., (w, r) = (u, A∇u). Hence, the standard argument ad absurdum allows us to conclude that (u ε , p ε ) weakly converges to (u, A∇u) in H 1 (Ω) × H div (Ω) as ε > 0 goes to zero. Due to [START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF], weak convergence implies strong convergence, which completes the proof.

Proposition 4.3. Suppose that the Cauchy problem of Definition 2.1 has no solution. Then,

lim ε→0 (u ε , p ε ) H 1 ×H div = ∞.
Proof. Assume that (u ε , p ε ) ≤ C, for some C > 0 independent of ε > 0. Then, there exists a sequence (ε n ) n∈N of positive real numbers converging to 0 and (w, r) ∈ H 1 (Ω) × H div (Ω) such that (u n , p n ) := (u εn , p εn ) weakly converges to (w, r) as n goes to infinity. Defining

ω = A∇w -r ∈ L 2 (Ω) d and ξ = ∇ • r -f ∈ L 2
(Ω), and taking the limit of ( 4), with ε = ε n and (u, p) = (u n , p n ), as n goes to infinity, it follows that ( 14)

       Ω ω • A∇v dx = 0 for all v ∈ V 0 , - Ω ω • q dx + Ω ξ(∇ • q) dx = 0 for all q ∈ D 0 .
As C ∞ 0 (Ω) d ⊂ D 0 , the second equation of ( 14) means, in particular, that ∇ξ = -ω ∈ L 2 (Ω) d in the sense of distribution, and thus ξ ∈ H 1 (Ω). In a similar manner, the first equation then implies that ∇ • A∇ξ = 0 in Ω, as A is symmetric by assumption and C ∞ 0 (Ω) ⊂ V 0 . We are therefore allowed to use the (generalized) Green's formula [START_REF] Nečas | Les méthodes directes en théorie des équations elliptiques[END_REF]

Ω (∇ • A∇ξ)v + A∇ξ • ∇v dx = A∇ξ • ν, v H -1/2 (∂Ω),H 1/2 (∂Ω)
for all v ∈ H 1 (Ω).

Together with the first part of [START_REF] Brezzi | Mixed and hybrid finite element methods[END_REF], this leads to

A∇ξ • ν, v H -1/2 (Γc), H1/2 (Γc) = 0 for all v ∈ V 0 ,
i.e., A∇ξ • ν = 0 on Γ c . By the Green's formula, we also have

Ω (∇ • q) ξ + q • ∇ξ dx = q • ν, ξ H -1/2 (∂Ω),H 1/2 (∂Ω)
for all q ∈ H div (Ω). Consequently, the second equation of ( 14) and the established connection between ξ and ω indicate that

q • ν, ξ H -1/2 (∂Ω),H 1/2 (∂Ω) = 0 for all q ∈ D 0 ,
which is just a more complicated way of writing ξ = 0 on Γ c . We have altogether concluded that ξ ∈ H 1 (Ω) satisfies

     ∇ • A∇ξ = 0 in Ω, ξ = 0 on Γ c , A∇ξ • ν = 0 on Γ c ,
and thus ξ must vanish by the unique solvability of this Cauchy problem. By the construction, we thus have ∇ • r = f and A∇w = r in Ω. Using the weak continuity of the trace operators

H 1 (Ω) ∋ v → v |Γ ∈ H 1/2 (Γ) and H div (Ω) ∋ q → (q • ν) |Γ ∈ H -1/2 (Γ), it also follows that w = lim n→∞ u n = g D on Γ, r • ν = lim n→∞ p n • ν = g N on Γ.
In other words, (w, r) is a solution of the reformulated Cauchy problem (3), which is a contradiction.

4.2. Monotonic convergence. In this section, we are interested in the following question: Assume that the original Cauchy problem of Definition 2.1 has a solution u and suppose (a bit unrealistically) that we have access to its noiseless Cauchy data. Consider two possible values for the regularization parameter ε 1 and ε 2 , with ε 1 < ε 2 . Which parameter should we use in the new mixed QR method of Definition 4.2? Indeed, even though we know that u ε tends to u and p ε to A∇u as ε goes to zero, we have not yet provided any result stating that the convergence is monotonic. We will tackle this imperfection and demonstrate that the function

R + ∋ ε → (u -u ε , A∇u -p ε ) H 1 ×H div is strictly increasing.
In the rest of this section, we continue to implicitly assume that the original Cauchy problem has a (unique) solution u ∈ H 1 (Ω).

4.2.1. Smoothness of the map ε → (u ε , p ε ). Let us define an auxiliary function

F : ε → (u ε , p ε ), R + → H 1 (Ω) × H div (Ω)
where (u ε , p ε ) is the unique solution of (4). We will demonstrate that F is smooth, starting with continuity.

Proposition 4.4.

It holds that F ∈ C 0 (R + , H 1 (Ω) × H div (Ω)).
Proof. Let ε ∈ R + and h ∈ R be such that also ε + h ∈ R + . Due to ( 6) and ( 7), we have

a ε+h (u ε+h , p ε+h ), (v, q) = a ε (u ε+h , p ε+h ), (v, q) + h (u ε+h , p ε+h ), (v, q) H 1 ×H div = (f, ∇ • q) L 2 , a ε (u ε , p ε ), (v, q) = (f, ∇ • q) L 2
for all v ∈ V 0 and q ∈ D 0 . Choosing v = u ε+hu ε , q = p ε+hp ε , and subtracting the two equalities, it follows that

A∇(u ε+h -u ε ) -(p ε+h -p ε ) 2 L 2 + ∇ • (p ε+h -p ε ) 2 L 2 + ε u ε+h -u ε , p ε+h -p ε 2 H 1 ×H div = -h (u ε+h , p ε+h ), (u ε+h -u ε , p ε+h -p ε ) H 1 ×H div .
Omitting the first two terms on the left-hand side and applying the Cauchy-Schwarz inequality, we have

(15) (u ε+h -u ε , p ε+h -p ε ) H 1 ×H div ≤ h ε (u ε+h , p ε+h ) H 1 ×H div ≤ h ε (u, A∇u) H 1 ×H div ,
where the latter inequality follows from [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 domains[END_REF].

Remark 4.5. Since (u ε , p ε ) converges to (u, A∇u) as ε goes to zero, we can extend F to be a continuous function on R + ∪ {0} by setting F (0) = (u, A∇u).

Let us then consider the following problem: For ε > 0, find (u, p)

∈ V 0 × D 0 such that (16) a ε (u, p), (v, q) = -(u ε , p ε ), (v, q) H 1 ×H div for all (v, q) ∈ V 0 × D 0 .
Proposition 4.5. The problem (16) has a unique solution (u

1 ε , p 1 ε ) ∈ V 0 × D 0 that satisfies (17) (u 1 ε , p 1 ε ) H 1 ×H div ≤ 1 ε (u, A∇u) H 1 ×H div .
Proof. The result follows straightforwardly from the Lax-Milgram theorem and the estimate [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 domains[END_REF].

Proposition 4.6. It holds that F ∈ C 1 (R + , H 1 (Ω) × H div (Ω)), with the corresponding derivative defined via F ′ (ε) = (u 1 ε , p 1 ε ), where (u 1 ε , p 1 ε ) ∈ V 0 × D 0
is the unique solution of (16). Proof. For all ε > 0, h ∈ R such that ε + h > 0, and (v, q) ∈ V 0 × D 0 , we have

a ε+h (u ε+h , p ε+h ), (v, q) = (f, ∇ • q) L 2 , -a ε (u ε , p ε ), (v, q) = -(f, ∇ • q) L 2 , -h a ε (u 1 ε , p 1 ε ), (v, q) = h (u ε , p ε ), (v, q) H 1 ×H div .
Choosing v = v ε,h := u ε+hu εhu 1 ε ∈ V 0 and q = q ε,h := p ε+hp εhp 1 ε ∈ D 0 , and summing the three equalities, one obtains that

A∇v ε,h -q ε,h 2 L 2 + ∇•q ε,h 2 L 2 +ε (v ε,h , q ε,h ) 2 H 1 ×H div = -h (u ε+h -u ε , p ε+h -p ε ), (v ε,h , q ε,h ) H 1 ×H div ,
and thus

1 |h| (u ε+h -u ε -hu 1 ε , p ε+h -p ε -hp 1 ε ) H 1 ×H div ≤ 1 ε (u ε+h -u ε , p ε+h -p ε ) H 1 ×H div h→0 ---→ 0.
It remains to be shown that the derivative ε → (u 1 ε , p 1 ε ) is continuous as a map from R + to H 1 (Ω) × H div (Ω). For all (v, q) ∈ V 0 × D 0 , it holds that

a ε+h (u 1 ε+h , p 1 ε+h ), (v, q) = -(u ε+h , p ε+h ), (v, q) H 1 ×H div a ε (u 1 ε , p 1 ε ), (v, q) = -(u ε , p ε ), (v, q) H 1 ×H div . Therefore, choosing v = v 1 ε,h := u 1 ε+h -u 1 ε ∈ V 0 , q = q 1 ε,h := p 1 ε+h -p 1 ε ∈ D 0 ,
and subtracting the two equalities, we obtain

a ε (v 1 ε,h , q 1 ε,h ), (v 1 ε,h , q 1 ε,h ) = -(u ε+h -u ε + h u 1 ε+h , p ε+h -p ε + h p 1 ε+h ), (v 1 ε,h , q 1 ε,h ) H 1 ×H div , meaning, in particular, that ε (v 1 ε,h , q 1 ε,h ) H 1 ×H div ≤ (u ε+h -u ε , p ε+h -p ε ) H 1 ×H div + h (u 1 ε+h , p 1 ε+h ) H 1 ×H div .
In consequence, the bounds ( 15) and ( 17) provide the estimate

(v 1 ε,h , q 1 ε,h ) H 1 ×H div ≤ 2 h ε 2 (u, A∇u) H 1 ×H div ,
which completes the proof.

For ε > 0, we now define a sequence of function pairs (u m ε , p m ε ) ∈ H 1 (Ω)×H div (Ω), m ∈ N 0 := N∪{0}, recursively:

• Set (u 0 ε , p 0 ε ) = (u ε , p ε ). • For m ∈ N 0 , define (u m+1 ε , p m+1 ε
) to be the unique element of V 0 × D 0 that satisfies [START_REF] Dardé | The 'exterior approach': a new framework to solve inverse obstacle problems[END_REF] a ε (u

m+1 ε , p m+1 ε ), (v, q) = -(m + 1) (u m ε , p m ε ), (v, q) H 1 ×H div for all (v, q) ∈ V 0 × D 0 .
As in the case of ( 16), the fact that the sequence (u m ε , p m ε ) is well defined follows from a simple application of the Lax-Milgram theorem, which also produces the estimate

(u m ε , p m ε ) ≤ m! ε m (u, A∇u) H 1 ×H div
as a recursive by-product. In particular, it turns out that the sequence (u m ε , p m ε ), m ∈ N 0 , defines the derivatives of F .

Theorem 4.6. It holds that F ∈ C ∞ (R + , H 1 (Ω) × H div (Ω)), and for all m ∈ N, F (m) (ε) = (u m ε , p m ε ), where (u m ε , p m ε ) ∈ V 0 × D 0
is the (recursively defined) unique solution of (18).

Proof. The claim follows from induction and similar arguments as those in the proof of Proposition 4.6.

4.2.2.

Proof of monotonic convergence. To begin with, note that the unique solvability of (4) and the linear dependence of (u m+1 ε , p m+1 ε ) on (u m ε , p m ε ), m ∈ N 0 , provide us with the following lemma.

Lemma 4.7. Suppose that the (compatible) data of the Cauchy problem verifies (f, g D , g N ) = (0, 0, 0). Then (u m ε , p m ε ) = (0, 0) for all m ∈ N and ε > 0. Now, we are ready to formulate and prove the main result of this section.

Theorem 4.8. The function

R + ∋ ε → (u ε -u, p ε -A∇u) H 1 ×H div is strictly increasing.
Proof. Obviously, it is sufficient to prove that g :

R + ∋ ε → 1 2 (u ε -u, p ε -A∇u) 2 H 1 ×H div is strictly increasing.
A simple computation shows that

g ′ (ε) = (u ε -u, p ε -A∇u), (u 1 ε , p 1 ε ) H 1 ×H div and g ′′ (ε) = (u ε -u, p ε -A∇u), (u 2 ε , p 2 ε ) H 1 ×H div + (u 1 ε , p 1 ε ) 2 H 1 ×H div .
Because (u 2 ε , p 2 ε ) solves ( 18) with m = 1, and

(u ε -u, p ε -A∇u) ∈ V 0 × D 0 , we have ε (u ε -u, p ε -A∇u), (u 2 ε , p 2 ε ) H 1 ×H div = -2 (u 1 ε , p 1 ε ), (u ε -u, p ε -A∇u) H 1 ×H div -(A∇u 2 ε -p 2 ε , A∇u ε -p ε ) L 2 -(∇ • p 2 ε , ∇ • p ε -f ) L 2 .
Since (u ε , p ε ) solves ( 4) and (u

2 ε , p 2 ε ) ∈ V 0 × D 0 , we can further deduce that ε ((u ε -u, p ε -A∇u), (u 2 ε , p 2 ε )) H 1 ×H div = -2 g ′ (ε) + ε (u 2 ε , p 2 ε ), (u ε , p ε ) H 1 ×H div .
Altogether, we have thus far obtained that

ε g ′′ (ε) + 2 g ′ (ε) = ε (u 1 ε , p 1 ε ) 2 H 1 ×H div + ε (u 2 ε , p 2 ε ), (u ε , p ε ) H 1 ×H div .
Now, utilizing the fact that (u 1 ε , p 1 ε ) is the solution of ( 18) with m = 0, it follows that

(u 2 ε , p 2 ε ), (u ε , p ε ) H 1 ×H div = -a ε (u 1 ε , p 1 ε ), (u 2 ε , p 2 ε ) . Since (u 2 ε , p 2 ε
) is in turn the solution of problem ( 18) with m = 1, we obtain

(u 2 ε , p 2 ε ), (u ε , p ε ) H 1 ×H div = 2 (u 1 ε , p 1 ε ) 2 H 1 ×H div .
In consequence,

ε 2 g ′ (ε) ′ = ε 2 g ′′ (ε) + 2ε g ′ (ε) = 3ε 2 (u 1 ε , p 1 ε ) 2 H 1 ×H div . Because (u 1 ε , p 1 ε ) = (0, 0), it holds that ε 2 g ′ (ε) ′ > 0, i.e., ε 2 g ′ (ε) is a strictly increasing function. Moreover, |ε 2 g ′ (ε)| = ε 2 (u ε -u, p ε -A∇u), (u 1 ε , p 1 ε ) H 1 ×H div ,
and thus the Cauchy-Schwarz inequality and (17) imply that

|ε 2 g ′ (ε)| ≤ ε (u ε -u, p ε -A∇u) H 1 ×H div (u, A∇u) H 1 ×H div ε→0 ---→ 0.
Hence, we have ε 2 g ′ (ε) > 0 for all ε > 0, indicating that also g ′ (ε) > 0. This completes the proof.

4.3. Noisy data. In this section, we focus on the important case of noisy data. Throughout this section it is assumed that the (reformulated) Cauchy problem of Definition 4.1 has a unique solution (u, p) ∈ H 1 (Ω) × H div (Ω), but one has only access to the noisy data

f δ ∈ L 2 (Ω), g δ D ∈ H 1/2 (Γ) and g δ N ∈ H -1/2 (Γ) that satisfy f δ -f L 2 (Ω) ≤ δ, g δ D -g D H 1/2 (Γ) ≤ δ, g δ N -g N H -1/2 (Γ) ≤ δ
for some δ > 0. Let us denote by (u δ ε , p δ ε ) the solution of the mixed QR problem (4), when the exact data (f, g D , g N ) are replaced by their noisy counterparts (f δ , g δ D , g δ N ); the pair (u ε , p ε ) still denotes the mixed QR solution for the noiseless data. The aim is to answer the following question: For a certain amplitude of noise δ > 0, how should the regularization parameter ε := ε(δ) be chosen in order to assure the convergence of (u δ ε(δ) , p δ ε(δ) ) to (u, p) when the amplitude of noise goes to zero.

Proposition 4.7. It holds that

(19) (u δ ε -u, p δ ε -p) H 1 ×H div ≤ C(ε) δ √ ε + C ′ δ + (u ε -u, p ε -p) H 1 ×H div , where C(ε) := 1 + C ′ 1 + 2 A ∞ + A 2
∞ + ε and C ′ > 0 depends only on the geometry of Ω.

Proof. Let (u D , p N ) and (ũ ε , pε ) be as in the proof of Proposition 4.1 and let us introduce a similar decomposition for the mixed QR solution corresponding to the noisy data:

(20) u δ ε = ũδ ε + u δ D , p δ ε = pδ ε + p δ N , with (21) u δ D H 1 (Ω) ≤ c g δ D H 1/2 (Γ) , p δ N H div (Ω) ≤ c g δ N H -1/2 (Γ) and (ũ δ ε , pδ ε ) ∈ V 0 × D 0 satisfying a ε (ũ δ ε , pδ ε ), (v, q) = Ω f δ (∇ • q) dx -a ε (u δ D , p δ N ), (v, q)
for all (v, q) ∈ V 0 × D 0 . By subtracting [START_REF] Ben Belgacem | Extended-domain-Lavrentiev's regularization for the Cauchy problem[END_REF], it follows that 22), we obtain

(22) a ε (ũ δ ε -ũε , pδ ε -pε ), (v, q) = Ω (f δ -f )(∇ • q) dx -a ε (u δ D -u D , p δ N -p N ), (v, q) . Choosing (v, q) = (ũ δ ε -ũε , pδ ε -pε ) in (
(ũ δ ε -ũε , pδ ε -pε ) 2 ε = Ω (f δ -f )(∇ • (p δ ε -pε )) dx -a ε (u δ D -u D , p δ N -p N ), (ũ δ ε -ũε , pδ ε -pε ) ,
where (•, •) ε denotes the ε-dependent norm on H 1 (Ω) × H div (Ω) induced by the bilinear form a ε . Due to the definition of a ε and the assumption on the noise level, it obviously holds that

Ω (f δ -f )(∇ • (p δ ε -pε )) dx ≤ f δ -f L 2 (Ω) ∇ • (p δ ε -pε ) L 2 (Ω) ≤ δ (ũ δ ε -ũε , pδ ε -pε ) ε , and
(u δ D -u D , p δ N -p N ) 2 ε ≤ (1 + 2 A ∞ + A 2 ∞ + ε) (u δ D -u D , p δ N -p N ) 2 H 1 ×H div ≤ 2 c 2 (1 + 2 A ∞ + A 2 ∞ + ε) δ 2 , (23)
where the last step follows from the fact that an estimates of the type ( 5) and ( 21) naturally holds also for the differences u δ Du D and p δ Np N by virtue of the linearity of the right inverses for the associated trace maps.

Together with the Cauchy-Schwarz inequality, the previous three formulas induce the estimate

(ũ δ ε -ũε , pδ ε -pε ) 2 ε ≤ δ 1 + √ 2 c 1 + 2 A ∞ + A 2 ∞ + ε (ũ δ ε -ũε , pδ ε -pε ) ε , which in turn leads to √ ε (ũ δ ε -ũε , pδ ε -pε ) H 1 ×H div ≤ (ũ δ ε -ũε , pδ ε -pε ) ε ≤ δ 1 + √ 2 c 1 + 2 A ∞ + A 2 ∞ + ε .
Combining this with [START_REF] Geuzaine | Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[END_REF] and the corresponding decomposition for (u ε , p ε ), we finally have

(u δ ε -u ε , p δ ε -p ε ) H 1 ×H div ≤ (ũ δ ε -ũε , pδ ε -pε ) H 1 ×H div + (u δ D -u D , p δ N -p N ) H 1 ×H div ≤ δ √ ε 1 + √ 2 c 1 + 2 A ∞ + A 2 ∞ + ε + √ 2 c δ,
where the second term on the right-hand side of the first inequality is estimated as in [START_REF] Hadamard | Lectures on Cauchy's Problem in Linear Partial Differential Equations[END_REF]. The claim now follows from the triangle inequality.

The following corollary is an immediate consequence of Proposition 4.7 and Theorem 4.3.

Corollary 4.9. For any choice of the regularization parameter ε = ε(δ) such that

(24) lim δ→0 ε(δ) = lim δ→0 δ 2 ε(δ) = 0, it holds that u δ ε(δ) , p δ ε(δ) → (u, p) in H 1 (Ω) × H div (Ω)
as δ goes to zero.

According to Corollary 4.9, any choice of ε = ε(δ) that satisfies ( 24) is admissible. For example, one could set ε(δ) := δ α with any α ∈ (0, 2). However, for a fixed noise level, there naturally exist choices of regularization parameter that are better (in some sense) than others. The task is then to choose the optimal parameter -and also the considered optimality criterion. In the following, we will briefly describe two methods that could be used to set the regularization parameter.

In [START_REF] Cao | A Carleman estimate and the balancing principle in the quasireversibility method for solving the Cauchy problem for the Laplace equation[END_REF], a method based on a balancing principle is proposed to choose the regularization parameter for the standard QR method described in Section 3.1. If translated to the setting of the new mixed QR method and Proposition 4.7, the idea of [START_REF] Cao | A Carleman estimate and the balancing principle in the quasireversibility method for solving the Cauchy problem for the Laplace equation[END_REF] can be described as follows: The right-hand side of the error estimate ( 19) is composed of two terms, namely

f 1 (ε) = C(ε) δ/ √ ε + C ′ δ and f 2 (ε) = (u ε -u, p ε -p) H 1 ×H div , which have opposite behaviors with respect to ε. Indeed, lim ε→0 f 1 (ε) = ∞, lim ε→∞ f 1 (ε) = 2C ′ δ and lim ε→0 f 2 (ε) = 0, lim ε→∞ f 2 (ε) = (u, p) H 1 ×H div ,
where the first inequality follows from Proposition 4.7 and the second one straightforwardly from the estimates in the proof of Proposition 4.1. Therefore, if δ is sufficiently small, it is clear that there exists a unique ε opt > 0 such that f 1 (ε opt ) = f 2 (ε opt ) by virtue of Theorem 4.8 and the monotonicity of f 1 .

In a sense, such a parameter choice balances the error due to the noise in the data and the discrepancy between the exact and the regularized solution for the noiseless case. Unfortunately, ε opt cannot be determined directly since f 2 is unknown. The authors of [START_REF] Cao | A Carleman estimate and the balancing principle in the quasireversibility method for solving the Cauchy problem for the Laplace equation[END_REF] use Carleman inequalities to obtain an estimate for f 2 inside the domain, i.e., far from the complementary boundary Γ c where the boundary conditions are unknown. They are then in position to propose an iterative algorithm that produces εopt ≈ ε opt , even without knowing explicitly the constants appearing in the estimate [START_REF] Fernandes | Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions[END_REF]. A close look at [START_REF] Cao | A Carleman estimate and the balancing principle in the quasireversibility method for solving the Cauchy problem for the Laplace equation[END_REF] shows that the balancing principle could be adapted to our mixed formulation, using in particular the present result of monotonic convergence (theorem 4.8).

Another technique for finding an optimal regularization parameter is proposed in [START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF]. The method is based on the well-known Morozov discrepancy principle, which states that ε opt > 0 should be the (largest) regularization parameter for which the discrepancy in the data fit equals the noise level. The authors of [START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF] propose an algorithm based on duality in optimization for computing such ε opt exactly (up to numerical errors). The method has been tested successfully with the standard QR formulation. However, a closer look at [START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF] demonstrates that this dual optimization algorithm could also be adapted without difficulty to our mixed QR setting, leading to a method for automatically obtaining the regularization parameter that satisfies the Morozov discrepancy principle. The exact formulation of the algorithm, together with a proof of convergence and numerical tests, is left for future studies. 4.4. Discretization. The mixed QR method of Definition 4.2 can obviously be discretized with conforming finite elements. Indeed, the considered Sobolev spaces, namely H 1 (Ω) and H div (Ω), are standard spaces appearing, e.g., in mathematical analysis of fluid mechanics and electromagnetism. Consequently, their finite element discretization has been studied extensively; see, e.g., [START_REF] Brezzi | Mixed and hybrid finite element methods[END_REF][START_REF] Ciarlet | The finite element method for elliptic problems[END_REF] and the references therein.

In this section, we assume that Ω is a two-dimensional polygonal domain (resp. a three-dimensional polyhedral domain). We define T h to be a regular triangulation of Ω in the sense of [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF], such that the diameter of each triangle (resp. each tetrahedron) is bounded by h > 0. We assume that Γ is the union of edges (resp. faces) of some triangles (resp. tetrahedra) of T h . For k ∈ N and a triangle/tetrahedron K of T h , we define P k (K) to be the space of polynomial functions of degree lower or equal to k in K. The standard Lagrange finite element space L k h is then defined to be the set of functions v h ∈ C 0 (Ω) such that v h|K ∈ P k (K) for any K of T h . It is well-known that L k h ⊂ H 1 (Ω), and thus we can use Lagrange finite elements to approximate V and V 0 .

To be more precise, we assume that g D ∈ H 1/2 (Γ) ∩ C 0 (Γ), and define g D,h to be its interpolant over the traces of L k h -functions on Γ with some fixed k ∈ N; here and in what follows, we define an interpolant as the element of some (context-dependent) finite element subspace with the same degrees of freedom as the function that is interpolated. We then set

V k h := v h ∈ L k h : v h = g D,h on Γ , V k 0,h := v h ∈ L k h : v h = 0 on Γ ⊂ V 0 .
By assumption V k h is non-empty, as is V k 0,h .

In order to obtain conforming approximations of D and D 0 , which are subsets of H div (Ω), we use the well-known Raviart-Thomas RT k h finite elements (cf., e.g., [START_REF] Brezzi | Mixed and hybrid finite element methods[END_REF]), which are defined as follows:

RT k h := q h ∈ H div (Ω) : q h|K ∈ P k (K) d + xP k (K) for all K ∈ T h , k ∈ N 0 ,
with x ∈ R d being the spatial variable. If the Neumann data g N is assumed to be in L 2 (Γ), we can introduce its interpolant g N,h over the space of the normal components on Γ of vector fields in RT k h , and then define

D k h := q h ∈ RT k h : q h • ν = g N,h on Γ , D k 0,h := q h ∈ RT k h : q h • ν = 0 on Γ ⊂ D 0 .
Again, D k h = ∅ = D k 0,h . With the help of these finite element spaces, we can now define the discretized version of the mixed QR problem. Definition 4.10 (Mixed QR h problem). For ε > 0 and some k ∈ N, find (u

h , p h ) ∈ V k h × D k-1 h such that for all (v, q) ∈ V k 0,h × D k-1 0,h (25)
       Ω (A∇u h -p h ) • A∇v dx + ε Ω (∇u h • ∇v + u h v) dx = 0, - Ω (A∇u h -p h ) • q dx + Ω (∇ • p h )(∇ • q) dx + ε Ω (∇ • p h )(∇ • q) + p h • q dx = Ω f (∇ • q) dx.
The unique solvability of [START_REF] Hanke | Convex source support and its application to electric impedance tomography[END_REF] follows in exactly the same way as that of (4).

Proposition 4.8. For all ε > 0, the mixed QR h problem of Definition 4.10 admits a unique solution

(u ε,h , p ε,h ) ∈ V k h × D k-1 h .
The main theorem of this section provides an estimate for the convergence of (u ε,h , p ε,h ) towards (u ε , p ε ). Theorem 4.11. Suppose that the solution of (4), i.e., (u ε , p ε ), belongs to (H k+1 (Ω) ∩ C 0 (Ω)) × H k+1 (Ω) d . Then,

(u ε -u ε,h , p ε -p ε,h ) H 1 ×H div ≤ c h k 1 + 2 A ∞ + A 2 ∞ + ε ε u ε 2 H k+1 (Ω) + p ε 2 H k+1 (Ω) d ,
where c > 0 is independent of h > 0 and ε > 0.

Proof. It follows directly from the Cea's lemma [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF] and the estimates in the proof of Proposition 4.1 that

(u ε -u ε,h , p ε -p ε,h ) H 1 ×H div ≤ C(ε) inf (v,q)∈V k h ×D k-1 h (u ε -v, p ε -q) H 1 ×H div ,
where

C(ε) = 1 + 2 A ∞ + A 2 ∞ + ε ε .
As u ε ∈ H k+1 (Ω) ∩ C 0 (Ω), we can define its interpolant ũε,h in L k h . By definition, the trace of ũε,h on Γ is equal to g D,h , making it an element of V k h . Furthermore, it holds that [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF]

u ε -ũε,h H 1 (Ω) ≤ ch k |u ε | H k+1 (Ω) ,
with a constant c > 0 that is independent of h and ε. Accordingly, if p ε is an element of H k+1 (Ω) d , we can define its interpolant pε,h in RT k-1 h , which by definition is an element of D k-1 h , and we have [START_REF] Brezzi | Mixed and hybrid finite element methods[END_REF]

p ε -pε,h H div (Ω) ≤ ch k p ε H k+1 (Ω) ,
where c > 0 is again independent of h and ε. The assertion now follows by combining the above three estimates.

Remark 4.12. Theorem 4.11 concretizes the well-known fact that the choice of the regularization parameter ε > 0 cannot be independent of the mesh size h > 0. More precisely, it is useless to choose a very small ε if the size of the mesh h is not small as well.

Numerical experiments

In this section, we test the new mixed QR method numerically. We start with the Cauchy problem and subsequently move on to the inverse obstacle problem for the Laplacian. All forward solutions are computed with FreeFem++ [START_REF] Hecht | Freefem++ manual[END_REF] coupled with the mesh generator Gmsh [START_REF] Geuzaine | Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[END_REF]. For the Cauchy problem, we choose Ω = C g \ C s ⊂ R 3 , i.e., the body of interest is the great cylinder C g := D(0, 1)×]0, 2[ without the small cylinder C s := D(0, 0.4)×]0.7, 1.4[. In all examples, Γ is a subset of ∂C g , for different parts of which we introduce the following shorthand notations:

Γ l = ∂D(0, 1)×]0, 2[, Γ t = D(0, 1) × {2}, Γ b = D(0, 1) × {0} .
In the first experiment, we set A ≡ I, i.e., consider the Cauchy problem for the Laplacian. The data are simulated by solving the boundary value problem ( 26)

       ∆u = 0 in Ω, ∂u ∂ν = w on ∂C g , u = 0 on ∂C s ,
where w ≡ 1, -1 and 0 on Γ t , Γ b and Γ l , respectively, using the FEM with L 2 h Lagrange elements. The corresponding solution is visualized in Figure 1.

Let us first consider the case of exact data and Γ = ∂C g . In other words, the Dirichlet and Neumann traces of the FEM solution to (26) on the whole exterior boundary of Ω are used directly as the latter two components of the data (0, g D , g D ) for the new mixed QR method introduced in Section 4. We choose the value ε = 10 -4 for the regularization parameter. In order to avoid an inverse crime, we use different meshes to solve the direct and inverse problems. The discrepancy between the first component of the solution to [START_REF] Hanke | Convex source support and its application to electric impedance tomography[END_REF] and the FEM solution of ( 26) is visualized in Figure 2, both inside Ω and on the lateral boundary of the void C s . Apparently, the method works as desired. Here and in all the following numerical tests, we have used k = 1 for the discretized problem of Definition 4.10.

In practise, the measurements always contain uncertainties, and thus it is essential to also test the new mixed QR method with noisy data. In addition, we make the Cauchy problem more demanding by

D -g δ D L 2 (Γ) = 0.05 u L 2 (Γ) , g N -g δ N L 2 (Γ) = 0.05 w L 2 (Γ)
due to introduction of additive noise (defined in a suitable way). We still use ε = 10 -4 as the regularization parameter. The corresponding discrepancy between the regularized solution provided by the new mixed QR method and the target solution of ( 26) is shown in Figure 3. Obviously, the reduction in the amount of data and the addition of noise have increased the discrepancy, but the proposed QR method still seems to function relatively well. We continue to consider the same geometrical setting and the Laplacian, but next choose u = x 2 + y 2 -2 z 2 to be the exact solution of the Cauchy problem. In other words, the explicit expressions for the Dirichlet and Neumann traces of this potential are employed to compute the latter two components of the (noiseless) Cauchy data (0, g D , g N ) on Γ = ∂C g ; note that once again the first component of the data triplet vanishes as the chosen target function satisfies the Laplace equation. Figure 4 shows the logarithmic discrepancy [START_REF] Lascaux | Some nonconforming finite elements for the plate bending problem[END_REF] log (u ε,hu, p ε,h -∇u) H1 (Ω)×H div (Ω)

between the chosen target potential and the solution of the mixed QR h problem of Definition 4.10, with h ≈ 0.05, as a function of log ε. As predicted by Theorem 4.8 for the exact QR solution of (4), for fairly large values of the regularization parameter the discrepancy decreases monotonically when ε gets smaller. However, for small ε > 0 the error between the solutions of ( 4) and ( 25) seems to dominate (cf. Theorem 4.11), and the convergence stalls. It is to be expected that for a finer discretization, i.e., for a smaller mesh parameter h > 0, the monotonic convergence would continue for even smaller regularization parameters ε > 0.

In our final numerical experiment with the Cauchy problem, we assume that the object of interest Ω = C g \ C s is anisotropic. To be more precise, we choose the 'diffusion matrix' in the Cauchy problem of Definition 2.1 to be which is a randomly picked symmetric and positive definite matrix with the eigenvalues 0.179, 0.8855, and 1.1355. The latter two components of the data triplet (0, g D , g N ) on Γ = Γ l are simulated by solving the boundary value problem [START_REF] Hecht | Freefem++ manual[END_REF] with the Laplacian and the standard normal derivative replaced by ∇ • A∇ and the conormal derivative ∂u/∂ν A from (1), respectively. Subsequently, 5% of noise is added to the data in the sense of ( 27). The corresponding target potential is shown in Figure 5.

The discrepancy between the target potential and the QR h solution of (25), corresponding to ε = 10 -4 and the Cauchy data with 5% of noise on Γ = Γ l , is visualized in Figure 6. Even in this noisy anisotropic case, the difference between the target solution and the one provided by our mixed QR method is of an order of magnitude smaller than the values of the target potential itself. assuming that there exists accurate enough information on the amount of measurement noise. Further considerations of this matter are left for future studies.

Theorem 4 . 3 .

 43 If the Cauchy problem of Definition 2.1 has a unique solution

Figure 1 .

 1 Figure 1. The target solution of (26).

5. 1 .

 1 Cauchy problem. Let us denote by D(x, r) the open disk of radius r > 0 centered at x ∈ R 2 .

Figure 2 .-Figure 3 .

 23 Figure 2. The discrepancy u εu in Ω (left) and on the lateral boundary of C s (right) for Γ = ∂C g and the Laplace operator. Relative L 2 (Ω)-error: 3.6%

Figure 4 .

 4 Figure 4. The logarithmic discrepancy (28) as a function of log ε for u = x 2 + y 2 -2 z 2 , Γ = ∂C g and h ≈ 0.05.

5. 2 .Figure 5 .

 25 Figure 5. The target solution of (26) with I replaced by A of (29).

Figure 6 .Figure 7 .

 67 Figure 6. The discrepancy u ε -u in Ω (left) and on the lateral boundary of C s (right) for Γ = Γ l , the diffusion matrix from (29) and 5% of noise in the data. Relative L 2 (Ω)-error: 8.7% that Ω \ O is connected and the problem

In fact, we could deal with partial Cauchy data, i.e., data defined only on a part of ∂Ω, in exactly the same way.

This work was supported by the Academy of Finland (project 135979).

(1) Reconstruct a potential u in Ω \ ω m from the Cauchy data available on ∂Ω, i.e., solve the Cauchy problem This algorithm builds a sequence of open sets (ω m ) m∈N verifying O ⊂ ω m , and it is proved in [START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF][START_REF] Dardé | The 'exterior approach': a new framework to solve inverse obstacle problems[END_REF] that this sequence converges to the searched obstacle if, e.g., the boundaries of the open sets are smooth enough. For more information on the details of the exterior approach, such as the stopping criterion or the choice of the free parameters, we refer to [START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF][START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF][START_REF] Dardé | The 'exterior approach': a new framework to solve inverse obstacle problems[END_REF].

In our implementation of the exterior approach, we use the mixed QR method introduced in Section 4 for solving [START_REF] Nečas | Les méthodes directes en théorie des équations elliptiques[END_REF]. More precisely, we set ε = 10 -4 and utilize the combination of first order Lagrange and zeroth order Raviart-Thomas elements, i.e., we choose k = 1 in Definition 4.10. The Poisson problem (32) is solved by a FEM employing first order Lagrange elements.

Let Ω be the open unit ball; the initial guess ω 0 is chosen to be a smaller concentric ball of radius 0.8 in all our tests. Figure 7 presents the reconstructions produced by the above outlined exterior approach algorithm for three different target obstacles O: a small ball, a union of two balls, and a torus. The corresponding Cauchy data were simulated by solving (with first order Lagrange elements) the well-posed boundary value problem obtained by deleting the second equation of [START_REF] Rousseau | Applications to unique continuation and control of parabolic equations[END_REF], setting g N ≡ 1, and choosing O accordingly. Afterwards, g D was defined to be the Dirichlet trace of the corresponding FEM solution.

In each of the three cases, the qualitative shape of the obstacle is reproduced accurately; in particular, the homotopy classes of the obstacle and the corresponding reconstruction are the same in all three tests. Here, we considered only exact Cauchy data (not accounting for numerical inaccuracies), but according to our experience the performance of the method with noisy data is comparable to what is presented in [START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF][START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF][START_REF] Dardé | The 'exterior approach': a new framework to solve inverse obstacle problems[END_REF] for the corresponding two-dimensional setting.

Concluding remarks

We have introduced a novel mixed QR method for regularizing the ill-posed Cauchy problem of Definition 2.1 without resorting to optimization schemes. Our method provides an approximate solution that converges monotonically to the exact one, if such exists; the technique for proving the monotonicity property in Section 4.2 can also be applied to the original QR method of [START_REF] Lattès | Méthode de Quasi-réversibilité et Applications[END_REF]. Like the first mixed QR method in [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation Inverse Problems[END_REF], our new formulation can be discretized using standard finite elements that are often available in numerical solvers. Furthermore, both components of the solution (u ε , p ε ) ∈ H 1 (Ω) × H div (Ω) to the new mixed QR problem (4) provide information about the solution u of the original Cauchy problem: while u ε approximates u, the vector field p ε gives an estimate for the corresponding flux A∇u. The functionality of our new method was demonstrated via three-dimensional numerical studies considering both the Cauchy problem and a related inverse obstacle problem.

In our numerical experiments the choice of the regularization parameter ε > 0 was not considered in detail, but its value was just picked so that the resulting reconstructions appeared reasonable (ε = 10 -4 in most tests). It should be noted, however, that the new QR method seems surprisingly insensitive to the size of a smallish ε. Moreover, the method of [START_REF] Bourgeois | A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data[END_REF] for choosing the regularization parameter as a function of the noise amplitude for the standard QR formulation can be straightforwardly adapted for our new method; see Section 4.3 for more information. As a consequence, it seems that the proposed mixed QR method can be coupled with a systematic technique for choosing the regularization parameter,


Myriam Lewkowicz

Manuel Zacklad

A guide through the construction of a groupware for efficient knowledge management

The main idea in this article is that groupware benefits organizational memory because it focuses on communication and coordination, but is inadequate for an efficient knowledge management. It has to be completed by a model for exchange structuring in order to improve dialog quality and to enable a conversation classification that would not be simply chronological. Our aim is to build a groupware (MEMO-Net) enriched with such a model. This model (DIPA), which uses and simplifies the concepts of Problem-Solving methods, comes from a review of existing Design Rationale formalisms that gave rise to the ABRICo formalism.

Introduction

The result presented in this article belongs to a research project whose aim is to manage knowledge used in design project for capitalization and reusability. In accordance to Zacklad and Grundstein ([24]), knowledge capitalization research can be classified in three categories: social and cooperative approaches, top-down modeling approaches and bottomup modeling approaches. In the first category, one considers that organizations' critical knowledge comes within a collective competence that is not enough or bad formalized. The development of systematic use of groupware, electronic-mail services, newsgroups, workflow, embodied in Intranets and particularly used in design projects would seem to explain that these tools are indeed considered potential aids for the knowledge capitalization process.

If groupware mediatizes designers' interactions, the best way to locate and protect crucial knowledge (in the sense of M. Grundstein, [START_REF] Grundstein | La capitalisation des connaissances de l'entreprise, une problématique de management[END_REF]) exchanged through these nets is to study the interactional structure and to suggest information structuring tools and models for highlighting exchanged knowledge and enabling easier future access. Most of the time i n knowledge capitalization projects, there is a lack of quality rather than quantity of information; quality regarding the structure of memorized materials.

In "social and cooperative" knowledge capitalization approaches, there seem to be two types of approach. Some work will aim for a posteriori structure information by reconstituting, from the traces of intellectual transactions ([25]), the concept structure previously elaborated collectively. Others will aim for a priori structure transactions to guarantee a better quality of both interactions and write track of these interactions that will enable an easier re-exploitation by the "knowledge managers". This interest in a priori structuring of problem-solving processes in order to guarantee a exploitation is not recent. In CSCW (Computer-Supported Cooperative Work) research, several authors ([6], [START_REF] Conklin | A Process-Oriented Approach to Design Rationale[END_REF]) have already expressed wish to switch from a "object-centered paradigm to a "process-centered" paradigm. In the last one, designers' interaction (that is to say questions, decisions and conversations that form the elaboration environment of the objects) would be memorized as well as objects and design process results.

Following this paradigm, our previous works dealt with Design Rationale, criticized its classical methods and proposed a new formalism, ABRICo ( [START_REF] Lewkowicz | A formalism for the rationalization of decision-making processes in complex collective design situations[END_REF]). In this paper, we present DIPA, an evolution of this formalism thanks to our interpretation of knowledge engineering results on problem-solving methods, and its implementation in a groupware, MEMO-Net.

CSCW and Groupware

Groupware offers multi-users interfaces to access electronic mail services, forum, and workflow, and to put into practice CSCW (Computer-Supported Cooperative Work) methods, or, as Malone says ( quoted in [START_REF] Coleman | Defining Groupware. Special Advertising Section to Network World[END_REF]), it is "information technology used to help people work together more effectively". CSCW may be seen as the scientific discipline that guides thoughtful and appropriate design and development of groupware [START_REF] Greenberg | Computer-Supported Work and Groupware[END_REF]. These technologies represent a changing paradigm in computer science, because they deal more with problems due to human-human coordination and communication than with defining human-computer dialogues for automated procedures.

There are two ways of viewing the variety of groupware, a time/space taxonomy and an application-level taxonomy. They are embodied in this classical 2x2 matrix in [START_REF] Ellis | Groupware some issues and experiences[END_REF] or [START_REF] Bullen | Groupware in practice: An interpretation of work experiences[END_REF]:

Same time Different times

Same place face to face interaction

Meeting Rooms GDSS (Group Decision

Support System) asynchronous interaction

Project Management tools

Different places distributed synchronous interaction

Videoconferences Shared screens distributed asynchronous interaction E-mail Forum Cooperative Writing As Grudin ([12]) stresses, technology alone cannot provide an efficient introduction of this kind of tools in organizations. It is essential to understand how groups and organizations function and evolve. This comprehension enables the choice of the right tools to fit existing communication flows and to crystallize organizational memory elements without penalizing users [START_REF] Conklin | Capturing Organizational Memory[END_REF].

But, if a groupware is capable of recording solution elaboration processes, it is not sufficient for efficient knowledge management ( [START_REF] Grudin | Groupware and Cooperative Work: Problems and Prospects[END_REF]). It has to be completed by a method that structures the exchanges for a better quality of dialogue and for a management of ideas that would not be simply chronological. These reflections have led to the IBIS method [START_REF] Burgess-Yakemovic | Report on a Development Project Use of an Issue-Based Information System[END_REF], connected afterwards to Design Rationale research.

Design Rationale

The IBIS (Issue-Based Information System) method [START_REF] Burgess-Yakemovic | Report on a Development Project Use of an Issue-Based Information System[END_REF] aims to improve the quality of design dialogue processes by structuring discussion in complex problems. It uses several categories: issues, positions that are possible solutions for the issues, and arguments for or against these solutions. This method has been implemented in a graphics tool (gIBIS) and a textual one (itIBIS). Work with IBIS falls within Design Rationale research from the Human-Computer Interaction community. Design Rationale research usually concerns problems appearing in the capture of the rationale followed by the articulation, representation and use of this rationale made explicit. In a first synthesis of these works, edited in a special issue of Human-Computer Interaction, Carroll and Moran [START_REF]Special Issue on Design Rationale[END_REF] described the importance of this field by stressing the following points:

According to them, constructing explicit Design Rationale could:

• support reasoning processes in design,

• facilitate communication among the various players in the design processes (designers, implementers, maintainers, users, etc.), • further the accumulation and development of design knowledge throughout design projects and products. IBIS and its different implementations have been experimented many times more or less successfully. In a general way, there are two reservations about this method [START_REF] Fischer | Making Argumentation Serve Design[END_REF] : (a) the model used to represent argumentation is too schematic and (b) its nature is exclusively «dialogal».

The first reservation concerns the non-representation of interdependency between issues. Relations between several options can not be represented; everything is done as if each discussion mediatized by the tool would correspond to an independent sub-problem.

Secondly, the non-deliberated issues are ignored. If a question is not debated, the tool will not stress it, although it could correspond to an important point that could influence the design process.

We made somewhat similar criticisms of the QOC (Questions Options Criteria) method ( [START_REF] Maclean | Questions, Options and Criteria : Elements of Design Space Analysis[END_REF]) and classical Design Rationale formalisms in our previous work ([15], [START_REF] Lewkowicz | A formalism for the rationalization of decision-making processes in complex collective design situations[END_REF]). According to us, formalisms like QOC seem to be relevant for simple, short-lasting design situations where one has to choose between several options and where the shape of the final solution is known. But they are not suitable for collective design situations that we have called complex (unknown solution shape, progressive elaboration of a unique longlasting solution), where we need to take into account the process dynamics and the participants' roles. We then proposed to represent these complex processes with an original formalism that we called ABRICo (from the French words Accords, Buts, pRopositions, Interprétations en Conception, that mean agreement, goals, propositions, interpretations through design). Its static model is described below (figure 1). We tested this formalism by representing real complex collective decision-making situations. Seeing that this experimentation was conclusive, we built a first version of a tool based on these concepts, MEMO-Net, and we submitted it to professional groups for a first evaluation. Unfortunately, the model proposed to users seemed to be too abstract for easy comprehension, a conclusion that re-opened the whole question of the ABRICo model and the first version of the tool.

One reason for the difficulty of implementing ABRICo was, from our point of view, that the complex decision process formulation was too far from the design situations which people were confronted with. Where the traditional but simplistic Design Rationale models enable an easy appropriation, the ABRICo model, although more realistic from a cognitive point of view, appeared them both too theoretical and too far from concrete implementation conditions in the work situations for which we built it.

DIPA, a collective problem-solving model

In order to approach the cognitive dimension of reasoning, we then enriched ABRICo with problem-solving method concepts from knowledge engineering. This enrichment was translated into a new model, DIPA (from the French words Données, Interprétations, Propositions, Accord, meaning facts, interpretations, propositions, agreement). This model has itself two declinations according to the situations that lead the actors to give priority to either analysis or synthesis processes (for example as in KADS methodology [START_REF] Wielinga | Modelling Expertise[END_REF]). This link with problem-solving methods seems to us a natural evolution in our researches of more realistic Design Rationale models suited to the complexity of real projects. Below is an outline of the characteristics of the three types of design meeting dialogues modeling that we studied or built.

1) Description of design meeting dialogues with classical argumentation models from

Design Rationale : Actors participate in an argumentative dialogue; they defend contradictory or exclusive options, quoting criteria or arguments in favor of their position.

2) Description of design meeting dialogues with an argumentation model corresponding to complex situations in ABRICo sense.

At the same time as putting forward p ropositions, the actors debate different possible interpretations that may justify these propositions. Interpretations, like propositions, are not necessarily contradictory or exclusive. An interpretation, regarded as belonging to a more abstract level, may be evoked without necessarily being part of the defence of a point of view. At the same time, the argument semantic is not considered as entirely independent of the roles of the persons who introduce them.

3) Description of design meeting dialogues with an argumentation model inspired by problem-solving models from knowledge engineering (DIPA)

Problem-solving models replace decision-making processes, even complex ones. Each argument is categorized by its role in the problem-solving method. The model comes in two forms, according to the kind of process that it assists (analysis or synthesis). This reference to problem-solving models allows the integration of an important knowledge category that was not taken into account in the two previous models, the "problem data". Actually, we could say that Design Rationale models have neglected the "information" phase of Simon's decision-making process [START_REF] Simon | Administrative Behavior, trad. Fçse[END_REF], and have only taken into account the solutions selection phase. Models from Artificial Intelligence do not have this fault. In the DIPA model, the reasoning progresses in three major steps:

• i. a problem description step plus collecting of data, considered as symptoms in analysis situations and as needs in synthesis situations; • ii. an abstraction step going from the collecting of problem data to their interpretation corresponding to a possible cause in analysis situations, and to a functionality in synthesis situations; • iii. an implementation step that going from an interpretation (cause or functionality) to the elaboration of a proposition that is a corrective action removing the symptom's cause (analysis) or the means suitable for the expressed functionality (synthesis)

Relevance of a unique model for analysis and synthesis

The fact that we had to present both analysis and synthesis models to designers teams may seem amazing. Actually, it might appear natural at first glance to propose only synthesis models and their variants (routine design, configuration…). But our practical experience of design meetings showed us that analysis activities are frequent. For example, as soon as a prototype has been developed, its function analysis will give important information that will be reintroduced in the process of solution finding.

These observations are also in accordance with cognitive ergonomic psychology ( [START_REF] Darses | Gestion des contraintes dans la résolution des problèmes de conception[END_REF]) results that teach us that design situations in the organizational sense in fact generate two distinct phases of activity: solution generation and then evaluation of these solutions. The first corresponds to synthesis problems in a KADS sense and is close to design models in this method. The second corresponds to analysis problems whose diagnosis models are well-known.

This idea of a unique model (figure 2) to represent the two types of activity is quite close to some interpretations of the heuristic classification [START_REF] Clancey | Heuristic Classification[END_REF]. Whereas Clancey compares, for example, methods coming within heuristic classification (where one select a solution and then tries to prove that it is the one that fits best), with methods coming within heuristic construction (where structure and behavior models are used to construct new solutions), Zacklad and Fontaine ( [START_REF] Zacklad | L'acquisition des connaissances classificatoires pour les systèmes à bases de connaissances[END_REF]) take a different position.

According to these authors, in both types of situation, there is both exploitation of knowledge from previous solved cases, and construction of an original solution. This solution is a proof or a justification in analysis case and a constraints-compatible approximate solution in synthesis cases. These authors defend the idea that, at a certain abstraction level, it is a matter of the same heuristic reasoning form. This reasoning is characterized (i) by the use of qualitative inferences using the componential semantic of the concepts and (ii) the non-hierarchical contact of concepts coming from different classification hierarchies (this second idea comes directly from Clancey's definition of the heuristic classification [START_REF] Clancey | Heuristic Classification[END_REF], p.294). We feel that this notion of non-hierarchical contact of concepts, that corresponds to the "heuristic matching" in the original heuristic classification model, is at the bottom base of heuristic reasoning used in design meetings, both in analysis and synthesis phases. Several authors have proposed convergent conceptualizations of this kind of reasoning. For example, for Hoc ([14]) it corresponds to exploiting the "implementing" hierarchy. The hierarchical relation does not correspond to running through a class hierarchy, but running through different levels that enables a physical implementation while starting from systematic description of a system's aims via different functional views.

In PAGIC 1 ([23]), this reasoning form is described as a navigation between different points of view about the same system that comes into action when designers take into account the "optical" dimension of the distinct "abstraction" of the hierarchical dimension of the "generalization".

Whereas, in the DIPA model, the abstraction and implementation inference steps are two symmetrical aspects of the same heuristic reasoning, applicable both in analysis and synthesis 2 . In abstraction cases, for example, the point of view about any one system will change according to whether the symptoms or their causes are considered most important, or even the requirements of internal system functions.

The formalism used t o describe DIPA was inspired by KADS ( [START_REF] Wielinga | Modelling Expertise[END_REF], [START_REF] Schreiber | Model Construction[END_REF]) but does not strictly follow the KADS conventions as to how to represent inference structures. Contributions are classified chronologically or according to DIPA model categories, or to the authors' names, their roles, or their department.

The following example concerns a team of designers of a trading application. The malfunction noticed is that the users of the application make many mistakes when they put in an order for products. The first symptom seemed to be caused by interface faults, and improvements are suggested, as well as a constraint: the interface has already been modified twice. Another possible cause is lack of training, and two corrective actions have been suggested: training seminars or free books of trading rules. The screen copy below (figure 4) shows the chronological view of contributors' propositions. Figure 5 presents a summarized view in which these contributions are classified by concepts. The second example that presented below, with, as previously, a chronological view (figure 6) and a summarized view (figure 7) concerns a team of researchers wanting to construct an Intranet for their laboratory, using Lotus Notes. Different needs are mentioned, such as announcing seminars or accessing particular documents. These needs refer to functionalities e.g. managing a documentary fund. Finally, various means to obtain these functionalities are suggested, e.g. a "News module" in a Lotus base or the creation of a Lotus library base. When users have already discussed a problem, one of their propositions may be submitted to others, in order to collect their opinion and take a decision. This last step corresponds to "selection" inference of DIPA model, which enables a definitive agreement on the best possible solution.

Conclusion

Newsgroups between experts working on a project or requests for "help desk" activities are often considered as important for knowledge capitalization. However, we think that they are not sufficient to enable an efficient management of this knowledge because the lack of structuring of the material makes it difficult to exploit.

Our idea is that the enrichment of groupware with knowledge-structuring models could enable both guiding collective work processes and obtaining reusable or exploitable knowledge.

The next steps of our research project will be to test the DIPA model via the MEMO-Net groupware, both in real work situations and with systematic experiments. In our experiments, we will evaluate the MEMO-Net influence in two ways: (a) the speed with which the participants reach an agreement and (b) the quality of the proposed solutions. To achieve this goal, we will ask two groups to solve the same problems several times, sometimes by using low-structured groupware (like a newsgroup) and other times using our tool.

Figure 1 :

 1 Figure 1 : static model ABRICo

Figure 2 :

 2 Figure 2 : DIPA, a heuristic model of design reasoning for analysis and synthesis

Figure 4 :Figure 5 :

 45 Figure 4: Chronological view of a diagnosis problem-solving process

Figure 6 :

 6 Figure 6 : Chronological view of a design problem-solving process

Figure 7 :

 7 Figure 7 : Summarized view of a design problem-solving process

Figure 8 :

 8 Figure 8 : A form to gather opinion plus the « choice » view visualizing previously opinions gathered

PAGIC is a French acronym for Partie-tout, Abstraction, Généralisation, Interaction, Cybernétique.

In KAL ontology that formalizes heuristic classification's reasoning ([START_REF] Zacklad | Principes de modélisation qualitative pour l'aide à la décision dans les organisations[END_REF]), the abstraction step corresponds to abductive propositions and the implementation step to "constructive-deductive" propositions.

[25] Zacklad, M. (1999) La théorie des transactions intellectuelles : une approche gestionnaire et cognitive pour le traitement du COS, Intellectica, numéro spécial sur le COS, à paraître.

8.

Références

Implementation of DIPA model : MEMO-Net groupware

We implemented the DIPA model to build the MEMO-Net groupware. This system consists of two modules, one for synthesis phases (named "design" in the interface), and the other for analysis phases (named "diagnosis" in the interface). Its goal is to allow a project team to solve problems met during design by alternating the two types of activity on a cooperative way. The exchange structure allows both to guide the solution process and to organize the arguments, particularly in argument capitalization aspects.

In the diagnosis module, members of the project team identify a dysfunction and evoke symptoms, causes or corrective actions. In design, once the goal is known, the actors evoke requirements, functionalities and means. To contribute, people click on the following signs (indicating a malfunction, symptom, cause, and corrective action) and then create the corresponding forms: