Giovanni S Alberti

Yves Capdeboscq

Combining The

Keywords: 2010 Mathematics Subject Classification. 35J25, 35B38, 35R30 Runge approximation, Whitney reduction, hybrid imaging, coupledphysics inverse problems, boundary control, non-zero constraints, photoacoustic tomography, thermoacoustic tomography. 1

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Introduction

We consider a general second-order elliptic equation [START_REF] Alberti | Generalized N-property and Sard theorem for Sobolev maps[END_REF] Lu := -div(a∇u + bu) + c • ∇u + qu = 0 in Ω,

where Ω ⊆ R d , d ≥ 2, is a bounded and smooth domain. We assume that L is uniformly elliptic, namely,

(2)

a (x) ξ • ξ ≥ λ |ξ| 2 , a.e. x ∈ Ω, ξ ∈ R d ,
for some λ > 0. The parameters of equation ( 1) are assumed to satisfy mild regularity assumptions, namely either (3)

a ∈ C -1,α Ω; R d×d , b ∈ C -1,α (Ω; R d ), c ∈ W -1,∞ (Ω; R d ), q ∈ W -1,∞ (Ω; R),
with ≥ 1 and α ∈ (0, 1), or

(4) a ∈ L ∞ Ω; R d×d , b, c ∈ L ∞ Ω; R d , q ∈ L ∞ Ω; R d ,
which will be referred to as = 0. By classical elliptic regularity theory [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF][START_REF] Giaquinta | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs[END_REF][START_REF] Morrey | Multiple integrals in the calculus of variations[END_REF], the solutions to (1) belong to C ,α loc (Ω; R) and, provided that the boundary conditions are chosen in the appropriate trace space, such a regularity extends up to the boundary, namely u ∈ C ,α Ω; R . In the case = 0, the Hölder exponent α is for example the one given by the De Giorgi-Nash-Moser theorem.

This paper focuses on how to enforce pointwise constraints on the solutions of (1). Our motivation for studying such a question comes from hybrid imaging. Hybrid, or multi-physics, imaging problems are a type of parameter identification problems that in many cases involve the reconstruction of the coefficients of a PDE from the knowledge of some internal functional of its solutions [START_REF] Ammari | An introduction to mathematics of emerging biomedical imaging[END_REF][START_REF] Widlak | Hybrid tomography for conductivity imaging[END_REF][START_REF] Bal | Hybrid inverse problems and internal functionals[END_REF][START_REF] Kuchment | Mathematics of hybrid imaging: a brief review[END_REF][START_REF] Ammari | Mathematics of superresolution biomedical imaging[END_REF][START_REF] Alberti | Lectures on elliptic methods for hybrid inverse problems[END_REF]. Amongst all these constraints, the most ubiquitous one is the non-vanishing Jacobian problem. It can be reworded as follows: given L as in [START_REF] Alberti | Generalized N-property and Sard theorem for Sobolev maps[END_REF] and a compact set K ⊆ Ω, how can one choose boundary conditions g 1 , . . . , g N such that [START_REF] Alberti | Absence of Critical Points of Solutions to the Helmholtz Equation in 3D[END_REF] rank (∇u 1 , . . . , ∇u N ) = d everywhere in K,

where Lu i = 0 in Ω, u i = g i on ∂Ω, i = 1, . . . , N ?

The difficulty here is that, apart from the fact that a, b, c and q are relatively smooth and coercive, nothing is known about these coefficients, which are the unknowns of the inverse problem. When d = 2, b = 0, c = 0 and q = 0, that is, L is simply Lu = -div(a∇u), it turns out that the Radó-Kneser-Choquet Theorem can be extended to this setting (without regularity assumptions) [START_REF] Alessandrini | Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions[END_REF][START_REF] Alessandrini | Univalent σ-harmonic mappings[END_REF][START_REF] Alessandrini | Estimates for the dilatation of σ-harmonic mappings[END_REF][START_REF] Alessandrini | Quantitative estimates on Jacobians for hybrid inverse problems[END_REF]. Only two boundary conditions, independent of the matrix valued function a, are required for the constraint to be satisfied globally. This result cannot be extended to higher dimensions [START_REF] Wood | Lewy's theorem fails in higher dimensions[END_REF][START_REF] Laugesen | Injectivity can fail for higher-dimensional harmonic extensions[END_REF][START_REF] Capdeboscq | On a counter-example to quantitative Jacobian bounds[END_REF][START_REF] Alberti | Critical points for elliptic equations with prescribed boundary conditions[END_REF][START_REF] Alberti | Lectures on elliptic methods for hybrid inverse problems[END_REF], even locally: it is not possible to find suitable boundary conditions independently of the (unknown) coefficient. For more general models, when b, c or q are not null, such as the Helmholtz equation, no solace can be found in any dimension, since the Radó-Kneser-Choquet Theorem, whose proof uses the maximum principle, does not apply.

One is therefore drawn to ask whether using a large number of boundary conditions would help. Again, counter-examples to several claims can be derived [START_REF] Alberti | Lectures on elliptic methods for hybrid inverse problems[END_REF]Corollary 6.18]; nevertheless, it is possible to construct open sets of boundary conditions valid for open sets of parameters for the relevant elliptic operator L. Two main strategies have been used to achieve this goal: complex geometrical optics (CGO) solutions [START_REF] Triki | Uniqueness and stability for the inverse medium problem with internal data[END_REF][START_REF] Bal | Inverse diffusion theory of photoacoustics[END_REF][START_REF] Bal | Quantitative thermo-acoustics and related problems[END_REF][START_REF] Bal | Multi-source quantitative photoacoustic tomography in a diffusive regime[END_REF][START_REF] Ammari | Microwave imaging by elastic deformation[END_REF][START_REF] Kocyigit | Acousto-electric tomography and CGO solutions with internal data[END_REF][START_REF] Monard | Inverse diffusion problems with redundant internal information[END_REF][START_REF] Bal | Hybrid inverse problems and internal functionals[END_REF][START_REF] Monard | Inverse anisotropic diffusion from power density measurements in two dimensions[END_REF][START_REF] Bal | Inverse diffusion from knowledge of power densities[END_REF][START_REF] Bal | Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions[END_REF][START_REF] Habib Ammari | Quantitative thermoacoustic imaging: an exact reconstruction formula[END_REF][START_REF] Bal | Ultrasound-modulated bioluminescence tomography[END_REF][START_REF] Bal | Inverse anisotropic conductivity from internal current densities[END_REF] and the Runge approximation property [START_REF] Bal | Inverse anisotropic conductivity from internal current densities[END_REF][START_REF] Monard | Inverse anisotropic conductivity from power densities in dimension n ≥ 3[END_REF][START_REF] Bal | Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions[END_REF][START_REF] Bal | Ultrasound-modulated bioluminescence tomography[END_REF]. Both strategies had earlier been used in the context of electrical impedance tomography [START_REF] Kohn | Determining conductivity by boundary measurements. II. Interior results[END_REF][START_REF] Sylvester | A global uniqueness theorem for an inverse boundary value problem[END_REF]. Other approaches based on frequency variations [START_REF] Alberti | On multiple frequency power density measurements[END_REF][START_REF] Alberti | On multiple frequency power density measurements II. The full Maxwell's equations[END_REF][START_REF] Alberti | Enforcing local non-zero constraints in PDEs and applications to hybrid imaging problems[END_REF][START_REF] Alberti | Absence of Critical Points of Solutions to the Helmholtz Equation in 3D[END_REF][START_REF] Alberti | On local non-zero constraints in PDE with analytic coefficients[END_REF][START_REF] Alberti | Multi-frequency acoustoelectromagnetic tomography[END_REF][START_REF] Habib Ammari | Admittivity imaging from multi-frequency micro-electrical impedance tomography[END_REF] or dynamical systems [START_REF] Bal | Boundary control of elliptic solutions to enforce local constraints[END_REF] were also developed: these are not discussed here.

CGO solutions are only available for isotropic coefficients a, that is, a = γI d where γ is a real-valued function. The CGO solution method provides a nonvanishing Jacobian globally inside the domain for a suitable choice of (d complexvalued) boundary conditions. This approach requires high regularity assumptions on the coefficients. On the other hand, the Runge approximation property holds provided that the unique continuation property holds [START_REF] Lax | A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations[END_REF], such a property being enjoyed by a much larger class of problems [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF]. A drawback is that the argument is local, applied on a covering of the domain by small balls, and so many boundary conditions are needed. Further, while CGO solutions are constructed (depending on the coefficients), the Runge approximation provides an existence result of suitable solutions, but not a constructive method to derive them.

In this work, we combine a Whitney projection argument [START_REF] Whitney | Analytic coordinate systems and arcs in a manifold[END_REF], as described in [START_REF] Greene | Whitney's imbedding theorem by solutions of elliptic equations and geometric consequences[END_REF][START_REF] Greene | Embedding of open Riemannian manifolds by harmonic functions[END_REF], with the Runge approximation. Conditions of the form (5) are related to the embeddings of manifolds; using Withney's argument to reduce the ambient dimension iteratively, we reduce the number of solutions needed. Not only do we provide an explicit bound on the number of boundary conditions to be considered, but we also obtain that these constitute an open and dense set. For instance, the set of 2d boundary conditions such that [START_REF] Alberti | Absence of Critical Points of Solutions to the Helmholtz Equation in 3D[END_REF] is satisfied is open and dense in H 1/2 (∂Ω; R) 2d . Our result applies to more general constraints than [START_REF] Alberti | Absence of Critical Points of Solutions to the Helmholtz Equation in 3D[END_REF], so that it is in particular applicable to a variety of imaging problems (see section 2 for details). Our result confirms what has been observed in numerical simulations in the setting of scalar (isotropic) diffusion coefficients, where good reconstructions are obtained for a relatively small set of boundary conditions [START_REF] Monard | Inverse diffusion problems with redundant internal information[END_REF][START_REF] Monard | Inverse anisotropic conductivity from power densities in dimension n ≥ 3[END_REF][START_REF] Bal | Boundary control of elliptic solutions to enforce local constraints[END_REF][START_REF] Bal | Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions[END_REF]. After the first version of this manuscript was published, we were made aware of the recent preprint [START_REF] Cekić | The Calderón problem for the fractional Schrödinger equation with drift[END_REF], where similar techniques are used for the fractional Calderón problem.

This paper is structured as follows. In section 2, we state our main results and discuss some open problems. Section 3 is devoted to the Runge approximation property. Finally, in section 4 we provide the proof of the main result.

Main results

Let K ⊆ Ω be a smooth compact set and

ζ : C ,α (K) → C 0,α (K) n
be a continuous linear map, with n ≥ 1. Let H(K) denote the set of solutions to (1) that are smooth in K, namely ( 6)

H(K) = u ∈ C ,α (K) ∩ H 1 (Ω) : Lu = 0 in Ω , equipped with the norm u H(K) = u C ,α (K) + u H 1 (Ω)
. We are interested in solutions u i ∈ H(K) satisfying the constraint [START_REF] Alberti | Critical points for elliptic equations with prescribed boundary conditions[END_REF] det

   ζ (u 1 ) . . . ζ (u n )    (x) = 0,
in K, locally or globally.

Example 1. Constraints of the form [START_REF] Alberti | Critical points for elliptic equations with prescribed boundary conditions[END_REF] appear in various problems.

• When n = 1, = 0 and ζ (u) = u, the constraint corresponds to avoiding nodal points, namely u 1 (x) = 0. This is useful whenever a division by u 1 is required. • When n = d, = 1, and ζ (u) = ∇u (taken as row vector), the constraint imposes a non-vanishing Jacobian. In that case, (u 1 , . . . , u d ) defines a local C 2 diffeomorphism. This is the case discussed in the introduction. andζ (u) = u ∇u (taken as a row vector) the constraint imposes a non-vanishing "augmented" Jacobian. The additional potential u may represent a scaled time derivative, in a time harmonic model. This constraint may also correspond to the non-vanishing Jacobian for v i = ui+1 u1 , i = 1, . . . , d. Such constraints appear in quantitative photoacoustic tomography [START_REF]Explicit Reconstructions in QPAT, QTAT, TE, and MRE[END_REF][START_REF] Bal | Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions[END_REF][START_REF] Bal | Inverse diffusion theory of photoacoustics[END_REF][START_REF] Bal | Multi-source quantitative photoacoustic tomography in a diffusive regime[END_REF], in quantitative thermoacoustic tomography [START_REF] Bal | Quantitative thermo-acoustics and related problems[END_REF][START_REF] Habib Ammari | Quantitative thermoacoustic imaging: an exact reconstruction formula[END_REF][START_REF] Alberti | Enforcing local non-zero constraints in PDEs and applications to hybrid imaging problems[END_REF], in acousto-electric tomography (also known as electrical impedance tomography by elastic deformation or ultrasound modulated electrical impedance tomography) [START_REF] Ammari | Electrical impedance tomography by elastic deformation[END_REF][START_REF] Capdeboscq | Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements[END_REF][START_REF] Monard | Inverse anisotropic diffusion from power density measurements in two dimensions[END_REF][START_REF] Bal | Inverse diffusion from knowledge of power densities[END_REF][START_REF] Kuchment | Stabilizing inverse problems by internal data[END_REF][START_REF] Monard | Inverse anisotropic conductivity from power densities in dimension n ≥ 3[END_REF][START_REF] Kuchment | Stabilizing inverse problems by internal data. II: non-local internal data and generic linearized uniqueness[END_REF], in microwave imaging by elastic deformation [START_REF] Ammari | Microwave imaging by elastic deformation[END_REF][START_REF] Alberti | On multiple frequency power density measurements[END_REF][START_REF] Alessandrini | Global stability for a coupled physics inverse problem[END_REF][START_REF] Triki | Uniqueness and stability for the inverse medium problem with internal data[END_REF], in current density imaging [START_REF] Hasanov | A new approach to current density impedance imaging[END_REF][START_REF] Lee | A reconstruction formula and uniqueness of conductivity in MREIT using two internal current distributions[END_REF][START_REF] Nachman | Current density impedance imaging. Tomography and Inverse Transport Theory[END_REF][START_REF] Bal | Inverse anisotropic conductivity from internal current densities[END_REF], in dynamic elastography [START_REF]Explicit Reconstructions in QPAT, QTAT, TE, and MRE[END_REF][START_REF] Bal | Reconstructions for some coupled-physics inverse problems[END_REF][START_REF] Bal | Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions[END_REF] and in other hybrid imaging modalities. We refer to [START_REF] Alberti | Lectures on elliptic methods for hybrid inverse problems[END_REF] for additional methods and further explanations on some of the models we have mentioned.

• When n = d + 1, = 1,
We introduce the following notation. Definition 2. Let K ⊆ Ω be a smooth compact set in R d and ζ : C ,α (K) → C 0,α (K) n be a continuous linear map, with l ≥ 0, n ≥ 1 and α ∈ (0, 1). The candidate set C(K) is the set of all x ∈ K for which there exist u 1 , . . . , u n ∈ H(K) so that det

   ζ (u 1 )
. . .

ζ (u n )    (x) = 0. The admissible set E(K) is the set of all u = (u 1 , . . . , u [ d+n α ] ) ∈ H(K) [ d+n α ] such that [ d+n α ] i1,...,in=1 | det    ζ (u i1 ) . . . ζ (u in )    (x)| > 0, for all x ∈ K.
Here,

d+n α = max N ∈ N : N ≤ d+n α denotes the integer part of d+n α . Remark 1. In other words, u ∈ H(K) [ d+n α ] belongs to E(K) if and only if for every x ∈ K there exist i 1 , . . . , i n ∈ {1, . . . , d+n α } such that (8) det    ζ (u i1 ) . . . ζ (u in )    (x) = 0,
namely, if and only if the desired constraint is satisfied everywhere in K for a suitable subset of the solutions u 1 , . . . , u [ d+n α ] . The candidate set is the subset of K where satisfying the constraint pointwise is possible at all. If C(K) = K, then E (K) is empty.

Remark 2. If the coefficients of the PDE are smooth enough, so that α > d+n d+n+1 , then the number of solutions is d+n = d+n α . Furthermore, the number of solutions increases as the coefficients become rougher, which reflects the irregular structure of the zero level-sets of [START_REF] Alberti | Critical points for elliptic equations with prescribed boundary conditions[END_REF].

In this general setting, we have the following result.

thm 3. Take a smooth compact set K ⊆ Ω. The admissible set E (K) is open in H (K) [ d+n α ] . If the candidate set satisfies (9) C (K) = K, then E (K) is dense in H (K) [ d+n α ] .
Remark 3. Similar admissibility sets were previously proved to be open in [START_REF] Monard | Inverse anisotropic diffusion from power density measurements in two dimensions[END_REF]. Since a finite intersection of open and dense sets is open and dense, Theorem 3 immediately extends to the case when finitely many constraints are imposed simultaneously.

Remark 4. As observed in Remark 1, the solutions in the admissible set E(K) satisfy the constraint (8) in K. It would be interesting to consider a quantitative version of this condition, namely an estimate of the form

| det    ζ (u i1 ) . . . ζ (u in )    (x)| ≥ C
for some constant C > 0 given a priori. However, in this case the corresponding admissible set would be neither open nor dense. The study of this scenario would need to make the argument presented below more quantitative.

In section 3 we observe, using the Runge Approximation Property, that assumption ( 9) is satisfied for a large class of examples, since C(Ω) = Ω.

Our initial focus was on boundary value problems, since such problems are relevant for non-invasive imaging methods, as explained in the introduction. The following corollary is a rewording of our result for boundary value problems. cor 4. Take a compact set K ⊆ Ω. Suppose that for every g ∈ H 1/2 (∂Ω) the problem

(10) Lu = 0 in Ω, u = g on ∂Ω,
admits a unique solution u g ∈ H 1 (Ω). If (9) holds true, then the set

g 1 , . . . , g [ d+n α ] ∈ H 1/2 (∂Ω) [ d+n α ] : u g1 , . . . , u g [ d+n α ] ∈ E(K)
is open and dense in H 1/2 (∂Ω) [ d+n α ] .

Proof. Consider the map

ψ : H 1/2 (∂Ω) [ d+n α ] → H 1 (Ω) [ d+n α ] , g 1 , . . . , g [ d+n α ] → u g1 , . . . , u g [ d+n α ] ,
where u gi is defined by [START_REF] Alessandrini | Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions[END_REF]. Because problem [START_REF] Alessandrini | Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions[END_REF] is well-posed, we have u g H 1 (Ω) ≤ C(L) g H 1/2 (∂Ω) for every g ∈ H 1/2 (∂Ω). Further, because of our outstanding regularity assumptions on the coefficients (3) we also have u g C ,α (K) ≤ C(L) g H 1/2 (∂Ω) . This shows that the map ψ :

H 1/2 (∂Ω) [ d+n α ] → H (K) [ d+n α
] is continuous. Its inverse is given by the trace operator acting component-wise, and is also continuous. In other words, ψ is an isomorphism, and the result immediately follows from Theorem 3, since the set under consideration is ψ -1 (E(K)).

Remark 5. Corollary 4 was stated for simplicity only if K is a proper subset of Ω. When K touches ∂Ω, for instance if K = Ω, the same result holds, provided that H 1/2 (∂Ω) is replaced with a suitable trace space consisting of smoother functions, e.g., H +α+ 1 2 (d-1) (∂Ω) or C ,α (∂Ω).

Future perspectives. The regularity assumptions we made are important in all generality, because we use a Unique Continuation Principle argument. For a specific problem, with a given geometry and/or coefficient structure, appropriate extension can often be envisioned (see e.g. [START_REF] Ball | On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients[END_REF] for a strategy on how to handle a large class of piecewise regular coefficients). We have limited ourselves to elliptic PDE with real coefficients. Considering the case of complex valued coefficients (which appear in thermo-acoustic tomography [START_REF] Bal | Quantitative thermo-acoustics and related problems[END_REF][START_REF] Habib Ammari | Quantitative thermoacoustic imaging: an exact reconstruction formula[END_REF]) is a natural extension of this work. Maxwell's equation [START_REF] Seo | Electrical tissue property imaging using MRI at dc and Larmor frequency[END_REF][START_REF] Guo | Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields[END_REF][START_REF] Bal | Hybrid inverse problems for a system of Maxwell's equations[END_REF][START_REF] Alberti | On multiple frequency power density measurements II. The full Maxwell's equations[END_REF][START_REF] Alberti | Enforcing local non-zero constraints in PDEs and applications to hybrid imaging problems[END_REF] and linear elasticity [START_REF] Barbone | Adjointweighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data[END_REF][START_REF] Mclaughlin | Calculating tissue shear modules and pressure by 2D log-elastographic methods[END_REF][START_REF] Lai | Uniqueness and stability of Lamé parameters in elastography[END_REF][START_REF] Bal | Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements[END_REF][START_REF] Bal | Reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields[END_REF] are not considered here and are natural frameworks where this method could be applied. With the current argument, based on Lemma 7, the number of solutions d+n α depends monotonically on the regularity of the solutions. It would be interesting to establish whether that is necessary, and if the number of solutions d+n α is optimal. Finally, note that while a rough description of our result could be that a "random" choice of d+n α boundary condition suffices, we have not established such a claim. It would be interesting to move from an open and dense set of admissible boundary conditions to a random choice of boundary conditions with high probability (or indeed probability 1). This may allow handling the setup discussed in Remark 4.

The Runge approximation property and assumption (9)

For simplicity of exposition, in this section we restrict ourselves to considering only the constraints associated to the maps ζ given in Example 1, namely:

• n = 1, = 0, ζ (u) = u; • n = d, = 1, ζ (u) = ∇u; • or n = d + 1, = 1, ζ (
u) = u ∇u . However, with minor modifications to the argument, many other constraints can be considered, since this approach is very general. The main tool to satisfy (9), namely to show that there always exist global solutions satisfying the desired constraints locally, is the following result: it is sufficient to build suitable solutions of the PDE with constant coefficients, and without lower order terms. prop 5. Let L be the elliptic operator defined in [START_REF] Alberti | Generalized N-property and Sard theorem for Sobolev maps[END_REF]. In addition to (2), (3) if = 1 and (4) if = 0, assume that a(x) is a symmetric matrix for every x ∈ Ω and, if d ≥ 3, a ∈ C 0,1 Ω; R d×d . Take x 0 ∈ Ω. Let r > 0 and u 1 , . . . , u n ∈ C ,α (Ω; R) be solutions to the constant coefficient problem -div (a (x 0 ) ∇u i ) = 0 in B(x 0 , r), i = 1, . . . , n.

If (11) det    ζ (u 1 ) . . . ζ (u n )    (x 0 ) = 0, then x 0 ∈ C(Ω).
Remark 6. This result can in some cases be extended to operators L with piecewise Lipschitz coefficients with possibly countably many pieces, following the strategy given in [START_REF] Ball | On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients[END_REF].

Proof. This result, even though not in this exact form, was first derived in [START_REF] Bal | Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions[END_REF], and later discussed in [9, Section 7.3] (only in the case x 0 ∈ Ω). Here we provide only a sketch of the proof in order to highlight the main features; the reader is referred to the references mentioned for the details of the argument. The proof is split into three steps.

Step 1: approximation of u i with local solutions v i to Lv i = 0. Using standard elliptic regularity estimates, it is possible to find r ∈ (0, r] and v i ∈ H 1 (B(x 0 , r)∩Ω) such that Lv i = 0 in B(x 0 , r) ∩ Ω and u i -v i C 1 (B(x0,r)∩Ω) is arbitrarily small (provided that r is chosen small enough). It is worth observing that, even if in [START_REF] Bal | Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions[END_REF] the lower order terms are kept in the PDE with constant coefficients, that is not needed [START_REF] Alberti | Lectures on elliptic methods for hybrid inverse problems[END_REF]Proposition 7.10].

Step 2: approximation of v i with global solutions w i to Lw i = 0. Thanks to the regularity assumptions on the coefficients, the elliptic operator L enjoys the unique continuation property [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF]. This is equivalent to the Runge approximation property [START_REF] Lax | A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations[END_REF], by which it is possible to approximate local solutions with global solutions. Thus, in our setting, there exist w i ∈ H 1 (Ω) solutions to Lw i = 0 in Ω such that w i -v i H 1 (B(x0,r)∩Ω) is arbitrarily small. By elliptic regularity, we can ensure that w i -v i C 1 (B(x0,r/2)∩Ω) is arbitrarily small too.

Step 3: (w 1 , . . . , w n ) satisfy the constraint in x 0 . Combining the previous steps, we have that u i -w i C 1 (B(x0,r/2)∩Ω) is arbitrarily small. For the maps ζ considered above, this immediately implies that |ζ(u i ) -ζ(w i )|(x 0 ) is arbitrarily small. Thus, by [START_REF] Alessandrini | Global stability for a coupled physics inverse problem[END_REF], r and w i may be chosen in such a way that det

   ζ (w 1 )
. . .

ζ (w n )    (x 0 ) = 0, which shows that x 0 ∈ C(Ω).
Let us now verify that for the maps ζ mentioned above, we always have C(Ω) = Ω; in other words, the assumptions of Theorem 3 are satisfied with K = Ω. cor 6. Let L be the elliptic operator defined in [START_REF] Alberti | Generalized N-property and Sard theorem for Sobolev maps[END_REF]. In addition to (2), (3) if = 1 and (4) if = 0, assume that a(x) is a symmetric matrix for every x ∈ Ω and, if

d ≥ 3, a ∈ C 0,1 Ω; R d×d . If ζ is one of the maps considered in Example 1, then C(K) = K for any K ⊆ Ω.
Proof. We consider the three constraints separately:

• n = 1, = 0, ζ (u) = u: set u 1 = 1. • n = d, = 1, ζ (u) = ∇u: set u 1 = x 1 , . . . , u n = x n . • n = d + 1, = 1, ζ (u) = u ∇u : set u 1 = 1, u 2 = x 1 , . . . , u n+1 = x n .
Given x 0 ∈ Ω, for any a(x 0 ), there holds

-div(a(x 0 )∇u i ) = 0 in R d , i = 1, . . . , n, and det    ζ (u 1 ) . . . ζ (u n )    (x 0 ) = 0.
The conclusion follows from Proposition 5.

Proof of Theorem 3

We need two lemmata. For k ≥ 2 and a ∈ R k-1 let P a : R k → R k-1 denote the linear map given by P a (y) = (y 1 -a 1 y k , . . . , y k-1 -a k-1 y k ) .

In the following, we shall identify the matrices in R k×n with k rows and n columns with the linear maps from R n into R k . We shall denote the Lebesgue measure in R m by | • | m . lem 7. Take a smooth and compact set K ⊆ R d and a positive integer k > d+n α . Let F : K → R k×n be of class C 0,α and such that

F x : R n → R k is injective for all x ∈ K. Let G ⊆ R k-1 be the set of those a ∈ R k-1 for which P a • F x : R n → R k-1 is injective for all x ∈ K, namely G = x∈K a ∈ R k-1 : P a • F x : R n → R k-1 is injective . Then |R k-1 \ G| k-1 = 0.
Proof. Note that ker P a = span{(a 1 , . . . , a k-1 , 1)}. Thus, since F x is injective, we have for

x ∈ K P a • F x is injective ⇐⇒ ran F x ∩ ker P a = {0} ⇐⇒ (a 1 , . . . , a k-1 , 1) / ∈ ran F x .
Then, a ∈ R k-1 \ G if and only if there exists x ∈ K such that (a 1 , . . . , a k-1 , 1) ∈ ran F x , namely (a 1 , . . . , a k-1 , 1) ∈ ∪ x∈K ran F x . Therefore, using the projection

π : R k → R k-1 , (b 1 , . . . , b k ) → (b 1 , . . . , b k-1 ), we can express R k-1 \ G as R k-1 \ G = π (B) , B = x∈K ran F x ∩ b ∈ R k : b k = 1 .
Hence, it remains to prove that

H k-1 (B) = 0,
where H k-1 denotes the Hausdorff measure of dimension k -1.

Consider the map

f : K × R n → R k , (x, v) → F x v.
By construction, ran f = x∈K ran F x . Note that dim(K × R n ) = d + n < αk, so that H αk (K × R n ) = 0. Thus, since f is of class C 0,α , by Proposition 10 applied to f N = f | K×B(0,N ) for N ∈ N, we obtain

H k x∈K ran F x = H k (ran f ) = H k N ∈N ran f N ≤ N ∈N H k (ran f N ) = 0.
By linearity of v → F x v, the set x∈K ran F x is closed under scalar multiplication, and so x∈K ran

F x ⊇ R + • B, which implies H k (R + • B) = 0.
Using the change of variables formula and Tonelli theorem, we deduce H k-1 (B) = 0, as desired.

The following result is in the spirit of Whitney's projection argument, and states that, given k solutions satisfying the necessary constraints, it is possible to reduce the number to k -1 by taking suitable linear combinations, provided that k is big enough. . . .

ζ (u k )    (x) = n, x ∈ K. Let G ⊆ R k-1 be the set of those a ∈ R k-1 such that rank      ζ (u 1 -a 1 u k ) ζ (u 2 -a 2 u k ) . . . ζ (u k-1 -a k-1 u k )      (x) = n, x ∈ K. Then |R k-1 \ G| k-1 = 0.
Proof. Let F : K → R k×n be the map of class C 0,α defined by

F x =    ζ (u 1 ) . . . ζ (u k )    (x) : R n → R k .
By assumption, we have that F x is injective for all x ∈ K. Observe that, by linearity of ζ, we have

P a • F x =    ζ (u 1 ) (x) -a 1 ζ (u k ) (x) . . . ζ (u k-1 ) (x) -a k-1 ζ (u k ) (x)    =    ζ (u 1 -a 1 u k ) . . . ζ (u k-1 -a k-1 u k )    (x)
and so the conclusion immediately follows by Lemma 7.

We are now ready to prove Theorem 3.

Proof of Theorem 3.

Step

1: E(K) is open in H(K) [ d+n α ] . Take u ∈ E(K). By definition, we have [ d+n α ] i1,...,in=1 | det    ζ (u i1 ) . . . ζ (u in )    (x)| > 0, x ∈ K.
Since ζ(u ij ) are C 0,α maps in K, and in particular continuous, and K is compact, we have that

[ d+n α ] i1,...,in=1 | det    ζ (u i1 ) . . . ζ (u in )    (x)| ≥ C, x ∈ K, for some constant C > 0. Finally, since the map ζ is continuous itself, if v ∈ H(K) [ d+n α ] is chosen close enough to u we have [ d+n α ] i1,...,in=1 | det    ζ (v i1 ) . . . ζ (v in )    (x)| ≥ C 2 , x ∈ K,
which implies v ∈ E(K). This concludes the first step.

Step 2:

E(K) is dense in H(K) [ d+n α ] . Take H = (h 1 , . . . , h [ d+n α ] ) ∈ H(K) [ d+n α ] .
By assumption, for all x ∈ K there exist u 1,x , . . . , u n,x ∈ H(K) such that

| det    ζ (u 1,x ) . . . ζ (u n,x )    (x)| > 0.
By continuity of ζ (u i,x ), there exist neighbourhoods U x x such that

| det    ζ (u 1,x ) . . . ζ (u n,x )    (y)| > 0, y ∈ U x ∩ K.
Since K ⊆ ∪ x∈K U x , by compactness there exist x 1 , . . . , x N ∈ K such that K ⊆ ∪ N j=1 U xj . Thus, for all x ∈ K there exists j ∈ {1, . . . , N } such that

| det    ζ u 1,xj . . . ζ u n,xj    (x)| > 0.
Consider all the M = nN corresponding solutions

(u 1 , . . . , u M ) = (u 1,x1 , . . . , u n,x1 , u 1,x2 , . . . , u n,x2 , . . . , u 1,x N , . . . , u n,x N ) , so that rank    ζ(u 1 ) . . . ζ(u M )    (x) = n, x ∈ K.
In particular, we have rank

          ζ(h 1 ) . . . ζ(h [ d+n α ] ) ζ(u 1 ) . . . ζ(u M )           (x) = n, x ∈ K.
By Lemma 8, since k = d+n α + M > d+n α , we have k > d+n α , and so for almost every a ∈ R k-1 we have rank

           ζ(h 1 -a 1 u M ) . . . ζ(h [ d+n α ] -a [ d+n α ] u M ) ζ(u 1 -a [ d+n α ]+1 u M ) . . . ζ(u M -1 -a k-1 u M )            (x) = n, x ∈ K.
Repeating this argument M times (as long as k > d+n α ) with very small weights a, we obtain that there exist ξ i,j ∈ R (i = 1, . . . , d+n α , j = 1, . . . , M ) which can be chosen arbitrarily small such that rank    ζ(h 1 -ξ 1,j u j ) . . .

ζ(h [ d+n α ] -ξ [ d+n α ],j u j )    (x) = n, x ∈ K,
where we used Einstein summation convention of repeated indices. This implies that (h 1 -ξ 1,j u j , . . . , h [ d+n α ] -ξ [ d+n α ],j u j ) ∈ E(K), which, since the weights ξ i,j are chosen arbitrarily small, concludes the proof.

Appendix A. Hausdorff measure and Hölder functions

We recall an elementary property of Hausdorff measures that is used in the proof of Lemma 7. For further details, the reader is referred to [1, In particular, if H αs (A) = 0, then H s (f (A)) = 0.

Proof. Fix δ > 0 and let Taking infima over all such sets {C i } ∞ i=1 , we find H s cδ α (f (A)) ≤ c s 2 αs γ(s) 2 s γ(αs) H αs δ (A).

{C i } ∞ i=1 ⊆ R m be such that diam C i ≤ δ, A ⊆ ∪ ∞ i=1 C i . Then diam f (C i ∩ B) ≤ c (diam C i ) α ≤ c δ α and f (A) ⊆ ∪ ∞ i=1 f (C i ∩ B) . Thus
Taking the limit as δ → 0 the result follows.

lem 8 .

 8 Take a smooth and compact set K ⊆ Ω and a positive integer k > d+n α . Let u 1 , . . . , u k ∈ H(K) be such that rank

2 Γ( s 2

 22 section 4.1] or [41, section 2.4.1].Definition 9.Let A ⊆ R m , 0 ≤ s < ∞, 0 < δ ≤ ∞. We write +1) and Γ is the gamma function. The quantityH s (A) := lim δ→0 H s δ (A) = sup δ>0 H s δ (A)is the s-dimensional Hausdorff measure of A.prop 10. Take B ⊆ R m , s ∈ [0, +∞), c > 0 and α ∈ (0, 1]. Let f : B → R k be such that |f (x 1 ) -f (x 2 )| ≤ c|x 1 -x 2 | α , x 1 , x 2 ∈ B.ThenH s (f (A)) ≤ c s 2 αs γ(s) 2 s γ(αs) H αs (A), A ⊆ B.

  H s cδ α (f (A)) ≤ ∞ i=1 γ(s) diam f (C i ∩ B) 2 s ≤ c s 2 αs γ(s) 2 s γ(αs) ∞ i=1 γ(αs) diam C i 2 αs .

Acknowledgements

The authors are grateful for the anonymous referees' comments and suggestions. GSA is member of the "Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni" (GNAMPA) of the "Istituto Nazionale per l'Alta Matematica" (INdAM) and is supported by a UniGe starting grant "curiosity driven".


André Bontemps

Bruno Agostini

Nadia Caney

FLOW BOILING IN MINICHANNELS

Introduction

The use of mini-channel heat exchangers (hydraulic diameter about 1 mm) in compact heat exchangers improves heat transfer coefficients, and thermal efficiency while requiring a lower fluid mass. They are widely used in condensers for automobile air-conditioning and are now being used in evaporators, as well as in other applications such as domestic air-conditioning systems. However, more general use requires a better understanding of boiling heat transfer in confined spaces. Many definitions of micro and minichannel hydraulic diameter are used throughout the literature. [START_REF] Kandlikar | Evolution of microchannel flow passages -thermohydraulic performance and fabrication technology[END_REF] proposed the following classification: conventional channels (D h > 3 mm), minichannels (200 µm < D h < 3 mm), micro-channels (D h < 200 µm), that will be used throughout this paper.

These definitions rely upon the molecular mean free path in a single-phase flow, surface tension effects and flow patterns in two-phase flow applications. In recent studies in minichannels the hydraulic diameter ranges from 100 µm to 2-3 mm. The channel cross sections were either circular or rectangular and much of the research concerned boiling. Commonly, classical correlations have been used with or without modifications to predict flow boiling results in minichannels. However agreement was poor and the need for new correlations was evident.

It has been shown through a number of experiments that boiling is controlled by two additive components: nucleate boiling and convective boiling. Nucleate boiling is due to nucleating bubbles and their subsequent growth and removal from the heated surface. Convective boiling is due to heated fluid moving from the heated surface to the flow core. These two mechanisms cannot be separated with any precision since they are closely interconnected. Figure 1 [START_REF] Collier | Convective boiling and condensation[END_REF].

(i) In subcooled boiling the average fluid temperature is below the saturation temperature while the fluid at the tube wall has already reached it and therefore can boil. The heat transfer coefficient rises and depends on the heat flux, until the core of the flow, which is colder, reaches the saturation temperature. Bubbles formed at 1 the wall move and condense in the flow core and increase its temperature.

(ii) In saturated boiling the flow core has reached saturation. Nucleate and convective boiling compete. In nucleate boiling the heat transfer coefficient depends on the heat flux which is the driving force of bubble generation (dashed lines). In convective boiling the heat transfer coefficient is independent of the heat flux but depends on the liquid quality and mass velocity which are the driving forces of convection. The combination of both shows almost horizontal and parallel lines at low vapour quality (nucleate boiling) which merge into a single increasing line at higher vapour quality (convective boiling). The smaller the heat flux the sooner (in terms of vapour quality) convective boiling will take over from nucleate boiling. This is further highlighted by figure 2 which represents experimental results on flow boiling regimes in tubes.

(iii) At high quality and heat flux, dry-out can occur. This is a dramatic outcome of boiling. The liquid layer wetting the wall and providing heat transportation is totally vaporised. Only gas remains which severely decreases the heat transfer from the wall. With imposed heat flux this can lead to tube meltdown.

(iv) Finally, when all the liquid is vaporised, single-phase gaseous flow governs the heat transfer with, of course, a heat transfer coefficient smaller than for a single-phase liquid flow.

The main difficulty is to establish the dependence of the heat transfer coefficient on vapour quality in relation to different mechanisms controlling flow boiling. Some correlations do not take into account the two mechanisms. Others account for convective and nucleate boiling. To the present author's knowledge, none take into account the influence of channel size. The aim here is to summarise recent work on flow boiling, to describe an experiment on the phenomenon in minichannels and to compare the results with classical correlations. (1996).

Review of selected flow boiling correlations for minichannels

For an extended review of experimental work on mini and microchannels, the reader is refered to the [START_REF] Thome | Boiling in microchannels: a review of experiment and theory[END_REF] and [START_REF] Kandlikar | Fundamental issues related to flow boiling in minichannels and microchannels[END_REF] papers. This brief review covers a representative selection of heat transfer studies in minichannels and its aim is to illustrate the tendencies observed in the presented data. Recently [START_REF] Kandlikar | An extension of the flow boiling correlation to transition, laminar, and deep laminar flows in minichannels and microchannels[END_REF] developed a new general correlation adapted to minichannels which gives very good results for low qualities but fails to take dry-out into account, as noted by the author in question. Lately [START_REF] Thome | Heat transfer model for evaporation in microchannels. part I: presentation of the model[END_REF] and [START_REF] Dupont | Heat transfer model for evaporation in microchannels. part II: comparison with the database[END_REF] proposed a semi-empirical three zone model which is the only published work to predict the unique trends observed in minichannels. In this model the dominant boiling mechanism is the evaporation of the liquid film pressed under confined bubbles.

A few studies on boiling in minichannels are available in the literature. The experimental conditions are gathered in table 1. [START_REF] Tran | A correlation for nucleate flow boiling in small channels[END_REF], [START_REF] Aritomi | Thermohydraulics of boiling two-phase flow in high conversion light water reactors (thermohydraulics at low velocities)[END_REF] and [START_REF] Kew | Correlations for the prediction of boiling heat transfer in small diameter channels[END_REF] established correlations of heat transfer coefficient for various refrigerants. All noted that the local heat transfer coefficient was only dependent on the heat flux. Accordingly they concluded that the governing mechanism was nucleate boiling and no dependence on quality was considered. Recently, [START_REF] Huo | Flow boiling and flow regimes in small diameter tubes[END_REF] studied boiling of refrigerant R134a in minichannels and highlighted the prevalence of nucleate boiling and the occurrence of dry-out at low vapour quality. [START_REF] Kew | Correlations for the prediction of boiling heat transfer in small diameter channels[END_REF] defined a non dimensional confinement number, Co, and proposed that microscale boiling should prevail for Co > 0.5 while macroscale boiling would occur for Co < 0.5. However some experimental studies in similar geometries show a dependence of the heat transfer coefficient on vapour quality. The [START_REF] Feldman | De l'ébullition en convection forcée dans des canaux d'échangeurs à plaques et ailettes[END_REF], [START_REF] Oh | Heat transfer characteristics of R134a in a capillary tube heat exchanger[END_REF] and [START_REF] Kandlikar | Evolution of microchannel flow passages -thermohydraulic performance and fabrication technology[END_REF] correlations illustrate a clear evolution of the heat transfer coefficient with vapour quality. These works indicate that nucleate boiling may not be the only mechanism governing boiling in minichannels and that new mechanisms may happen too. Another phenomenon experimentally identified in flow boiling is the oscillatory nature of the flow. Some intermittent local dry-out can occur in confined spaces. This occurrence certainly influences the evolution of the heat transfer coefficient at high vapour quality. [START_REF] Brutin | Experimental study of unsteady convective boiling in heated minichannels[END_REF] experimentally investigated two-phase flow instabilities in narrow channels. They observed vapour slug formation blocking the two-phase flow and pushing it back to the inlet. [START_REF] Kandlikar | Evolution of microchannel flow passages -thermohydraulic performance and fabrication technology[END_REF] observed periodic slug flow with quick dry-out and re-wetting. This phenomenon occurs faster and lingers longer when the heat flux increases. However the exact influence on the heat transfer coefficient has not been quantified yet.

Consequently there is no clear indication that boiling phenomena in small diameter channels is either dominated by nucleate boiling or convective boiling or any new mechanism. Yet there is strong evidence that these mechanisms are not interconnected as in conventional tubes. The motivation for the present work is therefore to get a more accurate vision of boiling in minichannels, to establish a correlation for flow boiling in minichannels and consequently to identify the most adapted correlation in the literature. To this end, experimental results of ascendant boiling flow of refrigerant R134a obtained with two minichannels, whose hydraulic diameters are 2 mm and 0.77 mm, will be presented and discussed.

Experimental set-up

Figure 3 is a schematic of the R134a experimental facility and test section. The test loop included a liquid pump and a mixed glycol-water circuit for heat evacuation. Subcooled liquid enters the bottom inlet manifold, is then vaporised in the test section and condensed further on in the heat exchanger. The test section consisted of a vertical industrial MPE (MultiPort Extruded) aluminium tube composed of parallel rectangular channels. The whole test section was thermally insulated with wrapping foam. For heat transfer measurements, a section of the tube was heated by Joule effect with the passage of an electric current from two brased electrodes through the tube wall. Upstream of the heated region there was an adiabatic zone to ensure the flow was hydrodynamically developed. Experimental conditions are summarised in table 2.

The determination of the channel dimensions was carried out using scanning electron microscopy. The hydraulic diameter was calculated with the total flow area and wet perimeter measured from electron microscope images in order to take into account the effect of the first and last channels which are rounded. Roughness measurements were also carried-out.

Figure 3 shows the test section and instrumentation. Ten wall temperatures on the tube external surface were measured with 0.5 mm diameter calibrated type E thermocouples electrically insulated from the aluminium. Fluid inlet and outlet temperatures were measured with 1 mm diameter calibrated type K thermocouples. Calibration was carried out with a Rosemount 162-CE platinum thermometer. Due to the high thermal conductivity of the aluminium and the low thickness of the tube walls the measured temperature is very close to the wall temperature in contact with the fluid (the difference less than 0.01 K). The inlet fluid pressure was measured with a calibrated Rosemount type II absolute pressure sensor. Two calibrated differential pressure sensors measured the pressure loss through the test section. A Rosemount Micro-motion coriolis flowmeter was used to measure the mass flow rate of R134a downstream of the pump. The heating voltage and current were measured directly through a HP 3421A multiplexer. The overall system was tested with single-phase flow runs in order to check heat losses [START_REF] Agostini | Friction factor and heat transfer coefficient of R134a liquid flow in mini-channels[END_REF] and [START_REF] Agostini | Étude expérimentale de l'ébullition en convection forcée de fluide réfrigérant dans des minicanaux[END_REF]). Classical turbulent single-phase flow correlations agreed with measurements to within ±10%.

The heat flux was varied for every fixed mass flow rate in order to obtain a series of outlet vapour qualities between 0.2 and 1 with a step of 0.05. Steady state values were monitored using a Hewlett Packard 3421A with a 30 minutes time lapse between each mass flow rate or heat flux change. Averaging was carried out after every 20 values and uncertainties were calculated according to the [START_REF] Kline | Describing uncertainties in single-sample experiments[END_REF] method. The total electrical power dissipated in the test section was calculated as the product of voltage and current. The variations of R134a thermophysical properties with temperature were calculated with the REFPROP 6.01 software.

∆P ∆ ∆

± 3 % T w T T , T fl T T (K) 276-308 ± 0.1-3 K 281-307 ± 0.2-2 % p i (kPa) 405 & 608 ± 4 % 517 ± 2.7 % ∆p (kPa)
9.5-37.5 ± 1.2-17 % 153-1570 ± 0.3-3.7 %

Heat transfer results

In this section the general trends of the measurements will be presented. Electrical power, refrigerant mass flow rate, in and outward fluid temperatures and wall temperatures were measured. From these data physical parameters of interest were computed as a function of tube length: heat flux q(z), fluid temperature T fl T T (z), vapour quality x(z) and heat transfer coefficient α(z). Table 3 presents uncertainties on some calculated parameters.

Figures 4 and5 show the local heat transfer coefficient as a function of local quality for D h = 2 mm and D h = 0.77 mm. Two general trends are observed. On figure 4 a strong decrease in the heat transfer coefficient with vapour quality is visible when the vapour quality is greater than a "critical quality" whereas on figure 5 the heat transfer coefficient starts decreasing before increasing with vapour quality. Figure 6 shows the local heat transfer coefficient as a function of local quality, for a given heat flux and mass velocity, for D h = 2 mm and D h = 0.77 mm. It is clear that the heat transfer coefficient increases when the hydraulic diameter decreases. Thus the local heat transfer coefficient is increased by 74 % ± 26 % when the hydraulic diameter is decreased by 62 %.

The enhancement ratio can also be written D -0.6±0.1 h which is, given the uncertainties, close to the values proposed by [START_REF] Ishibashi | Saturated boiling heat transfer in narrow spaces[END_REF] (D -0.67 h ) and [START_REF] Aritomi | Thermohydraulics of boiling two-phase flow in high conversion light water reactors (thermohydraulics at low velocities)[END_REF] (D -0.75 h ). It is not clear how [START_REF] Tran | A correlation for nucleate flow boiling in small channels[END_REF], who proposed D -1 h for the confinement effect, established this expression. The range of tested hydraulic diameters, 2.4-2.92 mm, is too small to build a correlation. It is possible, from their articles, that it comes from an analogy with single-phase flow classical theory and has no experimental basis. On the contrary the D -0.4 h term proposed by [START_REF] Steiner | Flow boiling heat transfer in vertical tubes correlated by an asymptotic model[END_REF] is based on some experimental data. Nevertheless the data concerning refrigerant fluids (R11 and R113) were performed for hydraulic diameters from 7 to 20 mm only. The lowest hydraulic diameters (from 1 to 5 mm) were tested with Helium I only so that this D -0.4 h factor is difficult to compare with the present results. Figure 7 represents the ratio of the measured average heat transfer coefficient to that predicted by the Shah (1976) correlation for conventional tubes, as a function of the heat flux. This figure shows the global intensification of heat transfer in MPE minichannels compared with conventional tubes. This intensification ranges from 0 to 400 % depending on the heat flux and occurs up to 35 kW/m 2 for D h = 2 mm and 20 kW/m 2 for D h = 0.77 mm.

α (W/m α 2 K) 2 x x m = m = q = q = • • • • kg/m s kg/m s kW/m kW/
(kW/m ) q (kW/m ) q 2 2 q • ) (kW/ q m m • (kg/m s) (kg/m s) 2 2 • ) (k /

Heat transfer analysis

An analysis and a physical interpretation of these observations will now be proposed. In order to analyse and classify the different heat transfer coefficient behaviours observed on figures 4 and 5 it is useful to represent, for a given vapour quality, the heat transfer coefficient as a function of the heat flux and the heat flux as a function of the wall-fluid temperature difference. For D h = 2 mm, since Co < 0.5, the results were analysed in terms of macroscale boiling. This was done on figure 8, which exhibits two trends:

(i) For T w T T -T sat T T < 3 K and q < 14 kW/m 2 , q is proportional to T w T T -T sat T T . Thus α is independent of q and moreover decreases with ṁ. This region may correspond to a convective boiling regime and, as will be further highlighted, the decrease with ṁ may be due to the occurrence of partial dry-out.

(ii) For T w T T -T sat T T > 3 K and q > 14 kW/m 2 , q is proportional to (T w T T -T sat T T ) 3 , therefore α is proportional to q2/3 , and the heat transfer coefficient depends only weakly on ṁ. This second region may be identified as a nucleate boiling regime.

However, as Co is very close to 0.5, these results may also be interpreted in terms of microscale boiling with the film evaporation mechanism proposed by [START_REF] Thome | Heat transfer model for evaporation in microchannels. part I: presentation of the model[END_REF].

For D h = 0.77 mm q is always proportional to T w T T -T sat T T and α is independent of ṁ. Since Co is greater than 0.5, microscale boiling should prevail and according to the three zone model of [START_REF] Thome | Heat transfer model for evaporation in microchannels. part I: presentation of the model[END_REF] film evaporation would be the boiling mechanism occurring in this tube. From an analysis conducted on figures 4 and 5 with the dimensionless boiling number, the following tendencies can be outlined. For D h = 2 mm: (i) for Bo > 4.3 • 10 -4 and x < 0.3-0.4, the heat transfer coefficient is weakly dependent on x and proportional to q2/3 . Thus the nucleate boiling regime might governs this region.

α (W/m K) α W W 2 q q T • • w T T (W/m ) q (W/m ) q -T w W W W W fl T T (K) l x = 0.2 2 2 m • (kg/m s) m
(ii) for Bo > 4.3 • 10 -4 and x > 0.3-0.4, the heat transfer coefficient decreases with x but is still proportional to q2/3 . This suggests that partial dry-out occurs with nucleate boiling which is confirmed on figure 9 where the wall temperature and the statistical uncertainty on T w T T suddenly rise for x > 0.3-0.4. Most of the data belong to these two regions.

(iii) for Bo < 4.3 • 10 -4 the heat transfer coefficient is weakly dependent on x and proportional to q2/3 for low qualities. It then starts increasing with vapour quality when x is greater than a transition value. This transition value is all the greater since the heat flux is high for a given mass velocity. This behaviour may correspond to competition between a convective boiling and a dry-out regime where partial dry-out and regeneration of the liquid layer occur. Furthermore it was found that this transition occurred for a constant value of the product Bo • (1x) equal to 2.2 × 10 -4 . These results are in agreement with the [START_REF] Huo | Flow boiling and flow regimes in small diameter tubes[END_REF] study which highlighted the prevalence of heat flux dependent boiling and early dry-out in a 2 mm diameter tube with refrigerant R134a. In their work the boiling number was always greater than 8 × 10 -4 which is coherent with the present results.

For D h = 0.77 mm, Bo is always smaller than 2.2 • 10 -4 and (i) for x < 0.1-0.2, the heat transfer coefficient increases weakly with x and is independent of of q. Film evaporation seems to dominate and the thinning of the liquid layer could explain the increase.

(ii) for x > 0.1-0.2, the heat transfer coefficient always decreases with x and is still independent of q. This suggests that intermittent dry-out governs the boiling alongside with film evaporation.

It was found that the critical vapour quality x cr did not depend on q and ṁ. This does not mean that x cr does not depend on q or ṁ but simply that such a variation is less than the uncertainty. The present study highlights a decrease in x cr from 0.3-0.4 to 0.1-0.2 when D h decreases from 2 to 0.77 mm. For comparison, [START_REF] Huo | Flow boiling and flow regimes in small diameter tubes[END_REF] found that x cr decreased from 0.4-0.5 to 0.2-0.3 when D h decreased from 4.26 to 2.01 mm. Given that the uncertainty on x cr in the present study is about 30 % the results are quite close.

Examination of figure 9 strongly suggest the occurrence of early dry-out. When the liquid layer disappears from the tube wall, the heat transfer coefficient suddenly decreases because of the lesser heat transport properties of the gas. This implies a wall temperature rise since heat removal is less efficient. Moreover, it is well know that dry-out is an intermittent phenomenon and that liquid drops regularly hit the tube wall, so that the tube wall temperature is submitted to quick changes. For the authors this explains why the uncertainty on the tube wall temperature also increases tenfold when dry-out occurs. Furthermore, the occurrence of early dry-out may be explained by the thinning of the liquid layer due to bubbles confinement. This hypothesis also explains why the critical vapour quality decrease from 0.4 to 0.2 when the hydraulic diameter decreases from 2 to 0.77 mm since this increases bubble confinement. Moreover, this thinning of the liquid layer also explains why the heat transfer coefficient increases when the hydraulic diameter decreases because the heat transfer resistance due to this liquid layer also decreases as long as dry-out does not occur.

The occuring of dry-out may explain why, when combined with heat flux dependent boiling, the heat transfer coefficient decreases with the mass velocity. The greater the mass velocity, the more probable dry-out should be, because the liquid film is increasingly dragged from the wall due to shear stress. Thus dry-out should cause a decrease in the heat transfer coefficient with the mass velocity. To explain why the boiling number seems to govern the transition between heat flux increasing α and vapour quality increasing α, the following interpretation is proposed, based on macroscale boiling mechanisms. From the [START_REF] Rohsenow | A method of correlation heat-transfer data for surface boiling of liquids[END_REF] and [START_REF] Kew | Correlations for the prediction of boiling heat transfer in small diameter channels[END_REF] analysis, an inertial characteristic time τ cv for the liquid layer and a characteristic time τ b for bubbles leaving the wall can be defined. Then, from the [START_REF] Kutaleladze | Principal equations of thermodynamics of nucleate boiling[END_REF] and [START_REF] Rohsenow | A method of correlation heat-transfer data for surface boiling of liquids[END_REF] analysis it can be shown that the ratio of these two characteristic times can be written:

τ cv τ b = f (θ, g, ρ l , ρ g , σ) • Bo • f (x). (1)
This ratio is a comparison of convective effects in the liquid layer (causing α to increase with x) and bubble dynamics at the wall (causing α to increase with q). Thus τ cv /τ b is proportional to Bo and a function of vapour quality so that the boiling number is the appropriate dimensionless number to study the transition between these two boiling regimes.

Finally, figure 10 illustrates the different boiling regimes in minichannels. This work suggests: (i) when Bo > 4.3•10 -4 and Co < 0.5, nucleate boiling and dry-out seems to govern boiling in minichannels. Moreover dry-out occurs at low vapour quality. Most of data for D h = 2 mm were in this case.

(ii) when 2.2 • 10 -4 < Bo < 4.3 • 10 -4 and Co < 0.5, nucleate boiling, convective boiling and dry-out seem to compete. The frontier between the nucleate boiling and convective boiling is Bo • (1x) = 2.2 • 10 -4 . Furthermore dry-out occurs also at low vapour quality and its effects are superimposed on those of nucleate boiling and convective boiling and competes with them.

(iii) when Bo < 2.2 • 10 -4 and Co > 0.5, boiling directly starts in the film evaporation regime with no heat transfer dependance since Bo • (1 -0) = 2.2 • 10 -4 . The heat transfer coefficient increases with vapour quality, and does not depend upon the mass velocity and heat flux, until dry-out occurs. Then boiling is totally governed by dry-out and the heat transfer coefficient decreases sharply with vapour quality and remains independent of the mass velocity and heat flux.

This scheme illustrates the difference between the classical boiling regimes representations like figures 1 and 2, and the present results on figure 10.

Correlating data

Most of the present data points belong to the heat flux dependent regime so that it has been possible to correlate the heat transfer coefficient in this region with ṁ, q and x. Finally, the following expressions were obtained.

For D h = 2 mm and Bo > 4.3 • 10 -4 ,

x < 0.3 -0.4 : α = 28 • q2/3 • ṁ-0.26 • x -0.10 and x > 0.3 -0.4 : α = 28 • q2/3 • ṁ-0.64 • x -2.08 , (2)
for D h = 0.77 mm and Bo < 2.2 • 10 -4 ,

x < 0.1 -0.2 : α = 10260 • x 0.15 and x > 0.1 -0.2 : α = 10260 • (1 -x) 1.57 .
(3) Equation ( 2), obtained by linear least squares fitting over 723 data points, predicts 95 % of our data in the ±30% range. Equation (3), obtained by linear least squares fitting over 825 data points, predicts 85 % of our data in the ±30% range.

Figure 11 compares the ability of various correlations to predict the present data for D h = 2 mm. The analytical expressions of these correlations have been reported in table 4. The [START_REF] Tran | A correlation for nucleate flow boiling in small channels[END_REF] and [START_REF] Kandlikar | An extension of the flow boiling correlation to transition, laminar, and deep laminar flows in minichannels and microchannels[END_REF] correlations, proposed for minichannels predicts the present data rather well in the pure heat flux dependent regime but fails as soon as dry-out occurs. The [START_REF] Steiner | Flow boiling heat transfer in vertical tubes correlated by an asymptotic model[END_REF] correlation over-predicts the present data since it includes a D -0.4 h diameter correction term which is not well fitted for such small diameters. On the contrary the [START_REF] Shah | A new correlation for heat transfer during boiling flow through pipes[END_REF], [START_REF] Liu | A general correlation for saturated and subcooled flow boiling in tubes and annuli, based on a nucleate pool boiling equation[END_REF] correlations under-predict the present data because they do not take into account any confinement phenomenon as suggested by Cornwell andKew (1992, 1995).

The [START_REF] Thome | Heat transfer model for evaporation in microchannels. part I: presentation of the model[END_REF] and [START_REF] Dupont | Heat transfer model for evaporation in microchannels. part II: comparison with the database[END_REF] three zone model was able to predict most of the trends observed, in particular for the 0.77 mm tube. It is not represented here since it implies the optimization of minimum and maximum liquid layer thickness and bubble generation frequency parameters. However the reader is invited to refer to the cited articles where this model predictions are compared with the present data. Shah ( 1976) [START_REF] Kandlikar | An extension of the flow boiling correlation to transition, laminar, and deep laminar flows in minichannels and microchannels[END_REF] eq. ( 2) In different studies identifying flow configurations in minichannels, fundamental configurations specific to minichannels are observed: isolated bubbles, confined bubbles and annular slug flow [START_REF] Kew | Correlations for the prediction of boiling heat transfer in small diameter channels[END_REF]). However, some authors observed flow regimes typical of macroscale tubes: bubbly, plug, slug, wavyannular and annular flow [START_REF] Kuwahara | Flow pattern of pure refrigerant HFC134a evaporating in a horizontal capillary tube[END_REF]). [START_REF] Triplett | Gas liquid two-phase flow in microchannels. Part I : two-phase flow patterns[END_REF] measured pressure drop and void fraction in minichannels with air-water adiabatic flows. They observed bubbly, churn, slug, slug-annular and annular flows as in conventional tubes, but the transitions were very different. Moreover they highlighted that the homogeneous model best predicted their pressure drop measurements for every flow configuration except the annular one. [START_REF] Huo | Flow boiling and flow regimes in small diameter tubes[END_REF] established a flow map for refrigerant R134a flowing in 2.01 mm and 4.26 mm diameter round vertical tubes. They observed six typical flow patterns, i.e. dispersed bubbles, bubbly, slug, churn, annular and mist. Significant differences were found with the existing models for normal size tubes. For example the churn flow pattern becomes a more important flow pattern compared to classical models where it shrinks to a very small area. The authors also observed that reducing the diameter shifted the transition of slug to churn and churn to annular to higher values of the gas velocity. Figure 12 presents the flow map proposed by [START_REF] Huo | Flow boiling and flow regimes in small diameter tubes[END_REF] for a 2.01 mm diameter tube. The data of the present work are reported and are all in the annular flow region. As a conclusion, the analysis in terms of flow chart does not allow us to find a clear relation with the heat transfer coefficient measurements.

measurements (a) (b) (c) (d) (e) (f) (g) (h) m = q = 30229

Pressure drop

A time averaging method was used in order to reduce wild pressure oscillations. In order to avoid non uniform distribution of coolant fluid only subcooled liquid entered the inlet manifold. Furthermore the engineering rule that the manifold diameter should be at last five times greater than the channel hydraulic diameter to equalise the fluid distribution was used. However, even if non uniform distribution occurs it will not affect the inlet and outlet measurements which are performed outside of the manifolds and it should not affect the local temperature measurements because of the averaging of wall temperatures across the N channels due to the very high thermal conductivity of the aluminium.

Figure 13 shows the two-phase pressure drop gradient versus the outlet quality. The solid lines represent the pressure gradient modelled with the homogeneous model. As shown in figure 13, the present measured pressure gradient is linear with x o . This is characteristic of preponderant frictional pressure losses since integration of the homogeneous model for uniform longitudinal heating, constant thermophysical properties and friction factor shows that the frictional part of the two-phase flow pressure drop is linear with the outlet quality.

However this result and the prevalence of heat flux dependent boiling for D h = 2 mm seem contradictory with the prevalence of the annular flow regime (see section 7.) which would rather suggest a separated phases model to calculate pressure losses. Yet, it is well known that the parietal heat transfer deeply disturbs the flow configuration because of the generation of bubbles at the wall. Thus the flow configuration might not be pure annular but actually slug-annular or churn flow. That would promote the mixing of liquid and vapour thus explaining the good predictions of the homogeneous model. Nevertheless this issue will be resolved only with a test section allowing flow visualisation and heat transfer at the same time.

Conclusions

Forced flow boiling heat transfer in minichannels in similar conditions as encountered in automobile air conditioners has been studied. Higher heat transfer coefficients than in conventional tubes are achieved but dry-out occurs at low vapour qualities thus decreasing performances. However the average heat transfer coefficient remains higher than in conventional tubes. These observations support literature studies which predict that bubble confinement leads to higher heat transfer coefficients and dry-out at low vapour quality in minichannels. The new [START_REF] Kandlikar | An extension of the flow boiling correlation to transition, laminar, and deep laminar flows in minichannels and microchannels[END_REF] general correlation for flow boiling in tubes was found to predict the present results before dry-out occurs. The [START_REF] Thome | Heat transfer model for evaporation in microchannels. part I: presentation of the model[END_REF] and [START_REF] Dupont | Heat transfer model for evaporation in microchannels. part II: comparison with the database[END_REF] three zone model for microscale boiling predicted most of the observed trends, including dry-out and the lack of mass velocity influence. Using it for predictions still requires testing over a consequent database. The effect of confinement on the heat transfer coefficient before dry-out was found to be an increase of 74% when the hydraulic diameter decreased from 2 to 0.77 mm. The effect of confinement on dry-out was found to be a decrease in the critical quality from 0.3-0.4 to 0.1-0.2 for the same reduction of the hydraulic diameter. Heat flux dependent boiling prevailed in the 2 mm hydraulic diameter tube while quality dependent boiling prevailed in the 0.77 hydraulic diameter tube because of the difference in boiling and confinement numbers. The transition from one regime to another occurred for Bo • (1x) ≈ 2.2 • 10 -4 regardless of the heat and mass velocity. Moreover it was found that dry-out could even be the dominant boiling mechanism at low qualities. The results obtained with the 2 mm hydraulic diameter tube were in total agreement with [START_REF] Huo | Flow boiling and flow regimes in small diameter tubes[END_REF]'s work. Finally frictional pressure losses seem to dominate up to mass velocities of 469 kg/m 2 s.

The choice of MPE tubes for the test section allowed easier measurements and results closer to industrial reality. Further studies should put the stress on the accurate influence of channel geometry and confinement on heat transfer with diabatic flow visualisation and a large variety of channels configurations.

Nomenclature

A fl total flow area (m 2 ) Bo = q/( ṁ • h lv )

Boiling number Co = (σ/(g • (ρ l -ρ v )) 0.5 /D h Confinement number Cv = ((1x)/x)

Figure 1 :

 1 Figure 1: Boiling regimes from Collier and Thome (1994).

Figure 2 :

 2 Figure 2: Boiling regimes observed[START_REF] Feldman | De l'ébullition en convection forcée dans des canaux d'échangeurs à plaques et ailettes[END_REF].

Figure 3 :

 3 Figure 3: Test loop and test section.

D

  h = 2 mm).

Figure 6 :

 6 Figure 6: Influence of confinement on heat transfer coefficients.

Figure 7 :

 7 Figure 7: Average heat transfer coefficient versus heat flux.

Figure 8 :

 8 Figure 8: α versus q and q versus T w T T -T sat T T (D h = 2 mm).

Figure 9 :

 9 Figure 9: Wall temperature and its uncertainty versus local quality (D h = 2 mm).

Figure 10 :

 10 Figure 10: Summing-up of boiling regimes in minichannels observed in the present work.

Figure 11 :

 11 Figure 11: Comparaison with litterature correlations.

Figure 12 :

 12 Figure 12: Flow regimes observed by Huo et al. (2004).

Figure 13 :

 13 Figure 13: Pressure drop versus outlet quality (D h = 2 mm).

Table 1 :

 1 Summary of experimental conditions of some studies on refrigerants flow boiling in minichannels

	Author	Fluid	q (kW/m 2 ) ṁ (kg/m 2 s) x o	D h (mm)
	Aritomi et al. (1993)	R113	10-100	31-620	0-0.8	1-4
	Feldman (1996)	R114	1.4-3.5	20-45	0 .1-0.6 1.66-2.06
	Kew and Cornwell (1997) R113,141b	9.7-90	188-212	0-0.8	1.04-3.69
	Tran et al. (1997)	R12,113,134a 0.75-129	44-832	0.2-0.8 2.46-2.92
	Oh et al. (1998)	R134a	10-20	240-720	0-1	1 -2
	Huo et al. (2004)	R134a	13-150	100-500	0-0.9	2.01-4.26
	Present study (2004)	R134a	2.8-31.6	90-469	0-1	0.77-2.01

Table 2 :

 2 Operating conditions and uncertainties

		value	error	value	error
	D h (mm)	2 .01	± 3 %	0.77	± 7 %
	ṁ (kg/m 2 s) 90-295	± 1.7-8.6 % 214-469	± 3.3-8.6 %
	q (kW/m 2 )	6-31.6	± 2-4.1 %	2.8-19.5

Table 3 :

 3 Uncertainties on calculated parameters

						value	error	value		error
		Co				0.46		± 3 %	1.14		± 7 %
		Bo × 10 -4		2.2-7.9	± 3.7-10.4 % 0.7-2.2	± 6.3-11.6 %
		α (kW/m 2 K) 0.8-10.3 ± 6-30 %	0.6-32.5 ± 5-30 %
		x o				0.26-1	± 1-7 %	0.23-0.9 ± 2-9 %
	D = 2 mm h								D = 0.77 mm h
	Bo > 4.3 10 -4			m • = 83 kg/m 2 s		Bo < 4.3 10 -4			m •	= 347 kg/m s 2
								2 10.8 kW/m 11.6		10000 11000				4.4 kW/m 2 5.3		7.6 8.4
								12.7		9000				6.1				8.7
	α α (W/m 2 K)							13.8 14.9 16.2	α α K) 2 (W/m	5000 6000 7000 8000				6.8				10.2 11.8 12.6 13.1 14.6
											4000
											3000
				x cr						2000	x cr
											1000
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8 0.9	1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8
						x									x

Figure 4: Local heat transfer coefficient versus local quality.

Table 4 :

 4 Expressions of correlations represented on figure 11.

  0.8 • (ρ v /ρ l ) 0.5 Convection number D h = 4A 4 4 fl /P fl

	P fl P P	total wet perimeter	(m)
	p	pressure	(Pa)
	p r	reduced pressure	(Pa)
	∆p	pressure loss	(Pa)
	q	heat flux	(W/m 2 )
	Re = ( ṁ • D h )/µ l	Reynolds number
	T	temperature	(K)
	v	velocity	(m/s)
	x	vapour quality
	z	z coordinate	(m)
	Greek letters
	α	heat transfer coefficient	(W/m 2 K)
	χ tt	Lockhart-Martinelli parameter
	ρ	mass density	(kg/m 3 )
	σ	surface tension	(N/m)
	τ	characteristic time	(s)
	θ	wetting angle
	Subscripts
	b	bubble
	cv	convective
	fl	fluid
	go	all gas flow
	i	inlet
	j	joule heated
	l	liquid
	lo	all liquid flow
	nb	nucleate boiling
	npb	nucleate pool boiling
	o	outlet
	onb	onset of nucleate boiling
	sat	saturation
	TP	two-phase
	v	vapour
	w	wall
	P P	hydraulic diameter	(m)
	L	tube length	(m)
	M	molecular weight	g/mol
	ṁ	mass velocity	(kg/m 2 s)
	N	number of channels

Author

Expression [START_REF] Shah | A new correlation for heat transfer during boiling flow through pipes[END_REF] α TP /α l = f (Bo, Cv) [START_REF] Cooper | Saturation nucleate boiling. A simple correlation[END_REF] α TP = 55 • p 0,12 r • q2/3 (log 10 p r ) -0,55 M-0,5 [START_REF] Liu | A general correlation for saturated and subcooled flow boiling in tubes and annuli, based on a nucleate pool boiling equation[END_REF] α 2 TP = α 2 lo + α 2 Cooper [START_REF] Steiner | Flow boiling heat transfer in vertical tubes correlated by an asymptotic model[END_REF] [START_REF] Kandlikar | An extension of the flow boiling correlation to transition, laminar, and deep laminar flows in minichannels and microchannels[END_REF] α TP = max(α nb , α cb ) Present study (2004) eq. ( 2)

Flow regimes

Different authors have identified various flow regimes in large channels. In both vertical and horizontal configurations these include bubbly, dispersed bubbly, slug, pseudo-slug, churn, annular, annular mist and dispersed droplet flows. An important difference in minichannels is that the liquid flow is preferentially laminar. Surface tension effects have more and more influence as the hydraulic diameter is reduced. Gravity becomes negligible compared to surface tension so that the orientation is less influential.


Benneth Sturm

Fernando Soldevila

Enrique Tajahuerce

Sylvain Gigan

Herve Rigneault

Hilton B De Aguiar
email: h.aguiar@phys.ens.fr

S Harmsen

R Huang

M A Wall

H Karabeber

J M Samii

M Spaliviero

J R White

S Mon- Ette

R O'connor

K L Pitter

S A Sastra

M Saborowski

E C Holland

S Singer

K P Olive

High-sensitivity high-speed compressive spectrometer for Raman imaging

Compressive Raman is a recent framework that allows for large data compression of microspectroscopy during its measurement. Because of its inherent multiplexing architecture, it has shown imaging speeds considerably higher than conventional Raman microspectroscopy. Nevertheless, the low signal-to-noise (SNR) of Raman scattering still poses challenges for high-sensitivity bio-imaging exploiting compressive Raman: (i ) the idle solvent acts as a background noise upon imaging small biological organelles, (ii ) current compressive spectrometers are lossy precluding high-sensitivity imaging. We present inexpensive high-throughput spectrometer layouts for high-sensitivity compressive hyperspectroscopy. We exploit various modalities of compressive Raman allowing for up to 80X reduction of data storage and 2X microspectroscopy speed up at a 230-nm spatial resolution. Such achievements allowed us to chemically image sub-diffraction-limited biological specimens (lipid bilayers) in few seconds.

I. INTRODUCTION

Raman microspectroscopy allows probing a chemically heterogeneous system with high spatial resolution and superb molecular selectivity. In particular, Raman-based imaging holds promising applications for biological specimens studies 1 , such as for cancer 2,3 and bacteria detection 4,5 , molecular biology imaging [6][7][8][9] , to cite few examples. Nevertheless, hyperspectroscopy (i.e. spatiallyresolved spectroscopy), in general, generates large data sizes which will eventually become a hurdle when bringing Raman spectroscopic imaging as a valuable tool for clinical tests, for example, for issues related to data storage and access. As Raman microspectroscopy acquisition speeds continuously advance 10 , it will become increasingly important to achieve data compression, in particular for dynamic and/or large-scale imaging at high-resolution.

Single-pixel-based compressive sensing technology has attracted a large interest recently 11 . The idea behind compressive sensing is to obtain the same information as in multi-pixel technologies approaches, however through a designed optical experiment that will compress the data at the acquisition stage, by undersampling. In a post-processing step, efficient computational methods retrieve the desired information accurately. Interestingly, single-pixel approaches are particularly powerful where multi-pixel technology is still lacking or challenging, such as in terahertz and pumpprobe spectroscopy methods 12,13 . Single-pixel compressive sensing is a very appealing approach for Raman imaging as it is inexpensive and may have very fast computational reconstruction times, allowing for real-time applications. Compressive microspectroscopy is based on spectral, or spatiospectral multiplexing 14 , which is achieved by fast programmable spatial light modulators, such as digital micromirror devices (DMD), operating at 10's kHz.

While compressive microspectroscopy [14][15][16][17][18][19][20][21][22][23][24][25][26] is a promising technology, it still lacks sensitivity for bio-imaging when compared to sensitive CCD cameras. High-sensitivity imaging is of particular relevance in the sub-micron spatial scale, as it is the size range of many organelle in a cell 7-9,27 , including lipid droplets, cell membranes, and membrane-less organelles. The challenges for highsensitivity compressive microspectroscopy are two-fold. First, apart from Raman scattering being a weak effect, the Raman signal (I R ) scales unfavourably with spatial resolution: I R ∝ ω 2 0 for objects that fill the focus (ω 0 being the focus size of the excitation laser). Therefore, upon lowresolution imaging, the overall signal increases but the sensitivity decreases for detecting objects that have a size smaller than the focus, as the surrounding solvent adds as a strong background.

Second, and related to compressive spectrometer layouts, there is a strong size mismatch (100X) between sensitive detector sizes and the size of the spectrum spread in the Fourier plane of a spectrometer (i.e. on the DMD). This mismatch is the main cause of sensitivity loss in compressive spectrometers.

Here, we exploit novel high-throughput and inexpensive spectrometer designs for high-resolution and high-sensitivity compressive Raman imaging of biological specimens. This is achieved by dispersion-controlled layouts to focus a large spectral bandwidth on a small single-pixel detector.

After discussion of the methods we propose, we showcase two compressive Raman modalities exploiting different levels of a priori knowledge from the system being imaged. We then demonstrate chemical imaging of sub-wavelength Raman sources using the compressive spectrometer.

II. METHODS

Confocal microscope

The compressive Raman microscope is based on a standard layout of an epi-detection Raman microscope (Fig. 1). The excitation laser (λ = 532 nm, 150 mW, Oxxius LCX-532) is cleaned up by a bandpass filter (FLH05532-4, Thorlabs), and expanded to about 7.2 mm, before entering an inverted microscope (Nikon Eclipse Ti-U). An additional half-waveplate and quarter-waveplate (not shown) pre-compensate and generate circularly polarized light 28 after the beam being reflect off from the dichroic filter (F38-532 T1, AHF) into a high-numerical aperture (NA), oil-immersion objective (Nikon 60X, 1.4 NA). Specimens are mounted on a translation-stage (Physik Instrumente P-545.3R7) for raster scanning. The inelastically backscattered Raman light (epi configuration) is focussed by the microscope tube lens on a multimode fiber (M50L02S-A, Thorlabs, 50 µm coresize), with a notch filter blocking residual pump light (NF533-17, Thorlabs), thus guiding the signal to the home-built compressive spectrometer.

Single-pixel compressive spectrometer

The home-built spectrometer is based on a traditional Czerny-Turner design. The signal coming out of the fiber is collimated by an achromatic lens L1 (f=19mm, AC127-019-A-ML, Thorlabs), which then impinges on a high-efficiency transmission grating (FSTG-VIS1379-911, Ibsen Photonics, 1379 l/mm) providing about 85 % diffraction efficiency around λ=600 nm. After the grating, the beam is spectrally dispersed and different colors are focused with an off-axis parabolic mirror PM (f=101.6 mm, 90 • off-axis angle, MPD149-P01, Thorlabs). The DMD module (V-7001, ViALUX, 0.7" XGA resolution, with V4395 controller board) selects the various wavelength to be detected. Typically, we bin about 8 pixels, thus having a macropixel size that is slightly below the spectral resolution of the spectrometer.

The signal from the DMD is steered into an avalanche photodiode operating in photon counting mode (SPCM-AQRH-44, Excelitas Technologies). This module combines high quantum efficiency from Si avalanche photodiodes (70 %), large dynamic range (low dark and high maximum count rate), with an active area diameter of 180 µm. The small area, while keeping high quantum efficiency, is of particular importance in the design because focusing the complete spectrum to the size of the detector-chip efficiently is not straightforward. We note other single-photon counters exist with mm's active area size, however, with lower quantum efficiency.

Sample preparation

Polystyrene particles (1 µm and 0.088 µm-diameter, Polysciences Inc.) were drop cast on a coverslip, then filled with water. Multilamellar vesicles (MLV) were prepared by drop-casting a film of lipid solution (10 mg/ml dipalmitoylphosphatidylcholine in chloroform) on a vial, then adding water and heat up at 38 • for about 30 min 29 . After spontaneous formation of the vesicles, an aliquot of the dispersion is then sandwiched between two coverslips.

RESULTS

Dispersion management for single-pixel compressed spectroscopy

The main challenge of a single-pixel compressive spectrometer is as follows. In a conventional spectrometer, a CCD camera is used to detect the dispersed input light, which size is typically few mm's. In contrast, the compressive spectrometer consists of a DMD, in place of the CCD, which steers wavelength bins onto a small single-pixel detector (100's µm) to achieve high-sensitivity.

Therefore, one has to considerably demagnify the beam reflected off by the DMD. Hence, we studied two strategies that could keep the interesting aspects (costs and throughput) of singlepixel detection.

We investigated two approaches that are presented in dashed box insets of Fig. 1. The first approach is based solely on lenses (lens-based), motivated by the fact that transmission of lenses are highest compared to other optical elements. To aid the discussion, we recall the Smith-Helmholtz invariant (yσ = y σ ) relating planes y (grating 1), ray angles σ (arising from grating dispersion before PM), to y (L3) and σ (arising from focusing with L2) of an optical system. As the grating plane y is imaged onto plane y by a PM and L2, we have:

σ σ = y y = f 2 f 1 = m. (1)
Eq. 1 shows that dispersions and magnifications can be interchanged. For instance, σ can be reduced by simply increasing the magnification m. However, the maximum magnification is limited by the clear aperture of L3. The second approach is based on a second grating to cancel the dispersion of the first one (Fig. 1, grating-based), as typically done in pulse shapers of ultrafast spectroscopy, if they have the same groove density. Note that if the second grating is of different dispersion, one can adjust f 2 to match the dispersion given by σ .

We have compared the two methods. For the lens-based approach, we used an effective 4X demagnification (in the figure it is presented as a single lens L2). For the grating-based method, we used a second grating with the same dispersion as the first grating. We have inspected the spectrum of the lens-based approach (17 nm-bandwidth) and grating-based approach (77 nmbandwidth). While both methods coupled large bandwidths (over 100's cm -1 ), the lens-based had too much losses originating from clipping in the edge of lenses. Therefore, we used the gratingbased design. Nevertheless, we believe the lens-based throughput could overcome the grating-based if a low-dispersion grating is used.

Compressive Raman without a priori knowledge

We exploited the spectrometer for two different types of compressive Raman schemes. Basically, there are two ways of performing compressive Raman: with (supervised) [18][19][20]22,23 or without (unsupervised) 16,17 a priori knowledge about the principal components (or eigenvectors) of the hyperspectrum. In this section, we present compressive Raman results without a priori knowledge.

The measurement process of the hyperspectrum is done by a combined use of the piezo-scanner and the DMD, described elsewhere 30 . Briefely, the sample is illuminated in a raster scan manner over the image plane. For each spatial position, the Raman signal generated by the sample is measured using the compressive spectrometer. Instead of performing a wavelength scan over the spectral domain, the spectral content is measured using multiplexed detection with Hadamard functions generated with the DMD. One key advantage of using Hadamard basis is for increasing the SNR 14,31,32 because half the pixels are always on. This sampling scheme can be expressed in matrix form as y = Sx, where the vector y contains each one of the integrated measurements,

x is the hyperspectral object (in vector form), and the whole sensing process is described by the linear operator S, which is the Kronecker product of the two sampling bases used here: S = I ⊗ H (I being the canonical spatial-scanning basis, and H the Hadamard spectral-scanning basis). In order to reduce the number of measurements, we select a random subset of rows of S to measure the hyperspectral dataset. Then, we use compressive sensing algorithms to solve the following optimization problem:

arg min x = 1 2 y -Sx 2 2 + τ x T V (2)
based on the Total Variation as the regularizing function, which has been previously reported to provide good results in multispectral imaging scenarios 33,34 . The reconstructions can be seen in Fig. 2 of a relatively small region of the sample (64x64 spatial positions and 128 spectral channels). From the spectral information, we can distinguish between the water content of the sample (coloured green), and the polystyrene beads (red).

Compressive Raman with a priori knowledge

In supervised compressive Raman, one estimates the proportions of a linear combination of M eigenspectra (P), from N measurements (x). This is achieved by projecting the input spectrum onto N optimized spectral filters (F) with the DMD. Each spatial pixel can then be written as

x = F P c, where proportions c are estimated from simple inversion of F P . The key step for proportion estimation is to develop filters that are robust against noise 18,21 . We use as a metric the variance of the Cramer-Rao bound (CRB) to determine the optimized spectral filters. A thorough description of the method can be found elsewhere 21,22 .

Fig. 3 presents results of large-scale and high-sensitivity chemical imaging. Fig. 3A shows an example of the compressive detection approach where a large-scale high-resolution image (0.5 Mpixel) is presented with chemical selectivity. After generation of the optimized filters (Fig. 3A, optimized filters are presented above spectra in the top-right inset), we obtain x from which we estimate c in a pixel-wise manner. Chemically selective images are then presented in pseudo-color images (Fig. 3A)

or analysed separately in their respective channels (Fig. 3A insets shows water and polystyrene proportions). Due to high-resolution imaging, the pseudo-color images are highly contrasted (lack of yellow color).

To further quantify the resolution and demonstrate high-sensitivity detection, we present images of 88-nm polystyrene beads, sizes below the diffraction limit of the system, in Fig. 3B. Despite the low SNR, one observes regions of aggregated beads with high intensities, and isolated ones with much lower intensities. To estimate the resolution of the system, we use a simple deconvolution using the measured line profiles of the low-intensity beads (≈250 nm). We obtain a resolution of ≈230 nm, a value close to the diffraction limit of the system. 8,9 , and also because the eigenspectrum is not expected to considerably change for different environments. Fig. 4A shows the estimated proportions of the lipid and water response. In the lipid channel, there are clearly various bilayers wrapped around each other, a topology common to thin multilamellar vesicles 35 . In the water channel, there is a slight decrease in the overall water response due to the finite thickness of the lipid bilayers.

Sensitivity to model membranes

The drop in intensity in the water channel allow us to estimate the water depletion thickness, as the Rayleigh length is much smaller than the vesicle curvature. From the intensity drop of the water channel (Fig. 4B), we estimate a maximum multilayer thickness of 54 nm (the strongest water proportion intensity dip), corresponding to 11 bilayers at that location.

DISCUSSION

We have presented a new spectrometer layout designed for single-pixel spectroscopy. With this layout, we showcase two types of compressive Raman modality, either without a priori knowledge or with a priori knowledge of the eigenspectra of the hyperspectrum. The main purpose of the high-throughput spectrometer is to enable sensitive measurements of specimens that are smaller than the microscope resolution; Because the solvent background generates a strong background, we reduce its intensity by tight focusing, and also increase the dynamic range of the detection system with low-noise and high-sensitivity detection.

Remarkably, we could detect with molecular selectivity few bilayers with the current design.

Previously, vesicle systems were mostly imaged by coherent Raman scattering techniques 29,[36][37][38][START_REF] Cheng | Proc. Natl. Acad. Sci. U. S. A[END_REF][START_REF] Chen | [END_REF][41] , which are considerably more expensive and complex than the layout presented here. We estimated that the strongest region in Fig. 4A contains about 11 bilayers. Given the linearity of the spontaneous Raman process on the local concentration, we expect single-bilayer 8,9,27 imaging with slight improvements in the current design, for instance, by addition of a second detector 42 and with larger clear-aperture gratings to reduce losses due to clipping.

for single-pixel compressive Raman microspectroscopy, and demonstrated its application with two different levels of a priori information. Such layout allowed for high-sensitivity imaging of fewbilayers vesicle in few seconds, therefore paving the way of compressive Raman for cell biology applications.

  FIG. 1. Simplified layout of the compressive Raman microscope. A CW laser source excited Raman scattering process, which is detected in epi configuration. The sample is raster-scanned with a high-resolution piezo-scanner. The Raman inelastically scattered light is steered into a home-built spectrometer designed for compressive single-pixel experiments. The dashed boxes depicts two strategies (grating or lens telescope) for dispersion management, introduced by the first grating, to focus large bandwidths in the high sensitivity single-pixel detector. DMD=digital micromirror device, PM=off-axis parabolic mirror, L=lens.

FIG. 2 .

 2 FIG. 2. Compressive Raman imaging without a priori knowledge. Left: Raman pseudocolored image of polystyrene beads obtained by means of computational reconstruction. Right: Spectral content of both species present in the sample, water (green), and polystyrene (red). Compression: 2X.

Finally, we useFIG. 3 .

 3 FIG. 3. Compressive Raman imaging with a priori detection. (A) Large scale imaging with 80X compression (axis are in pixel count). Raman pseudo-color images of 1 µm-polystyrene beads (red, upper left inset) and water (green, lower left inset). Spectra are presented in the inset with the optimized filters shown on top. Scale bar: 20 µm. Effective pixel dwell time: 640 µs. (B) High-sensitivity imaging of 88 nm-polystyrene beads. Only the polystyrene channel is presented. Scale bar: 1.5 µm. Effective pixel dwell time: 3 ms

  FIG. 4. High-sensitivity detection of lipid membranes in vesicles. (A) Proportions of lipids (left panel) and water (right panel). The water channel has been scaled to enhance the appearance of the signal dip. Scale bar: 10 µm. Total acquisition time: 30 s. Effective pixel dwell time: 2.4 ms. (B) Normalized (by the image mean) line profiles of the water proportion. The signal dip is due to a depletion of water from the focal volume, which allows estimation of the number of bilayers in the focal volume (see text for further details).

ACKNOWLEDGEMENTS

We thank Tom Sperber for fruitful discussions. H.B.A. was supported by LabEX ENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*. This works was supported by European Research Council (ERC) (724473), Universitat Jaume I (PREDOC/2013/32), Generalitat Valenciana (PROMETEO/2016/079), Ministerio de Economia y Competitividad (MINECO, FIS2016-75618-R), Agence Nationale de la Recherche (ANR) France Bio Imaging (ANR-10-INSB-04-01) and France Life Imaging (ANR-11-INSB-0006) infrastructure networks and Plan cancer INSERM (PC201508). S.G. is a member of the Institut Universitaire de France.

There have been various demonstrations to accelerate Raman imaging for biological applications. These include line scan methods 6 , spatial focii multiplexing 43 , and widefield microscopy 25,44 . Highsensitivity in slit-based spectrometers 6,43 is challenging as non-confocal geometries are hard to reach diffraction-limited performance 45 . In that sense, light-sheet excitation-widefield imaging 44 could probably provide highest sensitivity, however the whole image plane has to be optically clear and static during acquisition because of the spectral/temporal sampling. The abovementioned methods all use sensitive cameras, contraty to single-pixel approaches, thus avoiding slow and noisy measurements at high speeds (mostly read-out noise) 22 . In addition, the combination of singlepixel-spectroscopy with tight-focusing (or confocal) allows for low-power CW lasers to be used.

Nevertheless, the main disadvantage of compressive Raman with a priore information is its need for at least one long scanning to obtain the eigenspectra "dictionary", although a "blind" sampling approach to learn the dictionary has been proposed 19 . Note that even in biological applications one expects only a few eigenspectra 7 , which supports that compressive Raman approaches may have the fastest imaging speeds.

Previous compressive spectrometers used less sensitive detection. Quantum efficiency and dark counts of the detector are two key parameters for high-sensitivity detection as they determine the overall throughput and dynamic range for proportion estimation. The overall throughput scaling of the grating-based approach is φ 4 l φ DMD φ 2 g φ d , where φ i denotes the throughput of the lenses (l), DMD (DM D), gratings (g) and detector (d) (by detector throughput we mean quantum efficiency). Given that lenses and gratings may have negligible losses, the largest penalty is the DMD and detector quantum efficiency. In our design, we achieve a theoretical throughput of 34% (φ g = 85%, φ d = 70%, φ DMD = 70%). A higher throughput value (50%) is expected if operating in the near-IR as optical elements are commercially available. Note, however, while the sensitivity might be higher, the overall signal will be lower as the Raman cross-section scales as λ -4 . An alternative grating-based approach is to fold the design (back-reflect the light into the same grating) 15 , however this would be even more lossy as the DMD has the lowest throughput in the design. Finally, the lens-based approach may reach overall throughput of 50% after meticulous design of the optical elements. To address that, we have presented two inexpensive layouts of a high-throughput spectrometer