Jose Aguado

Domenico Borzacchiello

Kiran S Kollepara

Francisco Chinesta

Antonio Huerta

José V Aguado

Tensor representation of non-linear models using cross approximations

Keywords: linear Modeling, Reduced Order Modeling, Low-rank Tensor Approximation, Proper Generalized Decomposition, Cross Approximations, Parametrized PDE, Multi-dimensional Problems

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Tensor methods can be regarded as a specific class of reduced-order methods (ROM), designed for the solution of multi-dimensional and multi-parametric problems. ROM have proved in all its variants to be a key tool in simulationbased engineering sciences, providing a scientific and mathematical basis for fast simulation (sometimes even in real-time) of engineered systems. They have opened the path towards a better integration of numerical simulation in dynamic data-driven application systems [START_REF] Ch | Proper Generalized Decomposition based dynamic data-driven control of thermal processes[END_REF], rapid numerical prototyping [START_REF] Borzacchiello | Reduced Order Modelling for efficient optimisation of a hot-wall Chemical Vapour Deposition reactor[END_REF] and dedicated simulation applications [START_REF] Aguado | A Simulation App based on reduced order modeling for manufacturing optimization of composite outlet guide vanes[END_REF], among many other examples.

Reduced-order methods are designed to provide fast yet accurate approximations of the solution of a numerical model. The concept of reducibility is therefore central. It states that the solution of many models can be well approximated using few degrees of freedom, provided that a suitable representation basis is chosen. The two principal families of ROM differ essentially on how they build such appropriate representation basis. One one hand, a-posteriori ROM extract a low-dimensional subspace from a training simulation dataset [START_REF] Prud'homme | Reliable real-time solution of parametrized partial differential equations: Reduced-Basis Output Bound Methods[END_REF][START_REF] Willcox | Balanced model reduction via the Proper Orthogonal Decomposition[END_REF][START_REF] Beylkin | Algorithms for numerical analysis in high dimensions[END_REF][START_REF] Rozza | Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations -application to transport and continuum mechanics[END_REF][START_REF] Quarteroni | Reduced Basis Methods for Partial Differential Equations: An Introduction[END_REF]. On the other hand, tensor methods (also referred to as a-priori ROM) compute lowrank tensor subspaces by setting up an optimization problem [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: transient simulation using space-time separated representations[END_REF][START_REF] Nouy | A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations[END_REF][START_REF] Grasedyck | A literature survey of low-rank tensor approximation techniques[END_REF]. Both families of methods require a special treatment of non-linear problems in order to preserve their efficiency.

Literature on non-linear a-posteriori ROM is abundant. Most of the approaches propose building a secondary low-dimensional subspace, other than the solution's subspace, in order to approximate the non-linear term. Then, the Empirical Interpolation Method (EIM) [START_REF] Barrault | An "empirical interpolation method": application to efficient reduced-basis discretization of partial differential equations[END_REF][START_REF] Maday | A general multipurpose interpolation procedure: the Magic Points[END_REF], as well as its discrete counterpart (DEIM) [START_REF] Chaturantabut | Nonlinear model order reduction via Discrete Empirical Interpolation[END_REF], can be used to give interpolation properties to the non-linear term's subspace. As opposed to projection, interpolation allows approximating the non-linear term by evaluating only few points, which can be done inexpensively during the on-line stage. A number of variants have been proposed, such as the unassembled DEIM [START_REF] Tiso | Discrete Empirical Interpolation Method for Finite Element Structural Dynamics[END_REF], which is specifically tailored for finite element computations, and the localized DEIM [START_REF] Peherstorfer | Localized discrete empirical interpolation method[END_REF], that proposes to switch between local subspaces in order to capture different regimes of the system, using clustering and supervised learning techniques. We shall give in Section 3.2 further details on the interpolation approach as a means to introduce cross approximations, which can be seen as a sort of extension of EIM to a tensor framework.

Interpolation-based techniques were described in [START_REF] Chapman | Accelerated mesh sampling for the hyper reduction of nonlinear computational models[END_REF] as approximate-thenproject approaches. Alternatively, project-then-approximate (also, hyper reduction) approaches propose roughly to approximate the projection of the nonlinear term directly [START_REF] Ryckelynck | A priori hypereduction method : an adaptive approach[END_REF][START_REF] Farhat | Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energybased mesh sampling and weighting for computational efficiency[END_REF][START_REF] Farhat | Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models[END_REF]. This ultimately leads to the definition of empirical quadrature rules [START_REF] Hernández | Dimensional hyper-reduction of nonlinear finite element models via empirical cubature[END_REF], which seem to provide a more consistent basis to address non-linear reduced order modeling.

As opposed to a-posteriori ROM, literature on tensor-based ROM for nonlinear problems is much scarcer and lacks, in general, of a systematic approach. For instance, only polynomial non-linearities can be efficiently addressed with the Asymptotical Numerical Method (ANM) [START_REF] Niroomandi | Model order reduction for hyperelastic materials[END_REF][START_REF] Leygue | Towards a framework for non-linear thermal models in shell domains[END_REF][START_REF] Niroomandi | Realtime simulation of biological soft tissues: a PGD approach[END_REF]. The LATIN method has been successfully applied to compute space-time (or pseudo-time) separated representations involving elasto-plasticity, contact and multi-scale problems [START_REF] Ladevèze | The LATIN multiscale computational method and the Proper Generalized Decomposition[END_REF][START_REF] Néron | Proper generalized decomposition for multiscale and multiphysics problems[END_REF][START_REF] Capaldo | The Reference Point Method, a hyperreduction technique: Application to PGD-based nonlinear model reduction[END_REF].

The objective of this paper is three-fold: in first place, we discuss the importance of tensorization for the efficiency of tensor-based ROM. By tensorization we mean the ability to enforce a tensor representation of the model itself 1 , which reveals crucial for the efficiency of tensor-based algorithms. Secondly, we investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce tensorization, and propose the use of cross approximations [START_REF] Tyrtyshnikov | Incomplete Cross Approximation in the Mosaic-Skeleton Method[END_REF][START_REF] Espig | Black box low tensor-rank approximation using fiber-crosses[END_REF] for models in moderate dimension. And finally, linearization of tensor problems is analyzed and several strategies for the tensor subspace construction are proposed. Specifically, the canonical tensor format is used here as a reference, in spite of some well-known limitations that shall be discussed in detail in Section 4.3. As a practical consequence of these, the use of canonical format should be restricted to second or third order tensors. However, in this paper we shall put emphasis on concepts rather than on technical details, and therefore the validity of the proposed approach should not be compromised by the choice of tensor format. Other tensor formats are likely to perform better in higher dimensional problems and might motivate future research.

The rest of the paper is organized as follows. In Section 2, we review the basic formulation of tensor methods and the importance of tensorization for their efficient implementation. To this end, tensor methods are formulated as an optimization problem. In Section 3, we consider a generic non-linear problem and show how tensorization is in general lost. We also analyze both projection and interpolation approaches as a means to recover tensorization. In Section 4, we present cross approximations as a means to enforce tensorization of non-linear models. Strategies for coupling a given linearization scheme with the incremental subspace construction are analyzed in Section 5. Finally, the performance of the proposed method is assessed in Section 6 and Section 7 by means of several numerical examples.

Basic formulation of tensor methods

Tensor methods build a low-rank tensor subspace for the solution of multidimensional and multi-parametric problems. To this end, the weak form of the model at hand is regarded as an optimization problem where the set of admissible solutions is constrained to a low-rank tensor subspace. The efficiency of tensor methods relies on the tensorization of the model at hand, as it allows splitting a multi-dimensional problem into a series of lower-dimensional ones. When dealing with non-linear models, tensorization is in general lost, thus compromising the overall efficiency of tensor methods.

In next lines, tensor-based reduced-order methods (ROM) are introduced from a-posteriori ones by drawing a parallel between both approaches.

From vector to tensor subspaces

To the purposes of this section, it is useful to consider a parametrized problem as a model case. Let u(µ) ∈ R N0 be its solution on a given representation basis of dimension N 0 , for a given D-tuple of parameters µ ∈ M. A-posteriori ROM split the computational cost into"offline-online" stages. In the offline stage, a subspace of lower dimension, M ≪ N 0 , is extracted from the statistical analysis of a training simulation dataset. Then, admissible solutions are constrained to this low-dimensional subspace:

∀µ ∈ M : u(µ) ≈ u M (µ) := W 0 α(µ), (1)
where α ∈ R M are the representation coefficients of the solution on the subspace spanned by the columns of W 0 ∈ R N0×M . For an efficient offline-online splitting of the computational cost, restricting the solution to a subspace of dimension M should yield a reduced system equations whose complexity does not depend on N 0 anymore. However, this is not always the case, specially when dealing with non-linear models. In such a case, it is well-known that the non-linear terms of the equation need, in principle, to be fully evaluated and then projected at each iteration of the linearization scheme [START_REF] Maday | A general multipurpose interpolation procedure: the Magic Points[END_REF][START_REF] Chaturantabut | Nonlinear model order reduction via Discrete Empirical Interpolation[END_REF]. A number of techniques, already reviewed in Section 1, have been proposed in order to address this situation.

In contrast with a-posteriori ROM, tensor methods are designed to compute the solution of parametrized problem as an explicit function of the parameters. Therefore, tensor methods do not produce a reduced system of equations but the parametric solution itself. In this regard, parameters are treated exactly the same as if they were coordinates, and consequently, the computational domain becomes of higher dimension. In practice, this implies that the solution is no longer represented by a one-dimensional array, as in Eq. ( 1), but by a multidimensional array (or tensor) instead.

Remark 1 (Notation for tensors)

Although sometimes it is convenient to use multi-indices for tensors, here we prefer linear indexing in order to take advantage of standard matrix algebra operations, such as the Kronecker and Khatri-Rao products that will be used below.

Let u ∈ R N , N := N 0 N 1 • • • N D ,
be the full tensor representation of the parametric solution after discretizing the parametric domain M with a N 1 × • • • × N D grid. Clearly, full representations are in general precluded even in moderate dimension. Therefore, tensor subspaces must be introduced to represent parametric solutions efficiently. For the sake of concretion, we shall only discuss here the canonical tensor subspace, which lies at the basis of the Proper Generalized Decomposition (PGD) method [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: transient simulation using space-time separated representations[END_REF][START_REF] Néron | Proper generalized decomposition for multiscale and multiphysics problems[END_REF][START_REF] Falcó | A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach[END_REF]. It is defined as follows:

T M := span{w m 0 ⊗ w m 1 ⊗ • • • ⊗ w m D } M m=1 , (2)
where w m d ∈ R N d stands for the m-th factor, or mode, in direction d, with 0 ≤ d ≤ D. Note that T M shall be understood throughout this paper as a non-linear structure, where w d can be chosen freely, unless indicated otherwise. A low-rank approximation u M ∈ T M writes as:

u ≈ u M := M ∑ m=1 α m w m 0 ⊗ w m 1 ⊗ • • • ⊗ w m D ,
commonly referred to as separated tensor representation. Using the Khatri-Rao product (i.e. column-wise Kronecker product), it can also be written as:

u M ≡ (W 0 ⊙ W 1 ⊙ • • • ⊙ W D ) α, ( 3
)
where columns of matrix W d ∈ R N d ×M span a subspace of dimension M , and its m-th column is the mode w m d . In Eq. (3), "⊙" stands for the Khatri-Rao product. The array α ∈ R M contains the representation coefficients of the parametric solution on the canonical tensor subspace of rank M . Notation based on the Khatri-Rao product will be preferred in this paper by its compactness.

Remark 2 (Tensor reconstruction and decomposition) Given a low-rank representation, Eq. ( 3), the reconstruction operation allows obtaining the full tensor representation, noted as u M ← recons(α, W 0 , . . . , W D ). Conversely, decomposition is defined as {α, W 0 , . . . , W D } ← decomp(u M ). While reconstruction is trivial, decomposition requires careful study [START_REF] Kolda | Tensor decompositions and applications[END_REF], as it might not even be well-posed in some cases.

Optimization problem for tensor subspace construction

Consider a multi-dimensional, linear, steady-state model. After discretization, it can be written as follows:

Au + f = 0, (4)
where u ∈ R N is a full tensor representation of some parametric solution, as it has been previously defined in Section 2.1. On the other hand, A ∈ R N ×N represents a linear operator and f ∈ R N is the independent term. Tensor methods are designed to build a tensor subspace by turning Eq. ( 4) into an optimization problem. No sampling of the parameter domain is in principle required.

In particular, PGD uses a greedy strategy to build a tensor subspace progressively by computing rank-one corrections, i.e. by building a series of nested subspaces:

T 1 ⊂ T 2 ⊂ • • • ⊂ T M where T M := T M -1 + T 1 . (5)
In practice, the actual rank is driven by some error estimate [START_REF] Ammar | An error estimator for separated representations of highly multidimensional models[END_REF][START_REF] Ladevèze | On the verification of model reduction methods based on the proper generalized decomposition[END_REF] able to determine when the solution subspace is accurate enough. In general, the greedy rank-one correction strategy will not yield a best rank-(M + 1) approximation, even if u M is a best rank-M approximation. Assuming that a rank-M tensor approximation of the solution is known, u M ∈ T M , we seek a rank-one correction δu ∈ T 1 such that:

u M +1 := u M + δu where u M = D ⊙ d=0 W d α with α ∈R M , W d ∈ R N d ×M and δu = D ⊗ d=0 w d . ( 6
)
Assuming that the operator A is symmetric positive definite (specific formulations of PGD exist for non-symmetric problems, see [START_REF] Nouy | A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations[END_REF][START_REF] Boucinha | Ideal minimal residualbased Proper Generalized Decomposition for non-symmetric multi-field models. Application to transient elastodynamics in space-time domain[END_REF][START_REF] Barbarulo | Proper generalized decomposition applied to linear acoustic: a new tool for broad band calculation[END_REF][START_REF] Billaud-Friess | A tensor approximation method based on ideal minimal residual formulations for the solution of highdimensional problems[END_REF]), Eq. ( 4) can be regarded as an optimization problem where the set of admissible solutions is constrained to T 1 [START_REF] Falcó | A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach[END_REF]:

δu := arg min v∈T1 1 2 ⟨Av, v⟩ + ⟨Au M , v⟩ + ⟨f , v⟩, (7)
where ⟨•, •⟩ stands for the scalar product in R N . Eq. ( 7) constitutes a nonlinear optimization problem due to the tensor multiplicative structure of the subspace, see Eq. [START_REF] Borzacchiello | Reduced Order Modelling for efficient optimisation of a hot-wall Chemical Vapour Deposition reactor[END_REF]. For the efficient solution of Eq. ( 7), tensorization of A and f is essential.

Tensorization for the efficient optimization

Let us suppose that the following tensor representations are known:

A = R ∑ r=1 D ⊗ d=0 A r d with A r d ∈ R N d ×N d and f = D ⊙ d=0 V d γ with γ ∈ R S , V d ∈ R N d ×S . ( 8
)
By inserting Eq. ( 8) into Eq. ( 7), and after some tedious but conceptually simple manipulations, we see that the scalar product in R N can in fact be computed as the product of lower-dimensional scalar products. For instance, the first term in Eq. ( 7) writes:

⟨Av, v⟩ ≡ R ∑ r=1 D ∏ d=0 ⟨A r d v d , v d ⟩ d , (9)
where ⟨•, •⟩ d stands for the scalar product in R N d . Eq. ( 9) defines the separation property of the scalar product. This property suggests applying an alternating directions algorithm in order to optimize each direction w d [START_REF] Chinesta | A short review on model order reduction based on Proper Generalized Decomposition[END_REF]. This can be achieved by simply projecting the functional in Eq. ( 7) onto each direction. Thus, let w * the current direction to be optimized, implying that w d , ∀d ̸ = * , are frozen at their most current update. The restriction of Eq. ( 7) onto the current direction yields:

w * := arg min v * ∈R N * R ∑ r=1 { 1 2 ⟨β r * A r * v * , v * ⟩ * + ⟨A r * W * αr * , v * ⟩ * } + ⟨V * γ * , v * ⟩ * , (10)
where β r * ∈ R, αr * ∈ R M and γ * ∈ R S are coefficients carrying the result of the scalar products in all directions except the current one. See Annex A for the definition of these coefficients as well as for a detailed solution of Eq. [START_REF] Nouy | A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations[END_REF].

In summary, the correction δu can be computed by an alternating optimization of each separated factor, using Eq. ( 10), until stagnation [START_REF] Ammar | An error estimator for separated representations of highly multidimensional models[END_REF][START_REF] Uschmajew | Local convergence of the alternating least squares algorithm for canonical tensor approximation[END_REF]. In this way, the algorithm splits a multi-dimensional problem into a series of low-dimensional ones. This has been possible thanks to the tensorization of the problem, introduced in Eq. ( 8). In many cases, tensorization of linear problems can be achieved quite easily. However, when dealing with non-linear models, tensorization is in general lost, thus compromising the overall efficiency of tensor methods.

Non-linear problem formulation

In this section we present a generic non-linear problem and discuss in detail the adequacy of interpolation rather than projection approaches to enforce tensorization. To this end, we simply transform the independent term in Eq. ( 4) into a non-linear function. This yields the following multi-dimensional, nonlinear, steady-state model:

Au + f (u) = 0, (11)
where

f (u) is a non-linear function evaluated component-wise, that is f (u) := [f (u 1 ) • • • f (u N )] ∈ R N .

Tensorization is lost in non-linear problems

Let us consider, without loss of generality, a fixed-point linearization. Assuming that a rank-M tensor approximation of the solution at iteration ℓ is known, u ℓ M ∈ T M , the next iterate can be computed from:

Au ℓ+1 + f (u ℓ M ) = 0. ( 12
)
We seek to compute u ℓ+1 ⋆ ∈ T ⋆ , a good approximation of u ℓ+1 , where the rank will be driven by some error estimate [START_REF] Ammar | An error estimator for separated representations of highly multidimensional models[END_REF][START_REF] Ladevèze | On the verification of model reduction methods based on the proper generalized decomposition[END_REF]. As explained in Section 2.2, T ⋆ can be built progressively by computing rank-one corrections. Assuming that u ℓ+1 M ′ ∈ T M ′ has already been computed, a rank-one correction δu ∈ T 1 can be computed from:

δu := arg min v∈T1 1 2 ⟨Av, v⟩ + ⟨Au ℓ+1 M ′ , v⟩ + ⟨f (u ℓ M ), v⟩. (13)
Even though u ℓ M is a tensor form, no tensorized form of f (u ℓ M ) is in general known. As a consequence, the non-linear term cannot be evaluated efficiently. Recall that it is essential for the efficiency of tensor-based algorithms to take advantage of the scalar product separation property. If it holds, higher-dimensional scalar products in Eq. ( 13) could be computed as a series of lower-dimensional ones. Hence we require methods to compute a rank-S approximation of the non-linear term:

f (u ℓ M ) ≈ f S ∈ T S ⇔ f S = D ⊙ d=0 V d γ with γ ∈ R S , V d ∈ R N d ×S , ( 14
)
where neither V d nor γ are a priori known, i.e. both have to be computed for a given u ℓ M . The rank of the non-linear term approximation must not be confused with the rank of the independent term in Eq. ( 8), although we use the same letter: "S". These have nothing to do with each other. Note that if Eq. ( 14) was available, the problem would be completely equivalent to the one already studied in Section 2. Indeed, Eq. ( 14) could be inserted into Eq. ( 13), yielding an appropriately tensorized problem, which can be solved applying an alternating directions algorithm, as explained in Section 2.3. Therefore, the correction δu could be computed by an alternating optimization of each separated factor, provided that we dispose of Eq. ( 14). Unfortunately, it is by no means trivial to achieve a tensor representation such as Eq. ( 14).

Tensorization could be, in principle, recovered by means of a three-step procedure, involving reconstruction-evaluation-decomposition (RED) operations:

1. Reconstruction of the current iterate, yielding a full representation, as defined in Remark 2: u ℓ M ← recons(α, W 0 , . . . , W D ).

2. Evaluation of the non-linear term:

f (u ℓ M ) ← nlfun(u ℓ M ).
3. Decomposition of the non-linear term into a rank-S tensor representation:

{γ, V 0 , . . . , V D } ← decomp(f (u ℓ M )).
However, RED steps compromise the overall efficiency of tensor-based algorithms, as the manipulation of full representations is required. The choice of the decomposition method is crucial here. Some of the classical techniques that have been used within the PGD framework are the Singular Value Decomposition (SVD) [START_REF] Chinesta | A short review on model order reduction based on Proper Generalized Decomposition[END_REF][START_REF] Ammar | On the space-time separated representation of integral linear viscoelastic models[END_REF] and its variant for higher dimensions, the high-order SVD [START_REF] Lathauwer | A multilinear Singular Value Decomposition[END_REF]. PGD itself has also been used as a decomposition technique [START_REF] Modesto | Proper Generalized Decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: application to harbor agitation[END_REF]. These approaches are expensive from both memory and computational standpoints, as they require operating on the full tensor representation. In this paper we shall use interpolation-based algorithms that are able to adaptively determine the entries of the tensor that have to be read in order to compute a low-rank approximation. Therefore, the full representation of the tensor is not required, but only its evaluation at specific points.

Projection versus interpolation approaches

Coming back to Eq. ( 14), we seek an approximation f S ∈ T S of f (u ℓ M ). Let us assume that we have somehow managed to compute factor matrices V d , i.e. T S can be now understood as a linear subspace, and we only need to compute the representation coefficients γ. Therefore, we dispose of

V := ⊙ D d=0 V d ∈ R N ×S .
Even in such a favorable scenario, computing γ using projection techniques is expensive. The projection problem is defined as follows:

γ := arg min ω∈R S ⟨V ω, V ω⟩ -⟨f (u ℓ M ), V ω⟩, ( 15
)
which yields the following normal equations:

D ⃝ d=0 V T d V d γ = V T f (u ℓ M ), (16)
where ⃝ D d=0 is performs the Hadamard (i.e. component-wise) product. While V T V can be efficiently computed, thanks to the tensor structure of the basis, as indicated in the left-hand side of Eq. ( 16), the right-hand side requires a full evaluation of the non-linear term. It is worth to recall that if matrices V d were orthonormal, the left-hand side computation could be encompassed. Therefore, we see that in spite of the technicality of the tensor framework, when it comes to the evaluation of the non-linear term, the same kind of issues are encountered in both a-posteriori ROM and tensor-based framework.

Inspired by EIM, interpolation-based approaches could be envisaged. In its discrete form, EIM returns a set of interpolation indices, ϱ ∈ N S , from the basis V. This would allow defining the following interpolation problem:

P T Vγ = P T f (u ℓ M ) → f (u ℓ M ) ≈ V(P T V) -1 P T f (u ℓ M ), (17)
where P ∈ R N ×S is an extractor matrix full of zeros, with the exception of entries P ϱi,i = 1. Indices ϱ are such that P T V ∈ R S×S is invertible. As opposed to projection, interpolation allows approximating the non-linear term by evaluating only few points, which can be done inexpensively. However, a detailed examination of the algorithm yielding the interpolation indices reveals practical issues for its implementation in a tensor framework. Without going into details, the algorithm selects the interpolation indices at the absolute maximum value of some approximation residual, see [START_REF] Barrault | An "empirical interpolation method": application to efficient reduced-basis discretization of partial differential equations[END_REF][START_REF] Chaturantabut | Nonlinear model order reduction via Discrete Empirical Interpolation[END_REF]. When working with tensors, the max function must be used carefully, in order to avoid full tensor complexity.

Additionally, recall that a suitable tensor approximation basis has been assumed to be available, which is often not the case. An alternative approach able to deliver both a tensor interpolation basis and the interpolation indices is required. Full inspection of the non-linear term is precluded.

Enforcing tensorization with cross approximations

In this section, cross approximations are presented as a means to tensorize the non-linear term. Cross approximations were developed first for matrix approximations [START_REF] Tyrtyshnikov | Incomplete Cross Approximation in the Mosaic-Skeleton Method[END_REF] and then extended to the tensor framework [START_REF] Espig | Black box low tensor-rank approximation using fiber-crosses[END_REF]. When working with canonical tensor subspaces, as it is the case of this paper, cross approximations are subsidiary of the properties of this tensor subset: linear complexity in the representation but ill-posedness of the best approximation problem [START_REF] Hastad | Tensor rank is NP-complete[END_REF][START_REF] Kolda | Tensor decompositions and applications[END_REF]. Limitations derived from this fact, as well as perspectives for future research based on other tensor formats, are discussed in Section 4.3. Although cross approximations were developed independently from EIM, the exposition is made in such a way that resemblances between both methods are highlighted. Some relevant features of the cross approximations method are:

• It builds a canonical tensor subspace with interpolation properties. The interpolation indices are provided too. Both are computed simultaneously using a fiber search algorithm.

• The tensor subspace is constructed progressively by adding rank-one corrections.

• Like the EIM, the interpolation indices are placed where the residual of the approximation is maximum. Furthermore, they are nested.

• The full tensor representation is not required, but only its evaluation at specific points.

Non-linear term approximation via interpolation

Cross approximations build both a rank-⋆ subspace, T ⋆ , and a set of interpolation indices

I ⋆ := {ϱ s } ⋆ s=1 , such that f ⋆ ∈ T ⋆ interpolates f (u ℓ M ) at I ⋆ .
Here, "⋆" is the rank of the approximation, to be determined adaptively by the cross approximations algorithm. Note that I ⋆ contains linear indices, i.e. 1 ≤ ϱ s ≤ N . However, for notational convenience, in this section we will also use multi-indices, i.e. H ⋆ := {ρ s } ⋆ s=1 , with ρ s ∈ N D+1 and 1 ≤ ρ d s ≤ N d . As stated in the beginning of this section, cross approximations build T ⋆ progressively by adding rank-one corrections. Therefore, let us assume that T S has already been computed, and by consequence, matrices V d ∈ R N d ×S and interpolation coefficients H S (also, I S ) are available. A rank-S approximation, f S := ( ⊙ D d=0 V d ) γ, can be readily computed by solving the following interpolation problem:

D ⃝ d=0 P T d V d γ = P T f (u ℓ M ), (18)
where P d ∈ R N d ×S are extractor matrices full of zeros, with the exception of entries (ρ d s , s), which are populated with ones. On the other hand,

P = ⊙ D d=0 P d ∈ R N ×S .
Remark 3 (Non-linear term evaluation) From the non-linear term definition in Section 3, we have that the right-hand side of Eq. ( 18) can be written as follows:

P T f (u ℓ M ) ≡ f (P T u ℓ M ) ≡ f ({ D ⊙ d=0 P T d W d } α ) .
It is therefore clear that u ℓ M has to be evaluated at only S indices.

Simultaneous construction of both interpolation basis and indices

From the knowledge of f S , we seek a rank-one correction δf ∈ T 1 as well as its corresponding interpolation index, ϱ, such that:

f S+1 := f S + δf I S+1 := I S ⊕ ϱ, ( 19
)
where δf = ⊗ D d=0 v d . Recall that ϱ can be readily converted into a multi-index ρ such that H S+1 := H S ⊕ ρ. Like EIM, cross approximations seek to place the interpolation index ϱ where the residual of the current rank-S approximation reaches its maximum value: Since searching for the maximum of the approximation's residual is computationally expensive, a fiber search algorithm (presented hereafter) is preferred instead. Starting with a random choice for the multi-index ρ, we loop over its components seeking for the maximum of the residual restricted to the current component. The location where the maximum of the residual is found allows updating the current multi-index component. Having completed this iteration over all components, we obtain a new multi-index ρ. Fig. 1 illustrates this process in a three-dimensional tensor. It shows the multi-index ρ optimized over a cross, see Fig. 1d, and its associated fibers, depicted in Fig. 1a, Fig. 1b and Fig. 1c for each one of the three directions.

ϱ := arg max |r| with r := f (u ℓ M ) -f S . ( 20
) 0 1 2 r 0 r 1 r 2 a) b) c) d) ⇢ = (2, 3, 2) ⇢ 0 = (•, 3, 2) ⇢ 1 = (2, •, 2) ⇢ 2 = (2, 3, •)
Let ρ * be the current direction, implying that ρ d , ∀d ̸ = * are frozen at their most current update. The restriction of Eq. ( 20) onto the current fiber yields:

ρ * := arg max |r * | with r * := f (W * α * ) -V * γ * . ( 21
)
See Annex B for the definition of α * ∈ R M and γ * ∈ R S . By iterating along all directions, Eq. ( 21) allows updating each component of the multi-index ρ, until stagnation. The number of evaluations to be performed at each alternated fiber search is relatively small, as the size of the residual's restriction on a fiber is r * ∈ R N * . The just described cross approximation algorithm will not succeed, in general dimension, to find the maximum of the residual. Therefore, ρ ≡ ϱ ̸ = arg max |r|. Further details are given in Section 4.3.

The interpolation basis is defined from the fiber cross associated to the interpolation index: we set v d := r d , with 0 ≤ d ≤ D. Usually, the modes are normalized such that the ρ d -th entry of v d is equal to one. However, no distinct notation for normalized modes is used for the sake of simplicity. With the interpolation basis and the interpolation index at hand, the definition of the rank-one correction δf is completed. Coefficients γ ∈ R S+1 can be readily computed by solving the interpolation problem defined in Eq. [START_REF] Ryckelynck | A priori hypereduction method : an adaptive approach[END_REF].

The final rank of the approximation is determined by looking at the maximum of the residual, which should in principle be located at the interpolation index. For some prescribed tolerance, the stop criterion is as follows:

f ({ D ⊙ d=0 P T d W d } α ) - { D ⊙ d=0 P T d V d } γ < tol, (22)
where P d ∈ R N d ×1 is an extractor matrices full of zeros, except the entry ρ d which is equal to one. Due to some issues discussed below, the criterion given in Eq. ( 22) may not behave consistently. In practice, we prefer to evaluate the residual at a test set of indices, and not only at the interpolation index, in order to ensure the approximation is below the specified tolerance.

Discussion and alternative tensor formats

In this section, limitations inherited from the canonical tensor format shall be discussed, and perspectives on the use of other tensor formats as a basis for future research shall be given. Compared to other formats, the canonical format suffers from ill-posedness of the best approximation problem in general dimension [START_REF] Hastad | Tensor rank is NP-complete[END_REF][START_REF] Kolda | Tensor decompositions and applications[END_REF]. As a consequence, it is not possible to recover exactly a low-rank tensor from samples [START_REF] Hackbusch | Tensor spaces and numerical tensor calculus[END_REF]. In two-dimensional problems, it can be proven that δf interpolates f (u ℓ M ) at the cross defined by the new interpolation index ρ [START_REF] Espig | Black box low tensor-rank approximation using fiber-crosses[END_REF][START_REF] Hackbusch | Tensor spaces and numerical tensor calculus[END_REF]. Equivalently, the residual vanishes at the cross. From a practical point of view, this property implies that the interpolation matrix defined in the left-hand side of Eq. ( 18) is lower triangular. However, in dimension higher than two, the interpolation property on the crosses is lost, i.e. the residual may not vanish on three or higher dimensional crosses. As a consequence, cross approximations as formulated in Section 4.2 are likely to yield tensor decompositions of an unnecessarily high rank, see [START_REF] Espig | Black box low tensor-rank approximation using fiber-crosses[END_REF][START_REF] Hackbusch | Tensor spaces and numerical tensor calculus[END_REF] for details. A PGD projection step can be used in order to control the rank of the approximation, as discussed in Section 4.4. Some refinements related to the fiber search algorithm are also provided in the previous references. In our experience, cross approximations as formulated in Section 4.2 deliver reasonably good results in moderate dimension (two or three order tensors). Potentially, this would allow addressing problems defined in up to 6-D or 9-D domains, assuming that each tensor direction corresponds to the discretization of a 3-D domain.

In order to overcome limitations of the canonical format, cross approximations were extended to other tensor formats, such as Tucker [START_REF] Tucker | The extension of factor analysis to three-dimensional matrices[END_REF], Tensor Train [START_REF] Oseledets | Tensor-train decomposition[END_REF] and more general tree-based Hierarchical Tucker tensors [START_REF] Hackbusch | A new scheme for the tensor representation[END_REF]. We refer the interested reader to [START_REF] Oseledets | Tucker dimensionality reduction of three-dimensional arrays in linear time[END_REF][START_REF] Oseledets | TT-cross approximation for multidimensional arrays[END_REF][START_REF] Bebendorf | Adaptive cross approximation of multivariate functions[END_REF][START_REF] Ballani | Black box approximation of tensors in hierarchical tucker format[END_REF]. Contrary to the canonical format, the aforementioned formats allow recovering exactly a low-rank tensor from samples in general dimension. These could be used as a means to enforce tensorization, as suggested in this paper, and coupled with efficient solvers for high-dimensional systems [START_REF] Oseledets | Solution of linear systems and matrix inversion in the TT-Format[END_REF][START_REF] Dolgov | Alternating minimal energy methods for linear systems in higher dimensions[END_REF], other than the PGD used in this paper. These are valuable references that might motivate future research and extension of the ideas discussed in this paper.

Compressed rank via PGD projection

Cross approximations as formulated in 4.2 may deliver tensor decompositions of an unnecessarily high rank. Therefore, in this section we propose to use a PGD projection step in order to compute a rank as small as possible. Note that, in general, the compressed rank shall not be the exact rank, since PGD relies on the canonical tensor format, see Section 4.3 for details. The projection problem is formulated as in Eq. ( 15), but letting the approximation basis as an unknown to be optimized with the PGD method. Another difference is that in this section we do not seek to project f (u ℓ M ), but f S ∈ T S instead, which has tensor structure. This yields a conveniently tensorized PGD projection problem, as it shall be shown below.

Therefore, we seek to build a lower-rank subspace, T ⋆ , with ⋆ ≤ S, such that f ⋆ ∈ T ⋆ is equal to f S , up to some desired tolerance. Let us assume that f Q has already been computed and we seek a rank-one correction δf ∈ T 1 such that

f Q+1 := f Q + δf where f Q = D ⊙ d=0 V ′ γ ′ with γ ′ ∈R Q , V ′ d ∈ R N d ×Q and δf = D ⊗ d=0 v ′ d . ( 23
)
The projection problem is formulated as follows:

δf := arg min z∈T1 ⟨z, z⟩ + ⟨f Q -f S , z⟩, (24)
Since all quantities in Eq. ( 25) are tensorized, an alternating directions algorithm can be applied straightforwardly in order optimize each v ′ d . Assuming that the current direction being optimized is v ′ * :

v ′ * := arg min z * ∈R N * ⟨a * z * , z * ⟩ * + ⟨V ′ * γ′ * -V * γ * , z * ⟩ * , ( 25
)
where coefficients a * ∈ R, γ′ * ∈ R Q and γ * ∈ R S can be deduced applying the same methodology already shown in Appendix A, and therefore, their definition is not given explicitly for the sake of brevity.

Linearization and incremental subspace construction

Both the solution and the non-linear subspaces are built incrementally. This yields several strategies for coupling a given linearization scheme with the incremental subspace construction. In next lines, we choose for the sake of clarity a fixed-point linearization and analyze several strategies. Consider the non-linear problem already introduced in Eq. ( 11). The following strategies for coupling the linearization scheme with the incremental subspace construction are possible.

The outer linearization strategy

The first and most intuitive option consists in first linearizing the problem at hand, then applying a tensor method to solve the resulting linearized problem. Therefore, we have essentially two nested loops: one for the linearization scheme (the outer loop) and another one for the PGD rank-one corrections (the inner loop). The outer loop iterations are indexed by ℓ.

For the sake of notational clarity, we denote in this section u ℓ := u M ℓ ∈ T M ℓ , that is, a rank-M ℓ approximation of the solution at iteration ℓ. Likewise the non-linear term is denoted as f ℓ := f S ℓ ∈ T S ℓ . Therefore, the previous iterate is u ℓ-1 . After tensorization of the non-linear term evaluated at the previous iteration, f ℓ-1 , the next iterate u ℓ can be computed from the solution of:

Au ℓ + f ℓ-1 = 0. (26)
A tensor method can be used to solve Eq. ( 26). Observe that here the tensor method is regarded merely as a linear solver. Algorithm 1 summarizes the just described strategy.

Observe that the outer linearization strategy as described in Algorithm 1 builds a tensor subspace T M ℓ from scratch. A possible variant is to reuse the subspace of the previous iterate by setting: T M ℓ := T M ℓ-1 + T M , where T M is a rank-M correction. As we converge, we expect u ℓ ≈ u ℓ-1 , and then we can hope M ≪ M ℓ . Consequently, much less terms per iteration would be need to be computed. This variant of the outer linearization scheme may tend to cumulate Set ℓ = ℓ + 1

5:

Tensorize non-linear term: f ℓ-1 ≈ f (u ℓ-1 ) ▷ Cross Approx.

6:

New iterate: u ℓ ← PGD on Eq. ( 26) ▷ Inner loop 7:

if Convergence then 8:

Set flag = false return u ℓ ∈ T M ℓ terms unnecessarily during intermediate iterations. A compressed rank can be obtained via PGD projection, as explained in Section 4.4.

The inner linearization strategy

In this strategy, the order of the outer and inner loops defined in Section 5.1 is interchanged: the linearization loop becomes the inner loop while the PGD rank-one corrections loop becomes the outer loop. Therefore, suppose that we have already built u M ∈ T M and we want to compute a rank-one correction:

u M +1 := u M + δu. ( 27
)
Introducing Eq. ( 27) into Eq. ( 11), we have

Aδu + Au M + f (u M + δu) = 0, (28)
that can be linearized around the correction. We denote by δu ℓ the ℓ-th iterate of the correction δu. The fixed-point linearization yields the following approximation:

f (u M + δu) ≈ f (u M + δu ℓ-1
), that is, the non-linear term is evaluated at the previous iterate of the rank-one correction. Next, the nonlinear term is tensorized:

f ℓ-1 := f S ℓ-1 ≈ f (u M + δu ℓ-1
). The current iterate of the rank-one correction can be computed from the solution of:

Aδu ℓ + Au M + f ℓ-1 = 0. (29)
Eq. ( 29) can be solved with an alternating direction algorithm. The just described strategy is summarized in Algorithm 2.

Observe that the inner linearization strategy as described in Algorithm 2 builds a subspace T M from rank-one corrections, each one of them being iterated until convergence of the linearization. In doing so, the non-linear term needs to be tensorized several times (until convergence of the linearization scheme) for each rank-one correction. A possible variant is to reuse the non-linear term subspace from one iteration to another by setting: T S ℓ := T S ℓ-1 + T S . As rank-one correction converges, we expect δu ℓ ≈ δu ℓ-1 and hope that S ≪ S ℓ . Set M = M + 1

Algorithm 2 The inner linearization strategy

5:

Set ℓ = 0 and flagNl = true

6:
Initialize δu ℓ 7:

while ( ℓ ≤ ℓ max & flagNl ) do ▷ Inner loop 8: Set ℓ = ℓ + 1 9: Compute f ℓ-1 ≈ f (u M + δu ℓ-1 )
▷ Using cross approximations Set flagNl = false 13:

u M ← u M + δu ℓ 14:
if Convergence then 15:

Set flagPgd = false return u M ∈ T M In the following lines, we briefly summarize some of the convergence properties observed from the numerical experiments carried out in Section 6. See Section 6.2 for a detailed comparison between outer and inner schemes. As a matter of fact, both schemes showed quite different behavior. While the outer scheme converged monotonically at a constant rate, the inner scheme behaved less smoothly, and in some cases, the correction did not even converge. However, in spite of this lack of robustness, the global convergence of the inner strategy seemed to be faster. Specifically, the inner scheme required a smaller number of calls to the alternating directions algorithm. The number of calls to the fiber search algorithm was also counted for both outer and inner schemes. Results showed that the outer scheme required significantly less calls. As the computational cost tends to be dominated by the alternating directions algorithm rather than by the fiber search algorithm, the inner strategy seemed to be globally faster.

A non-linear 2-D steady state problem

In this section we provide a first numerical example in order to illustrate and assess the performance of the algorithms discussed in previous sections.

Let u := u(x, y) be the solution the following non-linear 2-D steady state problem, defined in a unit square domain:

{ -∇ 2 u + σ sin(2πu) = s in Ω := [0, 1] 2 u = 0 on ∂Ω, ( 30
)
where the source term is defined as s := s(x, y) = 100 sin(2πx) sin(2πy). On the other hand, σ stands for the magnitude of the non-linear term. We shall consider four different values: σ ∈ {1, 5, 10, 15} as a means to analyze the convergence of different linearization strategies. In what follows, we associate x and y with subscripts 0 and 1, respectively.

A rank-M approximation u ≈ u M := (W 0 ⊙ W 1 ) α is sought, where u stands for the discretized version of u. Both directions x and y are discretized with N 0 = N 1 = 100 nodes, i.e. 99 two-nodes linear finite elements per direction. This yields an equivalent 2-D finite element mesh with 99 × 99 bi-linear, fournodes quadrangular elements (only used for visualization purposes). as follows:

C outer ADA = ℓ ⋆ ∑ ℓ=1 M ℓ and C inner ADA = M ⋆ ∑ M =1 ℓ ⋆ M , (31)
where ℓ ⋆ is the number of non-linear iterations until convergence and M ℓ is the number of modes computed at each iteration. On the other hand, M ⋆ is the number of modes at convergence and ℓ ⋆ M is the number of non-linear iterations for the convergence of each mode.

Fig. 5 depicts the total number of ADA calls against the relative residual reduction. Four different values of the non-linear term's magnitude are used. As expected, the greater is the magnitude of the non-linear term, the more ADA calls are required to attain the same error level. However, convergence plots for outer and inner strategies are quite different, see Fig. 5a and Fig. 5b, respectively. While the outer strategy behaves regularly and converges at a constant rate, the inner strategy is less predictable. In some cases, the computed correction does not converge (it stagnates but no divergence is observed), as indicated in red in Fig. 5b. A possible explanation may be that two kinds of non-linearities are addressed simultaneously inside the inner loop, see Algorithm 2: the computation of the multiplicative factors of the correction and the nonlinear term itself. Despite its lack of robustness, the global convergence of the inner strategy is faster, as less ADA calls are required to attain the predefined error level.

In order to evaluate the cost of cross approximation for both outer and inner approaches, the total number of calls to the fiber search algorithm (FSA) is evaluated. This quantity is defined, in each case, as follows:

C outer FSA = ℓ ⋆ ∑ ℓ=1 S ℓ and C inner FSA = M ⋆ ∑ M =1 ℓ ⋆ M ∑ ℓ M =1 S ℓM , ( 32
)
where S ℓ is the number of modes of the non-linear term at each non-linear iteration ℓ. On the other hand, S ℓM is the number of modes of the non-linear term at iteration ℓ of mode M . Therefore, the inner scheme is likely to require a higher cost in terms of FSA calls. Fig. 6 compares the number of ADA versus FSA calls for both inner and outer schemes, and several values of the non-linear term's magnitude.

Parametrized fully-developed flow

To further assess the performance of the cross approximations algorithm, in this section we consider a problem of practical interest. The problem at hand consist in the fully developed inertialess flow of two stratified non-Newtonian fluids in a channel with square cross section. The flow field is described in terms of the streamwise velocity component w(x, y). The governing equation is formulated as follows:

∇ • (η∇w) = - ∂p ∂z , (33)
in which ∂p ∂z is the pressure gradient in the streamwise direction, which is arbitrarily set to the unit value, while ∇ := i∂/∂x + j∂/∂y is the gradient operator in the cross section plane. The non-linear viscosity is modeled according to the Carreau-Yasuda law:

η( γ) = η ∞ + (η 0 -η ∞ )(1 + (τ γ) a ) n-1 a . ( 34
)
In the latter expression the viscosity is a function of the shear rate:

γ = √ ( ∂w ∂x ) 2 + ( ∂w ∂y ) 2 . ( 35
)
The material parameters η ∞ , η 0 , τ , denote respectively the infinite and zero shear viscosities and the characteristic relaxation time of the fluid, while the phenomenological constants n and a are adjusted to reproduce shear thinning behavior. In what follows, we shall assume n = 0.2 and a = 2. In practical applications stratified flow of shear thinning fluids can be found in simulation of multi-component coextrusion flows of molten polymers [START_REF] Borzacchiello | On the mechanism of viscoelastic encapsulation of fluid layers in polymer coextrusion[END_REF][START_REF] Borzacchiello | Three-dimensional finite volume computation of viscoelastic fluid encapsulation by phase-field modeling[END_REF]. In this section we assume a bicomponent three-layer flow with uniformly distributed thickness of the fluid layers as depicted in Fig. 7. In practice, for fixed layer thicknesses and pressure gradient, the difference in viscosity between the two fluids determines how the flow rate is distributed in the cross section. Indeed, the layer with smallest viscosity is characterized by the highest flow rate. The differences in the velocity profile between low viscosity and high viscosity core configuration are also shown in Fig. 7.

For the numerical solution, the equation of flow is considered in its dimensionless form. Therefore the following parameters are defined:

µ 1 = η 02 η 01 ∈ [0.1, 10] Zero shear viscosity ratio µ 2 = η ∞1 η 01 ∈ [0.1, 1]
Infinite to zero shear viscosity ratio (fluid 1)

µ 3 = η ∞2 η 02 ∈ [0.1, 1]
Infinite to zero shear viscosity ratio (fluid 2)

µ 4 = τ 2 τ 1 ∈ [0.

1, 10] Characteristic relaxation time ratio

When written in dimensionless form for two fluids, the expression for the viscosity given in Eq. ( 34) reads as:

Fluid 1: η 1 ( γ) = 2 1 + µ 1 ( µ 2 + (1 -µ 2 )(1 + γa ) n-1 a ) , Fluid 2: η 2 ( γ) = 2µ 1 1 + µ 1 ( µ 3 + (1 -µ 3 )(1 + (µ 4 γ) a ) n-1 a
) .

(

) 36
Both space and parametric dimensions are discretized using standard first order Finite Element approximation. Each dimension is discretized with 64 equally spaced nodes.

The problem is then linearized using Picard iterations. At iteration ℓ, the viscosity is estimated using the cross approximation scheme based on the velocity field at iteration ℓ-1. The iteration loop is started using the Newtonian fluid flow as initial solution guess and is stopped when the relative variation in the solution between two successive iterations is less than 10 -4 . The convergence diagram of this iterative procedure is presented in Fig. 8.

At each non-linear iteration, PGD sub-iterations are used to construct a separated variables format of the solution (outer linearization scheme, see Section 5.1). In particular the following tensor format is enforced:

w(x, y, µ 1 , µ 2 , µ 3 , µ 4 ) = M ∑ m=1 W m 0 (x, y) W m 1 (µ 1 , µ 2 ) W m 2 (µ 3 , µ 4 ). ( 37
)
The complexity of the original six-dimensional problem is broken into a succession of three two-dimensional problems. The final solution consists of M = 28 terms. In terms of memory, the reduced solution requires 2.63 MByte, whereas the full explicit storage of its equivalent 6D solution would take 512 GByte in double-precision format (recall there are 64 nodes per dimension). This represents a reduction factor in the order of 10 1867MHz DDR3, running under MacOS X 10.12.6. Total RAM consumption during execution never went over 410 MBytes, which could eventually further reduced in an optimized implementation. CPU time of a brute-force approach was not measured for practical reasons, but it can be estimated instead. Assuming an optimistic runtime per simulation of 1 msec, it would take about 5 hours to complete in a single-thread implementation. Runtime for a compression step to reduce the 512 GByte is excluded from this estimation. Finally, the PGD solution is compared to the FEM at randomly drawn points in the parametric space. Results shown in Fig. 9 refer to the velocity profile in the vertical centerline of the cross section and are in good agreement with the direct FEM reference solutions. Indeed, the computed relative error in the maximum norm is in all cases less than 10 -3 .

Conclusions

In this paper, we have shown the importance of tensorization for the efficiency of tensor-based algorithms. The main ideas have been presented in the Proper Generalized Decomposition framework, but they could also be applied to other tensor representations. We have discussed the adequacy of interpolation rather than projection-based approaches as a means to approximate non-linear equations as well as to enforce tensorization. The use of cross approximations reveals simple and efficient for an interpolation-based approximation of models in moderate dimension. Finally, the performance of several schemes for the linearization of tensor problems has been assessed.

A Coefficients definition for alternating directions optimization

Coefficients β r * ∈ R, αr * ∈ R M and γ * ∈ R S in Eq. ( 10) are defined as follows: the rank-S representation of the non-linear term. See Eq. ( 6) and Eq. ( 8), respectively.

Using the coefficients defined above, we can define the following quantities: The restriction of the approximation residual onto direction " * " can be written as follows:

r * := P T (f (u ℓ M ) -f S ) ≡ f (P T u ℓ M ) -P T f S . ( 39
)
Recalling the definition of both u ℓ M and f S , we arrive to:

r * = f (W * α * ) -V * γ * , ( 40
)
where the coefficients α * ∈ R M and γ * ∈ R S are defined as follows: Here "•" stands for the Hadamard (component-wise) product.

Figure 1 :

 1 Figure 1: Illustration of tensor fibers and crosses definition, for a given multiindex ρ. The residual restriction onto its corresponding fibers, r * , are also indicated.

Algorithm 1 1 :

 11 The outer linearization strategy Set ℓ = 0 and flag = true 2: Solution initialization: u ℓ 3: while ( ℓ ≤ ℓ max & flag ) do ▷ Outer loop 4:

1 :

 1 Set M = 0 and flagPgd = true 2: Solution initialization: u M ≡ 0 3: while ( M ≤ M max & flagPgd ) do ▷ Outer loop 4:

3 m=1Figure 2 :

 32 Figure 2: PGD solution (σ = 10) at last iteration (ℓ = 11), using an outer linearization strategy. Solution's rank is M = 19.

3 s=1Figure 3 :Figure 4 :

 334 Figure 3: Cross Approximation of the non-linear term (σ = 10) at last iteration (ℓ = 11), using an outer linearization strategy. Solution's rank is M = 19. Non-linear term's rank is S = 14.

Figure 5 :

 5 Figure5: Residual reduction of both outer and inner linearization strategies, varying the magnitude of the non-linear term with the coefficient σ. In Fig.5a, each dot stands for a non-linear iteration, while in Fig.5beach dot stands for a mode.

(a) σ = 1 (b) σ = 5 (

 15 c) σ = 10 (d) σ = 15

Figure 6 :

 6 Figure 6: Assessment of the algorithmic performances of both outer and inner strategies: ADA versus FSA calls, varying the magnitude of the non-linear term with the coefficient σ.

Figure 7 :

 7 Figure 7: Schematics of bicomponent three-layer flow of non-Newtonian fluids in a square cross section channel. Velocity profiles for the high viscosity core configuration (a) and low viscosity core configuration (b).

Figure 8 :

 8 Figure 8: Convergence of the non-linear problem measured in terms of the relative difference between two consecutive iterations (stagnation criterion).

µ 1 =Figure 9 :

 19 Figure 9: Comparison of the PGD solution against the direct FEM solution at the vertical centerline of the cross section, for four randomly drawn parameter combinations.

à 2 ⟨

 2 αr * -V * γ * , that can be introduced in Eq. (10), leading to the following minimization problem:w * = arg min v * ∈R N * 1 Ã * v * , v * ⟩ * + ⟨ b * , v * ⟩ * , (38)whose solution is w * = Ã-1 * b * .B Fiber search algorithm in detailLet P := ⊙ D d=0 P d ∈ R N ×1 be an extractor matrix, with P * := I ∈ R N * ×N * . Additionally, P d ∈ R N d ×1 , for d ̸ = * , are extractor matrices full of zeros, except the entry ρ d (see Section 4.1), which is equal to one.

We shall use the term tensorization throughout the entire paper for the sake of conciseness.

Illustrating cross approximations

To illustrate cross approximations, we consider σ = 10 and an outer linearization strategy, see Section 5.1. Convergence is measured with the relative norm-2 of the residual, with respect to the first iterate. Stopping tolerance is set to 10 -6 . In these conditions, the solution is obtained after ℓ = 11 non-linear iterations, and the final rank of the solution is M = 19. Fig. 2b and Fig. 2c show, respectively, the first three modes (unit norm) of the solution in 0-direction {w m 0 } 3 m=1 and 1-direction {w m 1 } 3 m=1 . Fig. 2a shows the 2-D reconstruction of the solution.

We illustrate now the tensorization using cross approximations of the nonlinear term in the last iteration before convergence, ℓ = 11. We denote by f S := (V 0 ⊙ V 1 ) γ an approximation of the non-linear term in Eq. [START_REF] Falcó | A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach[END_REF]. The convergence tolerance of cross approximations is set to 10 -9 , yielding an approximation rank S = 14. Fig. 3c and Fig. 3d show, respectively, the first three modes (unit norm) in 0-direction {v s 0 } 3 s=1 and 1-direction {v s 1 } 3 s=1 . The interpolation indices associated to all 14 modes are depicted in Fig. 3b. On the other hand, Fig. 3a shows the reconstruction of the non-linear term's approximation.

We shall now illustrate the incremental construction of the non-linear term subspace. Fig. 4a shows the 2-D reconstruction of the initial residual, that is, when S = 0. The maximum of the residual is sought using a fiber search algorithm which is initialized with a random index. After 4 iterations, the algorithm converges to the point ρ 0 = (0.54, 0.19), depicted in red in Fig. 4a, as well as its associated cross. Fig. 4b and Fig. 4c show the residual after one term and two terms have been computed, respectively, as well as their corresponding interpolation indices. Finally, Fig. 4d shows that the final residual is in the order of 10 -9 .

Comparison of linearization strategies

The outer and inner linearization strategies defined in Section 5 are discussed here. To this end, we introduce a quantity measuring the cumulated number of mode computations, or equivalently, the total number of calls to the alternating directions algorithm (ADA) within PGD. This quantity is defined, in each case,

About the authors

Antonio Huerta

Laboratori de Càlcul Numèric (LaCàN). Departament de Matemàtica Aplicada III. E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universitat Politècnica de Catalunya, BarcelonaTech, 08034 Barcelona, Spain. e-mail: antonio.huerta@upc.edu web: http://www.lacan.upc.edu/


Dacia Dressen-Hammouda

Adapting methods for teaching work-based literacies to a changing workplace context: Using protocol-assisted modeling

Introduction 1.1 Context for the study

The field of technical writing has undergone significant changes over the last fifteen years. No longer is it sufficient to have "good writing skills" and produce clear manuals. Changing business practices have profoundly transformed what traditionally were writing tasks into the full-scale management of a company's "information meta-products". Writing professionals must now design information by skillfully combining several semiotic resources for each audience, situation and context, including textual, visual, spatial, audio, video, and code. One question teachers can legitimately ask in this context is, how can university programs in professional writing best help prepare students for such complex work-based literacies?

While teaching particular professional genres (e.g., a set of instructions, a user manual) can be one way to start, it is highly unlikely that such knowledge alone will help students learn to function successfully as professionals in such a complex, dynamic and multi-semiotic context. This observation is made even more salient when we consider that since [START_REF] Ivanič | Writing and Identity: The Discoursal Construction of Identity in Academic Writing[END_REF] we know that mastering contextual discourses and genres is as much, if not more, a matter of taking on new -and often competing -identities as it is of knowing how to wield the relevant surface features of the genres students will need to produce as professionals. For this reason, students typically have a hard time taking on an authentic voice and professional identity in their productions, an entirely expected problem since they are not yet legitimate members of the workplace community they are aiming to participate in (Lave & Wenger 1992).

The problems engendered by taking on new identities have been central in writing research since [START_REF] Ivanič | Writing and Identity: The Discoursal Construction of Identity in Academic Writing[END_REF]. In effect, the difficulties involved in taking on the identity of a target community can impede a writer's legitimate participation in that community (see also Lea & Street 1999, Lillis 2002, Street 1995[START_REF] Schriver | Dynamics in Document Design: Creating Text for Readers[END_REF]. Today, much work still remains to be done on exploring the implications of identity for teaching professional and academic writing. In technical writing, for example, developing a relevant professional identity requires simultaneously fostering relevant attitudes and behaviors as well as skills in technical and professional writing, information design, visual design, user studies methods, project management, scientific research methods, web/mobile application design, tutorials and online help design, and communication strategies. This topic is also an important area of research for international networks of writing researchers, especially in terms of intercultural communication. In effect, students in professional and academic writing programs in Europe are most often required to navigate the conditions of learning to write not only in their first language, but often in one or several additional languages, as well. What this has to say about switching between the salient and implicit norms from one language to the next is very relevant to issues of identity in writing, as it adds to the complexity of the task. Students learning the indexicalities of professional writing in the context of technical communication and information design must also attend to the ways in which those indexicalities differ across cultures. In addition to writing, they must learn to pay attention to the indexes of community belonging -attitudes, behaviors, implicit mental representations -including navigating how these indexicalities are expressed differently in different cultures. This is certainly the case for professionals who were former students in our program. Today they must juggle managing technical communication projects across both languages and cultures, in French, English, and one or several additional languages (German, Italian, Spanish, Portuguese, Russian, Arabic, or Chinese). Given the complex pedagogical needs of such students, it may perhaps be somewhat unexpected that the primary claim to be made in this paper is that stimulating awareness and consideration of audience, long a priority in writing research and pedagogy, is still essential. In effect, to be effective, writers need to have "an image of the intended audience interacting with the text" (Schriver 1989, p. 4). This paper describes results from an ongoing study mapping the emergence of situated work-based literacy in French and Dutch technical writing students, using a method specifically designed to elicit more well-developed representations of the actual readers and users of their work. It uses a modified version of [START_REF] Schriver | Teaching writers to anticipate readers' needs: A classroom-evaluated pedagogy[END_REF] protocol-assisted modeling (PAM) approach, designed to help writers construct a more accurate mental model of how readers think and react to information. Fifteen students enrolled in a master's program in technical communication participated in a semester-long study in which they were asked to predict potential problems for readers in six poorly designed documents, then revise their evaluation based on a think-aloud protocol transcript showing how the document's actual users reacted to it. A post-test assessed changes in their ability to anticipate reader problems. Writing pedagogies such as these may better contribute to the long-term construction of students' professional identities. In addition, the approach could very well be a relevant addition to writing pedagogies by researchers in discourse studies, applied linguistics or English Language Teaching, in that it specifically helps students learn how to better target the specific audiences they are writing for, including audiences whose native language and culture are not those of the author.

2. Background: What is protocol-assisted modeling?

Protocol-aided revision

In the early 1980s, [START_REF] Swaney | Editing for comprehension: Improving the process through reading protocols[END_REF][START_REF] Hayes | Uncovering cognitive processes in writing: An introduction to protocol analysis[END_REF]) developed a technique for helping writers to revise texts based on a method they adopted from industrial performance testing: think-aloud protocols (also described at length by Carroll 1997). In this method, participants were asked to perform a task while thinking aloud as they interacted with a document and/or machine. Because they were asked to verbalize anything that came to mind while they were carrying out the requested task(s), their comments revealed the key difficulties they encountered while using the document, both in terms of problem type and placement.

The resulting technique adapted by [START_REF] Swaney | Editing for comprehension: Improving the process through reading protocols[END_REF][START_REF] Hayes | Uncovering cognitive processes in writing: An introduction to protocol analysis[END_REF], called "protocol-aided revision", provided think-aloud protocols to writers to help them revise their texts: "Evaluators make visual and/or audio recordings of readers as they think aloud while comprehending a text. The transcripts are then analyzed for evidence of readers' problemsolving strategies, comprehension, misuse and error recovery, access and retrieval behaviors, inferences and predictions, level of satisfaction and preferences. This information was then used to guide the revision of: insurance forms, apartment leases, computer manuals, or medical support forms." (Schriver 1989, p. 28) 2.2 Reader protocol teaching method [START_REF] Schriver | Teaching writers to anticipate readers' needs: A classroom-evaluated pedagogy[END_REF][START_REF] Schriver | Dynamics in Document Design: Creating Text for Readers[END_REF] adapted the protocol-aided revision approach to teach writing to university students, with the idea being that the approach would help writers develop a stronger image of readers. In a series of papers [START_REF] Schriver | Evaluating text quality: The continuum from text-focused to reader-focused methods[END_REF](Schriver , 1990[START_REF] Schriver | Teaching writers to anticipate readers' needs: A classroom-evaluated pedagogy[END_REF], she reviews a number of readercentered techniques that are typically used to help teach writers how to create "an image of the intended audience interacting with the text" (Schriver 1990, p. 4). Among them, she cites audienceanalysis heuristics, peer review, and role-playing, which she found to be less successful than another technique, "protocol-assisted modeling" (PAM, also known as the "reader protocol method", Schriver 1992, and "protocol-aided audience modeling " Schriver 1997, p. 475). In an experimental study [START_REF] Schriver | Teaching writers to anticipate readers' needs: A classroom-evaluated pedagogy[END_REF], she compared the effectiveness of traditional audience awareness methods and PAM, and found that students who were exposed to the PAM method were: "better able to diagnose problems from the readers' point of view; more sensitive to problems caused by omissions or information gaps; more aware of problems at the global level of text." (1992, p. 202) The PAM method was shown to improve her student writers' ability to anticipate readers' needs. She suggested that the method could be used as a helpful supplement to other methods for teaching the process of 'imagining the reader.' As she also argued, the method shows significant potential for teaching tacit and perceptual knowledge because no explicit instruction is given to students on how they are to evaluate the texts and anticipate reader problems. In other words, by analyzing samples of reader responses to various texts, students gradually create their own 'intuitive', but empirically grounded, understandings and mental representations about real reader information needs. This is important because Previous research and classroom experience tell us that examples can be powerful ways of teaching perceptual knowledge, that is, the "seeing" or "listening" abilities that allow people to recognize significant categories of objects or events in their environment (Bransford, 1979;Brown & Campione, 1981;Sweller & Cooper, 1985;VanLehn, 1986;Zhu & Simon, 1987). Perceptual knowledge is important for instigating or guiding action in a variety of contexts for learning and doing. […] In particular, I explored the hypothesis that extensive experience in interpreting readers' feedback (provided through transcripts of think-aloud reading protocols) would help writers to become more aware of how readers construct text. (Schriver 1992, p. 182)

A description of protocol-assisted modeling

In this teaching method, in which students are taught over ten lessons to better anticipate reader reactions, [START_REF] Schriver | Teaching writers to anticipate readers' needs: A classroom-evaluated pedagogy[END_REF][START_REF] Schriver | Dynamics in Document Design: Creating Text for Readers[END_REF] proposes a four-part approach:

(1) present students with a problematic document (i.e., a poorly written and poorly visualized document that will cause comprehension problems). Students are asked to read the text and identify problems for the reader;

(2) give students a guided writing task in which they describe the features of the text they think will pose problems for the reader;

(3) have students read a think-aloud protocol transcript of a user attempting to understand and use the document;

(4) give students a second guided writing task in which they describe any other problematic features they may have missed when first evaluating the document. A sample PAM from Schriver (1997, p. 477-478) is provided (Figures 1234). Figure 1 shows an excerpt taken from a problematic instructions text, while Figure 2 shows the writing prompts students received after reading the original text.

EMACS: Commands for creating text

EMACS enables you to manipulate words, sentences or paragraphs of text. In addition, there are commands to fill text, and convert case.

Editing files of text in a human language ought to be done using Text mode rather than Fundamental mode. After describing the document features they predicted would be problematic for readers, students then read the think-aloud protocol transcript of an actual user trying to navigate that same document (Figure 3). As the final task in the exercise, students are once again asked to respond to a set of written prompts (Figure 4), intended to help them identify any other problems in the original document after having read about how a particular user reacted upon reading the document. Despite their potential, using think-aloud protocols to create teaching materials for developing audience awareness does pose a number of problems, as described by Schriver (1989, p. 27):

-Think-aloud protocol transcripts inevitably have "gaps" when participants forget to speak. In effect, one of the challenges involved in using think-aloud protocols is keeping the participants talking throughout their entire interaction with the document and/or machine. Quite often, people

Writing task (1):

1. Read the draft document.

2. Predict the reader's problems with this document.

3. Write your answers below. Editing files of text in a human language human language? Boy that sounds strange, what could they be distinguishing here? Maybe computer language or machine language from human language? ought to be done using text mode rather than fundamental mode. Writing task (2):

Ok, now I'm going to try …

1. Now, read the think-aloud protocol transcript.

2. Use the reader's comments to identify additional problems in the draft.

3. Write your answers below.

forget to speak as they become engaged with the task at hand, causing the transcript to give an incomplete account of the problems they encountered. The final transcript might therefore fail to fully account for all the problems a particular user may have had, although it can still give students a very good idea of many of the problems actually encountered.

-The second point follows from the first: participants may make "mental leaps" and reach a conclusion without verbalizing the intermediate steps of how they reached their understanding, once again leading to potentially incomplete information in the protocol transcript.

-If the original document contains visuals or other nonverbal representations, participants may be unable to verbalize what they are thinking.

-Protocols are time-consuming to make.

This final point is of particular concern when deciding whether or not to integrate the method into a professional or academic writing class. In effect, a significant amount of time needs to be set aside for creating the teaching corpus, including finding suitable examples of documents. Ideally, documents chosen should show clear enough problems from a pedagogical point of view while, at the same time, not being overloaded with problematic features. Clear criteria on whether or not to include the documents in the teaching module need to be decided at the outset, so as to avoid going through the inconvenience of spending a considerable amount of time both recording a live participant user-testing the document and transcribing the session, only to conclude in the end that the document is pedagogically unsuitable. In addition, issues of pedagogical progression -in other words, deciding whether it would be interesting to have students analyze one document before another, and why -should also be addressed before running the module in a classroom setting.

Another drawback concerns whether or not, due to study size, it is feasible to evaluate the method's effectiveness in comparison to the other methods commonly used to develop audience awareness (audience-analysis heuristics, peer review, or role-playing). In [START_REF] Schriver | Teaching writers to anticipate readers' needs: A classroom-evaluated pedagogy[END_REF] rigorously constructed experimental study, the studied population included 117 students enrolled in ten professional writing classes. The population size made it feasible to compare methods, using five classes as an experimental group, and five as control. Depending on teaching conditions, replicating Schriver's study could prove unethical if participants are constrained to a single class, and are taught variously using either PAM or traditional audience awareness methods. Such would have been the case in the current study, which describes the implementation and results of using PAM in a single group of 15 students. As a consequence, this study takes Schriver's claims about the effectiveness of PAM as its starting point, in order to focus on how it might contribute to the development of the students' professional identity and workplace literacies.

The purpose of the study was to examine the implementation of the PAM method with a small group of Master's students in technical communication and information design. The results from the study are very much to be taken as preliminary within a larger study underway looking at how professional writing programs can better help students learn the indexicalities of professional writing, specifically in the context of technical documentation and information design in France. In the conclusion, the method is evaluated, looking specifically at whether it effectively prepares students for the complex workplace contexts they will encounter, in which their ability to frequently and rapidly adapt to user and contextual information needs is crucial.

Methods

Research questions

A corpus of texts was designed for a 15-hour, first-year Master's course on information design, where the topics ranged from a focus on message design and Plain Language principles, to design theories and graphic design principles, visual design (color, shape and typography) and information visualization. The PAM exercises were created specifically to reflect these aspects of the course.

All students had previously been introduced to traditional audience analysis techniques (using guidelines, peer review and audience analysis) in a technical communication course taught the previous semester. At the end of the course, they produced and user-tested a user guide. During the user tests, students were also initiated into carrying out and transcribing think-aloud protocol analysis, and so had been familiarized with the technique.

The current study was designed to answer the following two questions:

(1) Does the PAM teaching method impact how well students identify problems in a document?

(2) If yes, how so?

Participants

Fifteen first-year Master's students participated in the study. Of those fifteen, only nine participants were targeted for analysis because they provided a full set (for various reasons, the remaining six participants were unable to complete the entire set of PAM exercises). Of those nine participants, seven were native speakers of French, and two of Dutch. Five were male, and four were female. Schriver (1997, p. 477-78) was used as a model to create a corpus of eight PAM exercises. The corpus included Schriver's sample, plus seven further exercises, each of which consisted of the fourpart approach shown in Figures 1234. Of those, two exercises were eliminated retrospectively from the analysis due to faults in their design. The other five PAM exercises used the following original documents: (1) the online explanation originally provided by Amazon when it introduced its currency converter;

Materials

(2) an administrative university document providing instructions for setting up hiring committees;

(3) a set of instructions including both text and images on how to install a wall clock; (4) an online user guide for using plagiarism detection software; and (5) an educational brochure intended to help adolescents safely manage their interactions with the police. Two of the documents were in French (1 & 2), the remaining four were in English (3-5, plus the sample PAM shown in Figures 1234).

Procedure

Participants carried out a PAM exercise each week. Alternately, one was carried out in class, and the other was completed independently outside of class. Each PAM exercise was made available on the course Moodle, and participants were asked to upload their completed PAMs onto the course Moodle. All participants responded to the writing prompts solely in English.

Results

Comparing the results obtained in the current study to the original claim, we can recall that [START_REF] Schriver | Teaching writers to anticipate readers' needs: A classroom-evaluated pedagogy[END_REF] found that students who had been exposed to the PAM method had a greater ability to:

identify and propose solutions to the global aspects of text quality (i.e., in terms of rhetoric, organization, logic, cohesion, and tone);

predict readers' needs.

Progressive increase in the length of written responses

One of the first observations to be made in the current study is that the participants' analysis of the documents became more detailed over time. This observation is reflected first and foremost in the number of words, as seen in Figure 5. While at the outset, both the first and second written responses averaged the same length (i.e., about 40 words), as time went on, participants proposed more observations based on their own analysis of the documents, than they did after reading the think-aloud protocol transcript.

Early focus on surface features

Interestingly, not only did the written responses increase quantitatively, they also demonstrated a qualitative shift in focus. The earliest written responses showed a greater tendency to focus on surface-level features, such as word choice (e.g., specialist jargon) and inappropriate grammar (i.e., non-compliant with Plain Language principles). In the first PAM exercise, for example, participants used a "checklist" approach in their evaluation, attempting to match the principles they had previously studied with the problems they identified in the document:

"Complicated wording and syntax. References to items not present. Vocabulary."

"Problems with specific vocabulary (domain specific, no definitions…). Problems with wordiness: useless words."

"Problem with document: difficult to understand, technical language." "Sentence complexity."

Over the course of repeating the exercises, however, they increasingly moved toward a more reader-centered approach, making more predictions about the problems readers would have based on the documents' actual features, rather than a checklist. In turn, the evaluations they made in the first guided writing responses became comparatively longer than the second guided responses. They also appeared to integrate the 'before' and 'after' analyses from the previous exercises, and to use that to formulate their own, more detailed observations, with progressively less and less recourse to the think-aloud protocol transcripts to identify the problems.

Increasing focus on predicting reader reactions

For example, over time, participants steadily focused more on predicting actual reader reactions based on the problems they themselves had identified in the documents. They indicated how the actual document features might impede reader interaction, as seen in comments such as:

"A user that is skimming the document will have a hard time finding the chapter titles. Too many red arrows and red lines in the images make it difficult to discern what is important. The use of arrows is not linked to the text, making it difficult to establish connections between the text and the images. Sometimes the image is on the left of the page and sometimes on the right which makes it difficult to establish a structure." "Readers will face problems with document usefulness. The content is neither appropriate nor structured. Multiple negatives will cause confusion. Text complexity and the way it's written do not take into account the user's needs. The content focuses entirely on liability, and the reader will be naturally unmotivated. Amazon encourages this lack of motivation with obscure language. Users will quickly forget the content (lack of accessibility)."

Increased proposals revision (unprompted)

In addition, all participants gradually offered more diagnostic analyses of the documents, as well as solutions for revision. On both first and second response sheets, they increasingly commented on what, in their opinion, the document authors should have done differently. Such evaluative comments are particularly noteworthy because they were entirely unprompted by the research design (i.e., participants offered them freely, without being explicitly asked to do so by the guided writing tasks): "The guide is not effective: it contains the right information but not in the right order, it should be reorganized (like a forum, with QA format)."

"This paragraph should be situated at the end of the brochure because when people are looking at this they can think that the following information is not important and they won't read it." "Too much text will make the reader skim and miss important information. Warnings at the beginning risk being overlooked. Readers will only read relevant info (7. Operation); they may miss important information. Make an index and use short sentences, make steps. Put warnings in places where absolutely needed. Start the document with the instructions and add other information later." "The background is too stimulating, and risks tiring the reader before he even starts. The yellow box on the right has information in it. It's a menu I think and the tags… I don't know what it is supposed to mean. The fact there is a similar box on the right means that the reader does not know where to start scanning for useful information. Use a calm background. Restructure the entire page. Put the menu in a clearly identified position. Use only 1 box. Maybe put a long horizontal block at the top with menus like on a website. Remove unnecessary and unhelpful information and images."

Discussion and conclusion

Shortcomings of the study

One of the shortcomings in carrying out this study stems from the difficulties encountered in constituting the corpus. In addition to being time-consuming, questions were raised retrospectively about the pedagogical appropriateness of the exercise progression. Whether the order in which the exercises are presented make a difference in terms of how well participants perceive the imagined needs of a document's readers will need to be validated in a later study.

Relatedly, results from the two final exercises (from the original eight) were discarded during the analysis phase, as it became apparent that they were inappropriately placed in terms of pedagogical progression: the problems in the original documents were far too numerous and blatantly obvious to maintain credibility in the eyes of the students, who after completing several PAM exercises were showing substantial sophistication and thoroughness in their evaluations.

How the PAM method contributes to building an effective professional identity

The PAM method is an inductive method, meaning that it does not explicitly point out to students what aspects of the information are relevant to pinpointing difficulties readers may have with a document. In this sense, it allows them to develop -on their own -an 'intuitive' sense of the strengths and shortcomings of professional documents: their perceptual knowledge [START_REF] Schriver | Teaching writers to anticipate readers' needs: A classroom-evaluated pedagogy[END_REF].

Through its ability to develop students' mental representations of a document's readers, the PAM method helps them better adapt their attitudes, behaviors and abilities to a company's needs. There is indirect evidence that the method has already been successful in helping the students who participated in the study to develop a more appropriate professional identity. One line of evidence comes from the positive evaluation of the students' performance by their internship tutors over the course of a mandatory 12-to-16 week company internship. In addition, of the nine students who participated in the study, six have received work-study contracts for their second year as a direct result of the quality of their work during their internships. In France, a work-study contract is the much coveted contrat d'alternance, which allows students to work as salaried employees in a company throughout the second year of their Master's program (spending three days in class, two days in company training); the company covers tuition fees, as well as travel and housing expenses while on site.

The next step in the study will be to build on these initial findings by carrying out a qualitative exploration of the impact of the PAM method, tracking the six students over the course of the coming year during their work-study contracts. To further explore how the PAM method influences students' ability to rapidly adapt their projects to widely variable user/shareholder information needs, the following methods are planned as part of an ongoing project:

semi-structured interviews with the six students, their company tutors and colleagues; onsite observations of their work practices and interactions in the workplace; reflective journals on their changing abilities to design effective information, the difficulties they encounter when doing so, and the strategies they put into play in order to resolve them.

Study implications

Although relatively little attention was paid in the writing research community to Protocol-Assisted Modeling when it was first proposed, it is clearly time to re-assess its potential as a major contributor to professional and academic writing programs, especially in terms of fostering new identities and complex workplace literacies. The implications of using Protocol-Assisted Modeling are far-reaching:

(1) The perceptual knowledge that arises when writers learn how to construct valid mental representations of readers / users, once set in motion, appears to become a sort of procedural knowledge that crosses both genre and situational boundaries. When writers learn how to effectively imagine and predict reader reactions in one genre or situation, they appear able to apply that knowledge to other genres and situations. Schriver (1989, p. 250) raises this possibility when she notes that the students who participated in the control groups (i.e., were exposed to the PAM method) transferred their ability to detect document problems from one genre and audience (computer manuals for instruction) to another (elementary science texts written for lay people). In this respect, being exposed to Protocol-Assisted Modeling may impact the construction of professional writers' identities in ways which have yet to be uncovered. (2) The PAM method can also be used to help speakers of additional languages gain access to the hidden indexicalities that make their mastery of writing in a foreign language so problematical. Writing skillfully and with legitimacy is clearly not a question of just learning the relevant genres' surface features, but is also highly dependent on a writer's ability to access what is both implicit and inferred, and wield that in contextually appropriate ways. While gaining access to the indexical knowledge that acts as a 'gatekeeper' for institutional

Figure 1 :

 1 Figure 1: An excerpt from a problematic document

  Invoke M-X Text Mode to enter Text mode. See section 20.1 [Major Modes], p. 85. M-X Text Mode causes Tab to run the function ˆR Tab to Tab Stop, which allows you to set any tab stops with M-X Edit Tab Stops (See section 11.3 [Indentation], p. 46). Features concerned with comments in programs are turned off except when explicitly invoked. Automatic display of parenthesis matching is turned off, which is what most people want. Finally, the syntax table is changed so that periods are not considered part of a word, while apostrophes, backspaces and underlines are. If you are editing input for the text justifier TEX, you might want to use TEX mode instead of Text mode. See the file INFO: ETEX. INFO

Figure 2 .

 2 Figure 2. First guided writing task, given after students read the original document

Figure 3 .

 3 Figure 3. An excerpt from a think-aloud usability protocol transcript

Figure 4 .

 4 Figure 4. Second guided writing task, given after students read the think-aloud protocol transcript

  Commands for English text. EMACS enables you to manipulate words, sentences, or paragraphs of text. These commands sound like ones I'd use all the time, good. In addition, there're commands to fill text, and convert case. I don't know what it means to fill text, I guess it means putting data from one text into another … that is, filling the text with what you want in it. Well, I guess I'll soon find out.

  If you are editing input for the text justifier TEX you might want to use TEX mode instead of text mode. See the file info etex info.Well, I don't want to edit input for the text justifier mode, I just want to figure out how to manipulate words and paragraphs. Like it says up here (points to top of section). This stuff is too complicated and it's aggravating to read.

Figure 5 .

 5 Figure 5. Length of written responses over time Number of PAM exercises Number of words per response

Well, I don't know what text mode or fundamental mode is, so how will I know which I'm in? Let's see

… Invoke MX text mode to enter text mode. I won't

do that because I don't have time to see the other section. That's terrible to tell me to… See

  section 20.1 major modes page 85 MX text mode cause tab to run the function ˆR tab to tab stop. Now

wait a minute, what is all this stuff? Sounds important, but I have no idea what it could mean. I know what tabs on a typewriter are, and I see a tab key here, but what the heck does running the function ˆR tab to tab stop mean? Oh god, now they want me to look up another section…

which allows you to set any tab stops with MX edit tab stops see section 11.3 indentation page 46… Just to get started in this one. No I think I'll skip that. I'

ll deal with it when I have to use it. I thought this section was on how to edit. What does setting tab stops have to do with editing

? Features concerned with comments in programs are turned off except when explicitly invoked. Hmmm… a feature?

Wonder what a comment feature is? How do they get turned off? Do I have to turn them off? Well I guess I can't turn them off if I don't know what they are

. Automatic display of parenthesis matching is turned off, which is what most people want. What

does that mean? How do I know if I'm like most people and want them turned off? I wish they'd tell me why I need to know all these hacker terms!

  Finally the syntax table is changed so that periods are not considered part of a word, while apostrophes, backspaces and underlines are.

Syntax table… no idea. Of course, I do know what the syntax of sentences are, so maybe it means something to do with certain commands EMACS considers acceptable… maybe certain actions in a row produce a correct syntax.