Patrick Chlo É Bouard

Alain Warin

Laurent Marty

Van Tuong Vila

Jean-Philippe Pham

Attan

Magnetic properties of perpendicularly magnetized [Au/Co/Pd] n thin films and nanostructures with Dzyaloshinskii-Moriya interaction

Magnetic force microscopy imaging enable to extract the main magnetic parameters (saturation magnetization, anisotropy, domain size, exchange constant). The magnetoresistance and the extraordinary Hall effect of the material have been characterized using nanostructures patterned on the multilayer. Both the magnetization reversal mode and the magnetotransport properties are found to depend strongly on the nanowire width. The relatively low coercive field, the high anisotropy, and the possibility to control domain wall motion in sub-100 nm wires make this system an interesting alternative to Pt-based multilayers for spin-transfer torques experiments.

Materials with perpendicular magnetic anisotropy and high Dzyaloshinskii-Moriya interaction (DMI) have recently attracted a large interest for spin-transfer torque applications based on domain walls (DW) and skyrmions. Indeed, the possibility to repel the Walker breakdown 1,2 and to stabilize Néel walls 3,4 or skyrmions in films 5 or nanostructures [6][7][8] could be the basis of new memory and logic devices. 9,10 Most experiments have been realized in bilayers or trilayers, including a layer of heavy metal and an ultrathin Co 8 or Fe layer. 11 In the case of single bilayers or trilayers, the magnetic texture has been found to be stable only at very low temperature, 12,13 and magnetic fields are required to stabilize a single metastable skyrmion. 14 Another route is the development of thick [metal 1/ferromagnet/metal 2] n multilayers, or of [metal/ferromagnet/oxide] n stacks, such as [Ir/Co/Pt] 10 , 15 [Pt/Co/Ta] 15 and [Pt/CoFeB/MgO] 15 . 16 In such materials, the interface effects could provide both the DMI and the perpendicular anisotropy, while conserving the lack of inversion symmetry at the vicinity of the ferromagnetic layer. It becomes possible to reach thicknesses large enough to facilitate skyrmions or DWs observation, to enhance their thermal stability, 15,16 and to enable the measurement of the topological Hall effect. 17,20 The use of multilayers could help tuning precisely the magnetic parameters such as the DMI, the anisotropy and the saturation magnetization. [18][19][20][21] More importantly, it would allow developing stacks in which most of the current used to induce DW or skyrmion motion would go through the magnetic part. 20 In this context, ab initio calculations 22 have recently predicted that [Au/Co/Pd] n multilayers are an interesting system to perform spin-transfer experiments, thanks to the opposite DMI signs of the Co/Pd and Co/Au interfaces.

In this article, we study the magnetization reversal and magnetotransport properties of [Au (10 Å)/Co (6.6 Å)/Pd (10 Å)] 10 thin films and nanostructures. The magnetoresistance (MR) a Jean-philippe.attane@cea.fr 2158-3226/2018/8(9)/095315/6 and the extraordinary Hall effect (EHE) of the material have been measured using nanostructures patterned on the multilayer. Both the magnetization reversal, which occurs by DW propagation, and the magnetotransport properties, are found to depend strongly on the nanowire width.

[Au (10 Å)/Co (6.6 Å)/Pd (10 Å)] 10 multilayers have been grown by DC magnetron sputtering on a silicon oxide substrate, with a Ti (5 Å)/Au (50 Å) buffer layer, and a Au (30 Å) capping layer to prevent oxidation. The magnetic properties of the sample have been measured using a SQUID magnetometer. The hysteresis loops for an in-plane and an out-of-plane applied field are shown in Fig. 1a. The multilayer is found to possess an out-of-plane magnetization, with a saturation magnetization of 1400 ± 100 kA/m and a large perpendicular anisotropy Keff = 150 ± 10 kJ/m 3 (estimated from the area between in-plane and perpendicular magnetization curves), comparable to that of Co/Pt 23 or Co/Ni 24 multilayers. The saturation magnetization has been calculated considering the ten 6.6 Å-thick layers of cobalt. The hysteresis loop for out-of-plane fields exhibit a remanence of around 30 %, and a relatively low coercive field of 14.5 ± 0.2 mT.

Fig. 1b shows the magnetic configuration of an as-grown thin film, observed by magnetic force microscopy (MFM). The structure of interlaced up and down domains is characteristic of layers with perpendicular magnetization, with a mean domain width of around 135 ± 5 nm (evaluated from the Fourier transform of the MFM images). Note that the mean domain width is also found to be around 135 nm after demagnetization, so that we can suppose that the observed domain width is close to the equilibrium domain width.

The equilibrium width results from the competition between the DW energy and the demagnetizing field energy. Using the analytical model proposed by Kaplan and Gehring, 25 and modified by Gehanno et al., 26 it is possible to extract the DW energy σ DW = 2.6 ± 0.2 mJ/m 2 . The energy of a Bloch DW is σ DW = 4 AK eff , with A the exchange stiffness. Considering A = 5 -15 pJ/m (values reported in literature for Co), we find σ DW = 3.5 -6.0 mJ/m 2 , which is larger than the value calculated from the equilibrium width of magnetic domains. This could indicate the presence of the DMI, as the DMI is known to lower the DW energy by a value equal to πD, 27 D being the DMI constant. The DMI constant value extracted from the DW energy would thus be of 0.7 ± 0.4 mJ/m 2 , slightly smaller than the theoretical prediction of 1.25 mJ/m 2 obtained by adding the two Co/Pd and Co/Au contributions calculated ab-initio, 22 and smaller than what can be obtained using Pt. 15 Note however that this value FIG. 1. a) In-plane and out-of-plane hysteresis loops at room temperature, measured using a SQUID magnetometer. 5×5 µm 2 MFM image of the magnetic domains b) in an as-grown sample and c) in the remanent state after magnetic saturation.

has to be taken cautiously. Further experiments could allow determining more precisely the strength of the DMI. 28,29 Fig. 1c shows the magnetic configuration observed by MFM in the remanent state, after saturation. The reversed domains appear in black. The DWs are less smooth than what was observed in the asgrown state: there is indeed a competition during the growth between the disorder (i.e., the pinning of DWs on structural defects), which tends to roughen the DW, the domain wall energy, which tends to smooth it, and the demagnetizing field effects, which try to impose the equilibrium domain size. 30 The resulting domain width of the growth process remains close to the equilibrium (135 ± 5 nm), which means that the domain structure is mostly governed by demagnetizing field effects. The coercivity, which is due to DW pinning, is relatively small (H C = 14.5 ± 0.2 mT). It is comparable to that of typical [Co/Pt] n multilayers with perpendicular anisotropy, 31 so that one can expect to observe current-induced DW motion at zero field.

The samples have been patterned into double Hall crosses separated, using e-beam lithography on a negative resist and ion milling (cf. Fig. 2a), with wire widths of 100, 200 and 500 nm. The wires are connected to large areas serving both as electrical contacts and domain nucleation pads. The devices have then been wire-bonded to perform magnetotransport measurements. The longitudinal and Hall resistances have been measured using a lock-in amplifier, working at 107 Hz, and with an applied current of 100 µA.

The temperature dependence of the longitudinal resistivity of the entire stack is given on Fig. 2b. The resistivity, which depends on the quality of the sample, is reasonably small (ρ xx = 24.3 µΩ.cm at 300 K, with a RRR ρ xx (300 K)/ρ xx (10 K) = 1.3). The longitudinal MR and the EHE have been measured in thin films and in the double Hall crosses, the magnetic field being applied perpendicular to the specimen. The evolution of these magnetotransport properties as a function of the width of the wires is presented on Fig. 3.

The measurement of the EHE on the thin film is consistent with the SQUID measurements of Fig. 1a. The EHE measurements in the Hall crosses allow to extract the EHE angle of the whole stack. The EHE angle is found to be θ EH = ρ EHE /ρ xx = 0.44 % at room temperature and at 10K, close to [Pt/Co] n multilayers (1.3 % for similar thicknesses). 31 Note that, since part of the current is shunted by the Au buffer layer, this value has to be considered as a lower bound of the EHE angle of the [Au/Co/Pd] 10 multilayer.

The MR curve presents at high fields a non-saturating linear decrease of the resistance, which can be attributed to the magnon magnetoresistance (MMR), i.e., to the contribution of magnons to the resistivity. 32 The value of dρ dB at room temperature is 0.04 µΩ.cm.T -1 or 0.2 %/T for the whole stack, comparable to that of FePt 34 and NiFe, 33 so that the MMR could be used to detect DW motion. 34 The MR curves of Fig. 3b also present a hysteretic behavior at low fields, with an increase of resistance due to the MMR, with possible additional contributions from the anisotropic MR and the DW resistance. Several features appear when measuring the MR and EHE of patterned nanowires. Concerning the Hall measurements of Fig. 3a, the EHE angle is not modified, but there is strong dependence of the hysteresis loop with the wire width. The decrease of the wire width leads to an increase of the remanence, which is of 100 % for 500 nm-wide crosses, an increase of the coercive field, and a decrease of the saturation field. The reasons for this evolution will be discussed later.

As the MMR contribution is proportional to the product M • B, 32 these evolutions of the hysteresis loop are responsible for the evolutions of the MR loops of Fig. 3b, with in particular the sharpening of the MR loop for narrow widths.

Fig. 3c exhibits MFM observations performed on nanostructures of different widths. The imaging being made right after the nanoprocessing, the sample is still in the as-grown magnetic state. For the 200 nm-wide crosses, the DWs tend to align perpendicularly to the nanowire edges, in order to minimize the DW surface, and thus the anisotropy and exchange energy cost. At 100 nm, the magnetic state consists of magnetic domains separated by DWs orthogonal to the wire. This is the ideal configuration to realize current-induced DW propagation experiments. This effect can also be seen in the nanostructure of Fig. 3d, which consists in a single nanowire of varying width. The transition occurs at a width comprised between 100 and 150 nm: when the wire width becomes smaller than the equilibrium domain width, the intertwined domain configuration is obviously unstable.

Let us now discuss the evolution of the loops observed in Fig. 3a for thin wires. A large decrease of the saturation field can be observed in thin wires, from 132 mT in thin films to 62 mT in 100 nm wide nanowires. This can be attributed to the decrease of demagnetizing field effects: while the demagnetizing field tends to prevent the magnetic saturation, in a very thin wire the effect is limited by the geometry.

Conversely, the coercivity increases in thin wires, from 15 mT to 56 mT. As discussed in Ref. 36, the main origin of this increase is the suppression of the available paths for DW propagation. The DW is moving in a disordered medium. In thin wires, the DW is forced to cross all the defects. In large wires, the DW somehow follows the easiest path during its propagation, since it can get around the biggest defects.

Note that at low temperature it becomes possible to observe Barkhausen jumps (cf. Fig. 3e and3f), i.e., the discrete motion of the DW from one pinning site to another. 36 The minor loop obtained in a 100 nm-wide cross shows 4 jumps. The EHE amplitudes of these jumps indicate that the DW reverses approximately a fifth of the cross for each jump, and thus that the typical distance between pinning sites is around 20 nm.

The increase of the coercivity in thin wires can also be seen in MFM images performed in remanent states, after magnetic saturation and partial reversal of the magnetization. Fig. 4a shows that the reversed magnetic domain, appearing in black and coming from a nucleation pad on the right side, does not propagate into the thinnest part of the wire. It can also be seen in Fig. 4b, where the injection and the propagation are more difficult in thinner nanowires.

Note that another effect can contribute to the enhancement of the coercivity: as nucleation centers are distributed randomly, the probability of a narrow area to contain a nucleation center is small. This effect tends to increase the nucleation field of smaller areas. For instance, in Fig. 4b there have been nucleation events in the large wires, but none in the two smaller wires.

In the 500 nm-wide nanowire of Fig. 4b, a single domain propagates in the wire. Interestingly, the DW does not switch the magnetization of the whole wire: as the width is larger than the equilibrium domain width, a multidomain magnetic configuration can be stabilized by the demagnetizing field.

This coercivity increase leads to the increase of the remanence seen in Fig. 3a, as it becomes difficult for a DW to reach the Hall cross. It is also responsible for the abruptness of the magnetization switching: at high fields, it becomes possible for the DW to reverse the magnetization of the whole cross.

To conclude, [Au (10 Å)/Co (6.6 Å)/Pd (10 Å)] 10 multilayers are promising candidates for current-induced DW experiments. Indeed, this new system exhibits a high perpendicular anisotropy, a reasonable coercivity and a low resistivity. As the spin-orbit coupling of Pd is lower than that of Pd, one can expect lower damping constants in Pd-based systems, which could thus be an interesting alternative to Pt-based multilayers for current-induced DW propagation. The extraction of the DMI constant from DW energy estimation suggests that the DMI could be large enough to stabilize Néel walls or skyrmions. Furthermore, in sub-150 nm wires the magnetic configuration consists in series of parallel DWs, which is the ideal configuration to perform current induced DW motion experiments. Further experiments should allow quantifying more precisely the DMI strength, the spin transfer torque efficiency, 1 and the maximal DW velocity that can be obtained under current.

This work was partly supported by the French Agence Nationale de la Recherche (ANR) through Projects SPINHALL (2010-2013) and SOSPIN (2013-2016).

35

 35

FIG. 2

 2 FIG. 2. a) Scanning electron microscopy image of a 500 nm wide cross (tilted sample). b) Temperature dependence of the longitudinal resistivity.

FIG. 4

 4 FIG. 4. a) MFM image of the structure consisting of wires of different widths, after saturation and partial magnetization reversal. b) MFM image of wires of different widths connected to a nucleation pad, after saturation and partial magnetization reversal.


Yassine Ariba

Frédéric Gouaisbaut

Alexandre Seuret

Kun Liu

Stability of distributed delay system with infinite support

published or not. The documents may come

I. INTRODUCTION

For several years, we have experienced a renewal in the study of the stability and stabilisation of delay systems [START_REF]Recent Results on Time-Delay Systems[END_REF]. Indeed, the study of stability is mainly carried out by using a Lyapunov-Krasovskii functional whose parameters are determined by means of a convex optimization scheme. For several years now, these functionals have become very efficient thanks to the relevant choices of their structure [START_REF] Zeng | New results on stability analysis for systems with discrete distributed delay[END_REF] which has become more complex and incorporates new states related to the partitioning of the delay [START_REF] Gouaisbaut | Delay-dependent stability analysis of linear time delay systems[END_REF], integral of states [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF] or the projection of the infinite-dimensional delay state on an orthogonal polynomial basis [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF]. This last feature also allows the use of very precise quadratic inequalities leading to very efficient LMI criteria. These inequalities, called Bessel inequalities, are all primarily based on the relevant choice of an inner product in L 2 and the use of a polynomial orthogonal basis. Thus, Legendre polynomials have been used to develop stability criteria for pointwise delay systems in a Lyapunov-Krasovskii functionals framework but also in a robust analysis setup [START_REF] Ariba | Bessel inequality for robust stability analysis of time-delay system[END_REF]. Notice that these inequalities include the inequalities of Jensen [START_REF] Gu | An integral inequality in the stability problem of time-delay systems[END_REF] and Wirtinger [START_REF] Seuret | Wirtinger-based integral inequality: Application to time-delay systems[END_REF] widely used in the literature on delay systems. Concerning distributed time-delay systems with a finite support, several papers have been devoted to the construction of Lyapunov functionals [START_REF] Chen | Delay-dependent robust stabilization for uncertain neutral systems with distributed delays[END_REF], [START_REF] Feng | Integral partitioning approach to robust stabilization for uncertain distributed time-delay systems[END_REF], [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF] (and references therein). The methodology involving orthogonal Legendre polynomials has also led to results reducing the conservatism of Lyapunov methods and keeping a relatively low numerical complexity [START_REF] Seuret | Complete quadratic Lyapunov functionals for distributed delay systems[END_REF], [START_REF] Gouaisbaut | Stability of distributed delay systems via a robust approach[END_REF].

Attempts to study systems with distributed delay with an infinite support are more scarce. For instance, frequency approaches have been developed to study the stability of Y. Ariba, F. Gouaisbaut, A. Seuret are with LAAS -CNRS, Université de Toulouse, CNRS, UPS, France e-mail: (yariba,fgouaisb,aseuret,@laas.fr).

Y. Ariba is also with the school of engineering Icam, Toulouse, France. Kun Liu is with the Key Laboratory of complex system control and decision, School of Automation, Beijing Institute of Technology, Beijing 100081, China mail: (kunliubit@bit.edu.cn).

This work is supported by the ANR project SCIDiS contract number 15-CE23-0014.

the traffic flow dynamics [START_REF] Morarescu | Stability crossing curves of shifted gamma-distributed delay systems[END_REF] or linear systems with gamma distributed delays [START_REF] Sipahi | Stability of traffic flow behavior with distributed delays modeling the memory effects of the drivers[END_REF]. These systems have also been studied under the angle of an extension of the functional [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF]. The interest of such methods is mainly the development of new quadratic integral inequalities with infinite support of integration. This has led to the construction of new Lyapunov functionals, which, combined with optimization algorithms, allowed at least on examples, to obtain very encouraging results [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF]. It should be noted that this approach has recently been extended by using terms that depend on a double integral in order to significantly improve the results [START_REF] Liu | Generalized jensen inequalities with application to stability analysis of systems with distributed delays over infinite time-horizons[END_REF].

The paper's objective is to study this class of delay systems using robust analysis, a method that has already led to the development of robust stability conditions for constant delay systems (pointwise or distributed). We focus more specifically on the quadratic separation approach whose principle is to model the distributed delay system as a closed loop system between an uncertain dynamical system and a linear transformation. The novelty of the approach lies in the use of a new inner product taking into account the infinite support of the distributed delay system. The kernel of the distributed part is thus, within this framework, approximated by a sum of Laguerre polynomials which are orthogonal for the selected inner product. Notice that the approximation error is also taken into account in the rewriting of the original model. A new comparison model is therefore obtained, in which the uncertainty set, gathering the delay elements, can be precisely described by using a Bessel-Laguerre inequality. Using the quadratic separation theorem, a sequence of LMIs tests, in which conservatism decreases as the degree of Laguerre polynomials increases is then proposed. This approach has to be viewed as the robust counterpart of [START_REF] Liu | Bessel-Laguerre inequality and its application to systems with infinite distributed delays[END_REF], where Laguerre polynomials have been also used in a Lyapunov framework.

Notations: Throughout the paper, the following notations are used. The sets N, R, C represent the set of non negative integer numbers, of real and complex number, respectively. The set R n×m is the set of matrices of n rows and m columns. 1 n and 0 m×n denote respectively the identity matrix of size n and null matrix of size m × n. If the context allows it, the dimensions of these matrices will be omitted. For two symmetric matrices, A and B, A > (≥) B means that A -B is (semi-) positive definite. A T denotes the transpose of A. A ⊥ stands for a full rank matrix whose columns span the null-space of A, i.e. AA ⊥ = 0. diag(A, B) stands for the block diagonal matrix diag(A,

B) = [ A 0 0 B ] = [ A B ]
. The notation A * represents the conjugate transpose of the complex matrix A. L 2 can be viewed as the space of square integrable functions with a finite energy:

f 2 = ∞ 0 |f (t)| 2 dt
. H 2 is the function space which contains the Laplace transform of signals in L 2 . H ∞ is the set of all functions F (s) such that sup Re(s)≥0 σ(F (s)) ≤ ∞, where σ stands for the maximum singular value.

II. PROBLEM STATEMENT

Let us consider a distributed delay system of the form:

ẋ(t) = Ax(t) + A d ∞ 0 K(θ)x(t -θ -h)dθ, (1)
where x(t) ∈ R n is the state vector and is assumed to belong to L 2 , A and A d ∈ R n×n are constant matrices. The constant delay h is unknown but belongs to an interval [h min , h max ].

The initial condition is given by φ ∈ C 1 (-∞, 0], where C 1 (-∞, 0] denotes the space of continuously differentiable functions φ : (-∞, 0] → R n with the norm

φ C 1 = sup s∈(-∞,0] |φ(s)| + sup s∈(-∞,0] | φ(s)|.
The kernel function K is assumed to be a scalar function of the form

K(θ) = f (θ)e -αθ ,
where the scalar function f is continuous and verifies ∞ 0 f (θ)e -αθ dθ < ∞ with α a positive scalar. Following [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF],

this specific inequality allows us to ensure that the integral in the distributed delay part is convergent. Taking the Laplace transform of (1), we obtain:

s1 n -A -A d ∞ 0 K(θ)e
-sθ e -sh dθ x(s) = 0.

We aim at assessing the stability of system (1) for a given kernel function K and a given maximal allowable constant gap delay h max . More precisely, following [START_REF] Gu | Stability of time-delay systems[END_REF], the system is asymptotically stable for all delays h belonging [0, h max ] if and only if

∀h ∈ [0, h max ], ∀s ∈ C, Re(s) ≥ 0, det(s1 n -A -A d ∞ 0 K(θ)e -sθ e -sh dθ) = 0.
This paper proposes the use of a classical robust analysis approach to address this problem. Firstly, we will model the given system as an uncertain system [START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF]. Then in a second stage, the stability of the uncertain system will be studied using the Quadratic Separation approach [START_REF] Iwasaki | Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations[END_REF], [START_REF] Peaucelle | Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation[END_REF], recalled in the following subsection.

III. PRELIMINARIES

A. Quadratic separation Approach

The quadratic separation principle is a specific approach to the robust analysis/control [START_REF] Iwasaki | Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations[END_REF], [START_REF] Peaucelle | Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation[END_REF]. It consists of analyzing the stability of a specific feedback system as illustrated in Figure 1. This latter is composed of a linear singular system, with two real constant matrices E and A, connected to a linear operator, with a complex matrix ∇. Basically, this operator encompasses the uncertain feature of the system, which can stem from uncertain parameters, nonlinear dynamics, delay dynamics [START_REF] Ariba | Stability analysis of time-delay systems via Bessel inequality: A quadratic separation approach[END_REF]. It is assumed to belong to an uncertain set ∇ ∇. Hence, the methodological principle relies on the modelling of the system under study so as to obtain an interconnected system as shown in Figure 1. Then a stability result, stated in Theorem 1, is applied to derive a wellposedness condition for the original system. This last condition allows also to prove the robust stability if the uncertain set ∇ ∇ is well chosen.

E -A ⊥ * Θ E -A ⊥ > 0, (2)
1 ∇ * Θ 1 ∇ ≤ 0 , ∀∇ ∈ ∇ ∇ . (3)
In practice, inequality ( 3) is defined at the modelling step, according to the choice of operators in the feedback system. Then, the second inequality (2) is studied so as to design a stability condition to be tested.

B. Laguerre Polynomials

In [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF], a generalized integral inequality has been introduced thanks to the use of the Bessel inequality together with the orthogonal sequence of Legendre polynomials. This formulation was adapted to the situation of distributed delay on finite support [START_REF] Seuret | Complete quadratic Lyapunov functionals for distributed delay systems[END_REF], but cannot be directly extended to the case of infinite support for the distributed delay term. Therefore, there is a need to formulate a new Bessel-like inequality dedicated to this situation, which is made possible through the use of Laguerre polynomials.

We define by M the vector space of complex valued square integrable functions on [0, ∞) with an exponential weight. For any functions in M, we define the Hermitian inner product useful for deriving some stability tests:

x, y = ∞ 0 x(θ)ȳ(θ)e -αθ du, (4)
where x and y belong to M. As in the definition of the kernel K(θ), the same coefficient α is used to define this specific inner product. Associated to this inner product, the Laguerre polynomials are also introduced. Definition 1 (Laguerre polynomials): The Laguerre polynomials are defined by

∀k ∈ N, L k (u) = e u k! d k du k e -u u k ,
with L k (0) = 1, for all k ∈ N. These polynomials are orthogonal with respect to the following inner product,

∀(k, l) ∈ N 2 , ∞ 0 L k (u)L l (u)e -u du = 0, k = l, 1, k = l. (5)
In order to adapt the Laguerre polynomials to the inner product (4), the polynomial definition is slightly modified by a change of variable u = αθ. The new polynomials considered in this paper are then of the form:

∀k ∈ N, L k (θ) = e αθ k! d k dθ k e -αθ θ k . (6)
Notice that in this case, we have:

L k , L l = 1 α κ kl ,
where κ kl is the Kronecker function. We now propose a first result that will be used later in the modelling part.

Lemma 1: The derivative of polynomials L k , defined in (6), can be expressed as a recurrence relation:

d dθ L k (θ) = -αL k-1 (θ) + d dθ L k-1 (θ), L 0 (θ) = 1.
Hence, we also have

d dθ L k (θ) = -α l-1 l=0 L l (θ).
Proof: Let us express the derivative of ( 6):

d dθ L k (θ) = d dθ ( e αθ k! d k dθ k e -αθ θ k ) d dθ L k (θ) = e αθ (k -1)! d k dθ k e -αθ θ k-1 = e αθ (k -1)! d k-1 dθ k-1 -αe -αθ θ k-1 -αL k-1 + e αθ (k -2)! d k-1 dθ k-1 e -αθ θ k-2 d dθ L k-1
, which concludes the proof.

IV. STABILITY ANALYSIS A. Modeling of the original system

A first step is to model the original system into an interconnected uncertain comparison system whose stability ensures the stability of the original time delay system. In order to benefit from the Laguerre polynomials properties, function f is reworded as

f (θ) = r k=0 a k L k (θ) + g r (θ)
where r is the degree of the polynomial and the coefficients a i are computed with the orthogonal projection of the function on the associated Laguerre polynomials,

a k = f, L k L k , L k , ∀k = 0, . . . , r.
The function g r is called the remainder at order r. The original time-delay system (1) can then be rewritten as:

ẋ(t) = Ax(t) + r k=0 A dk ∞ 0 L k (θ)e -αθ x(t -θ -h)dθ + A d ∞ 0 g r (θ)e -αθ x(t -θ -h)dθ, (7) with A dk = a k A d .
Let us define new extra-states as:

x k (t) = ∞ 0 L k (θ)e -αθ x(t -θ -h) dθ, ∀k = 0, . . . , r x gr (t) = ∞ 0 g r (θ)e -αθ x(t -θ -h) dθ.
The dynamics of the augmented states x k is written as:

ẋk (t) = ∞ 0 L k (θ)e -αθ d dt x(t -θ -h) dθ, = - ∞ 0 L k (θ)e -αθ d dθ x(t -θ -h) dθ.
An integration by parts and the use of Lemma 1 lead to

ẋk (t) = x(t -h) -α k l=0 x l (t).
This auxiliary dynamic is then appended to the initial system [START_REF] Gouaisbaut | Delay-dependent stability analysis of linear time delay systems[END_REF] to obtain the following extended system:

           ẋ(t) = Ax(t) + r k=0 A dk x k (t) + A d x gr (t), ẋk (t) = x(t -h) -α k l=0
x l (t), k = 0, . . . , r.

(8) with the particular initial conditions

   x(s) = φ(s), s ∈ (-∞, 0], x k (0) = ∞ 0 L k (θ)e -αθ φ(-θ -h) dθ, ∀k = 0, . . . , r.
It is worth noting that the augmented system (8), omitting the constraints on the initial conditions of the x k 's, represents a wider class of system than the original distributed delay system (1). However, the objective of the latter developments is to assess stability of the (8) with the constrained initial conditions in order to guarantee the stability of the initial system [START_REF] Ariba | Bessel inequality for robust stability analysis of time-delay system[END_REF].

Notice that in the Laplace domain, the original dynamical system can be written as

sx(s) = Ax(s) + r k=0 A dk ∞ 0 L k (θ)e -sθ e -αθ dθ δ k (s) e -sh x(s) + A d ∞ 0 g r (θ)e -sθ e -αθ dθ δg r (s)
e -sh x(s), emphasizing the relationship between the infinite part of the system and the extra states x k and x gr . To this end, we introduce a set of transfer functions as

δ k (s) = ∞ 0 L k (θ)e
-sθ e -αθ dθ, k = 0, . . . , r, δ gr (s) = ∞ 0 g r (θ)e -sθ e -αθ dθ.

At this stage, the extended time delay system (8) can be modeled as an uncertain interconnected system. It can be done by considering the following complex valued matrix

∇(s) =            s -1 1 (r+2)n e -sh 1 n 1-e -sh s 1 n δ 0 1 n . . . δ r 1 n δ gr 1 n            . ( 9
) interconnecting the signals

w(t) =                   x(t) x 0 (t) . . . x r (t) x(t -h) x(t) -x(t -h)
x 0 (t) . . .

x r (t) x g (t)                   ; z(t) =              ẋ(t) ẋ0 (t) . . . ẋr (t) x(t) ẋ(t) x(t -h) x(t -h)             
with W (s) = ∇(s)Z(s), in the Laplace domain. Using the definition of the augmented system (8), a linear transformation of the form EZ(s) = AW (s) linking W (s) and Z(s) can be built with

E = 1 (r+6)n 0 (r+2)n×(r+6)n , (10)
A =             A Ād 0 0 0 A d 0 U E 0 0 0 1 0 0 0 0 0 A Ād 0 0 0 A d 0 0 1 0 0 0 0 0 1 0 0 0 1 0 -1 -1 0 0 0 1 0 0 -1 0             , (11)
and

E =      1 1 . . . 1      , U =      -α1 0 . . . 0 -α1 -α1 0 . . . . . . -α1 -α1 . . . -α1      , Ād = A d0 A d1 . . . A dr .

B. Construction of quadratic constraints

The original time-delay has been modeled as an interconnected system depicted in Figure 1 and the next step is to find a so-called separator Θ such the inequality (3) is satisfied. Since, the uncertain complex valued matrix ∇(s) is a block diagonal matrix, a separator of the same structure is developed, that is a separator for each component of ∇(s). The next Lemmas recall some separators classically used in the literature of robust analysis of time-delay systems [START_REF] Ariba | Stability analysis of time-delay systems via Bessel inequality: A quadratic separation approach[END_REF].

Lemma 2: A quadratic constraint for s -1 is given by the following inequality for any positive definite matrix P in R n×n , ∀s ∈ C, Re(s) ≥ 0, s = 0,

1 n s -1 1 n * 0 -P -P 0 1 n s -1 1 n ≤ 0.
Lemma 3: A quadratic constraint for e -hs is given by the following inequality for any positive definite matrix Q 1 in R n×n , ∀s ∈ C, Re(s) ≥ 0,

1 n e -hs 1 n * -Q 1 0 0 Q 1 1 n e -hs 1 n ≤ 0.
Lemma 4: A quadratic constraint for µ = (1 -e -hs )s -1 is given by the following inequality for any positive definite matrix

Q 2 in R n×n , ∀s ∈ C, Re(s) ≥ 0, 1 n µ1 n * -h 2 max Q 2 0 0 Q 2 1 n µ1 n ≤ 0.
Proofs of Lemma 2, 3 and 4 can be found in [START_REF] Ariba | Stability analysis of time-delay systems via Bessel inequality: A quadratic separation approach[END_REF]. The next Lemma is dedicated to the design of a separator for the new operator δ 0 1 n . . . δ r 1 n T .

Lemma 5: A quadratic constraint for δ 0 1 n . . . δ r 1 n T is given by the following inequality for any positive definite matrix

R 1 in R n×n ,      1 n δ 0 1 n . . . δ r 1 n      T      -1 α 2 R 1 0 . . . 0 0 R 1 . . . . . . 0 R 1           1 n δ 0 1 n . . . δ r 1 n      ≤ 0.
Proof: Let us apply the Bessel inequality to the delay transfer function e -sθ with the orthonormal sequence {L 0 , . . . , √ αL k , . . . , √ αL r }:

e -sθ , e -sθ ≥ r k=0 | e -sθ , √ αL k | 2 ≥ r k=0 αδ k δ * k .
Since e -sθ , e -sθ is bounded by 1/α, we have

r k=0 αδ k δ * k ≤ 1 α , r k=0 δ k δ * k ≤ 1 α 2 .
This last inequality can then be directly extended to δ 0 1 n . . . δ r 1 n T using a matrix R 1 > 0 following the approach proposed in [START_REF] Ariba | Stability analysis of time-delay systems via Bessel inequality: A quadratic separation approach[END_REF], which concludes the proof. The next step is therefore to propose a separator for δ gr . Lemma 6: A quadratic constraint for δ gr is given by the following inequality for any positive definite matrix R 2 in R n×n , ∀s ∈ C, Re(s) ≥ 0,

1 n δ gr 1 n * -λ r R 2 0 0 R 2 1 n δ gr 1 n ≤ 0, with λ r =   ∞ 0 |g(θ)|e -αθ dθ   2 .
Proof: Let us express the norm of δ g :

δ gr δ * gr = ∞ 0 g r (θ)e -αθ e -sθ dθ 2 ≤   ∞ 0 |g r (θ)e -αθ e -sθ | dθ   2 ≤   ∞ 0 |g r (θ)|e -αθ dθ   2 = λ r .
This last inequality can then be extended straightforwardly to δ gr 1 n introducing a weighting matrix R 2 [START_REF] Ariba | Stability analysis of time-delay systems via Bessel inequality: A quadratic separation approach[END_REF].

Finally, concatenating all the separators for each operator, we proposed the following separator for ∇(s).

Lemma 7: Assume there exist positive definite matrices

P, Q 1 , Q 2 , R 1 , R 2 , then a matrix Θ of the form Θ = Θ 11 Θ 12 Θ T 12 Θ 22 (12)
with

Θ 11 = diag(0, -Q 1 , -h 2 max Q 2 , -1 α 2 R 1 , -λ r R 2 ), Θ 12 = diag(-P, 0, 0, 0, 0), Θ 22 = diag(0, Q 1 , Q 2 , R 1 , . . . , R 1 , R 2 ),
is a separator for ∇(s) defined by equation [START_REF] Gu | Stability of time-delay systems[END_REF]. The inequality (3) is then fulfilled, for all s ∈ C, Re(s) ≥ 0, s = 0.

Proof: The proof is omitted because it is a direct combination of previous Lemmas.

C. Stability results

Having modeled the system (1) as an uncertain interconnected system and having characterized the uncertainty ∇(s) with Lemma 7, a stability theorem is proposed.

Theorem 2: For a given constant h max , if there exist positive definite matrices P ∈ R n(r+2)×n(r+2) , Q 1 ,Q 2 , R 1 and R 2 ∈ R n×n , such that the following LMI is satisfied:

E -A ⊥ * Θ E -A ⊥ > 0,
where matrices E, A and Θ are defined in [START_REF] Iwasaki | Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations[END_REF], [START_REF] Liu | Generalized jensen inequalities with application to stability analysis of systems with distributed delays over infinite time-horizons[END_REF] and ( 12), then the system (1) is asymptotically stable for any constant gap h ≤ h max .

Proof: It has been proved that using extended states x k , the original system (1) could be modeled as in Figure 1 with the uncertain matrix ∇(s) defined in [START_REF] Gu | Stability of time-delay systems[END_REF] and the linear transformation defined by [START_REF] Iwasaki | Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations[END_REF], [START_REF] Liu | Generalized jensen inequalities with application to stability analysis of systems with distributed delays over infinite time-horizons[END_REF]. Using Lemma 7, a separator Θ defined in [START_REF] Liu | Bessel-Laguerre inequality and its application to systems with infinite distributed delays[END_REF] allows to satisfy the condition [START_REF] Chen | Delay-dependent robust stabilization for uncertain neutral systems with distributed delays[END_REF]. Hence, Applying Theorem 1, the feedback system is well-posed if the first condition ( 2) is also satisfied with E, A and Θ defined in [START_REF] Iwasaki | Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations[END_REF], [START_REF] Liu | Generalized jensen inequalities with application to stability analysis of systems with distributed delays over infinite time-horizons[END_REF] and [START_REF] Liu | Bessel-Laguerre inequality and its application to systems with infinite distributed delays[END_REF]. Following therefore [START_REF] Ariba | Stability analysis of time-delay systems via Bessel inequality: A quadratic separation approach[END_REF], it can be proved that the well-posedness of the feedback system implies that there do not exist eigenvalues of the distributed delay system on the right half plane, that is

∀h ≤ h max , ∀s ∈ C, Re(s) ≥ 0, det(s1 n -A -A d ∞ 0 K(θ)e -sθ e -sh dθ) = 0.
which concludes the proof.

Remark 1: The stability condition as stated in Theorem 2 is delay-dependent since it proves the stability of (1) for any constant gap h lower than a prescribed upper bound h max . If the third operator (1 -e -sh )/s in ( 9) is removed, adjusting correspondingly1 matrices E, A and Θ in ( 10)-( 11)-( 12), an independent-of-delay (IOD) stability condition is provided.

Remark 2: Theorem 2 can be viewed as the robust approach counterpart of a Lyapunov-Krasovskii functional proposed recently in [START_REF] Liu | Bessel-Laguerre inequality and its application to systems with infinite distributed delays[END_REF] defined by

V r (x t ) = ζ T r (t)P ζ r (t) + h 0 -h t t+θ ẋT (s)Q 2 ẋ(s)dsdθ + ∞ 0 t t-1 α θ-h e -θ x T (s)R 1 x(s)dsdθ,
where P, Q 2 , R 1 are postive definite matrices and ζ r (t) is an extended state which takes into account the projections of the state on Laguerre polynomials

ζ r (t) =          x(t) +∞ 0 L 0 (θ)x(t - 1 α θ -h)e -θ dθ . . . +∞ 0 L r (θ)x(t - 1 α θ -h)e -θ dθ          . V. NUMERICAL EXAMPLES A. Example 1
The model in this first example is extracted from [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF], and represents the traffic flow dynamic of cars on a ring [START_REF] Morarescu | Stability crossing curves of shifted gamma-distributed delay systems[END_REF]. The system is expressed as (1) with A = -2 0 0 -2 and A d = 0 2 2 0 . The kernel is a gamma distribution

K(θ) = θ N -1 e -θ T T N (N -1)! ,
for a given N and a given T . For N = 1 and N = 2, [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF] has shown that the system is stable for (T, h) ∈ [0 , 1000] × [0 , 1000]. This result is recovered with Theorem 2. It can be proved that it is actually IOD stable, for both N = 1 and N = 2, with a slight modification in Theorem 2 (see Remark 1). Note that for N = 1, it is sufficient to set r = 0 since in that case the function f is constant: f (θ) = 1 T . When N = 2, a zero r is not enough for this example and a first order Laguerre polynomial (r ≥ 1) is necessary to prove the stability, the remainder operator bound λ being too high otherwise. For instance, for T = 2 and h = 100, when r = 0 the bound λ equals 0.54 while it vanishes to 0 when r = 1.

B. Example 2

Let consider the following scalar example: ẋ(t) = 0.2 x(t) -5 ∞ 0 1 + θ 5 + θ e -4θ x(t -θ -h) dθ. [START_REF] Morarescu | Stability crossing curves of shifted gamma-distributed delay systems[END_REF] Notice that removing the delayed part, the system is unstable. According to our modeling, we have f (θ) = 1+θ 5+θ and α = 4. The function f being a rational fraction, the projection on the Laguerre polynomials sequence won't be exact. Theorem 2 allows to establish the stability of (13) for any value of the gap h lower than a certain upperbound as stated in Table I. One can observe that the higher the order r of the Laguerre polynomial basis is, the lower the remainder bound λ r is. Hence, conservatism is reduced as r increases and a larger maximum allowable gap is found.

VI. CONCLUSION

In this paper, we have studied the stability of a class of distributed delay systems with infinite support. Unlike the classical approach based on the construction of Lyapunov functionals, we have transform the problem as a robust stability problem. Within this framework, using quadratic separation, all the distributed delay transfer functions are embedded into uncertainties. These uncertainties are then characterized by quadratic inequalities. To this end, we proposed to develop new inequalities based on the use of Laguerre polynomials and the Bessel-Laguerre inequality. This last inequality makes it possible to obtain LMI conditions whose conservatism depends on the chosen degree for the Laguerre polynomials.

Fig. 1 .

 1 Fig. 1. Feedback system.

That is, removing the fourth and seventh rows from matrices E and A, and removing the fourth column in A and the two rows/columns associated to the matrix Q

in the separator Θ.


Gabriel Marzinotto

Frederic Bechet

Geraldine Damnati
email: geraldine.damnati@orange.comfrederic.bechet

Alexis Nasr
email: alexis.nasr@lif.univ-mrs.fr

Sources of Complexity in Semantic Frame Parsing for Information Extraction

Keywords: Frame Semantic Parsing, LSTM, Information Extraction

This paper describes a Semantic Frame parsing System based on sequence labeling methods, precisely BiLSTM models with highway connections, for performing information extraction on a corpus of French encyclopedic history texts annotated according to the Berkeley FrameNet formalism. The approach proposed in this study relies on an integrated sequence labeling model which jointly optimizes frame identification and semantic role segmentation and identification. The purpose of this study is to analyze the task complexity, to highlight the factors that make Semantic Frame parsing a difficult task and to provide detailed evaluations of the performance on different types of frames and sentences.

Introduction

Deep Neural Networks (DNN) with word embeddings have been successfully used for semantic frame parsing [START_REF] Hermann | Semantic frame identification with distributed word representations[END_REF]. This model extends previous approaches [START_REF] Hermann | Semantic frame identification with distributed word representations[END_REF] where classifiers are trained in order to assign the best possible roles for each of the candidate spans of a syntactic dependency tree. On the other hand, recurrent neural networks (RNN) with Long Short Memory (LSTM) cells have been applied to several semantic tagging tasks such as slot filling [START_REF] Mesnil | Using recurrent neural networks for slot filling in spoken language understanding[END_REF] or even frame parsing (Hakkani-Tür et al., 2016;[START_REF] Tafforeau | Joint syntactic and semantic analysis with a multitask deep learning framework for spoken language understanding[END_REF] for Spoken Language Understanding. Currently, there is an important amount of research addressed to optimize architecture variants of the recurrent neural networks for the different semantic tasks. In SRL the current state of the art (He et al., 2017) uses an 8 layers bidirectional LSTM with highway connections [START_REF] Srivastava | Highway networks[END_REF], that learns directly from the word embedding representations and uses no explicit syntactic information. More recently, [START_REF] Yang | A joint sequential and relational model for frame-semantic parsing[END_REF] proposed a combined approach that learns a sequence tagger and a span classifier to perform semantic frame parsing making significant performance gains on the FrameNet test dataset. However, there is little work in analyzing the sources of error in Semantic Frame parsing tasks. This is mainly due to the size of the SemEval07 corpus that contains 720 different frames and 754 Frame Elements (FE), with a lexicon of 3,197 triggers, for only 14,950 frame examples in the training set. Hence the size of the dataset, as well as number of examples per frame tends to be too small to perform this type of analysis. For this reason the analyses done by researchers in the domain focus mainly on the performance of their model on rare frames [START_REF] Hermann | Semantic frame identification with distributed word representations[END_REF]. In [START_REF] Marzinotto | Semantic frame parsing for information extraction : the calor corpus[END_REF] a new corpus of French texts annotated following the FrameNet paradigm (Baker et al., 1998a) (Fillmore et al., 2004) is introduced. This new corpus has been partially annotated using a restricted number of Frames and triggers. The purpose was to obtain a larger amount of annotated occurrences per Frame with the counterpart of a smaller amount of Frames.

In this paper we focus on analyzing the factors that make a Frame hard to predict, describing which Frames are intrinsically difficult, but also which types of frame triggers are more likely to yield prediction errors and which sentences are complex to parse.

Sequence labeling model

Highway bi-LSTM approach

Following the previous work of ( He et al., 2017), in this study, we propose a similar architecture, a 4 layer bidirectional LSTM with highway connections. For this model we use two types of LSTM layers, forward (F ) layers and backward (B) layers which are concatenated and propagated towards the output using highway connections [START_REF] Srivastava | Highway networks[END_REF]. A diagram of our model architecture is shown in Figure 1. There are 2 main differences between the model proposed in (He et al., 2017) and ours. First, we do not implement A* decoding of the output probabilities, second, our system not only relies on word embeddings as input features, but we also include embeddings encoding: syntactic dependencies, POS, morphological features, capitalization, prefixes and suffixes of the input words. We have observed these features to be useful for the FE detection and classification task. In order to deal with both the multi-label and linking problems we have built training samples containing only one predicate. More precisely a sentence containing N predicates provides N training samples. The downside of this approach is that during prediction time, parsing a sentence with N predicates requires N model applications. At decoding time each pair { sentence , predicate } is processed by the network and a distribution probability on the frames and frame elements for each word is produced. To these probabilities we apply a coherence filter in which we take as ground truth the frame prediction (represented as the label assigned by the tagger to the trigger) and we discard frame element labels that are incompatible to the predicted frame.

The CALOR Semantic Frame Corpus

The experiments presented in this paper were carried out on the CALOR corpus, which is a compilation of doc-Figure 1: Highway bi-LSTM Model Diagram uments in French that were hand annotated in frame semantics. This corpus contains documents from 4 different sources: Wikipedia's Archeology portal (WA, 201 documents), Wikipedia's World War 1 portal (WGM, 335 documents), Vikidia's1 portals of Prehistory and Antiquity (VKH, 183 documents) and ClioTexte's2 resources about World War one (CTGM, 16 documents). In contrast to full text parsing corpus, the frame semantic annotations of CALOR are limited to a small subset of frames from FrameNet (Baker et al., 1998b). The goal of this partial parsing process is to obtain, at a relatively low cost, a large corpus annotated with frames corresponding to a given applicative context. In our case this applicative context is Information Extraction (IE) from encyclopedic texts, mainly historical texts. Beyond Information Extraction, we attempt to propose new exploration paradigms through collections of documents, with the possibility to link documents not only through lexical similarity but also through similarity metrics based on semantic frame structure. The notion of document is more central in our study than in other available corpora. This is the reason why we have chosen to annotate a larger amount of documents on a smaller amount of Frames. Precisely, while Framenet proposes 1,223 different frames, 13,635 LUs, and 28,207 frame occurrences on full text annotations, CALOR is limited to 53 different frames, 145 LUs (among which 13 are ambiguous and can trigger at least two frames) and 21,398 frame occurrences. This means that the average number of examples per frame in CALOR is significantly higher than in the full-text annotations from FrameNet.

Results

In order to run the experiments we divided the CALOR corpus into 80% for training and 20% for testing. This partition is done ensuring a similar frame distribution in training and test.

In the CALOR corpus, ambiguity is low, with only 53 different Frames. Most triggers have only 1 possible Frame. This makes the performance of our model in the frame selection subtask as high as 97%. For this reason we focus our analysis on the FE detection and classification subtask. We trained our model on the CALOR corpus and we evaluated it by thresholding the output probabilities in order to build the FE detection and classification precision recall curves shown in Figure 2. The three curves correspond to three possible precision-recall metrics: soft spans, weighted spans and hard spans. When evaluating using soft spans, a FE is considered correct when at least one token of its span is detected. In this case we achieve an F measure of 69,5%. If we use the weighted span metric, a FE is scored in proportion to the size of the overlap between the hypothesis and the reference segments. Using this metric we observe a 60,9% F-measure. Finally, the hard span metric considers a FE correct only if the full span is correctly detected. In this case, the performance degrades down to 51,7%. This experience shows that the model detects most of the FEs but it rarely finds the full spans. It should be possible to boost the model performance by +18pts of F-measure by expanding the detected spans to its correct boundaries. In the following subsections we present the results using the soft span metrics for the FE detection and classification task and we focus on analyzing several complexity factors in frame semantic parsing. We divide these factors into Frame Intrinsic (Section 4.1.) and Sentence Intrinsic 4.2.). In Section 4.3. we analyze the performance of the model at a document level using correlation analysis and regression techniques to retrieve relevant parameters that allow to predict the model performance on test documents.

Frame Intrinsic Complexity Factors

Some frames are intrinsically more difficult than others, this is due to their number of possible FEs, to the syntactic and lexical similarities between FEs and to the type of semantic concepts they represent. In Figure 3 we analyze the performance of our model on each FE with respect to their

Sentence Intrinsic Complexity Factors

We have identified three sentence intrinsic complexity factors, the Trigger POS, the Trigger Syntactic Position and the sentence length.

Trigger POS

As shown in table 3 the model performance varies more than 17pts of F measure depending on whether the triggers are nouns or verbs. This is due to the variety, in French, of the syntactic nature of Verb dependents when compared to Nouns. Verb arguments can be realized as Subjects, Objects, indirect Objects (introduced by specific prepositions), adverbs and Prepositional Phrases. Nouns arguments, in contrast, are usually realized as Prepositional Phrases and adjectives. Since FEs are mostly realized as arguments of their Frame trigger, Verb triggered Frames offer a wider range of syntactic, observable, means to distinguish its FE, making them easier to model.

Trigger Syntactic Position

We observe that the model's performance varies significantly depending on whether the sentence's triggers are at the root of the syntactic dependency tree or not. The results for this experience are summed up in Table 3. We observe that the easiest triggers are at the root of their syntactic tree and there is a difference of 14pts of F measure between them and the triggers that occupy other positions in the syntactic tree.

Sentence Length

Another important factor is sentence length. In general, longer sentences are harder to parse as they often present more FEs and a more complex structure. Also, it is in Table 3: Performance for different types of triggers longer sentences that we find the most non-root triggers.

In this experiment our model yields 74.0% F-measure for sentences with less than 27 words and 65.7% F-measure for sentences with 27 words or more. However, we also observed that parsing a long sentence (with 27 words or more) with the trigger at the root of the syntactic tree can still be done with a fairly good performance of 76.6%. While, when parsing a long sentence with non root triggers the performances degrade down to 62.8%. All of these factors add up. Short sentences with a verbal trigger at the root of the syntactic tree can be parsed with an F-measure of 82.6% while long sentences with noun triggers that are not root of the syntactic tree are parsed with an F-measure of 52.8%.

Document Intrinsic Complexity Factors

In the previous sections we have presented the main factors that influence semantic frame parsing. In order to quantify the impact of each factor we have compared so far the performance of our model in subsets of the test set corresponding to the different modalities of the complexity factors. In this section, we try to directly evaluate the impact of these factors on the model performance. The analysis is performed here at the document level, the applicative motivation being to be able to predict the semantic parsing performances for a given new document.

We address the analysis of the complexity factors as a regression problem where we describe a dependent variable y (that quantifies the model performance) using a set of explanatory variables X = (X 1 , ..., X n ) which are our candidate complexity factors. First, we use our model to generate hypothesis predictions of frame semantic parsing on the entire CALOR corpus, with a 5-fold protocole. For each fold, we train on 80% of the corpus, and generate predictions for the remaining 20%. Then, we evaluate the model's predictions using the gold annotations to compute the model's performance for each of the 735 documents in the CALOR corpus (Section 3.).

For each of these documents, we also compute the set of features (complexity factors candidates) listed below:

• Percentage of root / non-root triggers.

• Percentage of verbal / nominal triggers.

• Mean phrase length.

• Mean trigger syntactic depth.

• Mean trigger position in sentence.

• Part of Speech (POS) distribution1 .

• Syntactic dependency relation (DEP) distribution 1 .

To avoid taking into consideration parts of the document that were not processed by our semantic frame parser, these features are computed only using the sentences that contain at least 1 trigger. In order to make the analysis more robust to outliers we discarded the documents with less than 30 triggers, yielding a total amount of 327 documents. Unidimensional statistics show that the model's F-measure follows a Gaussian distribution across documents. This Gaussian distribution is centered at 69 pts of F-measure and has an standard deviation of 6.5 pts. This value of standard deviation shows that the model is fairly robust and has a stable performance across documents. Finally, we used this set of document's features and model's performances in two experiments:

• To compute the Pearson correlation coefficient between the F-measure and each feature. • To train a linear regression model that attempts to predict the model's performance on a document given a small set of parameters.

Pearson Correlation

We computed the Pearson correlation between each feature and the F-measure of the system and verified that the correlation coefficient passes the Student's t-Test. In table 5 we observe that the most important parameter is the syntactic depth of the trigger. As we have previously shown, triggers at the root of the syntactic tree have the best performances (also, root triggers are often verbs). The second most correlated parameter is the position of the trigger in the sentence. Triggers that are far from the beginning of the sentence show lower performance, as they are prone to errors in syntax. Our third and fourth parameters are the percentage of verbal triggers and the average sentence length. As shown in previous experiences, verbal triggers and short sentences are, in general, easier to parse. This study also reveals morpho-syntactic parameters that are correlated with the model's performances: documents with a large amount of punctuation marks, adverbs, and conjunctions are more complex and harder to parse. On the other hand, documents with a large proportion of proper nouns are simpler, as proper nouns correspond to places, institutions and persons' names, which often appear as Frame Elements. The same observation can be made with Numbers, that correspond to dates and quantities. Prepositions also facilitate parsing, as they are associated to specific FEs.

As concerns dependency parsing related features, the highest correlation is observed for Oblique Nominal (OBL) dependency. OBL dependencies attach a noun phrase functioning as a non-core argument to the syntactic head. Documents with a large proportion of oblique nominal groups are positively correlated with the F-measure. OBL arguments are often annotated as FEs (Time, Place, Purpose...) and when they contain a Prepositional Case they are easy to associate to their corresponding FE. This also explains the positive correlation of the Prepositional Case. Surprisingly, documents with a large proportion of sentences in passive voice are correlated with better performances, while copula verbs degrade the results. In the CALOR corpus, some copula verbs are annotated as triggers se nommer (to be named), être élu (to be elected), devenir (to become)) . Thus the negative correlation may be due to low performances for these lexical units. Finally, Multi Words Expressions (MWE) are also associated with low performances, as the meaning of unseen MWE is harder to be inferred, misleading the semantic parser.

Performance Inference

In this experience we trained a linear regression model with incremental feature selection using cross validation. The objective is to predict the performances of our frame semantic parser on a document given a small set of parameters. In incremental feature selection, we start with an empty set of selected features. At each iteration of the algorithm we test all the unselected feature candidates and we pick the feature that minimizes the cross validation mean square error (MSE) given all the previously selected features. The algorithm's stopping criterion finishes the process when the MSE no longer evolves. Unlike the previous experience where all features are evaluated independently, this experimet allows to select a smaller feature set that is not redundant. We evaluate the usefulness of our linear regression models by comparing them with a naive constant model that always predicts the average document performance observed on the training corpus. Note that the training corpus here means the training corpus for regression estimation but not for the semantic frame parsing model estimation (each document is parsed in a k-fold protocole). Incremental Fea-ture Selection determines that the optimal linear regression model is trained using 8 features and the insertion of more parameters does not reduce the MSE. Table 6 shows the MSE for the naive model (Mean F-measure) and compares it with each step of our linear regression with incremental feature selection. Each row in Table 6 4 shows a scatter plot of the documents with their predicted F-measure and their true F-measure. We can clearly observe that both scores are correlated and the variance that can be explained by the linear regression is R 2 = 0.46. However, there is still more than half of the variance that remains unexplained by the linear regression. This is because frame semantic parsing is very complex task and the model's performances depend on many other phenomena such as the lexical coverage, lexical units and frames that appear within a document, the type of FEs that are evocated and the degree of ambiguity at each level.

Conclusion

In this paper we proposed to identify complexity factors in Semantic Frame parsing. To do so we ran experiments on the CALOR corpus using a frame parsing model that considers the task as a sequence labeling task. In our case only partial annotation is considered. Only a small subset of the FrameNet lexicon is used, however the amount of data annotated for each frame is larger than in any other corpora, allowing to make more detailed evaluations of the error sources on the FE detection and classification task. The main contribution of this work is to characterize the principal sources of error in semantic frame parsing. We divide these sources of error into two main categories: Frame intrinsic and sentence intrinsic. Examples of Frame intrinsic factors are the number of possible FE, and the syntactical similarity between them. As for the sentence intrinsic factors, we enhanced the position of the trigger in the syntactic tree, the POS of the trigger and the sentence length. In this work we showed that some morpho-syntactic categories and syntactic relations have an impact on the complexity of the frame semantic parsing. Finally, we showed that it is possible to make a fair prediction of the model's performance on a given document thanks to a regression estimation knowing its sentence intrinsic parameters. The features selected for the regression estimation confirm the observations regarding the task complexity but also enhance new assertions. The complexity factors presented in this article may allow further work on feature engineering to improve the frame semantic parsing models. Fillmore, C. J., Baker, C. F., and Sato, H. (2004). Framenet as a "net". In LREC. Hakkani-Tür, D., Tur, G., Celikyilmaz, A., Chen, Y.-N., Gao, J., Deng, L., and Wang, Y.-Y. (2016). Multidomain joint semantic frame parsing using bi-directional rnn-lstm. In Proceedings of The 17th Annual Meeting of the International Speech Communication Association. He, L., Lee, K., Lewis, M., and Zettlemoyer, L. (2017).

Bibliographical References

Deep semantic role labeling: What works and what's next. In Proceedings of the Annual Meeting of the Association for Computational Linguistics.

Figure 2 :

 2 Figure 2: Model's Precision Recall curves using 3 different metrics: soft spans, weighted spans and hard spans

Figure 3 :

 3 Figure 3: Model's performance for different FEs w.r.t the number of training samples

Figure 4 :

 4 Figure 4: Document Scatter plot showing True F-measure vs Predicted F-measure using a Linear Regression with 8 features

Table 1 :

 1 List of Semantic Frames annotated in the CALOR corpus number of occurrences. In general, the more examples of a class we have, the better its performance should be. However, there are some ambiguity and complexity phenomena that must be taken into account.4.1.1. Number of Frame ElementsThe number of possible FEs is not the same for each Frame. Intuitively, a Frame with more FEs should be harder to parse. In table 2 we divide Frames into 3 categories Small, Medium and Large depending of their number of possible FEs. From this experience we observed that this is not such a relevant factor and that the number of possible FEs must be really large (above 10) in order to see some degradation in the model's performance. to assign the correct FE. This explains the low performances observed in FEs such as Teacher. Some FEs are very similar up to a small nuance. For example, the Frame Education Teaching has 6 possible FEs to describe what is being studied Course, Subject, Skill, Fact, Precept and Role and their slight difference relies in the type of content studied. This kind of FE are very prone to confusions even for human annotators. Moreover, if the FE is a pronoun, finding the correct label may not be possible without the sentence context. Another type of FE similarity appears in symmetric actions, for example, the Frame Hostile Encounter , has FEs Side1, Side2 or Sides to describe the belligerents of an encounter. Such FEs are prone to confusions and for this reason, our model has a low performance on them.

	Accomplishment	Activity-start	Age	Appointing	Arrest
	Arriving	Assistance	Attack	Awareness	Becoming
	Becoming-aware	Buildings	Change-of-leadership	Choosing	Colonization
	Coming-to-believe	Coming-up-with	Conduct	Contacting	Creating
	Death	Deciding	Departing	Dimension	Education-teaching
	Existence	Expressing-publicly	Finish-competition	Giving	Hiding-objects
	Hostile-encounter	Hunting	Inclusion	Ingestion	Installing
	Killing	Leadership	Locating	Losing	Making-arrangements
	Motion	Objective-influence	Origin	Participation	Request
	Scrutiny	Seeking	Sending	Shoot-projectiles	Statement
	Subjective-influence	Using	Verification

étudier (study) , enseigner(teach) and apprendre (which can translate both into learn and teach) as potential triggers and Student , Teacher among their FEs. When the trigger is étudier, the syntactic subject is Student; when the trigger is enseigner, the subject is the Teacher; finally, when the trigger is apprendre, the subject could be either Student or Teacher and further disambiguation is needed in order

Table 4

 4

			presents the
	model performance with respect to trigger syntactic posi-
	tion and sentence length.
		Root	Non-Root
		Triggers	Triggers
	Short (< 27 words)	81.6	68.2
	Long (≥ 27 words)	76.5	62.8
	Table 4: F-measure for different sentence lengths (Above
	and Below the Median) and trigger positions (Root and
	Non-Root)

  Table5shows the 15 parameters that have the highest absolute correlation with the F-measure.

		Rank	Pearson
			Correlation
	Mean Trigger Depth	1	-0.44
	Mean Trigger Position	2	-0.36
	Verbal Trigger Percentage	3	+0.31
	Mean Sentence Length	4	-0.30
	DEP Oblique Nominal	5	+0.30
	DEP Passive Auxiliary	6	+0.29
	POS Punctuation	7	-0.28
	POS Proper Noun	8	+0.27
	POS Adverbs	9	-0.20
	Multi Words Expressions	10	-0.20
	DEP Prepositional Case	11	+0.20
	POS Preposition	12	+0.15
	POS Conjunction	13	-0.14
	DEP Copula	14	-0.14
	POS Number	15	+0.11
	Table 5: Pearson Correlation between the best 15 Docu-
	ment Features and the F-measure

  adds a new feature to the linear regression model, up to the last row that contains the final set of 8 selected parameters.

	We observe that the naive prediction algorithm has a
	MSE of 42.7. A linear regression model with only one
	feature (Mean Trigger Depth) reduces MSE by
	16% relative and the best linear regression model with 8
	features (Mean Trigger Depth, DEP Oblique
	Nominal, Verbal Trigger Percentage,
	DEP Passive Auxiliary, DEP Copula, DEP
	Fixed Multi Words, POS Punctuation, POS
	Proper Noun) yields a 41% relative MSE reduction.
	Figure

Table 6 :

 6 Mean Squared Error (MSE) for Linear Regression with Incremental Feature Selection

		# Features	MSE
	Mean F-measure	0	42.7
	Mean Trigger Depth	1	35.9
	DEP Oblique Nominal	2	33.5
	Verbal Trigger Percentage	3	30.5
	DEP Passive Auxiliary	4	29.1
	DEP Copula	5	27.4
	Multi Words Expressions	6	26.3
	POS Punctuation	7	25.6
	POS Proper Noun	8	25.1

  Baker, C. F., Fillmore, C. J., and Lowe, J. B.(1998a). The berkeley framenet project. In Proceedings of the 36th Annual Meeting of the Association for Computational Linguistics and 17th International Conference on Computational Linguistics-Volume 1, pages 86-90. Association for Computational Linguistics. Baker, C. F., Fillmore, C. J., and Lowe, J. B. (1998b). The berkeley framenet project. In Proceedings of the 36th Annual Meeting of the Association for Computational Linguistics and 17th International Conference on Computational Linguistics -Volume 1, ACL '98, pages 86-90, Stroudsburg, PA, USA. Association for Computational Linguistics. Das, D. (2014). Statistical models for frame-semantic parsing. In Proceedings of Frame Semantics in NLP: A Workshop in Honor of Chuck Fillmore, volume 1929, pages 26-29.

https://fr.vikidia.org

https://clio-texte.clionautes.org/

POS and DEP from the Universal Dependencies project (http://universaldependencies.org/)


Emmanuel Agullo

Luc Giraud

Stéphane Lanteri

Gilles Marait

Anne-Cécile Orgerie

Louis Poirel

Energy analysis of a solver stack for frequency-domain electromagnetics

Keywords: Energy consumption, HPC Consommation énergétique, HPC

High-performance computing (HPC) aims at developing models and simulations for applications in numerous scientific fields. Yet, the energy consumption of these HPC facilities currently limits their size and performance, and consequently the size of the tackled problems. The complexity of the HPC software stacks and their various optimizations makes it difficult to finely understand the energy consumption of scientific applications. To highlight this difficulty on a concrete use-case, we perform an energy and power analysis of a software stack for the simulation of frequency-domain electromagnetic wave propagation. This solver stack combines a high order finite element discretization framework of the system of three-dimensional frequency-domain Maxwell equations with an algebraic hybrid iterative-direct sparse linear solver. This analysis is conducted on the KNL-based PRACE-PCP system. Our results illustrate the difficulty in predicting how to trade energy and runtime.

Introduction 2 Introduction

The advent of numerical simulation as an essential scientific pillar has led to the co-development of hardware and software capable of solving larger and larger problems at a tremendous level of accuracy. The focus of performance-at-any-cost computer operations has led to the emergence of supercomputers that consume vast amounts of electrical power and produce so much heat that large cooling facilities must be constructed to ensure proper performance https://www.top500. org/green500. To address this trend, the Green500 list puts a premium on energy-efficient performance for sustainable supercomputing. In the meanwhile, the high-performance computing (HPC) community has faced the challenge with the emergence of energy-efficient computing. While many innovative algorithms [START_REF] Balaprakash | Multi Objective Optimization of HPC Kernels for Performance, Power, and Energy[END_REF][START_REF] Gholkar | PShifter: Feedback-based Dynamic Power Shifting Within HPC Jobs for Performance[END_REF] have been proposed to reduce the energy consumption of the main kernels used in the numerical simulations, fewer studies [START_REF] Li | The Power-Performance Tradeoffs of the Intel Xeon Phi on HPC Applications[END_REF] have tackled the overall energy analysis of a complex numerical software stack.

This work aims at conducting an energy and power analysis of the simulation of frequencydomain electromagnetic wave propagation, as detailed in Section 3. Such a simulation is representative of a numerical simulation involving the usage of a complex software stack. In the present case, we study the combined HORSE/MaPHyS numerical software stack. The HORSE (High Order solver for Radar cross Section Evaluation) simulation software implements an innovative high order finite element type method for solving the system of three-dimensional frequencydomain Maxwell equations, as presented in Section 4. From the computational point of view, the central operation of a HORSE simulation is the solution of a large sparse and indefinite linear system of equations. High order approximation is particularly interesting for solving high frequency electromagnetic wave problems and, in that case, the size of this linear system can easily exceed several million unknowns. In this study, we adopt the MaPHyS [START_REF] Giraud | Parallel scalability study of hybrid preconditioners in three dimensions[END_REF] - [START_REF] Agullo | Parallel hierarchical hybrid linear solvers for emerging computing platforms[END_REF] hybrid iterative-direct sparse system solver, which is based on domain decomposition principles [START_REF] Mathew | Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations[END_REF]. MaPHyS is representative of fully-featured adaptive sparse linear solvers [START_REF] Li | Factorization-based sparse solvers and preconditioners[END_REF][START_REF] Rajamanickam | ShyLU: A hybrid-hybrid solver for multicore platforms[END_REF][START_REF] Gaidamour | A parallel direct/iterative solver based on a Schur complement approach[END_REF][START_REF] Jolivet | Scalable Domain Decomposition Preconditioners for Heterogeneous Elliptic Problems[END_REF] involving multiple numerical linear steps combining the usage of dense and sparse direct numerical linear algebra kernels as well as iterative methods, as further discussed in Section 5.

Energy consumption is a major concern putting a strain on HPC infrastructures' budget [START_REF] Balaprakash | Multi Objective Optimization of HPC Kernels for Performance, Power, and Energy[END_REF]. Additionnaly, peak power consumption hampers the size of these infrastructures, challenging the electricity provisioning [START_REF] Gholkar | PShifter: Feedback-based Dynamic Power Shifting Within HPC Jobs for Performance[END_REF]. Power capping techniques have been developed to control this power consumption [START_REF] Kelley | Adaptive Power Profiling for Many-Core HPC Architectures[END_REF][START_REF] Khan | RAPL in Action: Experiences in Using RAPL for Power Measurements[END_REF]. While these techniques dynamically limit the power consumption of running nodes, we argue here that a better understanding of HPC applications behavior could be more beneficial in finding an adequate trade-off between performance, power and energy consumption. The objective of this paper is to study the energy profile of a complex software stack that is representative of numerical simulations. To perform the energy and power analysis, we rely on two tools, Bull Energy Optimizer (BEO) and High Definition Energy Efficiency Visualization (HDEEVIZ). We first study the behavior of the MaPHyS hybrid solver in a standalone fashion, showing the energy profiles of each of its numerical steps. We then study the behavior of the overall HORSE/MaPHyS simulation stack on a realistic 3D test case.

The rest of the paper is organized as follows. Section 3 briefly presents the high order discretization technique employed while Section 4 presents the HORSE simulation tool that implements it. Section 5 presents the MaPHyS sparse hybrid solver that is used to solve the underlying sparse linear system. Section 6 presents the methodology employed to conduct our energy analysis and discusses the energy profile of the considered software stack, before concluding in Section 7.

Inria 3 Numerical approach

During the last 10 years, discontinuous Galerkin (DG) methods have been extensively considered for obtaining an approximate solution of Maxwell's equations. Thanks to the discontinuity of the approximation, this kind of methods has many advantages, such as adaptivity to complex geometries through the use of unstructured, possibly non-conforming, meshes, easily obtained high order accuracy, hp-adaptivity and natural parallelism. However, despite these advantages, DG methods have one main drawback particularly sensitive for stationary problems: the number of globally coupled degrees of freedom (DoF) is much larger than the number of DoF required by conforming finite element methods for the same accuracy. Consequently, DG methods are expensive in terms of both CPU time and memory consumption, especially for time-harmonic problems. Hybridization of DG (HDG) methods [START_REF] Li | A hybridizable discontinuous Galerkin method combined to a Schwarz algorithm for the solution of 3d time-harmonic Maxwell's equations[END_REF] is devoted to address this issue while keeping all the advantages of DG methods. HDG methods introduce an additional hybrid variable on the faces of the elements, on which the definition of the local (element-wise) solutions is based. A so-called conservativity condition is imposed on the numerical trace, whose definition involves the hybrid variable, at the interface between neighbouring elements. As a result, HDG methods produce a linear system in terms of the DoF of the additional hybrid variable only. In this way, the number of globally coupled DoF is reduced. The local values of the electromagnetic fields can be obtained by solving local problems element-by-element. We have recently designed such a high order HDG method for the system of 3D time-harmonic Maxwell's equations [START_REF] Li | A hybridizable discontinuous Galerkin method combined to a Schwarz algorithm for the solution of 3d time-harmonic Maxwell's equations[END_REF].

4 Simulation software HORSE is a computational electromagnetic simulation software for the evaluation of radar cross section (RCS) of complex structures. This software aims at solving the full set of 3D timeharmonic Maxwell equations modeling the propagation of a high frequency electromagnetic wave in interaction with irregularly shaped structures and complex media. It relies on an arbitrary high order HDG method that is an extension of the method proposed in [START_REF] Li | A hybridizable discontinuous Galerkin method combined to a Schwarz algorithm for the solution of 3d time-harmonic Maxwell's equations[END_REF]. This HDG method designed on an unstructured possibly non-conforming tetrahedral mesh, leads to the formulation of an unstructured complex coefficient sparse system of linear equations for the DoF of the hybrid variable, while the DoF of the components of the electric and magnetic fields are computed element-wise from those of the hybrid variable. This software is written in Fortran 95. It is parallelized for distributed memory architectures using a classical SPMD strategy combining a partitioning of the underlying mesh with a message-passing programming model using the MPI standard. One important computational kernel of this software is the solution of a large sparse linear system of complex coefficients equations. In a preliminary version of this software, this system was solved using parallel sparse direct solvers such as MUMPS [START_REF] Amestoy | Multifrontal parallel distributed symmetric and unsymmetric solvers[END_REF] or PaStiX [START_REF] Hénon | PaStiX: A High-Performance Parallel Direct Solver for Sparse Symmetric Definite Systems[END_REF]. However, sparse direct solvers may have a limited scalability when it comes to solve very large linear systems arising from the discretization of large 3D problems. In this paper, we instead rely on a hybrid iterative/direct solver. We employ the MaPHyS solver, discussed below, to do so.

MaPHyS algebraic solver

The solution of large sparse linear systems is a critical operation for many academic or industrial numerical simulations. To cope with the hierarchical design of modern supercomputers, hybrid solvers based on algebraic domain decomposition methods have been proposed, such as PDSLin [START_REF] Li | Factorization-based sparse solvers and preconditioners[END_REF], ShyLU [START_REF] Rajamanickam | ShyLU: A hybrid-hybrid solver for multicore platforms[END_REF], HIPS [START_REF] Gaidamour | A parallel direct/iterative solver based on a Schur complement approach[END_REF] or HPDDM [START_REF] Jolivet | Scalable Domain Decomposition Preconditioners for Heterogeneous Elliptic Problems[END_REF]. Among them, approaches consisting of solving the problem on the interior of the domains with a sparse direct method and the problem on their interface with a preconditioned iterative method applied to the related Schur Complement have shown an attractive potential as they can combine the robustness of direct methods and the low memory footprint of iterative methods. MaPHyS (Massively Parallel Hybrid Solver) [START_REF] Giraud | Parallel scalability study of hybrid preconditioners in three dimensions[END_REF] - [START_REF] Agullo | Parallel hierarchical hybrid linear solvers for emerging computing platforms[END_REF] is a parallel linear solver, which implements this idea. The underlying idea is to apply to general unstructured linear systems domain decomposition ideas developed for the solution of linear systems arising from PDEs. The interface problem, associated with the so-called Schur complement system, is solved using a block preconditioner with overlap between the blocks that is referred to as Algebraic Additive Schwarz. Although MaPHyS has the functionality of exploiting two level parallelism using threads within an MPI process, this feature was not used in this study and focus was made on flat MPI performance only. MaPHyS makes use of a sparse direct solver as a subdomain solver such as MUMPS or PaStiX. The parallelization of the direct solver relies on a specific partitioning of the matrix blocks. PaStiX and MUMPS make extensive use of highly optimized dense linear algebra kernels (e.g., BLAS kernels).

A representation of the software architecture is given in Figure 1.

6 Numerical and performance result subdomains (see "sparse direct step" below) and MUMPS 5.0.2 is used for processing the coarse space for the dense+CSC variant (see "dense+CSC" below). We rely on Intel MKL 2017.0.0 for achieving high performance BLAS and LAPACK. In particular, the Intel MKL DSYGVX routine is used for computing the eigenvalue problem in the subdomains in the case of the dense+CSC variant (see "dense+CSC" below).

The energy measurements are performed with two tools developed by Atos-Bull: BEO v1.0 and HDEEVIZ. The former has been used to obtain accurate measures of the total energy spent for the MaPHyS computation; the latter to visualize with a fine time resolution the variations of the energy consumption during the execution.

MaPHyS used in standalone mode

The weak scalability of the MaPHyS solver is first investigated in a standalone mode. For these experiments, we solve a 3D Poisson problem on a 2.5D domain that corresponds to a beam and a 1D decomposition, as shown in Figure 2. Each subdomain has at most two neighbors and is essentially a regular cube of size 40 3 (i.e., each subdomain has approximately 64,000 unknowns). The energy performance has been measured with BEO as the total energy consumed by the job. We have also used the Bull HDEEVIZ graphic tool for a detailed visualization of the energy consumption over time. In Figure 5, we display the energy profile provided by this tool on a typical run of MaPHyS. The main steps of the hybrid solver can be observed by matching the execution time given by HDEEVIZ with the MaPHyS output timers. Thus we can identify these four steps:

• analysis: corresponds to the initialization of MaPHyS, reading the subdomain from dumped files and performing some preliminary index computations and communications;

• sparse direct: corresponds to the factorization of the local matrix associated with internal unknowns and the computation of the Schur complement associated with the interface unknowns;

• precond: the preconditioning step consists of the computation of the local preconditioner and involves neighbor to neighbor communications plus the factorization of the assembled local Schur complement. This factorization might be either dense or sparse depending on the preconditioning strategy further described below;

• iterative: the iterative step aims at computing the solution of the Schur complement system using a preconditioned Krylov subspace method; that is the Conjugate Gradient for the Poisson problem or GMRES for the electromagnetic computation.

The local matrices are read from files, which is both time and energy consuming but not really relevant to MaPHyS performance, since the matrices are usually computed locally and directly provided to the solver by the application codes such as HORSE for the experiments reported in the next section. This file access appears on the profile before the analysis starts.

We have considered five test cases whose properties are displayed in Table 1 and assessed three numerical variants of the solver that are referred to as:

• dense: we consider the fully assembled local Schur complements to build the additive Schwarz preconditioner; only dense linear algebra kernels, cpu intensive with very regular memory access patterns, are then used in the preconditioner application at each iteration of the iterative step.

• sparse: the entries of the local dense Schur complements that are smaller than a given relative threshold (10 -5 ) are discarded; the resulting sparse matrices are then used to build the additive Schwarz preconditioner; mostly sparse linear algebra kernels, less cpu demanding with very irregular memory access patterns, are then used when applying the preconditioner in the iterative part.

• dense+CSC: in addition to the previously described dense preconditioner, a coarse space correction [START_REF] Agullo | Robust coarse spaces for Abstract Schwarz preconditioners via generalized eigenproblems[END_REF] (CSC) is applied to ensure that the convergence will be independent from the number of subdomains. In this experiment we compute five vectors per subdomain to create the coarse space. The coarse space being relatively small compared to the global problem, computations are centralized onto one process and solved by the direct solver (MUMPS in the experiments reported here).

As it can be seen in Figure 5 (power profile of the dense variant on a single node execution), the memory energy consumption represents a significant part of the total (45.7%), which is consistent with the current trend: floating point operations are cheap while memory accesses are relatively costly. One can furthermore observe that the setup and analysis parts of the run have a low energy consumption. The three numerical steps of the MaPHyS solver can be clearly identified through the combined observation of the memory and CPU power profiles. The iterative solve step appears quite clearly as a large plateau, where the power consumption is high for memory and low for CPU. It is consistent with the fact that this step is memory bound with many communications and relatively few computations. The total energy consumed by the node is 5.6 Wh = 20,160 J, which corresponds to the results given by BEO for this case.

We also report the total energy of the job provided by BEO, the total time to solution, the number of iterations and the time spent in the iterative step, in figures 4a, 4b, 4c and 4d, respectively. The time of each step of the solver is represented Figure 3. A value is missing for the "sparse" curve with 128 subdomains due to measurement error. Step by step time performance for MaPHyS. q q q q q q q q q q q q q q 0 50000 100000 150000 200000 64 128 192 256 320

Number of domains

Energy consumed(J)

Preconditioner type: q q q Dense Dense+CSC Sparse (a) Energy to solution.

q q q q q q q q q q q q q q 0 50 100 64 128 192 256 320

Number of domains

Total time (s)

Preconditioner type: q q q Dense Dense+CSC Sparse (b) Total time to solution in MaPHyS.

q q q q q q q q q q q q q q 0 50 100 Dense Dense+CSC Sparse (c) Number of iterations. q q q q q q q q q q q q q q 0 10 20 We observe that, despite the extra-memory traffic due to indirections, the fact that it induces much fewer floating point operations than the dense variant leads to a lower overall energy consumption. The high energy required by the dense+CSC preconditioner is mainly due to the setup of the CSC, which is both memory and CPU demanding. Because the dense and sparse preconditioners do not implement any global coupling numerical mechanisms, the number of iterations is expected to grow as the number of subdomains for the 1D decomposition of the domain on the Poisson test example. This poor numerical behavior can be observed in Figure 4c, while it can be seen that the coarse space correction (dense+CSC variant) plays its role and ensures a number of iterations independent from the number of domains (see [START_REF] Agullo | Robust coarse spaces for Abstract Schwarz preconditioners via generalized eigenproblems[END_REF] for further insights on the numerical properties of the method). This nice numerical behavior translates in terms of time to solution for the iterative part where the dense+CSC method outperforms the two other variants. However, the overhead of the setup phase for the construction of the coarse space, which requires the solution of generalized eigenproblems, is very high and cannot be amortized at that intermediate scale if only a single right-hand side has to be solved (which would not be the case for, e.g., radar cross section evaluation as considered in Section 6.3, where multiple right-hand sides must be solved in real-life test cases). Nonetheless, the relative ranking in terms of power requirements are different. Through a simple linear regression, we can observe that the average power requirements are about 328, 326 and 321 W/node for the dense, sparse and dense+CSC preconditioners, respectively.

Scattering of a plane wave by a PEC sphere

We now consider a more realistic problem that consists in the scattering of a plane wave with a frequency F=600 MHz by a perfectly electric conducting (PEC) sphere. The contour lines of the x-component of the electric field are displayed in Figure 6a, and the obtained RCS is plotted in Figure 6b together with a comparison with a reference RCS obtained from a highly accurate (but costly) BEM (Boundary Element Method) calculation. This problem is simulated using the coupled HORSE/MaPHyS numerical tool. The underlying tetrahedral mesh contains 37,198 vertices and 119,244 elements. We have conducted a series of calculations for which the number of iterations of the MaPHyS interface solver has been fixed to 100. Simulations are performed using a flat MPI mode, binding one MPI process per physical core without any shared memory parallelism. We consider the following two situations: (a) the interpolation order in the HDG discretization method is uniform across the cells of the mesh; (b) the interpolation order is adapted locally to the size of the cell based on goal-oriented criterion. In the latter situation, we distribute the interpolation order such that there are at least 9 integration points (degrees of freedom of the Lagrange basis functions) per local wavelength. For the particular tetrahedral mesh used in this study, we obtain the following distribution of polynomial approximation within the mesh elements: 12,920 P 1 elements, 70,023 P 2 elements, 31,943 P 3 elements and 4,358 P 4 elements. For a given mesh, a uniform interpolation order is not necessarily the best choice in terms of computational cost versus accuracy, especially if the mesh is unstructured as it is the case here. Increasing the interpolation order allows for a better accuracy at the expense of a larger sparse linear system to be solved by MaPHyS. Distributing the interpolation order according to the size of the mesh cells allows for a good trade-off between time to solution and accuracy. The performance in terms of both time and energy consumption are reported in Table 2. In this table, the number of subdomains also corresponds to the total number of core or MPI processes. The number of MPI processes per node can be deduced from the number of nodes. First of all, in most of the tested configurations, we observe a superlinear speedup, as a result of the reduction of the size of the local discretization matrices within each subdomain. The size n of the local matrices roughly decreases linearly with the number of domains but the arithmetic complexity for their factorization is O(n 2 ) that leads to this superlinear effect. We first note, as expected, that the energy consumption with higher values of the interpolation order increases, since the size of the HDG sparse linear system increases drastically. For a given number of subdomains, a second noticeable remark is that the energy consumption increases when the number of MPI processes per node decreases; this leads to a higher network traffic and moving data through the network is more costly than moving them in memory. For instance, for the HDG-P 1 method, Using a decomposition of the tetrahedral mesh in 64 subdomains, the energy consumption is equal to 38 kJ on 4 nodes (i.e., with 16 MPI processes per node) and 68 kJ on 8 nodes (i.e., with 8 MPI processes per node). A similar behavior is observed for the HDG-P 2 method and a 64 subdomain decomposition.

The power consumption increases with the number of nodes due to their idle power consumption (i.e. power consumption when being switched one but performing no useful task). Yet, this characteristic does not order the energy consumption. Indeed, for the HDG-P 1 method, the energy consumption is higher with 1 node than with 2 or 4 nodes, but lower than with 8 or 16 nodes. This situation highlights the fact that the most energy-efficient solution is not always the faster time-to-solution or the least number of nodes: the trade-off lies in-between these two performance metrics and is specific to the considered solving method.

A final comment is that the use of a locally adapted distribution of the interpolation order allows for a substantial reduction of the energy consumption for a target accuracy. This is in fact the result of lowering the computing time solution because of the reduction of the size of the problem, as it can be seen by comparing the figures for the HDG-P 4 and HDG-P k methods with a 256 subdomain decomposition. Finally, for a given test example, the best computing time does not correspond to the best energy to solution; the policies to determine the best trade-off between these two criteria will be the subject of future work.

Conclusion

Through the experiments conducted within this work, several lessons have been learnt on the numerical software designer side. For the numerical linear algebra solvers, we have seen that the best time to solution solver is also the best in terms of energy to solution, but not necessarily in terms of power requirement. From the application perspective, we have observed that the local polynomial order adaptivity, i.e., HDG-P k not only provides the best trade-off in terms of amount of memory and number of floating point operations, but also a nice trade-off in terms of Inria energy consumption. As for the linear solver, the best option in terms of energy might not be the best one in terms of power.

Figure 1 :

 1 Figure 1: Software architecture of the solver stack.

Figure 2 : 2 .

 22 Figure 2: 2.5D test case with 6 subdomains along a 1D decomposition, used for the standalone MaPHyS experiments.

Figure 3 :

 3 Figure 3: Step by step time performance for MaPHyS.

  Time in the iterative step of MaPHyS.

Figure 4 :

 4 Figure 4: Time and energy performance for MaPHyS .

Figure 5 :

 5 Figure 5: Power dissipation over the execution time of MaPHyS, visualized with HDEEVIZ.

  (a) Contour lines of the x-component of the electric field. (b) Contour lines of the x-component of the RCS.

Figure 6 :

 6 Figure 6: Scattering of a plane wave by a perfectly electric conducting sphere.

Table 1 :

 1 Main characteristics of the numerical example considered in the standalone MaPHyS experiments.

	Num. of Num. of	Global	Global
	nodes	domains matrix size Schur size
	1	64	4,305,041	211,806
	2	128	9,033,444	426,974
	3	192	14,202,169	642,142
	4	256	19,826,576	857,310
	5	320	25,922,025	1,072,478

Table 2 :

 2 Performance of the coupled HORSE/MaPHyS software stack. Scattering of a plane wave by a PEC sphere. Timings for 100 iterations of the interface solver of MaPHyS.

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Acknowledgment

This work was partially financially supported by the PRACE project funded in part by the EU's Horizon 2020 research and innovation programme (2014-2020) under grant agreement 653838.