Sophie Chemarin
email: sophie.chemarin@polytechnique.edu

Caroline Orset
email: caroline.orset@univ-tlse1.fr

Pierre Picard

François Salanié

Sandrine Spaeter-Loehrer

Innovation and information acquisition under time inconsistency and uncertainty *

Keywords: innovation, information acquisition, uncertainty, self-control, time inconsistency, liability rules. JEL Classification: D81, D83, D92

This paper analyzes the impact of hyperbolic discounting preferences on the agent's information acquisition decision who wants to undertake a potential dangerous activity for human health or the environment. We find that below certain discount rate threshold, an agent prefers ignoring information and continuing his project. On the other hand, above this threshold, it is optimal for him to acquire information, and the investment for acquiring the information is increasing with the discount rate. We then conclude that hyperbolic discounting preferences limit the information acquisition. Moreover, we explain that the lack of self-control induced by hyperbolic discounting preferences also restraints the information acquisition. Finally, we analyze the efficiency of the strict liability rule and the negligence rule to motivate the agent to acquire information.

Introduction

A recent report of the European Commission [START_REF] Aho | Creating an innovative Europe[END_REF] emphasizes the large gap between the scientific and technological knowledge of European countries and the relatively low level of innovation. Actually, the non-stability and/or the lack of information characterizing innovations constitute one of the main barriers to innovation. As an example of scientific innovation, one could point out genetically modified organisms (GMO) ([29]).

The general principle of producing GMO is to add genetic material into an organism's genome to generate new traits. Examples of GMOs are highly diverse several fish species, transgenic plants (e.g. tomatoes), medicines (e.g. gene therapy), or agricultural products (e.g. golden rice). As for many innovations, we do not perfectly know the effects that GMOs may entail on people's health and/or on the environment. The recent debate on the toxicity of transgenic maize plants MON863 emphasizes the difficulties to define what should be the adequate and proportionate decision to take under such a context of scientific uncertainty. If recent scientific studies seem to point out a potential danger due to the consumption of this kind of transgenic plants, maize MON863 is not forbidden in France while it is, for example, in Netherlands (Seralini, Cellier and Spiroux de Vendomoix, 2007). Indeed, the French "Commission du Génie biomoléculaire" (CGB) concludes in their experts report that "the results of the toxicological study did not point out any toxic effects on rats kidney due to the consumption of maize 863" (CGB, June 15th 2007). In such a context, should we limit technological and scientific innovation as it is done regarding GMOs production in California (United States) or in Prince Edward Island (Canada), which ban all of them to prevent a possible risk? On the other hand, would not it be more relevant to encourage innovation's and research's efforts at the same time in order to reduce scientific uncertainties and to behave according to the information they could obtain from it?

The European report on innovation concludes that there is a real need for action to implement what it defines as a "pact for research and innovation" [START_REF] Aho | Creating an innovative Europe[END_REF]. Indeed, investing in research and development allows a reduction of the scientific uncertainty which characterizes the innovations. In this regard, the precautionary principle, as a public decision criteria, proposes to combine innovation, security and information acquisition (Pouillard, 1999, Henry and[START_REF] Henry | L'essence du principe de précaution: la science incertaine mais néanmoins fiable[END_REF]. It inspires many European directives and propositions that aim at protecting health and environment. Thus, the directive on environmental liability proposes to apply the 'pollutant-payer' principle: if a damage happens, the pollutant has to pay for it. The investors are then liable for eventual incident due to their activities. The goal of this principle is to act directly on the investors behaviours to increase their prevention and research efforts. Mechanisms based on lia-bility rules, such as a strict liability rule (which use is in the spirit of the application of the precautionary principle) or a negligence rule, are used to protect consumers as well as the environment by improving the control and the prevention of risks induced by firms' activities and products. However, uncertainty limits their efficiency but also induces contradictory effects on both innovation and security. For example, regarding technological risks, Sinclair-Desgagné and Vachon (1999) note that limited liability leads to less prevention and the extension of the liability to all the firm's partners may limit innovating efforts. To go further in the current debate on precaution and economic development, one should investigate what determines, under existing liability framework, the innovation decision of an investor regarding activities for which scientific knowledge is still incomplete.

Individual risks perception and thus risks assessment is never exempt of subjectivity.

For example, according to [START_REF] Kahneman | Judgment under uncertainty: heuristics and biases[END_REF], people undervalue or overvalue small probabilities in proportion to the importance of potential damages, and according to their past experiences. In a more general way, as emphasized by several empirical studies, risk perception and thus, individual preferences change over time (Frederick, Loewenstein and O'Donoghue, 2002). [START_REF] Strotz | Myopia and inconsistency in discounting utility maximization[END_REF] is the first to suggest an alternative to exponential discounting. [START_REF] Phelps | On second-best national saving and gameequilibrium growth[END_REF] introduce the functional form of this kind of changing preferences. Let D(k) represent a discount function such that:

D(k) = 1 if k = 0; D(k) = βδ k if k > 0.
We can define a hyperbolic discounted utility function as follows:

U t = T -t k=0 D(k)u k .
Elster (1979) applies this (β, δ) formalization to a decision problem in characterizing time inconsistency by a decreasing discount rate between the present and the future and a constant discount rate between two future periods. [START_REF] Laibson | Golden eggs and hyperbolic discounting[END_REF][START_REF] Laibson | Life cycle consumption and hyperbolic discount functions[END_REF] then uses this formulation to saving and consumption problems, while other economists like [START_REF] Bénabou | Self confidence and personal motivation[END_REF], 2004) and [START_REF] Carrillo | Strategic ignorance as a self disciplining device[END_REF] apply it into the problem of information acquisition. In particular, [START_REF] Carrillo | Strategic ignorance as a self disciplining device[END_REF] describe intertemporal consumption's decisions involving a potential risk in the long run. They show that hyperbolic discounting can favour strategic ignorance. Indeed, a person with time inconsistent preferences might choose not to acquire free information in order to avoid over consumption or engagement in activities, which may require much more fundamental research on potential social costs or externalities they could involve in the long term. However, if ignorance is a self-disciplining device when an agent is confronted to uncertainty and hyperbolic discounting, is it then useless for him to acquire information to develop a project?

In this paper, we propose to analyze the impact of hyperbolic discounting preferences on the agent's information acquisition decision who wants to undertake a activities with potential risks on health or the environment. We choose to both define information acquisition as an investment in research1 to reduce the uncertainty on the potential risk of damage, and to study innovator's intertemporal choices as a joint product of many conflicting psychological motives. Since empirical studies [START_REF] Frederick | Time discounting and time preference: a critical review[END_REF] suggest it, the innovator's preferences are described by a hyperbolic discounting function.

In such a case, preferences are said to be time inconsistent. The discount rate β gathers, as in [START_REF] Akerlof | Procrastination and obedience[END_REF] and [START_REF] Carrillo | Strategic ignorance as a self disciplining device[END_REF], all the psychological motives of the agent's investment choice such that anxiety, confidence or impatience. If β = 1, the psychological motives have no influence on his choice, and his preferences are time consistent. On the other hand, if β < 1, the innovator is temporally inconsistent.

Under uncertainty, we first study the optimal decision-making of an innovator with hyperbolic discounting preferences. This allows us to analyze the direct impact of the hyperbolic preferences on the information acquisition decision. We then study the lack of self-control induced by the hyperbolic discounting preferences on the acquisition information. In this regard, we define the self-control effect as the agent's ability to commit in the future. Finally, we study how two main forms of liability -strict liability and negligence rule -may give incentives the innovator to acquire information. We choose to provide the investment in research (i.e. information acquisition) as a measure of care, in so far as it leads to a reduction of the uncertainty linked to the innovation, and to decisions that may reduce the cost of damages suffered by both the innovator (according to the liability rule enforced) and the environment in case of accident.

As a result, we show that below a certain discount rate threshold, acquiring information is not optimal. The innovator prefers ignoring information in order to never stop his project and to protect his innovation's ability. On the other hand, above this threshold, the investment in research is increasing with the value of β. We then conclude that the hyperbolic discounting preferences limit the information acquisition and favours the strategic ignorance. Moreover, we explain that the lack of self-control induced by hyperbolic discounting preferences also favours the strategic ignorance. Finally, because such ignorance behaviours have to be regulated to both protect people and environment, we show that, due to the hyperbolic discounting preferences and uncertainty, the impacts of liability rules on the innovator's ability to acquire information is not so clear. Indeed, hyperbolic discounting preferences limit the efficiency of the strict liability rule to motivate the agent to acquire information. Nevertheless, we show that the negligence rule may be an alternative tool to avoid the hyperbolic discounting preferences effect and then may incentive the agent to acquire information.

The paper is organized as follows. Section 1 presents the model. Section 2 investigates the optimal decision-making. Section 3 studies the self-control effect. Finally, section 4 proposes solutions, the strict liability and the negligence rule, to incentive the innovator to acquire information. All proofs are in appendix.

The model

We consider a three periods model. At period 0, the innovator invests a given amount of money I in a project that may create damage on the environment or on people health.

There are two possible states of the world: a dangerous state noted H and a less dangerous state, L. The probability that a damage happens is θ H in state H and θ L in state L.

Since H is the most dangerous state of the world, we obtain

θ L < θ H .
At the beginning, the prior beliefs of the innovator are p 0 on the state H and 1p 0 on the state L. We then define the expected probability of the damage by

E(θ) = p 0 θ H + (1 -p 0 )θ L .
At period 0, the innovator invests C ≥ 0 in research, to obtain information at period 1, through a signal σ ∈ {h, l} on the true state of the world. We define the precision of the signal as an increasing and concave function f (C) such that:

P (h|H, C) = P (l|L, C) = f (C) and P (h|L, C) = P (l|H, C) = 1 -f (C) and f (0) = 1 2 ; f ′ (0) = +∞ and f ′ (+∞) = 0.
Hence, the signal is not informative when the innovator does not invest in research and it becomes more and more precise, i.e. P (h|H, C) increases and P (l|L, C) decreases when C increases.

According to Bayes' rule,

P (h|H, C) = p 0 f (C) p 0 f (C) + (1 -p 0 )(1 -f (C)) and P (H|l, C) = p 0 (1 -f (C)) p 0 (1 -f (C)) + (1 -p 0 )f (C) .
At period 1, according to the perceived signal the innovator decides to stop his project with a probability 1x, or to continue it with a probability x, with x = x σ for signal σ ∈ {l, h} . If the innovator stops the project, he recovers a part of his investment D such that 0 < D < I . Nevertheless, if he decides completely to achieve it, he wil receive a positive return R 2 at the next period (period 2).

An accident may occur at period 2. If it occurs while the innovator has prematurely stopped his project, he suffers a financial cost K ′ related to the damages. But if it occurs while he has decided to complete the project, the financial cost K is higher. So, we assume 0 < K ′ < K.

We define by B S the undiscounted benefit to continue the project instead of stopping it at period 1, when the state of nature is S ∈ {L, H}, i.e.

B S = (R 2 -θ S K) -(D -θ S K ′ ).
We assume that the undiscounted benefit is positive in state L while it is negative in state H so

B H < 0 < B L .
A collection of risk-neutral incarnations with conflicting goals represents the innovator's preferences. 2 At each period t, there is only one incarnation called "self-t". Each self-t depreciates the following period with a discount rate β < 1. 3 In our model, we consider that β represents the salience of current payoffs relative to the future stream of returns.

Remember our GMOs farmer's example, his confidence with GMOs may decrease over time because of a better knowledge of their potential negative effects (i.e. reduction of the scientific uncertainty), which could imply period-to-period different decisions.

Hence, intertemporal expected payoffs of self-t= 0, 1, 2 may be expressed recursively. If signal σ has been perceived, self-2's intertemporal expected payoff is written as

V 2 (x σ , σ, C) = x σ [P (H|σ, C)(R 2 -θ H K) + (1 -P (H|σ, C))(R 2 -θ L K)] -(1 -x σ )[P (H|σ, C)θ H K ′ + (1 -P (H|σ, C))θ L K ′ ]. Likewise, self-1's intertemporal expected payoff is V 1 (x σ , σ, C) = (1 -x σ )D + βV 2 (x σ , σ, C).
Finally, self-0's intertemporal expected payoff is

V 0 (x h , x l , C) = -I -C + β[p 0 f (C) + (1 -p 0 )(1 -f (C))](V 2 (x h , h, C) + (1 -x h )D) +β[(1 -p 0 )f (C) + p 0 (1 -f (C))](V 2 (x l , l, C) + (1 -x l )D).
We define θ(β) as the damage probability threshold for which self-1 is indifferent between stopping or carrying on the project to the end. Since

θ(β) is such that D -βK ′ = β(R 2 -K), we obtain θ(β) = βR 2 -D β(K -K ′ ) .
We also assume that

E(θ) ≤ θ(β) which is equivalent to βR 2 -βE(θ)K ≥ D -βE(θ)K ′ . (1)
Hence, if no research effort is made (C = 0), i.e. with No Learning, self-1 continues the project to the end.

So, self-0's expected payoff with No Learning is written as

V N L 0 (β) = -I + β[p 0 (R 2 -θ H K) + (1 -p 0 )(R 2 -θ L K)]. = -I + β[R 2 -E(θ)K].

The optimal decision-making

We now turn to the innovator's optimal decision-making. Subsection 2.1 studies self-1's optimal decision to continue the project or to stop it, and subsection 2.2 determines self-0's optimal decision to acquire information.

Stopping or continuing the project

At period 1, if self-1 is uninformed he always continues his project. However, if he gets information, he receives a signal σ ∈ {h, l} on the probability of damage and updates his beliefs. According to this information, he has to choose either to complete or not his project. Formally, for σ ∈ {h, l} and for all C ≥ 0, self-1 continues the project if his expected payoff when he continues the project is higher than when he stops it, i.e.

V 1 (1, σ, C) ≥ V 1 (0, σ, C).
After signal σ ∈ h, l has been perceived, let E(θ|σ, C) = P (H|σ, C)θ H +(1-P (H|σ, C))θ L be the expected probability of a damage and x * σ the equilibrium strategy.

Proposition 1 If E(θ|σ, C) ≤ θ(β), then x * σ = 1: the innovator continues the project; if θ(β) < E(θ|σ, C), then x * σ = 0: the innovator stops the project.

Proposition 1 characterizes the conditions on the revised expected probability of the damage, E(θ|σ, C), under which self-1 decides to partially or completely achieve his project.

These two cases may arise. If E(θ|σ, C) is lower (resp. larger) than the threshold θ(β),

self-1's optimal decision is to carry on (resp. to stop) the project.

We easily verify that θ(β) is increasing with β:

θ ′ (β) = D β 2 (K -K ′ ) is positive. So, for all β ∈]0, 1], θ(β) ≤ θ(1).
So according to Proposition 1, the range of values in which the hyperbolic agent (β < 1) may decide to stop his project at period 1 is higher than the one of the exponential agent (β = 1), also defined as an innovator with consistent preferences.

Lemma 1 E(θ|l, C) ≤ E(θ) ≤ E(θ|h, C) for all C ≥ 0. And E(θ|h, C) is increasing with C while E(θ|l, C) is decreasing with C.
Then the second part of Lemma 1 suggests that from a certain value of C, the signal have an impact on the decision.

Moreover, according to Lemma 1 and condition (1), when self-1 receives signal l, he always decides to continue the project while when he receives signal h, if θ(β) is higher (lower) than E(θ|h, C) self-1 continues (stops) his project.

Overall, two possible cases may occur: either self-1 always continues the project whatever the signal, or self-1 behaves according to the information he gets and continues (stops) the project if he receives signal l (h).

Information acquisition

At period 0, self-0 chooses his optimal research investment C * (β) to acquire information on the risk of accident, knowing that at period 1, self-1 either always continues the project whatever the signal (case 1) or stops (continues) it if he received the signal h (l) (case 2). However, self-0 has no commitment power, he then cannot control the future self's decision.

We first study case 1, i.e. self-1 always continues the project. Self-0's expected payoff is

V 0 (1, 1, C) = -I -C + β[p 0 (R 2 -θ H K) + (1 -p 0 )(R 2 -θ L K)].
Since V 0 (1, 1, C) is decreasing with C, among all the possible levels of investment, it is obvious that the optimal solution is

C * 11 (β) = 0.
Since, the signal does not have any influence on self-1's behaviour (i.e. whatever the signal self-1 continues the project), it is then not surprising that optimally at period 0, self-0 does not want to invest in research and then does not acquire any information.

Overall, optimally case 1 is equivalent to the situation where the innovator is uninformed.

We now turn to case 2. Self-1 stops (continues) the project according to the signal he receives. Self-0's expected payoff is

V 0 (0, 1, C) = -I -C + β[p 0 (1 -f (C))(R 2 -θ H K) + (1 -p 0 )f (C)(R 2 -θ L K)] +β[p 0 f (C)(D -θ H K ′ ) + (1 -p 0 )(1 -f (C))(D -θ L K ′ )].
Define by C 01 (β) the investment in research which solves the following problem max

C≥0 V 0 (0, 1, C). (2)
We get:

Proposition 2 C 01 (β) is characterized by f ′ (C 01 (β)) = 1 β[(1 -p 0 )B L -p 0 B H ] . (3)
C 01 (β) is increasing with β.

However, in this case the investment in research chosen by self-0 leads self-1 to choose the strategy [x * l = 1, x * h = 0]. We then define by C(β) the smallest C ≥ 0 which satisfies the following condition

E(θ|l, C) ≤ θ(β) ≤ E(θ|h, C). Proposition 3 C(β) is characterized by f ( C(β)) = p 0 (θ H -θ(β)) p 0 (θ H -θ(β)) + (1 -p 0 )( θ(β) -θ L ) .
C is decreasing with β.

Finally, to define the optimal investment in information C * 01 (β) that provides useful information (i.e. a signal that influence self-1's decision), self-0 has to solve the following expected payoff maximization problem

max C V 0 (0, 1, C) C(β) ≤ C.
Proposition 4 characterizes self-0 's optimal investment in research C * 01 (β) under the assumption that self-1 behaves according to [x * l = 1, x * h = 0] strategy.

Proposition 4 There exists a unique β ∈]0, 1] such that C 01 (β) = C(β) and for all

β ∈ [0, β] then C * 01 (β) = C(β); for all β ∈]β, 1] then C * 01 (β) = C 01 (β).
We note that β = 0. Indeed, if we suppose that β = 0 then:

• f ′ (C 01 (0)) = +∞ and since f is concave, C 01 (0) = 0; • f ( C(0)) = p 0 2p 0 -1 . If C 01 (0) = C(0) = 0 then f (0) = p 0 2p 0 -1 = 1 2 which is impossible because -1 = 0.
Although self-0 invests in research and self-1 behaves according to the information he gets, one should underline that the level of this investment depends on the discount value β.

We define by C * (β) the optimal research investment. To find C * (β), we compare the self-0's expected payoff of both strategies and we select the level of research investment that leads, from self-0's perspective, to the highest expected payoff. This gives the following proposition.

Proposition 5 The optimal research investment C * (β) is such that for all β ∈ [0, β] then C * (β) = 0; for all β ∈]β, 1] then C * (β) = C 01 (β).
Figure 1 illustrates Proposition 5. To obtain the shape of the function C 01 (β), we first differentiate equation (3) with respect to β, we obtain:

f ′′ (C 01 (β))C ′ 01 (β) = -1 β 2 [(1 -p 0 )B L -p 0 B H ] .
And then since f is concave, when β tends towards infinity C ′ 01 (β) tends towards infinity and when β tends towards zero C ′ 01 (β) tends towards zero. So, we get the following figure: for the present may lead him to stop prematurely his project. To avoid stopping the project, the innovator may prefer not obtaining information and always continuing.

On the other hand, for β ∈]β, 1] , the optimal investment in information is increasing with β. It tends toward the optimal investment reached when the agent has time consistent preferences and a commitment power (i.e. C * (1)).

Moreover, we note a discontinuity for the optimal research investment function C * (β).

Actually, since the information precision is increasing with the research investment, the research investment must be rather large so that the information precision is valuable for the agent and he then decides to invest in research.

According to Proposition 5, the exponential agent (β = 1) always invests in research and his investment is C * 01 (1). On the other hand, the hyperbolic agent (β < 1) only gets information when β ∈]β, 1].

Hence, Result 1 may arise:

Result 1 The hyperbolic discounting effect limits the information acquisition.

This result fits into the literature on hyperbolic discounting. [START_REF] Akerlof | Procrastination and obedience[END_REF] points out that a time inconsistent innovator (β < 1) always postpones a costly activity (or investment). In our model since the innovator has only the choice to invest in research at period 0, postponing this investment is equivalent to not doing it. Now, we may take a look on Figure 2. For β ∈ [0, β], a low investment in research has no influence on self-1's behaviour, the information precision is not large enough. Indeed, for We make a last remark. If there is no recovered investment D or if D is given at period 2, the selves 1 with β = 1 and with β < 1 have the same condition to stop or not their project. Actually, since a hyperbolic innovator (β < 1) much prefers earning money now than tomorrow, when D is given at period 1, he has more incentive than the exponential innovator to obtain D now, and to stop the project, than to wait for future gains.

all C ≥ 0, V 0 (1, 1, 0) > V 0 (0, 1, C).
3 The effect of self-control

According to [START_REF] Salanié | Over-savings and hyperbolic discounting[END_REF], a self-control effect is induced by the hyperbolic discounting preferences. The self-control effect may be defined as the agents ability to commit in the future. Hence, a lack of self-control implies that there is no possibility of commitment between the future selves of the innovator (i.e. β < 1 between the period 1 and the period 2). On the other hand, self-control leads to a commitment power (i.e.

β = 1 between the period 1 and the period 2). Such effect does not depend on whether the innovator's preferences display a 'bias for the present' or not, it is then not straightforward that a lack of self-control leads to less research investment.

Self-control problems characterized situations where investors may have difficulties to undertake long run strategies. For example, the development of chemicals or medicines might suffer from a possible lack of commitment power. If the research and development stage concerning a new drug is only realized at the time when the firm invests in this new product, the decision to develop it is taken according to the information provided by the R& D stage. Since there is no others information available in the future, there is no reason for the firm to modify her decision. In others words, as soon as the decision is taken according to the ex-ante available information, there is self-control. In contrast, if there is a stock of information available during all the periods of the new product development, it is difficult to implement a strategy in the long run, without modifying this strategy according to the information. In such a case, since the firm might not be able to commit in the long run, there might be a lack of self-control.

According to [START_REF] Salanié | Over-savings and hyperbolic discounting[END_REF], to analyze the self-control effect on the innovator's behaviour, we isolate it in comparing the decision of an innovator with a lack of self-control (see previous section), 4 to the decision of an innovator with self-control.

The innovator with self-control follows the same timing of the intra-personal game. His intertemporal payoffs are similar to those of an innovator with a lack of self-control. We only switch β < 1 to β = 1 between periods 1 and 2.

We define x * * σ the equilibrium strategy after signal σ has been perceived. Thus, we immediately obtain:

Lemma 2 For all σ ∈ {h, l}, if E(θ|σ, C) < θ(1), then x * * σ = 1: the innovator continues the project; if θ(1) < E(θ|σ, C), then x * * σ = 0: the innovator stops the project.

Lemma 2 characterizes the conditions on the expected probability of the damage according to the signal, E(θ|σ, C), under which self-1 with self-control stops or carries on his project.

According to Lemma 1 and condition (1), we obtain two cases again. Self-1 always continues the project whatever the received signal; or self-1 continues (stops) the project if he receives the signal l (h).

Self-0 gets the same intertemporal payoff than the one of an innovator with a lack of selfcontrol. Only the damage probability threshold that condition self-1's decision changes:

θ(β) is switching in θ(1). We define C * * (β) the optimal research investment for an innovator with self-control. We then obtain:

Lemma 3 There exists β ∈ [0, 1] such that C 01 (β) = C(1). Then C * * (β) is such that for all β ∈ [0, β] then C * * (β) = 0; for all β ∈]β, 1] then C * * (β) = C 01 (β).
We note that β may be equal to zero. Indeed, if β = 0 then f ′ (C 01 (0)) = +∞.

Since f is concave that implies C 01 (β) = 0. If C 01 (0) = C(1) = 0, we obtain that

f (0) = p 0 (θ H -θ(1))
p 0 (θ H -θ(1))+(1-p 0 )( θ(1)-θ L ) = 1 2 . Thus, E(θ) = θ(1) which is impossible by assumption.

It is then not surprising that β influences the innovator's behaviour again. For low value of β, the innovator ignores the information and continues the project. Nevertheless, for high values of β, linking innovation and information acquisition seems possible.

We turn now to the self-control effect and compare the behaviour of the innovator with a lack of self-control to the behaviour of the innovator with self-control.

First, according to Proposition 1 and Lemma 3 when the self-0 gets information, the innovator with self-control has more opportunity to achieve the project than the innovator with a lack of self-control.

Moreover, according to Proposition 5 and Lemma 3, the innovator with a lack of selfcontrol (respectively with self-control) does not invest in research when β ≤ β (respectively. β ≤ β) and starts investing after this threshold. β and β are respectively charac-terized by C 01 (β) = C(β) and C 01 (β) = C(1). We then obtain:

Lemma 4 β < β.
So, according to Proposition 5 and Lemmas 3 and 4, we obtain that: When β ∈ [0, β], both innovators (i.e. with self control and with a lack of self-control)

• If β ∈ [0, β], then C * (β) = C * * (β) = 0; • if β ∈]β, β], then C * (β) = 0 and C * * (β) = C * 01 (β); • if β ∈]β, 1], then C * (β) = C * * (β) = C * 01 (β).
prefer ignoring the information on the probability of damage because they strongly discount the cost of damage. But, when β ∈]β, β], the innovator with self-control starts investing in research to acquire information, while the innovator with a lack of selfcontrol prefers staying ignorant. Finally, when β ∈]β, 1] both types of innovators acquire information and choose the same research investment level. As a result, we finally obtain:

Result 2 The self-control effect limits the information acquisition.

Moreover, we again point out that if there is no recovered investment D, or if D is given at period 2, there is no self-control effect. 54 Solutions to incentive the information acquisition

Firms are constrained by a legal framework, in which liability rules specify how to allocate damages from an accident. Regarding innovation as well as other risky activities, when they have to decide whether to start and continue their activities, they should receive the correct incentives not to neglect risk and information acquisition. Under scientific uncertainty, the 'precautionary principle' should lead to such behaviours, but it remains difficult to apply. In this regard, this section proposes to analyze how, at the innovator level, through the existing regulatory framework of risk prevention, in particular strict liability rule and negligence rule, we may find a way to implement such a principle by developing incentives to both enhance innovation and information acquisition.

The strict liability rule

Under a strict liability rule, the innovator is fully liable and thus he must pay for the damages caused by his activities. Nevertheless, his responsibility is engaged only if the victims can demonstrate a causality link between the damage and the activity or the product sold. Such a rule implies that the innovator needs to consider the effect on accident losses of both his level of care, and his level of activity. Thus, he exercises optimal prevention efforts to reduce the risks and undertakes the optimal level of care (i.e. paying the optimal cost of care).

In our model, we investigate intertemporal investment behaviour in a potentially risky project. We suppose that if an accident happens, the innovator suffers the financial cost.

Thus, we implicitly suppose that the innovator is constrained by a strict liability rule.

Moreover, we define the level of care as the level of investment in research (i.e. information acquisition). By investing in research, the innovator gets information on the dangerousness of his activity and then he can decide to stop it or not. It is a way to underline the impact of the liability rule on the innovator behaviour, although both the ex-ante investment I and the cost of damages K which do not depend on C (like in the case in the usual models of liability). Indeed, if the innovator exercises the level of care C and receives a signal of high danger (i.e. σ = h), he stops his activity and then limits the cost of damages (i.e. K ′ < K). Otherwise, if he does not exercise C, he never stops his project and therefore he exposes the environment or the people health to a more severe risk, for which he is responsible. In such a case, he suffers a higher financial cost in case of accident.

A liability rule is efficient when it gives the right incentives to prevent risks and to reduce damages. In our model, the efficiency of the strict liability rule is linked to the innovator's ability to obtain information that may reduce at the end the financial cost of damages. In other words, the rule is efficient when the innovator invests in research and gets useful information. Moreover, when the agent's selves can perfectly internalize the externality they exert on future selves (i.e. β = 1), strict liability rule leads the innovator to undertake the optimal 'cost of care' [START_REF] Shavell | Strict liability versus negligence[END_REF], 1992[START_REF] Miceli | Economics of the law[END_REF]. Thus, we say that the rule is fully efficient when the innovator is willing to pay this optimal cost of care C * (1). Otherwise, if the investment in research is lower, the rule is less efficient, and if the investment is null, the rule is not efficient. We define the efficiency of a strict liability rule like the ability of an innovator to invest in research. We obtain

Result 3

The impact of time inconsistency is such that for β ∈ [0, β], the strict liability rule is inefficient; for β ∈]β, 1[, the strict liability rule efficiency is increasing with β; for β = 1, the strict liability rule is fully efficient.

Time inconsistency leads to a lower and decreasing investment in research. Moreover, for small values of β, innovators might prefer being liable and develop their activities, than getting information on the risk of damage of their project and being tempted to stop it prematurely. Thus, hyperbolic discounting preferences limits the efficiency of the strict liability rule to incite the innovator to acquire information.

The negligence rule

Under a negligence rule, it is said that "the injurer [innovator] is liable for the victim's only if he failed to take a minimum level of care" [START_REF] Miceli | Economics of the law[END_REF], also defined as a 'due standard of care' by [START_REF] Shavell | Strict liability versus negligence[END_REF][START_REF] Shavell | Liability and the incentive to obtain information about risk[END_REF]. In such a case, he has to pay for the damages.

On the other hand, if he exercises at least the level of care, he is not liable and the victims or the State pay the financial cost.

In our model, we define by C N R (β) the level of investment in research to exercise, not to be liable if an accident occurs. Since C N R (β) is an investment in information on the probability of damage to reduce the uncertainty linked to the project, it does not represent the ability of the innovator to exercise physical prevention efforts. Thus, we define

C N R (β) like a due standard of research. 6
According to the literature, we should define C N R (β) as the optimal cost of investment in research, from which the innovator is not liable for the damage. However, due to time inconsistency and uncertainty, for low β (i.e. β ∈ [0, β]), the optimal level of investment in research is null. Thus, if

C N R (β) = C * (β) = 0 for β ∈ [0, β]
, the innovator will never be liable when an accident occurs. Applying a negligence rule would have no impact on the innovator's behaviour, except if we define, for low level of β, C N R (β) as the imposed level of investment in research, in order to force the innovator to acquire information (i.e. to exercise a minimum level of care), whatever his intertemporal preferences, and to behave according to this information.

Nevertheless, to make the investor change his behaviour, it should also be on his interest to undertake such imposed investment. Thus, for β ∈]0, β], his intertemporal expected payoff needs to be higher when he is not liable than when he is. In such a case, he is then never tempted to invest less than C N R (β) and then to be at fault. Remember that according to Proposition 1, at period 1, the investor has to choose between two strategies:

to carry on his project whatever the signal he receives, or to continue (to stop) it only if he gets signal l (h). We define p σ as the probability to receive the signal σ:

p l (C) = p 0 (1 -f (C)) + (1 -p 0 )f (C) and p h (C) = p 0 f (C) + (1 -p 0 )(1 -f (C)).
Thus, at period 0, if the agent undertakes the due standard of research, his expected intertemporal payoffs according to the strategies chosen are

V N R 0 (1, 1, C) = V 0 (1, 1, C) | K=K ′ =0 = -I -C + βR 2 , V N R 0 (0, 1, C) = V 0 (0, 1, C) | K=K ′ =0 = -I -C + βp l R 2 + βp h D
Since we define C N R (β) as the minimum investment in research that makes him change his behaviour: C N R (β) is thus the minimum cost of research that provides him information that influence, when he is not liable, his decision to stop or continue the project at period 1. In others words,

C N R (β) is such that the innovator is at least indifferent between getting V N R 0 (1, 1, C) or V N R 0 (0, 1, C).
Moreover, we know that for low value of β, the agent is more concerned by present satisfaction, he optimally prefers not investing in information and getting V N L 0 . To avoid such effect, C N R (β) has also to be the cost that leads the innovator to be indifferent between "being informed but not liable", and "staying uniformed but being liable for damages". In others words, if he refuses to pay this due standard of research, he pays the financial cost of damages, if an accident occurs, and he gets self-0's expected payoff, when self-0 decides not to obtain information, i.e.

V N L 0 . Otherwise, if he invests C N R (β), he does not have to pay for damages, but the information acquisition on the risk of damage and not in physical prevention measures. Nevertheless, in a general approach, they are equivalent notions.

information is useful to limit the cost of these damages (i.e. even if the innovator does not pay for damages, if he receives a signal of high danger (σ = h), he stops his activity).

Therefore, for low level of β, the due standard of research is such that

V N R 0 (1, 1, C) = V N R 0 (0, 1, C) and V N R 0 (0, 1, C) = V N L 0 (β). which is equivalent to C N R (β) = βE(θ)K (4)
Thus, if the investors invests less than C N R (β), he is liable for damages and gets a lower payoff, than the one he could obtain if he invests at least C N R (β). Moreover, he also obtains useful information and then limits people and environment's exposition to the dangerousness of his activity.

On figure 4 below, we represent, for a given β ∈ [0, β], C N R (β), as well as the investor expected payoff whereas he respects or not the due standard of research (black line).

For low value of β, we also could have imposed

C N R (β)) = C * 01 (β) | K=K ′ =0
as the due standard of research. Indeed, as we force the investor to change his behaviour and to choose the strategy [x * l = 1, x * h = 0], he should then prefer investing C 01 (β) | K=K ′ =0 in information, which allows him to maximize his expected payoff V N R 0 (0, 1, C), and always provides him a higher payoff than the one he gets without any information if he is liable for damages (V N L 0 (β)). Thus for β ∈ [0, β], the due standard of research defined by [START_REF] Bénabou | Self confidence and personal motivation[END_REF] is the smallest value of the due standard of research to impose.

Otherwise, for β ∈]β, 1], under a strict liability rule, the innovator optimally invests C 01 (β) to obtain information . Thus, the due standard of research under negligence should be equal to the optimal level research C * 01 (β), when the investor is not liable (i.e. K = K ′ = 0). Overall, we obtain:

Proposition 6 The due standard of research C N R (β) is such that for β ∈ [0, β[, C N R (β) ∈ [C N R (β), C * 01 (β) | K=K ′ =0 ], with C N R (β) = βE(θ)K ; for β ∈ [β, 1], C N R (β) = C * 01 (β) | K=K ′ =0 .
We remark that for all β, C N R (β) is strictly positive. In term of information acquisition, the negligence rule is thus more efficient than the strict liability rule; it leads to an investment in research under time inconsistency closer to the one exercised under time consistency. Furthermore, a negligence rule may limit innovation. An informed innovator has more opportunities to stop his project. In particular, for low β, while a negligence rule lays down a minimum investment in research, a strict liability rule leads the innovator to be uniformed and always to continue the project. We then obtain the following result Result 4 Comparing the impact of strict liability rule and negligence rule on innovation and information acquisition, we obtain for β ∈ [0, β[, a negligence rule favours information acquisition but limits innovation, while a strict liability rule favours innovation but limits information acquisition; For β ∈ [β, 1], both rules are equivalent and they allow to a combination between innovation and information acquisition.

However, from a social perspective, the negligence rule is more efficient. It proposes a minimum research investment that a hyperbolic investor is willing to undertake, and provides information that may influence the innovator's decision to continue or not his project. Thus ,it allows to limit damages in case of an accident happens. Hence, we obtain the following result:

Result 5 Under hyperbolic discounting preferences and uncertainty, the negligence rule is socially more efficient than the strict liability rule.

Nevertheless, under hyperbolic discounting preferences and uncertainty, both liability rules are difficult to apply and can have pervert effect on innovation and information acquisition: combining both of them remain difficult.

Conclusion

In this paper, we study the information acquisition of an agent with hyperbolic discounting preferences who wants to undertake a potential dangerous activities. Possible examples of application of this model include innovators in new technologies (e.g. nanotechnologies, mobile phones), pharmaceutical firms (e.g. development and production of new drugs) or chemical firms (e.g. production of new fertilizers). Since for the GMOs plants farm's example, in all those cases innovators produce while they may have an incomplete knowledge on the dangerousness of their activities in the long run. We find that for large enough values of the discount rate, the innovator acquires information on the damages risks. His investment in research is increasing with the value of β: it tends towards the level of investment reached when the agent has time consistent preferences and is allowed to commit in the long run. On the other hand, for small values of the discount rate the innovator may ignore the information: he refuses to invest in research to obtain information on the dangerousness of his activity in order to achieve his project.

Moreover, if the hyperbolic discounting preferences limit the innovator's ability to acquire information, the lack of self-control (i.e. no commitment between the different future selves of the innovators) strengthens this limitation.

We also observe that if the investor does not have the possibility to recover a part of his initial investment when he gives up his project (i.e. no salvage value (D)), or if he can only obtain this salvage value at the project's date of maturity (i.e. at the period 2), both the discounting and the self-control effects would not have any impact on his decisions to completely or partially achieve his project. Actually, the innovator with time inconsistency preferences is more concerned by current rewards than by future ones and in particular when there is a possible delayed cost. Since D is given at period 1, the hyperbolic innovator is more likely than the exponential one to earn D now and to stop his project, than to wait for future gains and maybe to suffer the delayed cost of damages.

Moreover, we underline that hyperbolic discounting preferences also limit the efficiency of the legal framework, and in particular of the strict liability rule. If there are situations in which an innovator prefers being uninformed in order to develop his activity, one should define others rules and incentives to ensure that investments in innovation's projects are not done to the detriment of information acquisition and conversely. In this regard, even if it does not always lead to an optimal choice, we consider the application of a complete negligence rule as a possible alternative solution.

However, lessons have to be learned from practical examples. Regarding GMO crops and sales, the current legislation does not seem to be completely efficient to prevent potential risks, to ensure a safe use of such organisms and to identify who should take responsibility for them. For example, it is particularly the case when non-modified organisms are contaminated by modified organisms through pollens scattering. This confirms our results: if scientific uncertainty limits the effect of a strict liability on producers' decisions, time inconsistent preferences strengthen such limitation. Nevertheless, experiences also underline the persuasive role that strict liability can play on producers behaviour, even if its application remains difficult. Weill (2005) notes that when the 'burden of the proof' is in on the potential injurer, and not on the victims, as it is the case under a negligence rule, producers are more likely to stop potentially dangerous products from the market.

The recent European legislation on chemicals (REACH directive) 7 tackles the challenging issue related to the application of the precautionary principle to both enhance innovation as well as people and environment protection. It is based on a strict liability rule, under which the 'burden of the proof' is on the industry, but it also requires manufacturers and importers to take the responsibility "to gather information on the properties and risks of all substances produced or imported". 8 This legislation proposes an interesting way to implement the precautionary principle to deal with chemicals, by combining the positive effects of a strict liability rule, to a research obligation for firms that should avoid the negative ones. This approach should provide relevant elements in the current debate on the regulation of others kind of scientific and/or technological innovation.

Appendix Proof of Proposition 1

At period 1, the innovator receives the signal σ ∈ {h, l}. For all C ≥ 0, he chooses to continue i.e. Thus, E(θ|l, C) ≤ E(θ) ≤ E(θ|h, C).

x σ = 1 if V 1 (1, σ, C) ≥ V 1 (0, σ, C) i.e. E(θ|σ, C) ≤ θ(β) ≡ βR 2 -D β(K -K ′ ) .
We differentiate E(θ|h, C) with respect to C, we obtain

(1 -p 0 )p 0 f ′ (C)(θ H -θ L ) [(1 -p 0 )(1 -f (C)) + p 0 f (C)] 2
which is positive. Thus, E(θ|h, C) is increasing with C.

We differentiate E(θ|l, C) with respect to C, we obtain (1p 0 )p 0 f ′ (C)((θ Lθ H ) [p 0 (1f (C)) + (1p 0 )f (C)] 2 which is negative. Thus, E(θ|l, C) is decreasing with C.

Proof of Proposition 2

Concavity of V 0 (0, 1, C): We differentiate V 0 (0, 1, C) with respect to C, we obtain

-1 + β[(1 -p 0 )B L -p 0 B H ]f ′ (C). ( 5
)
We differentiate equation ( 5) with respect to C, we obtain

β[(1 -p 0 )B L -p 0 B H ]f ′′ (C).
Since f is concave, B L is positive and B H is negative then V 0 (0, 1, C) is concave.

The solution C 01 (β) to problem (2) is characterized by

-1 + β[(1 -p 0 )B L -p 0 B H ]f ′ (C 01 (β)) = 0 ⇔ f ′ (C 01 (β)) = 1 β[(1 -p 0 )B L -p 0 B H ] .
Since B L is positive and B H is negative, we verify that f ′ (C 01 (β)) > 0.

We easily check that f ′ (C 01 (.)) is decreasing with β. Since f is concave then C 01 is increasing with β. Since f is increasing to compare C 1 (β) and C 2 (β) we compare f ( C 1 (β)) and f ( C 2 (β)).

We obtain that if E(θ) < θ(β) then C(β) = C 1 (β); and if E(θ) > θ(β) then C(β) = C 2 (β).

According to condition (1), we obtain that C(β) = C 1 (β). Moreover, we easily show that

f ( C(β)) ≥ 1 2 .
We differentiate f ( C(β)) with respect to β we obtain

-p 0 (1 -p 0 ) θ ′ (β)(θ H -θ L ) [p 0 (θ H -θ(β)) + (1 -p 0 )( θ(β) -θ L )] 2
that is negative. According to Lemma 1 and condition (1), since f is increasing then

C(β) is decreasing with β.
that is positive by assumption.

We differentiate V 0 (0, 1, C) with respect to β we obtain

p 0 (1 -f (C))(R 2 -θ H K) + (1 -p 0 )f (C)(R 2 -θ L K) +p 0 f (C)(D -θ H K ′ ) + (1 -p 0 )(1 -f (C))(D -θ L K ′ )
which is positive by assumption.

We suppose that V 0 (0, 1, C 01 (β)) = V 0 (1, 1, 0) this implies that

f (C 01 (β)) = β(1 -p 0 )B L + C 01 (β) β[(1 -p 0 )B L -p 0 B H ] . (6)
We differentiate V 0 (0, 1, C 01 (β)) and V 0 (1, 1, 0) with respect to β and we replace f (C 01 ) by the right hand side of equation [START_REF] Carrillo | Strategic ignorance as a self disciplining device[END_REF]. Since for all β ∈ [0, 1] C 01 (β) ≥ 0, we obtain that ∂V 0 (0, 1, C 01 (β) ∂β ) ≥ ∂V 0 (1, 1, 0) ∂β .

Thus, there exists β ∈ [0, 1] such that V 0 (0, 1, C 01 ( β)) = V 0 (1, 1, 0) and

• for all β ∈ [0, β] then V 0 (0, 1, C 01 (β)) ≤ V 0 (1, 1, 0);

• for all β ∈] β, 1] then V 0 (0, 1, C 01 (β)) > V 0 (1, 1, 0).

We notice that for β = 0, we obtain C 01 (0) = 0 and then V 0 (0, 1, C 01 (0)) = V 0 (1, 1, 0) = -I.

Thus, β = 0 and for all for all β ∈]0, 1] then V 0 (0, 1, C 01 (β)) > V 0 (1, 1, 0). Since β = 0 then for all for all β ∈]β, 1] C * (β) = C 01 (β).

Overall we obtain for all β ∈ [0, β] then C * (β) = 0; for all β ∈]β, 1] then C * (β) = C 01 (β).

Proof of Lemma 2

Similar to the proof of Proposition 1, thus omitted.

Proof of Lemma 3

Similar to the proof of Proposition 5, thus omitted.

Figure 1 :

 1 Figure 1: Optimal research investment.

  The investor has strong preferences for the present and do not really care about what can happen in the future, there is no value of C that makes him change his behaviour. On the other hand, for β ∈]β, 1], V 0 (0, 1, C * 01 (β)) > V 0 (1, 1, 0) and C * 01 (β) gives a valuable information to self-1.

Figure 2 :

 2 Figure 2: Optimal research investment with β ∈ [0, β] and β ∈]β, 1].

Figure 3

 3 Figure 3 illustrates this result.

Figure 3 :

 3 Figure 3: The effect of self-control on the investment in research.

Figure 4 :

 4 Figure 4: Due standard of research.

  Since θ H > θ L and for all C ≥ 0, f (C) ≥1 2 we obtain thatE(θ|l, C) -E(θ) = (1p 0 )p 0 (θ Hθ L )(1 -2f (C)) (1p 0 )f (C) + p 0 (1f (C)) ≤ 0 and E(θ) -E(θ|h, C) = (1p 0 )p 0 (θ Lθ H )(2f (C) -1) p 0 f (C) + (1p 0 )(1f (C)) ≤ 0.

Proof of Proposition 3

 3 According to Lemma 1 and condition[START_REF] Aho | Creating an innovative Europe[END_REF], C(β) the smallest C ≥ 0 which satisfiesE(θ|l, C) ≤ θ(β) ≤ E(θ|h, C) is such that either E(θ|l, C(β)) = θ(β) < E(θ|h, C(β)) or E(θ|l, C(β)) < θ(β) = E(θ|h, C(β)) Define C 1 (β) which verifies that E(θ|l, C(β)) = θ(β) we obtain f ( C 1 (β)) = p 0 (θ Hθ(β)) p 0 (θ Hθ(β)) + (1p 0 )( θ(β)θ L ) .Define C 2 (β) which verifies that E(θ|h, C(β)) = θ(β) we obtainf ( C 2 (β)) = (1p 0 )( θ(β)θ L )p 0 (θ Hθ(β)) + (1p 0 )( θ(β)θ L ) .

  Since for β ∈]β, 1] it is optimal to invest in research. The innovator solves the followingproblem max C V N R 0 (0, 1, C) E(θ|l, C) < θ(β) < E(θ|h, C).Thus, we obtain thatC N R (β) = C * 01 (β) | K=K ′ =0 . Overall we obtain that ∀β ∈ [0, β[, C N R (β) ∈ [C N R (β), C * 01 (β) | K=K ′ =0 ]; et ∀β ∈ [β, 1], C N R (β) = C * 01 (β) | K=K ′ =0 .

Such investment in research could be considered as a resort to private experts like private laboratory or to any other private party able to provide scientific knowledge on the dangerousness or more generally on the characteristics on the innovator's activities

Following Strotz (1956), this conflict is captured by assuming that the innovator's preferences are dynamically inconsistent.

[START_REF] Akerlof | Procrastination and obedience[END_REF] defines β as the "salience of current payoffs relative to the future stream of returns", but it is also interpreted in the literature as a lack of willpower[START_REF] Bénabou | Self confidence and personal motivation[END_REF], of foresight[START_REF] Masson | Risque et horizon temporel : quelle typologie des consommateurs -pargnants?[END_REF], O'Donoghue and Rabin (1999)) or as impatience or impulsiveness[START_REF] Ainslie | Picoeconomics: The Strategic Interaction of Successive Motivational States within the Person[END_REF]. Moreover, to simplify the (β, δ) formalization, we assume that δ = 1.

We already studied the behaviour of an innovator with a lack of self-control in the previous section which analyzes the impact of the discounting effect on the investor's decision to invest in research.

θ(β) = θ(1) and β = β.

We define C N R (β) like a due standard of research and not a due standard of care, because it is an

REACH stands for Registration, Evaluation, Authorization and Restriction of Chemicals

for more details on REACH, see European Commission [28].

Proof of Proposition 4

Since C(β) is decreasing with β and C 01 is increasing with β there exists a β ∈]0, 1] such that C 01 (β) = C(β).

Since C(β) is the smallest C ≥ 0 which satisfies the following condition

Proof of Proposition 5

The optimal research investment C * (β) is such that

We compare V 0 (0, 1, C * 01 (β)) and V 0 (1, 1, 0). According to Proposition 4, we have for all

We first compare V 0 (0, 1, C(β)) and V 0 (1, 1, 0). We obtain

We replace f ( C(β)). We obtain

which is negative because of condition [START_REF] Aho | Creating an innovative Europe[END_REF]. Thus for all β ∈ [0, β] we obtain that

Then C * (β) = 0. Now we compare V 0 (0, 1, C 01 (β)) and V 0 (1, 1, 0). It is easily verified that V 0 (1, 1, 0) and V 0 (0, 1, C) are both increasing with β. We differentiate V 0 (1, 1, 0) with respect to β we obtain

By definition

Since C is decreasing with β then

Since C 01 is increasing with β we obtain β < β.

Proof of Proposition 6

For

Thus, we get C N R (β) = βE(θ)K.

However, if the agent invests C N R (β), his intertemporal pay-off is V N R 0 (0, 1, C), and he solves the following maximisation problem:


Jocelyn Chauvet
email: jocelyn.chauvet@umontpellier.fr

Catherine Trottier

Xavier Bry

Regularisation of Generalised Linear Mixed Models with autoregressive random effect

Keywords: Regularised EM algorithm, Generalised Linear Mixed Model, Autoregressive random effect, Panel data analysis

We address regularised versions of the Expectation-Maximisation (EM) algorithm for Generalised Linear Mixed Models (GLMM) in the context of panel data (measured on several individuals at different time-points). A random response y is modelled by a GLMM, using a set X of explanatory variables and two random effects. The first one introduces the dependence within individuals on which data is repeatedly collected while the second one embodies the serially correlated time-specific effect shared by all the individuals. Variables in X are assumed many and redundant, so that regression demands regularisation. In this context, we first propose a L2-penalised EM algorithm, and then a supervised component-based regularised EM algorithm as an alternative.

Introduction

One of the main purposes of panel data analysis is to account for the dependence induced by repeatedly measuring an outcome on each individual over time. Besides, due to the fact that it is nowadays increasingly possible to collect large amounts of data, the potentially high level of correlation among explanatory variables should be taken into account. To this end, ridge-, lasso-and component-based regularisations have recently been highlighted. In the Linear Mixed Models (LMM) framework, [START_REF] Eliot | Ridge Regression for Longitudinal Biomarker Data[END_REF] proposed to extend the classical ridge regression to longitudinal biomarker data. They suggested a variant of the EM algorithm to maximise a ridge-penalised likelihood. This variant includes a new step to find the best shrinkage parameter -in the Generalised Cross-Validation (GCV) sense -at each iteration. With a view towards variable selection, [START_REF] Schelldorfer | Glmmlasso: an algorithm for high-dimensional generalized linear mixed models using l 1penalization[END_REF] proposed a L 1 -penalised algorithm for fitting a high-dimensional Generalised Lin-ear Mixed Models (GLMM), using Laplace approximation and an efficient coordinate gradient descent. In the GLM framework, in order both to regularise the linear predictor and to facilitate its interpretation, [START_REF] Bry | Supervised component generalized linear regression using a pls-extension of the fisher scoring algorithm[END_REF] developed a PLStype method -Supervised Component-based Generalised Linear Regression (SCGLR) -which yields explanatory components. [START_REF] Chauvet | Extension to mixed models of the Supervised Component-based Generalised Linear Regression[END_REF] extended SCGLR to GLMM by using an adaptation of Schall's algorithm [START_REF] Schall | Estimation in generalized linear models with random effects[END_REF]).

To the best of our knowledge, the random effects in the previous strategies are assumed normally distributed with independent levels. However, in the panel data framework, the question naturally arises of the autocorrelation of the time-specific random effect. Consequently, our objective is twofold: on the one hand, to extend the Mixed Ridge Regression of [START_REF] Eliot | Ridge Regression for Longitudinal Biomarker Data[END_REF] to the GLMMs with an AR(1) random effect; and on the other hand, to present the main ideas of a new version of SCGLR which handles the high dimensional case.

Model hypotheses

In this section, we recall the main hypotheses of the GLMM framework and we introduce the random effect distributions. For the sake of simplicity, we consider balanced panel data with N individuals, each of them observed at the same T time-points. We denote by n = N × T the total number of observations. Let X be the n × p fixed effects design matrix, and U the n × q random effects design matrix. Let also Y be the n-dimensional random response vector, β the p-dimensional vector of fixed effects, and ξ the q-dimensional vector of random effects. We observe a realisation y of Y , but ξ is not observed. We conventionally assume that:

(i) the Y i | ξ, i ∈ {1, .
. . , n} are independent and their distribution belongs to the exponential family;

(ii) the conditional mean µ i = E(Y i | ξ) depends on β and ξ through the link function g and the linear predictor

η i = x T i β + u T i ξ, with η i = g(µ i ).
Less conventionally, we consider two random effects ξ 1 and ξ 2 with different roles and distributions:

(i) ξ 1 is the individual-specific random effect. Assuming individuals are independent, we suppose:

ξ 1 ∼ N N 0, σ 2 1 I N ,
with σ 2 1 the unknown "individual" variance component.

(ii) ξ 2 is the serially correlated time-specific effect common to all the individuals, which can be viewed as some latent phenomenon not measured in the explanatory variables. As these effects tend to persist over time, we model them with a stationary order 1 autoregressive process (AR(1)), i.e. for each t ∈ {1, . . . , T -1},

ξ 2,t+1 = ρξ 2,t + ν t , ν t iid ∼ N 0, σ 2 2 ,
where ρ is the unknown parameter of the AR(1) and σ 2 2 the unknown "temporal" variance component. Such time-specific effects arise naturally for instance in an economic context (e.g. all companies share a common economic climate which tend to persist over time), or in biology (e.g. the ecological environment is often too complex to be directly observed through the explanatory variables).

Finally, ξ 1 and ξ 2 are assumed independent. Denoting ξ

= (ξ T 1 , ξ T 2 ) T , U 1 = I N ⊗ 1 T , U 2 = 1 N ⊗ I T and U = [U 1 |U 2 ]
, linear predictor η can be matricially written:

η = Xβ + U ξ.

Methods

Owing to the GLMM dependence structure, the Fisher scoring algorithm was adapted by [START_REF] Schall | Estimation in generalized linear models with random effects[END_REF]. We, in turn, adapt Schall's algorithm by introducing a regularised EM at each step in order to take into account the high level of correlation in X and the unconventional random effects distributions. Two steps appear in our method: the linearisation step and the estimation step.

Linearisation step. For each i ∈ {1, . . . , n}, a classic order 1 linearisation of y i around µ i is given by: g(y i ) z i = g(µ i ) + (y i -µ i )g (µ i ). Matricially, this approximation provides a working variable z entering the following linearised model

M : z = Xβ + U ξ + e, with Var(e | ξ) = Diag [g (µ i )] 2 Var(Y i | ξ) i=1,...,n = Γ.
Estimation step. Instead of solving Henderson's system associated with M seen as a LMM (as proposed by [START_REF] Schall | Estimation in generalized linear models with random effects[END_REF]), we rather propose a regularised EM step. We suggest an adaptation of the L 2 -penalised EM algorithm of [START_REF] Eliot | Ridge Regression for Longitudinal Biomarker Data[END_REF] for low dimensional data (p < n), and a supervised component-based regularised EM algorithm for the high dimensional case (p n), because then, interpretable dimension reduction is needed.

The low dimensional case

Our estimation step is based on [START_REF] Green | On use of the EM for penalized likelihood estimation[END_REF], who popularised the use of the EM algorithm for penalised likelihood estimation, and [START_REF] Golub | Generalized cross-validation as a method for choosing a good ridge parameter[END_REF], who encouraged the use of the GCV for efficiently choosing the ridge parameter λ. However, contrary to the homoskedastic LMM considered in [START_REF] Eliot | Ridge Regression for Longitudinal Biomarker Data[END_REF], M contains heteroskedastic errors. We will then opt for the modified GCV criterion suggested by [START_REF] Andrews | Asymptotic optimality of generalized CL, crossvalidation, and generalized cross-validation in regression with heteroskedastic errors[END_REF], p. 372. Denoting θ = β, σ 2 1 , σ 2 2 , ρ , Algorithm 1 describes the current iteration of our L 2 -penalised EM algorithm for GLMM with AR(1) random effect.

(1) Linearisation step. Set:

M [t] : z [t] = Xβ + U ξ + e, with Var(e | ξ) = Γ [t] .
(2) Estimation step.

(2.a) Denoting L the complete log-likelihood of the linearised model, define the associated complete penalised log-likelihood L pen by:

L pen (θ; z, ξ) = L(θ; z, ξ) - λ 2 β T β.
(2.b) Denoting z [t] the fitted values and S

[t]

λ the "hat-matrix" satisfying the equality z

[t] = S [t]
λ z [t] , set:

λ [t] ←-arg min λ      GCV(λ) = n -1 z [t] -S [t] λ z [t] 2 Γ [t] -1 1 -n -1 tr S [t] λ 2      .
(2.c) EM step. Set: t+1] , and update working variable z [t+1] and variance-covariance matrix Γ [t+1] .

Q pen θ, θ [t] = E ξ|z L pen (θ; z [t] , ξ) | θ [t] , λ [t] , θ [t+1] ← arg max θ Q pen θ, θ [t] . (3) Updating step. Set ξ [t+1] = E ξ|z ξ | θ [
Steps (1)-( 3) are repeated until stability of parameters β, σ 2 1 , σ 2 2 and ρ is reached.

Algorithm 1: Current iteration of the L 2 -penalised EM algorithm for GLMM with AR(1) random effect.

The high dimensional case

In the p n case, we need to decompose the linear predictor on a small number of interpretable dimensions. To that end, we propose to iteratively maximise a component-based regularised Q-function. Let C = XU be the set of principal components of X with non-zero eigenvalues and f = Cw the component we currently seek. Let also φ denote a structural relevance (SR) criterion (see [START_REF] Bry | THEME: THEmatic model exploration through multiple co-structure maximization[END_REF]):

φ(w) =   p j=1 cor 2 x j , f l   1 l , l 1.
s ∈ [0, 1] being a parameter tuning the relative importance of the SR with respect to L, the Q-function would then be:

Q reg θ, θ [t] = E ξ|z L reg (θ; z, ξ) | θ [t] , with L reg (θ; z, ξ) = (1 -s)L(θ; z, ξ) + sφ(w).
Parameters s and l are tuned by cross-validation and higher rank components are computed like rank 1 component, after adding extra orthogonality constraints.

Numerical results

In order to evaluate the performance of our L 2 -penalised EM algorithm, we conducted simulation studies in the canonical Poisson case. We present some graphical diagnoses in FIGURE 1, which aim at answering three questions: (1) Is the convergence assured? (2) How good are the estimations? (3) Are they sensitive to the value of ρ? The answers to these questions is given in the figure's caption.

FIGURE 1 .

 1 FIGURE 1. Graphical diagnoses relative to the L2-penalised EM algorithm. (a),(b): 40 trajectories of the L2-convergence criterion for parameters β and σ 2 1 (A similar behaviour is observed for parameters σ 2 2 and ρ). About a hundred iterations is necessary to achieve convergence. (c): MSEs of parameters β, σ 2 1 , σ 2


Andrei Constantinescu

Nicolas Tardieu
email: tardieu@lms.polytechnique.fr

ON THE IDENTIFICATION OF ELASTOVISCOPLASTIC CONSTITUTIVE LAWS FROM INDENTATION TESTS

This paper addresses the identi®cation of the parameters of a nonlinear constitutive law from indentation tests. The identi®cation problem is considered as a constrained minimization problem and the gradient is computed using the adjoint state method, in spite of the diculties of the underlying contact problem. This provides a general framework to perform optimization in some problems involving contact conditions and nonlinear material behaviour. The case of a Maxwell viscoelastic and a Norton-Ho elastoviscoplastic constitutive law are treated extensively and a series of numerical identi®cation examples are shown.

INTRODUCTION

The indentation test is performed by pressing a punch into a material sample. It was initialy used to evaluate the hardness of metals and is now being considered as an ecient non-destructive method for determining mechanical characteristics of materials [START_REF] Taljat | New analytical procedure to determine stress-strain curve from spherical indentation data[END_REF].

The constitutive law should be identi®ed from the knowledge of the indentation curve (see Fig. 3), which represents the load applied on the punch versus the penetration depth. The applicability of the indentation tests depends at this point on the precision and generality of this identi®cation.

The identi®cation strategies currently used in the experiments are based on semi-empirical formulas dedicated to given constitutive behaviour: elasticity, perfect plasticity [START_REF] Johnson | Contact mechanics[END_REF], power laws [START_REF] Jayaraman | Determination of monotonic stress-strain curve of hard metals from ultra-low-load indentation tests[END_REF], FFF. These formulas are based on elastoplastic ¯ow theories as presented in the classical monograph of Tabor [START_REF] Tabor | Hardness of metals[END_REF].

Only a few studies present a general method applicable to a large class of constitutive behaviours, like de®ning the identi®cation as the minimization of a cost functional.

Koguchi [START_REF] Koguchi | Determination of mechanical properties of thin ®lms and functional gradient materials using an inverse technique[END_REF] identi®ed the elastic coecients of a layered half-space indented with a sphere, minimizing a least squares distance between measurements and computations using a closed form solution to the direct problem.

More generally, for a non-linear elastic constitutive law and a least squares cost functional Hasanov [7 ± 9] proved some existence results and proposed a numerical procedure based on trial and error.

The previous methods are dedicated essentially to some elastoplastic behaviours and the minimizations do not include gradient computations of the cost functional. Gradient computations are dicult due to the mathematical complexity of the contact description [START_REF] Hilding | Optimization of structures in unilateral contact[END_REF][START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and ®nite element methods[END_REF], appearing independently of the constitutive behaviour of the material. As the direct problem is numerically costly, the knowledge of the gradient could speed up the minimization algorithm.

The computation of a gradient of a least squares functional in the case of linear elasticity has been given in [START_REF] Constantinescu | The adjoint state method applied to material identi®cation through indentation tests[END_REF][START_REF] Tardieu | On the determination of elastic coecients from indentation tests[END_REF]. It has been shown that the adjoint state method [START_REF] Lions | Contr ole optimal des syst emes gouvern es par des equations aux d eriv ees partielles[END_REF] can be extended to variational inequalities. The eciency of the method has been illustrated by numerical examples in the identi®cation of linear elastic moduli from indentation tests.

The goal of this paper is to show that the gradient computation by an adjoint state method can be used in combination with a nonlinear elastoviscoplastic constitutive law under the assumption of small strains and rotations. Moreover, the method applies to any suciently regular standard generalized constitutive law (for a presentation of these behaviours, see [START_REF] Bourgeois | Contr ole optimal et probl emes inverses en plasticit e[END_REF][START_REF] Halphen | Sur les mat eriaux standards g en eralis es[END_REF]). This ensures the generality of the approach.

As a consequence the parameters of nonlinear elastoviscoplastic constitutive laws can be identi®ed from indentation tests by minimizing least squares functionals with gradient algorithms.

The accuracy and robustness of the method is illustrated in this paper through numerical examples for a Maxwell viscoelastic and a Norton-Ho viscoplastic constitutive law.

THE DIRECT PROBLEM

Let us consider an axisymmetric body, with its section occupying in its reference con®guration an open subset &R 2 with smooth boundary À (see Fig. 1). The boundary is partioned in three parts À À D À F À C : the part À D where displacements are imposed, the free surface À F , and the surface À C where contact might occur. n and t denote the normal and tangent vector to the boundary À.

The axisymmetric hypothesis is taken in order to simplify the presentation and the computational burden and does not restrict the generality of the method.

The problem will be treated within the theory of small strains and rotations. The validity of this hypothesis will be discussed later. Therefore, let us denote respectively by u, " and s the vector ®eld of displacements and the tensor ®elds of small strains and stresses.

The problem considered in the sequel is the indentation of the body by a rigid punch whose pro®le is characterized by the gap g. For a FIGURE 1 The con®guration of the direct problem for a conical punch.

complete presentation of contact computations see [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and ®nite element methods[END_REF]. The contact is considered without friction. An indentation experiment is driven either by the vertical displacement U or by the force F applied to the punch. The force F can be expressed as integral of the contact pressure:

F Z À C n Á s Á n dÀ
An experiment provides an indentation curve (see Fig. 3), representing a displacement-force history (U exp , F exp ) over a given time interval [0, T ].

In this work, we have always expressed the problem as driven by the punch displacement. The governing equations can be written using: a time continous expression, where the intervening quantities are the velocities of the ®elds, or a time discretized expression, where the intervening quantities are small increments of the ®elds between two time steps.0 Describing contact conditions using velocities of the ®elds is a complicated task demanding care in the choice of the functional spaces of the mathematical formulations [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and ®nite element methods[END_REF]. Therefore the time discretized expression, which permits to avoid some of these diculties, will be used in this paper.

Constitutive Law

A standard generalized material without work hardening [START_REF] Halphen | Sur les mat eriaux standards g en eralis es[END_REF] is considered here. This constitutive behaviour is completely determined by the elasticity tensor S(c) and by the pseudo-potential of dissipation È È(s, c). The later is supposed to be twice dierentiable with respect to s. c is the vector of the material parameters characterizing the material behaviour (Young's modulus, elasticity limit, FFF).

Time Continuous Expression

In a time continuous description, the constitutive law is expressed by the classical set of equations:

" uSc X s " p 1 " p @Ès; c @s 2
where the dot . denotes the time derivative and " p is the viscoplastic strain.

Time Discretized Expression

In a time discretization, the previous equations are expressed as

"Áu i S XÁs i Á" p i 3 Á" p i @Ès i ; c @s Át 4
Examples The following classical constitutive laws can be expressed under this formalism:

The Maxwell viscoelastic material : the pseudo potential È is given by È(s i , c) (1/2)s i : M : s i where M is a fourth order tensor. The plastic strain increment is determined by : Á" p i M X s i Át. The Norton-Ho viscoplastic material: the pseudo potential È is given by Ès i ; cK=m 1hs i eq À Y =Ki m1 where Y is the elasticity limit, hÁi is the positive part operator and (Á) eq is the equivalent Mises stress. The plastic strain increment is determined by: Á" p i 3=2h i eq À Y =Ki m si = i eq Át; si is the deviator of s i .0

Equations of the Direct Problem

The governing equations of the direct problem consist of the equilibrium and constitutive equations, the boundary and contact conditions and a set of initial values. The contact conditions on À C are expressed using the Lagrange multipliers p i P N, where N {q P (H 1/ 2 (À C )) H jq 0} is a closed convex set and (H 1/2 (À C )) H denotes the dual of H 1/2 (À C ) (see [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and ®nite element methods[END_REF]).

The punch is driven by its vertical displacement U i at time t i . At the same moment, the gap g i between the surface À C and the punch is expressed by g i g U i À u n i .

Equilibrium and Constitutive Equation in V divÁs i 0 5

"Áu i S XÁs i @Ès i ; c @s Át 6
Boundary Conditions

Ás i Á n 0o nÀ F 7 Áu i 0o nÀ D 8
Contact Conditions

Áu n i À g i À ÁU i q À p i1 !0 Vq P N 9
Á nn i Ás i Á n Á n Áp i 10
Á nt i Ás i Á n À Á nn i Á n Á t 0 11 Initial Conditions s 0 0i n 12
u 0 0o n 13
" p 0 Á n 0o n 14 15

THE INVERSE PROBLEM À1

In the present inverse problem, one wants to identify the parameters of the material behaviour c from the knowledge of the indentation curve (U exp , F exp ). c is supposed to evaluate in a closed convex of R n (n ! 2). This inverse problem can be expressed as a minimization problem of a well chosen cost functional. Since the direct problem is driven by the imposed displacement of the punch U, it is natural to express the cost functional as a function of the resultant force F. A possible formulation of the inverse problem À1 is:

Find c P minimizing tc 1 2 X I i0 F comp i cÀF exp i 2 1 2 X I i0 Z À C p i cdÀ À F exp i 2 16
where, F comp is the computed resultant force from the direct problem driven by U exp . One can remark that the cost functional t depends implicitly on the material parameters c through the intermediate of the pressure distribution p. The resolution of the direct problem permits the determination of the Lagrange multiplier p i and then the calculation of F calc i . In consequence, this minimization problem can be considered as a constrained one, the constraint being the resolution of .

In the case of an elastoplastic constitutive law existence results for this problem are given in [7 ± 9].

Resolution

The resolution of a constrained minimization problem is equivalent, under some regularity conditions, to ®nding the saddle point of a Lagrangian functional v. In the case of the inverse problem À1 , the Lagrangian v is introduced as a sum between the cost functional and a variational formulation of the direct problem .

For each variable of the direct problem, an adjoint variable, denoted by a ? superscript, is introduced. These adjoint variables are the Lagrange multipliers of the constraints, the equations of the direct problem.

According to the optimal control theory, all direct and adjoint variables will be considered mutually independent. The relationships between them will be recovered from the stationarity conditions of the Lagrangian v, characterizing the saddle point.

The Lagrangian functional has the following form:

; p ? i 1 2 Z À C p i dÀ À F exp i 2 Z divÁs i Áu ? i d Z À F Ás i Á n Á u ? i dÀ Z À D Áu i Á s ? i Á ndÀ Z À C Áu n i À ÁU i À g i Áp ? i dÀ À Z "Áu i ÀS XÁs i @Ès i ; c @s Át X s ? i d Z À C Áp i À Ás nn i Áu n? i dÀ Z À C Á nt i Á u t? i dÀ and u i , u ? i PH 1 2 s i , s ? i PL 2 4 p i , p ? i P N i fq PH 1=2 H À C jq 0o nÀ C =À C i g
and where À C i is the eective contact surface at time t i .0

The complex form of this Lagrangian does not permit to draw any conclusions with regard to the existence and uniqueness of its saddle point. Nevertheless, necessary conditions of stationarity can be formally written in order to characterize this eventual saddle point.

The stationarity conditions of v are given by the following expressions:

X I i0 @v i @u ; dw i 0 Vdw i PH 1 2 18
X I i0 @v i @s ; dt i 0 Vdt i PL 2 4 19 X I i0 @v i @p ; dq i 0 Vdq i P N i 20
X I i0 @v i @u ? ; dw i 0 Vdw i PH 1 2 21 X I i0 @v i @s ? ; dt i 0 Vdt i PL 2 4 22 X I i0 @v i @p ? ; dq i 0 Vdq i P N i 23
X I i0 @v i @c ; d À c ! 0 Vd P 24
where hÁ, Ái represents in each equation the duality pairing for the corresponding functional spaces.

Calculating the derivatives with respect to the adjoint variables (Eqs. (21) ± ( 23)) leads to the set of equations:

divÁs i 0i n 25 "Áu i ÀS XÁs i À @Ès i ; c @s Át 0i n 26
Ás i Á n 0o nÀ F 27 Áu i 0o nÀ D 28 Áu n i À g i À ÁU i 0 Ás nn i Áp i Á nt i 0 9 = on À C i 29
The preceding calculation leads in classical Lagrangian theory to the equations of the direct problem. In the present case, opposite to the classical frame, the equations do not represent exactly the direct problem. However, if (u, s, p) are the solutions to , they obviously verify the above relations.

The derivation of v with respect to the direct variables (Eqs. ( 18) ± ( 20)) and series of calculations: spatial integration by parts and use of the ®rst order approximation

f i Á Ág i % f i 1 Á g i 1 À f i Á g i À Áf i Á g i , gives
the following set of equations: This set of equations and the ®nal conditions de®ne a well-posed incremental problem with Dirichlet conditions on a part of the boundary and will be called the adjoint problem ? .

divÁs i 0i n 30 "Áu ? i S XÁs ? i À @ 2 Ès i ; c @s 2 Át X s ? i in 31
Ás ? i Á n 0o nÀ F 32
Áu ? i 0o nÀ D 33
Áu n? i F calc i cÀF exp i Á nn? i Áp ? i Á nt? i 0 9 = on À C i 34
and
The preceding calculations can be summarized as:

Stationarity Result

If (u, s, p) and (u ? , s ? , p ? ) are respectively the solutions to the incremental direct and adjoint problem and ? , then the Conditions (18) ± (23) of stationarity of the Lagrangian v are veri®ed.

Moreover, if (u, s, p) are the solutions to , one can notice that the Lagrangian v is reduced to the cost functional t . Together with the expression of stationarity conditions (24) this implies that:

Gradient Computation

If (u, s, p) and (u ? , s ? , p ? ) are respectively the solutions to the incremental direct problem and to the incremental adjoint problem à , then the gradient of the cost functional t is given by

r c t X I i0 Z Ás i X @S @c X s ? i @ 2 È @s@c Át X s ? i d 40
Some remarks can be done about the preceeding results:

The adjoint problem is not a contact problem. Its loading is a Dirichlet conditions (imposed displacement) on À C i , the eective contact surface of the direct problem. The adjoint problem is a time dependent system of partial dierential equations on [0, T] and a ®nal condition given by a well-posed elasticity problem. Therefore the adjoint problem will be integrated from T to 0 in the reversed time.

The adjoint constitutive law is viscoelastic considered in the reversed time i H 2 I À i:

"Áu ? i H S XÁs ? i H R X s ? i H Át
where R is the forth order tensor:

R @ 2 Ès i H ; c @s 2 i H
The pseudo potential È has to be twice dierentiable to ensure the existence of the adjoint state (For example in the Norton-Ho constitutive law, m > 1 is needed ). From a numerical point of view, this leads to a linear problem at each time step and therefore a rapid integration.

The parameters to the adjoint constitutive law depend on the parameters of the direct constitutive law, but also on the solution of the direct problem. Hence the solution to the adjoint problem is implicitly dependent on the solution of the direct problem. The equations of the adjoint problem in the reversed time describe a linear viscoelastic problem with Dirichlet boundary conditions and initial conditions, and therefore well posed. This method allows the computation of the gradient of the cost functional t using the solutions to the direct and adjoint problems, independently of the number of parameters involved. A rapid evaluation of the computational burden shows that a gradient calculation takes %1.5 the time for solving the direct problem due to the simplicity of the adjoint behaviour and the elimination of the contact condition. This is extremely interesting for problems with a large number of parameters. However, the intervention of the solution to the direct problem in the resolution of the adjoint problem, demands large memory space for keeping track of all the ®elds.0

NUMERICAL EXAMPLE FOR THE MAXWELL EQUATION

In order to illustrate the presented method, let us consider the identi®cation of the parameters of a Maxwell viscoelastic material:

"Áu i S XÁs i Á" p i where Á" p i M X s i Át 41 S ijkl 1 E 1 ik jl À kl ij 42 M ijkl 1 ik jl À 1 3 kl ij 43
where E, , denote respectively the Young's modulus, the Poisson coecient and the viscosity. The tensor M computes the deviatoric part of stress.

As explained before, this constitutive law enters the formalism of the standard generalized materials without work-hardening.

The identi®cation problem consists of determining E and from an indentation curve (U exp , F exp ). The Poisson coecient was considered as known, but could be identi®ed in the same way.

In this work the indentation curve is obtained from numerical experiments as explained in the sequel.

Direct Calculations

The experiment simulations have been realized through ®nite elements computations using the CASTEM2000 code [START_REF]Castem2000 Finite Element[END_REF]. The body was a cylinder with radius 10 mm and height 10 mm and the punch was a rigid cone with a 68 half angle at the apex. The mesh of the body had 20 Â 20 quadratic elements. At this coarness the solution of the direct problem is independent of the mesh size.

The indentation process has been displacement controlled and consists of a loading, maintain and unloading parts (see Fig. 2). A typical indentation curve is represented on Figure 3.

The hypothesis of small strains and rotations has been the keypoint in the development of the calculations of the adjoint state method. It was therefore important to validate this assumption. A series of direct computations have been done in three dierent cases: small strain and rotations, large strains and large strains and rotations, with E 2 10 4 MPa and 310 4 MPa Á s. The results show a good agreement of the indentation curves (see Fig. 3). It is important to remark that the small dierence is due in part to the simple constitutive law assumed. This hypothesis should be checked before applying this method for other constitutive laws.

Identi®cation Procedure

The identi®cation procedure presented next is based on minimization of the cost functional t (16) using a gradient descent method. The ``experimental'' curve was simulated by ®nite element calculations as stated in the previous section with E 21 0 4 MPa, 0.3, 3 10 4 MPa Á s.

The gradient has been computed using the adjoint state method with the expression (40) after solving the direct problem and the adjoint problem ? . In the case of the Maxwell material behaviour the adjoint behaviour is also a Maxwell law, which we might call a self adjoint material behaviour. This is due to the quadratic viscoelastic potential È.

The numerical gradient computation by the adjoint method has been compared with a computation by ®nite dierences. The results for several points and directions showed less than 10% dierence between the two methods.

The minimization algorithm was the quasi Newton BFGS algorithm with a line search obeying the Armijo selection rule [START_REF] Gill | Practical optimization[END_REF].

The shape of the cost functional has been plotted in Figure 4 from a series of direct computations. We remark a smooth ®at valley which should not pose special diculties to the identi®cation.

Identi®cation Using Exact Measurements

A ®rst series of identi®cations have been performed with exact measurements. The results for dierent initial points are presented in Table I. The starting values for the algorithm have been at maximum 5 times smaller or 3 times larger than the real values. In all cases the ®nal value was less than 0.02% from the value to be identi®ed, after about 15 iterations. Some typical evolution path of the algorithm on the isovalues of the cost functional are plotted in Figure 5.

Figure 6 shows the real indentation curve in comparison to the initial and converged indentation curve. In terms of cost functional the algorithm brought its value from % 10 8 to % 10 1 .

Identi®cation Using Measurements with Random Error

In order to check the robustness of the identi®cation procedure the simulated measurements have been perturbed by a 10% random noise. The results of several identi®cations using perturbed measurement data are presented in Table II. The ®rst pair and the last pair of data are results coming from identi®cation with dierent starting points but with the same measurement perturbation. The identi®ed values lie at 2.5% from the real values for both measurement perturbations.

The path of the algorithm on the isovalues of the perturbed cost functional is shown on Figure 7.

Figure 8 shows the real indentation curve (E 21 0 4 MPa, 3 10 4 MPa Á s) in comparison to the initial (E 610 4 MPa, 910 4 MPa Á s) and converged (E 1.95 10 4 MPa, 2.91 10 4 MPa Á 10 8 to % 10 4 . This is minimal value of the cost functional where the algorithm could descent. However, there is a good agreement between the experimental and identi®ed indentation curve.

NUMERICAL EXAMPLE FOR THE NORTON-HOFF MATERIAL

We recall that the viscoplastic Norton-Ho constitutive law is expressed as:

"Áu i S XÁs i Á" p i where Á" p i M X s i Át 44 S ijkl 1 E 1 ik jl À kl ij 45 Á" p i 3 2 i eq À Y K m si i eq Át 46
As explained before, this constitutive law enters the formalism of the standard generalized materials without work-hardening. In comparaison with the Maxwell viscoelasticity two diculties have been adjoined a yield limit and a nonlinearity represented by the power law.

Direct Calculations

The direct computations have been programmed in the same framework as previously presented and similar meshes and loading histories (Fig. 2) have been used. The integration of the direct constitutive law was performed using a build in forward Euler scheme (implicit) under the hypothesis of small strains.

Identi®cation Procedure

The identi®cation procedure is based on minimization of the cost functional t (16) using the same algorithm as before.

The ``experimental'' curve was simulated by ®nite element calculations with E 10 5 MPa, 0.3, K 1500 MPa Á s 1/m , m 5, Y 500 MPa.

The gradient has been computed using the adjoint state method with the expression (40) after solving the direct problem and the adjoint problem ? .

Opposite to the Maxwell case, the adjoint constitutive law is dierent from the direct one and can be expressed as:

"Áu à i S XÁs à i M X s à i 47
where:

M À 3 2 i eq À Y K m 1 s i eq J si si s i 2 eq 9 4 i eq À Y K m 1 s i 2 eq m K i eq À Y K mÀ1 J ijkl ik jl À 1 3 kl ij
This formula already expresses a forward Euler integration scheme which has been programmed as such in the code. One can remark that the adjoint constitutive law is a viscoelastic, nevertheless anisotropic and nonhomogenous.

Identi®cation Using Exact Measurements

A series of identi®cations have been performed with exact measurements on two meshes with dierent coarsness (tests 1 ± 5 and 6 ± 8) and the results for dierent initial points are presented in Table III.

The starting values for the identi®cation are in this case just % 30% from the real values. This choice has been motivated by the slow convergence of the direct computation for the Norton-Ho material.

One can remark that the parameters have been identi®ed within % 5% of the real values for tests 1, 2, 7 and 8. In the other cases, only the Young's modulus E and the viscosity K have been reasonably identi®yed (1% error for E and 1 À 10% for K ). The power coecient m and the yield limit Y are in these cases still far away from the real values. However the ®nal computed indentation curves are always superposed with the experimental curve, meaning that the minimization of the geometrical distance between them has been achived.

The initial and ®nal parameters in tests 5 present a peculiar property. As can be seen in Figure 9, both parameters sets produce the same loading curve as the experimental one. The best identi®ed parameter in this case, was the Young's modulus. This result is a new con®rmation of the importance of the unloading part of the experimental curve in the identi®cation of the Young's modulus as already observed in a series of previous papers [START_REF] Loubet | Vickers indentation curve for a magnesium oxide (MgO)[END_REF][START_REF] Oliver | An improved technique for determining hardness and elastic moduli used load and displacement sensing indentation experiments[END_REF].

The previous results are a direct consequence of the poor sensitivity of the cost functional in this region of the parameter space. In order to illustrate this property we have plotted the values of the cost Using a coordinate parametrization given by two orthogonal vectors, the parameter sets in this plane can be generated using the following formula:

CONCLUSION

The identi®cation of the parameters of a standard generalized non hardening constitutive law from indentation tests was presented. It has been shown that the adjoint state method can be extended to contact problems using Lagrange multipliers.

The eciency of the method has been studied on numerical examples for an indentation problem in the case of Maxwell viscoelatic and Norton-Ho viscoplastic constitutive laws.

Even if no precise proof of uniqueness and stability has been provided, one can remark that for the Maxwell material the cost functional is almost convex and presents a unique minimum (see Fig. 4). For other values of the material parameters c (E, )wehave obtained similar shapes of the cost functional. It is important to state that for the Maxwell material identi®cations started with completely dierent values did converge to the same parameter set. All these remarks suggest that this inverse identi®cation problem has a unique solution.

In the case of the Norton-Ho material the numerical examples presented a lack of sensitivity of the cost functional with regards to certain parameters. However the Young moduli and the viscosity are well identi®ed. In spite of the lack of sensitivity it seems that the identi®cation problem has a unique minimum.

In order to improve the presented solution, dierent possibilities are available: to increase the number of experimental measures for example by varying the indentation rate, to change the expression of the cost functional like the error on constitutive law or crossed dierences (t U exp À U comp ÂF comp À F exp see [START_REF] Tardieu | On the determination of elastic coecients from indentation tests[END_REF]). These questions do not depreciate the presented approach as the technique for the gradient computations remains valid. They rather demand a study of the strategy of identi®cation and of the identi®ability of a certain family of constitutive laws.

FIGURE 2

 2 FIGURE 2 Loading history of the indentor.

FIGURE 3

 3 FIGURE 3 Indentation curve for small and large strains for the Maxwell material.

FIGURE 4

 4 FIGURE 4 Shape of the cost functional t for the Maxwell material.

FIGURE 7

 7 FIGURE 7 Path of the algorithm (measurements with random error) for the Maxwell material.

FIGURE 8

 8 FIGURE 8 Evolution of the indentation curve from measurements with random error for the Maxwell material.

FIGURE 9

 9 FIGURE 9 Evolution of the indentation curve with exact measurements for the Norton-Ho material.

E

  100000 À 1315:4a À 80362:7b K 1500 273:917a 714:075b m 5 2:93482a 1:08013b y 500 À 394:32a 142:208b 48 where a, b are two real parameters. In Figure 11 we represented the values of the cost functional for the parameter sets generated with (a, b) P [ À 1, 1] Â [ À 0.5, 0.5]. Two

FIGURE 10

 10 FIGURE 10 Convergence of the normalized parameters for the Norton-Ho material (test 7).

FIGURE 11

 11 FIGURE 11 The values of the cost functional t on the ``critical'' plane (see Eq. (cc)) of the parameter space.

  the following ®nal conditions at time T

	divs ? I 0i n	35
	"Áu ? I S XÁs ? I in	36
	u ? I 0o nÀ D	37
	s ? I Á n 0o nÀ F	38
	u n? i 0 p ? I I nn?	on À C	39

TABLE I

 I Identi®cation results with exact measurements for the Maxwell material for the reference values E ref 20000. MPa, ref 30000. MPa Á s (E ini , ini )( E ®nal , ®nal ) (MPa, MPa Á s)( MPa, MPa Á s)

	(4000., 70000.)	(20002., 30004.)
	(60000., 10000.)	(20002., 30009.)
	(10000., 5000.)	(20002., 30006.)
	(60000., 90000.)	(19999., 29999.)

FIGURE 5 Path of the algorithm (exact measurements) for the Maxwell material.

TABLE II

 II

	Identi®cation results from measurements with random error for the
	Maxwell material
	(E ini , ini )( E ®nal , ®nal )
	(MPa, MPa Á s)( MPa, MPa Á s)
	(4000., 70000.)	(20580., 30490.)
	(10000., 5000.)	(20611., 30457.)
	(60000., 10000.)	(19509., 29130.)
	(60000., 90000.)	(19509., 29136.)

FIGURE 6 Evolution of the indentation curve with exact measurements.

TABLE III

 III Identi®cation results with exact measurements for the Norton-Ho material for the reference values E ref 100000. MPa, K ref 1500. MPa. s 1/5 m ref 5. and Y ref 500. MPa

	Test E ini MPa (MPa Á s) 1/m K ini	m ini	Y ini MPa	E ®nal MPa (MPa Á s) 1/m K ®nal	m ®nal	Y ®nal MPa
	1 130000.	800.	2.	1000.	99664.3	1533.36 5.38417 446.414
	2 150000.	1000.	3.	1000. 100228.0	1524.64 5.00296 491.795
	3 150000.	2000.	3.	200.	99429.3	1598.24 6.18529 343.533
	4 150000.	2000.	8.	1000.	99288.9	1657.20 6.98908 246.959
	5 70000.	2000.	8.	200.	99126.5	1682.47 6.94556 238.366
	6 70000.	3000.	8.	250. 100598.0	1451.36 4.24316 602.651
	7 70000.	1000.	3.	250.	99630.1	1521.18 4.94161 497.229
	8 150000.	1000.	3.	250.	99778.0	1516.32 5.28676 464.255

Acknowledgments

The authors would like to thank Said Taheri, St ephane Andrieux and Laurent Bourgeois for helpful discussions and acknowledge the ®nancial support of EDF ± Electricit e de France for this work.