Jean Charles

Nonlinear optimization and large-scale problems 1

Keywords: conjugate gradient, large-scale optimization, limited memory quasi-Newton method, partitioned quasi-Newton method, unconstrained optimization

This paper recalls some basic concepts of unconstrained optimization techniques and reviews some useful methods for solving large-scale problems: conjugate gradient methods without critical line-searches, limited memory quasi-Newton methods and the partitioned quasi-Newton method. We shall discuss their efficiency, their convergence properties and their respective advantages and disadvantages.

Introduction

Optimization problems are frequent in engineering, physics (minimization of energy), economics (minimization of costs) and mathematics (variational problems, least-squares methods), etc. When they are related to real-life problems, very often, a large number of variables is involved. Then, we talk about large-scale optimization.

The notion of large-scale problems does not depend only on the number of independent parameters that describe the problem. Actually, it would be more suitable to say that a large-scale problem is a problem hard to solve by "traditional" techniques. In this respect, the notion of large-scale problem is time dependent: the increase in storage capacities and calculation speed of today's computers has raised the limit separating small and large-scale problems. This notion is also dependent on the nature of the problem: a quadratic function whose Hessian has clustered eigenvalues is easier to minimize than a general nonlinear function. To fix the ideas, however, we may say that a problem without any particular structure is large today if its number n of variables is greater than 500. To fix this number, we considered mainly the updating cost and memory "clutter" of O(n 2 ) elements of storage.

Methods that are good for problems of small or medium scale need to use information on the curvature of the function to minimize, i.e., on its Hessian matrix. For large-scale problems, such methods become unusable because they require too much storage (this is the case with the standard Newton method). They can also become inefficient, because they require too many iterations to collect the proper information (this is the case with quasi-Newton (qN) methods). Therefore, one possibility is to use methods that are less efficient but that require fewer locations in memory. The conjugate gradient methods are of this type. They are considered in Section 3. Another possibility is to adapt fast methods to large-scale problems. This is the case of limited memory qN methods (discussed in Section 4) and the case of the partitioned qN method (examined in Section 5).

With these three classes of methods, we shall not cover all the possibilities. In particular, almost nothing will be said on the discrete Newton method, which can be good alternative (references are given below). Also, we shall limit our discussion to unconstrained optimization and to descent-direction methods.

To conclude this introduction, let us mention some optimization books and review papers. The books by [START_REF] Gill | Practical Optimization[END_REF], by [START_REF] Mccormick | Nonlinear Programming. Theory, Algorithms and Applications[END_REF], by [START_REF] Minoux | Programmation Mathématique. Théorie et Algorithmes[END_REF], also available in English, by [START_REF] Dennis | Numerical Methods for Unconstrained optimization and nonlinear equations[END_REF] and by [START_REF] Fletcher | Practical methods of optimization[END_REF] are now classical, see also [START_REF] Stoer | Méthodes numériques d'optimisation. Collection didactique[END_REF]. They are recommended as introduction to optimization techniques. The book by [START_REF] Bertsekas | Constrained Optimization and Lagrange Multiplier Methods[END_REF] is also recommended but is more specialized. The book by [START_REF] Gruver | Algorithmic Methods in Optimal Control[END_REF] deals with optimization algorithms for optimal control. The book by [START_REF] Bertsekas | Parallel and Distributed Computation[END_REF] describes some optimization algorithms that are suitable for parallelization. The references [START_REF]Nonlinear Optimization. Theory and Algorithms[END_REF], [START_REF]Nonlinear Programming 4[END_REF], [START_REF]Nonlinear Optimization 1981[END_REF], [START_REF]Mathematical Programming. The State of the Art[END_REF], [START_REF]The State of the Art in Numerical Analysis[END_REF], [START_REF]Optimization[END_REF], [START_REF]New Methods in Optimization and their Industrial Uses[END_REF] and [START_REF]Optimization[END_REF] are collections of papers giving the state of the art in optimization. Finally, let us mention the review paper by Toint (1986) on large-scale optimization, which contains an important bibliography.

Basic concepts

f (x * ) ≤ f (x), ∀ x ∈ N (x * ) , (1)
where N (x * ) denotes some neighborhood of x * . A point x * satisfying the above inequalities is called a local minimizer of f . If (1) holds for N (x * ) = Ω, then x * is called a global minimizer of f on Ω. Finding a global minimizer is a hard task and for this subject, we refer the reader to the collections of papers by Dixon andSzegö (1975, 1978); see also [START_REF]Optimization[END_REF]. The algorithms considered in this paper are only able to find local minimizers.

If x * is a local minimizer, it satisfies the first order optimality conditions:

f ′ (x * ) • h = 0, ∀ h ∈ IR n ,
where f ′ (x * ) is the Fréchet derivative of f at x * . These conditions can be rewritten as follows:

g(x * ) = 0 , (2)
where g(x) := ∇f (x) denotes the gradient of f at x. It is important to keep in mind that the gradient depends on the scalar product chosen on IR n . It is indeed defined by means of the Riesz theorem: 2) is called a stationary point of f . A minimizer is a stationary point and the converse is true for convex function.

∃ g(x) ∈ IR n , unique : g(x), h = f ′ (x) • h, ∀ h ∈ IR n . If the Euclidean scalar product is used, u, v = n i=1 u i v i , then g i (x) is the i-th partial derivative of f at x. A point x * satisfying (
Here, we have to be still more modest as to what the algorithms described in this paper can do. In general, they can only find stationary points. This is due to the fact that they will use only the value of f and the value of its gradient. If for some reasons, a local minimizer has to be found (and not only a stationary point) and if there are many stationary points that are not minimizers, something more has to be done. A good idea is to use second order information to locate directions of negative curvature. Along this line, see the papers by [START_REF] Gill | Newton-type methods for unconstrained and linearly constrained optimization[END_REF], [START_REF] Mccormick | A modification of Armijo's step-size rule for negative curvature[END_REF], [START_REF] Moré | On the use of directions of negative curvature in a modified Newton method[END_REF] and [START_REF] Goldfarb | Curvilinear path steplength algorithms for minimization which use directions of negative curvature[END_REF].

Finding a stationary point of a function is necessarily an iterative process. Indeed, in the simplified case where n = 1 and f is a polynomial of degree p + 1, the problem is equivalent (by the optimality condition (2)) to finding the roots of a polynomial of degree p. Since Galois, it is known that this problem cannot be solved exactly by using simple arithmetic operations, as soon as p ≥ 5. Therefore, the only thing that is reasonable to expect from an optimization method is that it generates a sequence {x k } k≥1 of points that converges to a solution.

An important class of methods generating sequences of approximations of a solution is formed by the descent-direction methods. They proceed as follows. It is assumed that some approximation x k is known at the beginning of iteration k, k ≥ 1 (x 1 has to be given at the beginning of the run). Then, an iteration consists of calculating a new approximation x k+1 that is better than the previous one, in the sense that

f (x k+1 ) < f (x k ) . (3)
To obtain this descent property, the first stage consists in choosing a descent direction at x k , i.e., a direction

d k verifying f ′ (x k ) • d k < 0 . (4)
The second stage consists in selecting a positive step-size ρ k along d k and taking

x k+1 = x k + ρ k d k
such that x k+1 satisfies, at least, the descent property (3). This stage is called the line-search.

Let us summarize this method as follows.

The descent-direction algorithm:

1. choose x 1 ∈ Ω; 2. k := 1; 3. until convergence do { 3.1. choose a descent direction d k ; 3.2.
select a positive step-size ρ k along d k such that (3) holds; 3.3.

x k+1 := x k + ρ k d k ; 3.4. k := k + 1 3. }
The way of calculating the descent direction characterizes a descent-direction method. For example, when g k := g(x k ) = 0, an obvious descent direction is:

d k = -g k . (5)
The method that uses this descent direction at each iteration is the gradient method. More generally, if H k is a positive definite matrix of order n,

d k = -H k g k (6)
will be a descent direction. Many practical algorithms for minimizing a function have this form of descent direction: the Newton method (near a solution), the bfgs method (or more generally, algorithms in the family of quasi-Newton methods) and even conjugate gradient methods, as we shall see in Section 4.

The descent condition (4) implies that for sufficiently small positive step-size ρ, we have f (x k + ρd k ) < f (x k ). Therefore, to satisfy (3), a step-size selection procedure could simply choose a small value for ρ. This is not satisfactory, however, because examples can be given for which taking arbitrarily small positive step-sizes forces a descent algorithm to generate a sequence {x k } k≥1 that converges to a nonstationary point (false convergence). This phenomenon is dangerous and hopefully, it can be avoided in several ways. We mention two of them.

A first idea is to try to get the maximal decrease of f in the direction d k , in other words to minimize the function ξ k (ρ) := f (x k + ρd k ) for positive ρ. We shall say that a step-size selection procedure realizes an optimal line-search if it selects an optimal step-size, that is to say a positive step-size ρ k such that

x k+1 = x k + ρ k d k satisfies f (x k+1 ) ≤ f (x k + ρk d k ) , (7)
where ρk is the smallest positive stationary point of the function ξ k . A step-size that is a stationary point of ξ k will be said to be critical and is obtained by a critical line-search. The first local minimizer and the global minimizer of { ξ k (ρ) : ρ > 0 }, if they exist, are optimal and critical. Actually, the optimal and critical line-search strategies have two drawbacks. First, the step-size ρk may not exist at some iteration although problem (1) has a solution. Secondly, finding a stationary point of ξ k is a nonlinear minimization problem in itself and this cannot be solved exactly, except in particular cases (for instance, when f is quadratic). Therefore, although this procedure is theoretically suitable (see Theorem 1 below), it is useless in practice.

If an optimal step-size is not computable, it is possible to calculate almost optimal stepsizes, in a sense to precise. In fact, the convergence analysis of descent-direction methods has shown that to be acceptable a step-size has to define a point x k+1 satisfying some conditions. First, the descent property (3) is not strong enough: it does not force a sufficient decrease of the objective function. Instead, the following inequality with ω 1 ∈ ] 0, 1[ is suitable:

f (x k+1 ) ≤ f (x k ) + ρ k ω 1 g k , d k . ( 8
)
It is easy to verify that this inequality is always satisfied for small positive ρ k . If this is comforting, it means also that it does not prevent the step-size from being too small. As we said, this can lead to false convergence. Therefore, another condition is needed. For example, the following inequality g(x k+1 ),

d k ≥ ω 2 g k , d k (9)
with ω 2 ∈ ] ω 1 , 1[ can be used. Remark that it does not accept small step-sizes and that in particular, ρ k = 0 is rejected. Actually, this inequality expresses that the slope of the function ξ k at ρ k has to be sufficiently larger than the slope at the origin. Conditions ( 8) and ( 9) are called the Wolfe conditions and a point x k+1 satisfying them is called a Wolfe point. It can be proved (see [START_REF] Wolfe | Convergence conditions for ascent methods[END_REF]) that if f is bounded below and if 0 < ω 1 < ω 2 < 1, it is always possible to find a Wolfe point. Typical values for the parameters are ω 1 = 10 -4 . . . 10 -2 and ω 2 = 0.5 . . . 0.99. In any case, it is healthy to take ω 1 < 0.5, so that the step-size minimizing the quadratic approximation of ξ k will be accepted by the inequality [START_REF] Dennis | Quasi-Newton methods, motivation and theory[END_REF]. Calculating a Wolfe point requires a specific algorithm. The one proposed by [START_REF]A view of line-searches[END_REF] has been proved to find a Wolfe point in a finite number of trials. For some algorithms, we shall need a stepsize satisfying the strong Wolfe conditions, that is to say, satisfying [START_REF] Dennis | Quasi-Newton methods, motivation and theory[END_REF] and an inequality stronger than [START_REF] Dennis | Numerical Methods for Unconstrained optimization and nonlinear equations[END_REF], namely

| g(x k+1 ), d k | ≤ ω 2 | g k , d k | . (10)
An algorithm for finding a strong Wolfe point can be found in [START_REF] Al-Baali | An efficient line search for nonlinear least squares[END_REF] and [START_REF] Moré | On line search algorithms with guaranteed sufficient decrease[END_REF].

The preceding two line-search techniques allow us to obtain the following result that can be derived from the papers by [START_REF] Zoutendijk | Nonlinear programming, computational methods[END_REF] and [START_REF] Wolfe | Convergence conditions for ascent methods[END_REF][START_REF] Wolfe | Convergence conditions for ascent methods II: some corrections[END_REF]. Before stating the theorem, let us specify some notation. We shall denote by θ k the angle between d k and -g k , that is

cos θ k := - g k , d k |g k | |d k | .
If {ǫ k } k≥1 is a sequence of real numbers, the notation "lim inf ǫ k " will mean "lim k→∞ inf 1≤i≤k ǫ i ", which differs from the notation in topology.

Theorem 1

Let f be a differentiable function defined on the open set Ω, with a Lipschitz continuous derivative. Suppose that a descent-direction method generates a sequence of points {x k } k≥1 in Ω and a sequence of directions {d k } k≥1 verifying (4). Suppose also that at each iteration the line-search procedure finds either an optimal point satisfying [START_REF] Dembo | Truncated-Newton algorithms for large-scale unconstrained optimization[END_REF] or a Wolfe point satisfying [START_REF] Dennis | Quasi-Newton methods, motivation and theory[END_REF] and [START_REF] Dennis | Numerical Methods for Unconstrained optimization and nonlinear equations[END_REF]. Suppose finally that {f (x k )} k≥1 is bounded below. Then, the following conclusions hold:

(i) k≥1 |g k | 2 cos 2 θ k is convergent; (ii) |g k | cos θ k → 0; (iii) if k≥1 cos 2 θ k is divergent, then lim inf |g k | = 0.
The conclusions (ii) and (iii) are clear consequences of the conclusion (i).

The theorem says nothing on the convergence of the sequence of points {x k } k≥1 . Only the sequence of gradients is concerned, although it is not claimed that g k converges to zero. Actually, Theorem 1 expresses the contribution of the line-search procedure to force convergence. Now, to make its conclusions useful, some more information concerning the way the descent directions d k are chosen is needed. For instance, if the angle θ k remains bounded away from ±π/2, then cos θ k remains bounded away from 0 and the conclusion (ii) of Theorem 1 implies that the sequence of gradients converges to zero and therefore that any limit point of {x k } k≥1 is stationary. This is the case for the gradient method, because θ k = 0, ∀k ≥ 1. This is also the case for the method (6), if the condition numbers κ(H k ) of the positive definite matrices H k remain bounded; that is, if there exists a positive constant κ such that

κ(H k ) := ||H k || ||H -1 k || ≤ κ, ∀k ≥ 1. Indeed, we have cos θ k ≥ κ(H k ) -1 .
These remarks indicate that to be sure to have a convergent algorithm, it is enough to use method [START_REF] Buckley | QN-like variable storage conjugate gradients[END_REF] with constant matrices H k (the identity matrix, for instance) or with matrices having bounded condition number. Our task is not finished, however. Indeed, such directions can be very poor descent directions in the sense that the decrease of the objective function may be small along them. Therefore, in the subsequent sections, we shall play the dangerous game of trying to define "good" descent directions by taking information from the function with the risk of violating the conditions of convergence stated in Theorem 1. For some of the methods below, it is still not clear whether they can fail to converge when they aim at minimizing general functions.

3 Conjugate gradient methods without optimal line-searches

The conjugate gradient (cg) method was introduced to solve linear systems of equations [START_REF] Hestenes | Methods of conjugate gradients for solving linear systems[END_REF] and later to minimize strongly convex quadratic functions [START_REF] Fletcher | Function minimization by the conjugate gradients[END_REF]. For these problems, the method has the outstanding feature to terminate in at most n iterations. This is a dramatic improvement in comparison with the gradient method, which usually requires an infinite number of iterations to converge.

The basic idea of the cg method is to generate a set of conjugate directions, i.e., directions that are orthogonal with respect to the scalar product induced by the Hessian matrix A of f . As a result, f is minimized on the larger and larger affine subspace spanned by the successive directions, the Krylov subspace. When the first direction is -g 1 , the conjugacy of the directions can be obtained by the following simple formula:

d k = -g k + β HS k d k-1 ,
where β HS k is the Hestenes-Stiefel beta (see [START_REF] Hestenes | Methods of conjugate gradients for solving linear systems[END_REF]):

β HS k := y k-1 , g k y k-1 , d k-1 .
We note y k := g k+1 -g k and s k := x k+1 -x k . To have conjugacy, it is necessary to do optimal line-searches, which does not pose any difficulty in case of quadratic functions. It is worth noting that if the cg method is better than the gradient method, it also uses information from previous iterates: d k-1 is used in the definition of d k . This feature is typical of efficient algorithms.

The CG algorithm for quadratic functions:

1. choose a point x 1 ∈ Ω; 2. d 1 := -g 1 ; k := 1; 3. repeat { 3.1. ρ k := -g k , d k / Ad k , d k ;
/ * the optimal step-size * / 3.2.

x k+1 := x k + ρ k d k ; 3.3.
if convergence then stop; 3.4.

d k+1 := -g k+1 + β HS k+1 d k ; 3.5. k := k + 1 3. }
Usually, real-life large-scale models are not quadratic. The question is therefore to know whether, for minimizing a general nonlinear function, it still makes sense to use a cg-like method defined for some scalar β k by

d 1 = -g 1 d k = -g k + β k d k-1 , for k ≥ 2 . ( 11
)
Note that for non quadratic functions, it is still possible to define a local notion of conjugacy by using the average Hessian of f :

Bk = 1 0 ∇ 2 f (x k + ts k ) dt .
Indeed, because y k = Bk s k , equation [START_REF]Towards Global Optimization[END_REF] with

β k = β HS k implies that Bk d k , d k+1 = 0 for k ≥ 1.
Therefore, d k and d k+1 are conjugate with respect to Bk . However, it does not seem that this concept is very helpful in the non quadratic case and therefore, on this basis only, it is hard to claim that the HS beta is better than any other beta.

Another point is to know whether the directions defined by [START_REF]Towards Global Optimization[END_REF] are downhill: do we have

g k , d k negative? As g k , d k = -|g k | 2 + β k d k-1 , g k ,
an obvious way of getting descent directions is to do critical line-searches: d k-1 , g k = 0. We have said that such strategy is not realistic. Hopefully, it can be avoided. The following theorem (see [START_REF] Gilbert | Global convergence properties of conjugate gradient methods for optimization[END_REF]Theorem 3.2]) is an extension of a result of [START_REF] Al-Baali | Descent property and global convergence of the Fletcher-Reeves methods with inexact line search[END_REF]. It shows that provided |β k | is not too large, a linesearch procedure realizing the strong Wolfe conditions automatically ensures descent of the direction d k . Furthermore, global convergence can be proved. Note that this result strongly relies on the fact that the first search direction is -g 1 . In this theorem, the Fletcher-Reeves beta is used to control the magnitude of β k . It is defined by (see [START_REF] Fletcher | Function minimization by the conjugate gradients[END_REF]):

β FR k := |g k | 2 |g k-1 | 2 .
For quadratic functions and optimal line-searches, β HS k = β FR k .

Theorem 2

Let f be a twice continuously differentiable function defined on the open set Ω = IR n and let x 1 be a point in Ω. Suppose that the level set { x ∈ Ω : f (x) ≤ f (x 1 ) } is bounded. If the nonlinear cg method [START_REF]Towards Global Optimization[END_REF] with |β k | ≤ β FR k starts from x 1 and uses the strong Wolfe line-search with 0 < ω 1 < ω 2 < 1/2, then the directions d k are downhill and lim inf |g k | = 0. This result could let think that the cg method [START_REF]Towards Global Optimization[END_REF] with

β k = β FR k
and the strong Wolfe line-search is a good algorithm. Unfortunately, it is not the case. The point is that this theorem only ensures the global convergence of the method, i.e., the convergence from any starting point, but not its fast convergence. In fact, numerical experiments with the Fletcher-Reeves formula have shown that the method can converge very slowly. A partial clarification of this phenomenon has been given by [START_REF] Powell | Restart procedures for the conjugate gradient method[END_REF][START_REF] Powell | Convergence properties of algorithms for nonlinear optimization[END_REF] who gave a reason why the Polak-Ribière beta, defined by (see [START_REF] Polak | Note sur la convergence de méthodes de directions conjuguées[END_REF]):

β PR k := y k-1 , g k |g k-1 | 2 ,
should by used instead of β FR k (for critical line-searches, β HS k = β PR k ). Numerical experiments have confirmed the superiority of the PR beta over the FR beta.

The misfortune is that the PR method is not as safe as the FR method. If it can be proved that the method is convergent for convex functions (see [START_REF] Polak | Note sur la convergence de méthodes de directions conjuguées[END_REF]), [START_REF] Powell | Nonconvex minimization calculations and the conjugate gradient method[END_REF] has given an example of a nonconvex function where the PR method diverges: the generated sequence cycles and each of its limit points is nonstationary! Two globally convergent remedies to this situation have been proposed in [START_REF] Gilbert | Global convergence properties of conjugate gradient methods for optimization[END_REF]. The first one is inspired by Theorem 2. It consists of using the PR beta only when it lies in the interval [-β FR k , +β FR k ]. Specifically,

Method PR|FR : β k =      -β FR k if β PR k < -β FR k β PR k if |β PR k | ≤ β FR k β FR k if β PR k > β FR k .
This strategy avoids one of the main disadvantages of the FR method, as discussed in [START_REF] Gilbert | Global convergence properties of conjugate gradient methods for optimization[END_REF].

Another globally convergent remedy consists of restarting the PR method in the opposite direction of the gradient, each time β PR k is negative. Synthetically, this corresponds to the following choice of β k :

Method PR + : β k = max{0, β PR k }. For this method, a specific line-search assuring descent of the directions has to be implemented.

FR

PR|FR PR PR + > 4.07 1.55 1.02 1.00

Table 1: Comparison of the methods FR, PR|FR, PR and PR + .

Table 1 allows us to compare the performance of several cg methods: FR, PR, PR|FR and PR + . The results come from tests made in [START_REF] Gilbert | Global convergence properties of conjugate gradient methods for optimization[END_REF] on a collection of 26 large-scale problems with a number of variables between 100 and 10000. The table gives the total number of function/gradient calls used for all the test-problems divided by the 18792 calls required by method PR + . The sign > means that the method was stopped on some test-problems because the number of function/gradient calls exceeds 9999. The actual number should therefore be greater than the one given in the table. Methods PR and PR + appear to be the best one, while the FR method is on average more than four times slower than method PR + ! 4 Limited memory quasi-Newton methods

The Newton method

The Newton method plays a major part in optimization. It is obtained by considering the local quadratic model of the function given by a second order Taylor expansion of f about the current point x k :

f (x) = f (x k + d) = m k (d) + o(|d| 2 ), m k (d) = f (x k ) + g k , d + 1 2 ∇ 2 f (x k ) d, d ,
where ∇ 2 f (x k ) denotes the Hessian matrix of f at x k . This matrix also depends on the scalar product. The displacement of the method is obtained by minimizing the model m k instead of f . When ∇ 2 f (x k ) is positive definite, this gives:

d k = -∇ 2 f (x k ) -1 g k . (12)
The direction has the form (6) and it is a descent direction when ∇ 2 f (x k ) is positive definite. This is the case when x k is close to a solution x * satisfying the second order sufficient conditions of optimality, i.e., with ∇ 2 f (x * ) positive definite.

An attractive feature of Newton's method is its rate of convergence. If the function f is sufficiently smooth and if x 1 is close enough to x * , then the method generates a quadratically convergent sequence {x k } k≥1 , that is

||x k+1 -x * || ≤ C ||x k -x * || 2 , ∀k ≥ 1 ,
for some positive constant C. This property implies that if x k is close enough to x * , its number of correct significant digits grows very quickly. Quadratic convergence is typical of algorithms using the second order information contained in the Hessian of f . If convergence occurs with the Newton method when the initial point x 1 is close to a solution, divergence is often observed if the guess is not good enough. This drawback of the method can be tempered by line-search techniques as described in the introduction or by a trust region approach (see the survey of the topic by [START_REF] Moré | Recent developments in algorithms and software for trust region methods[END_REF]). Other difficulties may occur. For example, when the Hessian is not positive definite, the Newton direction may no longer be downhill. One remedy to this situation is to modify the Hessian in equation ( 12) to obtain a positive definite matrix: one of the most effective way of doing this is due to [START_REF] Gill | Newton-type methods for unconstrained and linearly constrained optimization[END_REF]. Another remedy is to solve the linear system (12) within a trust region.

Let us come back to our main concern: large-scale optimization. Can we use the Newton method in this context? The stumbling block is, of course, the memory requirement to store the Hessian matrix. The basic idea is to solve the linear system (12) by a cg method when the Hessian is positive definite or by a Lanczos tridiagonalization otherwise and whenever the quantity ∇ 2 f (x k ) v is required, it is evaluated by finite differences:

∇ 2 f (x k ) v ≃ ∇f (x k + tv) -∇f (x k ) t ,
for some small positive t. This approach is known as the discrete Newton method, see the papers by [START_REF] Dembo | Truncated-Newton algorithms for large-scale unconstrained optimization[END_REF] and by O'Leary (1983).

The quasi-Newton methods

The Newton method requires the calculation of the Hessian matrix of f , which can be in certain cases, either too expensive or impossible. This is precisely the domain where the quasi-Newton (qN) methods can be useful. See the review paper by [START_REF] Dennis | Quasi-Newton methods, motivation and theory[END_REF].

The descent direction of a qN method has the form [START_REF] Buckley | QN-like variable storage conjugate gradients[END_REF], where H k is a matrix that is updated at each iteration by some formula. Hence, the method generates two sequences: a sequence of points {x k } k≥1 and a sequence of matrices {H k } k≥1 . At the beginning of the run, the algorithm is given a point x 1 and a matrix H 1 . Often, H 1 is the identity matrix (we shall give better choices later) or a diagonal matrix, like the diagonal of the Hessian ∇ 2 f (x 1 ), when it is available.

The general scheme of a qN method is the following.

The QN algorithm: 1.

choose x 1 ∈ Ω and a positive definite matrix H 1 ; 2.

k := 1; 3.

until convergence do { 3.1.

d k := -H k g k ; 3.2.
select a step-size ρ k by the Wolfe procedure ( 8) and ( 9); 3.3.

x k+1 := x k + ρ k d k ; 3.4. calculate H k+1 ; 3.5. k := k + 1 3.
}

The only thing to examine in this algorithm is the way the matrix H k is updated. If we set B k = H -1 k (we shall see how to maintain H k positive definite, hence nonsingular), we see that B k plays the part of ∇ 2 f (x k ) in Newton's method. On the other hand, with s k := x k+1 -x k and y k := g k+1 -g k , we have by Taylor's theorem:

y k = 1 0 ∇ 2 f (x k + ts k ) dt s k .
These two remarks show that it makes sense to impose the matrix B k+1 , which has to be calculated at the end of iteration k, to satisfy:

y k = B k+1 s k .
This equation is called the secant or quasi-Newton equation. If n ≥ 2, the equation does not determine B k+1 completely and some other conditions can be imposed. Generally, they make B k+1 keeping information from B k . There is today a wide consensus that one of the best update formulae for minimization is the bfgs update. To write this formula, it is convenient to use the following tensor product of two vectors u and v of IR n : u ⊗ v is the rank one matrix defined by (u ⊗ v)h = v, h u , ∀h ∈ IR n . If the Euclidean scalar product is used, u ⊗ v is the matrix uv T . See the appendix in [START_REF] Gilbert | Some numerical experiments with variable storage quasi-Newton algorithms[END_REF] for a derivation of the bfgs formula in this setting, which differs from the usual one by the use of a general scalar product. The bfgs formula can be written like this:

B k+1 = B k + y k ⊗ y k y k , s k - (B k s k ) ⊗ (B k s k ) B k s k , s k =: U (B k , y k , s k ) . (13)
Note that the inverse matrices H k can also be updated by using the inverse bfgs formula:

H k+1 = I - s k ⊗ y k y k , s k H k I - y k ⊗ s k y k , s k + s k ⊗ s k y k , s k =: Ū (H k , y k , s k ) . (14)
Precisely, if 13) and ( 14) have the nice property of transmitting the positive definiteness of B k to B k+1 (respectively of H k to H k+1 ), if and only if y k , s k is positive. Therefore to have H k positive definite, it will suffice to take H 1 positive definite, to update the matrices by the inverse bfgs formula and to ensure the positivity of y k , s k at each iteration. Note that for strictly convex functions, y k , s k is automatically positive and that for nonconvex functions, the second inequality (9) of the Wolfe line-search implies the positivity of y k , s k . This important remark makes the Wolfe step-size selection procedure the most suitable for the qN framework in optimization. It justifies also Statement 3.2 of the qN algorithm.

H k = B -1 k , B k+1 = U (B k , y k , s k ) and H k+1 = Ū (H k , y k , s k ), then H k+1 = B -1 k+1 . Formulae (
For problems having small or medium dimension, the bfgs method is very efficient. It is globally convergent for convex problems (see [START_REF] Powell | Some global convergence properties of a variable metric algorithm for minimization without exact line search[END_REF]) and for general functions, when convergence occurs, it is fast, namely superlinearly convergent. This means that

||x k+1 -x * ||/||x k -x * || → 0 .
Although, this is less good than the quadratic rate of convergence of the Newton method, it is generally satisfactory in practice.

The limited memory quasi-Newton methods

There are many proposals to adapt qN methods to large-scale problems and many of them fit into the following framework.

Although n may be large, the initial matrix H 1 takes generally little space in memory: it is most commonly a positive multiple of the identity matrix. On the other hand, H k is formed from H 1 and k -1 couples {(y i , s i )} 1≤i<k by using formula [START_REF] Fletcher | Function minimization by the conjugate gradients[END_REF]. Therefore, to take less storage, one can think of storing these elements instead of H k and computing H k g k by an appropriate algorithm. Of course, when the number of iterations increases, these pieces of information become more and more cumbersome in memory and we must get rid of some of the couples (y i , s i ). A method will be called an m-storage qN method if only m of these couples are used to form H k from a starting matrix H 0 k . In this type of method, the inverse update formula ( 14) is preferable to the direct formula [START_REF] Fletcher | Practical methods of optimization[END_REF], because the inversion of B k may be problematic.

The most famous limited memory methods can be seen from the preceding point of view: see [START_REF] Gilbert | Some numerical experiments with variable storage quasi-Newton algorithms[END_REF] for the details. For example, the algorithm conmin of Shanno and Phua (1976) is a 2-storage qN method (see also [START_REF] Shanno | Conjugate gradient methods with inexact searches[END_REF]). The algorithm of [START_REF] Buckley | QN-like variable storage conjugate gradients[END_REF] is an m-storage qN method where m varies with the iteration index. Finally, the method proposed by [START_REF] Todd | Updating quasi-Newton matrices with limited storage[END_REF] is also of this type. We describe it in more detail because we feel it is important.

Nocedal proposed to use always the last m couples (y, s), i.e., {(y i , s i )} k-m≤i≤k-1 , to form the matrix H k : going from iteration k to iteration k + 1, the couple (y k-m , s k-m ) is discarded and cyclically replaced by the new couple (y k , s k ). Therefore, H k is always formed from the most recent information and, as a result, it should be more appropriate.

Description of Nocedal's proposal to calculate H

k : 1.
choose a positive definite matrix H 0 k ; 2.

for i

:= 0 to m -1 do H i+1 k := Ū (H i k , y k-m+i , s k-m+i ); 3. H k := H m k ;
Of course, this scheme is formal: in practice, the matrices H i k and H k are not stored (they are too large). Instead, just H 0 k and the couples {(y i , s i )} k-m≤i≤k-1 are kept in memory and an elegant algorithm, described in [START_REF] Todd | Updating quasi-Newton matrices with limited storage[END_REF], is provided to compute H k g k .

The above description of Nocedal's proposal outlines the need of a choice for the starting matrices H 0 k . Contrary to the standard qN methods where this choice has to be made only once at the beginning of a run, here it has to be made at each iteration. Its role is therefore crucial. One of the aims of the work [START_REF] Gilbert | Some numerical experiments with variable storage quasi-Newton algorithms[END_REF] is to examine several possible starting matrices. From this study, two choices emerge.

A first choice is simply to take a judicious multiple of the identity matrix:

H 0 k = δ k-1 I , δ k-1 := y k-1 , s k-1 |y k-1 | 2 . ( 15
)
The factor δ k-1 is known as the Oren-Spedicato factor (see [START_REF] Oren | Optimal conditioning of self-scaling variable metric algorithms[END_REF]) and aims at giving H 0 k a good scaling. Some justifications for using this factor are given in [START_REF] Gilbert | Some numerical experiments with variable storage quasi-Newton algorithms[END_REF].

The other possibility is to take a diagonal matrix and to update it at each iteration. In this case, the algorithm generates and stores two sequences, {x k } k≥1 and {H 0 k } k≥1 , and a circular order array for the couples (y i , s i ). Then H k is formed by updating m times H 0 k , using the m couples {(y i , s i )} k-m≤i≤k-1 . At the beginning, H 0 2 = δ 1 I. Then, for k ≥ 2, the formula for updating the i-th diagonal element of H 0 k is the following:

(H 0 k+1 ) (i) = H 0 k y k , y k y k , s k (H 0 k ) + y k , e i 2 y k , s k - H 0 k y k , y k ( s k , e i /(H 0 k ) (i) ) 2 y k , s k (H 0 k ) -1 s k , s k -1 , (16)
where {e i } 1≤i≤n is an orthonormal basis for the given scalar product on IR n . It was obtained by scaling a diagonalized version of the bfgs formula and proved to be very effective. [START_REF] Gilbert | Some numerical experiments with variable storage quasi-Newton algorithms[END_REF], which were obtained with 8 real-life test-problems having a dimension between 34 and 1559 (the average value of n is 692). A comparison is done between the standard bfgs method implemented by [START_REF] Buckley | QN-like variable storage conjugate gradients[END_REF], the cg method of [START_REF] Hestenes | Methods of conjugate gradients for solving linear systems[END_REF] implemented in the code va14 from the Harwell library and three algorithms from the modulopt library (INRIA): m1gc3 is the code of [START_REF] Buckley | QN-like variable storage conjugate gradients[END_REF], m1qn2 and m1qn3 use Nocedal's proposal with the scaling ( 15) and ( 16) respectively. Table 2 gives the number of function/gradient calls divided by the one required by the standard bfgs algorithm. The number m of updates ranges from 2 to 50. The sign > used for va14 means that the true number should be larger than the one given because the algorithm failed on one of the test-problems. Observe that the performance of the algorithms increases with m, except for va14, which is not a variable storage method. Observe also that m1qn2 and m1qn3 are better than the bfgs method as soon as m is greater than 10. . . 20 and 2. . . 5, respectively. This improvement seems mainly due to the scaling made by the initial matrix in these algorithms, which acts as a dynamic preconditioner. An algorithm similar to m1qn2 has been tested by [START_REF] Liu | On the limited memory BFGS method for large scale optimization[END_REF] who report similar results in their comparison with the method of Buckley and LeNir.

m = 2 m = 5 m = 10 m = 20 m = 50 va14 > 1.
The convergence of m1qn2 on strongly convex functions has been proved in [START_REF] Liu | On the limited memory BFGS method for large scale optimization[END_REF]. However, because of the counter-examples of [START_REF] Powell | Nonconvex minimization calculations and the conjugate gradient method[END_REF], there is some doubt about the possibility of extending this result for nonconvex functions. To make this remark clear, observe that, with critical line-searches, the limited memory methods can be seen as a generalization of the cg method of Hestenes-Stiefel or Polak-Ribière. Indeed, let m = 1 and H 0 k = I in Nocedal's proposal and suppose that critical line-searches are done, i.e., s k-1 , g k = 0. Then,

y k-1 , d k-1 = -g k-1 , d k-1 = |g k-1 | 2
and using formula [START_REF] Fletcher | Function minimization by the conjugate gradients[END_REF], we obtain:

d k = -H k g k = -Ū (I, y k-1 , s k-1 ) g k = -g k + y k-1 , g k y k-1 , d k-1 d k-1 = -g k + β PR k d k-1 .
As the PR method with critical line-searches does not converge on nonconvex functions, the limited memory method (at least for m = 1 and without scaling) does not converge either.

The partitioned quasi-Newton method

The algorithms we have seen so far are suitable for any objective function f . The partitioned quasi-Newton (pqN) method proposed by [START_REF] Griewank | On the unconstrained optimization of partially separable functions[END_REF] is dedicated to the minimization of functions having a particular structure.

A function f is said partially separable if it can be written as a sum of p elements functions f i :

f (x) = p i=1 f i (x) , ∀x ∈ Ω
and if each of the element functions f i is invariant on some subspace E i ⊂ IR n :

f i (x + e) = f i (x) , ∀e ∈ E i .
A typical case is when the subspaces E i are spanned by some basic vectors of IR n , which means that the f i 's only depend on few of the n variables. Note that the E i 's need not be disjoint, hence the name of partial separability.

In the pqN method, the Hessian of f , ∇ 2 f (x) = p i=1 ∇ 2 f i (x), is approximated at iteration k by a matrix of the form

B k = p i=1 B i k .
The matrix B i k approximates the Hessian of the i-th element function and is updated to satisfy an element-wise secant condition:

B i k+1 s k = y i k := ∇f i (x k+1 ) -∇f i (x k ) .
This approach has some weak points:

• If the element functions are nonconvex, their Hessian is not positive definite and y i k , s k is not necessarily positive; therefore the overall approximation is not necessarily positive definite either and may even be singular although the true Hessian is positive definite.

• The preceding point implies that different update formulae (bfgs and sr1) are used, depending on the element function, which is a lack of homogeneity and an additional complexity of the method. • It requires more effort from the user who has to supply separately the gradient of the element functions. • The computation of the step by solving ( p i=1 B i k ) d k = -g k can be expensive. But it has also important nice features:

• if the dimension of the subspaces E i is large, the element Hessian are identified very quickly because the B i 's have low rank and therefore the overall approximation is rapidly correct; • the overall approximation is updated by a consistent high rank update;

• the notion of partial separability is basis independent, in contrast with the notion of sparsity.

Liu and Nocedal (1988) have made some numerical comparison between the pqN approach and m1qn2. They report excellent results for the pqN method when the dimension n i of the subspaces E i is small. What is disappointing is that the n i 's have to be very small (less than 4. . . 10!), otherwise m1qn2 becomes competitive. The comparison was made on the computing time and not on the number of function/gradient calls, which generally favors the pqN code. This seems due to the iteration time of the pqN method that can be 2. . . 5, sometimes 10. . . 15, times larger than the one of m1qn2.

To conclude this section, let us mention that the pqN method has been implemented by [START_REF] Ph | VE08AD, a routine for partially separable optimization with bounded variables[END_REF] in the Harwell routine ve08.

Final remarks

It should be clear at the end of this review that practical optimization remains an art or, at least, a matter for experts. If there is no miracle recipes, it is, however, possible to outline some rules to follow:

• the cg method remains a efficient algorithm for very large-scale problems, say, when only 5n memory locations are available; it is also a method to use for large-scale problems when the cost of a simulation (the calculation of the function and its gradient) is very cheap: in this case the calculation cost of the algorithm becomes relevant, which favors the very low cost of a cg iteration; • if the cost of a simulation is important, it is crucial to have a number of function/gradient calls as small as possible; in this case, it is a good idea to use the codes m1qn2 and m1qn3 (Nocedal's approach), which are available in the library modulopt of INRIA; • if the function to minimize has the partial separability property with element functions depending on a small number of variables, then the pqN algorithm is appropriate.

Let

  IR n be equipped with a scalar product •, • and its associated norm | • |. We shall use the following matrix norm on IR n : ||A|| := sup{|Au| : |u| = 1}. Given a smooth function f : Ω → IR, defined on an open set Ω of IR n , the unconstrained minimization problem consists in finding a point x * ∈ Ω, such that

Table 2 :

 2 Comparison between the limited memory, cg and bfgs methods.

		92 > 1.92 > 1.92 > 1.92 > 1.92
	m1gc3	2.07	1.60	1.46	1.37	1.23
	m1qn2	1.49	1.14	1.07	0.97	0.92
	m1qn3	1.00	0.85	0.85	0.76	0.74

Table 2

 2

gathers the numerical results from

Presented at the second conference on Mathematics for Engineers: "Mathematical Programming: a Tool for Engineers",May 17-19, 1989, Mons (Belgium). To appear in Engineering Optimization.


Laurent Goffart
email: laurent.goffart@cnrs.fr

Critique of the cerebralization of notions belonging to the physical sciences and mathematics

Confusions

Over the past decades, confusions have arisen in the brain sciences between -the concepts -the measurements (and their subsequent numerical transformations) and -the physiological processes that generate the measured phenomenon.

The theoretical notions associated with some measurements led some authors to imagine options such as a "coding" in the brain of space, time, numbers, movement duration, speed, acceleration, geometry and arithmetic. Over the past decades, confusions have arisen in the brain sciences between -the concepts -the measurements (and their subsequent numerical transformations) and -the physiological processes that generate the measured phenomenon.

The theoretical notions associated with some measurements led some authors to imagine options such as a "coding" in the brain of space, time, numbers, movement duration, speed, acceleration, geometry and arithmetic.

In the neuroscientific study of movement generation, kinematic parameters were imagined to be "encoded" in the activity of neurons. While they emit spikes, the neurons would "speak a coded language" imbued with notions of classical mechanics.

Yet the movement of a body segment (like the eyeball) is the ultimate measured product, the outcome of multiple processes occurring in parallel in the brain and converging on the groups of neurons which are responsible for muscle contractions.

To the totally ordered structure of numerical values (measured positions of the moving segment), a radically different structure is associated: dynamic, firstly diffluent then confluent, whose elements are spatially distributed and whose activities temporally overlap in the brain. At the ends of the optic nerves, the retinal excitation triggers a flow of activity that manifests in the brain as a kind of tree-like explosion (diffluence). The evoked activity propagates spatially. It is distributed in multiple groups of neurons whose activities are more or less persistent.

As the activity spreads and joins the groups of cells at the origin of the motor nerves that innervate the muscles, this explosion is transformed into a kind of compression (confluence), of which only the duration matters (longer muscle contractions make longer movement) in addition to the recruitment of new groups of muscles (e.g. extraocular + neck muscles  larger gaze shift).

What happens in the brain?

The eye "All human brains receive in sharing, from birth, sophisticated mental representations, independent of language that can be described as "proto-mathematical".

During its evolution, our brain has been endowed with elementary representations of space, time and number, which we share with many other animal species, and which are the basis of our mathematical intuitions."

Brain and Mathematics

Helen De Cruz ( 2006):

"The philosophy of mathematics has not paid much attention to the biological context of mathematics, although it goes without saying that mathematicians are biological organisms and that they produce mathematical theories and practices with their brains."

Jean-Pierre Changeux (1989):

"Mathematics is in the mathematician's brain, especially in his long-term memory."

Alain Berthoz (2005):

Mathematics is "the expression of laws and mechanisms which are in operation in the brain" ... "a language of the brain on the brain itself, that is to say, since the brain is part of the physical world, [a language] about the Universe and its laws."

Stanislas Dehaene (2010):

"Numbers, like all mathematical objects, are mental constructions whose roots lie … in the adaptation of the human brain to the of the universe."

By reducing the mind to the brain and the intellectual development to brain maturation, many authors have confused an individual's brain functioning with the historical and collective evolution of scientific discourse, which is memorized in books (and stored in libraries) and obeying Boolean logic.

Culture and Mathematics

Cognitive structures are external to the individual. They "depend on external instruments (books, libraries, calculation and observation instruments, etc.) as well as on social structures for the production and accumulation of knowledge (encyclopedias, learned societies, etc.) … knowledge is a whole process at the same time material, social and collective, never limited, locked up, preserved or produced by individual skills which concern only moments and fragments of it"

Sylvain Auroux (1998)

"the men [and women] who ... managed to constitute abstract objects, and thereby constituted mathematical theories, are ... men [and women] in the plural, concrete men [and women] acting in historically determined conditions, and not sorts of excrescences of a "cognitive subject"".

Fernand Doridot & Marco Panza 2004b

"The constitution of a mathematical theory is a historical fact … it is the business of a few mathematicians working under determined conditions rather than the realization of a cognitive process specific to the species. And if, …, it is so, then only history can account for this constitution and explain it ... The constitution of a mathematical theory seems to us difficult to conceive other than as a historical process, at least historically situated and determined." The probability that a post-synaptic neuron (C) emits spikes depends on the quasi-simultaneity of pre-synaptic spikes (when the influence is excitatory). There exist also inhibitory influences. The correlation between the firing rate of neurons and an external event (number of items in the visual field) does not necessarily imply that the spikes contribute to the behavioral response.

Fernand

Fallacious logic

The slight drops of performance 8% (color discrimination) and 4% (quantitative discrimination) may result from a mild visual impairment (diplopia, ocular divergence, etc.?).

Simpler explanation

Inactivation of LIP There is no evidence of an involvement of the parietal cerebral cortex in numerical "cognition" in monkeys.

In humans, brain imaging studies say nothing about the activity of neurons. The fMRI-BOLD signal reflects pre-synaptic activity and not the emission of action potentials (Lippert et al. Magnetic Resonance Imaging 2010).

During the last decades, confusion has arisen in brain sciences between -the measurements (and their numerical transformations), -the physiological processes that generate the measured phenomenon and -the concepts

The limitations of measurements and the associated theoretical notions led many cognitive neuroscientists (authors, reviewers, popularizers and teachers) to promote a nativist and evolutionary option, such as coding in the brain of space, time, numbers, Geometry and Arithmetic.

Neuroscience has neglected the fact that a multitude of cognitive structures are external to the individual, i.e., the "external instruments (books, libraries, calculation and observation instruments...) and the social structures of production and knowledge accumulation (encyclopedias, learned societies, etc.)" (Sylvain Auroux 1998).

"Mathematics is a collective work of art that derives its objectivity through social interactions" (Yehuda Rav 1988)

"The locus of scientific rationality is not the individual scientist but the scientific community" (Miriam Solomon 1996) Conclusion

  consider the case of a spot of light in the visual field.

  of the optic nerves Explosion of visual signals: spatial and temporal diffluence Felleman & Van Essen Cerebral Cortex 1991 Wallisch & Movshon Neuron 2008 6/20/2023 The eye is not only a retina Projection of the target on the fovea IF & only IF proper contraction of extraocular muscles (direction of gaze) Projection of the target on the fovea IF & only IF proper contraction of neck muscles (same muscles and motor nuclei BUT largely distinct central circuits Horizontal Vertical values > 0 : rightward / upward value = 0 : straight ahead values < 0 : leftward / downward Two types of eye movement in response to the motion of a visual target Goffart, Quinet & Bourrelly Encylopedia Human Brain submitted velocity H . values > 0 : rightward / upward value = 0 : straight ahead values < 0 : leftward / downward Two types of eye movement in response to the motion of a visual target values > 0 : rightward / upward value = 0 : straight ahead values < 0 : leftward / downward Two types of eye movement in response to the motion of a visual target H eye -H target : H (position) error quantities and arithmetic in the brain ? QUESTIONING THE BRAIN-MODEL ISOMORPHISM Measuring and modelling the movement of the eyeball (a rigid body) with kinematic notions does not imply equivalence in the brain, and notably that the firing of neurons ought to "encode" (i.e., emit signals related to) amplitude, velocity or acceleration of the eye movement.Thinking otherwise does not lead to a dead end.

  quantities and arithmetic in the brain ? New theory No movement = (poly)-equilibrium Movement = restoration of equilibrium Journal of Neurophysiology 120: 3234-3245, 2018 Progress in Brain Research, vol. 249, 375-384, 2019 https://cnrs.hal.science/ tel-02968991 https://cnrs.hal.science/ tel-04070171 6/20/2023 Stanislas Dehaene (2012):

  C) is contacted by several pre-synaptic neurons (A, E and F) 1 neuron (A) contacts several post-synaptic neurons (B, C and D)

?

  C) is contacted by several pre-synaptic neurons (A, E and F) 1 neuron (A) contacts several post-synaptic neurons (B, C and D) Q=4 D(3,4) < 2 spikes/s 0,2 spike/100 ms 1,2 spike/600 ms Out of the numerous spikes they receive from other neurons during 600 milliseconds, how do the post-synaptic neurons manage to distinguish the spikes emitted by this neuron and enable the monkey to produce a different response according to whether 2, 3 or 4 stimuli were presented in its visual field ? do post-synaptic neurons distinguish the spikes emitted by the "counting" neurons (LIP) from those emitted by a multitude of other (non-LIP) neurons ? ? modified from Roitman, Brannon & Platt PLOS Biol 2007 Firing rate (sp/s) Firing rate (sp/s) Population of 14 neurons "accumulator" Population of 17 neurons "decumulator" 400 ms are necessary to distinguish the curves The post-synaptic neurons receive distinct signals IF they accumulate the number of spikes from LIP neurons for 400 ms, provided that the spikes they receive from other (non LIP) neurons do not interfere with the integration.

  , Luo, Brannon & Platt PLOS One 2017 LIP area does not participate to visual "counting" either According to DeWind, Peng, Luo, Brannon & Platt (2017), If LIP and VIP areas are specialized in processing the number of visual items, THEN inactivation of either of these areas should cause a greater deficit in a numerical discrimination task than in a color discrimination task. We have seen that this is not the case Instead of rejecting their premises and concluding that LIP and VIP areas play no role in processing the number of visual items, they propose another inference :IF inactivation leads to similar deficits in both tasks (numerical and color discrimination) THEN the numerical processing in LIP and VIP areas is the mere signature of a general purpose circuit for ordinal comparison, visual attention or perceptual decision making

Neuron A : During task 8, the post-synaptic neurons (targeted by neuron A) cannot distinguish whether 8-16 or 32 stimuli were presented in the response field of the recorded neuron. They receive cannot distinguish either whether 2 or 4 stimuli were presented. During task 4, the post-synaptic neurons cannot distinguish whether 8 versus 16 (or 2 versus 4) stimuli were presented.

Neuron B :

During task 32, the post-synaptic neurons cannot "distinguish" whether 2-4-8 or 16 stimuli were presented in the response field of the recorded neuron. During task 2, they receive no signal enabling them to "distinguish" whether 4, 8 or 16 stimuli were presented.


Ghida Ismail

Malak Issa

Taghrid El-Hajj

Hassan Tfaily

Mohamad Dassouki

Elie Assaf

Halim Abboud

Pascale Salameh

Amal Al-Hajje

Farid Boumediene

Bassem Yamout

Maya Zeineddine

Quality of Life, Knowledge and Access to Treatment in Times of Economic Crisis Among Lebanese Multiple Sclerosis Patients

INTRODUCTION

• Multiple sclerosis (MS) is a degenerative disease of the central nervous system that can lead to lifelong disabilities, and thus, significantly impacts patient's quality of life (QOL). • Knowing that MS requires a long-term treatment, one of the major goals in treating MS is to reduce its negative effects on QOL. • Health-related QOL (HRQOL) is an important concept in which the patient's health is evaluated through physical, emotional, mental, and social components. • Knowledge and accessibility to treatment are factors that may affect the QOL of MS patients. • Lebanon is almost now three years into an economic and financial crisis which has been affecting the QOL of Lebanese citizens in general and to a greater extent the QOL of patients with chronic diseases including MS.

AIM

To our knowledge, this is the first study conducted in Lebanon, during the economic crisis, which aims to:

RESULTS

• A total of 350 MS patients were included in the study. • Most participants (92.3%) were diagnosed with relapsing-remitting MS and were females (65.1%) with a mean age of 39.45 ±11.43 years.

Quality of Life

• The mean MusiQol global index was 69.25 ± 11.53.

Access to Treatment During Economic Crisis

• 61.7% (n=216) of Lebanese MS patients had no access to MS treatment due to the economic crisis.

Table 2: Multivariate regression analysis for identification of predictors of MusiQoL

.

Knowledge

• The mean MSKQ-25 score is 15.11 ± 3.997.

Factors associated with the MSKQ-25 score:

• Higher MSKQ-25 score was significantly associated with higher level of education, being employed, receiving treatment at a specialized MS center and residency in Beirut.

DISCUSSION

• This study showed that Lebanese MS patients have a moderate HRQOL and emphasized that the HRQOL of MS patients is multifactorial. • The socio-demographic and clinical characteristics identified in our study as significant determinants of HRQOL were similar to those recognized in other international and regional QOL studies 2-3 .

• Physical disability was the only clinical factor significantly associated with HRQOL. However, we were unable to confirm the impact of physical disability on HRQOL because the Expanded Disability Status Scale (EDSS) was not collected.

• As for knowledge about the disease, the mean MSKQ-25 score was above average, indicating that Lebanese MS patients have good knowledge about their disease. This could be mainly explained by the availability of specialized MS centers across Lebanon, trained neurologists and other allied health care professionals, and the recently increasing number of awareness activities and media campaigns about MS. • The economic recession in Lebanon negatively impacted the accessibility to MS therapies as the majority of MS patients had no access to their medications during the crisis; and this was significantly associated with lower HRQOL.

CONCLUSION

The Lebanese Ministry of Public Health should urgently intervene to improve the QOL of MS patients by (1) securing the MS therapies and guaranteeing their accessibility to all patients regardless of their socio-economic status, (2) establishing social support systems, and (3) encouraging all types of patients' educational activities.

Figure 2 :

 2 Figure 2: Consequences of Inaccessibility to MS therapies

Figure 1 :

 1 Figure 1: The frequency of MS patients in each category 2 of MusiQol global index

Table 1 : Global MusiQol score and its dimensions

 1

	Dimensions of MusiQol	Mean ± SD
	Activities of daily living	65.0 ± 28.7
	Psychological wellbeing	66.1 ± 26.4
	Symptoms	69.0 ± 24.1
	Relationship with friends	42.76 ± 38.2
	Relationship with family	77.2 ± 26.9
	Sentimental and sexual life	51.6 ± 35.1
	Coping	80.5 ± 28.6
	Rejection	93.8 ± 18.1
	Relationship with the healthcare	77.1 ± 24.1
	system
	Global MusiQol score	69.2 ± 11.5
	Factors associated with the MusiQol global
	index:
	• Higher MusiQol scores were significantly
	associated with younger age, higher level of
	education, being employed, being married, living
	outside Beirut, and having no physical disability.
	• Lower MusiQol scores were significantly
	associated with discontinuing MS treatment
	and being self-payers with no access to any
	insurance.

Table 3 : Linear regression analysis for identification of associated factors with MSKQ-25

 3

	Unstandardized	p-value	95%
	Beta Coefficient		Confidence
			Interval


Thermographie Instrumentale

Olivier Riou
email: olivierriou@hotmail.com

JOURNEES D'ETUDES THERMOGRAM' 2005

Le secteur du contrôle thermographique des installations électriques est un grand utilisateur de caméras thermiques. Ce secteur, très concurrentiel, dispose d'une très grande variété de caméras dont le rendu visuel est un atout commercial. Cependant, il ne suffit pas à garantir une mesure exacte de température. C'est dans ce contexte que se pose le choix de la caméra thermique la mieux adaptée. Après avoir présenté les caractéristiques des caméras, nous développerons la notion de résolution spatiale et nous en présenterons les limites en analysant le comportement fin d'un système otique simple (lentille mince). Pour une caméra utilisant une matrice de détecteurs, la tache de diffusion est associée à la réponse impulsionnelle du détecteur. Nous proposerons, dans ce cadre, une méthode d'évaluation de l'incertitude systématique en température basée sur le test de réponse à une fente d'un système de thermographie. La méthode est appliquée à une caméra dans le contexte des inspections électriques.

INTRODUCTION

Le secteur du contrôle thermographique des installations électriques est un grand utilisateur de caméras thermiques. Ce secteur, très concurrentiel, dispose d'une très grande variété de caméras dont le rendu visuel est un atout commercial. Cependant, il ne suffit pas à garantir une mesure exacte de température. C'est dans ce contexte que se pose le choix de la caméra thermique la mieux adaptée.

Parmi les critères de choix, la résolution spatiale et l'exactitude de mesure des caméras sont privilégiées. La résolution spatiale est associée au pouvoir de séparation des caméras thermiques. Il s'agit ici de pouvoir distinguer sur l'objet deux traits voisins sans risque de confusion. L'exactitude de mesure résulte en partie des qualités de métrologies du système, en particulier de sa sensibilité, de son étendue de mesure, de sa reproductibilité et de sa stabilité temporelle. L'autre partie est liée à l'utilisation des caméras par un opérateur de thermographie. Ce dernier se doit en effet de maîtriser sa caméra et la situation de mesure, en particulier, il doit savoir prendre en compte les grandeurs d'influence que sont l'émissivité de l'objet et la température d'environnement.

Nous devons toujours garder en mémoire que L'objet de la mesure est un flux. De part sa nature, il existe un lien étroit entre la résolution spatiale du système (aptitude à le percevoir) et son exactitude de mesure (aptitude à l'évaluer correctement).

CARACTERISTIQUES DES CAMERAS THERMIQUES

Les caméras thermiques associent les fonctions de radiomètre et d'imageur. Nous classons communément leurs caractéristiques dans deux espaces distincts, l'espace de l'imagerie et l'espace de la mesure thermique. Les deux espaces superposés réunissent les caractéristiques des caméras [START_REF] Pajani | Technologie et applications[END_REF] .

Rappelons-en les principales distinctions. L'imageur thermique a pour vocation de présenter sur un écran des images thermiques de bonne qualité, au standard vidéo et pour des scènes aux températures ordinaires. Ces imageurs sont principalement caractérisés en termes de « résolution » thermique, spatiale et temporelle. Il n'a pas la prétention ni les capacités à quantifier ces images en carte de luminance ou à les transcrire en température. Ces fonctions sont dédiées au radiomètre. Un radiomètre est conçu pour la mesure de flux ou de températures quelconques, non plus seulement au températures ordinaires. Les radiomètres satisfont aux critères généraux de métrologie : sensibilité, d'étendue de mesure, exactitude de mesure et stabilité temporelle, etc… Il existe alors une correspondance unique entre le photosignal délivré par le système et la température apparente de l'objet.

Nous pouvons rappeler sur le tableau 1, les caractéristiques des caméras thermiques et leurs méthodes de détermination selon la norme NF A09 420 2 .

La notion récente de « Spot Measurement Size » [START_REF] Warren | applying distance to spot ratio values to infrared imagers for accurate temperature measurement[END_REF]4 est à rapprocher du pouvoir de résolution spatiale de mesure (PRSM).

RESOLUTION SPATIALE

Comme pour tout système optique, la résolution spatiale est associée à son pouvoir de séparation. Il s'agit ici de pouvoir distinguer deux traits voisins sans risque de confusion. Le pouvoir de séparation est usuellement décrit en trait par millimètre. Pour les systèmes fonctionnant dans le visible, le pouvoir de séparation atteint 400 traits au millimètre (2,5 10 -3 mrad). Comme nous allons le montrer, les systèmes fonctionnant dans la bande spectrale 8-12µm présentent au mieux un pouvoir de séparation de l'ordre de 20 traits par millimètre (0,05 mrad).

Pour une caméra à matrice de détecteur, le pouvoir de séparation est à distinguer du champ élémentaire instantané. Ce champ élémentaire contient le flux reçu par chaque détecteur. Il est défini par l'angle solide sous tendu par la surface sensible du détecteur depuis l'optique principale de la caméra. Le champ élémentaire est associé à l'IFOV (instantaneous field of view). Ce nombre est calculé en divisant le champ total de la caméra par le nombre de détecteur en ligne.

La nature ondulatoire de la lumière impose une limite à la finesse des détails fournis par une chaîne optique. Principalement à cause des phénomènes de diffraction et d'aberration, l'image d'un point lumineux n'est pas ponctuelle. Ces phénomènes dégradent le pouvoir séparation des systèmes optiques. L'étendue des taches de diffusion dépendra principalement de l'ouverture optique F/N du système optique. Parfois aux effets antagonistes, elles pourront être corrigées. Les procédés de corrections sont bien connus des opticiens.

Espace imagerie

Dans quelques cas simples, il est possible de déterminer l'étendue angulaire de la tache de diffusion d'un système. Projetée dans le plan objet, on donnera le pouvoir de séparation du système. L'étude est donnée pour une lentille simple en Germanium, très utilisé dans l'infrarouge en raison de sa transmission spectrale étendue et de sa dispersion très faible sur le domaine 8-12µm.

La diffraction

   
Pour un système optique de diamètre de pupille 2h = 50 mm, de focale f = 100 mm et fonctionnant dans la bande 8-12 µm (= 10 µm), le diamètre de la tache de diffraction est, à la distance focale, de l'ordre de 50 µm (d=0,49 mrad). A noter que ces dimensions augmentent avec l'ouverture optique N (N = f/2h).

Les aberrations

Comme pour la diffraction, les aberrations affectent la qualité des images. On distingue deux types d'aberrations : les aberrations chromatiques et les aberrations géométriques. Elles projettent l'image d'un point sous la forme d'une tache de révolution dont le diamètre est mesuré par un angle solide de diffusion d.

1f d 2N R    
Pour le système précédent, l'ordre de grandeur du de la tache de diffusion est, au plan focal, de 22 µm (la constringence du Germanium est donnée sur la bande spectrale 8-12µm : = 1112). Ce terme reste du même ordre de grandeur de la tache principale de diffraction.

Pour des ouvertures plus faibles (F/1 ou moins), Le chromatisme peut être si important qu'il interdise toute observation. Bien entendu, il existe plusieurs moyens de réduire ces aberrations. La correction des aberrations chromatiques se fait par association de plusieurs lentilles telles que la dispersion de l'une compense la dispersion de l'autre. Le système corrigé est dit achromatiques ou apochromatiques (corrections chromatiques d'ordre supérieur). Les systèmes optiques ne comportant que des miroirs ne sont pas affectés par les aberrations chromatiques.

En dehors des phénomènes de diffraction et d'aberrations chromatiques, Les aberrations géométriques sont dues à ce que les rayons éloignés de l'axe ne convergent pas au même point que les rayons paraxiaux (écarts à la condition de Gauss). Il en résulte un flou de l'image obtenue, préjudiciable là aussi à sa qualité.

On distingue quatre aberrations fondamentales caractérisées par un développement du troisième ordre de l'aberration géométrique transversale : les aberrations sphériques (étendue géométrique  (f/N) [START_REF] Warren | applying distance to spot ratio values to infrared imagers for accurate temperature measurement[END_REF] ), le coma ( R(f/N) 2 ), L'astigmatisme et la courbure de champ (R 2 (f/N) ) et la distorsion ( R 3 ).

De part sa dépendance en (f/N) [START_REF] Warren | applying distance to spot ratio values to infrared imagers for accurate temperature measurement[END_REF] , Il est intéressant de regarder le comportement des aberrations sphériques avec l'ouverture optique N. Le résultat est donné pour une lentille simple en germanium au plan focal image 5 : Germanium n=4,0

3 3 8,7 d 10 N     (radian)
Pour le système précédent, le calcul indique un angle de diffusion de 1 mrad (diamètre de diffusion : 100µm). L'angle de diffusion diminue rapidement quand l'indice de réfraction augmente.

La coma et l'astigmatisme affectent exclusivement l'image des points les plus excentrés. La coma donne une tache de diffusion qui ne possède pas de symétrie de révolution. La figure obtenue est un glissement de taches circulaires (rayon 2c) sur la distance 3c [START_REF] Gaussorgues | la thermographie infrarouge : principes, technologies[END_REF] . La figure de la coma et son ouverture angulaire maximale sont présentées sur la figure 4.

DISCUSSION DANS LE CAS D'UNE LENTILLE MINCE

Evaluons la diffusion optique d'une lentille simple en Germanium (n=4). La lentille présente une focale caractéristique de 100mm et un diamètre 2h=50mm. Elle est ouverte à F/2 pour un champ total de 12°. Le diamètre de diffusion est, donné à la distance focale pour les bandes spectrales 3-5 µm (constringence =88) et 3-5 µm (constringence 1112). Le résultat est présenté sur le tableau 3.

Tableau 3. Evaluation des taches de diffusion pour une lentille mince

AUTRES LIMITES A LA RESOLUTION SPATIALE DES CAMERAS THERMIQUES

Dans le cas de caméras à matrice de détecteurs, le pouvoir de séparation est aussi limité par la dimension du détecteur, par la variation de sa sensibilité locale et par l'électronique de lecture.

REPONSE IMPULSIONNELLE LINEAIRE

Tous ces facteurs donnent la figure de réponse d'un détecteur, encore appelée réponse impulsionnelle linéaire D(x) du système. On modélise la réponse impulsionnelle par une fonction Gaussienne dans laquelle  représente le rayon de la tache globale de diffusion correspondant à 84% du flux reçu. Les caméras thermiques à matrices sont conçues de telle manière à ce que 4  couvre la surface moyenne occupée par un détecteur (environ 4d 2 si on tient compte de l'espacement entre deux détecteurs du même ordre que leur dimension d). Il est inutile de faire mieux. Ainsi,  est du même ordre que d. L'angle solide associé /f s'identifie au champ élémentaire instantané de la caméra (IFOV)

FONCTION DE REPONSE A UNE FENTE

Dans ce formalisme, il est possible de mesurer le pouvoir séparateur du système en donnant à l'objet la structure d'une fente de fort contraste thermique de largeur a ou d'ouverture angulaire a/D (D : distance fentes-caméras). On procède ici au test normalisé de fonction de réponse à une fente (NF A09-421)

La répartition normalisée de luminance de l'objet L 0 (x) est alors définie par la fonction rectangle : Lorsque la largeur de fente est réduite de telle façon que l'amplitude du signal est la moitié de celle fente ouverte, la résolution spatiale dite à 50% de modulation est représentée par la dimension  (ou l'ouverture angulaire /f = IFOV de la caméra).

Enfin, en fermant la fente jusqu'à une largeur bien inférieure à , le profil du signal correspondant est la réponse impulsionnelle linéaire D(x) encore appelée LSF (Line Spread Function ou fonction d'étalement de ligne). Sa détermination est normalisée (NF A 09-421). En raison de la faiblesse des signaux, cette caractéristique est en pratique difficilement accessible pour les cameras thermiques commerciales.

RESOLUTION SPATIALE ET EXACTITUDE EN TEMPERATURE

Dès qu'il s'agit de mesure, le système doit présenter les caractéristiques générales d'un instrument de mesure : reproductibilité et exactitude.

Nous avons montré que, lorsque la largeur de fente est comparable à la tache globale de diffusion, chaque point élémentaire de la matrice reçoit au mieux la moitié du flux total ! Pour en mesurer l'intégralité, Il faudrait une fente d'ouverture supérieure à 4. Bien entendu, il n'est pas toujours possible ni souhaitable de s'imposer de telle contrainte. On cherchera plutôt à concilier une image correctement résolue dont au moins un Pel donnera la température d'émission dans une « bonne » exactitude.

EXACTITUDE EN TEMPERATURE

Pour chiffrer l'exactitude en température, il faut donner à l'objet une dimension angulaire qui donne à la SRF une valeur égale à 1-,  étant le taux d'erreur consenti dans la mesure. Un écart de % sur la valeur relative du signal correspond à une erreur systématique en température donnée par la relation max I T s

    

où Imax est l'écart consenti en niveau sur la SRF et <s> en niveaux/°C est la sensibilité de la caméra sur l'intervalle [T1,T2] (T1 : température de fente, T2 :température de corps). L'erreur systématique T est comptée négative car la température radiométrique est estimée par défaut. La sensibilité est déduite de la courbe d'étalonnage de la caméra. La procédure est illustrée sur la figure 8. En pratique, la température de corps noir est associée à la température objet tandis que la température de fente est associée à la température de fond (environnement radiométrique situé derrière l'objet).

L'erreur systématique T, de même que le PRSM est donnée pour ces températures.

DISCUSSION DANS LE CAS DES INSPECTIONS ELECTRIQUES

Le secteur du contrôle thermographique des installations électriques est un grand utilisateur de caméras thermiques. Ce secteur, très concurrentiel, utilise une très grande variété de caméras dont le rendu visuel est un atout commercial.

C'est dans ce contexte que se pose le choix de la caméra thermique la mieux adaptée au contrôle thermographique. Ce choix doit répondre à deux problématiques :

 La première est de disposer d'un imageur capable de scruter l'installation pour avoir une vue d'ensemble du dispositif et pouvoir situer correctement les anomalies de température en toute sécurité ;

 La seconde est de disposer d'un radiomètre permettant de quantifier correctement le flux émis en réduisant les erreurs systématiques inhérentes à une mauvaise utilisation de l'appareil afin d'estimer la température de câble dans la limite de l'exactitude instrumentale de 2°C, sans avoir des à priori sur la géométrie et la position des câbles par rapport au champ de vision.

Contexte du contrôle thermographique

REMARQUES COMPLEMENTAIRES

L'erreur systématique inhérente au choix du PRSM se rajoute à l'incertitude instrumentale. Etant découplées, l'erreur globale est la somme quadratique des deux termes. Dans l'exemple, l'erreur globale est de l'ordre de 3°C. Si on souhaite réduire l'erreur globale à l'incertitude expérimentale, on recherchera un PRSM de l'ordre de 0,5 °C seulement.

La valeur moyenne de la sensibilité repose sur la transcription flux-température. Elle est constante si courbe d'étalonnage est linéaire sur les températures de test [T1,T2]. Dans le cas contraire, la sensibilité moyenne et l'erreur systématique varient avec les températures de test.

Le résultat est donné pour une disposition verticale des fentes. De part l'asymetrie des matrices, le PRSM peut se dégrader dans la disposition horizontale ou si l'objet à la forme d'un disque. Il est donc tout aussi important de reproduire les résultats du test dans ces dispositions et de convenir du choix d'un PRSM dans la situation la moins favorable [START_REF] Warren | applying distance to spot ratio values to infrared imagers for accurate temperature measurement[END_REF] .

CONCLUSION

Une analyse fine du processus de formation des images a montré les limites de la résolution spatiale chez le système optique constitué d'une lentille mince. Une étude chiffrée de la tache de diffusion est donnée et indique une très forte sensibilité avec l'ouverture optique du système et la bande spectrale du système. Appliquée à une caméra à matrice focale, la tache de diffusion est associée à la réponse impulsionnelle D(x) d'un détecteur dont l'étendue est comparable au champ élémentaire de mesure ou IFOV du système. Dans ce formalisme, nous proposons une interprétation du signal fourni par l'observation d'une fente présentant un fort contraste thermique. L'intensité maximale relative du signal (SRF) est étudiée pour des différentes ouvertures angulaires. La connaissance de la SRF se révèle être un outil indispensable dans l'évaluation de l'erreur systématique de mesure. Appliquée à un contrôle par thermographie des installations électriques, nous montrons qu'une erreur systématique de -1°C sur la température d'émission correspond à une ouverture angulaire de l'ordre de 4 IFOV.

Fig. 2 .

 2 Fig.2. Aberrations chromatiques. Un système optique d'ouverture N=f/2h produit une tache chromatique issue des variations de la focale du système sur la bande spectrale du rayonnement.

Fig. 3 .

 3 Fig. 3. Aberrations sphériques. Le système produit une tache géométrique issue de la variation de la focale avec l'ouverture des optiques (écarts aux conditions de Gauss)

Fig. 4 . 5 .

 45 Fig. 4. Figure de la Coma d'un système optique Fig. 5. Figure d'astigmatisme d'un système optique

Fig. 6 .

 6 Fig. 6. Réponse impulsionnelle linéaire

Tableau 5 .Fig. 7 .

 57 Fig. 7. Valeur maximale relative sur le signal donné par une fente de largeur a.

Fig. 8 .

 8 Fig.8. Résolution spatiale et exactitude en température (gauche : SRF ; à droite : courbe d'étalonnage). a/D est l'ouverture angulaire de la fente (T1 : température des fentes, T2 : température de corps noir). La sensibilité moyenne est estimée aux valeurs limites du niveau maximum relatif par (N(T2)-N(T1))/(T2-T1) L'ouverture angulaire correspondant à 1- du niveau maximum relatif est appelée pouvoir de résolution spatiale de mesure (PRSM).

Tableau 1. Méthodes de caractérisation des systèmes de thermographie Méthodes de caractérisation Désig. Norme Caractéristiques déduites Fonction d'étalement de ligne (ligne spread function)

		FEL / LSF NF A 09-421	Angle élémentaire limite
	Fonction de réponse à une fente (Slit	FRF / SRF NF A 09-421	Angle
	Response function)

élémentaire de mesure Pouvoir de résolution spatiale de mesure (PRSM) : angle d'ouverture à FRF=0,98 par exemple Fonction de transfert de contraste

	FTC	NF A 09-421

Angle élémentaire d'observation à n% Fonction de transfert géométrique

	FTG	NF A 09-421	Angle d'

observation suivant les balayages trame et ligne Fonction de transfert de modulation (Transformée de Fourier de la Fonction d'Etalement de Ligne)

	FTM	NF A 09-421	Pouvoir

de résolution spatiale limite (PRSL) Pouvoir de résolution spatiale d'observation (PRSO)

	Résolution thermique	NETD ou	NF A 09-421	«
		DETB

Différence de température du corps thermorayonnant équivalent à la valeur efficace du bruit de l'appareillage » Espace de la mesure

	Réponse spectrale relative	RSR	NF A 09-421	Longueur d'

onde au maximum de la réponse spectrale (pic) Bande spectrale [1,2] définie à 5% du maximum de la réponse relative Courbes d'étalonnages NF A 09-421 Calibres et étendue de mesure Sensibilité moyenne et différentielle Linéarité Temps de stabilisation Dérive temporelle Influence de la température ambiante Résolution thermique (Uniformité de réponse dans l'image thermique) = bruit spatial fixe résiduel / sensibilité

	Notes : les écarts minima de température détectable (Minimum Detectable Temperature Difference, MRTD) et écart
	minimum De température séparable (Minimum Resolvable Temperature Difference, MRTD) sont des
	Caractéristiques subjectives et leur détermination est facultative.

2.1. LES LIMITES DE LA RESOLUTION SPATIALE

	La diffraction et les aberrations géométriques sont des phénomènes physiques incontournables dans la
	construction des images. Ils sont présents dans tous les systèmes utilisant des pupilles (lentille,
	diaphragme…). A cause de la diffraction, l'image d'un point lumineux fournie par un système optique
	parfait à pupille circulaire n'est pas ponctuelle. Les aberrations chromatiques et géométriques agissent de
	la même façon : elles transforment l'image d'un point en une tache de diffusion.

  A cause de la diffraction, l'image d'un point lumineux fournie par un système optique parfait à pupille circulaire n'est pas ponctuelle. La figure de diffraction obtenue, dite tâches d'Airy, présente une symétrie de révolution avec un maximum d'amplitude central entouré d'anneaux d'amplitude nulle alternés Le maximum central contenant la presque totalité du rayonnement, le diamètre du premier anneau « obscur » 2r0 donne la dimension de la tâche image. Il est défini par le premier zéro de la fonction de Bessel du premier ordre :

		diffraction		2,44		f 2h
	Pour de faibles angles, cette dimension est associée à un angle solide de diffraction donné par la relation
	suivante :
	d	f	2,44	2h

  n'est pas rare de rencontrer des ouvertures au-delà de F/1. Nous avons vu qu'une lentille simple en Germanium de 100 mm de focale, ouverte à F/2 donne une tache de diffusion centrale de l'ordre de 1,8 mrad pour un champ total de 12°. La coma et l'astigmatisme, affectant les points images les plus excentrés (demi champ =6°), dégradent le pouvoir de séparation tandis que le chromatisme, important en ondes courtes, interdirait une observation fine de l'objet ! Un système optique parfaitement corrigé donnerait un champ élémentaire voisin de la limite angulaire de diffraction. Le calcul d'un système constitué par une lentille simple « de meilleur forme » en Germanium sur la bande 8-12 µm donne un résultat plus proche de la réalité. Le calcul porte sur l'évaluation des aberrations sphériques, du coma et de l'astigmatisme. Il est fait en plaçant le plan image dans le plan du cercle de moindre diffusion de l'aberration sphérique (la défocalisation à donner est de -1,035 mm ramenant la focale du système à 98,96 mm au lieu de 100 mm). Le résultat, issu de la référence[START_REF] Gaussorgues | la thermographie infrarouge : principes, technologies[END_REF] , est présenté sur le tableau 4.

	Tableau 4. Evaluation des taches de	Bande spectrale 8-12 µm
	diffusion pour une lentille mince de	Ouverture angulaire (mrad)
	« meilleur forme »
	Aberration sphérique	1,6
	Coma	0,6
	Astigmatisme	5,4
	Etendue de diffusion hors coma et	2,3 mrad (230µm)
	astigmatisme
	Etendue totale de diffusion	8,3 mrad (830 µm)
		Bande spectrale 3-5 µm	Bande spectrale 8-12 µm
		Ouverture	Diamètre de	Ouverture	Diamètre de
		angulaire (mrad)	diffusion (µm)	angulaire (mrad)	diffusion (µm)
	Diffraction à 10µm	0,194	19,4	0,488	48,8
	Chromatisme (bande spectrale 8-12µm)	2,84	284	0,225	22,5
	Aberration sphérique	1,087	108,7	1,087	108,7
	Coma	0,26	26	0,26	26
	Astigmatisme	2,5	250	2,5	250
	Etendue de diffusion hors coma et	4,12 mrad (412 µm)		1,8 mrad (180µm)
	astigmatisme
	Etendue totale de diffusion	6,88 mrad (688 µm)		4,56 mrad (456 µm)

Le pouvoir séparateur des optiques infrarouges est en général bien inférieur à celui des optiques conçues pour le domaine visible. Il est très dépendant de l'ouverture optique du système. Dans le domaine infrarouge, il Une telle lentille donnerait un champ élémentaire paraxial de 2,3 mrad et une tache image d'environ 230µm. Le pouvoir de séparation correspondant est d'un peu plus de 4 traits par millimètre. Ce résultat est à comparer au pouvoir de séparation de 20 traits par millimètre pour une lentille limitée uniquement par la diffraction (bande 8-12 µm).

  L'expérience des contrôles thermographiques des installations électriques démontre que les échauffements anormaux concernent tous les types de conducteurs petits ou gros. Ainsi, beaucoup d'échauffements anormaux sont constatés sur des câbles de 1,5 mm 2 (2,5 mm de diamètre).La distance d'inspection en régime base tension est prescrite par la publication UTE C 18-510 qui impose d'être à plus de 30 cm des pièces nues sous moyenne tension. En régime haute tension, cette distance est portée à 60 cm. Dans le cas d'un contrôle d'une armoire électrique, cette prescription fixe la distance minimale de sécurité à 50 cm si l'on tient compte de la profondeur moyenne de l'armoire (environ 20 cm, profondeur qui reste à normaliser).Concernant la température des câbles, nous pouvons prendre comme référence les valeurs de la norme C15 100 (70°C pour un câble gainé PVC et 90°C pour ceux gainées PR). Les températures de fond communément recensées varient de 20 à 50°C. En l'absence d'indication spécifique, la température de 20°C est réaliste car peu contraignante.Examinons par exemple une caméra thermique 320240 dont la résolution spatiale à partir de l'observation du signal donné par une fente présentant un fort contraste thermique. Les fentes sont éclairées par un corps noir dont on règle la température à 70°C. Les fentes, rendues émissives, sont à la température de 20°C. Elles sont placées à la distance minimale de mise au point de la caméra, soit D=30 cm. L'ouverture angulaire d'observation est définie par le rapport a/D où a est l'ouverture de fente. Le test est donné pour une optique standard (24°18°) après le temps de stabilisation du système. Le résultat est présenté sur la figure 9 et quelques valeurs particulières dans le tableau 6.Compte tenu des valeurs limites relevées, on peut déterminer la sensibilité moyenne de la camera sur les températures de test : <s>=66 niveaux/°C. Sa connaissance nous permet maintenant de convertir les valeurs maximales relatives du signal en température. Le résultat est montré sur le tableau 7. systématique de mesure varie grandement avec le PRSM. Elle est de l'ordre de l'incertitude instrumentale pour un PRSM de l'ordre de 4 mrad. Projeté sur le plan objet, la taille caractéristique de la zone de mesure est, à 50 cm, de l'ordre de 2 mm. Cet imageur est, à première vue, qualifié pour la mesurer d'un câble de 2,5 mm de diamètre.

	La valeur maximale relative du signal reproduit bien l'IFOV de la caméra (24° 0,419 rad/320= 1,3 mrad
	horizontal).
		4.2.3. Analyse des données
				Tableau 7. Erreur systématique en température
				PRSM (mrad)	1,3	3,1	3,9	4,4	5
				Imax (niveau)	1640 328	164	66	33
				T (°C)		-24,8 -5	-2,5	-1	-0,5
	L'erreur
		4.2.2. Caméra
				SRF (Niveaux)
		10000					Tableau 6. Valeur maximale relative du signal donné par une fente
		9570
							Imax (%)	0,5	0,9	0,95 0,98 0,99
		9000
							PRSM (mrad)	1,3	3,1	3,9	4,4	5
	Niveaux	8000
		7000		Température de corps noir: 70°C Températ ure de corps noir : 70°C Température de fentes: 20°C Températ ure de fentes : 20°C
				Optique 20° F/1,8 Optique 24°18°
		6290
		6000
		0	2	4	6	8	10
				Ouverture angulaire
	Fig. 9. SRF