ON THE CONVERGENCE TO EQUILIBRIA OF A SEQUENCE DEFINED BY AN IMPLICIT SCHEME

. We present numerical implicit schemes based on a geometric ap-proach of the study of the convergence of solutions of gradient-like systems given in [3]. Depending on the globality of the induced metric, we can prove the convergence of these algorithms.

1. Introduction. In this paper we deal with the asymptotic behaviour of discretized dynamical gradient systems on manifolds. The interplay between continuous analysis and numerical approximation analysis is the keystone of modelization. This is in the sense that continuous models are commonly issued from (at least formal) limits of the application of laws to a large number of studied objects. It is usually much more comfortable (this does not mean easiness) to deal with properties in the continuous framework than in the discrete world. But in order to completely deal with modelization, one has to understand the discretized version of the continuous situation, at least to understand behaviors observed in silico. In the present work, we try to reconnect the behavior of solutions of a gradient system on a manifold to the behavior of an implicit scheme constructed on this gradient system. This is done in section 3. Gradient systems are natural from the point of view of modelization and there properties are usually tied to the physical interpretation of them (e.g. minimizing energies behaviors). In the last section of the paper, we try to reconnect the properties of the gradient-like systems implicit discretization to the continuous systems ones by means of the striking and nice, but difficult to handle, introduction of Riemannian metrics performed in [3] in order to apply the results of section 3. Unfortunately we are only able to give partial insights on the expected behaviors but without any proofs.

2.
Notation. For a Riemannian manifold (M, g) of dimension N we denote ·, · g the scalar product defined on each tangent space. The induced norm is denoted · g (or · when there is no risk of confusion). For a local system of coordinates on M , g ij will denote the coefficient of the matrix defining the scalar product above.
Let us recall that a C 1 curve x : [0, 1] → M is called a geodesic between x(0) and x(1) iff it is a critical point of the functional For a differentiable function f : M → R and p ∈ M we denote ∇ g f (p) the unique element of the tangent space T p M to M at p such that 3. An implicit numerical scheme and main result of the paper. Let us consider (M, g) a complete connected (non necessarily compact) Riemaniann manifold and E a smooth real function. Associated to E, it is quite natural to consider the following gradient systemẊ (t) + ∇ g E(X(t)) = 0. (1) In the paper [12] the authors Merlet & Pierre consider the situation when (M, g) is the standard R N with its natural Euclidian structure and prove the convergence of a sequence defined by an implicit scheme associated to (1). It is quite natural to extend the scheme there introduced to the case of more general manifolds. Such insights were initially considered in [13] provided (M, g) is a submanifold of R N . However the specific case of the backward Euler scheme was not considered in this paper under the intrinsic point of view, i.e. the backward scheme is constructed ex post in [13], considering the embedded situation. Here we try to focus on the intrinsic geometry given by g even if we will use the existence of isometric embeddings in some Euclidean space. Of course comparing the backward algorithm given in [13] and the scheme constructed in the present paper is certainly of interest but we have not considered it yet for the moment being.
We will assume in this section that M is complete, i.e. for any pair of distinct points of M there exists a minimizing geodesic between them. Without loss of much generality, according to Nash's theorem (see [7] and [14]) we can always assume that M is isometrically embedded in R P for a large enough P . The induced distance on M will be denoted by d.
For some δt > 0, we consider the following sequence : Assume that X 0 , .., X n are constructed, we consider Definition 3.1. The sequence (X n ) is the implicit Euler scheme associated to (1), for the given time step δt.
Let us note that given X n , the existence and uniqueness of X n+1 depends on different hypothesis. A natural assumption is that E is coercive and semi-convex. Under the coercivity assumption of E, the semi-convexity assumption can be released provided δt is small enough.
The convergence of the solutions of (1) has been extensively studied either in finite or infinite dimensions. In the situation when E is analytic the convergence was firstly studied by S. Lojasiewicz in [10,11] (see also [8,6]) A major sufficient assumption for proving the convergence is the fact that E satisfies the so-called Lojasiewicz's inequality at critical points: In the following section we will prove the following theorem, main result of this paper. Before stating it, we recall that E is said to be coercive if there exists some y ∈ M such that E(x) tends to +∞ as x ∈ M and d(x, y) → ∞. We note that, due to the completeness of M , this definition does not depend on the choice of y. When M is compact, the coercivity is automatically satisfied. Finally, let us note that the coercivity assumption implies that the set in the right hand side of (2) is nonempty.
Assume that E is coercive and satisfies the Lojasiewicz's inequality then any sequence (X n ) that complies with (2) converges to a critical point of E.
As we said, the convergence of the sequence defined by discretized schemes associated to dynamical systems has been recently studied. The pioneering work in that direction is given in [1]. The case of implicit schemes for dynamical systems was considered quite simultaneously in [12] and [4], though some implicit scheme in the framework of Lojasiewicz's inequality was already considered in [2].
In order to deal with this result, it is required to get some informations from the Euler-Lagrange equation satisfied by X n+1 .
The computation is similar to the one made in order to derive the geodesic equations. We closely follow it (see for example [9]). In order to do so, let For any s ∈ [0, 1] we have The derivative of s → d(X n , φ(1, s)) with respect to s is thus Now we take s = 0. Let us recall that, for s = 0, we have a constant speed geodesic, thus we get Thus, due to this and integrating by part in (4), we get Now using the fact that t → φ(t, 0) is a geodesic, the integral term in (5) vanishes. Thus we have d ds (s → d(X n , φ n (1, s))) s=0 = 1 d(X n , X) Finally, the infinitesimal variation associated to (2) with respect to the variation given by φ n is given by 1 δt ∂φ n ∂t (0, 1), ∂φ n ∂s (0, 1) + ∇ g E, ∂φ n ∂s (0, 1) = 0.
The Euler-Lagrange equation associated to (2) is thus : Let us point out that this condition is natural. Indeed when M = R N with the standard Euclidean metric, the constant speed geodesic joining X n to X n+1 in a time duration 1 is t → X n + t(X n+1 − X n ) which gives the standard Euler implicit scheme associated to (1).
Let us give some estimates that will be useful in the sequel.
Since we can assume that M is isometrically embedded in R P for some P , we deduce that there is a universal constant A such that A||X n+1 − X n || ≤ d(X n , X n+1 ).
Since t → φ n (t, 0) is a constant speed geodesic joigning X n to X n+1 for t ∈ [0, 1] we have, according to (8) and || ∂φ n ∂t (1, 0)|| g = d(X n , X n+1 ), Let us remark that we can take A = 1 but we keep the letter A in order to follow further computations. Let us also note that, on compact subsets of M we have also for some B which does depend only on this compact. We will not use such an inequality. Due to the left hand side of this estimate and by following the proof of the Merlet & Pierre results of [12], we will show that the sequence (X n ) converges to some critical point of E.

4.
Proof of the main result. In this section we give the proof of our main result, namely theorem 3.2.
We now closely follow [12]. Let us note that since The sequence (E(X n )) n∈R N is therefore non increasing, converges due to our assumptions to a value which can be chosen equal to 0 without any loss of generality, and thus there exists a subsequence of (X n ) that converges to some X ∞ . Note that we also have lim n→∞ d(X n , X n+1 ) = 0. Note also that according to (7), (9), (10) ∇E(X ∞ ) = 0.
Assume now that E(X n+1 ) ≤ 1 2 E(X n ). We have, for Thus in both cases, we get that for all n such that d(X n+1 , X ∞ ) < σ, we have LetĒ > 0 small enough such that Letn large enough such that ||Xn − X ∞ || < σ/3 and E(Xn) <Ē and N the largest integer such that ||X n − X ∞ || < 2σ/3 for all n such thatn ≤ n ≤ N . Assume that N is finite. We have

This implies
which is a contradiction if N is finite. As a consequence the sequence (X n ) converges which ends the proof of the main result.
5. Some generalizations and partial extensions. In the work by Chill & al [5], the equation (1) is also considered, as well as the so-called quasi-gradient system on R Nẍ +ẋ + ∇F(x) = 0, (12) as a particular case of a more general system on R Ṁ x + F (x) = 0.
In [3], it is shown that if there exists a continuously differentiable, strict Lyapunov function E for (13), then there exists a Riemannian metric g on the open set M = {x ∈ R M , F (u) = 0} such that F = ∇ g E. We will here assume the existence of this function E. Some fundamental properties of the metric g are strongly related to the so-called compatibility condition (C) and angle condition (AC).
Let us recall the following definitions. The first is given in [5] (see also [3]). This angle condition (AC) has first appeared in [1].
We will say that E and F satisfies the angle condition (AC) iff there exists a > 0 such that We will also need the following one, given in [3]. We will say that E and F satisfies the compatibility condition (C) iff there exist c 1 , c 2 > 0 such that The following result is proven in [3] Theorem 5.1. The Euclidean metric and the metric g are equivalent on M if and only if E and F satisfy the conditions (AC) and (C).
Though this property has a very nice appearance, it is not clear that it can be used according to the first section of the present paper. Indeed, in order to do so, one has to check that this metric g can be extended or not to R M to a geodesic convex metric. If so, the results of part (1) can be applied.
Otherwise the situation is not clear. In this case we modify the algorithm given in section one the following way.
We will moreover assume that E is non-negative, that its infimum is 0 and that {x, F (x) = 0} is compact (for the initial topology). We choose R > 0 such that We take ε > 0 such that ε < R. Let M ε/2,2R be the manifold {x, ε/2 < E(x), x < 2R} and let g be the metric constructed in [3]. By compactness, it is standard that there exists ρ > 0 such that for any x ∈ M 2ε/3,3R/2 , the geodesic ball is geodesic convex (see [15]). We can moreover assume that ρ < 1 and, moreover, if x ∈ M ε,R B g (x, ρ) ∈ M 2ε/3,3R/2 .
We choose x 0 ∈ M ε,R and consider the following minimization problem: By compactness of the ball, the existence of a minimizer is obvious. We denote x 1 such a minimizer. Assume that we have constructed x 1 , ..., x N . We have two possibilities. Either x N ∈ M ε,R so that we take We then go on by replacing ε by ε/2, and this defines the sequence. Now let us study the convergence of this sequence. Assume that for some ε > 0 we have ∀n ∈ N, x n ∈ M ε,R .
Let n k an increasing injection of N such that (x n k ) converges and let l denotes the limit. Let x n k and x n k such that l ∈ B g (x n k , ρ/4) ∩ B g (x n k , ρ/4).
This is impossible since this would imply E(x n k ) < E(x n k ) < E(x n k ).
The same proof implies that either there exists a N such that E(x N ) = 0 or ∀ε > 0 there exists N such that ∀n > N 0 < E(x n ) < ε.
Indeed, if there exists ε > 0 such that for any N , there is n N > N such that x n N ∈ M ε,R and we can apply our preceding argument. The only other possibilities are that the sequence (x n ) is stationary from a certain rank. This in fact does not imply the convergence of the sequence which is for the moment being unknown to us.
Let us remark that if the metric g is globally defined, then the same argument as the proof of the main theorem shows that the sequence converges if we moreover assume (without loss of generality) that ρ is small enough in order to have for every N min Indeed, in this case, ρ can be chosen globally on the set on the open set B(0, 2R) and the same strategy applies as in the proof of the main theorem.