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arXiv:math/0502317v1 [math.PR] 15 Feb 2005
An adaptive scheme for the approximation of
dissipative systems
Vincent Lemaire
Laboratoire d’Analyse et de Math´ ematiques Appliqu´ ees, UMR 8050,
Universit´ e de Marne-la-Vall´ ee, 5 boulevard Descartes, Champs-sur-Marne,
F-77454 Marne-la-Vall´ ee Cedex 2, France.
Abstract
We propose a new scheme for the long time approximation of a diffusion when the
drift vector field is not globally Lipschitz. Under this assumption, regular explicit
Euler scheme –with constant or decreasing step– may explode and implicit Euler
scheme are CPU-time expensive. The algorithm we introduce is explicit and we
prove that any weak limit of the weighted empirical measures of this scheme is
a stationary distribution of the stochastic differential equation. Several examples
are presented including gradient dissipative systems and Hamiltonian dissipative
systems.
Key words: diffusion process; dissipative system; invariant measure; stochastic
algorithm; Euler method; simulation
1991 MSC: 65C30, 60J60
1Introduction
We consider the following stochastic differential equation
dxt= b(xt)dt + σ(xt)dBt,x(0) = x0∈ Rd, (1)
where b : Rd→ Rdis a locally Lipschitz continuous vector field and σ is locally
Lipschitz continuous on Rd, with values in the set of d × m matrices and B
is an m-dimensional Brownian motion. Assume that (xt)t≥0has a Lyapounov
function V i.e. a positive regular function decreasing along trajectories (precise
conditions are given by Assumption 1 in Section 2), so that there exists at least
one invariant measure.
Email address: vincent.lemaire@univ-mlv.fr (Vincent Lemaire).
Preprint submitted to Elsevier Science1 February 2008
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Until recently, the approximation of the stationary mode of the diffusion has
been studied under the assumption that V is essentially quadratic i.e.
|∇V |2= O(V )andsup
x∈Rd?D2V ? < +∞, (2)
and |b|2= O(V ) (which implies sublinear growth for b). When σ is bounded
and the diffusion is uniformly strictly elliptic, the invariant measure ν is unique
and Talay proposed in [12] a method for the computation of ν based on the
constant step Euler scheme. He proved the convergence of the invariant mea-
sure of the scheme to ν. On the other hand, Lamberton and Pag` es studied in
[5] the ergodic properties of the weighted empirical measures (νη
creasing step Euler scheme. They proved the almost sure tightness of (νη
and that any weak limit is a stationary distribution for the diffusion.
n)n≥0of a de-
n)n≥1
However, the conditions (2) and |b|2= O(V ) are too restrictive for studying
systems used in random mechanics (see Soize [10]). Indeed, the drift vector
field b is generally locally Lipschitz and in many cases V is not essentially
quadratic. This framework has been recently investigated by Talay in [13]
and by Mattingly et al. in [9]. In these papers, implicit Euler schemes with
constant steps are used for the approximation of the diffusion. In recent work,
Lamba, Mattingly and Stuart have introduced on finite time interval [0;T] an
adaptive explicit Euler scheme (see [4] and [8]). The step is adapted according
to the error between the Euler and Heun approximations of the ODE ˙ x = b(x).
They prove strong mean-quadratic convergence of the scheme on over finite
time intervals and ergodicity when the noise is non-degenerate. We propose a
completely different explicit scheme based on a stochastic step sequence and
we obtain the almost sure convergence of its weighted empirical measures to
the invariant measure of (1).
The key to prove the almost sure tightness of the weighted empirical measures
of the decreasing Euler scheme (Xn)n≥0introduced in [5] is that the scheme
satisfies a stability condition i.e. there exist ˜ α > 0 and˜β > 0 such that
E
?
V (Xn+1)|Fn
?
− V (Xn)
γn+1
≤ −˜ αV (Xn) +˜β,(3)
where (γn)n≥0is the deterministic decreasing step sequence. Without assump-
tions (2) we can no longer prove the stability condition (3) for this scheme.
Our scheme is built in order to satisfy (3). We proceed as follows. Firstly, we
start from a deterministic X0= x0∈ Rdand set
Xn+1= Xn+ ˜ γn+1b(Xn) +
?
˜ γn+1σ(Xn)Un+1,n ≥ 0, (4)
where (Un)n≥1 is a Rm-white noise more precisely defined in Section 2 and
˜ γn+1 = γn+1∧ χn with (γn)n≥0a positive nonincreasing sequence and χn a
σ(U1,...,Un)–measurable random variable.
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The basic main idea is to choose χnsmall when the scheme starts to explode.
In this case the discretization is finer and the stability condition of the diffusion
prevents the explosion. Furthermore we prove that the scheme satisfies a sim-
ilar condition with (3). A non-optimal –although natural– choice for χnmay
be χn=
the Lyapounov function (see (29) and (53)).
1
|b(Xn−1)|2∨1. For the two studied examples, optimal choices depend on
A crucial feature of our algorithm is the existence of an almost surely finite
time n1such that for every n ≥ n1, ˜ γn= γni.e. the event {χn−1< γn} does
not occur any more.
Numerically the algorithm is very simple to implement and the complexity is
the same as that of a regular Euler scheme. Another interest is that the scheme
is explicit, which is a big advantage on implicit schemes for high dimensional
problems. Indeed a fixed point algorithm is not needed is our case. Moreover,
we will see that wrong convergence problem due to fixed point algorithm may
be avoided using our algorithm.
The paper is organized as follows. We introduce the framework and the algo-
rithm in Section 2. In Section 3 are presented some preliminary results about
the approximation scheme of (Xn)n≥0defined in (4). In Section 4 we extend
some results of [5] and give conditions for the almost sure tightness of the
empirical measure and for its weak convergence to an invariant measure of
(1). Section 5 is devoted to the study of monotone systems and Section 6 of
stochastic Hamiltonian dissipative systems. The numerical experiments are
in Section 7 including some comparaison with recently introduced implicit
scheme. We confirm the non-explosion and the convergence of the scheme.
2 Framework and algorithm
We will denote by A the infinitesimal generator of (1). The following assump-
tion will be needed throughout the paper.
Assumption 1 There is a C2function V on Rdwith values in [1,+∞[ such
that lim
|x|→+∞V (x) = +∞ and satisfying
∃α > 0, ∃β > 0, ?∇V,b? ≤ −αV + β, (5)
∃CV,σ> 0, ∃a ∈ (0,1],
and∃C > 0,
Tr
?
?
σ∗(∇V )⊗2σ
σ∗D2V σ
?
≤ CV,σV2−a,
(x) ≤ C sup sup
x∈RdTr
?
x∈RdTr(σ∗σ)(x).
(6)
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Remark 1 If V is essentially quadratic (i.e. satisfies (2)) then (6) is satisfied
as soon as there exists Cσ> 0 and a ∈ (0,1] such that Tr(σ∗σ) ≤ CσV1−a.
Under this assumption, there exists a global solution to equation (1) and
(at least) one invariant measure. An important point to note here is that all
invariant measures have exponential moments. Indeed, an easy computation
shows that for all λ <
α
aCV,σwe have
∃˜ α > 0, ∃˜β > 0,Aexp
?
λVa?
≤ −˜ αexp(λVa) +˜β,
and this implies ν
?
exp(λVa)
?
is finite for all invariant measures ν.
For the approximation of the diffusion, we assume that (Un)n≥1is a sequence
of i.i.d. random variables defined on a probability space (Ω,A,P), with values
in Rm, and such that U1is a generalized Gaussian (see Stout [11]) i.e.
∃κ > 0, ∀θ ∈ Rd,
E
?
exp
?
?θ,U1?
??
≤ exp
?κ|θ|2
2
?
. (7)
and that var(U1) = Idm. We will call (Un)n≥1a Rm-valued generalized Gaus-
sian white noise. The condition (7) implies that U1is centered and satisfies
∃τ > 0,
E
?
exp
?
τ|U1|2??
< +∞. (8)
Moreover, the condition var(U1) = Idm implies that κ ≥ 1. In the sequel,
Fndenotes, for n ≥ 1, the σ-field generated on Ω by the random variables
U1,...,Un, and F0the trivial σ-field.
Remark 2 The assumptions made on the white noise (Un)n≥1are not restric-
tive for numerical implementation. Indeed, centered Gaussian and centered
bounded random variables satisfy (7).
The stochastic step sequence ˜ γ = (˜ γn)n≥0is defined by
∀n ≥ 1,˜ γn= γn∧ χn−1,˜ γ0= γ0,(9)
where (γn)n≥0is a deterministic nonincreasing sequence of positive numbers
satisfying
limand
nγn= 0
?
n≥0
γn= +∞,
and (χn)n≥1is an (Fn)n≥0–adapted sequence of positive random variables. It
is important to note that the step sequence ˜ γ is (Fn)n≥0–predictable.
Now we introduce the weighted empirical measures like Lamberton and Pag` es
in [5]. Given a sequence η = (ηn)n≥1of positive numbers satisfying?
n≥1ηn=
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+∞, we denote by νη
nthe random probability measure on Rddefined by
νη
n=
1
Hn
n
?
k=1
ηkδXk−1, withHn=
n
?
k=1
ηk.
Throughout the paper, |.| denotes the Euclidean norm and ?.? denotes the
natural matrix norm induced by |.| i.e. for every square matrix A, ?A? =
sup|x|=1|Ax|. The letter C is used to denote a positive constant, which may
vary from line to line.
3 Preliminary results
In this section, we prove results which are the keys to study the Euler scheme
with predictable random step defined in the introduction. Proposition 3 con-
tains two results: the first one (11) provides a substitute for the Lp–bounded-
ness of (V (Xn))n≥0used in [5]. The second one (12) is a new consequence of
the stability condition (10) and is used to prove the fundamental proposition
5 which ensures the existence of an almost surely finite time n1such that for
every n ≥ n1, ˜ γn= γn.
Proposition 3 Let W be a nonnegative function and (˜ γn)n≥0be a (Fn)n≥0–
predictable sequence of positive and finite random variables satisfying: there
exist α > 0, β > 0, n0∈ N, such that
?
˜ γn+1
∀n ≥ n0,
E
W(Xn+1)|Fn
?
− W(Xn)
≤ −αW(Xn) + β. (10)
Suppose (θn)n≥1is a positive nonincreasing sequence such that E
is finite, then
?
n≥n0+1
If, in addition, limnθn= 0 then
??
n≥0θn˜ γn
?
E
?
θn˜ γnW(Xn−1)|Fn0
?
< +∞.(11)
lim
nθnW(Xn) = 0 a.s.(12)
The above proposition is related to Robbins-Siegmund’s theorem (see Theorem
1.3.12 in [1] and the references therein).
Proof. Let Rn=
n
?
k=0
θk˜ γkand R∞= lim
nRn∈ L1(P).
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We consider the sequence (Zn)n≥n0defined by
∀n ≥ n0, Zn+1= Zn+ θn+1
We first prove that for every n ≥ n0, Zn≥ 0. Indeed, an Abel transform yields
for every n ≥ n0,
n−1
?
n−1
?
?
W(Xn+1) − W(Xn)
?
,Zn0= θn0W(Xn0).
Zn=
k=n0
θk+1
?
W(Xk+1) − W(Xk)
?
+ θn0W(Xn0),
=
k=n0
?
θk− θk+1
?
W(Xk) + θnW(Xn).
The sequence (θn)n≥0 is nonincreasing and the function W is nonnegative,
then (Zn)n≥n0is positive.
Let (Sn)n≥n0denote the process defined for every n ≥ n0by
n
?
Sn= Zn+ α
k=n0+1
θk˜ γkW(Xk−1) + β
?
E
?
R∞|Fn
?
− Rn
?
.
Since E
as W satisfies (10) we have
?
R∞|Fn
?
− Rn≥ 0, the sequence (Sn)n≥n0is nonnegative. Moreover,
∀n ≥ n0,
E
?
Zn+1|Fn
?
≤ Zn− αθn+1˜ γn+1W(Xn) + βθn+1˜ γn+1.
Then it follows from this and from the Fn–measurability of ˜ γn+1that
∀n ≥ n0,
E
?
Sn+1|Fn
?
≤ Sn.
Thus (Sn)n≥0converges a.s. to a nonnegative finite random variable S∞, and
we have
E
?
?
n≥n0+1
θn˜ γnW(Xn−1)|Fn0
?
< +∞.
From the almost sure convergence of (Sn)n≥n0, we also deduce the almost sure
convergence of the series
?
n≥1
θn
?
W(Xn) − W(Xn−1)
?
.
Since (θn)n≥0is nonincreasing and converges to 0, Kronecker’s lemma implies
the almost sure convergence ofθnW(Xn)
??
n≥0to 0.
2
Remark 4 A substitute for the Lp-boundedness of (V (Xn))n≥0 has been al-
ready found in Lemma 4 of [6] but does not apply in our case. Indeed, we have
no information on the expectation of the random variable ˜ γn.
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haviour when γ0is equals to 2−4or 2−5. For bigger values of γ0, the rate of
convergence is poor but the Lorenz problem is a difficult numerical problem
and the parameter γ0is hard to fix. Other numerical methods have the same
problem. The important point to note here is that the scheme does not explode
(for any γ0) and appears convergent to the same limit.
Figure 2 gives a representation of the stochastic step sequence (˜ γn)n≥1when
γ0= 0.5. We show that the bigger n is and the less the stochastic part χ(Xn−1)
is used. For this path, after 20000 iterations we have ˜ γn= γn(at least until
n = 107). Moreover before the 20000th iteration the event {χ(Xn−1) < γn}
occurs only 924 times (4.62% of time).
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 5000 10000 1500020000
˜ γn
Fig. 2. Stochastic step sequence (˜ γn)n≥1.
We compare our results with the approximation of E
regular Euler scheme (figure 3(a)) or a implicit Euler scheme (figure 3(b)). We
represent the results only for h ≤ 2−6because for bigger values the empirical
expectation (based on 10000 paths) explodes whith the regular Euler scheme.
For the implicit scheme the expectation remains bounded but the behaviour
is very poor when h ≤ 2−8. The parameters h, T and the number of paths
used for the Monte-Carlo procedure are hard to fix.
?
f(¯ XT
h)
?
where (¯ XT
h) is a
For the implicit Euler scheme (figure 3(b)), the jump between h = 2−8and h =
2−9is due to the fact that the scheme remains trapped in the neighborhood of
only one attractor (Lorenz equation has two attractors). This behavior does
not occur with the regular Euler scheme and with our scheme.
7.2Perturbed Hamiltonian system
The second example is a perturbed Hamiltonian system derived from a multi-
dimensional linear oscillator under external random excitation. It is a 3-DOF
(degree of freedom) system studied by Ibrahim and Li (see [3]) and Soize (see
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760
780
800
820
840
860
880
900
012345
h = 2−6
h = 2−7
h = 2−8
h = 2−9
h = 2−10
(a) Euler scheme:¯ XT
k+1,h=¯ XT
k,h+ hb(¯ XT
k,h) +√hσUk+1
760
780
800
820
840
860
880
900
012345
h = 2−6
h = 2−7
h = 2−8
h = 2−9
h = 2−10
(b) Implicit Euler scheme:¯ XT
k+1,h=¯ XT
k,h+ hb(¯ XT
k+1,h) +√hσUk+1
Fig. 3. Computation of E?f(¯ XT
h)?by a Monte-Carlo procedure on 10000 paths and
for T = 5
[10] chap. XIII.6). We have thus the following equation in R6
?dqt= ∂pH(qt,pt)dt,
dyt= −∂qH(qt,pt)dt − f0D0∂pH(qt,pt)dt + g0S0dWt
where H(qt,pt) =1
2?M(q)−1p,p? +1
2?K0q,q? and
D0= S0=
1 0 0
0 0 0
0 0 0
,K0=
1.300
0 0.1540
000.196
,g0= 0.5,f0= 0.9965,
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and
M(q) =
1.3v2+ 0.3v3
v2+ 0.3 v2
2+ 2v2+ 0.314 v2v3+ v3+ 0.0375
v3
v2v3+ v3+ 0.0375v2
3+ 0.1
,
with v2(q) = −1.61(0.375q2+q3) and v3(q) = −1.61q2. This numerical example
is taken from [10] (page 257–264). This is the first damping model case with
the external excitation applied to DOF 1 and the system parameter equal to
0.7. In this case we have an analytic expression of the density of the invariant
measure and we can calculate the mean-square response for the DOF 1. We
consider the function f1: (q,p) ?→ q2
1and we have
?
R6f1(x)ν(dx) ≃ 0.0965.
The stochastic sequence used in the following simulations is defined by ˜ γ0= γ0
and ∀n ≥ 1
˜ γn=γ0n−1/3?
The results for different value of γ0are given in figure 4. The convergence seems
better when γ0is big. Our scheme behaves very well and a representation of
the stochastic step sequence (˜ γn)n≥1is given in figure 5.
?
∧
?
1
|b(Xn−1)|2∨ 1
?
.
0.06
0.07
0.08
0.09
0.1
0.11
0.12
02000004000006000008000001e+06
γ0= 2−1
γ0= 2−2
γ0= 2−3
γ0= 2−4
Fig. 4. One path of?νη
n(f)?
1≤n≤106for different value of γ0.
We do not represent the approximation of E
plicit Euler scheme because the empirical expectation (based on 10000 paths)
explodes for h < 2−3.
?
f(¯ XT
h)
?
where (¯ XT
h) is the ex-
Figure 6 gives results using the approximation of E
implicit Euler scheme.
?
f(¯ XT
h)
?
where (¯ XT
h) is the
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0
0.05
0.1
0.15
0.2
0.25
0.3
0200400600800100012001400
˜ γn
Fig. 5. One representation of the stochastic step sequence (˜ γn)n≥1.
0.06
0.07
0.08
0.09
0.1
0.11
0.12
012345
h = 2−3
h = 2−4
h = 2−5
h = 2−6
Fig. 6. Computation of E?f(¯ XT
h)?by a Monte-Carlo procedure on 10000 paths and
for T = 5 where¯ XT
his a implicit Euler scheme.
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