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A micro/macro algorithm to accelerate MonteCarlo simulation of stochastic differential
equations
Kristian Debrabant
Scientific Computing Research Group,Katholieke Universiteit Leuven, Belgium
Innsbruck, October 29, 2010
1
Outline
1 Introduction
2 Accelerated Monte Carlo simulation
3 A convergence result
joint work with Giovanni Samaey (K.U. Leuven)
2
Introduction Accelerated Monte Carlo simulation A convergence result
Stochastic differential equations (SDEs)
dX (t) = g0(X (t)
)dt +
m∑l=1
gl(X (t)
)? dWl(t), X (t0) = X0
X (t) =X0 +
∫ t
t0g0(X (s)
)ds +
m∑
l=1
∫ t
t0gl(X (s)
)? dWl(s)
︸ ︷︷ ︸, t0 ≤ t ≤ T
0 t
W (t)lim
∆s→0
∑j
gl
(X(ξj )
)(Wl (sj+1)−Wl (sj )
)
ξj =sj : Itô-integral,∫ tt0
gl
(X(s))
dWl (s)
ξj =12 (sj +sj+1): Stratonovich-integral,∫ t
t0gl
(X(s))◦dWl (s)
W (t): standard Wiener-processW (0) = 0 a. s.W (t2)−W (t1) ∼ N(0, t2 − t1) for 0 ≤ t1 < t2 ≤ TW (t2)−W (t1) and W (t4)−W (t3) are independent for0 ≤ t1 < t2 ≤ t3 < t4 ≤ T
4
Introduction Accelerated Monte Carlo simulation A convergence result
Model problem: Immersed polymers
dX (t) =
[κ(t) X (t)− 1
2WeF(X (t)
)]dt +
1√We
dW (t)
X - polymer’s length vectorκ(t) - fluid’s velocity gradientF (X ) - entropic force, here:finitely extensible nonlinearlyelastic (FENE),
F (X ) =X
1− ‖X‖2/γ
Function of interest: non-Newtonian stress tensor
τp(t) =ε
We
(E(
X (t)⊗ F(X (t)
))− 1l
)
5
Introduction Accelerated Monte Carlo simulation A convergence result
Euler-Maruyama method
Aim: discrete approximation Y ∆t =(Y ∆t (t)
)t∈I∆t on
I∆t = {t0, t1, . . . , tN}, t0 < t1 < · · · < tN ≤ T ,
such that Yn = Y ∆t (tn) ≈ X (tn)
X (tn+1) = X (tn) +
∫ tn+1
tng0(X (s)
)ds +
m∑
l=1
∫ tn+1
tngl(X (s)
)dWl(s)
Yn+1 = Yn + g0(Yn)
∫ tn+1
tnds +
m∑
l=1
gl(Yn)
∫ tn+1
tndWl(s)
= Yn + g0(Yn) (tn+1 − tn)︸ ︷︷ ︸=∆nt
+m∑
l=1
gl(Yn)(Wl(tn+1)−Wl(tn)
)︸ ︷︷ ︸
=∆nWl
6
Introduction Accelerated Monte Carlo simulation A convergence result
Convergence
0 0.2 0.4 0.6 0.80
1
2
3
4
5
t
individual paths
expectation
Strong convergence:
maxt∈I∆t
E ‖Y ∆t (t)−X (t)‖ ≤ C∆tp
Weak convergence:∀f ∈ C2(p+1)
P (IRd , IR)
maxt∈I∆t|E(
f(Y ∆t (t)
)−f(X (t)
))| ≤ Cf ∆tp
Euler-Maruyama: Strong concergence order 0.5, weakconvergence order 1
7
Introduction Accelerated Monte Carlo simulation A convergence result
Discretization of the model problem
E. g. Euler-Maruyama scheme:
Yk+1 = Yk +
[κ(tk ) Yk −
12We
F(Yk)]
∆t +1√We
∆kW
Accept-reject strategy (e.g. Öttinger):
‖Yk+1‖ >√
(1−√
∆t)γ ⇒ reject Yk+1 and try again
⇒ ∆t has to be chosen very small
8
Introduction Accelerated Monte Carlo simulation A convergence result
Idea of accelerated Monte Carlo simulation
t∗ t∗ + Kδt t∗ + ∆t
1. simulate
4. project
5. simulate
t
Y(t,ω
)
t∗ t∗ + Kδt t∗ + ∆t
2.restrict
3. extrapolate
4.pr
ojec
t
t
Mac
rosc
opic
stat
es
10
Introduction Accelerated Monte Carlo simulation A convergence result
Simulation and restriction
SimulationDo K ≥ 1 microsteps with one-step method ϕ: For k = 1, . . . ,K
Y (j)(t? + kδt) = ϕ(t? + (k − 1)δt ,Y (j)(t? + (k − 1)δt); δt)
Restriction
Map ensemble Y = (Y (j))Jj=1 to a number L of (macroscopic)
state variables U = (Ul)Ll=1,
U(t) = R(Y(t)
), with Rl
(Y(t)
)=
1J
J∑
j=1
ul(Y (j)(t)
).
Example: ul(x) = x l yields standard empirical moments of thedistribution in a one-dimensional setting.
11
Introduction Accelerated Monte Carlo simulation A convergence result
Extrapolation and Projection
Extrapolation
U(t? + ∆t) =K∑
k=0
lkU(t? + kδt)
Simplest form: linear extrapolation,
U(t? + ∆t) =
U(t? + K δt) + (∆t − K δt)U(t? + K δt)− U
(t? + (K − 1)δt
)
δt
ProjectionE. g. by
Y(t? + ∆t) = argminZ: R(Z)=U(t?+∆t)
‖Z − Y(t? + K δt)‖2
13
Introduction Accelerated Monte Carlo simulation A convergence result
Projection
Corresponding Lagrange equations:
Y(t? + ∆t) = Y(t? + K δt) +L∑
l=1
λl∇YRl(Y(t? + ∆t)
),
with Λ = {λl}Ll=1 such that R(Y(t? + ∆t)
)= U(t? + ∆t)
⇒ Expensive. Cheap alternative:
Y(t? + ∆t) = Y(t? + K δt) +L∑
l=1
λl∇YRl(Y(t? + K δt)
),
with Λ = {λl}Ll=1 such that R(Y(t? + ∆t)
)= U(t? + ∆t)
14
Introduction Accelerated Monte Carlo simulation A convergence result
Projection
Lemma (Conditions for local solvability)For standard empirical moments Ul :
det(Ui+j−2
)i,j=1,...,L 6= 0
Neglecting statistical error:
det(
E X i+j−2)
i,j=1,...,L= 0
only possible if pdf has finite support
15
Introduction Accelerated Monte Carlo simulation A convergence result
Projection - numerical results for FENE dumbbells I
0
0.2
0.4
0.6
ϕ(x)
0 2 4 6
x
ϕ[2]
ϕ[5]
ϕ[8]
ϕ∗
ϕ−
10−16
10−12
10−8
10−4
100
(Ul−U
∗ l)/U
∗ l0 5 10 15 20
l
L = 2
L = 5
L = 8
L = 10
1d, κ = 2, γ = 49, δt = 2 · 10−4, J = 105, t− = 1, t∗ = 1.15
16
Introduction Accelerated Monte Carlo simulation A convergence result
Projection - numerical results for FENE dumbbells II
10−5
10−4
10−3
10−2
rel.errorin
τ p
0.001 0.01
∆t
L = 3
L = 4
L = 5
O(∆t)
1d, κ = 2, γ = 49, δt = 2 · 10−4, J = 105, t− = 1.5, 100 realizations
17
Introduction Accelerated Monte Carlo simulation A convergence result
Numerical example: FENE dumbbells
0
10
20
Eτ p(t)−Eτ p(t)
0 1 2 3 4 5 6
t
0
200
400
Eτ p(t),Eτ p(t)
0
1
2
3
4
Stdev(τ
p(t))
0 1 2 3 4 5 6
t
L = 2
L = 3
L = 4
reference
1d, κ(t) = 2 ·(1.1 + sin(πt)
), γ = 49, δt = 2 · 10−4, ∆t = 1 · 10−3, J = 5000, 500
realizations
18
Introduction Accelerated Monte Carlo simulation A convergence result
Idealized restriction and projection operators
limit J →∞restriction:
U(t) = R(Y (t)
), with Rl
(Y (t)
)= E ul
(Y (j)(t)
).
projection:Y ∗ = P(Y ,U∗) with R(Y ∗) = U∗.
Self-consistency: Y = P(Y , R(Y )
)
Sequences of projection and restriction operators
U[L] = (Ul)Ll=1
Corresponding projection and restriction operators: P [L] andR[L].
20
Introduction Accelerated Monte Carlo simulation A convergence result
Properties of projection operators
Uniform continuity in U∗:
|E g(P [L](Z ,U∗[L])
)− E g
(P [L](Z ,U
+[L]))| ≤ Cg‖U∗[L] − U+
[L]‖
Consistency:
|E g(P [L](Z ∗,U[L])
)− E g
(P [L](Z +,U[L])
)|
≤ Cg,L|E g(Z ∗)− E g(Z +)|with Cg,L → 0 for L→∞
LemmaFor normally distributed random variables and sequences(
U∗[L]
)L=1,2,...
of (centralized) moment values consistent with
normal distributions, the mentioned projection operators arecontinuous and consistent.
21
Introduction Accelerated Monte Carlo simulation A convergence result
Convergence
TheoremSuppose the following conditions hold:
(i) The SDE-coefficients are sufficiently smooth.
(ii) The one step method ϕ is weakly consistent of order pϕ.(iii) The sequence of (self-consistent) projection operators is
continuous and consistent for the numerical approximationprocess.
(iv) The extrapolation is consistent of order pe ≥ 1.Then for all t ∈ I∆t , all L ≥ L0, and all ∆t ∈ [0,∆t0]
|E f(Y[L](t)
)− E f
(X (t)
)| ≤ CL + CL(∆t)min{pe,pϕ}
with CL → 0 for L→∞.
22
Introduction Accelerated Monte Carlo simulation A convergence result
Conclusion
acceleration technique for Monte-Carlo simulationconvergence in the absence of statistical errorfor more details and references, see arXiv:1009.3767
Open problems:prove consistency and convergence of projection step forgeneral random variables,study stability and propagation of statistical error,study possibilities for variance reduction,construct an efficient adaptive error control, controlling thenumber of moments to extrapolate, the microscopic andmacroscopic time step, and the number of SDE realizations,couple FENE-SDE to Navier-Stokes equations
Thank you very much for your attention!23