Probabilistic representations and numerical methods for the

Probabilistic representations and numerical methods for the
Navier-Stokes equations
´ Mardones1
Hernan
´ TOSCA, Nancy
Journee
5 June 2014
1
´ Chile
PhD student in Mathematical Engineering at University of Concepcion,
Context
Forward-backward stochastic differential systems
Context
Incompressible Navier-Stokes equations
Burgers equation
Probabilistic representations
Forward-backward stochastic differential systems
FBSDS associated with Navier-Stokes equations
Approximation of the Navier-Stokes equations
Numerical method
Algorithm
Burgers equation
Taylor-Green Vortex
Numerical method
Context
Forward-backward stochastic differential systems
Context
Incompressible Navier-Stokes equations
Burgers equation
Probabilistic representations
Forward-backward stochastic differential systems
FBSDS associated with Navier-Stokes equations
Approximation of the Navier-Stokes equations
Numerical method
Algorithm
Burgers equation
Taylor-Green Vortex
Numerical method
Context
Forward-backward stochastic differential systems
Numerical method
Incompressible Navier-Stokes equations
Consider the following Navier-Stokes equations for the velocity field
u : (T0 , T ] × Rd → Rd of an incompressible, viscous fluid:

 ∂ u + ν ∆u + (u · ∇)u + ∇p + f = 0, t ≤ T ;
t
2
 ∇ · u = 0, u ( T ) = G .
A terminal condition G is given at T > 0, ν > 0 is the kinematic viscosity, p is the
pressure field, and f is the external force field.
• Navier-Stokes equations were introduced by Navier (1822) [26] and Stokes
(1849) [30] (see e.g. Chorin and Marsden (1979) [9] for details or Fefferman
(2000) [18] for open problems).
(1)
Context
Forward-backward stochastic differential systems
Numerical method
Burgers equation
The d-dimensional Burgers equation
�
∂t v + ν2 ∆v + (v · ∇)v + f = 0, t ≤ T ;
v (T ) = G ,
(2)
is associated to the following coupled forward-backward stochastic differential
equation (FBSDE):





dXs (t , x ) = Ys (t , x ) ds +
Xt (t , x ) = x ;
√
ν dWs ,
s ∈ [t , T ];
√

−dYs (t , x ) = f (s, Xs (t , x )) ds − νZs (t , x ) dWs ;



YT (t , x ) = G(XT (t , x )).
(3)
They are related by:
Ys (t , x ) = v (s, Xs (t , x )), Zs (t , x ) = ∇v (s, Xs (t , x )),
(see e.g. Ma, Protter and Yong (1994) [23]).
s ∈ [ t , T ] × Rd
(4)
Context
Forward-backward stochastic differential systems
Numerical method
In the context of the Navier-Stokes equations (1), supposing a divergence free f we
have the Poisson equation
∆p = −div div (u ⊗ u )
(see e.g. Chorin (1967) [7], Majda and Bertozzi (2002) [24]). Then (1) becomes:
�
∂t u + ν2 ∆u + (u · ∇)u + ∇(−∆)−1 div div (u ⊗ u ) + f = 0, t ≤ T ;
u (T ) = G .
(5)
Context
Forward-backward stochastic differential systems
Numerical method
Probabilistic representations
• Representing solutions of PDEs as the expected functionals of stochastic
processes, probabilistic approaches are an open field of development for the
study of deterministic models (see [1, 3, 20, 25]).
• The Vortex method considers the Navier-Stokes equations in the vorticity form
(see Chorin (1973) [8]).
• The Fourier transform method interpreters the velocity field in a Fourier space in
terms of a backward branching process and a composition rule along the
associated tree (see Le Jan and Sznitman (1997) [21]).
• The Lagrangian flows method gives probabilistic Lagrangian representations for
the velocity field (see Constantin and Iyer (2008) [10]).
Context
Forward-backward stochastic differential systems
Numerical method
• Forward-backward stochastic differential systems (FBSDS) associated to the
Navies-Stokes equations is a very recent approach (see Cruzeiro and
Shamarova (2009) [14], Delbaen, Qiu and Tang (2014) [17]).
• One common setback in all the above methods is the inability to deal with
boundary conditions. To this context, see Bouchard and Menozzi (2009) [4] and
Constantin and Iyer (2011) [11].
• Numerical implementation of probabilistic approaches is a recent field of
´ Rodr´ıguez-Rozas and Spigler (2010) [2], Chorin
research (see e.g. Acebron,
`
(1973) [8], Henry-Labordere,
Tan and Touzi (2014) [20], Ramirez (2006) [27]).
Context
Forward-backward stochastic differential systems
Context
Incompressible Navier-Stokes equations
Burgers equation
Probabilistic representations
Forward-backward stochastic differential systems
FBSDS associated with Navier-Stokes equations
Approximation of the Navier-Stokes equations
Numerical method
Algorithm
Burgers equation
Taylor-Green Vortex
Numerical method
Context
Forward-backward stochastic differential systems
Numerical method
Forward-backward stochastic differential systems
• FBSDEs are well-known to be connected to systems of PDEs (see e.g. Cheridito
et al. (2007) [6]).
• For the numerical approximation theory of FBSDEs, we refer to Delarue and
Menozzi [15, 16] and references therein.
• Because the estimation of conditional expectations, variance reduction
` (2010)
techniques reduce the computational effort (see e.g. Corlay and Pages
[12], Lejay and Reutenauer (2012) [22] for general theory).
• Recently, Henry-Labordere,
`
Tan and Touzi (2014) [20] introduced a branching
process approach to solve decoupled FBSDEs associated to semi-linear
parabolic PDEs.
Context
Forward-backward stochastic differential systems
Numerical method
FBSDS associated with Navier-Stokes equations
Delbaen, Qiu and Tang (2014) [17] associate the Navier-Stokes equations (1) to the
following coupled FBSDS:
√

dXs (t , x ) = Ys (t , x ) ds + ν dWs , s ∈ [t , T ];





Xt (t , x ) = x ;


�
�

√


�0 (s, Xs (t , x )) ds − νZs (t , x ) dWs ;

−
dYs (t , x ) = f (s, Xs (t , x )) + Y





YT (t , x ) = G(XT (t , x ));
�
��
�
d


j
j

�s (t , x ) = ∑ 27 Yti Ytj (t , x + Bs ) B i2s − B is

−d Y
Bs − B 2s B s ds

3

3
3
3
2s
3

i ,j = 1






− Z�s (t , x )dBs , s ∈ (0, ∞);



�∞ (t , x ) = 0.
Y
(6)
• This formulation is new in the literature because both BSDEs in FBSDS (6) are
defined on two different time-horizons [t , T ] and (0, ∞).
Context
Forward-backward stochastic differential systems
Numerical method
Approximation of the Navier-Stokes equations
By truncating the infinite time interval, the class of FBSDSs
√

dXs (t , x ) = Ys (t , x ) ds + ν dWs , s ∈ [t , T ];





Xt (t , x ) = x ;


�
�

√


�0 (s, Xs (t , x )) ds − νZs (t , x ) dWs ;

−dYs (t , x ) = f (s, Xs (t , x )) + Y





YT (t , x ) = G(XT (t , x ));
�
��
�
d


j
j

�s (t , x ) = ∑ 27 Yti Ytj (t , x + Bs ) B i2s − B is

−
d
Y
B
−
B
B s I� 1 � (s) ds

2s
s

3
,N
3
3
2s3
3

N
i
,
j
=
1






− Z�s (t , x )dBs , s ∈ (0, ∞);



�∞ (t , x ) = 0,
Y
(7)
is associated to the PDEs
�
∂t u N + ν2 ∆u N + (u N · ∇)u N + PN (u N ⊗ u N ) + f = 0, T0 < t ≤ T ;
u N (T ) = G .
�0 (t , x ) := limε→0 E Y
�ε (t , x ) and
Here Y
�
�
�1 −Y
�N , ∀N ∈ (1, ∞).
PN ( u N ⊗ u N ) = E Y
N
(8)
Context
Forward-backward stochastic differential systems
Numerical method
By the representation Ys (t , ·) = Ys (s, Xs (t , ·)), the solution u N : (T0 , T ] × Rd → Rd
of PDE (8) is given by
u N (t , x ) := Yt (t , x ).
(9)
Then, u N is used to approximate the velocity field u of Navier-Stokes equations (1).
Moreover, there exists a constant C independent of N such that
�
�
�
�
�u − u N �
C (t ,T ;C k ,α )
(see [17] for details).
≤
C
α
N4
(10)
Context
Forward-backward stochastic differential systems
Context
Incompressible Navier-Stokes equations
Burgers equation
Probabilistic representations
Forward-backward stochastic differential systems
FBSDS associated with Navier-Stokes equations
Approximation of the Navier-Stokes equations
Numerical method
Algorithm
Burgers equation
Taylor-Green Vortex
Numerical method
Context
Forward-backward stochastic differential systems
Numerical method
Algorithm
It is possible to approximate numerically the strong solution of Navier-Stokes
equations (1) by solving the PDE (8). For this purpose, FBSDS (7) is rewritten into the
following form
√

dXs (t , x ) = Ys (s, Xs (t , x )) ds + ν dWs , s ∈ [t , T ];





Xt ( t , x ) = x ;



�
�

√

 −dYs (s, Xs (t , x )) = f (s, Xs (t , x )) + PN (Ys ⊗ Ys )(s, Xs (t , x )) ds − νZs (t , x ) dWs ;


YT (T , x ) = G(x );



�

d


N


 P (Ys ⊗ Ys )(s, x ) = ∑ E
i ,j =1
N
3
1
3N
3
2r 3
j
j
¯r + B
˜r + B
ˆ r )B
¯ri B
˜r B
ˆr dr ,
Ysi Ys (s, x + B
¯ B
˜ and B
ˆ are three independent d-dimensional Brownian motions.
where B,
(11)
Context
Forward-backward stochastic differential systems
Numerical method
In the spirit of Delarue and Menozzi [15, 16], it’s defined the following algorithm:
∀ x ∈ Rd , u¯N (T , x ) = G(x ),
˜ − 1] ∩ Z, ∀ x ∈ Ξ,
∀ k ∈ [ 0, N
√
J (tk , x ) = u¯N (tk +1 , x )h + ν∆Wtk ,
d
P N (tk , x ) = ∑ E
i ,j = 1
�
N
3
1
3N
3
2r 3
¯r + B
˜r + B
ˆ r )B
¯ri B
˜rj B
ˆr dr ,
(¯
u N )i (¯
u N ) j ( tk + 1 , x + B
�
�
¯N (tk , x ) = E u¯N (tk +1 , x + J (tk , x )) + h f (tk , x ) + P N (tk , x ) ,
u
˜ − 1] ∩ Z) and Ξ = δZd is the infinite Cartesian grid of
where h = T˜ , tk = kh (k ∈ [0, N
N
step δ > 0.
Context
Forward-backward stochastic differential systems
Numerical method
• Approximation of SDEs.
• Projection mapping onto the grid.
• Estimation of the expectations (Monte-Carlo, Quantization, variance reduction).
• The approximations for the diffusion coefficient of the BSDEs are not considered.
• In the context of the Burgers equation, the algorithm converges strongly in terms
¨
of the Holder
continuity of the parameters. It involves the optimality of the
Gaussian quantization (see Delarue and Menozzi (2008) [16]).
• Numerical estimation of the Poisson equation.
• Convergence of the algorithm to the incompressible Navier-Stokes equations.
Context
Forward-backward stochastic differential systems
Numerical method
Burgers equation
We study the 2-dimensional Burgers equation (Delarue and Menozzi (2008) [16]):



∂t u (x , t ) − (u .∇x u ) (x , t ) +
ε2
u ( T , x ) = H ( x ) , x ∈ R2 ,
2
∆u (x , t ) = 0, (t , x ) ∈ [0, T [ × R2 , ε > 0;
(12)
with ((u ∇x u ))i = u ∇x ui for i = 1, 2. The explicit solution is known when H = ∇H0 ,
where H0 is a real-valued funtion.
We choose a spatially periodic H0 (x ) = ∏i =1,2 sin2 (πxi ), T = 3/8, h = 2.5 × 10−2 ,
δ = 0.01 and ε2 = 0.4.
Context
Forward-backward stochastic differential systems
Numerical method
4
3
u 1 (T , x, y)
2
1
0
−1
−2
−3
−4
1
0.9
0.8
1
0.7
0.9
0.6
0.8
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
y
0
x
Figure: Terminal condition u1 (T , ·, ·).
Context
Forward-backward stochastic differential systems
Numerical method
4
3
u 2 (T , x, y)
2
1
0
−1
−2
−3
−4
1
0.9
0.8
1
0.7
0.9
0.6
0.8
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
0
y
x
Figure: Condition u2 (T , ·, ·).
Context
Forward-backward stochastic differential systems
Numerical method
1
0.9
0.8
0.7
y
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
Figure: 2-dimensional velocity field u (T , ·, ·).
0.8
0.9
1
Context
Forward-backward stochastic differential systems
Numerical method
0.08
0.06
u 1 (0, x, y)
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
1
0.9
0.8
1
0.7
0.9
0.6
0.8
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
y
0
x
` and
Figure: Reference u1 (0, ·, ·) via quantization with M = 600 points (see Corlay, Pages
Printems (2005) [13]).
Context
Forward-backward stochastic differential systems
Numerical method
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1
0.9
0.8
1
0.7
0.9
0.6
0.8
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
y
0
x
Figure: Absolute point-wise error for u1 (0, ·, ·) using M = 150 points for quantization.
Context
Forward-backward stochastic differential systems
Numerical method
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1
0.9
0.8
1
0.7
0.9
0.6
0.8
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
y
0
x
Figure: Absolute point-wise error for u1 (0, ·, ·) using Quantization (M = 4) as a control variate
variable to the Monte-Carlo estimations (4 realizations).
Context
Forward-backward stochastic differential systems
Numerical method
Taylor-Green Vortex
Finally, we study the incompressible Navier-Stokes equations

 ∂ u − ν ∆u + (u · ∇)u + ∇p = 0, t ≤ T ;
t
2
 ∇ · u = 0, u (0) = u .
0
In two dimensions, the Taylor-Green vortex has the exact solution
u1 (x , t ) = − exp (−νt ) cos (x1 ) sin (x2 ) ,
u2 (x , t ) = exp (−νt ) sin (x1 ) cos (x2 ) ,
1
p (x , t ) = − exp (−2νt ) (cos (2x1 ) + cos (2x2 )) ,
4
for x ∈ [0, 2π]2 (see e.g. Canuto et al. (2007) [5]).
(13)
Context
Forward-backward stochastic differential systems
Numerical method
1
0.8
u 1 (0, x, y)
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
7
6
7
5
6
4
5
3
4
3
2
2
1
1
0
0
y
x
Figure: u1 (0, ·, ·) for ν = 0.1.
Context
Forward-backward stochastic differential systems
Numerical method
0.8
0.6
u 1 (T , x, y)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
7
6
7
5
6
4
5
3
4
3
2
2
1
1
0
y
0
x
Figure: Reference u1 (T , ·, ·) at time T = 3 for ν = 0.1.
Context
Forward-backward stochastic differential systems
Numerical method
1
u 1 (T , x, y)
0.5
0
−0.5
−1
−1.5
7
6
7
5
6
4
5
3
4
3
2
2
1
1
0
y
0
x
Figure: Estimation u1 (T , ·, ·) at time T = 3 for ν = 0.1 (time-step h = 0.02, spatial discretization
δ = 2π/40, N = 6, NR = 18 subintervals and M = 4 quantization points).
Context
Forward-backward stochastic differential systems
Numerical method
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Context
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Numerical method

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