Filtering with Lagrangian Floaters in
Stochastic Navier Stokes Flow
Xin Thomson Tong
(Joint work with Ramon van Handel)
ORFE - Princeton
February 21, 2013
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
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Lagrangian Floaters
Stochastic Navier Stokes Equations (SNS):
Incompressible periodic velocity field ut ∈ H1 (T2 ):
dut = ν4ut + (u · ∇)u + ∇p dt + dWt ;
∇ · ut = 0.
Wt stands for some divergence free Gaussian process.
r floaters are dispatched into ut . Their locations xit follow the
following equation:
Z t
xit = xi0 +
us (xis )ds, i = 1, . . . , r.
0
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February 21, 2013
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Oceanography
Data that Oceanographers receive:
Yni = xitn + ξni ;
Oceanographers want to recover ut based on Yni .
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Floaters Ergodicity
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Hidden Markov Model
This model is a Hidden Markov Chain:
Zn = (utn , x1tn , . . . , xrtn ) is a Markov Chain;
Yni is a perturbed partial observation of Zn .
Want to recover Πn = P(Zn ∈ · |Y0 , . . . , Yn ).
···
/ Zn−1
···
Yn−1
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/ Zn
/ Zn+1
Yn
Floaters Ergodicity
Yn+1
/ ···
···
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Nonlinear Filtering
The evolution of Πn = P(Zn ∈ · |Y0 , . . . , Yn ):
Denote joint distribution of Z0 as µ, let Πµ0 ( · ) = µ(Z0 ∈ · |Y0 ) ;
Compute Πµn = Pµ (Zn ∈ · |Y0 , . . . , Yn ) using the recursive Bayes’
formula:
R
µ
µ
A Πn (dzn )p(zn , dzn+1 )g(zn+1 , Yn+1 )
R
.
Πn+1 (A) =
Πµn (dzn )p(zn , dzn+1 )g(zn+1 , Yn+1 )
We assume the transition kernel from Zn to Zn+1 is p(zn , dzn+1 );
and the transition kernel from Zn to Yn has density g(Zn , Yn ).
This is process is called nonlinear filtering or optimal filtering.
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
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Nonlinear Filtering
The evolution of Πn = P(Zn ∈ · |Y0 , . . . , Yn ):
Denote joint distribution of Z0 as µ, let Πµ0 ( · ) = µ(Z0 ∈ · |Y0 ) ;
Compute Πµn = Pµ (Zn ∈ · |Y0 , . . . , Yn ) using the recursive Bayes’
formula:
R
µ
µ
A Πn (dzn )p(zn , dzn+1 )g(zn+1 , Yn+1 )
R
.
Πn+1 (A) =
Πµn (dzn )p(zn , dzn+1 )g(zn+1 , Yn+1 )
We assume the transition kernel from Zn to Zn+1 is p(zn , dzn+1 );
and the transition kernel from Zn to Yn has density g(Zn , Yn ).
This is process is called nonlinear filtering or optimal filtering.
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
5 / 16
Significance in Practice
The selection of the prior distribution µ appears to be a problem,
We need to know the distribution of u0 , the velocity field at time
0, which is implausible;
We can plug in reasonable approximation;
How does the selection of µ impact the result?
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
6 / 16
Significance in Practice
The selection of the prior distribution µ appears to be a problem,
We need to know the distribution of u0 , the velocity field at time
0, which is implausible;
We can plug in reasonable approximation;
How does the selection of µ impact the result?
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
6 / 16
Significance in Practice
The selection of the prior distribution µ appears to be a problem,
We need to know the distribution of u0 , the velocity field at time
0, which is implausible;
We can plug in reasonable approximation;
How does the selection of µ impact the result?
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
6 / 16
Significance in Practice
The selection of the prior distribution µ appears to be a problem,
We need to know the distribution of u0 , the velocity field at time
0, which is implausible;
We can plug in reasonable approximation;
How does the selection of µ impact the result?
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
6 / 16
Experimental Convenience
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Filter Stability
Desired: nonlinear filtering is self correcting:
n→∞
|P µ (Zn ∈ · |Y0 , . . . , Yn ) − P ν (Zn ∈ · |Y0 , . . . , Yn )| −→ 0
a.s.
Intuition: when there is no observation:
n→∞
kP µ (Zn ∈ · ) − P ν (Zn ∈ · )kT V −→ 0.
Intuition: true if the unconditioned chain has ergodic behavior.
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
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Filter Stability
Desired: nonlinear filtering is self correcting:
n→∞
|P µ (Zn ∈ · |Y0 , . . . , Yn ) − P ν (Zn ∈ · |Y0 , . . . , Yn )| −→ 0
a.s.
Intuition: when there is no observation:
n→∞
kP µ (Zn ∈ · ) − P ν (Zn ∈ · )kT V −→ 0.
Intuition: true if the unconditioned chain has ergodic behavior.
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
8 / 16
Filter Stability
Desired: nonlinear filtering is self correcting:
n→∞
|P µ (Zn ∈ · |Y0 , . . . , Yn ) − P ν (Zn ∈ · |Y0 , . . . , Yn )| −→ 0
a.s.
Intuition: when there is no observation:
n→∞
kP µ (Zn ∈ · ) − P ν (Zn ∈ · )kT V −→ 0.
Intuition: true if the unconditioned chain has ergodic behavior.
Xin Tong (Princeton)
Floaters Ergodicity
February 21, 2013
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Conditional Ergodicity with Asymptotic Coupling:
Asymptotic coupling:
The space of Zn has a metric d;
Assume for any two initial condition z0 = (u0 , x10 , . . . , xr0 ),
z̃0 = (ũ0 , x̃10 , . . . , x̃r0 ), there is a coupling Q such
X
Q(
d(Zn , Z̃n )2 < ∞) > a > 0.
n
Theorem (van Handel and Tong, 2012)
If a Hinden Markov Chain has asymptotic coupling, assume also the
observation density g > 0, then for any d-Lip function f ,
|Πµn (f ) − Ππn (f )| → 0,
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Floaters Ergodicity
π-a.s.
February 21, 2013
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Coupling
Coupling is useful to show memoryless of initial condition. e.g.,
dZt = dWt ,
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with Z0 = 0 and 1.
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Coupling
Coupling is useful to show memoryless of initial condition. e.g.,
dZt = dWt ,
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with Z0 = 0 and 1.
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Asymptotic Coupling
For infinite dimensional systems, e.g. SNS, exact coupling is
impossible;
A weaker version, asymptotic coupling, can be achieved:
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Heuristic behind Asymptotic Coupling
At each time t, with prob 1 − q(zt ) the distance decreases to its
half;
The prob that you can repeat this coupling forever is about
1
0
[1 − q(z0 )][1 − q( z0 )] · · · ∼ O(e−q (0) )
2
It will be strictly positive if we can verify the differentiability of
the Kolmogorov transition density: ∇z P (t, z).
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February 21, 2013
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Known Facts
The case where the observations are done at fixed points has
asymptotic coupling. (M. Hairer, J.C. Mattingly and
M.Scheutzow. 2009)
We have verified the same mechanic works for single floater case;
Multiple floater remain unknown.
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Interesting Consequence of Asymptotic Coupling
Multiple floaters will separate no matter how close they are in the
beginning;
Fracture behavior of the flow map.
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February 21, 2013
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Thank You for Your Attention!
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February 21, 2013
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