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 1 / 16 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 Xin Tong (Princeton) Floaters Ergodicity February 21, 2013 2 / 16 Oceanography Data that Oceanographers receive: Yni = xitn + ξni ; Oceanographers want to recover ut based on Yni . Xin Tong (Princeton) Floaters Ergodicity February 21, 2013 3 / 16 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 Xin Tong (Princeton) / Zn / Zn+1 Yn Floaters Ergodicity Yn+1 / ··· ··· February 21, 2013 4 / 16 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 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 Xin Tong (Princeton) Floaters Ergodicity February 21, 2013 7 / 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 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 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 8 / 16 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, Xin Tong (Princeton) Floaters Ergodicity π-a.s. February 21, 2013 9 / 16 Coupling Coupling is useful to show memoryless of initial condition. e.g., dZt = dWt , Xin Tong (Princeton) with Z0 = 0 and 1. Floaters Ergodicity February 21, 2013 10 / 16 Coupling Coupling is useful to show memoryless of initial condition. e.g., dZt = dWt , Xin Tong (Princeton) with Z0 = 0 and 1. Floaters Ergodicity February 21, 2013 11 / 16 Asymptotic Coupling For infinite dimensional systems, e.g. SNS, exact coupling is impossible; A weaker version, asymptotic coupling, can be achieved: Xin Tong (Princeton) Floaters Ergodicity February 21, 2013 12 / 16 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). Xin Tong (Princeton) Floaters Ergodicity February 21, 2013 13 / 16 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. Xin Tong (Princeton) Floaters Ergodicity February 21, 2013 14 / 16 Interesting Consequence of Asymptotic Coupling Multiple floaters will separate no matter how close they are in the beginning; Fracture behavior of the flow map. Xin Tong (Princeton) Floaters Ergodicity February 21, 2013 15 / 16 Thank You for Your Attention! Xin Tong (Princeton) Floaters Ergodicity February 21, 2013 16 / 16