6 th Workshop, Random Dynamical Systems Bielefeld, 1.11.2013 3DVAR for 2D-NS Dirk Bl¨ omker Accuracy and Stability Introduction of the Continuous-Time 3DVAR Filter Set-Up Navier-Stokes for 2D Navier-Stokes Equation 3DVAR noisy observer Numerics attractivity Dirk Bl¨ omker stability Forward accuracy stability joint work with: Pull-back transformation Andrew Stuart, Kody Law (Warwick) Birkhoff accuracy Konstantinos Zygalakis (Southhampton) stability Outlook other filter todo summary
Filtering 3DVAR for Methods used for high dimensional data assimilation 2D-NS Dirk Bl¨ omker problems (for example in weather forecasting) Introduction Widely applied, but not that well studied Set-Up Navier-Stokes in the nonlinear, stochastic & infinite dimensional setting. 3DVAR noisy observer Numerics attractivity stability Forward accuracy stability Pull-back transformation Birkhoff accuracy stability Outlook other filter todo summary
Filtering 3DVAR for Methods used for high dimensional data assimilation 2D-NS Dirk Bl¨ omker problems (for example in weather forecasting) Introduction Widely applied, but not that well studied Set-Up Navier-Stokes in the nonlinear, stochastic & infinite dimensional setting. 3DVAR noisy observer Numerics attractivity Basic Idea of Filtering stability Forward estimate time-evolution of a trajectory based on accuracy partial observation & knowledge of the model stability Pull-back use model to predict next step transformation Birkhoff use data to correct prediction accuracy stability Outlook Problem: other filter todo only partial and noisy observations/data summary
Our Approach: 3DVAR for 2D-NS Dirk Bl¨ omker Introduction As starting point simple 3DVAR-filter Set-Up Navier-Stokes (in this talk: not many details about filter) 3DVAR noisy observer Example for the underlying dynamical system: Numerics attractivity deterministic 2D-Navier-Stokes equation stability Forward Limit of high frequency noisy observations yields accuracy stability stochastic PDE (continuous time filters/noisy observer) Pull-back Study Accuracy & Stability = ⇒ Stochastic Dyn. Syst. transformation Birkhoff accuracy stability Outlook other filter todo summary
Accuracy & Stability 3DVAR for 2D-NS Dirk Bl¨ omker Introduction Set-Up Navier-Stokes 3DVAR noisy observer Numerics attractivity stability Forward accuracy stability Pull-back transformation Birkhoff accuracy stability Outlook other filter todo summary
Accuracy & Stability 3DVAR for 2D-NS Dirk Bl¨ omker Stability Introduction Trajectories from observer converge towards each other Set-Up Navier-Stokes 3DVAR ⇒ It does not matter where to initiate the filter noisy observer Numerics attractivity stability Forward accuracy stability Pull-back transformation Birkhoff accuracy stability Outlook other filter todo summary
Accuracy & Stability 3DVAR for 2D-NS Dirk Bl¨ omker Stability Introduction Trajectories from observer converge towards each other Set-Up Navier-Stokes 3DVAR ⇒ It does not matter where to initiate the filter noisy observer Numerics attractivity stability Accuracy Forward accuracy Trajectories from observer get close to the true trajectory stability (on the order of observational noise) Pull-back transformation Birkhoff accuracy ⇒ Filter gives the true answer stability Outlook (recover unknown solution from partial noisy observations) other filter todo summary
The Model 3DVAR for 2D-NS For simplicity: Dirk Bl¨ omker In this talk only an ODE instead of 2D Navier Stokes. Introduction Thus H = R n , n ≫ 1 very large. Set-Up Navier-Stokes 3DVAR noisy observer Numerics attractivity stability Forward accuracy stability Pull-back transformation Birkhoff accuracy stability Outlook other filter todo summary
The Model 3DVAR for 2D-NS For simplicity: Dirk Bl¨ omker In this talk only an ODE instead of 2D Navier Stokes. Introduction Thus H = R n , n ≫ 1 very large. Set-Up Navier-Stokes 3DVAR Let u : R �→ H be any bounded solution of noisy observer Numerics attractivity ∂ t u = − δ A u + B ( u, u ) + f stability Forward accuracy stability Pull-back transformation Birkhoff accuracy stability Outlook other filter todo summary
The Model 3DVAR for 2D-NS For simplicity: Dirk Bl¨ omker In this talk only an ODE instead of 2D Navier Stokes. Introduction Thus H = R n , n ≫ 1 very large. Set-Up Navier-Stokes 3DVAR Let u : R �→ H be any bounded solution of noisy observer Numerics attractivity ∂ t u = − δ A u + B ( u, u ) + f stability Forward accuracy stability A diagonal operator, A ≥ 1, δ > 0, Pull-back transformation B : H × H → H – symmetric bilinear map Birkhoff accuracy stability f deterministic forcing (could be time dependent) Outlook other filter todo summary
The Model 3DVAR for 2D-NS For simplicity: Dirk Bl¨ omker In this talk only an ODE instead of 2D Navier Stokes. Introduction Thus H = R n , n ≫ 1 very large. Set-Up Navier-Stokes 3DVAR Let u : R �→ H be any bounded solution of noisy observer Numerics attractivity ∂ t u = − δ A u + B ( u, u ) + f stability Forward accuracy stability A diagonal operator, A ≥ 1, δ > 0, Pull-back transformation B : H × H → H – symmetric bilinear map Birkhoff accuracy stability f deterministic forcing (could be time dependent) Outlook other filter todo trajectory u is the unknown we want to observe summary
Existence & Uniqueness (well known) 3DVAR for 2D-NS Dirk Bl¨ omker For 2D-Navier-Stokes see [Temam 95, 97], [Robinson 01]. Introduction Set-Up Navier-Stokes 3DVAR Theorem noisy observer Numerics Suppose �B ( u, u ) , u � ≤ 0 and f is bounded. attractivity stability Then for all initial conditions u (0) there exists a global Forward solution in C 1 ([0 , ∞ ) , H ). accuracy stability Pull-back Furthermore there is a global attractor in B R (0) ⊂ H transformation Birkhoff containing all bounded solutions. accuracy stability Outlook other filter todo summary
Very brief description of 3DVAR [Harvey 91, .... ] 3DVAR for 2D-NS Consider Dirk Bl¨ omker S h – one step in the model (of time h > 0) Introduction u j = u ( jh ) = S h ( u j − 1 ) – unknown true trajectory Set-Up Navier-Stokes y j = Pu j + N (0 , Γ) – observation (noisy & partial) 3DVAR noisy observer P – projection Numerics attractivity m j – estimation ˆ stability Forward accuracy stability Pull-back transformation Birkhoff accuracy stability Outlook other filter todo summary
Very brief description of 3DVAR [Harvey 91, .... ] 3DVAR for 2D-NS Consider Dirk Bl¨ omker S h – one step in the model (of time h > 0) Introduction u j = u ( jh ) = S h ( u j − 1 ) – unknown true trajectory Set-Up Navier-Stokes y j = Pu j + N (0 , Γ) – observation (noisy & partial) 3DVAR noisy observer P – projection Numerics attractivity m j – estimation ˆ stability Forward Prediction: m j +1 = S h ( ˆ m j ) accuracy stability Pull-back transformation Birkhoff accuracy stability Outlook other filter todo summary
Very brief description of 3DVAR [Harvey 91, .... ] 3DVAR for 2D-NS Consider Dirk Bl¨ omker S h – one step in the model (of time h > 0) Introduction u j = u ( jh ) = S h ( u j − 1 ) – unknown true trajectory Set-Up Navier-Stokes y j = Pu j + N (0 , Γ) – observation (noisy & partial) 3DVAR noisy observer P – projection Numerics attractivity m j – estimation ˆ stability Forward Prediction: m j +1 = S h ( ˆ m j ) accuracy stability u j | y 1 . . . y j ∼ N ( ˆ m j , C ) Pull-back Assume Gaussianity: transformation u j +1 | y 1 . . . y j ∼ N ( m j +1 , C ) Birkhoff accuracy stability Kalman mean update (Bayes’ rule + some work) Outlook other filter m j +1 = m j +1 + CP (Γ + PCP ) − 1 ( y j +1 − Pm j +1 ) ˆ todo summary
High Frequency Observation Limit 3DVAR for 2D-NS Limit ( h ↓ 0) of high frequency noisy observations for Dirk Bl¨ omker 3DVAR yields for sufficiently large observational noise a Introduction stochastic equation (noisy observer/continuous time filter) Set-Up Navier-Stokes 3DVAR noisy observer Key Point: Numerics attractivity The discrete time filter can be written as stability an Euler-Maruyama discretization of the observer Forward accuracy stability Pull-back The formal limit is true in a much more general setting transformation Birkhoff and for several filter (no rigorous result yet) accuracy stability Outlook other filter Discrete time case for 2D-Navier Stokes: [Law, Stuart, et. al. 11] todo summary
Noisy observer 3DVAR for 2D-NS Dirk Bl¨ omker m )+ f + ω A − 2 α P λ [ u − ˆ m + σ A − β ∂ t W ] ∂ t ˆ m = − δ A ˆ m + B ( ˆ m, ˆ Introduction P λ – proj. onto the observed low modes Set-Up Navier-Stokes (for 2D Navier-Stokes approximately λ 2 many) 3DVAR noisy observer Assume : Γ = 1 h σ 2 A − 2 β P λ covariance of the (given) Numerics attractivity observational noise (think of β = 0) stability Forward C = ωσ 2 A − 2( α + β ) how to weight data or the model accuracy stability W – standard cylindrical Wiener process Pull-back transformation (space-time white noise) Birkhoff accuracy stability λ and ω are free parameters of the filter (also C and α ) Outlook other filter todo summary
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