Filtering Sparse Regular Observed Linear and Nonlinear Turbulent System

Transcription

1 Filtering Sparse Regular Observed Linear and Nonlinear Turbulent System John Harlim, Andrew J. Majda Department of Mathematics and Center for Atmosphere Ocean Science Courant Institute of Mathematical Sciences, New York University July 22, 28

2 What is filtering (data assimilation)? A predictor-corrector method that includes observations (via Bayesian update) to improve the real time prediction.

3 Difficulties in Real Time Filtering and Prediction of Turbulent Signals from Partial Observations 1. For turbulent signals from nature with many scales, even with mesh refinement the model has inaccuracies from parametrization, under-resolution, etc. How to overcome them? Can judicious model error help filtering?

4 Difficulties in Real Time Filtering and Prediction of Turbulent Signals from Partial Observations 1. For turbulent signals from nature with many scales, even with mesh refinement the model has inaccuracies from parametrization, under-resolution, etc. How to overcome them? Can judicious model error help filtering? 2. Computational efficiency: how big of ensemble is needed in representing the uncertainty of billions of variables?

5 Difficulties in Real Time Filtering and Prediction of Turbulent Signals from Partial Observations 1. For turbulent signals from nature with many scales, even with mesh refinement the model has inaccuracies from parametrization, under-resolution, etc. How to overcome them? Can judicious model error help filtering? 2. Computational efficiency: how big of ensemble is needed in representing the uncertainty of billions of variables? 3. The most accurate ensemble filters is not immune from catastrophic filter divergence (beyond machine infinity).

6 Goal: Provide math guidelines and new numerical strategies thru modern applied math paradigm Modelling Turbulent Signals Filtering Stochastic Langevin Models Complex Nonlinear Dynamical Systems Extended Kalman Filter Classical Criteria: Observability Controllability Numerical Analysis Classical Von-Neumann stability analysis for frozen coef!cient linear systems

7 Filtering Linear Problem Real Space Fourier Space Simplest Tur6ule"t Model Constant Coef!cient Linear Stochastic PDE FT!"depe"de"t Fourier -oe. cie"t Langevin equation Classical Kalman Filter Innovative Strategy Fourier Domain Kalman Filter 8ois9 :6ser;atio"s FT Fourier -oe. cie"ts o. t=e "ois9 o6ser;atio"s

8 How to deal with Sparse Regularly Spaced Observations? ALIASING!! 1.8 m=25, l=3, k=2, N=5 sin(25x) sin(3x) sin(3x j )=sin(25x j ).6.4.2!.2!.4!.6!.8!

9 Example: 123 grid pts (61 modes) but only 41 observations (2 modes) available Physical Space sparse observations for P=3 Fourier Space aliasing set!(1) = {1,-4,42} for P=3 and M= aliasing set!(11) = {11,-3,52} for P=3 and M=2

10 Example: Stochastically forced advection-diffusion equation u(x, t) t = 2 u(x, t) µ x x 2 u(x, t)+ F (x, t)+σ(x)ẇ (t), x 2π

11 Example: Stochastically forced advection-diffusion equation u(x, t) t = 2 u(x, t) µ x x 2 u(x, t)+ F (x, t)+σ(x)ẇ (t), x 2π Fourier Domain Kalman Filter (FDKF) dû k (t) = [( µk 2 ik)û k (t) + ˆF k (t)]dt + σ k dw k (t), FDKF : v l (t) = u ki (t) + ηl o (t), k i A(l)

12 Example: Stochastically forced advection-diffusion equation u(x, t) t = 2 u(x, t) µ x x 2 u(x, t)+ F (x, t)+σ(x)ẇ (t), x 2π Fourier Domain Kalman Filter (FDKF) dû k (t) = [( µk 2 ik)û k (t) + ˆF k (t)]dt + σ k dw k (t), FDKF : v l (t) = u ki (t) + ηl o (t), k i A(l) Reduced Fourier Domain Kalman Filter (RFDKF) RFDKF : v l (t) = u l (t) + η o l (t), where η o l (t) N (, r o /2M + 1), k N, l M.

13 Example: Stochastically forced advection-diffusion equation u(x, t) t = 2 u(x, t) µ x x 2 u(x, t)+ F (x, t)+σ(x)ẇ (t), x 2π Fourier Domain Kalman Filter (FDKF) dû k (t) = [( µk 2 ik)û k (t) + ˆF k (t)]dt + σ k dw k (t), FDKF : v l (t) = u ki (t) + ηl o (t), k i A(l) Reduced Fourier Domain Kalman Filter (RFDKF) RFDKF : v l (t) = u l (t) + η o l (t), where η o l (t) N (, r o /2M + 1), k N, l M. Strongly Damped Approximate Filter (SDAF, VSDAF): Observation is modeled as in FDKF but we implement it with dynamic-less unresolved modes. e µk2 1 t = O(1), e µk2 i t = O(ɛ) 1, 2 i P.

14 Skill of cheaper approximate filters with model error depends on observation time at given wave number and how rough spectrum is for truth signal Decorrelation time vs observation time T corr = 1/! k =1/(µ k 2 ) "t 1 =.1 "t 2 =.1 Decorrelation time 1 1!1 1!2 1!3 1! wave number k

15 ETKF Filter Divergence (K = 15, r = 4%), observability is violated Extreme event, t 2 =.1, E k = k 5/3 RMS ETKF, <corr>= filtered unfiltered obs u(x,1!t) ETKF, at T=1!t, corr=.7692 true unfiltered filtered obs time!5! model space 25 ETKF, at T=5!t, corr= ETKF, at T=1!t, corr=.913 u(x,5!t) !5! u(x,1!t) 2 1!1!

16 SDAF high skill (observability is satisfied) Spontaneous development of extreme event for t 2 =.1 and E k = k 5/3 RMS SDAF, <corr>= filtered unfiltered obs u(x,1!t) SDAF, at T=1!t, corr= true unfiltered filtered obs time!5! model space 25 SDAF, at T=5!t, corr= SDAF, at T=1!t, corr= u(x,5!t) !5! model space u(x,1!t) 2 1!1! model space

17 Radical Filtering Strategy for Nonlinear System Stochastically forced linear PDE FT Uncoupled Langevin eqn Replace the Nonlinear terms with an Ornstein-Uhlenbeck process Nonlinear Chaotic Dynamical Systems FT Coupled nonlinear ODE through nonlinear terms

18 Filtering turbulent nonlinear dynamical systems L-96 model (Lorenz 1996), 4 modes. (absorbing ball property) du j dt = (u j+1 u j 2 )u j 1 u j + F, j =,..., J 1 Energy Rescaled Variables: F=6 weakly chaotic F=8 strongly chaotic F=16 fully turbulent 2 F=6 2 F=8 2 F= time

19 Climatological Variance and Correlation time x 1!3 Variance Spectrum F= F=5 F=6 F=8 F= Correlation times F= F=5 F=6 F=8 F=16 6 Variance Correlation time Wave Numbers Wave Numbers Climatological Stochastic Model (CSM): fit the damping coefficient and stochastic noise strength to these climatological statistical quantities.

20 Regularly spaced sparse observations: weakly chaotic regime F = 6, P = 2, r o = 1.96, t = !2!4 EAKF TRUE! space !2!4 EAKF C67! space !2!4 FDKF C67! space !2!4 RFDKF C67! space

21 Regularly spaced sparse observations: weakly chaotic regime F = 6, P = 2, r o = 1.96, t =.234 hrs Table: This is a regime where EAKF true is superior. scheme RMS corr. EAKF true EAKF CSM ETKF true - ETKF CSM FDKF CSM RFDKF CSM No Filter 2.8 -

22 Regularly spaced sparse observations: fully turbulent regime F = 16, P = 2, r o =.81, t =.78 hrs EAKF TRUE EAKF CSM !5!1! space !5!1! space 2 FDKF CSM 2 ETKF CSM !5!5!1!1! space! space

23 Regularly spaced sparse observations: fully turbulent regime F = 16, P = 2, r o =.81, t =.78 hrs Table: This is a regime where FDKF is superior. Scheme RMS corr. EAKF true - EAKF CSM ETKF true - ETKF CSM FDKF CSM No Filter 6.3 -

24 Summary:

25 Summary: For M < N sparse regular observations, FDKF reduces (2N+1)-dim filtering problem to M decoupled P-dim filtering problems with a single scalar observation.

26 Summary: For M < N sparse regular observations, FDKF reduces (2N+1)-dim filtering problem to M decoupled P-dim filtering problems with a single scalar observation. The poor-man s CSM model degrades the filtering skill in the weakly chaotic regime but suggests encouraging results in the strongly chaotic and fully turbulent regimes.

27 Summary: For M < N sparse regular observations, FDKF reduces (2N+1)-dim filtering problem to M decoupled P-dim filtering problems with a single scalar observation. The poor-man s CSM model degrades the filtering skill in the weakly chaotic regime but suggests encouraging results in the strongly chaotic and fully turbulent regimes. Practically, our radical strategy is independent of tunable parameters and ensemble size.

28 Summary: For M < N sparse regular observations, FDKF reduces (2N+1)-dim filtering problem to M decoupled P-dim filtering problems with a single scalar observation. The poor-man s CSM model degrades the filtering skill in the weakly chaotic regime but suggests encouraging results in the strongly chaotic and fully turbulent regimes. Practically, our radical strategy is independent of tunable parameters and ensemble size. Catastrophic filter divergence in a chaotic DS with absorbing ball property needs further mathematical theory.

29 References: MG Majda and Grote, Explicit off-line criteria for stable accurate time filtering of strongly unstable spatially extended systems, PNAS 14(4), , 27. CHM Castronovo, Harlim, and Majda, Mathematical test criteria for filtering complex systems: Plentiful observations, J. Comp. Phys., 227(7), , 28. HM1 Harlim and Majda, Filtering nonlinear dynamical systems with linear stochastic models, Nonlinearity 21(6), , 28. HM2 Harlim and Majda, Mathematical test criteria for filtering complex systems: Regularly sparse observations, J. Comp. Phys., 227(1), , 28. HM3 Harlim and Majda, Catastrophic filter divergence in filtering nonlinear dissipative systems, submitted to Comm. Math. Sci., 28.