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INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Data Assimilation: New Challenges in Random and Stochastic Dynamical Systems Daniel Sanz-Alonso & Andrew Stuart D Bl¨ omker (Augsburg), D Kelly (NYU), KJH Law (KAUST), A. Shukla (Warwick), KC Zygalakis (Southampton) EQUADIFF 2015 Lyon, France, July 6 th 2015 Funded by EPSRC, ERC and ONR

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Outline INTRODUCTION 1 THREE IDEAS 2 DISCRETE TIME: THEORY 3 CONTINUOUS TIME: DIFFUSION LIMITS 4 CONCLUSIONS 5

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Table of Contents INTRODUCTION 1 THREE IDEAS 2 DISCRETE TIME: THEORY 3 CONTINUOUS TIME: DIFFUSION LIMITS 4 CONCLUSIONS 5

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Signal � � Consider the following map on Hilbert space H , �· , ·� , | · | : Signal Dynamics v j +1 = Ψ( v j ) , v 0 ∼ µ 0 .

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Signal � � Consider the following map on Hilbert space H , �· , ·� , | · | : Signal Dynamics v j +1 = Ψ( v j ) , v 0 ∼ µ 0 . Assume dissipativity : Absorbing Set Compact B in H with the property that, for | v 0 | ≤ R , there is J = J ( R ) > 0 such that, for all j ≥ J , v j ∈ B .

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Signal � � Consider the following map on Hilbert space H , �· , ·� , | · | : Signal Dynamics v j +1 = Ψ( v j ) , v 0 ∼ µ 0 . Assume dissipativity : Absorbing Set Compact B in H with the property that, for | v 0 | ≤ R , there is J = J ( R ) > 0 such that, for all j ≥ J , v j ∈ B . Limited predictability : Global Attractor d ( v j , A ) → 0 , as j → ∞ .

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Signal and Observation Random initial condition : Signal Process v j +1 = Ψ( v j ) , v 0 ∼ µ 0 .

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Signal and Observation Random initial condition : Signal Process v j +1 = Ψ( v j ) , v 0 ∼ µ 0 . Observations, partial and noisy , P : H → R J : Observation Process E | ξ j | 2 = 1 , i . i . d . w / pdf ρ. y j +1 = Pv j +1 + ǫξ j +1 , E ξ j = 0 ,

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Signal and Observation Random initial condition : Signal Process v j +1 = Ψ( v j ) , v 0 ∼ µ 0 . Observations, partial and noisy , P : H → R J : Observation Process E | ξ j | 2 = 1 , i . i . d . w / pdf ρ. y j +1 = Pv j +1 + ǫξ j +1 , E ξ j = 0 , Filter : probability distribution of v j given observations to time j : Filter � � µ j ( A ) = P v j ∈ A |F j , F j = σ ( y 1 , . . . , y j ) .

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Signal and Observation: Control Unpredictability? Pushforward under dynamics : Signal Process µ j +1 = Ψ ⋆ µ j . �

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Signal and Observation: Control Unpredictability? Pushforward under dynamics : Signal Process µ j +1 = Ψ ⋆ µ j . � Incorporate observations via Bayes’ Theorem: Observation Process � � � ǫ − 1 ( y j +1 − Pv ) A ρ � µ j +1 ( dv ) µ j +1 ( A ) = � � � µ j +1 ( dv ) . ǫ − 1 ( y j +1 − Pv ) H ρ �

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Signal and Observation: Control Unpredictability? Pushforward under dynamics : Signal Process µ j +1 = Ψ ⋆ µ j . � Incorporate observations via Bayes’ Theorem: Observation Process � � � ǫ − 1 ( y j +1 − Pv ) A ρ � µ j +1 ( dv ) µ j +1 ( A ) = � � � µ j +1 ( dv ) . ǫ − 1 ( y j +1 − Pv ) H ρ � When is the filter predictable : Filter Accuracy µ j ≈ δ v † j as j → ∞ .

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Goal (Cerou [5], SIAM J. Cont. Opt. 2000) Key Question: For which Ψ and P does the filter µ j concentrate on the true signal, up to error ǫ , in the large-time limit?

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Goal (Cerou [5], SIAM J. Cont. Opt. 2000) Key Question: For which Ψ and P does the filter µ j concentrate on the true signal, up to error ǫ , in the large-time limit? Key Problem: Ψ may expand

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Goal (Cerou [5], SIAM J. Cont. Opt. 2000) Key Question: For which Ψ and P does the filter µ j concentrate on the true signal, up to error ǫ , in the large-time limit? Key Problem: Ψ may expand View P as a projection on H . Define Q = I − P . Key Idea: Q Ψ should contract

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS A Large Class of Examples Geophysical Applications dv dt + Au + B ( u , u ) = f . Dissipative with energy conserving nonlinearity ∃ λ > 0 : � Av , v � ≥ λ | v | 2 . � B ( v , v ) , v � = 0 . f ∈ L 2 loc ( R + ; H ) . Examples Lorenz ’63 Lorenz ’96 Incompressible 2D Navier-Stokes equation on a torus

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Table of Contents INTRODUCTION 1 THREE IDEAS 2 DISCRETE TIME: THEORY 3 CONTINUOUS TIME: DIFFUSION LIMITS 4 CONCLUSIONS 5

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Filter Accuracy Dynamical Probability Data Assimilation Systems Synchronization Filter Optimal 3DVAR Dissipative Weather Conditioning: Systems Prediction Galerkin

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Idea 1: Synchronization (Foias and Prodi [7], RSM Padova 1967 Pecora and Carroll [13], PRL 1990.) Truth v † = ( p † , q † ) Synchronization Filter m = ( p , q ) p † j +1 = P Ψ( p † j , q † p j +1 = p † j ) , j +1 , q † j +1 = Q Ψ( p † j , q † q j +1 = Q Ψ( p † j ) , j , q j ); − −− − −− v † j +1 = Ψ( v † m j +1 = Q Ψ( m j ) + p † j ) , j +1 .

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Idea 1: Synchronization (Foias and Prodi [7], RSM Padova 1967 Pecora and Carroll [13], PRL 1990.) Truth v † = ( p † , q † ) Synchronization Filter m = ( p , q ) p † j +1 = P Ψ( p † j , q † p j +1 = p † j ) , j +1 , q † j +1 = Q Ψ( p † j , q † q j +1 = Q Ψ( p † j ) , j , q j ); − −− − −− v † j +1 = Ψ( v † m j +1 = Q Ψ( m j ) + p † j ) , j +1 . Synchronization for various chaotic dynamical systems (including the three canonical examples above [8, 13, 4, 14]): | m j − v † j | → 0 , as j → ∞ .

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Idea 2: 3DVAR (Lorenc [12] Q. J. R. Met. Soc 1986) Cycled 3DVAR Filter. | · | A = | A − 1 2 · | . m j +1 = argmin m ∈H {| m − Ψ( m j ) | 2 C + ǫ − 2 | y j +1 − Pm | 2 Γ } . Solve Variational Equations (with C = ǫ 2 � η − 2 Γ P + Q )) K = (1 + η 2 ) − 1 P , m j +1 = ( I − K )Ψ( m j ) + Ky j +1 , Variance Inflation (from weather prediction) η ≪ 1 m j +1 = Q Ψ( m j ) + Py j +1 , η = 0 . Synchronization Filter .

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Inaccurate: η too large. (NSE torus) Law and S [10], Monthly Weather Review, 2012 ν =0.01, h=0.2 ν =0.01, h=0.2, Re(u 1,2 ) 3DVAR, 3DVAR, m 0.3 u + 0 0.2 y n 10 ||m(t n )−u + (t n )|| 2 0.1 tr( Γ ) tr[(I−B n ) Γ (I−B n ) * ] 0 −1 10 −0.1 −0.2 −2 10 5 10 15 20 0 1 2 3 4 step t

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Accurate: smaller η . (NSE torus) Law and S [10], Monthly Weather Review, 2012 ν =0.01, h=0.2, Re(u 1,2 ) ν =0.01, h=0.2 [3DVAR], [3DVAR], ||m(t n )−u + (t n )|| 2 m 0.3 u + tr( Γ ) 0.2 y n tr[(I−B n ) Γ (I−B n ) * ] 0 10 0.1 0 −0.1 −1 10 −0.2 −0.3 0 2 4 6 8 10 12 10 20 30 40 50 60 70 t step

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Idea 3: Filter Optimality (Folklore, but see e.g. Williams · · · ) Recall F j = σ ( y 1 , . . . , y j ) and define the mean of the filter: v j := E ( v j |F j ) = E µ j ( v j ) . ˆ Use Galerkin orthogonality wrt conditional expectation For any F j measurable m j : v j | 2 ≤ E | v j − m j | 2 . E | v j − ˆ Take m j from 3DVAR to get bounds on the mean of the filter. Similar bounds apply to the variance of the filter. (Not shown.)

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS Table of Contents INTRODUCTION 1 THREE IDEAS 2 DISCRETE TIME: THEORY 3 CONTINUOUS TIME: DIFFUSION LIMITS 4 CONCLUSIONS 5