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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¨

• mker (Augsburg), D Kelly (NYU), KJH Law (KAUST),
• A. Shukla (Warwick), KC Zygalakis (Southampton)

EQUADIFF 2015 Lyon, France, July 6th 2015 Funded by EPSRC, ERC and ONR

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Outline

1

INTRODUCTION

2

THREE IDEAS

3

DISCRETE TIME: THEORY

4

CONTINUOUS TIME: DIFFUSION LIMITS

5

CONCLUSIONS

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Table of Contents

1

INTRODUCTION

2

THREE IDEAS

3

DISCRETE TIME: THEORY

4

CONTINUOUS TIME: DIFFUSION LIMITS

5

CONCLUSIONS

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Signal

Consider the following map on Hilbert space

• H, ·, ·, | · |
• :

Signal Dynamics vj+1 = Ψ(vj), v0 ∼ µ0.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Signal

Consider the following map on Hilbert space

• H, ·, ·, | · |
• :

Signal Dynamics vj+1 = Ψ(vj), v0 ∼ µ0. Assume dissipativity: Absorbing Set Compact B in H with the property that, for |v0| ≤ R, there is J = J(R) > 0 such that, for all j ≥ J, vj ∈ B.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Signal

Consider the following map on Hilbert space

• H, ·, ·, | · |
• :

Signal Dynamics vj+1 = Ψ(vj), v0 ∼ µ0. Assume dissipativity: Absorbing Set Compact B in H with the property that, for |v0| ≤ R, there is J = J(R) > 0 such that, for all j ≥ J, vj ∈ B. Limited predictability: Global Attractor d(vj, A) → 0, as j → ∞.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Signal and Observation

Random initial condition: Signal Process vj+1 = Ψ(vj), v0 ∼ µ0.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Signal and Observation

Random initial condition: Signal Process vj+1 = Ψ(vj), v0 ∼ µ0. Observations, partial and noisy, P : H → RJ: Observation Process yj+1 = Pvj+1 + ǫξj+1, Eξj = 0, E|ξj|2 = 1, i.i.d. w/pdf ρ.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Signal and Observation

Random initial condition: Signal Process vj+1 = Ψ(vj), v0 ∼ µ0. Observations, partial and noisy, P : H → RJ: Observation Process yj+1 = Pvj+1 + ǫξj+1, Eξj = 0, E|ξj|2 = 1, i.i.d. w/pdf ρ. Filter: probability distribution of vj given observations to time j: Filter µj(A) = P

• vj ∈ A|Fj
• ,

Fj = σ(y1, . . . , yj).

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Signal and Observation: Control Unpredictability?

Pushforward under dynamics: Signal Process

• µj+1 = Ψ ⋆ µj.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Signal and Observation: Control Unpredictability?

Pushforward under dynamics: Signal Process

• µj+1 = Ψ ⋆ µj.

Incorporate observations via Bayes’ Theorem: Observation Process µj+1(A) =

• A ρ
• ǫ−1(yj+1 − Pv)
• µj+1(dv)
• H ρ
• ǫ−1(yj+1 − Pv)
• µj+1(dv).

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Signal and Observation: Control Unpredictability?

Pushforward under dynamics: Signal Process

• µj+1 = Ψ ⋆ µj.

Incorporate observations via Bayes’ Theorem: Observation Process µj+1(A) =

• A ρ
• ǫ−1(yj+1 − Pv)
• µj+1(dv)
• H ρ
• ǫ−1(yj+1 − Pv)
• µj+1(dv).

When is the filter predictable: Filter Accuracy µj ≈ δv†

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 µj 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 µj 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 µj 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: QΨ 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 ∈ L2

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

1

INTRODUCTION

2

THREE IDEAS

3

DISCRETE TIME: THEORY

4

CONTINUOUS TIME: DIFFUSION LIMITS

5

CONCLUSIONS

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Filter Accuracy Dynamical Systems Data Assimilation Probability Synchronization Dissipative Systems 3DVAR Weather Prediction Filter Optimal Conditioning: 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 = PΨ(p† j , q† j ),

pj+1 = p†

j+1,

q†

j+1 = QΨ(p† j , q† j ),

qj+1 = QΨ(p†

j , qj);

− −− − −− v†

j+1 = Ψ(v† j ),

mj+1 = QΨ(mj) + p†

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 = PΨ(p† j , q† j ),

pj+1 = p†

j+1,

q†

j+1 = QΨ(p† j , q† j ),

qj+1 = QΨ(p†

j , qj);

− −− − −− v†

j+1 = Ψ(v† j ),

mj+1 = QΨ(mj) + p†

j+1.

Synchronization for various chaotic dynamical systems (including the three canonical examples above [8, 13, 4, 14]): |mj − 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 · |.

mj+1 = argminm∈H{|m − Ψ(mj)|2

C + ǫ−2|yj+1 − Pm|2 Γ}.

Solve Variational Equations (with C = ǫ2 η−2ΓP + Q)) mj+1 = (I − K)Ψ(mj) + Kyj+1, K = (1 + η2)−1P, Variance Inflation (from weather prediction) η ≪ 1 mj+1 = QΨ(mj) + Pyj+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

5 10 15 20 10

−2

10

−1

10 step 3DVAR, ν=0.01, h=0.2 ||m(tn)−u+(tn)||2 tr(Γ) tr[(I−Bn)Γ(I−Bn)*] 1 2 3 4 −0.2 −0.1 0.1 0.2 0.3 3DVAR, ν=0.01, h=0.2, Re(u1,2) t m u+ yn

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Accurate: smaller η. (NSE torus) Law and S [10], Monthly Weather Review, 2012

10 20 30 40 50 60 70 10

−1

10 step [3DVAR], ν=0.01, h=0.2 ||m(tn)−u+(tn)||2 tr(Γ) tr[(I−Bn)Γ(I−Bn)*] 2 4 6 8 10 12 −0.3 −0.2 −0.1 0.1 0.2 0.3 [3DVAR], ν=0.01, h=0.2, Re(u1,2) t m u+ yn

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Idea 3: Filter Optimality (Folklore, but see e.g. Williams · · · )

Recall Fj = σ(y1, . . . , yj) and define the mean of the filter: ˆ vj := E(vj|Fj) = Eµj(vj). Use Galerkin orthogonality wrt conditional expectation For any Fj measurable mj : E|vj − ˆ vj|2 ≤ E|vj − mj|2. Take mj 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

1

INTRODUCTION

2

THREE IDEAS

3

DISCRETE TIME: THEORY

4

CONTINUOUS TIME: DIFFUSION LIMITS

5

CONCLUSIONS

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Assumptions

There are two equivalent Hilbert spaces:

• H, ·, ·, | · |
• and
• V, ·, ·V , ·
• :

Assumption 1: Absorbing Ball Property There is R0 > 0 such that: for B(R0) := {x ∈ H : |x| ≤ R0}, Ψ(B(R0)) ⊂ B(R0); for any bounded set S ⊂ H ∃J = J(S) : ΨJ(S) ⊂ B(R0). Assumption 2: Squeezing Property There is α(R0) ∈ (0, 1) such that, for all u, v ∈ B(R0), Q(Ψ(u) − Ψ(v))2 ≤ α(R0)u − v2.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Theorem (Sanz-Alonso and S, 2014, [15]) Let Assumptions 1,2 hold. Then there is a constant c > 0 independent of the noise strength ǫ such that lim sup

j→∞

E|vj − ˆ vj|2 ≤ cǫ2 . Idea of proof: Fix m0 ∈ B(R0) and let P denote the H−projection onto B(R0). Define the modified 3DVAR: mj+1 = P

• QΨ(mj) + yj+1
• .

Prove lim sup

j→∞

E|vj − mj|2 ≤ cǫ2. Use the L2 optimality of the filtering distribution.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Idea of proof (sketch, Ψ globally Lipschitz): mj+1 = QΨ(mj) +

yj+1

• PΨ(vj) + ǫξj+1,

vj+1 = QΨ(vj) +PΨ(vj). Subtract and use independence plus contractivity of QΨ: Evj+1 − mj+12 = EQ (Ψ(vj) − Ψ(mj)) − ǫξj+12 ≤ EQ (Ψ(vj) − Ψ(mj)) 2 + ǫ2Eξj+12 ≤ αEvj − mj2 + ǫ2Eξj+12. Use Gronwall.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Lorenz ’63 (uses noiseless synchronization filter analysis in Hayden, Olson and Titi [8], Physica D 2011.)

dv(1) dt + a(v(1) − v(2)) = 0 dv(2) dt + av(1) + v(2) + v(1)v(3) = 0 dv(3) dt

dv dt

+ bv(3)

• Av

− v(1)v(2)

• B(v, v)

= − b(r + a)

• f

Observation matrix P :=   1   . Theory applicable with · 2 := |P · |2 + | · |2 for h sufficiently small: [8], [11].

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Lorenz ’96 (Law, Sanz-Alonso, Shukla and S [14], arXiv 2014.)

Consider the following system, subject to the periodicity boundary conditions v0 = v3J, v−1 = v3J−1, v3J+1 = v1: dv(j) dt

dv dt

+ v(j)

• Av

+ v(j−1)(v(j+1) − v(j−2))

• B(v, v)

= F

• f

, j = 1, 2, . . . , 3J. Observation matrix P: observe 2 out of every 3 points. Theory applicable with · 2 := |P · |2 + | · |2 for h sufficiently small: [14].

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

2D Navier-Stokes Equation (again uses analysis in Hayden, Olson and Titi [8], Physica D 2011.)

Pleray denotes the Leray projector: Au = −νPleray∆u, B(u, v) = 1 2Pleray[u ·∇v]+ 1 2Pleray[v ·∇u]. Observation operator in (divergence-free) Fourier space: Pu =

• |k|≤kmax

uk k⊥ |k| eik·x. Theory applicable with H = V := H1

div(T2) and kmax sufficiently

large/h sufficiently small: [3], [8].

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Summary of Examples

Observations control unpredictability in these cases:

ODE Dimension of v Rank(P) Lorenz ’63 3 1 Lorenz ’96 3J 2J NSE on torus ∞ Finite

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Table of Contents

1

INTRODUCTION

2

THREE IDEAS

3

DISCRETE TIME: THEORY

4

CONTINUOUS TIME: DIFFUSION LIMITS

5

CONCLUSIONS

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Generalize 3DVAR: The EnKF. Evensen [6] Journal of Geophysical Research 1994.

Ensemble Kalman Filter. For n = 1, . . . , N: v(n)

j+1 = argminv∈H{|v − Ψ(v(n) j

)|2

Cj + ǫ−2|y(n) j+1 − Pv|2 Γ}.

Empirical Covariance ¯ vj = 1 N

N

• n=1

v(n)

j

, Cj = 1 N

N

• n=1
• v(n)

j

− ¯ vj

• ⊗
• v(n)

j

− ¯ vj

• .

Perturbed Observations y(n)

j+1 = yj+1 + ǫξ(n) j

, ξ(n)

j

i.i.d. w/pdf ρ.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

S(P)DE Limits with Brett et al [3], Bl¨

• mker et al 2012 [2], Kelly et al 2014 [9]

High Frequency Data Limit – 3DVAR dm dt + Am + B(m, m) + CP∗Γ−1 P(m − v) + ǫΓ

1 2 dW

dt

• = f

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

S(P)DE Limits with Brett et al [3], Bl¨

• mker et al 2012 [2], Kelly et al 2014 [9]

High Frequency Data Limit – 3DVAR dm dt + Am + B(m, m) + CP∗Γ−1 P(m − v) + ǫΓ

1 2 dW

dt

• = f

High Frequency Data Limit – Ensemble Kalman Filter dv(n) dt +Av(n)+B(v(n), v(n))+CP∗Γ−1 P(v(n)−v)+ǫΓ

1 2 dW (n)

dt

• = f ,

v = 1 J

J

• j=1

v(n), C = 1 J

J

• j=1

(v(n) − v) ⊗ (v(n) − v).

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

S(P)DE Accuracy see also Azouani, Olson and Titi 2014 [1] and Tong, Majda, Kelly 2015 [16] .

Theorem (3DVAR Accurate, with Bl¨

• mker 2012 et al [2])

Under similar assumptions to the discete case there is a constant c > 0 independent of the noise strength ǫ such that lim sup

t→∞ E|v − m|2 ≤ cǫ2

. Theorem (EnKF Well-Posed, with Kelly et al [9]) Let Assumptions 1 hold and P=I. Then there is a constant c > 0 independent of the noise strength ǫ such that sup

t∈[0,T] N

• n=1

E|v(n)(t)|2 ≤ C(T)

• 1 + E|v(n)(0)|2).

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

SPDE Inaccurate (NSE Torus) (Bl¨

• mker et al [2])

0.5 1 −0.02 −0.01 0.01 0.02 t 0.5 1 −0.2 −0.1 0.1 t 5 10 10 t u14,0 m(k)

14,0

u7,0 m(k)

7,0

|m(k)(t)−u(t)|/|u(t)| 5 10 −3 −2 −1 1 2 t u1,0 m(k)

1,0

0.05 0.1 0.15 −2 2 t

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

SPDE Accurate (NSE Torus) (Bl¨

• mker et al [2])

5 10 −4 −2 2 t 0.5 1 1.5 2 −0.02 −0.01 0.01 t 0.5 1 1.5 −0.1 −0.05 0.05 0.1 0.15 t 5 10 10

−2

10

−1

10 10

1

t u1,0 m(k)

1,0

u14,0 m(k)

14,0

u7,0 m(k)

7,0

|m(k)(t)−u(t)|/|u(t)| 0.02 0.04 −2 −1 1 t

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Table of Contents

1

INTRODUCTION

2

THREE IDEAS

3

DISCRETE TIME: THEORY

4

CONTINUOUS TIME: DIFFUSION LIMITS

5

CONCLUSIONS

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Summary

Chaos – and resulting unpredictability – is the enemy in many scientific and engineering applications.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Summary

Chaos – and resulting unpredictability – is the enemy in many scientific and engineering applications. Its study has led to a great deal of interesting mathematics

• ver the last century.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Summary

Chaos – and resulting unpredictability – is the enemy in many scientific and engineering applications. Its study has led to a great deal of interesting mathematics

• ver the last century.

Data – when combined with models – can have a massive positive impact on prediction in all of these scientific and engineering applications.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Summary

Chaos – and resulting unpredictability – is the enemy in many scientific and engineering applications. Its study has led to a great deal of interesting mathematics

• ver the last century.

Data – when combined with models – can have a massive positive impact on prediction in all of these scientific and engineering applications. The emerging new field, in which model and data are analyzed simultaneously, will lead to interesting new mathematics over the next century.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

Summary

Chaos – and resulting unpredictability – is the enemy in many scientific and engineering applications. Its study has led to a great deal of interesting mathematics

• ver the last century.

Data – when combined with models – can have a massive positive impact on prediction in all of these scientific and engineering applications. The emerging new field, in which model and data are analyzed simultaneously, will lead to interesting new mathematics over the next century. Data Assimilation needs input from Dynamical Systems.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

References I

• A. Azouani, E. Olson, and E.S. Titi.

Continuous data assimilation using general interpolant observables. Journal of Nonlinear Science, 24(2):277–304, 2014.

• D. Bl¨
• mker, K.J.H. Law, A.M. Stuart, and K.C. Zygalakis.

Accuracy and stability of the continuous-time 3DVAR filter for the Navier-Stokes equation. Nonlinearity, 26:2193–2219, 2013. C.E.A. Brett, K.F. Lam, K.J.H. Law, D.S. McCormick, M.R. Scott, and A.M. Stuart. Accuracy and Stability of Filters for the Navier-Stokes equation. Physica D: Nonlinear Phenomena, 245:34–45, 2013.

• A. Carrassi, M. Ghil, A. Trevisan, and F. Uboldi.

Data assimilation as a nonlinear dynamical systems problem: Stability and convergence of the prediction-assimilation system. Chaos: An Interdisciplinary Journal of Nonlinear Science, 18:023112, 2008.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

References II

• F. C´

erou. Long time behavior for some dynamical noise free nonlinear filtering problems. SIAM Journal on Control and Optimization, 38(4):1086–1101, 2000.

• G. Evensen.

Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics. Journal of Geophysical Research, pages 143–162, 1994.

• C. Foias and G. Prodi.

Sur le comportment global des solutions non statiennaires des equations de Navier–Stokes en dimension 2.

• Rend. Sem. Mat. Univ. Padova, 39, 1967.
• K. Hayden, E. Olson, and E.S. Titi.

Discrete Data Assimilation in the Lorenz and 2D Navier–Stokes equations. Physica D: Nonlinear Phenomena, 240:1416–1425, 2011.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

References III

D.T.B. Kelly, K.J.H. Law, and A.M. Stuart. Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time. Nonlinearity, 27:2579–2603, 2014. K.Law and A.M. Stuart. Evaluating data assimilation algorithms. Monthly Weather Review, 140:3757–3782, 2012. K.J.H. Law, A. Shukla, and A.M. Stuart. Analysis of the 3DVAR Filter for the Partially Observed Lorenz ’63 Model. Discrete and Continuous Dynamical Systems A, 34:1061–1078, 2014. A.C. Lorenc. Analysis methods for numerical weather prediction. Quarterly Journal of the Royal Meteorological Society, 112(474):1177–1194, 1986.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

References IV

L.M. Pecora and T.L. Carroll. Synchronization in chaotic systems. Physical review letters, 64(8):821, 1990.

• D. Sanz-Alonso, K.J.H. Law, A. Shukla, and A.M. Stuart.

Filter accuracy for chaotic dynamical systems: fixed versus adaptive

• bservation operators.

arxiv.org/abs/1411.3113, 2014.

• D. Sanz-Alonso and A.M. Stuart.

Long-time asymptotics of the filtering distribution for partially observed chaotic deterministic dynamical systems. arxiv.org/abs/1411.6510, 2014. X.T. Tong, A.J. Majda, and D.T.B. Kelly. Nonlinear stability and ergodicity of ensemble based Kalman filters. NYU preprint 2015.

INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS

References V

Matlab files and book chapters freely available: http://tiny.cc/damat http://www2.warwick.ac.uk/fac/sci/maths/people/staff/andrew stuart/

1

Law · Stuart · Zygalakis

Kody Law · Andrew Stuart · Konstantinos Zygalakis

Data Assimilation

A Mathematical Introduction

Data Assimilation

Kody Law · Andrew Stuart Konstantinos Zygalakis

Data Assimilation A Mathematical Introduction

• f the problem provides the bedrock for the derivation, development and analysis of

• f this interdisciplinary fi

• f other scientifi

Texts in Applied Mathematics 62

Texts in Applied Mathematics 62