Accuracy and Stability Introduction of the Continuous-Time 3DVAR - - PowerPoint PPT Presentation

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Accuracy and Stability Introduction of the Continuous-Time 3DVAR - - PowerPoint PPT Presentation

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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

slide-1
SLIDE 1

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

6th Workshop, Random Dynamical Systems Bielefeld, 1.11.2013

Accuracy and Stability

  • f the Continuous-Time 3DVAR Filter

for 2D Navier-Stokes Equation Dirk Bl¨

  • mker

joint work with: Andrew Stuart, Kody Law (Warwick) Konstantinos Zygalakis (Southhampton)

slide-2
SLIDE 2

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Filtering

Methods used for high dimensional data assimilation problems (for example in weather forecasting) Widely applied, but not that well studied in the nonlinear, stochastic & infinite dimensional setting.

slide-3
SLIDE 3

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Filtering

Methods used for high dimensional data assimilation problems (for example in weather forecasting) Widely applied, but not that well studied in the nonlinear, stochastic & infinite dimensional setting.

Basic Idea of Filtering

estimate time-evolution of a trajectory based on partial observation & knowledge of the model use model to predict next step use data to correct prediction Problem:

  • nly partial and noisy observations/data
slide-4
SLIDE 4

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Our Approach:

As starting point simple 3DVAR-filter (in this talk: not many details about filter) Example for the underlying dynamical system: deterministic 2D-Navier-Stokes equation Limit of high frequency noisy observations yields stochastic PDE (continuous time filters/noisy observer) Study Accuracy & Stability = ⇒ Stochastic Dyn. Syst.

slide-5
SLIDE 5

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Accuracy & Stability

slide-6
SLIDE 6

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Accuracy & Stability

Stability

Trajectories from observer converge towards each other

⇒ It does not matter where to initiate the filter

slide-7
SLIDE 7

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Accuracy & Stability

Stability

Trajectories from observer converge towards each other

⇒ It does not matter where to initiate the filter Accuracy

Trajectories from observer get close to the true trajectory

(on the order of observational noise)

⇒ Filter gives the true answer

(recover unknown solution from partial noisy observations)

slide-8
SLIDE 8

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

The Model

For simplicity: In this talk only an ODE instead of 2D Navier Stokes. Thus H = Rn, n ≫ 1 very large.

slide-9
SLIDE 9

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

The Model

For simplicity: In this talk only an ODE instead of 2D Navier Stokes. Thus H = Rn, n ≫ 1 very large. Let u : R → H be any bounded solution of ∂tu = −δAu + B(u, u) + f

slide-10
SLIDE 10

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

The Model

For simplicity: In this talk only an ODE instead of 2D Navier Stokes. Thus H = Rn, n ≫ 1 very large. Let u : R → H be any bounded solution of ∂tu = −δAu + B(u, u) + f A diagonal operator, A ≥ 1, δ > 0, B : H × H → H – symmetric bilinear map f deterministic forcing (could be time dependent)

slide-11
SLIDE 11

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

The Model

For simplicity: In this talk only an ODE instead of 2D Navier Stokes. Thus H = Rn, n ≫ 1 very large. Let u : R → H be any bounded solution of ∂tu = −δAu + B(u, u) + f A diagonal operator, A ≥ 1, δ > 0, B : H × H → H – symmetric bilinear map f deterministic forcing (could be time dependent)

trajectory u is the unknown we want to observe

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SLIDE 12

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Existence & Uniqueness

(well known)

For 2D-Navier-Stokes see [Temam 95, 97], [Robinson 01]. Theorem Suppose B(u, u), u ≤ 0 and f is bounded. Then for all initial conditions u(0) there exists a global solution in C1([0, ∞), H). Furthermore there is a global attractor in BR(0) ⊂ H containing all bounded solutions.

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SLIDE 13

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Very brief description of 3DVAR

[Harvey 91, .... ]

Consider Sh – one step in the model (of time h > 0) uj = u(jh) = Sh(uj−1) – unknown true trajectory yj = Puj + N(0, Γ) – observation (noisy & partial) P – projection ˆ mj – estimation

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SLIDE 14

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Very brief description of 3DVAR

[Harvey 91, .... ]

Consider Sh – one step in the model (of time h > 0) uj = u(jh) = Sh(uj−1) – unknown true trajectory yj = Puj + N(0, Γ) – observation (noisy & partial) P – projection ˆ mj – estimation Prediction: mj+1 = Sh( ˆ mj)

slide-15
SLIDE 15

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Very brief description of 3DVAR

[Harvey 91, .... ]

Consider Sh – one step in the model (of time h > 0) uj = u(jh) = Sh(uj−1) – unknown true trajectory yj = Puj + N(0, Γ) – observation (noisy & partial) P – projection ˆ mj – estimation Prediction: mj+1 = Sh( ˆ mj) Assume Gaussianity: uj|y1 . . . yj ∼ N( ˆ mj, C) uj+1|y1 . . . yj ∼ N(mj+1, C) Kalman mean update

(Bayes’ rule + some work)

ˆ mj+1 = mj+1 + CP(Γ + PCP)−1(yj+1 − Pmj+1)

slide-16
SLIDE 16

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

High Frequency Observation Limit

Limit (h ↓ 0) of high frequency noisy observations for 3DVAR yields for sufficiently large observational noise a stochastic equation (noisy observer/continuous time filter) Key Point: The discrete time filter can be written as an Euler-Maruyama discretization of the observer The formal limit is true in a much more general setting and for several filter (no rigorous result yet)

Discrete time case for 2D-Navier Stokes: [Law, Stuart, et. al. 11]

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SLIDE 17

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Noisy observer

∂t ˆ m = −δA ˆ m+B( ˆ m, ˆ m)+f+ωA−2αPλ[u − ˆ m + σA−β∂tW] Pλ – proj. onto the observed low modes

(for 2D Navier-Stokes approximately λ2 many)

Assume: Γ = 1

hσ2A−2βPλ covariance of the (given)

  • bservational noise

(think of β = 0)

C = ωσ2A−2(α+β) how to weight data or the model W – standard cylindrical Wiener process

(space-time white noise)

λ and ω are free parameters of the filter (also C and α)

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SLIDE 18

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Generation of SDS

Stochastic Dynamical System:

(not all details)

S(t, s, W) ˆ m0 solution of observer at time t > 0 given path {W (t)}t≥s given initial condition ˆ m(s) = ˆ m0 Flow Property: S(t, r, W)S(r, s, W) = S(t, s, W) Theorem:

(by standard methods, details later)

The noisy observer generates a SDS in H. Remark: No Random Dynamical System is generated as the observer is non-autonomous due to u and possibly f.

slide-19
SLIDE 19

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Numerical Results

∂t ˆ m = −δA ˆ m+B( ˆ m, ˆ m)+f +ωA−2αPλ[u− ˆ m+σA−β∂tW] Parameter: λ large ω = 100 α = 1/2 β = 0 σ = 0.005 Spit step method Pseudospectral method for Navier Stokes equation, using higher order method in time Euler-Maruyama discretization for the OU-process

(linear equation with noise)

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SLIDE 20

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Attractivity of the observer

3 Fourier modes & relative error

2 4 −0.5 0.5 1 1.5 t u1,0 m1,0 2 4 −0.02 −0.01 0.01 t u14,0 m14,0 2 4 −0.2 −0.1 0.1 t u7,0 m7,0 20 40 60 80 10

−2

10

−1

10 t |m(t)−u(t)|/|u(t)|

slide-21
SLIDE 21

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Stability of the observer

3 Fourier modes & relative error

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

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SLIDE 22

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Stability of the observer

2 4 6 8 10 10

−15

10

−10

10

−5

10 10

5

t |m(k)(t)−m(1)(t)|/|m(1)(t)|

Relative error of an ensemble of trajectories.

slide-23
SLIDE 23

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Accuracy & Stability

Results forward in time

mean square & in probability

slide-24
SLIDE 24

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Forward Accuracy

Only in mean square – no almost sure results expected E|u(t) − ˆ m(t)|2 = O(noise-strength2) for t → ∞

slide-25
SLIDE 25

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Forward Accuracy

Only in mean square – no almost sure results expected E|u(t) − ˆ m(t)|2 = O(noise-strength2) for t → ∞ Conjectures: lim inf

t→∞

P

  • ˆ

m(t) ∈ B

  • > 0

and P

  • ∃ t > 0 : ˆ

m(t) ∈ B

  • = 1

for all open B ⊂ H

Well known for stochastic Navier-Stokes [Hairer, Mattingly 06].

slide-26
SLIDE 26

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Forward Accuracy

Assumption on γ: γ|h|2 ≤ 2ω|A−αPλh|2 + δ|A1/2h|2 ∀ h.

slide-27
SLIDE 27

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Forward Accuracy

Assumption on γ: γ|h|2 ≤ 2ω|A−αPλh|2 + δ|A1/2h|2 ∀ h. Theorem [BLSZ12] Suppose γ = KRδ−1 + γ0 for some γ0 > 0, where K is a constant defined by B and the bound on u. Then E| ˆ m(t)−u(t)|2 ≤ e−γ0t| ˆ m(0)−u(0)|2+ω2σ2 1

γ0 ·tr

  • A−4α−2βPλ
  • .

Consequence: lim supt→∞ E| ˆ m(t) − u(t)|2 = O(σ2).

slide-28
SLIDE 28

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Forward Accuracy

Assumption on γ: γ|h|2 ≤ 2ω|A−αPλh|2 + δ|A1/2h|2 ∀ h. Theorem [BLSZ12] Suppose γ = KRδ−1 + γ0 for some γ0 > 0, where K is a constant defined by B and the bound on u. Then E| ˆ m(t)−u(t)|2 ≤ e−γ0t| ˆ m(0)−u(0)|2+ω2σ2 1

γ0 ·tr

  • A−4α−2βPλ
  • .

Consequence: lim supt→∞ E| ˆ m(t) − u(t)|2 = O(σ2). Proof based on Itˆ

  • -formula – SPDE for the error e = u − ˆ

m

slide-29
SLIDE 29

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Forward Stability

Assume R′ = supt∈R |f + ωA−2αPλu|2

H−1 < ∞

and define R′′ = K

δ2 R′ + K δ ω2σ2tr

  • A−4α−2βPλ
  • < ∞

Theorem [BLSZ12] ˆ mi trajectories of observer; initial condition ˆ m(0) = ˆ mi(0). Suppose γ = R′′ + γ0 for some γ0 > 0. Then for all η ∈ (0, γ0) | ˆ m1(t) − ˆ m2(t)|eηt → 0 in probability as t → ∞. Key: P( 1

t

t

0 ˆ

mi(s)2ds ≤ R′′) → 1 for t → ∞ .

slide-30
SLIDE 30

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Pull-Back Convergence

sending initial time to −∞

almost sure and pathwise results

slide-31
SLIDE 31

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Transformation to a random PDE

Define for Φ > 0 and W = ωσA−2α−βPλW ZΦ(W) =

−∞

es(−δA+Φ)dW(s) . The stationary OU-process Z(t) = ZΦ(ϑtW) = t

−∞

e(t−s)(−δA+Φ)dW(s) with measure preserving ergodic shift ϑtW = W(t + ·) − W (t) solving dZ = (−δA + Φ)Z dt + dW

slide-32
SLIDE 32

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Transformation to a random PDE

e.g. [Crauel, Debussche, Flandoli, 97]

Define v(t) = S(t, s, W) ˆ m0 − Z(t), which solves ∂tv = −δAv + B(v, v) + f +2B(v, Z) + B(Z, Z) + ωA−2α(u − v − Z) − φZ Existence & Uniqueness of solutions is standard using pathwise PDE-results (needed for generation of SDS) Bounds on v using standard energy type methods & Gronwalls inequality

slide-33
SLIDE 33

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Birkhoff’s ergodic theorem

For bounds on v we need bounds on integrals over Z(t) = ZΦ(ϑtW) inside exponentials. Theorem 1 t − s t

s

ZΦ(ϑτW)2dτ → EZΦ2 as s → −∞ where EZΦ2 → 0 for Φ → ∞

slide-34
SLIDE 34

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Accuracy

Assume Kδ−1(17EZφ2 + 16R) < γ . (not optimal) Theorem (Pull-Back Accuracy)

[BLSZ12]

There is a random radius r(W) > 0 such that for all ˆ m0 lim sup

s→−∞

|S(t, s, W) ˆ m0 − u(t) − Zφ(ϑtW)|2 ≤ r(ϑtW) . with an almost surely finite constant

slide-35
SLIDE 35

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Accuracy

Assume Kδ−1(17EZφ2 + 16R) < γ . (not optimal) Theorem (Pull-Back Accuracy)

[BLSZ12]

There is a random radius r(W) > 0 such that for all ˆ m0 lim sup

s→−∞

|S(t, s, W) ˆ m0 − u(t) − Zφ(ϑtW)|2 ≤ r(ϑtW) . with an almost surely finite constant (due to Birkhoff) r(W) = 4 δ

−∞

exp

τ

  • 16Kδ−1(Z2 + R) − γ
  • dη
  • T 2dτ ,

where T := KZ(Z + 2R1/2) + φ|Z| + ω|A−2αPλZ|. Idea of Proof: Bounds on v − u using the PDEs.

slide-36
SLIDE 36

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Conclusion

As r(W) ≈ O(Z2) ≈ O(σ2): S(t, s, W) ˆ m0 − u(t) = v(t) − u(t) + Zφ(ϑtW) = O(Z2) + Zφ(ϑtW) = O(σ)

Thus we verified accuracy for the observer.

slide-37
SLIDE 37

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Birkhoff bounds

For stability we assume: γ > 0 is sufficiently large that for some η > 0 lim sup

s→−∞

1 t − s t

s

S(τ, s, W) ˆ m(1)

0 2dτ < γ − 2η

4K δ . Not proved. Should be possible, by varying Φ = Φ(W)

[Stannat, Es-Sahir 11], [Flandoli, Gatarek 95]

Also method in [Chueshov, Duan, Schmalfuss 03] might work. Technical Problem: r(W) does not satisfy Birkhoff-theorem (Er = ∞)

slide-38
SLIDE 38

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Stability

Theorem (Pull-Back Accuracy)

[BLSZ12]

Assume one initial condition ˆ m(1) ∈ H satisfies Birkhoff-bounds. Let ˆ m(2) ∈ H be any other initial condition. Then lim

s→−∞ |S(t, s, W) ˆ

m(1) − S(t, s, W) ˆ m(2)

0 | · eη(t−s) = 0.

slide-39
SLIDE 39

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Stability

Theorem (Pull-Back Accuracy)

[BLSZ12]

Assume one initial condition ˆ m(1) ∈ H satisfies Birkhoff-bounds. Let ˆ m(2) ∈ H be any other initial condition. Then lim

s→−∞ |S(t, s, W) ˆ

m(1) − S(t, s, W) ˆ m(2)

0 | · eη(t−s) = 0.

Idea of proof: random PDE for e = ˆ m(1) − ˆ m(2) energy type estimates – Gronwall’s Lemma – Birkhoff bounds for integrals on Z and ˆ m(1) in the exponential....

slide-40
SLIDE 40

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Generalized Observer I

Other filter lead in the limit of high frequency observations to ∂t ˆ m = −δA ˆ m + B( ˆ m, ˆ m) + CPΓ−1P (u − ˆ m) + CPΓ− 1

2 ∂tW

∂tC = LC + CL∗ − CPΓ−1PC Γ operator determined by observational noise linear operators L determined as part of the filter P projects onto the observed modes C is a covariance operator (symmetric, trace-class)

slide-41
SLIDE 41

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Generalized Observer I

Other filter lead in the limit of high frequency observations to ∂t ˆ m = −δA ˆ m + B( ˆ m, ˆ m) + CPΓ−1P (u − ˆ m) + CPΓ− 1

2 ∂tW

∂tC = LC + CL∗ − CPΓ−1PC Γ operator determined by observational noise linear operators L determined as part of the filter P projects onto the observed modes C is a covariance operator (symmetric, trace-class) Observation: If the Ricatti-type equation has an attracting stable steady state for C, then this algorithm simply converges to 3DVAR algorithm.

slide-42
SLIDE 42

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Generalized Observer II

∂t ˆ m = −δA ˆ m + B( ˆ m, ˆ m) + CPΓ−1P (u − ˆ m) + CPΓ− 1

2 ∂tW

∂tC = LC + CL∗ − CPΓ−1PC Problem: For filters like the extended Kalman Filter we have L = −δA + 2B( ˆ m, ·) and thus coupled equations.

slide-43
SLIDE 43

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Remarks

Proof of the high frequency limit?

Convergence of the Euler-Maruyama scheme!?

General solutions u, non-autonomous f?

We only needs boundedness of u for t → ∞ (or s → −∞)

Other types of equations/models?

Proofs use only local and one-sided Lipschitz conditions.

Birkhoff bounds ??? stability & accuracy for generalized observer ???

  • nly partial ideas – work in progress
slide-44
SLIDE 44

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Summary

Study Filter in the high frequency limit Stability & Accuracy via continuous time filter/observer 2D Navier-Stokes & 3DVAR as first example

slide-45
SLIDE 45

3DVAR for 2D-NS Dirk Bl¨

  • mker

Introduction Set-Up

Navier-Stokes 3DVAR noisy observer

Numerics

attractivity stability

Forward

accuracy stability

Pull-back

transformation Birkhoff accuracy stability

Outlook

  • ther filter

todo summary

Summary

Study Filter in the high frequency limit Stability & Accuracy via continuous time filter/observer 2D Navier-Stokes & 3DVAR as first example

Thank you very much for you attention!