1. No category

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 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, h·, ·i, | · | :
Signal Dynamics
vj+1 = Ψ(vj ),
v0 ∼ µ 0 .
INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS
Signal
Consider the following map on Hilbert space H, h·, ·i, | · | :
Signal Dynamics
vj+1 = Ψ(vj ),
v0 ∼ µ 0 .
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, h·, ·i, | · | :
Signal Dynamics
vj+1 = Ψ(vj ),
v0 ∼ µ 0 .
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 ∼ µ 0 .
INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS
Signal and Observation
Random initial condition:
Signal Process
vj+1 = Ψ(vj ),
v0 ∼ µ 0 .
Observations, partial and noisy, P : H → RJ :
Observation Process
yj+1 = Pvj+1 + ξj+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 ∼ µ 0 .
Observations, partial and noisy, P : H → RJ :
Observation Process
yj+1 = Pvj+1 + ξj+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
µ
bj+1 = Ψ ? µj .
INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS
Signal and Observation: Control Unpredictability?
Pushforward under dynamics:
Signal Process
µ
bj+1 = Ψ ? µj .
Incorporate observations via Bayes’ Theorem:
Observation Process
R
ρ −1 (yj+1 − Pv ) µ
bj+1 (dv )
A
µj+1 (A) = R
.
−1
bj+1 (dv )
H ρ (yj+1 − Pv ) µ
INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS
Signal and Observation: Control Unpredictability?
Pushforward under dynamics:
Signal Process
µ
bj+1 = Ψ ? µj .
Incorporate observations via Bayes’ Theorem:
Observation Process
R
ρ −1 (yj+1 − Pv ) µ
bj+1 (dv )
A
µj+1 (A) = R
.
−1
bj+1 (dv )
H ρ (yj+1 − Pv ) µ
When is the filter predictable:
Filter Accuracy
µj ≈ δv † as j → ∞.
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 , 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 , 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 , 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
+ Au + B(u, u) = f .
dt
Dissipative with energy conserving nonlinearity
∃λ > 0 : hAv , v i ≥ λ|v |2 .
hB(v , v ), v i = 0.
f ∈ L2loc (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
3DVAR
Filter Optimal
Dissipative
Systems
Weather
Prediction
Conditioning:
Galerkin
INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS
Idea 1: Synchronization
Truth v † = (p † , q † )
(Foias and Prodi [7], RSM Padova 1967 Pecora and Carroll [13], PRL 1990.)
Synchronization Filter m = (p, q)
†
pj+1
= PΨ(pj† , qj† ),
†
pj+1 = pj+1
,
†
qj+1
= QΨ(pj† , qj† ),
qj+1 = QΨ(pj† , qj );
− −−
†
vj+1
=
Ψ(vj† ),
− −−
†
mj+1 = QΨ(mj ) + pj+1
.
INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS
Idea 1: Synchronization
Truth v † = (p † , q † )
(Foias and Prodi [7], RSM Padova 1967 Pecora and Carroll [13], PRL 1990.)
Synchronization Filter m = (p, q)
†
pj+1
= PΨ(pj† , qj† ),
†
pj+1 = pj+1
,
†
qj+1
= QΨ(pj† , qj† ),
qj+1 = QΨ(pj† , qj );
− −−
†
vj+1
=
Ψ(vj† ),
− −−
†
mj+1 = QΨ(mj ) + pj+1
.
Synchronization for various chaotic dynamical systems (including
the three canonical examples above [8, 13, 4, 14]):
|mj − vj† | → 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)
1
Cycled 3DVAR Filter. | · |A = |A− 2 · |.
mj+1 = argminm∈H {|m − Ψ(mj )|2C + −2 |yj+1 − Pm|2Γ }.
Solve Variational Equations (with C = 2 η −2 ΓP + Q))
mj+1 = (I − K )Ψ(mj ) + Kyj+1 ,
K = (1 + η 2 )−1 P,
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)
3DVAR, ν=0.01, h=0.2
Law and S [10], Monthly Weather Review, 2012
3DVAR, ν=0.01, h=0.2, Re(u1,2)
m
0.3
0
10
||m(tn)−u+(tn)||2
tr(Γ)
tr[(I−Bn)Γ(I−Bn)*]
−1
10
u+
yn
0.2
0.1
0
−0.1
−0.2
−2
10
5
10
15
step
20
0
1
2
3
t
4
INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS
Accurate: smaller η. (NSE torus)
Law and S [10], Monthly Weather Review, 2012
[3DVAR], ν=0.01, h=0.2, Re(u1,2)
[3DVAR], ν=0.01, h=0.2
||m(tn)−u+(tn)||2
0
10
m
0.3
tr(Γ)
tr[(I−Bn)Γ(I−Bn)*]
u+
yn
0.2
0.1
0
−0.1
−1
10
−0.2
−0.3
10
20
30
40
step
50
60
70
0
2
4
6
8
t
10
12
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:
vˆj := E(vj |Fj ) = Eµj (vj ).
Use Galerkin orthogonality wrt conditional expectation
For any Fj measurable mj :
E|vj − vˆj |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, h·, ·i, | · | and V, h·, ·iV , k · k :
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 ),
kQ(Ψ(u) − Ψ(v ))k2 ≤ α(R0 )ku − v k2 .
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 such that
lim sup E|vj − vˆj |2 ≤ c2
j→∞
.
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 E|vj − mj |2 ≤ c2 .
j→∞
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):
yj+1
mj+1 = QΨ(mj )
vj+1 = QΨ(vj )
z
}|
{
+ PΨ(vj ) + ξj+1 ,
+PΨ(vj ).
Subtract and use independence plus contractivity of QΨ:
Ekvj+1 − mj+1 k2 = EkQ (Ψ(vj ) − Ψ(mj )) − ξj+1 k2
≤ EkQ (Ψ(vj ) − Ψ(mj )) k2 + 2 Ekξj+1 k2
≤ αEkvj − mj k2 + 2 Ekξj+1 k2 .
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)
+ a(v (1) − v (2) )
=0
dt
dv (2)
+ av (1) + v (2) + v (1) v (3) = 0
dt
dv (3)
+
bv (3)
− v (1) v (2) = − b(r + a)
dt
| {z }
| {z } | {z } | {z }
dv
dt
Av
B(v, v)
f
Observation matrix


1 0 0
P :=  0 0 0  .
0 0 0
Theory applicable with k · k2 := |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)
+ v (j) + v (j−1) (v (j+1) − v (j−2) ) = F ,
dt
| {z } |{z} |
{z
} |{z}
dv
dt
Av
B(v, v)
j = 1, 2, . . . , 3J.
f
Observation matrix P: observe 2 out of every 3 points. Theory
applicable with k · k2 := |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,
1
1
B(u, v ) = Pleray [u · ∇v ] + Pleray [v · ∇u].
2
2
Observation operator in (divergence-free) Fourier space:
Pu =
X
|k|≤kmax
uk
k ⊥ ik·x
e .
|k|
1 (T2 ) and k
Theory applicable with H = V := Hdiv
max 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
Lorenz ’63
Lorenz ’96
NSE on torus
Dimension of v
3
3J
∞
Rank(P)
1
2J
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:
(n)
(n)
(n)
vj+1 = argminv ∈H {|v − Ψ(vj )|2Cj + −2 |yj+1 − Pv |2Γ }.
Empirical Covariance
v¯j =
N
1 X (n)
vj ,
N
n=1
Cj =
N
1 X (n)
(n)
vj − v¯j ⊗ vj − v¯j .
N
n=1
Perturbed Observations
(n)
(n)
yj+1 = yj+1 + ξj ,
(n)
ξj
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¨
omker et al 2012 [2], Kelly et al 2014 [9]
High Frequency Data Limit – 3DVAR
1 dW
dm
+ Am + B(m, m) + CP ∗ Γ−1 P(m − v ) + Γ 2
=f
dt
dt
INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS
S(P)DE Limits
with Brett et al [3], Bl¨
omker et al 2012 [2], Kelly et al 2014 [9]
High Frequency Data Limit – 3DVAR
1 dW
dm
+ Am + B(m, m) + CP ∗ Γ−1 P(m − v ) + Γ 2
=f
dt
dt
High Frequency Data Limit – Ensemble Kalman Filter
(n) 1 dW
dv (n)
+Av (n) +B(v (n) , v (n) )+CP ∗ Γ−1 P(v (n) −v )+Γ 2
=f,
dt
dt
v=
J
1 X (n)
v ,
J
j=1
C=
J
1 X (n)
(v − v ) ⊗ (v (n) − v ).
J
j=1
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¨
omker 2012 et al [2])
Under similar assumptions to the discete case there is a constant
c > 0 independent of the noise strength such that
lim sup E|v − m|2 ≤ c2
t→∞
.
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 such that
sup
N
X
t∈[0,T ] 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)
2
u1,0
m(k)
1,0
1
−2
u14,0
0.02
m(k)
14,0
0.01
0
−1
(Bl¨
omker et al [2])
2
0
0
−0.01
−3
0 −2
0
5
t
0.05 0.1 0.15
t
10
−0.02
0
0.5
1
t
|m(k)(t)−u(t)|/|u(t)|
0.1
0
u7,0
−0.1
−0.2
0
m(k)
7,0
0.5
1
t
0
10
0
5
t
10
INTRODUCTION THREE IDEAS DISCRETE TIME: THEORY CONTINUOUS TIME: DIFFUSION LIMITS CONCLUSIONS
SPDE Accurate (NSE Torus)
(Bl¨
omker et al [2])
2
u14,0
m(k)
14,0
0.01
0
0
u
−2 1
0
1,0
m(k)
1,0
−1
−4
0−2
0 0.02 0.04
t
5
t
−0.01
10
−0.02
0
0.5
1
t
1.5
2
1
u7,0
0.15
m(k)
7,0
0.1
10
|m(k)(t)−u(t)|/|u(t)|
0
10
0.05
0
−1
10
−0.05
−0.1
0
−2
0.5
1
t
1.5
10
0
5
t
10
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
over 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
over 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
over 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
over 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¨
omker, 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
observation 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/
Texts in Applied Mathematics 62
Data Assimilation
A Mathematical Introduction
This book provides a systematic treatment of the mathematical underpinnings of work in
data assimilation, covering both theoretical and computational approaches. Specifically
the authors develop a unified mathematical framework in which a Bayesian formulation
of the problem provides the bedrock for the derivation, development and analysis of
algorithms; the many examples used in the text, together with the algorithms which
are introduced and discussed, are all illustrated by the MATLAB software detailed in
the book and made freely available online.
TAM
62
Kody Law · Andrew Stuart
Konstantinos Zygalakis
The book is organized into nine chapters: the first contains a brief introduction to the
mathematical tools around which the material is organized; the next four are concerned
with discrete time dynamical systems and discrete time data; the last four are concerned
with continuous time dynamical systems and continuous time data and are organized
analogously to the corresponding discrete time chapters.
This book is aimed at mathematical researchers interested in a systematic development
of this interdisciplinary field, and at researchers from the geosciences, and a variety
of other scientific fields, who use tools from data assimilation to combine data with
time-dependent models. The numerous examples and illustrations make understanding of the theoretical underpinnings of data assimilation accessible. Furthermore, the
examples, exercises and MATLAB software, make the book suitable for students in
applied mathematics, either through a lecture course, or through self-study.
1
Data Assimilation
Mathematics
ISBN 978-3-319-20324-9
9
783319 203249
Texts in Applied Mathematics 62
Law · Stuart · Zygalakis
Kody Law · Andrew Stuart · Konstantinos Zygalakis
Data
Assimilation
A Mathematical Introduction