In this talk we present a framework for splitting data assimilation problems based upon the model dynamics. This is motivated by assimilation in the unstable subspace (AUS) and center manifold and inertial manifold techniques in dynamical systems. Recent efforts based upon the development of particle filters projected into the unstable subspace will be highlighted.

1. Projected Data Assimilation
Erik S. Van Vleck
Department of Mathematics
University of Kansas
SAMSI CLIM Transition Workshop
May 14, 2018
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 1 / 20

2. Bayesian Data Assimilation
The basic idea is to determine the conditional probability p(u0|y) of the
initial condition u0 given the observations y using Bayes’ rule:
p(u0|y) =
p(y|u0)p(u0)
p(y)
, p(y) = p(y|u0)p(u0)du0 = constant
Posterior is Proportional to Prior times Likelihood
For example, if the observational noise is Gaussian (ηi ∼ N(0, Ri)),
then
p(y|u0) ∝ exp(−J(u0, y)), J(u0, y) =
i
(Hi(ui)−yo
i )T
R−1
i (Hi(ui)−yo
i )
and if the prior p(u0) is Gaussian, then
p(u0) ∝ exp(−(u0 − ub
0)T
B−1
(u0 − ub
0)).
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 2 / 20

3. Difficulties
Gaussian vs Non-Gaussian Errors
Linear vs. Nonlinear
(Phase Space Model and Observation Operator)
Low vs. High Dimensional
(Phase Space Model and Observation Operator)
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 3 / 20

4. Consider a nonlinear evolution equation (solution operator of a model)
un+1 = Fn(un; α), n = 0, 1, ..., N − 1
where
un are the state variables at time n,
α are adjustable model parameters, e.g., global in time.
Write un = u
(0)
n + δn.
If we can decompose the time dependent tangent space into slow
variables and fast variables, then we write δn = Pnδn + (I − Pn)δn.
Rewrite original nonlinear evolution approximately as two subsystems
...
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 4 / 20

5. For k = 0, 1, 2, ... (k like Newton iterate)
P System: [u
(k)
n and u
(k)
n+1 known (d+
n := Pnδn, ∀n)]
u
(k)
n+1 + d+
n+1 = Pn+1Fn(u(k)
n + d+
n ),
Update: u
(k+1
2
)
n = u
(k)
n + d+
n , n = 0, 1, ..., N − 1,
I-P System: [u
(k+1
2
)
n and u
(k+1
2
)
n+1 known (d−
n := (I − Pn)δn, ∀n)]
u
(k+1
2
)
n+1 + d−
n+1 = (I − Pn+1)Fn(u
(k+1
2
)
n + d+
n ),
Update: u
(k+1)
n = u
(k+1
2
)
n + d−
n , n = 0, 1, ..., N − 1.
Pn can be P
(k)
n or also P
(k+1/2)
n .
Roughly speaking the first subsystem contains the slow variables
(positive, zero, and slightly negative Lyapunov exponents) and the
second subsystem contains the fast variables (strongly negative).
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 5 / 20

6. Expect convergence (at least locally) when there exists
a time dependent decoupling transformation/change of variables
(bounded, continuous)
that transforms wn+1 = Fn(un)wn ≡ Anwn to
vn+1 = Bnvn where
Bn =
Cn 0
0 Dn
.
Possible, for example, when system has exponential separation.
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 6 / 20

7. Importance of observations rich in unstable subspace
Assimilation in the Unstable Subspace (AUS) [Trevisan, D’Isidoro, Talagrand ’10
Q.J.R. Meteorol. Soc., Palatella, Carrassi, Trevisan ’13 J. Phys A, ...]
Error analysis in DA for hyperbolic system [Gonz´alez-Tokman, Hunt ’13 Phys D]
Adaptive observation operators and unstable subspace [K.J.H. Law, D.
Sanz-Alonso, A. Shukla, A.M. Stuart ’16 Phys D]
Convergence of covariances matrices in unstable subspace [Bosquet
etal. ’17 SIAM UQ]
...
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 7 / 20

8. Forming time dependent projections
Discrete QR algorithm for determining Lyapunov exponents, local in
time stability information, etc.:
For Q0 ∈ IRm×p random such that QT
0 Q0 = I,
Qn+1Rn = Fn(un)Qn ≈
1
[Fn(un + Qn) − Fn(un)], n = 0, 1, ...
where QT
n+1Qn+1 = I and Rn is upper triangular with positive diagonal
elements.
Orthogonal Projections:
Pn = QnQT
n , (I − Pn) = I − QnQT
n
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 8 / 20

9. Data Assimilation
For n = 0, 1, ..., N − 1,
Data Model yn+1 = H(un+1) + ηn+1
State Space Model un+1 = Fn(un) + ξn
Here yn are the observations, ηn observational noise, and ξn the model
noise.
Idea is to use the observations to determine solution, i.e., u0.
Common techniques (based on Bayes’ Rule): Variations on ensemble
Kalman filter, 4DVar, proposal density methods, particle filters, etc..
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 9 / 20

10. Slow/Fast Splitting and Techniques for P and I − P
Using the framework of a slow/fast splitting we
may employ different DA/parameter estimation techniques in each
subsystem,
obtain an explicit representations for the time dependent unstable
subspace.
Interested in systems that are hyperbolic in flavor (finite number of
positive Lyapunov exponents, few zero Lyapunov exponents,
potentially many negative Lyapunov exponents).
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 10 / 20

11. Slow/Fast Splitting and Techniques for P and I − P
Techniques for P System:
Particle filters that are effective for low dimensional problem,
Shadowing refinement (strong state space model constraint, solve
BVP with no BCs, look for small perturbations) [Grebogi, Hammel, Yorke, and
Sauer, Phys Rev Lett (1990)] ), ....
Techniques I-P System:
Techniques such as ETKF, LETKF, 4DVar (essentially a shooting
method starting from ub
0, trying to match observations, basin of
attraction shrinks in the presence of positive Lyapunov
exponents).
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 11 / 20

12. Shadowing Refinement and Parameter Estimation
We’ve developed
[B. de Leeuw, S. Dubinkina, J. Franks, A. Steyer, X. Tu, EVV, “Projected Shadowing Based Data Assimilation,” (2018) in revision
SIAM J. Appld. Dyn. Sys. ]
an interval sequential technique where
all observations over each subinterval are employed simultaneously,
using
Shadowing refinement and parameter estimation for P problem,
Insertion synchronization or EnKF/ETKF/LETKF for I − P
problem.
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 12 / 20

13. Example: Lorenz ’96 Model (N = 40, F = 8)
˙uk = (uk+1 − uk−2)uk−1 − uk + F , k = 0, 1, . . . , N − 1 , (mod N)
˙u = −Iu + N(u)
13 positive Lyapunov exponents, Lyapunov dimension ≈ 28.
Start from a high precision numerically generated solution, noisy
orbit using matlab command 2*(2*rand()-1) .
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 13 / 20

14. Projected P System
State Space Model un+1 = Fn(un) + ξn
Data Model yn+1 = Hun+1 + ηn+1
State Space Model (perturbed form) u
(0)
n+1 + δn+1 = Fn(u
(0)
n + δn) + ξn
Data Model (perturbed form) yn+1 = H(u
(0)
n+1 + δn+1) + ηn+1
Projected State Space Model
Pn+1δn+1 = Pn+1[Fn(u(0)
n + Pnδn) + ξn − u
(0)
n+1]
Projected Data Model
PH
n+1H†
yn+1 = PH
n+1(u
(0)
n+1 + δn+1) + PH
n+1H†
ηn+1
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 14 / 20

15. Projected P System
Projected State Space Model
Pn+1δn+1 = Pn+1[Fn(u(0)
n + Pnδn) + ξn − u
(0)
n+1]
Projected Data Model
PH
n+1H†
yn+1 = PH
n+1(u
(0)
n+1 + δn+1) + PH
n+1H†
ηn+1
where H† = HT (HHT )−1 [the pseudo inverse of H], and
PH
n+1 is the orthogonal projection onto the intersection of the
subspaces spanned by the columns of Qn+1 and the rows of H.
Note: PH = HT (HHT )−1H = H†H (we are assuming H full rank).
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 15 / 20

16. Projected P System
The von Neumannn alternating projection method gives
PH
n+1 = lim
k→∞
(Pn+1PH)k
.
To approximate we employ a finite k (or other algorithms) and since
Pn+1 = Qn+1QT
n+1,
Pn+1δn+1 and PH
n+1H†yn+1 can be written as a linear combination of
the p < m columns of Qn+1.
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 16 / 20

17. Projected Particle Filter (ongoing with Jack Maclean
...)
The Filtering step to update the particle weights:
wi
n = c wi
n−1R(ˆxi
n, Yn)
where
R(x, y)
.
= exp −
1
2
y − Hx, y − Hx R−1
Simple calculation (back to phase space):
y − Hx, y − Hx R−1 = H†
y − PHx, H†
y − PHx ˜R−1
where ˜R−1 = HR−1HT .
Initial results for 40D Lorenz ’96 (F = 20, ≈ 20 positive Lyapunov
exponents).
Observe every other variable every 0.2 time units, p = 10 (dimension of
projected phase space).
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 17 / 20

18. Projected Particle Filter
0 1 2 3 4 5 6 7 8
Time
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Meansquarederror
No DA
Standard Particle Filter
Projected Particle Filter
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 18 / 20

19. Projected Particle Filter
10 20 30 40
-5
0
5
t= 0
10 20 30 40
-5
0
5
t= 1
10 20 30 40
-5
0
5
t= 2
10 20 30 40
-5
0
5
t= 3
10 20 30 40
-5
0
5
t= 4
10 20 30 40
-5
0
5
t= 5
10 20 30 40
-5
0
5
t= 6
10 20 30 40
-5
0
5
t= 7
Weighted particle distributions in each variable
(blue is projected PF, magenta is standard PF)
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 19 / 20

20. Direction
Different techniques for P and I − P system with Newton like iteration
between the systems.
Challenges
Developing effective ways to “communicate” between techniques
in P and I − P systems.
Bayesian interpretation.
Bench marking, testing, and comparison with existing techniques.
Application to Applications (Noah-MP/Ameriflux data, ...)
Erik S. Van VleckDepartment of MathematicsUniversity of Kansas SAMSI CLIM Transition WorkshopProjected Data Assimilation May 14, 2018 20 / 20