Nonlinear data assimilation via hybrid particle …...2017/07/08 · Nonlinear data assimilation...
Transcript of Nonlinear data assimilation via hybrid particle …...2017/07/08 · Nonlinear data assimilation...
Nonlinear data assimilation via hybridparticle-Kalman filters and optimal coupling
Walter Acevedo, Sebastian Reich and Maria ReinhardtUniversität Potsdam, Germany
RIKEN International Symposium on Data Assimilation 201727 February – 2 March, 2017, AICS, Kobe, Japan
Sequential Data Assimilation Strategies
Traditional approachesÉ Ensemble Kalman Filters
+ Robust and moderately affordable– Biased for non-Gaussian PDFs
É Traditional Particle Filters+ Non-Gaussianity properly handled– Liable to the "Curse of
Dimensionality”
Some recent developmentsÉ Localized particle filters
[Lei and Bickel, 2011][Poterjoy, 2015]É Hybrid Kalman-particle filters
[Frei and Künsch, 2013][Chustagulprom et al., 2016]É Optimal transportation based particle filters
[Reich, 2013]É Intrinsic Dimension [Agapiou et al,
arXiv:1511.06196],
Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 2
Outline
Motivation
Linear Ensemble Transform Filters
Hybrid Ensemble Transform Filter
Single one-dimensional DA step
Non-spatially extended dynamical systems
Spatially extended systems
Conclusions and Prospect
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 3
Linear Ensemble Transform Filter (LETF)
É Forecast and Analysis ensembles:
zfi, za
i, i = 1, . . . ,M
É Update step via a General linear transformation[Reich and Cotter, 2015]
zaj
=M∑
i=1
zfidij
É Transform coefficients satisfy
M∑
i=1
dij = 1
M∑
j=1
dij = Mwi
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 5
Ensemble Square Root Filter [Tippett et al., 2003]
ESRF as a LETF:
zaj
=M∑
i=1
zfiwi +
M∑
i=1
(zfi− zf)sij =
M∑
i=1
zfidKF
ij
É Transform coefficients:
dKFij
= dKFij
({zfl},yobs) := sij + wi −
1
M,
É Square root matrix: S = {sij}
S =
�
I +1
M− 1(HAf)TR−1HAf
�−1/2
,
É ESRF “Weights”:
wi =1
M−
1
M− 1eT
iS2(HAf)TR−1(Hzf − yobs),
M∑
i=1
wi = 1
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 6
Ensemble Transform Particle Filter [Reich, 2013]
Analysis Mean
za =M∑
i=1
wizfi, wi ∼ exp
�
−1
2(Hzf
i− yobs)TR−1(Hzf
i− yobs)
�
ETPF transform update DF : {dPFij }
zaj
=M∑
i=1
zfidPF
ij
Transform constraints:M∑
i=1
dij = 1,M∑
j=1
dij = wiM, dij ≥ 0,
Minimal Transportation Cost:
DPF = argmin J(D), J(D) =M∑
i,j=1
dij‖zfi− zf
j‖2
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 7
Optimal Transportation (OT) Problem
5 10 15 200
0.02
0.04
0.06
0.08
0.1
Imp
ort
an
ce
we
igth
Particle Number
⇒ OptimalCoupling
⇒
5 10 15 200
0.02
0.04
0.06
0.08
0.1
Imp
ort
an
ce
we
igth
Particle Number
Optimal transportation cost: min J(D)
É also known as Earth Mover distanceÉ takes into account shape of the PDFs
Optimal coupling: DPF
É maximizes correlation between prior and posterior ensemblesÉ is deterministic, preserving the regularity of the state fieldsÉ consistent with Bayes’ formula [Reich 2013]É convergence (regarding ensemble size) is faster than SIR-PFs at the
cost of solving an optimal transportation problem
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 8
OT Solvers
Exact solution
É Computational complexity O(M3 ln(M))
É Efficient Earth Mover Distance algorithms available, e.g. FastEMD[Pele and Werman, 2009].
Entropic Regularized Approximation [Cuturi, 2013]
J(D) =M∑
i,j=1
�
dij‖zfi− zf
j‖2 +
1
λdij lndij
�
É Sinkhorn’s fixed point iteration can be used.É Computational complexity O(M2 ·C(λ))
1D Approximation(Each variable independently updated)É OT problem reduces to reordering.É Computational complexity O(M ln(M))
É No particle distance needed
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 9
Example: Single 1D Assimilation Step
Observation within the convex hull
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
0.0
0.5
1.0
1.5
2.0
2.5
3.0ETPF weights
Obser ation
Prior ensemble
ETPF analysis
ETKF analysis
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 10
Example: Single 1D Assimilation Step
Observation within the convex hull
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Pos
erior In
dex
ER ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Par icle Index
0.0
0.5
1.0
1.5
2.0
2.5
3.0ER ETPF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Pos
erior In
dex
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Par icle Index
0.0
0.5
1.0
1.5
2.0
2.5
3.0ETPF weights
Observa ion
Prior ensemble
ETPF analysis
En ropic Reg. ETPF analysis
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 11
Example: Single 1D Assimilation Step
Observation outside the convex hull
0 1 2 3 4 5 6
Prior Inde
0
1
2
3
4
5
6
Posterior Inde
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Inde
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Inde
0
1
2
3
4
5
6
Posterior Inde
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Inde
0
1
2
3
4
5
6ETPF weights
Observation
Prior ensemble
ETPF analysis
ETKF analysis
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 12
Example: Single 1D Assimilation Step
Likelihood splitting [Frei and Künsch, 2013]
πY(yobs|z) ∝exp�
−1
2(eyz)TR−1
eyz
�
, eyz = Hz − yobs
∝ exp�
α
�
−1
2(eyz)TR−1
eyz
��
× exp�
(1− α)
�
−1
2(eyz)TR−1
eyz
��
∝ exp
�
−1
2(eyz)T
�
R
α
�−1eyz
�
× exp
�
−1
2(eyz)T
�
R
1− α
�−1eyz
�
⇓ ⇓ETPF ESRF
Bridging parameter extrems:É α = 0 ⇒ Pure Kalman FilterÉ α = 1 ⇒ Pure Particle Filter
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 14
Hybrid Ensemble Transform Filter
Models explored:
É Single DA step
É Lorenz 63
É Lorenz 96
É Lorenz 96 coupled to a wave equation
É Shallow water equations (2D)
É Modified shallow water equations (1D)
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 15
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 1
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 17
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0.9
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 18
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0.8
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 19
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0.7
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 20
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0.6
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 21
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0.5
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 22
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0.4
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 23
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0.3
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 24
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0.2
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 25
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0.1
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 26
Single one-dimensional DA step
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETKF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETKF weights
0 1 2 3 4 5 6
Prior Index
0
1
2
3
4
5
6
Posterior Index
ETPF transform
−1.2
−0.9
−0.6
−0.3
0.0
0.3
0.6
0.9
1.2
0 1 2 3 4 5 6 7
Particle Index
−3
−2
−1
0
1
2
3
4
5ETPF weights
Bridging Parameter = 0
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 27
Single one-dimensional DA step
Bayesian Inference
for bimodal prior
and
Gaussian likelihood
-10 -5 0 5 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45Baysian inference
priorlikelihoodposterior
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 28
Single one-dimensional DA step
ETPF-ESRF
performance vs
ensemble size
101
102
103
RM
S e
rro
r
0
0.1
0.2
0.3
0.4ETPF-ESRF
ensemble size10
110
210
3
optim
alα
0
0.2
0.4
0.6
0.8
1
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 29
Fighting ensemble under-dispersion
Particle rejuvenation:
zaj→ za
j+
M∑
i=1
(zfi− zf)
βξijp
M− 1
É β: Rejuvenation parameterÉ {ξij}: i.i.d. Gaussian random variables with mean zero and
variance one
Properties:É Increase ensemble spread while preserving ensemble meanÉ Ameliorates particle degeneracyÉ Does not destroy spatial regularity
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 31
Example: Lorenz 63 model
Dynamical system:
x1 = 10(x2 − x1)
x2 = x1(28− x3)− x2
x3 = x1x2 − 8/3x3−20
−100 10
20−50
0
50
0
5
10
15
20
25
30
35
40
45
50
Perfect model DA experiments:É x1 observed every 12 time-stepsÉ Observation error variance R = 8
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 32
Example: Lorenz 63 model
PF-KF RMSE with exact OT solver
PF-KF CRPS with exact OT solver
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 33
Update ordering dependency
Possible updating sequences:É ETPF-ESRF
zhj
=M∑
i=1
zfidPF
ij, dPF
ij:= dPF
ij(α,{zf
l},yobs),
zaj
=M∑
i=1
zhidKF
ij, dKF
ij:= dKF
ij(α,{zh
l},yobs).
É ESRF-ETPF
zhj
=M∑
i=1
zfidKF
ij, dKF
ij:= dKF
ij(α,{zf
l},yobs),
zaj
=M∑
i=1
zhidPF
ij, dPF
ij:= dPF
ij(α,{zh
l},yobs).
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 34
Updating order dependency
PF-KF RMSE with exact OT solver
KF-PF RMSE with exact OT solver
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 35
OT solver dependency
PF-KF RMSE with exact OT solver
PF-KF RMSE solving entropic regularized OT problem
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 36
OT solver dependency
PF-KF RMSE with exact OT solver
PF-KF RMSE solving 1D OT problem
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 37
Localization
R-Localization
rLOCqq
(xk) =rqq
ρ�
‖xk−xq‖Rloc
� ,
with ρ a compactly supported tempering function.
É Directly applicable to Kalman FiltersÉ Directly applicable to importance weights
Particle distance Localization [Cheng and Reich, 2015]
cij(x) =
∫
R
ρ
�‖xk − xq‖
Cloc
�
‖zfi(xq)− zf
j(xq)‖2dxq
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 39
Example: Lorenz 96 model
Dynamical system:
xj = (xj+1 − xj−2)xj−1 − xj + F,
xj = xj+N
where F = 8 and N = 40.
Perfect model DA experiments:É Odd variables observed every 22 time-stepsÉ Observation error variance R = 8É Particle rejuvenation β = 0.2É Localization radius is Rloc = 4É OTP solved using FastEMD algorithm
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 40
Example: Lorenz 96 model
Skill dependence on
bridging parameter
for different
ensemble sizes
using fixed
likelihood splitting
bridging parameter α0 0.1 0.2 0.3 0.4 0.5 0.6
RM
S e
rror
1.55
1.6
1.65
1.7
1.75hybrid ETPF-LETKF
M=20M=25M=30
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 41
Adaptive likelihood splitting
Normalized effective sample size: θ(α) =1
M∑M
i=1 wi(α)2
θ vs α for different
obs. error levels
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
α
θ
R=1
R=2
R=4
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 42
Adaptive likelihood splitting
Normalized effective sample size: θ(α) =1
M∑M
i=1 wi(α)2
θ vs α for different
obs. error levels
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
α
θ
R=1
R=2
R=4
θinf
Adaptivity criterion: Set α to the maximal value for which θ ≥ θinf .
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 43
Adaptive likelihood splitting
Skill dependence on
minimum effective
sample size
for different
ensemble sizes
using adaptive
likelihood splitting
0.75 0.8 0.85 0.9 0.95 1
RM
S e
rror
1.55
1.6
1.65
1.7
1.75hybrid ETPF-LETKF
M=20M=25M=30
θinf
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 44
Adaptive likelihood splitting
Skill dependence on
ensemble size for
both update orders
using optimal fixed
bridging parameter
10 20 30 40 50 60 701.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Ensemble Size
Tim
e a
ve
rag
e R
MS
err
or
Hybrid ETKF−ETPF
hybrid ETPF−LETKF
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 45
Conclusions
É Proposed Hybrid scheme
allows consistent localization for both filters,
preserves model state regularity,
outperforms both ETKF and ETPF for a suitable α
É Approximated OT solution approaches are promising
cheaper options, in particular 1D approximation
É Update ordering sensitivity is model-dependent
É Adaptive likelihood splitting benefits spatially extended
systems.
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 47
Ongoing Research and Prospect
É ETPF and hybrid scheme being implemented for the
German Weather Service (DWD)
É Second order accurate Ensemble Transform Particle Filter
(under review)
É 1D OT approximation and then multivariate dependence
recovery via Ensemble copula coupling [Schefzik et al., 2013]
É Generation of unbalanced fields through localization
currently being addressed via mollification.
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 48
References I
Cheng, Y. and Reich, S. (2015).Nonlinear Data Assimilation, chapter Assimilating data into scientificmodels: An optimal coupling perspective, pages 75–118.
Chustagulprom, N., Reich, S., and Reinhardt, M. (2016).A hybrid ensemble transform particle filter for nonlinear and spatiallyextended dynamical systems.SIAM/ASA J. Uncertainty Quantification, 4(1):592–608.
Cuturi, M. (2013).Sinkhorn distances: Lightspeed computation of optimal transport.In Burges, C., Bottou, L., Welling, M., Ghahramani, Z., and Weinberger,K., editors, Advances in Neural Information Processing Systems 26,pages 2292–2300. Curran Associates, Inc.
Frei, M. and Künsch, H. R. (2013).Bridging the ensemble kalman and particle filters.Biometrika.
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 49
References II
Lei, J. and Bickel, P. (2011).A moment matching ensemble filter for nonlinear non-gaussian dataassimilation.Mon. Wea. Rev., 139(12):3964–3973.
Pele, O. and Werman, M. (2009).Fast and robust earth mover’s distances.In ICCV.
Poterjoy, J. (2015).A localized particle filter for high-dimensional nonlinear systems.Mon. Wea. Rev., 144(1):59–76.
Reich, S. (2013).A non-parametric ensemble transform method for bayesian inference.SIAM J Sci Comput, 35:A2013–A2014.
Reich, S. and Cotter, C. (2015).Probabilistic Forecasting and Bayesian Data Assimilation.Cambridge University Press.
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 50
References III
Schefzik, R., Thorarinsdottir, T. L., and Gneiting, T. (2013).Uncertainty quantification in complex simulation models usingensemble copula coupling.pages 616–640.
Tippett, M. K., Anderson, J. L., Bishop, C. H., Hamill, T. M., andWhitaker, J. S. (2003).Ensemble square root filters.Monthly Weather Review, 131(7):1485–1490.
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Thanks!
,Nonlinear DA via hybrid particle-Kalman filters and optimal coupling 52