[PPT] - The 7th International Symposium on Data Assimilation (ISDA2019) PowerPoint Presentation

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The 7th International Symposium on Data Assimilation (ISDA2019) Efficient Implementations of Ensemble Based Methods In Sequential Data Assimilation: Accounting for Localization

Elias D. Ni˜ no-Ruiz Applied Math and Computer Science Laboratory (AML-CS) Department of Computer Science Universidad del Norte BAQ 080001, Colombia

E. Ni˜

no-Ruiz, ISDA2019 - RIKEN R-CCS

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Outline I

Data Assimilation Components Ensemble Based Methods The Stochastic Ensemble Kalman Filter Localization Methods Precision Matrix Localization Efficient EnKF-MC Shrinkage Covariance Matrix Estimation Ensemble Kalman filter based on RBLW Efficient Implementation of the RBLW EnKF-RBLW EnKF-MC and EnKF-RBLW with the SPEEDY Model Accuracy of the EnKF-MC Local Estimation of B−1 Accuracy of the EnKF-RBLW Parallel Implementations of Ensemble Based Methods Recent References References

E. Ni˜

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Components in DA [BS12] I

◮ We want to estimate x∗ ∈ ❘n×1. n ∼ O

108

. ◮ Imperfect numerical model: xnext = Mtcurrent→tnext (xcurrent) , where x ∈ ❘n×1. ◮ Noisy observations: y = H (x) + ǫ ∈ ❘m×1, where H : ❘n → ❘m and ǫ ∼ N (0m, R).m ∼ O

106

. ◮ Prior estimate xb ∈ ❘n×1 with errors following N (0, B).

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Components in DA [BS12] II

20 40 60 80 100 120 20 40 60 80 100 120

(a) x∗

2000 4000 6000 8000 10000 12000 14000 16000 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

(b) H

20 40 60 80 100 120 20 40 60 80 100 120

(c) y = H · x∗ + ǫ

20 40 60 80 100 120 20 40 60 80 100 120

(d) xb

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Components in DA [BS12] III

◮ By Bayes’ Theorem we know that: P (x|y) ∝ P (x) · L (x|y) where P (x) ∝ exp

−1

2 ·

x − xb
2

B−1

L (x|y)

∝ exp

−1

2 · y − H · x2

R−1

and therefore,

xa = arg max

x

P (x|y) ,

E. Ni˜

no-Ruiz, ISDA2019 - RIKEN R-CCS

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Components in DA [BS12] IV

◮ It can be easily shown that: xa = xb + A · HT · R−1 · d = A ·

B−1 · xb + HT · R−1 · y
=

xb + B · HT ·

R + H · B · HT−1

· d where A =

B−1 + HT · R−1 · H

−1 ∈ ❘n×n, and d = y − H · xb ∈ ❘m×1. ◮ Posterior distribution: x ∼ N (xa, A) . ◮ How do we estimate xb and B?.

E. Ni˜

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Ensemble Based Methods

◮ We can make use of an ensemble of model realizations: Xb =

xb[1], xb[2], . . . , xb[N]

∈ ❘n×N ◮ Empirical moments of the ensemble: xb ≈ xb = 1 N · Xb · 1N ∈ ❘n×n , B ≈ Pb = 1 N − 1 · δX · δXT , and δX = Xb − xb · 1T

N ∈ ❘n×N.

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The Lorenz 96 Model - Toy Model I

◮ The Lorenz 96 model: dxj dt =      (x2 − xn−1) · xn − x1 + F for i = 1, (xi+1 − xi−2) · xi−1 − xi + F for 2 ≤ i ≤ n − 1, (x1 − xn−2) · xn−1 − xn + F for i = n, (1) where xi stands for the i-th model component, for 1 ≤ i ≤ n, usually n = 40. ◮ Each model component stands for a particle which fluctuates in the atmosphere. ◮ Exhibits chaotic behaviour when the external force F is set to 8.

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The Lorenz 96 Model - Toy Model II

2 4 6 8 10

10
5

5 10 15

(e) x5

2 4 6 8 10

10
5

5 10 15

(f) x10

2 4 6 8 10

10
5

5 10 15

(g) x20

2 4 6 8 10

10
5

5 10 15

(h) x30

2 4 6 8 10

10
5

5 10 15

(i) x35

2 4 6 8 10

10
5

5 10 15

(j) x40

E. Ni˜

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Estimation of B via N = 105.

5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40

(a) Structure (b) Surf

Figure: Estimation of B via N = 105.

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The Stochastic Ensemble Kalman Filter [Eve03, Eve06] I

◮ Sequential Monte Carlo method for parameter and state estimation. ◮ Analysis ensemble (posterior ensemble): Xa = Xb + Pb · HT ·

R + H · Pb · H
· ∆Y

Xa = Xb + Pa · HT · R−1 · ∆Y ∈ ❘n×N, Xa = Pa ·

HT · R−1 · Ys +
Pb−1

· Xb

∈ ❘n×N,

where Pa =

HT · R−1 · H +
Pb−1−1

∈ ❘n×n, and the e-th column of ∆Y ∈ ❘m×N and Ys ∈ ❘n×N are: d[e] = y + ǫ[e] − H

xb[e]

∈ ❘m×1, and ys[e] = y + ǫ[e] , respectively, for 1 ≤ e ≤ N, and ǫ[e] ∼ N (0m, R).

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L − 2 Error Norms in Time, N = 105

5 10 15

0.5

0.5 1 1.5

(a) p = 50%

5 10 15

2
1

1 2

(b) p = 100%

Figure: L − 2 error norms in time, N = 105.

But too many samples!!! In practice, model realizations are constrained by the hundreds...

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L − 2 error norms in time, N = 10

5 10 15 1.38 1.4 1.42 1.44 1.46 1.48 1.5

(a) p = 50%

5 10 15 1.35 1.4 1.45 1.5

(b) p = 100%

Figure: L − 2 error norms in time, N = 10.

What is going on here?...

E. Ni˜

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Estimation of B via N = 10

5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40

(a) Structure (b) Surf

Figure: Estimation of B via N = 10.

What can we do? Localization methods...

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Localization Methods

◮ Avoid the impact of spurious correlations. ◮ Increase the rank of Pb. ◮ Three different flavors:

1. Covariance Matrix Localization. (Precision Localization)

[NRSD15, NRSD17, NR17, NRSD18].

2. Spatial Domain Localization [OHS+04].
3. Observation Localization [AND07, AND09].
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Precision Matrix Localization I

◮ Component-wise products are prohibitive in high-dimensional spaces. ◮ When two model components are conditional independent, their corresponding entry in the precision covariance matrix is zero.

(a) r = 0 (b) r = 1 (c) r = 3

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Precision Matrix Localization II

◮ Modified Cholesky Decomposition [BL+08]:

B−1 = TT · D−1 · T

where the non-zero elements from T ∈ ❘n×n are given by fitting models of the form: x[i] =

q∈P(i,r)

x[q] · {−T}i,q + ǫ[i] ∈ ❘N×1, for 1 ≤ i ≤ n , and {D}i,i = var

ǫ[i]

.

(a) N(6, 1) (b) P(6, 1)

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Precision Matrix Localization III

◮ An estimate:

(a) Pb

10 20 30 40 nz = 160 5 10 15 20 25 30 35 40

(b) T

10 20 30 40 nz = 40 5 10 15 20 25 30 35 40

(c) D

10 20 30 40 nz = 298 5 10 15 20 25 30 35 40

(d) B−1 Str (e) B−1 (f) B

Results:

E. Ni˜

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Precision Matrix Localization IV

5 10 15

2
1

1 2

(a) N = 30, r = 1, p = 100%

5 10 15

2
1

1 2

(b) N = 30, r = 3, p = 100%

5 10 15

2
1

1 2

(c) N = 30, r = 5, p = 100%

5 10 15 0.2 0.4 0.6 0.8 1 1.2 1.4

(d) N = 30, r = 1, p = 50%

5 10 15

1
0.5

0.5 1 1.5

(e) N = 30, r = 3, p = 50%

5 10 15 0.6 0.8 1 1.2 1.4

(f) N = 30, r = 5, p = 50%

E. Ni˜

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Efficient EnKF-MC I

Consider Xa = Xb +

B−1 + HT · R−1/2 · R−1/2 · H

−1 · HT · R−1 · ∆Y = Xb +

B−1 + Z · ZT−1

· HT · R−1 · ∆Y = Xb +   B−1 +

m

j=1

z[j] ·

z[j]T

 

−1

· HT · R−1 · ∆Y , z[j] ∈ ❘n×1 is the j-th column of Z = HT · R−1/2 ∈ ❘n×m.

E. Ni˜

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Efficient EnKF-MC II

A(0) =

T(0)T

· D(0) ·

T(0)

= TT · D · T = B−1 , A(1) = A(0) + z[1] ·

z[1]T

=

T(1)T

· D(1) ·

T(1)

, A(2) = A(1) + z[2] ·

z[2]T

=

T(2)T

· D(2) ·

T(2)

, . . . A(m) = A(m−1) + z[m] ·

z[m]T

=

T(m)T

· D(m) ·

T(m)

= TT · D · T = A−1 ,

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Efficient EnKF-MC III

at any intermediate step j, for 1 ≤ j ≤ m, we have, A(j) =

T(j−1)T

· D(j−1) ·

T(j−1)

+ z[j] ·

z[j]T

=

T(j−1)T

·

D(j−1) + p(j) ·
p(j)T

·

T(j−1)

, where

T(j−1)T · p(j) = z[j] ∈ ❘n×1. By computing the Cholesky

decomposition of, D(j−1) + p(j) ·

p(j)T

=

T(j−1)T

· D(j) ·

T(j−1)

, therefore, A(j) =

T(j−1) · T(j−1)T

· D(j) ·

T(j−1) · T(j−1)

=

T(j)T

· D(j) ·

T(j)

,

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Efficient EnKF-MC IV

where T(j) = T(j−1) · T(j−1) ∈ ❘n×n. Based on the Dolittle’s method,

T(j−1)T

· D(j) ·

T(j−1)

i,k

= δi,k ·

D(j−1)

i,i +

p(j)

i ·

p(j)

k ,

from which,

D(j)

n,n =

p(j)

n

2 +

D(j−1)

n,n ,

T(j−1)

i,k =

1

D(j)

i,i

·  

p(j)

i ·

p(j)

k −

q∈P(i, r)
D(j)

q,q ·

T(j−1)

q,i ·

T(j−1)

q,k

  ,

and

D(j)

i,i =

p(j)

i

2 +

D(j−1)

i,i −

q∈P(i, r)
D(j)

q,q ·

T(j−1)

q,i

2 ,

for n − 1 ≥ i ≥ 1 and k ∈ P(i, r), where δi,j is the Kronecker delta

function. Hence:

Xa = Xb + Q where

TT ·

D · T

· Q = HT · R−1 · ∆Y .
E. Ni˜

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Shrinkage Covariance Matrix Estimation I

◮ Samples {si}N

i=1, where si ∼ N (0n, C)

◮ Structure of matrices:

C = γ · T + (1 − γ) · Cs ∈ ❘n×n ,
ptimal value of γ in squared loss sense E
C − C
2

F

where

C ∈ ❘n×n is the true covariance matrix. T = tr(Cs)

n

· I. ◮ Properties:

◮ Have been proven more accurate than the sample covariance matrix [CM14]. ◮ Better conditioned than the true covariance matrix [CWEH10]. ◮ They are strong under the condition n ≫ N [CWH11].

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Shrinkage Covariance Matrix Estimation II

◮ Ledoit and Wolf estimator [LW04, CWEH10]: γLW = min   N

i=1

Cs − si ⊗ sT

i

2

F

N2 ·

tr (C2

s) − tr2(Cs) n

, 1   ◮ Rao-Blackwell Ledoit and Wolf estimator [CWEH10]: γRBLW = min  

N−2 n

· tr

C2

s

+ tr2 (Cs)

(N + 2) ·

tr (C2

s) − tr2(Cs) n

, 1   ◮ It is proven that [CWH11]: E

CRBLW − C
2

F

≤ E
CLW − C
2

F

.
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RBLW in the EnKF context

◮ Replace Pb by a better estimator of B. ◮ RBLW estimator in the EnKF context:

B

= γ

B ·

µ

B · In×n

+
1 − γ

B

·

δX · δX

T ∈ ❘n×n .

where δX =

1 √ N−1 · δX ∈ ❘n×N.

◮ Parameters: µ

B

= tr

Pb

n γ

B

= min    

N−2 n

· tr

Pb2

+ tr2 Pb (N + 2) ·

tr
Pb2

−

tr2(Pb) n

, 1     ◮ The direct implementation is prohibitive, recall n ∼ O

108

.

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Efficient Implementation of the RBLW I

◮ Recall: tr

Pb

=

n

i=1

σi =

N−1

i=1

σi tr

Pb2

=

n

i=1

σ2

i = N−1

i=1

σ2

i .

◮ Note Pb =

δX ·

δX

T =

U

δX ·

Σ

δX · VT

δX
·
U

δX ·

Σ

δX · VT

δX

T = U

δX ·

Σ

2

δX · UT
δX
E. Ni˜

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Efficient Implementation of the RBLW II

this implies σi

Pb

= σi

2

δX
,

for 1 ≤ i ≤ N − 1. ◮ The estimator reads:

B

= γ

B ·

µ

B · In×n

+
1 − γ

B

·

δX · δX

T ∈ ❘n×n .

◮ Efficient computation of the parameters: µ

B

= N−1

i=1

σi

2

n , γ

B

= min    

N−2 n

· N−1

i=1

σi

4 +

N−1

i=1

σi

22

(N + 2) · N−1

i=1

σi

4 − [ N−1

i=1

σi 2]

2

n

, 1     .

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EnKF-RBLW Implementation

◮ EnKF model space, with ϕ = µ

B · γ B and δ = 1 − γ B:

Xa = Xb + E · Π · Z

B + ϕ · HT · Z B,

where E = √ δ · δX ∈ ❘n×N, Π = H · E ∈ ❘m×N, and Z

B ∈ ❘m×N:

Γ + Π · ΠT

· Z

B

=

Y − H
Xb

, Γ = R + ϕ · H · HT ∈ ❘m×m.

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EnKF-MC and EnKF-SC with the SPEEDY Model I

◮ We make use of FORTRAN 90 in order to code the EnKF-MC and the EnKF-RBLW (from now on EnKF-SC). ◮ 96 ensemble members were used for the experiments. ◮ The initial perturbation of the background state is 5% the true state of the system. ◮ The model is propagated for a period of 24 days, observations are taken every 2 days. ◮ The SPEEDY model is used with T-63 resolution (96 × 192) with 4 variables. 8 layers per variable. n ≈ 590, 000. ◮ Three sparse observational networks were used for the tests. ◮ We compare the results with the LETKF.

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EnKF-MC and EnKF-SC with the SPEEDY Model II

(g) p = 12% (h) p = 6% (i) p = 4%

Figure: Observational networks for different values of p.

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Accuracy of the EnKF-MC I

Time (days) 5 10 15 20 25 Root Mean Square Error (RMSE) 500 1000 1500 2000 2500 3000 3500 4000 4500 Zonal Wind Component (U), (m/s) Background EnKF-MC LETKF

(a) r = 3 and p = 12%

Time (days) 5 10 15 20 25 Root Mean Square Error (RMSE) 500 1000 1500 2000 2500 3000 3500 4000 4500 Zonal Wind Component (U), (m/s) Background EnKF-MC LETKF

(b) r = 5 and p = 6%

Figure: RMSE of the LETKF and EnKF-MC implementations for different model variables, radii of influence and observational networks.

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Accuracy of the EnKF-MC II

(a) Reference (b) Background (c) EnKF-MC (d) LETKF

Figure: 5-th layer of the meridional wind component (v).

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Accuracy of the EnKF-MC III

(a) Reference (b) Background (c) EnKF-MC (d) LETKF

Figure: 2-th layer of the zonal wind component (u).

E. Ni˜

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Local Estimation of B−1

(a) T (b) B−1 (c) B (d) B

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SLIDE 36

Accuracy of the EnKF-RBLW I

(e) r = 3 and p = 12% (f) r = 5 and p = 6%

Figure: RMSE of the LETKF and EnKF-RBLW implementations for different model variables, radii of influence and observational networks.

E. Ni˜

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SLIDE 37

Accuracy of the EnKF-RBLW II

(a) Reference (b) Background (c) EnKF-RBLW (d) LETKF

Figure: 5-th layer of the meridional wind component (v).

E. Ni˜

no-Ruiz, ISDA2019 - RIKEN R-CCS

SLIDE 38

Accuracy of the EnKF-RBLW III

(a) Reference (b) Background (c) EnKF-RBLW (d) LETKF

Figure: 2-th layer of the zonal wind component (u).

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Parallel implementations of ensemble based methods

◮ Blueridge Super Computer @ VT

◮ BlueRidge is a 408-node Cray CS-300 cluster. ◮ Each node is outfitted with two octa-core Intel Sandy Bridge CPUs and 64 GB of memory. ◮ Total of 6,528 cores and 27.3 TB of memory systemwide. ◮ Eighteen nodes have 128 GB of memory. ◮ In addition, 130 nodes are outfitted with two Intel MIC (Xeon Phi) coprocessors.

◮ The methods are coded in FORTRAN using MPI. ◮ LAPACK [ABD+90] and BLAS [BDD+01] are used in order to efficiently perform matrix computations. ◮ We vary the number of processors from 96 (16 computing nodes) to 2,048 (128 computing nodes)

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SLIDE 40

Parallel implementations of ensemble based methods I

◮ The approximations are based on domain decomposition

(a) 12 (b) 80

E. Ni˜

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Parallel implementations of ensemble based methods II

◮ Boundary information

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Parallel implementations of ensemble based methods III

◮ Accuracy (EnKF-MC): number of processors ranges from 96 (16 computing nodes) to 2,048 (128 computing nodes)

Time (days) 10 20 30 Root Mean Square Error (RMSE) 100 150 200 250 300 350 400 450

Specific Humidity (g/Kg)

Background EnKF-MC 6 EnKF-MC 16 EnKF-MC 32 EnKF-MC 48 EnKF-MC 64 EnKF-MC 96 EnKF-MC-128 LETKF 6 LETKF 16 LETKF 32 LETKF 48 LETKF 64 LETKF 96 LETKF 128

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SLIDE 43

Parallel implementations of ensemble based methods IV

◮ Accuracy (EnKF-MC): number of processors ranges from 96 (16 computing nodes) to 2,048 (128 computing nodes)

Time (days) 10 20 30 Root Mean Square Error (RMSE) 500 1000 1500 2000 2500 3000 3500 4000 4500

Zonal Wind Component (U), (m/s)

Background EnKF-MC 6 EnKF-MC 16 EnKF-MC 32 EnKF-MC 48 EnKF-MC 64 EnKF-MC 96 EnKF-MC-128 LETKF 6 LETKF 16 LETKF 32 LETKF 48 LETKF 64 LETKF 96 LETKF 128

E. Ni˜

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SLIDE 44

Parallel implementations of ensemble based methods V

◮ Accuracy (EnKF-RBLW): number of processors ranges from 96 (16 computing nodes) to 2,048 (128 computing nodes)

Time (days) 10 20 30 Root Mean Square Error (RMSE) 1000 2000 3000 4000 5000

Zonal Wind Component (U), (m/s)

Background EnKF-SC 6 EnKF-SC 16 EnKF-SC 32 EnKF-SC 48 EnKF-SC 64 EnKF-SC 96 EnKF-SC 128 LETKF 6 LETKF 16 LETKF 32 LETKF 48 LETKF 64 LETKF 96 LETKF 128

E. Ni˜

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SLIDE 45

Parallel implementations of ensemble based methods VI

◮ Accuracy (EnKF-RBLW): number of processors ranges from 96 (16 computing nodes) to 2,048 (128 computing nodes)

Time (days) 10 20 30 Root Mean Square Error (RMSE) 50 100 150 200 250 300 350 400 450

Specific Humidity (g/Kg)

Background EnKF-SC 6 EnKF-SC 16 EnKF-SC 32 EnKF-SC 48 EnKF-SC 64 EnKF-SC 96 EnKF-SC 128 LETKF 6 LETKF 16 LETKF 32 LETKF 48 LETKF 64 LETKF 96 LETKF 128

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SLIDE 46

Parallel implementations of ensemble based methods VII

◮ Computational time: number of processors ranges from 96 (16 computing nodes) to 2,048 (128 computing nodes)

Computing nodes (x 16 processors) 50 100 150 Time (s) 500 1000 1500 EnkF-MC EnKF-SC LETKF

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SLIDE 47

EnKF-MC Publications

1. Elias D. Nino-Ruiz, Adrian Sandu, and Xinwei Deng. ”An

Ensemble Kalman Filter Implementation Based on Modified Cholesky Decomposition for Inverse Covariance Matrix Estimation”, SIAM Journal on Scientific Computing 40:2, A867-A886 (2018).

2. Elias D. Nino-Ruiz, Adrian Sandu, and Xinwei Deng. ”A

parallel implementation of the ensemble Kalman filter based

n modified Cholesky decomposition”, Journal of

Computational Science, Elsevier, (2017).

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SLIDE 48

EnKF-SC Publications

1. Elias D. Nino-Ruiz, and Adrian Sandu. ”Efficient Parallel

Implementation of DDDAS Inference using an Ensemble Kalman Filter with Shrinkage Covariance Matrix Estimation”. Cluster Computing, Springer. (2017).

2. Cosmin G. Petraa, Victor M. Zavalab, Elias D. Nino-Ruiz, and

Mihai Anitescud. ”A high-performance computing framework for analyzing the economic impacts of wind correlation.” Electric Power Systems Research, Elsevier, 141 (2016): 372-380.

3. Nino-Ruiz, Elias D., and Adrian Sandu. ”Ensemble Kalman

filter implementations based on shrinkage covariance matrix estimation.” Ocean Dynamics, Springer, 65.11 (2015): 1423-1439.

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Bibliography I

[ABD+90]

E. Anderson, Z. Bai, J. Dongarra, A. Greenbaum, A. McKenney, J. Du Croz, S. Hammerling,
J. Demmel, C. Bischof, and D. Sorensen. LAPACK: A Portable Linear Algebra Library for

High-performance Computers. In Proceedings of the 1990 ACM/IEEE Conference on Supercomputing, Supercomputing ’90, pages 2–11, Los Alamitos, CA, USA, 1990. IEEE Computer Society Press. [AND07] JEFFREY L. ANDERSON. An adaptive covariance inflation error correction algorithm for ensemble

filters. Tellus A, 59(2):210–224, 2007.

[AND09] JEFFREY L. ANDERSON. Spatially and temporally varying adaptive covariance inflation for ensemble

filters. Tellus A, 61(1):72–83, 2009.

[BDD+01]

L. S. Blackford, J. Demmel, J. Dongarra, I. Duff, S. Hammarling, G. Henry, M. Heroux, L. Kaufman,
A. Lumsdaine, A. Petitet, R. Pozo, K. Remington, and R. C. Whaley. An Updated Set of Basic Linear

Algebra Subprograms (BLAS). ACM Transactions on Mathematical Software, 28:135–151, 2001. [BL+08] Peter J Bickel, Elizaveta Levina, et al. Regularized estimation of large covariance matrices. The Annals of Statistics, 36(1):199–227, 2008. [BS12]

M. Bocquet and P. Sakov. Combining Inflation-free and Iterative Ensemble Kalman Filters for

Strongly Nonlinear Systems. Nonlinear Processes in Geophysics, 19(3):383–399, 2012. [CM14] Romain Couillet and Matthew McKay. Large Dimensional Analysis and Optimization of Robust Shrinkage Covariance Matrix Estimators. Journal of Multivariate Analysis, 131(0):99–120, 2014. [CWEH10] Yilun Chen, A Wiesel, Y.C. Eldar, and AO. Hero. Shrinkage Algorithms for MMSE Covariance

Estimation. Signal Processing, IEEE Transactions on, 58(10):5016–5029, Oct 2010.

[CWH11] Yilun Chen, A Wiesel, and AO. Hero. Robust Shrinkage Estimation of High-Dimensional Covariance

Matrices. Signal Processing, IEEE Transactions on, 59(9):4097–4107, Sept 2011.

[Eve03] Geir Evensen. The ensemble kalman filter: Theoretical formulation and practical implementation. Ocean dynamics, 53(4):343–367, 2003. [Eve06] Geir Evensen. Data Assimilation: The Ensemble Kalman Filter. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006.

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Bibliography II

[LW04] Olivier Ledoit and Michael Wolf. A Well-conditioned Estimator for Large-dimensional Covariance

Matrices. Journal of Multivariate Analysis, 88(2):365 – 411, 2004.

[NR17] Elias D Nino-Ruiz. A matrix-free posterior ensemble kalman filter implementation based on a modified cholesky decomposition. Atmosphere, 8(7):125, 2017. [NRSD15] Elias D. Nino-Ruiz, Adrian Sandu, and Xinwei Deng. A parallel ensemble kalman filter implementation based on modified cholesky decomposition. In Proceedings of the 6th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems, ScalA ’15, pages 4:1–4:8, New York, NY, USA, 2015. ACM. [NRSD17] Elias D Nino-Ruiz, Adrian Sandu, and Xinwei Deng. A parallel implementation of the ensemble kalman filter based on modified cholesky decomposition. Journal of Computational Science, 2017. [NRSD18] Elias D Nino-Ruiz, Adrian Sandu, and Xinwei Deng. An ensemble kalman filter implementation based

n modified cholesky decomposition for inverse covariance matrix estimation. SIAM Journal on

Scientific Computing, 40(2):A867–A886, 2018. [OHS+04] Edward Ott, Brian R. Hunt, Istvan Szunyogh, Aleksey V. Zimin, Eric J. Kostelich, Matteo Corazza, Eugenia Kalnay, D. J. Patil, and James A. Yorke. A local ensemble kalman filter for atmospheric data

assimilation. Tellus A, 56(5):415–428, 2004.
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