THE DIFFUSION KERNEL

IN PREDICTION PROBLEMS

Paul Krause

Math. Arizona

8th ADJOINT WORKSHOPTannersville, PAMay, 18-22, 2009

thanks to organizers

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2

(1) GLANCE:

working with

• NON-GAUSSIAN FILTERING meth-ods for

• continuous-time deterministic models• discrete-time noisy data

Model :d

dtΦ = f (Φ)

Initial : Φ0 = random

Data : yk = h(Φtk) + noise

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issue is

• COST effectiveness

while preserving

• MATH consistency• PHYS consistency

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• MATH consistency

Are we closely solving for Φ|y ?

0 5 10 15 20−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

TIME

3rd

MO

ME

NT

VORTEX 3rd MOMENT in 2−VORTEX OCEAN LDA

EXACT 3rd MOMENT500K−DKF 3rd MOMENT500K−enKF 3rd MOMENT

Reference:The Diffusion Kernel Filter applied to Lagrangian Data Assimila-tion; w/ J.M. Restrepo; MWR, (2009)

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Is that needed?

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

X

Y

VORTEX PATH in 2−VORTEX OCEAN LDA

TRUE PATHEXACT AVG ESTIMATE500K−DKF AVG ESTIMATE500K−enKF AVG ESTIMATE

Reference:The Diffusion Kernel Filter applied to Lagrangian Data Assimila-tion; w/ J.M. Restrepo; MWR, (2009)

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• PHYS consistency

Are we closely sampling from the deter-ministic model dynamics ?

98 98.2 98.4 98.6 98.8 99 99.2 99.4 99.6 99.8 1000

0.5

1

1.5

2

2.5

3

3.5

4ESTIMATE ERROR in L63 FILTERING

ER

RO

R (

2−N

OR

M)

TIME

AVERAGE ESTIMATE ERRORAVG ENTROPY ESTIMATE ERRORMAX LIKEHD−WGHT ESTIMATE ERR

Reference:The Diffusion Kernel Filter; J. Stat. Phy., 134:2 (2009), pp. 365-380

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(2) METHOD:

• STEP 1: cast non-Gaussian initial un-certainty into Gaussian mixture of ini-tial uncertainties

• STEP 2: cast Gaussian initialuncertainty into model noise andsolve the filtering problem forGaussian branches with Kalmananalysis

• STEP 2’: parallelize reduced TangentOperator-based forecasts, not sample-based forecasts

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• STEP 2: cast Gaussian initialuncertainty into model noise

d

dtΦ = f (Φ)

Φ0 ∼ N (φ0, C0)

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dΦ = f (Φ) dt + g dw

(dw/dt = white)

Φ0 ∼ Dirac(φ0)

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(in distribution)

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RESULT: linear regime

Pt = Mt(tggT)MT

t, where

Mt := Tangent Operator obtained fromTLM about φt

Reference:The Diffusion Kernel Filter; J. Stat. Phy., 134:2 (2009), pp. 365-380

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SINCE: linear regime

Ct = MtC0MT

t, where

Mt := Tangent Operator obtained fromTLM about φt

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/——————–/τ

Cτ = Pτ for ggT := C0τ

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ANSATZ: REDUCED FORMULA

Ct := Mt(tge(t)ge(t)T)MT

t, where

ge(t) := (∫ ttk

Df (φs)ds) g + g

• ge models the effect of the unresolved compo-nents of Ct onto Ct

• τge(τ )ge(τ ) at filtering time tk+1 involves τggT =

diag(Ck,Ck), where Ck is the value at time tk ofthe unresolved block of Ct within the predictioninterval [tk, tk+1]

• Ck would be roughly estimated on a fast sideprocess. Here, it is set to R (w/ the corre-sponding analysis vars set to data values, in allbranches).

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(3) TESTS:

Lorenz-63 flows with 1 time-unit long pre-diction steps and observation function h :R×R×R → R×R, h(Φ1, Φ2, Φ3) = (Φ2

1+

Φ22, Φ3), with error ǫ ∼ N (0, diag([10−2, 10−4]))

STEP 1: launch C0 with different caliberfor different branches

STEP 2: TLM is solved just for the firsttwo columns of Mt

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0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

TIME

2−N

OR

M E

RR

OR

5−BRANCH VARS−1,2 ESTIMATE ERROR

MAX LIKEHD ESTIMATE ERROR ae−ESTIMATE ERROR AVG ESTIMATE ERROR

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2D periodic rotating shallow water flowswith 15 mins long prediction steps on a50 km×50 km domain covered by a 50×50grid with observation function h : R

50·50 ×R

50·50×R50·50 → R

50·50×R50·50, h(u, v, η) =

(u2+v2, η), with error ǫ ∼ N (0, diag(10−4))(similar results were obtained with 1 hr longprediction steps on a 70 × 70 grid)

STEP 1: launch C0 with different caliberfor different branches

STEP 2: TLM is solved just for the u, vcolumns of Mt

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0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1

0.15

0.2

0.25

0.3

TIME (hrs)

RM

S E

RR

OR

(m

/s)

(U,V)−FIELD ae−ESTIMATE RMS ERROR

2−BRANCH ae−ESTIMATE ERROR 4−BRANCH ae−ESTIMATE ERROR 5−BRANCH ae−ESTIMATE ERROR

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0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1

0.15

0.2

0.25

0.3

TIME (hrs)

RM

S E

RR

OR

(m

/s)

5−BRANCH (U,V)−FIELD ESTIMATE RMS ERROR

MAX LIKEHD ESTIMATE ERROR ae−ESTIMATE ERROR AVG ESTIMATE ERROR

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0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

X (km)

Y (

km)

INITIAL (U,V)−FIELD TRUTH

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

X (km)

Y (

km)

INITIAL (U,V)−FIELD ae−ESTIMATE

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0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

X (km)

Y (

km)

FINAL (U,V)−FIELD TRUTH

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

X (km)

Y (

km)

FINAL (U,V)−FIELD ae−ESTIMATE

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0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

TIME (hrs)

RM

S E

RR

OR

(m

/s)

(U,V)−FIELD ae−ESTIMATE RMS ERROR

5−BRANCH g

e=g−hat

5−BRANCH full ge

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(4) CONCLUSION:

we’ve got

• NON-GAUSSIAN “LINEAR” FILTER• COST effective? (reduced formula)• MATH consistent (within settings

where the formula applies)• PHYS consistent (ae-estimate)

THANKS!