A Sampling Filter forNon-Gaussian Data Assimilation

Ahmed Attia1 and Adrian Sandu1

1Computational Science Laboratory (CSL)Department of Computer Science

Virginia Tech{attia,sandu}@cs.vt.edu

[1/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Outline

I MotivationI Problem formulationI Sampling approach: HMCMCI Experiment and resultsI Future workI Questions

Outline [2/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Motivation:

I A Monte Carlo approach capable of finding a good analysis for DAproblems in both Gaussian and non-Gaussian frameworks!

I Why don’t we sample directly from the posterior PDF?I A robust, fast sampling strategy is required!I MCMC: popular and guaranteed to converge but may take forever!I Auxiliary variable MCMC: Hybrid Monte Carlo (HMCMC);

I Fast convergence rate,I High acceptance probability of generated states,I Fairly easy to tune the parameters and enhance performance.

Motivation [3/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Sequential DA:I From Bayes’ Theorem:

Pa(x) = P(x|y) =P(y|x)Pb(x)

P(y), (1)

∝ P(y|x)Pb(x) . (2)

I Gaussian framework:

Pb(x) ∝ exp(−1

2(x− xb)T B−1(x− xb)

), (3)

P(y|x) ∝ exp(−1

2(H(x)− y)T R−1(H(x)− y)

). (4)

I Posterior PDF:

Pa(x) ∝ exp (−J (x)) , (5)

J (x) =12

(x− xb)T B−1(x− xb) +12

(H(x)− y)T R−1(H(x)− y) . (6)

Problem Formulation [4/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Hybrid MC:I HMCMC: Analogy to Hamiltonian mechanical system.I To draw samples {x(e)} from a given probability distribution π(x):

- View the state variable as a position variable,- Add an auxiliary variable ”momentum“ p and sample from the joint PDF of

both variables (p, x).+ The Hamiltonian:

H(p, x) =12

pT M−1p− log(π(x)) =12

pT M−1p︸ ︷︷ ︸kinetic energy

+ J (x)︸ ︷︷ ︸potential energy

, (7)

+ The Hamiltonian dynamics (symplectic integrator needed):

dxdt

= ∇p H = M−1p ,dpdt

= −∇x H = −∇xJ (x). (8)

I The canonical PDF of (p,x):

exp (−H(p, x)) = exp(−1

2pT M−1p− J (x)

)= exp

(−1

2pT M−1p

)· π(x). (9)

I Generate a Markov chain with stationary distribution is exp (−H(p,x)).

Sampling Approach: HMCMC [5/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

HMCMC Sampling for Sequential DA:

Assimilation at time tkI Forecast: given an analysis ensemble {xa

k−1(e)}e=1,2,...,Nens at time tk−1;generate the forecast ensemble via the modelM:

xbk (e) =Mtk−1→tk

(xa

k−1(e)), e = 1,2, . . . , Nens. (10)

I Analysis: given the observation vector yk at time point tk , start samplingfrom Pa(x|yk ):

i- Initialize the MC? Use the best estimate available for (x0)! (e.g. Forecast,EnKF, 3D-Var),

ii- Optionally, calculate the ensemble-based forecast error covariance matrixBk ,

iii- Choose the mass matrix M! (e.g. constant diagonal, diagonal of B0 or Bk ),iv- Generate the MC and sample at stationarity,iv- Use the generated samples {xa

k (e)}e=1,2,...,Nens as an analysis ensemble andcalculate the best estimate of the state (e.g. the mean), and the analysiserror covariance matrix.

Sampling Approach: HMCMC [6/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

HMCMC Sampling Algorithm:Given the initial state of the chain:

I Draw a random vector pk ∼ N (0,M),I Use a symplectic numerical integrator (e.g. Verlet) to advance the current

state (pk ,xk ) by a time increment T to obtain a proposal state (p∗,x∗):

(p∗,x∗) = ΦT((pk ,xk )

).

I Evaluate the loss of energy based on the Hamiltonian function ∆H

∆H = H(p∗,x∗)− H(pk ,xk ). (11)

I Calculate the probability:

a(k) = 1 ∧ e−∆H . (12)

I Discard both p∗, pk .I (Acceptance/Rejection) Draw a uniform random variable u(k) ∼ U(0,1):

i- If a(k) > u(k) accept the proposal as the next sample: xk+1 := x∗;ii- If a(k) ≤ u(k) reject the proposal and continue with the current state:

xk+1 := xk .I Repeat steps 1 to 6 until sufficiently many distinct samples are drawn.

Sampling Approach: HMCMC [7/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Symplectic numerical integratorsOne step advance of the solution of the Hamiltonian equations from time tk totime tk+1 = tk + h as follows:

1. Position Verlet integrator

xk+1/2 = xk +h2

M−1 pk , (13a)

pk+1 = pk − h∇xJ (xk+1/2) , (13b)

xk+1 = xk+1/2 +h2

M−1 pk+1. (13c)

2. Two-stage integrator

x1 = xk + (a1h)M−1pk , (14a)

p1 = pk − (b1h)∇xJ (x1) , (14b)

x2 = x1 + (a2h)M−1p1 , (14c)

pk+1 = p1 − (b1h)∇xJ (x2) , (14d)

xk+1 = x2 + (a2h)M−1pk+1 , (14e)

a1 = 0.21132 , a2 = 1− 2a1 , b1 = 0.5 .

Sampling Approach: HMCMC [8/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Symplectic numerical integratorsOne step advance of the solution of the Hamiltonian equations from time tk totime tk+1 = tk + h as follows:

3. Three-stage integrator

x1 = xk + (a1h)M−1pk , (15a)

p1 = pk − (b1h)∇xJ (x1) , (15b)

x2 = x1 + (a2h)M−1p1 , (15c)

p2 = p1 − (b2h)∇xJ (x2) , (15d)

x3 = x2 + (a2h)M−1p2 , (15e)

pk+1 = p2 − (b1h)∇xJ (x3) , (15f)

xk+1 = x3 + (a1h)M−1pk+1 , (15g)

where:

a1 = 0.11888010966548 ,

a2 = 0.5− a1 ,

b1 = 0.29619504261126 ,

b2 = 1− 2b1.

Sampling Approach: HMCMC [9/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Symplectic numerical integratorsOne step advance of the solution of the Hamiltonian equations from time tk totime tk+1 = tk + h as follows:

4. Four-stage integratorx1 = xk + (a1h)M−1pk , (16a)

p1 = pk − (b1h)∇xJ (x1) , (16b)

x2 = x1 + (a2h)M−1p1 , (16c)

p2 = p1 − (b2h)∇xJ (x2) , (16d)

x3 = x2 + (a3h)M−1p2 , (16e)

p3 = p2 − (b2h)∇xJ (x3) , (16f)

x4 = x3 + (a2h)M−1p3 , (16g)

pk+1 = p3 − (b1h)∇xJ (x4) , (16h)

xk+1 = x4 + (a1h)M−1pk+1 , (16i)

where:

a1 = 0.071353913450279725904 ,

a2 = 0.268458791161230105820 ,

a3 = 1− 2a1 − 2a2 , b1 = 0.1916678 , b2 = 0.5− b1.

Sampling Approach: HMCMC [10/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Symplectic numerical integratorsOne step advance of the solution of the Hamiltonian equations from time tk totime tk+1 = tk + h as follows:

5. Infinite dimensional integrator

p1 = pk −h2

M−1∇xJ (xk ) , (17a)

xk+1 = cos (h)xk + sin (h)p1 , (17b)

p2 = − sin (h)xk + cos (h)p1 , (17c)

pk+1 = p2 −h2

M−1∇xJ (xk+1). (17d)

Here, the loss of energy is computed as:

∆H = φ(x∗)− φ(xk ) +h2

8

(|M−

12 (−∇φ(xk ))|2 − |M−

12 (−∇φ(x∗))|2

)+ h

m−1∑i=1

(pT

k (−∇φ(xk )))

+h2

(pT

k (−∇φ(xk )) + (p∗)T(−∇φ(x∗))

),

where φ(x) = − log (π(x))

Sampling Approach: HMCMC [11/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Experimental SettingsI Applied to 40-variables Lorenz-96, Observe each third component,I Observation operators:

1) Linear:H(x) = Hx = (x1, x4, x7, . . . , x37, x40)T ∈ R14 .

2) Quadratic:H(x) = (x2

1 , x24 , x2

7 , . . . , x237, x2

40)T ∈ R14.

3) Cubic:H(x) = (x3

1 , x34 , x3

7 , . . . , x337, x3

40)T ∈ R14.

4) Magnitude:

H(x) = (|x1|, |x4|, |x7|, . . . , |x37|, |x40|)T ∈ R14.

5) Quadratic with threshold:

H(x) = (x ′1, x ′4, x ′7, . . . , x ′37, x ′40)T ∈ R14 ,

where

x ′i =

{x2

i : xi ≥ 0.5−x2

i : xi < 0.5 ,6) Exponential:

H(x) = (er·x1 , er·x4 , er·x7 , . . . , er·x37 , er·x40 )T ∈ R14 ,

Experiment and Results [12/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Results: Quadratic H with a threshold 0.5

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Time

RMSE

Forecast

EnKF

MLEF

(a) Position Verlet integrator(13)

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Time

RMSE

Forecast

EnKF

MLEF

(b) Two-stage integrator (14)

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Time

RMSE

Forecast

EnKF

MLEF

(c) Three-stage integrator

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

Time

RMSE

Forecast

MLEF

EnKF

(d) Four-stage integrator (16)

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Time

RMSE

Forecast

EnKF

MLEF

(e) Integrator defined onHilbert space (17)

Figure : Data assimilation results with the quadratic observation operator with a thresh-old value (12). The integrator used is indicated under each panel. The time step for allintegrators is T = 0.1 with h = 0.01, m = 10, and 30 inter-chain steps.

Experiment and Results [13/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Results: Exponential H with r = 0.2

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Time

RMSE

EnKF

Forecast

MLEF

(a) Position Verlet integrator(13)

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Time

RMSE

Forecast

EnKF

MLEF

(b) Two-stage integrator (14)

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Time

RMSE

EnKF

Forecast

MLEF

(c) Three-stage integrator(15)

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Time

RMSE

Forecast

EnKF

MLEF

(d) Four-stage integrator (16)

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Time

RMSE

Forecast

EnKF

MLEF

(e) Integrator defined onHilbert space (17)

Figure : Data assimilation results with the exponential observation operator with r =0.2 (12). The integrator used is indicated under each panel. The time step for allintegrators is T = 0.1 with h = 0.01, m = 10, and 30 inter-chain steps.

Experiment and Results [14/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Results: Exponential H with r = 0.2

Figure : Scatter plot of the sampled joint pdf of selective components. Results areobtained from HMCMC ensembles with three-stage integrator.

Experiment and Results [15/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Results: Exponential H with a factor r = 0.5

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Time

RM

SE

Forecast

(a) Three-stage integrator; h = 0.01, m =60

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Time

RMSE

Forecast

(b) Four-stage integrator; h = 0.001, m =30

Figure : Data assimilation results with the exponential observation operator with r =0.5 (12). The integrator used is indicated under each panel. The step size h andthe number of steps m are indicated under each panel, and 30 inter-chain steps areconsidered.

Experiment and Results [16/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Tuning the parameters:

(a) RMSE mean: two-stageintegrator; h = 2/Nvar, m =Nvar

(b) RMSE standard devia-tion: two-stage integrator;h = 2/Nvar, m = Nvar

(c) RMSE mean: three-stageintegrator; h = 3/Nvar, m =Nvar/2

(d) RMSE standard devia-tion: three-stage integrator;h = 3/Nvar, m = Nvar/2

(e) RMSE mean: four-stageintegrator; h = 4/Nvar, m =Nvar/2

(f) RMSE standard deviation:four-stage integrator; h =4/Nvar, m = Nvar/2

Figure : The mean and standard deviation of RMS error for the quadratic observationoperator (12) for different MC number of inter-chain steps.

Experiment and Results [17/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Results: Quadratic H with tuned parameters

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Time

RMSE

Forecast

EnKF

MLEF

Figure : RMS error of the sampling filter using three-stage integrator with h =3/Nvar, m = Nvar/2. The number of inter-chain steps is 40.

Experiment and Results [18/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)

Future work

I Apply to larger models,I Extend to 4D case,I Enhance and parallelize,I ....

Future Work [19/19]October 13, 2014, UMD-VT Data Assimilation Day. Ahmed Attia, Adrian Sandu. (http://csl.cs.vt.edu)