EnKF-based particle filters

Transcription

1 EnKF-based particle filters Jana de Wiljes, Sebastian Reich, Wilhelm Stannat, Walter Acevedo June 20, 2017

2 Filtering Problem Signal dx t = f (X t )dt + 2CdW t Observations dy t = h(x t )dt + R 1/2 dv t Signal Obs 0.5 time Goal: determine π(x Y 0:t )

3 State of the art Linear Model: Kalman- Bucy Filter d x M t = A x M t dt + bdt P M t H T R 1 (H x M t dt dy t ) Non-linear Model: approximate with empirical measure, i.e., π(x Y 0:t ) 1 M M δ(x Xt i ). i=1 Ansatz: define modified evolution equation for particles X i t

4 Ensemble Kalman Filter (EnKF) P(x Y 0:t ) P(x) time t dx i t = f (X i t )dt + CdW i t 1 2 PM t H T R 1 ( HX i t dt + H x M t dt 2dY t ) x M t = 1 M M i=1 X i t P M t = 1 M 1 M (Xt i x t M )(Xt i x t M ) T i=1 Works remarkably well in practice: meteorology, oil reservoir exploration But: theoretical understanding is largely missing

5 Recent accuracy results for EnKF Interacting particle representation of the model error: dx i t = f (X i t )dt+cc (P M t ) 1 (X i t x M t )dt 1 2 PM t H T R 1 ( HX i t dt + H x M t dt 2dY t ) Setting: dy t = X t dt + R 1/2 dw t with R = εi Results: ([dwrs16]) Control of spectrum of covariance matrix Pt M over t [0, ). Control of estimation error e t = Xt ref x t M 2 in expectation and pathwise over fixed interval t [0, T ]. E[E t ] = O(ɛ 1/2 ) (1) P[sup E t c 1 ɛ q ] O(ɛ 1/2 η q ) (2) t T

6 Numerical confirmation for L time averaged mean squared error 10 0 error theoretical order measurement noise variance

7 EnKF-based particle filters Ansatz: modified evolution equation for the particles e.g., of the form dxt i = f (Xt i )dt + CdWt i Xt j dst ji j Aims: achieve first/second order accuracy to go beyond Gaussian assumption consistency for M hybrid formulation to combine different interacting particle filters ([CRR16])

8 Linear ensemble transform filters (LETF) Given: M samples x f i π(x) (prior) Analysis step: x a j = i x f i d ij (3) with transformation matrix D = {d ij } subject to M d ij = 1 (4) i=1 Examples: ENKF, ESRF, NETF, ETPF (see [RC15])

9 Ensemble transform Particle Filter (ETPF) Given: M samples xi f π(x) (prior ensemble) normalized importance weights w i π(y xi f ) (likelihood) Desired: M samples xi a π(x y) Ansatz: replace resampling step with linear transformation by interpreting it as discrete Markov chain given by transition matrix s.t. d ij 0 and D R M M d ij = 1, i 1 M d ij = w i. j Benefits: localization, hybrid

10 Ensemble transform Particle Filter (ETPF) Then the posterior ensemble members are distributed according to the columns of the transformation d 1j d 2j X a j. d Mj and x j a = E[ X a j ] = i x a i d ij (5) Example. Monomial resampling w 1 w 1 w 1 w 2 w 2 w 2 D Mono := w 1 = w M w M w M.

11 Ensemble transform Particle Filter (ETPF) Then the posterior ensemble members are distributed according to the columns of the transformation d 1j d 2j X a j. d Mj and x j a = E[ X a j ] = i x a i d ij (5) Example. Monomial resampling w 1 w 1 w 1 w 2 w 2 w 2 D Mono := w 1 = w M w M w M. Here: X f and X a are independent. Idea: increase correlation between X f and X a

12 Ensemble transform Particle Filter (ETPF) Solve optimization problem D ETPF = arg min M M i,j=1 t ij x f i z f j 2 to find transformation matrix D ETPF that increases correlation ([Rei13]). Remarks: consistent for M first-order accurate for finite M, i.e, x a = 1 M M i=1 x a i = M i=1 w i x f i (6) But: not second-order accurate

13 Second-order accuracy The analysis covariance matrix P a = 1 M M (xi a i=1 x a )(x a i x a ) T is equal to the covariance matrix defined by the importance weights, i.e. P a = M i=1 w i (t k )(x f i x a )(x f i x a ) T.

14 First-order accurate LETFs Notation: and analogously X f = (x1 f,..., xm), f X a = (x1 a,..., xm) a Nx M R w = (w 1,..., w M ) T R M 1 and W = diag (w) LETF is first-order accurate if This holds if D satisfies 1 M Xa 1 = X f w. 1 D1 = w. M

15 First-order accurate LETFs Class of first-order accurate LETFs D 1 = {D R M M D T 1 = 1, D1 = Mw } Examples: D EnKF / D 1 D ESRF / D 1 D ETPF D 1 D Mono D 1

16 Second-order accurate LETF Analysis covariance matrix: P a = 1 M Xf (D w1 T )(D w1 T ) T (X f ) T (7) Importance sampling covariance matrix: P a = X f (W ww T )(X f ) T. (8) Thus the following equation has to hold for second-order accuracy: (D w1 T )(D w1 T ) T = W ww T (9) Class of second-order accurate LETFs D 2 = {D D 1 (D w1 T )(D w1 T ) T = W ww T }. (10)

17 Second-order corrected LETF Given: D D 1 Goal: correct transformation D D 2 Ansatz: D = D + with R M M such that 1 = 0, T 1 = 0 ([dwar17]). Need to solve algebraic Riccati equation: M(W ww T ) (D w1 T )(D w1 T ) T =(D w1 T ) T + (D w1 T ) T + T.

18 Numerical example I Gaussian prior, non-gaussian likelihood: Figure: Prior and posterior distribution for the single Bayesian inference step

19 Numerical example I Figure: Absolute errors in the first four moments of the posterior distribution.

20 Numerical example II Lorenz-63 model, first component observed infrequently ( t = 0.12) and with large measurement noise (R = 8): Figure: RMSEs for various second-order accurate LETFs compared to the ETPF, the ESRF, and the SIR PF as a function of the sample size, M.

21 Numerical example II Hybrid filter: D := D ESRF D ETPF. Figure: RMSEs for hybrid ESRF (α = 0) and 2nd-order corrected LETF/ETPF (α = 1) as a function of the sample size, M.

22 Data Assimilation Center Potsdam I PhD and Postdoc positions in 11 projects I Mathematical foundation and applications in geophysics, biologie and cognitive sciences I Support for external projects and long term visitors

23 Data Assimilation Center Potsdam I PhD and Postdoc positions in 11 projects I Mathematical foundation and applications in geophysics, biologie and cognitive sciences I Support for external projects and long term visitors Thank you!

24 References I Nawinda Chustagulprom, Sebastian Reich, and Maria Reinhardt. A hybrid ensemble transform filter for nonlinear and spatially extended dynamical systems. SIAM/ASA J Uncertainty Quantification, 4: , J. de Wiljes, W. Acevedo, and S. Reich. A second-order accurate ensemble transform particle filter. accepted in SIAM Journal on Scientific Computing, (ArXiv: ), J. de Wiljes, S. Reich, and W. Stannat. Long-time stability and accuracy of the ensemble Kalman-Bucy filter for fully observed processes and small measurement noise. Technical Report S. Reich and C. Cotter. Probabilistic Forecasting and Bayesian Data Assimilation. Cambridge University Press, 2015.

25 References II Sebastian Reich. A non-parametric ensemble transform method for Bayesian inference. SIAM J Sci Comput, 35:A2013 A2014, 2013.