Multilevel ensemble Kalman filtering

Transcription

1 Multilevel esemble Kalma filterig Håko Hoel 1 Kody Law 2 Raúl Tempoe 3 1 Departmet of Mathematics, Uiversity of Oslo, Norway 2 Oak Ridge Natioal Laboratory, TN, USA 3 Applied Mathematics ad Computatioal Scieces, KAUST uiversity, Saudi Arabia UQAW 2016, Thuwal 1 / 27

2 Overview 1 Problem descriptio 2 Kalma filterig 3 Esemble Kalma filterig 4 Multilevel esemble Kalma filterig 5 Extesio of MLEKF ad Coclusio 2 / 27

3 Problem descriptio Cosider the uderlyig ad uobservable dyamics u +1 = u + a(u t ) dt + b(u t ) dw (t) } 1 {{ 1 } =:Ψ(u ) with u R d, ad Lipschitz cotiuous a : R d R d ad b : R d Rd ˆd. Ad oisy observatios y = Hv + γ, with i.i.d. γ N(0, Γ) ad H R k d. Objective: Let Y := (y 1, y 2,..., y ) ad let Y obs be a sequece of fixed observatios. Costruct a efficiet method for trackig u (Y = Y obs ). That is, approximate [ ] E φ(u ) Y = Y obs for a observable φ : R d R. Abuse of otatio: will write u Y obs to represet u (Y = Y obs ). 3 / 27

4 Problem descriptio Cosider the uderlyig ad uobservable dyamics u +1 = u + a(u t ) dt + b(u t ) dw (t) } 1 {{ 1 } =:Ψ(u ) with u R d, ad Lipschitz cotiuous a : R d R d ad b : R d Rd ˆd. Ad oisy observatios y = Hv + γ, with i.i.d. γ N(0, Γ) ad H R k d. Objective: Let Y := (y 1, y 2,..., y ) ad let Y obs be a sequece of fixed observatios. Costruct a efficiet method for trackig u (Y = Y obs ). That is, approximate [ ] E φ(u ) Y = Y obs for a observable φ : R d R. Abuse of otatio: will write u Y obs to represet u (Y = Y obs ). 3 / 27

5 Problem descriptio Cosider the uderlyig ad uobservable dyamics u +1 = u + a(u t ) dt + b(u t ) dw (t) } 1 {{ 1 } =:Ψ(u ) with u R d, ad Lipschitz cotiuous a : R d R d ad b : R d Rd ˆd. Ad oisy observatios y = Hv + γ, with i.i.d. γ N(0, Γ) ad H R k d. Objective: Let Y := (y 1, y 2,..., y ) ad let Y obs be a sequece of fixed observatios. Costruct a efficiet method for trackig u (Y = Y obs ). That is, approximate [ ] E φ(u ) Y = Y obs for a observable φ : R d R. Abuse of otatio: will write u Y obs to represet u (Y = Y obs ). 3 / 27

6 Overview 1 Problem descriptio 2 Kalma filterig 3 Esemble Kalma filterig 4 Multilevel esemble Kalma filterig 5 Extesio of MLEKF ad Coclusio 4 / 27

7 Kalma filterig Cosider the liear settig u +1 = Au + ξ +1, y +1 = Hu +1 + γ +1, ξ +1 N(0, Σ), γ +1 N(0, Γ). (1) Gaussiaity of the filterig distributio If u Y obs N(m, C ), the uder (1) both u +1 Y obs will also be Gaussias. ad u +1 Y obs +1 u +1 Y obs = (Au +ξ +1 ) Y obs = A(u Y obs )+ξ +1 N( Am }{{}, AC A T + Σ), }{{} =: m +1 =:Ĉ+1 Ca use Bayesia iferece to show u +1 Y obs +1 N(m +1, C +1 ), where m +1 = (I K +1 H) m +1 + K +1 y obs +1, C +1 = (I K +1 H)Ĉ+1, K +1 = Ĉ+1H T (Γ + HĈ+1H T ) 1. 5 / 27

8 Kalma filterig Cosider the liear settig u +1 = Au + ξ +1, y +1 = Hu +1 + γ +1, ξ +1 N(0, Σ), γ +1 N(0, Γ). (1) Gaussiaity of the filterig distributio If u Y obs N(m, C ), the uder (1) both u +1 Y obs will also be Gaussias. ad u +1 Y obs +1 u +1 Y obs = (Au +ξ +1 ) Y obs = A(u Y obs )+ξ +1 N( Am }{{}, AC A T + Σ), }{{} =: m +1 =:Ĉ+1 Ca use Bayesia iferece to show u +1 Y obs +1 N(m +1, C +1 ), where m +1 = (I K +1 H) m +1 + K +1 y obs +1, C +1 = (I K +1 H)Ĉ+1, K +1 = Ĉ+1H T (Γ + HĈ+1H T ) 1. 5 / 27

9 Kalma filterig Cosider the liear settig u +1 = Au + ξ +1, y +1 = Hu +1 + γ +1, ξ +1 N(0, Σ), γ +1 N(0, Γ). (1) Gaussiaity of the filterig distributio If u Y obs N(m, C ), the uder (1) both u +1 Y obs will also be Gaussias. ad u +1 Y obs +1 u +1 Y obs = (Au +ξ +1 ) Y obs = A(u Y obs )+ξ +1 N( Am }{{}, AC A T + Σ), }{{} =: m +1 =:Ĉ+1 Ca use Bayesia iferece to show u +1 Y obs +1 N(m +1, C +1 ), where m +1 = (I K +1 H) m +1 + K +1 y obs +1, C +1 = (I K +1 H)Ĉ+1, K +1 = Ĉ+1H T (Γ + HĈ+1H T ) 1. 5 / 27

10 Kalma filterig formulas 1 The predictio step for u +1 Y obs Predictio N( m +1, Ĉ+1): m +1 = Am Ĉ +1 = AC A T + Σ. 2 The update step. Update the predictio with the latest observatio y +1 ito u +1 Y +1 N(m +1, C +1 ): Update m +1 = (I K +1 H) m +1 + K +1 y obs +1, C +1 = (I K +1 H)Ĉ+1, K +1 = Ĉ+1H T (Γ + HĈ+1H T ) 1. 6 / 27

11 Overview 1 Problem descriptio 2 Kalma filterig 3 Esemble Kalma filterig 4 Multilevel esemble Kalma filterig 5 Extesio of MLEKF ad Coclusio 7 / 27

12 Esemble Kalma filterig (Evese 94) I more complex settigs, Kalma filterig has major drawbacks: I high dimesios, d 1, storage ad evolutio of the full d d covariace matrix C is very costly. Kalma filterig is ot desiged for oliear dyamics u +1 = Ψ(u ); liearizatio is the eeded, ad it is hard to estimate the errors from liearizatio. Esemble Kalma filterig (EKF) may overcome these drawbacks by propagatig a esemble of particles {v (ω i )} M i=1 {v +1(ω i )} M i=1 ad usig their sample momets i the predict ad update steps. 8 / 27

13 Esemble Kalma Filterig II Predict 1 Compute (umerical solutios of) M particle paths oe step forward v +1,i = Ψ(v,i, ω i ) for i = 1, 2,..., M. 2 Compute sample mea ad covariace m MC +1 = E M [ v +1 ] Ĉ MC +1 = Cov M [ v +1 ] where E M [ v +1 ] := 1 M M i=1 v +1,i ad Cov M [ v +1 ] := E M [ v +1 v T +1] E M [ v +1 ](E M [ v +1 ]) T. 9 / 27

14 Esemble Kalma Filterig III Update 1 Geerate sigal observatios for the esemble of particles ỹ +1,i = y obs +1 + γ +1,i for i = 1, 2..., M, with i.i.d. γ +1,1 N(0, Γ). 2 Use sigal observatios to update particle paths where v +1,i = (I K MC +1H) v +1,i + K MC +1ỹ +1,i, K MC +1 = Ĉ MC +1H T (HĈ MC +1H T + Γ) 1. Note: After the first step, all particles are correlated due to K MC / 27

15 From EKF to mea field EKF For studyig covergece properties of EKF it is useful to itroduce the mea field EKF (MFEKF) v +1,i MF = Ψ(v,i MF, ω i ) ] K+1 Pr m +1 MF MF = E [ v MF = Ĉ +1 MFHT (HĈ +1 MFHT + Γ) 1 +1,i Up ỹ +1,i = y +1 + γ +1,i Ĉ+1 MF = Cov[ v +1,i MF ], v+1,i MF = (I K+1 MF MF H)v+1,i + K +1ỹ+1,i. MF ad i compariso, EKF v +1,i = Ψ(v,i, ω i ) Pr m +1 MC = E M [ v +1 ] Up Ĉ+1 MC = Cov M [ v +1 ] K MC ỹ +1,i v +1,i +1 = Ĉ MC +1 HT (HĈ MC +1 HT + Γ) 1 = y +1 + γ +1,i = (I K+1 MCH) v +1,i + K+1ỹ+1,i. MC Whe uderlyig dyamics is liear with Gaussia additive oise ad u 0 Gaussia, it holds that µ MF (dx) = P ( u dx Y obs ), where µ MF = Law(v,i MF ). I oliear settigs, we use as approximatio goal φ(x)µ MF R d [ (dx)( E φ(u ) Y obs ] ) 11 / 27

16 From EKF to mea field EKF For studyig covergece properties of EKF it is useful to itroduce the mea field EKF (MFEKF) v +1,i MF = Ψ(v,i MF, ω i ) ] K+1 Pr m +1 MF MF = E [ v MF = Ĉ +1 MFHT (HĈ +1 MFHT + Γ) 1 +1,i Up ỹ +1,i = y +1 + γ +1,i Ĉ+1 MF = Cov[ v +1,i MF ], v+1,i MF = (I K+1 MF MF H)v+1,i + K +1ỹ+1,i. MF ad i compariso, EKF v +1,i = Ψ(v,i, ω i ) Pr m +1 MC = E M [ v +1 ] Up Ĉ+1 MC = Cov M [ v +1 ] K MC ỹ +1,i v +1,i +1 = Ĉ MC +1 HT (HĈ MC +1 HT + Γ) 1 = y +1 + γ +1,i = (I K+1 MCH) v +1,i + K+1ỹ+1,i. MC Whe uderlyig dyamics is liear with Gaussia additive oise ad u 0 Gaussia, it holds that µ MF (dx) = P ( u dx Y obs ), where µ MF = Law(v,i MF ). I oliear settigs, we use as approximatio goal φ(x)µ MF R d [ (dx)( E φ(u ) Y obs ] ) 11 / 27

17 Covergece of EKF Theorem 1 (Le Glad et al. (2009)) Cosider the dyamics ad observatios, u +1 = f (u ) + ξ +1, y +1 = Hu +1 + γ +1, ξ +1 N(0, Σ), γ +1 N(0, Γ), ad assume u 0 L p (Ω) for ay p 1, ad that max( f (x) f (x ), φ(x) φ(x ) ) C x x (1+ x s + x s ), for a s 0. The, for the EKF update esemble {v,i } M i=1, ( sup M E[ M 1 M i=1 φ(v,i ) M for ay order p 1 ad fiite. φ(x)µ MF R d p]) 1/p (dx) <. Extesio to further oliear settigs i [Law et al. (2014)]. 12 / 27

18 Cetral step i proof: M φ(v,i ) M φ(x)µ MF M (dx) i=1 R d p i=1 M + i=1 φ(v,i ) M φ(v,i MF ) M M φ(v,i MF ) M i=1 p φ(x)µ MF R d = O( v,i v MF,i ˆp + M 1/2 ) (dx) p 13 / 27

19 Computatioal cost of EKF To meet the costrait ( M E[ i=1 φ(v,i ) M φ(x)µ MF R d oe thus eeds esemble of size M = O(ɛ 2 ). p]) 1/p (dx) = O(ɛ), How does the computatioal cost icrease if the EKF dyamics has to be sampled usig a umerical solver for which E [ Ψ Ψ ] = O( t α )? Short aswer (uder additioal assumptios): the cost icreases to O(ɛ (2+1/α) ). 14 / 27

20 Overview 1 Problem descriptio 2 Kalma filterig 3 Esemble Kalma filterig 4 Multilevel esemble Kalma filterig 5 Extesio of MLEKF ad Coclusio 15 / 27

21 Multilevel EKF (MLEKF) Predictio Compute a esemble of particle paths o a hierarchy of accuracy levels v l 1 +1,i = Ψ l 1 (v l 1,i, ω l,i ), v l +1,i = Ψ l (v l,i, ω l,i ), for the levels l = 0, 1,..., L ad i = 1, 2,..., M l. Multilevel approximatio of mea ad covariace matrices: m ML +1 = Ĉ ML +1 = L l=0 L l=0 E Ml [ v l +1 v l 1 +1 ], Cov Ml [ v l +1] Cov Ml [ v l 1 +1 ] Notice the telescopig properties E [ m +1] ML ] = E [ v L ] +1 ad = Cov( v +1 L ) + O(1/M L). E[Ĉ ML / 27

22 MLEKF update step Update For l = 0, 1,..., L ad i = 1, 2,..., M l, ỹ l +1,i = y obs +1 + γ l +1,i, i.i.d. γ l +1,i N(0, Γ) v+1 l 1 ML = (I K+1H) v l 1 +1,i + K +1ỹ ML +1,i, l v+1,i l = (I K+1H) v ML +1,i l + K+1ỹ ML +1,i, l where K ML +1 = Ĉ ML +1H T (HĈ ML +1H T + Γ) / 27

23 Covergece of MLEKF For observables φ : R d R, itroduce otatio µ ML (φ) := L 1 M l M l l=0 i=1 φ(v l,i) φ(v l 1,i ). ad µ MF (φ) := φ(x)µ MF (dx). R d Questio: Uder what assumptios ad at what cost ca oe achieve µ ML (φ) µ MF (φ) L p (Ω) = O(ɛ)? 18 / 27

24 Assumptio 1 Cosider the dyamics u +1 = Ψ(u ) = u + +1 a(u t )dt + +1 b(u t )dw (t), = 0, 1,... with u 0 L p (Ω) for all p 1 ad a hierarchy of umerical solvers {Ψ l } L l=0. For ay p 2, l N, ad u, v L p (Ω), we assume that Ψ l (u) Ψ l (v) p C u v p, Ψ l (u) p C(1 + u p ). 19 / 27

25 Assumptio 2 Assume the observable φ : R d R satisfies φ(x) φ(x ) C x x (1 + x s + x s ), for a s 0. ad there exists costats α, β > 0 such that for u, v L p (Ω), (i) [ E φ(ψ l (u)) φ(ψ(v)) ] N α l, provided that E[u v] N α l, (ii) φ(ψ l (v)) φ(ψ l 1 (v)) p N β/2 l, for all p 2, (iii) Cost ( Ψ l (v) ) N l. Assume further (i) ad (ii) hold for that all moomials φ of degree / 27

26 Theorem 2 (MLEKF accuracy vs. cost) Suppose Assumptios 1 ad 2 hold. The, for ay ɛ > 0 ad p 2, there exists a L > 0 ad {M l } L l=0 such that µ ML (φ; (M l )) µ MF (φ) p ɛ. Ad ( log(ɛ) 1 ɛ) 2, if β > 1, Cost (MLEKF) ( log(ɛ) 1 ɛ) 2 log(ɛ) 3, if β = 1, (2) ( log(ɛ) 1 1 β (2+ ɛ) α ), if β < 1. I compariso, is achieved at cost O µ EKF ( ɛ (2+ 1 α) ). (φ) µ MF (φ) p ɛ, 21 / 27

27 Theorem 2 (MLEKF accuracy vs. cost) Suppose Assumptios 1 ad 2 hold. The, for ay ɛ > 0 ad p 2, there exists a L > 0 ad {M l } L l=0 such that µ ML (φ; (M l )) µ MF (φ) p ɛ. Ad ( log(ɛ) 1 ɛ) 2, if β > 1, Cost (MLEKF) ( log(ɛ) 1 ɛ) 2 log(ɛ) 3, if β = 1, (2) ( log(ɛ) 1 1 β (2+ ɛ) α ), if β < 1. I compariso, is achieved at cost O µ EKF ( ɛ (2+ 1 α) ). (φ) µ MF (φ) p ɛ, 21 / 27

28 Cetral idea i the proof Itroduce µ MLMF (φ) := µ MF,L (φ) := E L 1 M l M l l=0 i=1 [ φ(v MF,L ad boud MLEKF error by µ ML φ(v MF,l (ω i,l )) φ(v MF,l 1 (ω i,l )) ) Y obs ], (φ) µ MF (φ) p µ ML (φ) µ MLMF (φ) p + µ MLMF (φ) µ MF,L [ L c v l v MF,l ˆp + c l=0 ( ɛ + L l=0 (φ) p + µ MF,L v MF,l M 1/2 l N β/2 l + N α L v MF,l 1 M 1/2 l ) (φ) µ MF (φ) p ] ˆp + E [ φ(v MF,L ) φ(v MF ) Y obs ] 22 / 27

29 Numerical example Uderlyig dyamics Orstei Uhlebeck SDE with a set of observatios du = udt + 0.5dW (t), y = u + γ, i.i.d. γ N(0, 0.04) Solvers: Hierarchy of Milstei solutio operators {Ψ l } L l=0 with t l = O ( 2 l). Compare the approximatio errors for the observable φ(x) = x i terms of the RMSE 100 µ ML (φ) µ MF (φ) 2. =1 23 / 27

30 Numerical example Uderlyig dyamics Orstei Uhlebeck SDE du = udt + 0.5dW (t), with a set of observatios y = u + γ, i.i.d. γ N(0, 0.04) RMSE EKF MLEKF cs 1/3 cs 1/ Rutime [s] 23 / 27

31 Overview 1 Problem descriptio 2 Kalma filterig 3 Esemble Kalma filterig 4 Multilevel esemble Kalma filterig 5 Extesio of MLEKF ad Coclusio 24 / 27

32 Extesio of MLEKF to ifiite dimesioal state spaces Work i progress, together with Alexey Cherov, Law, Nobile ad Tempoe. Ifiite dimesioal stochastic dyamics: u +1 = Ψ(u ) where u L p (Ω; H) with H = Spa({ν i } i=1 ), ad Ψ : L p (Ω; H) L p (Ω; H). Ad fiite dimesioal observatios y = Hu + γ, with liear H : H R m Itroduce ested hierarchy of Hilbert spaces H 0 H 1... H = H, where H l = Spa({ν i } N l i=1 ) ad work with a hierarchy of solvers Ψ l : L p (Ω; H l ) L p (Ω; H l ). 25 / 27

33 Coclusio Exteded EKF to multilevel EKF. Verified asymptotic efficiecy gai for approximatios of expectatio of observables. We hope to improve result further! Further extesio of MLEKF to ifiite dimesioal state space is work i progress. Preprit available o Arxiv 26 / 27

34 Refereces Thak you! 1 Hoel, Håko, Kody JH Law, ad Raul Tempoe, Multilevel esemble Kalma filterig, arxiv preprit arxiv: (2015). 2 Fraçois Le Glad, Valérie Mobet, Vu-Duc Tra, et al., Large sample asymptotics for the esemble kalma filter, The Oxford Hadbook of Noliear Filterig, (2011), pp Kody JH Law, Hamidou Tembie, ad Raul Tempoe, Determiistic methods for oliear filterig, part i: Mea-field esemble kalma filterig, arxiv preprit arxiv: , (2014). 27 / 27