Particle filters for infinite-dimensional systems: combining - - PowerPoint PPT Presentation

Particle filters for infinite-dimensional systems: combining localization and optimal transportation

Sebastian Reich

University of Potsdam and University of Reading

MCMSki 2014 Chamonix 7 January 2014

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 1 / 22

Talk Synopsis

Introduction from a “weather prediction perspective”

Ensemble prediction McKean data analysis cycle

Linear ensemble transform filters (LETFs)

Sequential importance resampling (SIR) particle filter Ensemble Kalman filter (EnKF)

Ensemble transform particle filter (ETPF)

Optimal transportation Example: Lorenz-63

Spatially extended systems

Localization Example: Lorenz-96

Future work and references

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 2 / 22

Introduction Ensemble prediction

We consider a state space model in form of an iteration zn+1 = Ψ(zn), n ≥ 0, with zn ∈ RNz. The initial states z0 are realizations of a random variable (RV) Z 0 : Ω → RNz with PDF πZ 0. Hence all zn’s become realizations of associated RVs Z n with marginal PDFs πZ n. Ensemble prediction relies on M independent realizations z0

i = Z 0(ωi) (MC

• r quasi-MC) from the initial RV and associated trajectories

zn+1

i

= Ψ(zn

i ),

n ≥ 0, i = 1, . . . , M.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 3 / 22

Introduction Ensemble prediction

We consider a state space model in form of an iteration zn+1 = Ψ(zn), n ≥ 0, with zn ∈ RNz. The initial states z0 are realizations of a random variable (RV) Z 0 : Ω → RNz with PDF πZ 0. Hence all zn’s become realizations of associated RVs Z n with marginal PDFs πZ n. Ensemble prediction relies on M independent realizations z0

i = Z 0(ωi) (MC

• r quasi-MC) from the initial RV and associated trajectories

zn+1

i

= Ψ(zn

i ),

n ≥ 0, i = 1, . . . , M.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 3 / 22

Introduction Ensemble prediction

We consider a state space model in form of an iteration zn+1 = Ψ(zn), n ≥ 0, with zn ∈ RNz. The initial states z0 are realizations of a random variable (RV) Z 0 : Ω → RNz with PDF πZ 0. Hence all zn’s become realizations of associated RVs Z n with marginal PDFs πZ n. Ensemble prediction relies on M independent realizations z0

i = Z 0(ωi) (MC

• r quasi-MC) from the initial RV and associated trajectories

zn+1

i

= Ψ(zn

i ),

n ≥ 0, i = 1, . . . , M.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 3 / 22

Introduction Ensemble prediction

Ensemble prediction for Lorenz-63 model with initial distribution N(0, 0.01I) and M = 1000 ensemble members.

20 20 40 20 20 40 10 10 20 30 40 50 initial conditions 20 20 40 20 20 40 10 10 20 30 40 50 solutions at t = 10 20 20 40 20 20 40 10 10 20 30 40 50 solutions at t = 100

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 4 / 22

Introduction McKean data analysis cycle

We now aim to shadow or track an unknown reference solution zn+1

ref

= Ψ(zn

ref),

which is accessible to us only through partial and noisy observations yn

• bs = h(zn

ref) + ξn,

n ≥ 1. Here h : RNz → RNy is the forward operator and the ξn’s are realizations of independent Gaussian RVs with mean zero and covariance matrix R. Bayesian inference is used to merge ensemble predictions and partial

• bservations in order to obtain an optimal shadowing of the reference

trajectory. This is called the analysis step in weather forecasting.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 5 / 22

Introduction McKean data analysis cycle

We now aim to shadow or track an unknown reference solution zn+1

ref

= Ψ(zn

ref),

which is accessible to us only through partial and noisy observations yn

• bs = h(zn

ref) + ξn,

n ≥ 1. Here h : RNz → RNy is the forward operator and the ξn’s are realizations of independent Gaussian RVs with mean zero and covariance matrix R. Bayesian inference is used to merge ensemble predictions and partial

• bservations in order to obtain an optimal shadowing of the reference

trajectory. This is called the analysis step in weather forecasting.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 5 / 22

Introduction McKean data analysis cycle

Summary of the McKean approach to the analysis step: PDFs RVs MC Ref.: Del Moral (2004), CJC & SR (2013), YC & SR (2014).

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 6 / 22

Introduction McKean data analysis cycle

(A) Fit a Gaussian N(¯ zf, Pf) to the forecast ensemble {zf

i } and assume that

the forward operator can be linearized. Then the analysis is also Gaussian N(¯ za, Pa) with ¯ za = ¯ zf − K(H¯ zf − yobs), Pa = Pf − KHPf. Here K denotes the Kalman gain matrix. (B) Use the empirical measure πf(z) = 1 M

M

• i=1

δ(z − zf

i )

to define the analysis measure πa(z) =

M

• i=1

wiδ(z − zf

i )

with importance weights wi = exp

• − 1

2(h(zf i ) − yobs)TR−1(h(zf i ) − yobs)

• M

j=1 exp

• − 1

2(h(zf j ) − yobs)TR−1(h(zf j ) − yobs)

• Sebastian Reich (UP and UoR)

Particle filters for infinite-dimensional systems 7 January 2014 7 / 22

Introduction McKean data analysis cycle

(A) Fit a Gaussian N(¯ zf, Pf) to the forecast ensemble {zf

i } and assume that

the forward operator can be linearized. Then the analysis is also Gaussian N(¯ za, Pa) with ¯ za = ¯ zf − K(H¯ zf − yobs), Pa = Pf − KHPf. Here K denotes the Kalman gain matrix. (B) Use the empirical measure πf(z) = 1 M

M

• i=1

δ(z − zf

i )

to define the analysis measure πa(z) =

M

• i=1

wiδ(z − zf

i )

with importance weights wi = exp

• − 1

2(h(zf i ) − yobs)TR−1(h(zf i ) − yobs)

• M

j=1 exp

• − 1

2(h(zf j ) − yobs)TR−1(h(zf j ) − yobs)

• Sebastian Reich (UP and UoR)

Particle filters for infinite-dimensional systems 7 January 2014 7 / 22

Introduction McKean data analysis cycle

Implementation of the McKean approach then either requires coupling two Gaussians (approach A) or two empirical measures (approach B). Approach A leads to the family of ensemble Kalman filters (Evensen, 2006), while Approach B leads to particle filters (Doucet et al, 2001). Optimal couplings in the sense of minimizing some cost function are known in both cases (CJC & SR, 2013). We now provide a unifying mathematical framework in form of linear ensemble transform filters (LETFs) (YC & SR, 2014) instead of focusing on the McKean Markov process aspect.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 8 / 22

Linear ensemble transform filters

The analysis steps of a standard particle filter as well as an ensemble Kalman filter (EnKF) are of the form za

j = M

• i=1

zf

i sij,

where {zf

i }M i=1 is the forecast ensemble and {za i }M i=1 is the analysis ensemble.

The matrix S ∈ RM×M with entries sij depends on yobs and the forecast ensemble. In some cases, S is the realization of a matrix-valued RV S : Ω → RM×M, i.e. S = S(ω).

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 9 / 22

Linear ensemble transform filters SIR particle filter

Sequential importance resampling (SIR) particle filters rely on S : Ω → RM×M satisfying the following properties sij ∈ {0, 1}, M

i=1 sij = 1,

E

• 1

M

M

j=1 sij

• = wi.

One often adds a rejuvenation step: za

j = M

• i=1

zf

i sij + ξa j .

The ξa

j ’s are realizations of M independent Gaussian RVs with mean zero and

appropriate covariance matrix.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 10 / 22

Linear ensemble transform filters SIR particle filter

Sequential importance resampling (SIR) particle filters rely on S : Ω → RM×M satisfying the following properties sij ∈ {0, 1}, M

i=1 sij = 1,

E

• 1

M

M

j=1 sij

• = wi.

One often adds a rejuvenation step: za

j = M

• i=1

zf

i sij + ξa j .

The ξa

j ’s are realizations of M independent Gaussian RVs with mean zero and

appropriate covariance matrix.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 10 / 22

Linear ensemble transform filters Ensemble Kalman filter (EnKF)

The EnKF with perturbed observations leads to sij = δij + 1 M − 1(zf

i − ¯

zf)THT(HTPfH + R)−1(yobs − Hzf

j + ηj)

with ηj a realization of N(0, R). The coefficients satisfy M

i=1 sij = 1,

¯ za = M

j=1 ˜

wizf

i with “importance weights”

˜ wi = 1 M

M

• j=1

sij, The coefficients sij’s can take negative values! Ensemble inflation or rejuvenation is often applied.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 11 / 22

Linear ensemble transform filters Ensemble Kalman filter (EnKF)

The EnKF with perturbed observations leads to sij = δij + 1 M − 1(zf

i − ¯

zf)THT(HTPfH + R)−1(yobs − Hzf

j + ηj)

with ηj a realization of N(0, R). The coefficients satisfy M

i=1 sij = 1,

¯ za = M

j=1 ˜

wizf

i with “importance weights”

˜ wi = 1 M

M

• j=1

sij, The coefficients sij’s can take negative values! Ensemble inflation or rejuvenation is often applied.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 11 / 22

Linear ensemble transform filters Ensemble Kalman filter (EnKF)

The EnKF with perturbed observations leads to sij = δij + 1 M − 1(zf

i − ¯

zf)THT(HTPfH + R)−1(yobs − Hzf

j + ηj)

with ηj a realization of N(0, R). The coefficients satisfy M

i=1 sij = 1,

¯ za = M

j=1 ˜

wizf

i with “importance weights”

˜ wi = 1 M

M

• j=1

sij, The coefficients sij’s can take negative values! Ensemble inflation or rejuvenation is often applied.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 11 / 22

Linear ensemble transform filters Optimal transportation

The ensemble transform particle filter (ETPF) (SR, 2013) is determined by a coupling T ∈ RM×M between the discrete random variables Z f : Ω → {zf

1, . . . , zf M}

with P[zf

i ] = 1/M

and Z a : Ω → {zf

1, . . . , zf M}

with P[zf

i ] = wi,

respectively. A coupling T has to satisfy tij ≥ 0,

M

• i=1

tij = 1/M,

M

• j=1

tij = wi, where the wi’s are the importance weights characterizing the fit of the forecasts zf

i to the observed yobs.

Any coupling T determines a matrix S in the linear ensemble transform step

• f a particle filter via

S = M T.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 12 / 22

Linear ensemble transform filters Optimal transportation

The ensemble transform particle filter (ETPF) (SR, 2013) is determined by a coupling T ∈ RM×M between the discrete random variables Z f : Ω → {zf

1, . . . , zf M}

with P[zf

i ] = 1/M

and Z a : Ω → {zf

1, . . . , zf M}

with P[zf

i ] = wi,

respectively. A coupling T has to satisfy tij ≥ 0,

M

• i=1

tij = 1/M,

M

• j=1

tij = wi, where the wi’s are the importance weights characterizing the fit of the forecasts zf

i to the observed yobs.

Any coupling T determines a matrix S in the linear ensemble transform step

• f a particle filter via

S = M T.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 12 / 22

Linear ensemble transform filters Optimal transportation

The ensemble transform particle filter (ETPF) (SR, 2013) is determined by a coupling T ∈ RM×M between the discrete random variables Z f : Ω → {zf

1, . . . , zf M}

with P[zf

i ] = 1/M

and Z a : Ω → {zf

1, . . . , zf M}

with P[zf

i ] = wi,

respectively. A coupling T has to satisfy tij ≥ 0,

M

• i=1

tij = 1/M,

M

• j=1

tij = wi, where the wi’s are the importance weights characterizing the fit of the forecasts zf

i to the observed yobs.

Any coupling T determines a matrix S in the linear ensemble transform step

• f a particle filter via

S = M T.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 12 / 22

Linear ensemble transform filters Optimal transportation

Chosing a coupling that maximizes the correlation between forecast and analysis leads to an optimal transport problem with cost J({tij} =

• i,j

zf

i − zf j 2tij.

This leads to the celebrated Monge-Kantorovitch problem: π∗

Z f Z a(zf, za) = arg

inf

πZf Za(zf ,za)∈Π(πZf ,πZa) EZ f Z a

• zf − za2

as M → ∞ (McCann, 1996, SR, 2013). Let us denote the minimize by T ∗, then the ETPF is given by za

j = M M

• i=1

zf

i t∗ ij .

The coefficients sij = Mt∗

ij are now entirely deterministic.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 13 / 22

Linear ensemble transform filters Optimal transportation

Chosing a coupling that maximizes the correlation between forecast and analysis leads to an optimal transport problem with cost J({tij} =

• i,j

zf

i − zf j 2tij.

This leads to the celebrated Monge-Kantorovitch problem: π∗

Z f Z a(zf, za) = arg

inf

πZf Za(zf ,za)∈Π(πZf ,πZa) EZ f Z a

• zf − za2

as M → ∞ (McCann, 1996, SR, 2013). Let us denote the minimize by T ∗, then the ETPF is given by za

j = M M

• i=1

zf

i t∗ ij .

The coefficients sij = Mt∗

ij are now entirely deterministic.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 13 / 22

Linear ensemble transform filters Optimal transportation

Chosing a coupling that maximizes the correlation between forecast and analysis leads to an optimal transport problem with cost J({tij} =

• i,j

zf

i − zf j 2tij.

This leads to the celebrated Monge-Kantorovitch problem: π∗

Z f Z a(zf, za) = arg

inf

πZf Za(zf ,za)∈Π(πZf ,πZa) EZ f Z a

• zf − za2

as M → ∞ (McCann, 1996, SR, 2013). Let us denote the minimize by T ∗, then the ETPF is given by za

j = M M

• i=1

zf

i t∗ ij .

The coefficients sij = Mt∗

ij are now entirely deterministic.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 13 / 22

Linear ensemble transform filters Optimal transportation

Chosing a coupling that maximizes the correlation between forecast and analysis leads to an optimal transport problem with cost J({tij} =

• i,j

zf

i − zf j 2tij.

This leads to the celebrated Monge-Kantorovitch problem: π∗

Z f Z a(zf, za) = arg

inf

πZf Za(zf ,za)∈Π(πZf ,πZa) EZ f Z a

• zf − za2

as M → ∞ (McCann, 1996, SR, 2013). Let us denote the minimize by T ∗, then the ETPF is given by za

j = M M

• i=1

zf

i t∗ ij .

The coefficients sij = Mt∗

ij are now entirely deterministic.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 13 / 22

Linear ensemble transform filters Example: Lorenz-63

Consider Lorenz-63 model with outputs generated every 0.12 units of time. Only the x variable is observed with measurement error variance equal to R = 8. Each DA algorithm is implemented either with ensemble inflation or particle

• rejuvenation. A total of 20,000 assimilation steps are performed.

We compare the resulting time-averaged RMSEs:

• 20000
• n=1

¯ za,n − zn

ref2.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 14 / 22

Linear ensemble transform filters Example: Lorenz-63

10 20 30 40 50 60 70 80 1 1.02 1.04 1.06 1.08 1.1 a) EnKF Ensemble size Inflation factor 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 20 30 40 50 60 70 80 0.15 0.2 0.25 0.3 0.35 0.4 b) ETPF Ensemble size Rejuvenation 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 20 30 40 50 60 70 80 0.15 0.2 0.25 0.3 0.35 0.4 c) ETPF_R0 Ensemble size Rejuvenation 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 10 20 30 40 50 60 70 80 1.8 2 2.2 2.4 2.6 2.8 3 Ensemble Size d) RMSE EnKF ETPF_R0 ETPF SIR

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 15 / 22

Spatially extended systems

Spatially extended dynamical systems We now consider spatially extended system with x ∈ R taking the role of the spatial variable. The forecast ensemble is now {zf

i (x)} and the LETF

becomes za

i (x) = M

• i=1

zf

i (x) sij

for all x. This does not work unless M is huge. Instead one uses localization, which gives rise to x-dependent coefficients sij(x), i.e. za

i (x) = M

• i=1

zf

i (x) sij(x).

Note that analysis fields need to have sufficient spatial regularity! This rules out localization for standard SIR particle filters.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 16 / 22

Spatially extended systems

Spatially extended dynamical systems We now consider spatially extended system with x ∈ R taking the role of the spatial variable. The forecast ensemble is now {zf

i (x)} and the LETF

becomes za

i (x) = M

• i=1

zf

i (x) sij

for all x. This does not work unless M is huge. Instead one uses localization, which gives rise to x-dependent coefficients sij(x), i.e. za

i (x) = M

• i=1

zf

i (x) sij(x).

Note that analysis fields need to have sufficient spatial regularity! This rules out localization for standard SIR particle filters.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 16 / 22

Spatially extended systems Localization for ETPF

R-localization for the ETPF: Define the localization function ρ(x − x′) = 1 − |x − x′|/rloc for |x − x′| ≤ rloc, else. The error variance R of an observation at x′ is modified to ˜ R−1(x) = ρ(x − x′) R−1 and gives rise to importance weights wi(x) ∝ exp

• −1

2(h(zf

i ) − yobs)T ˜

R−1(x)(h(zf

i ) − yobs)

• .

An optimal transport problem is now solved for each x with localized transport cost d(zf, za)(x) :=

• R

ρ(x − x′)zf(x) − za(x)2 dx.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 17 / 22

Spatially extended systems Localization for ETPF

R-localization for the ETPF: Define the localization function ρ(x − x′) = 1 − |x − x′|/rloc for |x − x′| ≤ rloc, else. The error variance R of an observation at x′ is modified to ˜ R−1(x) = ρ(x − x′) R−1 and gives rise to importance weights wi(x) ∝ exp

• −1

2(h(zf

i ) − yobs)T ˜

R−1(x)(h(zf

i ) − yobs)

• .

An optimal transport problem is now solved for each x with localized transport cost d(zf, za)(x) :=

• R

ρ(x − x′)zf(x) − za(x)2 dx.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 17 / 22

Spatially extended systems Localization for ETPF

R-localization for the ETPF: Define the localization function ρ(x − x′) = 1 − |x − x′|/rloc for |x − x′| ≤ rloc, else. The error variance R of an observation at x′ is modified to ˜ R−1(x) = ρ(x − x′) R−1 and gives rise to importance weights wi(x) ∝ exp

• −1

2(h(zf

i ) − yobs)T ˜

R−1(x)(h(zf

i ) − yobs)

• .

An optimal transport problem is now solved for each x with localized transport cost d(zf, za)(x) :=

• R

ρ(x − x′)zf(x) − za(x)2 dx.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 17 / 22

Spatially extended systems Example: Lorenz-96

The Lorenz-96 ODE model duj dt = −uj−1uj+1 − uj−2uj−1 3∆x − uj + F, j = 1, . . . , 40, can be thought of as the discretization of the forced-damped advection equation ∂u ∂t = −1 2 ∂(u)2 ∂x − u + F. Every other grid point is observed in intervals of ∆t = 0.12. The error variance is R = 8. A total of 10,000 assimilation steps are performed and the resulting time-averaged RMSEs are compared.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 18 / 22

Spatially extended systems Example: Lorenz-96

The Lorenz-96 ODE model duj dt = −uj−1uj+1 − uj−2uj−1 3∆x − uj + F, j = 1, . . . , 40, can be thought of as the discretization of the forced-damped advection equation ∂u ∂t = −1 2 ∂(u)2 ∂x − u + F. Every other grid point is observed in intervals of ∆t = 0.12. The error variance is R = 8. A total of 10,000 assimilation steps are performed and the resulting time-averaged RMSEs are compared.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 18 / 22

Spatially extended systems Example: Lorenz-96

Time averaged spatial correlation of solutions to the Lorenz-96 ODE:

−20 −15 −10 −5 5 10 15 −0.4 −0.2 0.2 0.4 0.6 0.8 1 correlation factor distance

The localization radius varies between rloc = 1 and rloc = 8. The EnKF allows for slightly larger radii than the ETPF .

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 19 / 22

Spatially extended systems Example: Lorenz-96

10 20 30 40 50 60 70 80 1 1.02 1.04 1.06 1.08 1.1 1.12 a) EnKF Ensemble size Inflation factor 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 10 20 30 40 50 60 70 80 0.1 0.15 0.2 0.25 0.3 0.35 0.4 c) ETPF_R1 Ensemble size Rejuvenation 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 10 20 30 40 50 60 70 80 0.1 0.15 0.2 0.25 0.3 0.35 0.4 b) ETPF_R0 Ensemble size Rejuvenation 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 10 20 30 40 50 60 70 80 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 d) Ensemble Size RMSE EnKF ETPF_R0 ETPF_R1 ETPF_R2

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 20 / 22

Future work and references

Topics for future work: systematic exploration of DA/stochastic filtering for multi-scale processes, mathematical theory for DA/stochastic filtering on infinite-dimensional state spaces, stochastic filtering/DA under systematic model and representation errors (e.g. numerical approximation or parametrization errors) combined state and parameter estimation replace linear transport by approximations such as Sinkhorn (Doucet, Cuturi, 2013) time-continuous LETF formulations dzi =

M

• i=1

zidsij + dΞi (Crisan et al, 2010, Sean Meyn et al, 2013, CR, 2013).

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 21 / 22

Future work and references

References:

1

C.J. Cotter and S. Reich, Ensemble filter techniques for intermittent data assimilation, in: M. Cullen et al (eds.), Large scale inverse problems, Radon Ser. Comput. Appl. Math. Vol. 13, pages 91-134, 2013.

2

• S. Reich, A non-parametric ensemble transform method for Bayesian

inference, SIAM SISC, Vol. 35, pages A2013-A2014, 2013.

3

• Y. Cheng and S. Reich, A McKean optimal transportation perspective on

Feynman-Kac formulae with application to data assimilation, Frontiers in Dynamical Systems, Springer-Verlag, to appear.

4

• S. Reich and C.J. Cotter, Uncertainty quantification and Bayesian data

assimilation, Cambridge University Press, to appear.

Sebastian Reich (UP and UoR) Particle filters for infinite-dimensional systems 7 January 2014 22 / 22