On the convergence of (ensemble) Kalman filters and smoothers onto the unstable subspace
|
|
- Brendan Foster
- 6 years ago
- Views:
Transcription
1 On the convergence of (ensemble) Kalman filters and smoothers onto the unstable subspace Marc Bocquet CEREA, joint lab École des Ponts ParisTech and EdF R&D, Université Paris-Est, France Institut Pierre-Simon Laplace Based on collaborations with Alberto Carrassi (NERSC), Karthik Gurumoorthy (ICTS), Amit Apte (ICTS), Colin Grutzen (UNC), and Christopher K.R.T.Jones (UNC) M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
2 Context Outline 1 Context 2 Theoretical results for linear dynamical systems The filter case The smoother case From linear to nonlinear models 3 Numerical results for nonlinear systems The Lorenz-95 model Eigenspectrum Geometry 4 Conclusions 5 References M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
3 Context Assimilation in the unstable subspace (AUS paradigm) Several numerical results suggest that the skills of ensemble-based data assimilation methods in chaotic systems are related to the instabilities of the underlying dynamics [Ng et al., 2011]. Numerical evidence that some asymptotic properties of the ensemble-based covariances (rank, range) relate to the unstable modes of the dynamics [Sakov and Oke, 2008; Carrassi et al., 2009]. This behaviour is at the basis of algorithms known as Assimilation in the Unstable Subspace [Trevisan and Uboldi, 2004; Palatella et al. 2013], and exploited in the Iterative Ensemble Kalman Filter/Smoother [Bocquet and Sakov, ]. But we need formal proof with a view to a better design of reduced-order methods, and a better understanding of these results. A formal proof could be obtained in the linear model case, while numerical evidence could be obtained in the non-gaussian/nonlinear case. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
4 Context Characterization of model dynamics: reminder State and (infinitesimal) error dynamics: dx(t) dt = M t (x(t)), de(t) dt The time integration of the linear error dynamics yields the resolvent: = M x(t),t e(t). (1) e(t 1 ) = M(t 1,t 0 )e(t 0 ). (2) The Oseledec theorem tells that the limiting matrix (far future) { } S(t 0 ) = lim M(t 1,t 0 ) T 1 2(t M(t 1,t 0 ) 1 t 0 ). (3) t 1 exists, has eigenvalues e λ 1 e λ 2... e λ n where the λ i are called the Lyapunov exponents that do not depend on t 0, and has eigenvectors that are called the forward Lyapunov vectors (which depend on t 0 ). Symmetrically (far past) S(t 1 ) = lim {M(t 1,t 0 )M(t 1,t 0 ) T} 1 2(t 1 t 0 ). (4) t 0 exists, has the same eigenvalues that do not depend on t 1, and eigenvectors that are called the backward Lyapunov vectors (which depend on t 1 ). M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
5 Context Characterization of model dynamics: reminder The forward and backward Lyapunov vectors are orthonormal. They are norm-dependent. The positive Lyapunov exponents correspond to exponentially growing error/unstable modes. The negative Lyapunov exponents correspond to exponentially decaying error/stable modes. The zero Lyapunov exponents correspond to neutral modes. The backward Lyapunov vectors generate a sequence of embedded subspaces of R n for each t 1 such that F 1 (t 1) F 2 (t 1) F n (t 1 ) = R n (5) where for e Fi (t 1 )\Fi 1 (t 1), M 1 (t 1,t 0 )e t 0 e λ i (t 1 t 0 ) e. We define the unstable-neutral subspace U t1 F n 0 (t 1 ) as the space generated by the n 0 backward Lyapunov vectors that are related to positive and zero Lyapunov exponents. Here, the stable subspace is defined as the orthogonal U t 1 of U t1 in R n. See [Legras and Vautard, 1996] for a topical introduction. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
6 Theoretical results for linear dynamical systems Outline 1 Context 2 Theoretical results for linear dynamical systems The filter case The smoother case From linear to nonlinear models 3 Numerical results for nonlinear systems The Lorenz-95 model Eigenspectrum Geometry 4 Conclusions 5 References M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
7 Theoretical results for linear dynamical systems The filter case Linear case: Degenerate Kalman filter equations Model dynamics and observation model: x k = M k x k 1 + w k, (6) y k = H k x k + v k. (7) The model and observation noises, w k and v k, are assumed mutually independent, unbiased Gaussian white sequences with statistics E[v k vl T ] = δ k,l R k, E[w k wl T ] = δ k,l Q k, E[v k wl T ] = 0. (8) Forecast error covariance matrix P k recurrence of the Kalman filter (KF) P k+1 = M k+1 (I + P k Ω k ) 1 P k M T k+1 + Q k+1, (9) where Ω k H T k R 1 k H k (10) are the precision matrices and P 0 can be of arbitrary rank. In the case Q k 0, Gurumoorthy at al. (2016) proved rigorously that the full-rank KF P k collapses onto the unstable subspace. Still in the case Q k 0, this is going to be generalised [Bocquet at al., 2016] in the following in several ways and for degenerate P 0 required to connect to reduced-order methods such as the ensemble Kalman filter (EnKF). M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
8 Theoretical results for linear dynamical systems The filter case Result 1: Bound of the covariance free forecast Simple inequality in the set of the semi-definite symmetric matrices P k M k:0 P 0 M T k:0 + Ξ k. (11) where Ξ 0 0 and fork 1 k Ξ k M k:l Q l M T k:l (12) l=1 is known as the controllability matrix [Jazwinski, 1970]. In the absence of model noise (Q k 0 for the rest of this talk), it reads Assuming the dynamics is non-singular P k M k:0 P 0 M T k:0. (13) Im(P k ) = M k:0 (Im(P 0 )). (14) If n 0 is the dimension of the unstable-neutral subspace, it can further be shown that lim rank(p k) min{rank(p 0 ),n 0 }. (15) k M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
9 Theoretical results for linear dynamical systems The filter case Result 2: Collapse onto the unstable subspace Let σi k, for i = 1,...,n denote the eigenvalues of P k ordered as σ1 k σ 2 k σ n k. We can show that ( ) σi k α i exp 2kλi k (16) where kλi k is a log-singular value of M k:0 and lim k λi k = λ i. This gives an upper bound for all eigenvalues of P k and a rate of convergence for the n n 0 smallest ones. If P k is uniformly bounded, it can further be shown that the stable subspace of the dynamics is asymptotically in the null space of P k, i.e. for any vector u k:0 in the stable subspace lim P ku k:0 = 0. (17) k M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
10 Theoretical results for linear dynamical systems The filter case Numerical illustration and verification Magnitude 10 0 (a) Eigenvalues of analysis error covariance matrix Linearized Lorenz-95 model around a Lorenz-95 trajectory. (a) Evolution of the eigenvalues of P k. Convergence rate Assimilation count (b) Convergence rates of eigenvalues 10 Estimated convergence rates 2* Lyapunov exponents Stable dimension (least to most) (b) Convergence rate to 0 of the eigenvalues related to the stable modes. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
11 Theoretical results for linear dynamical systems The filter case Result 3: Explicit dependence of P k on P 0 Using either analytic continuation or the symplectic symmetry of the linear representation of covariances, we have proven that where An alternative is P k = M k:0 P 0 M T k:0 k 1 Γ k l=0 ( I + Γ k M k:0 P 0 M T k:0) 1. (18) M T k:l Ω l M 1 k:l. (19) P k = M k:0 P 0 [I + Θ k P 0 ] 1 M T k:0 (20) where Θ k M T k 1 k:0 Γ km k:0 = M T l:0 Ω l M l:0. (21) l=0 is the information matrix, directly related to the observability of the DA system. Formulas theoretically enlightening! M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
12 Theoretical results for linear dynamical systems The filter case Result 4: Asymptotics of P k Questions: Under which conditions does P k forget about P 0 = X 0 X T 0? Can we analytically compute its asymptotics? We proposed a sufficient set of conditions Condition 1: Assume the forward Lyapunov vectors at t 0 associated to the unstable and neutral directions are the columns of V +,0 R n n 0. The condition reads ( ) rank X T 0 V +,0 = n 0. (22) Condition 2: The model is sufficiently observed so that the unstable and neutral directions remain under control, that is U T +,k Γ ku +,k > εi (23) where U +,k is a matrix whose columns are the backward Lyapunov vectors related to non-negative exponents and ε > 0 is a positive number. Condition 3: For any neutral backward Lyapunov vector u k, we have i.e. the neutral modes should be sufficiently observed. lim k ut k Γ ku k =, (24) M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
13 Theoretical results for linear dynamical systems The filter case Result 4: Asymptotics of P k Under these three conditions, we obtain ] } 1 lim {P k U +,k [U T+,k Γ ku +,k U T+,k = 0. (25) k The asymptotic sequence does not depend on P 0, only Γ k! Peculiar role of the neutral modes (arithmetic convergence). Numerical illustration and verification Frobenius norm of the difference 10 5 (b) Universality of sequence r 0 = r' 0 = n n 0 < r 0 = r' 0 < n r r' 0 0 r = r' 0 < n Assimilation count Linearized Lorenz-95 model around a Lorenz-95 trajectory. Frobenius norm of the difference between two different P 0 when the conditions are satisfied, i.e. P a k P a k. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
14 Theoretical results for linear dynamical systems The filter case From the degenerate KF to the square-root EnKF Normalised anomaly decomposition Square-root formulation; right-transform update formula P k = X k X T k. (26) X k = M k:0 X 0 [I + X T 0 Θ kx 0 ] 1/2 Ψk (27) where Ψ k is an orthogonal matrix. Square-root formulation; left-transform update formula [ 1/2 X k = I + M k:0 P 0 M T k:0 k] Γ Mk:0 X 0 Ψ k. (28) With linear models, Gaussian observation and initial errors, the (square-root) degenerate KF is equivalent to the square-root EnKF and can serve as a proxy to the EnKF applied to nonlinear models. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
15 Theoretical results for linear dynamical systems The smoother case Degenerate square root Kalman smoother t L 2 y L 3 y L 2 S t t L 3 t L 2 y L 1 y L t L S t t L 1 t L y L+1 y L+2 t L+2 L t t L+1 t L+2 The scheme at a glance, variational correspondence (x = x k + X k w) : J (w) = 1 k+l 2 y l H l M l:k (x k + X k w) 2 R l w 2 (29) l=k+l S+1 From the Hessian of J, we infer I N + X T k Ω k X k where Ωk k+l M T l:k Ω l M l:k, (30) l=k+l S+1 ( X k+s = M k+s:k X k I N + X T k Ω ) 1 2 k X k Ψ k. (31) M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
16 Theoretical results for linear dynamical systems The smoother case Degenerate square root Kalman smoother: dependence on X 0 From the recurrence, we can obtain the explicit expression of the anomalies as a function of the initial anomalies. Left-transform update; if k = ps, p = 0,1,... : [ X k = M k:0 X 0 I N + X T 0 Θ ] 1 2 k X 0 Ψ k (32) where p 1 Θ k M T qs:0 Ω qs M qs:0. (33) q=0 Right-transform update; if k = ps, p = 0,1,... : ] X k = [I n + M k:0 P 0 M T 1 2 k:0 Γ k M k:0 X 0 Ψ k (34) where p 1 Γ k = q=0 M T k:qs Ω qs M 1 k:qs. (35) M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
17 Theoretical results for linear dynamical systems The smoother case Degenerate square root Kalman smoother: convergence onto the unstable-neutral subspace The convergence rate of the collapse of P k of the smoother is not expected to be faster than the filter s: the bounding rate is the same. However the accuracy of the smoother for re-analysis is expected to be better which should impact the asymptotic sequences. Indeed we have, for k = ps, p = 0,1,...: lim {X k U +,k [U T+,k Γ ] 1 } 2 k U +,k Ψ k = 0. (36) k The only difference is in the observability matrix Γ k, for k = ps, p = 0,1,...: which guarantees that k+l S Γ k = Γ k + l=k M T k:l Ω l M 1 k:l. (37) U +,k [ U T +,k Γ k U +,k ] 1 U T +,k U +,k [U T +,k Γ ku +,k ] 1 U T +,k. (38) M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
18 Theoretical results for linear dynamical systems From linear to nonlinear models From linear to nonlinear models Nonlinear model dynamics and nonlinear observation model: x k = M k:k 1 (x k 1 ), (39) y k = H k (x k ) + v k (40) Square root degenerate Kalman filter Square root ensemble Kalman filter (EnKF) Square root degenerate Kalman smoother Iterative ensemble Kalman smoother (IEnKS) The IEnKS follows the square root degenerate Kalman smoother that we inferred from a variational principle, but from the cost function J (w) = 1 k+l 2 y l H l M l:k (x k + X k w) 2 R l w 2. (41) l=k+l S+1 The this archetype of the so-called four-dimensional EnVar method (4DEnVar) which avoid the need to use an adjoint model, but with a proper ensemble update, and an outer loop. The IEnKS is systematically more accurate for smoothing and filtering than 4D-Var, the EnKF, and the EnKS. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
19 Numerical results for nonlinear systems Outline 1 Context 2 Theoretical results for linear dynamical systems The filter case The smoother case From linear to nonlinear models 3 Numerical results for nonlinear systems The Lorenz-95 model Eigenspectrum Geometry 4 Conclusions 5 References M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
20 Numerical results for nonlinear systems The Lorenz-95 model Nonlinear chaotic models: the Lorenz-95 low-order model x Time It represents a mid-latitude zonal circle of the global atmosphere. Set of M = 40 ordinary differential equations [Lorenz and Emmanuel 1998]: dx m = (x m+1 x m 2 )x m 1 x m + F, (42) dt where F = 8, and the boundary is cyclic. Conservative system except for a forcing term F and a dissipation term x m. Chaotic dynamics, 13 positive and 1 neutral Lyapunov exponents, a doubling time of about 0.42 time units. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
21 Numerical results for nonlinear systems Eigenspectrum Spectrum of the analysis error covariance matrix Normalized mean eigenvalues EnKF IEnKS, L=30, S=1, filtering IEnKS, L=30, S=1, smoothing σ = 1 Normalized mean eigenvalues EnKF IEnKS, L=30, S=1, filtering IEnKS, L=30, S=1, smoothing σ = Eigenvalue rank r Eigenvalue rank r Normalized mean eigenvalues EnKF IEnKS, L=30, S=1, filtering IEnKS, L=30, S=1, smoothing σ = 1 Normalized mean eigenvalues EnKF IEnKS, L=30, S=1, filtering IEnKS, L=30, S=1, smoothing σ = Eigenvalue rank r Time-average spectra of P a k : A visible transition at r = Eigenvalue rank r M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
22 Numerical results for nonlinear systems Geometry Geometry of the anomaly simplex and unstable-neutral subspace Anomaly to perturbation: Anomaly: vk n = [X k] n with 1 n N. Lyapunov vector: u p k at t k, 1 p P. Projection of vk n onto up k : v n cos(θ n,p k k ) where θ n,p k is the angle between the vectors (recall u p k = 1.) Anomaly to the unstable-neutral subspace: Consider the relative position of an anomaly with respect to the unstable-neutral subspace U k, where U k = Span { u 1 } k,u2 k,...,un 0 k. Square cosine of the angle between the anomaly vk n and U k is obtained as { cos 2 (θk n n 0 ) = cos 2 (θ n,p n 0 (u p k k ) = )T v n } 2 k. (43) p=1 p=1 v n 2 k Anomaly simplex to the unstable-neutral subspace: A complete characterization of the relative orientation can be achieved by a set of angles, called principal angles, whose number is given by the minimum of the dimensions of both subspaces. These angles are intrinsic and do not depend on the parameterization of both subspaces. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
23 Numerical results for nonlinear systems Geometry Nonlinear chaotic model Mean angle with the unstable-neutral subspace EnKF IEnKS, L=30, S=1, filtering IEnKS, L=30, S=1, smoothing Observation error standard deviation Average angle (in degrees) between an anomaly (from the ensemble) and the unstable-neutral subspace as a function of the observation error (EnKF and IEnKS, Lorenz-95, t = 0.05, R = σ 2 I N = 20). M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
24 Numerical results for nonlinear systems Geometry Nonlinear chaotic model Mean angle with the unstable-neutral subspace EnKF IEnKS, L=5, S=1, filtering IEnKS, L=5, S=1, smoothing Time interval between updates Average angle (in degrees) between an anomaly (from the ensemble) and the unstable-neutral subspace as a function of the interval between updates (EnKF and IEnKS, Lorenz-95, R = 10 4 I, N = 20). M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
25 Numerical results for nonlinear systems Geometry Nonlinear chaotic model 5 Mean angle with the unstable-neutral subspace RMSE Angle Ensemble size Average root mean square error Average angle (in degrees) between an anomaly (from the ensemble) and the unstable-neutral subspace as a function of the DAW length (IEnKS, Lorenz-95), as well as the corresponding RMSE of the analysis. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
26 Numerical results for nonlinear systems Geometry Nonlinear chaotic model Mean angle with the unstable-neutral subspace EnKF, σ=0.01, t=0.05, N=20 EnKF, σ=1, t=0.20, N=20 IEnKS, σ=1, t=0.05, N=20, L=50, S=30, smoothing IEnKS, σ=1, t=0.05, N=20, L=50, S=30, filtering EnKF, σ=1, t=0.05, N= Principal angle index Average principle angles (in degrees) between the anomalies simplex and the unstable-neutral subspace as a function of the angle index (IEnKS, Lorenz-95, S = 1, R = I, N = 15) at the end of the DAW (filtering). M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
27 Conclusions Outline 1 Context 2 Theoretical results for linear dynamical systems The filter case The smoother case From linear to nonlinear models 3 Numerical results for nonlinear systems The Lorenz-95 model Eigenspectrum Geometry 4 Conclusions 5 References M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
28 Conclusions Conclusions We proved that if the models are linear and the initial and observation error statistics are Gaussian, the (degenerate) KF, or (deterministic) EnKF in this case, collapse onto the unstable subspace. We provided a rate of convergence. We showed that under specific observability conditions and for P 0 of sufficiently large column space, P k asymptotics is independent from P 0 and can be computed analytically. These results can be extrapolated to the case of smoothers. These degenerate KF/KS serve as proxies for the EnKF and the IEnKS (for filtering and smoothing). We numerically studied the collapse onto the unstable-neutral subspace of those nonlinear filter/smoother using a geometrical description of the relative position of the unstable-neutral subspaces with the set of filter anomalies. M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
29 Conclusions Final word Thank you for your attention! M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
30 References References [1] M. Bocquet, Ensemble Kalman filtering without the intrinsic need for inflation, Nonlin. Processes Geophys., 18 (2011), pp [2] M. Bocquet and A. Carrassi, Ensemble variational filters and smoothers, and the unstable subspace, in preparation, (2016). [3] M. Bocquet, K. S. Gurumoorthy, A. Apte, A. Carrassi, C. Grudzien, and C. K. R. T. Jones, Degenerate Kalman filter error covariances and their convergence onto the unstable subspace, arxiv preprint arxiv: , (2016). [4] M. Bocquet, P. N. Raanes, and A. Hannart, Expanding the validity of the ensemble Kalman filter without the intrinsic need for inflation, Nonlin. Processes Geophys., 22 (2015), pp [5] M. Bocquet and P. Sakov, An iterative ensemble Kalman smoother, Q. J. R. Meteorol. Soc., 140 (2014), pp [6] A. Carrassi, S. Vannitsem, D. Zupanski, and M. Zupanski, The maximum likelihood ensemble filter performances in chaotic systems, Tellus A, 61 (2009), pp [7] K. S. Gurumoorthy, C. Grudzien, A. Apte, A. Carrassi, and C. K. R. T. Jones, Rank deficiency of Kalman error covariance matrices in linear perfect model, arxiv preprint arxiv: , (2015). [8] G.-H. C. Ng, D. McLaughlin, D. Entekhabi, and A. Ahanin, The role of model dynamics in ensemble Kalman filter performance for chaotic systems, Tellus A, 63 (2011), pp [9] L. Palatella, A. Carrassi, and A. Trevisan, Lyapunov vectors and assimilation in the unstable subspace: theory and applications, J. Phys. A: Math. Theor., 46 (2013), p [10] L. Palatella and A. Trevisan, Interaction of Lyapunov vectors in the formulation of the nonlinear extension of the Kalman filter, Phys. Rev. E, 91 (2015), p [11] A. Trevisan and F. Uboldi, Assimilation of standard and targeted observations within the unstable subspace of the observation-analysis-forecast cycle, J. Atmos. Sci., 61 (2004), pp M. Bocquet 11 th EnKF workshop, Bergen, Norway, June / 30
Data assimilation in the geosciences An overview
Data assimilation in the geosciences An overview Alberto Carrassi 1, Olivier Talagrand 2, Marc Bocquet 3 (1) NERSC, Bergen, Norway (2) LMD, École Normale Supérieure, IPSL, France (3) CEREA, joint lab École
More informationFinite-size ensemble Kalman filters (EnKF-N)
Finite-size ensemble Kalman filters (EnKF-N) Marc Bocquet (1,2), Pavel Sakov (3,4) (1) Université Paris-Est, CEREA, joint lab École des Ponts ParisTech and EdF R&D, France (2) INRIA, Paris-Rocquencourt
More informationRelationship between Singular Vectors, Bred Vectors, 4D-Var and EnKF
Relationship between Singular Vectors, Bred Vectors, 4D-Var and EnKF Eugenia Kalnay and Shu-Chih Yang with Alberto Carrasi, Matteo Corazza and Takemasa Miyoshi 4th EnKF Workshop, April 2010 Relationship
More informationEstimating model evidence using data assimilation
Estimating model evidence using data assimilation Application to the attribution of climate-related events Marc Bocquet 1, Alberto Carrassi 2, Alexis Hannart 3 & Michael Ghil 4,5 and the DADA Team (1 CEREA,
More informationRelationship between Singular Vectors, Bred Vectors, 4D-Var and EnKF
Relationship between Singular Vectors, Bred Vectors, 4D-Var and EnKF Eugenia Kalnay and Shu-Chih Yang with Alberto Carrasi, Matteo Corazza and Takemasa Miyoshi ECODYC10, Dresden 28 January 2010 Relationship
More informationData assimilation; comparison of 4D-Var and LETKF smoothers
Data assimilation; comparison of 4D-Var and LETKF smoothers Eugenia Kalnay and many friends University of Maryland CSCAMM DAS13 June 2013 Contents First part: Forecasting the weather - we are really getting
More informationarxiv: v4 [math.oc] 26 Jul 2018
ASYMPTOTIC FORECAST UNCERTAINTY AND THE UNSTABLE SUBSPACE IN THE PRESENCE OF ADDITIVE MODEL ERROR COLIN GRUDZIEN, ALBERTO CARRASSI, AND MARC BOCQUET, arxiv:1707.08334v4 [math.oc] 6 Jul 018 Abstract. It
More informationOn the Kalman Filter error covariance collapse into the unstable subspace
Nonlin. Processes Geophys., 18, 243 250, 2011 doi:10.5194/npg-18-243-2011 Author(s) 2011. CC Attribution 3.0 License. Nonlinear Processes in Geophysics On the Kalman Filter error covariance collapse into
More informationMultiple-scale error growth and data assimilation in convection-resolving models
Numerical Modelling, Predictability and Data Assimilation in Weather, Ocean and Climate A Symposium honouring the legacy of Anna Trevisan Bologna, Italy 17-20 October, 2017 Multiple-scale error growth
More informationSmoothers: Types and Benchmarks
Smoothers: Types and Benchmarks Patrick N. Raanes Oxford University, NERSC 8th International EnKF Workshop May 27, 2013 Chris Farmer, Irene Moroz Laurent Bertino NERSC Geir Evensen Abstract Talk builds
More informationRevision of TR-09-25: A Hybrid Variational/Ensemble Filter Approach to Data Assimilation
Revision of TR-9-25: A Hybrid Variational/Ensemble ilter Approach to Data Assimilation Adrian Sandu 1 and Haiyan Cheng 1 Computational Science Laboratory Department of Computer Science Virginia Polytechnic
More informationA new Hierarchical Bayes approach to ensemble-variational data assimilation
A new Hierarchical Bayes approach to ensemble-variational data assimilation Michael Tsyrulnikov and Alexander Rakitko HydroMetCenter of Russia College Park, 20 Oct 2014 Michael Tsyrulnikov and Alexander
More informationEnKF and Catastrophic filter divergence
EnKF and Catastrophic filter divergence David Kelly Kody Law Andrew Stuart Mathematics Department University of North Carolina Chapel Hill NC bkelly.com June 4, 2014 SIAM UQ 2014 Savannah. David Kelly
More informationPredictability and data assimilation issues in multiple-scale convection-resolving systems
Predictability and data assimilation issues in multiple-scale convection-resolving systems Francesco UBOLDI Milan, Italy, uboldi@magritte.it MET-NO, Oslo, Norway 23 March 2017 Predictability and data assimilation
More informationProf. Stephen G. Penny University of Maryland NOAA/NCEP, RIKEN AICS, ECMWF US CLIVAR Summit, 9 August 2017
COUPLED DATA ASSIMILATION: What we need from observations and modellers to make coupled data assimilation the new standard for prediction and reanalysis. Prof. Stephen G. Penny University of Maryland NOAA/NCEP,
More informationThe Ensemble Kalman Filter:
p.1 The Ensemble Kalman Filter: Theoretical formulation and practical implementation Geir Evensen Norsk Hydro Research Centre, Bergen, Norway Based on Evensen 23, Ocean Dynamics, Vol 53, No 4 p.2 The Ensemble
More informationEnsemble Kalman Filter
Ensemble Kalman Filter Geir Evensen and Laurent Bertino Hydro Research Centre, Bergen, Norway, Nansen Environmental and Remote Sensing Center, Bergen, Norway The Ensemble Kalman Filter (EnKF) Represents
More informationData assimilation : Basics and meteorology
Data assimilation : Basics and meteorology Olivier Talagrand Laboratoire de Météorologie Dynamique, École Normale Supérieure, Paris, France Workshop on Coupled Climate-Economics Modelling and Data Analysis
More informationA Note on the Particle Filter with Posterior Gaussian Resampling
Tellus (6), 8A, 46 46 Copyright C Blackwell Munksgaard, 6 Printed in Singapore. All rights reserved TELLUS A Note on the Particle Filter with Posterior Gaussian Resampling By X. XIONG 1,I.M.NAVON 1,2 and
More informationState and Parameter Estimation in Stochastic Dynamical Models
State and Parameter Estimation in Stochastic Dynamical Models Timothy DelSole George Mason University, Fairfax, Va and Center for Ocean-Land-Atmosphere Studies, Calverton, MD June 21, 2011 1 1 collaboration
More informationEnsemble Data Assimilation and Uncertainty Quantification
Ensemble Data Assimilation and Uncertainty Quantification Jeff Anderson National Center for Atmospheric Research pg 1 What is Data Assimilation? Observations combined with a Model forecast + to produce
More informationEnKF and filter divergence
EnKF and filter divergence David Kelly Andrew Stuart Kody Law Courant Institute New York University New York, NY dtbkelly.com December 12, 2014 Applied and computational mathematics seminar, NIST. David
More informationErgodicity in data assimilation methods
Ergodicity in data assimilation methods David Kelly Andy Majda Xin Tong Courant Institute New York University New York NY www.dtbkelly.com April 15, 2016 ETH Zurich David Kelly (CIMS) Data assimilation
More informationLocal Ensemble Transform Kalman Filter: An Efficient Scheme for Assimilating Atmospheric Data
Local Ensemble Transform Kalman Filter: An Efficient Scheme for Assimilating Atmospheric Data John Harlim and Brian R. Hunt Department of Mathematics and Institute for Physical Science and Technology University
More information4 Derivations of the Discrete-Time Kalman Filter
Technion Israel Institute of Technology, Department of Electrical Engineering Estimation and Identification in Dynamical Systems (048825) Lecture Notes, Fall 2009, Prof N Shimkin 4 Derivations of the Discrete-Time
More informationEnsemble prediction and strategies for initialization: Tangent Linear and Adjoint Models, Singular Vectors, Lyapunov vectors
Ensemble prediction and strategies for initialization: Tangent Linear and Adjoint Models, Singular Vectors, Lyapunov vectors Eugenia Kalnay Lecture 2 Alghero, May 2008 Elements of Ensemble Forecasting
More informationWhat do we know about EnKF?
What do we know about EnKF? David Kelly Kody Law Andrew Stuart Andrew Majda Xin Tong Courant Institute New York University New York, NY April 10, 2015 CAOS seminar, Courant. David Kelly (NYU) EnKF April
More informationMaximum Likelihood Ensemble Filter Applied to Multisensor Systems
Maximum Likelihood Ensemble Filter Applied to Multisensor Systems Arif R. Albayrak a, Milija Zupanski b and Dusanka Zupanski c abc Colorado State University (CIRA), 137 Campus Delivery Fort Collins, CO
More informationAsynchronous data assimilation
Ensemble Kalman Filter, lecture 2 Asynchronous data assimilation Pavel Sakov Nansen Environmental and Remote Sensing Center, Norway This talk has been prepared in the course of evita-enkf project funded
More informationAddressing the nonlinear problem of low order clustering in deterministic filters by using mean-preserving non-symmetric solutions of the ETKF
Addressing the nonlinear problem of low order clustering in deterministic filters by using mean-preserving non-symmetric solutions of the ETKF Javier Amezcua, Dr. Kayo Ide, Dr. Eugenia Kalnay 1 Outline
More informationConvergence of the Ensemble Kalman Filter in Hilbert Space
Convergence of the Ensemble Kalman Filter in Hilbert Space Jan Mandel Center for Computational Mathematics Department of Mathematical and Statistical Sciences University of Colorado Denver Parts based
More informationThe Kalman Filter. Data Assimilation & Inverse Problems from Weather Forecasting to Neuroscience. Sarah Dance
The Kalman Filter Data Assimilation & Inverse Problems from Weather Forecasting to Neuroscience Sarah Dance School of Mathematical and Physical Sciences, University of Reading s.l.dance@reading.ac.uk July
More informationOrganization. I MCMC discussion. I project talks. I Lecture.
Organization I MCMC discussion I project talks. I Lecture. Content I Uncertainty Propagation Overview I Forward-Backward with an Ensemble I Model Reduction (Intro) Uncertainty Propagation in Causal Systems
More informationHandling nonlinearity in Ensemble Kalman Filter: Experiments with the three-variable Lorenz model
Handling nonlinearity in Ensemble Kalman Filter: Experiments with the three-variable Lorenz model Shu-Chih Yang 1*, Eugenia Kalnay, and Brian Hunt 1. Department of Atmospheric Sciences, National Central
More informationConvergence of Square Root Ensemble Kalman Filters in the Large Ensemble Limit
Convergence of Square Root Ensemble Kalman Filters in the Large Ensemble Limit Evan Kwiatkowski, Jan Mandel University of Colorado Denver December 11, 2014 OUTLINE 2 Data Assimilation Bayesian Estimation
More informationGaussian Process Approximations of Stochastic Differential Equations
Gaussian Process Approximations of Stochastic Differential Equations Cédric Archambeau Dan Cawford Manfred Opper John Shawe-Taylor May, 2006 1 Introduction Some of the most complex models routinely run
More informationData assimilation with and without a model
Data assimilation with and without a model Tyrus Berry George Mason University NJIT Feb. 28, 2017 Postdoc supported by NSF This work is in collaboration with: Tim Sauer, GMU Franz Hamilton, Postdoc, NCSU
More information4. DATA ASSIMILATION FUNDAMENTALS
4. DATA ASSIMILATION FUNDAMENTALS... [the atmosphere] "is a chaotic system in which errors introduced into the system can grow with time... As a consequence, data assimilation is a struggle between chaotic
More informationGaussian Filtering Strategies for Nonlinear Systems
Gaussian Filtering Strategies for Nonlinear Systems Canonical Nonlinear Filtering Problem ~u m+1 = ~ f (~u m )+~ m+1 ~v m+1 = ~g(~u m+1 )+~ o m+1 I ~ f and ~g are nonlinear & deterministic I Noise/Errors
More informationFundamentals of Data Assimila1on
014 GSI Community Tutorial NCAR Foothills Campus, Boulder, CO July 14-16, 014 Fundamentals of Data Assimila1on Milija Zupanski Cooperative Institute for Research in the Atmosphere Colorado State University
More informationFundamentals of Data Assimilation
National Center for Atmospheric Research, Boulder, CO USA GSI Data Assimilation Tutorial - June 28-30, 2010 Acknowledgments and References WRFDA Overview (WRF Tutorial Lectures, H. Huang and D. Barker)
More informationHOW TO IMPROVE FORECASTING WITH BRED VECTORS BY USING THE GEOMETRIC NORM
HOW TO IMPROVE FORECASTING WITH BRED VECTORS BY USING THE GEOMETRIC NORM Diego Pazó Instituto de Física de Cantabria (IFCA), CSIC-UC, Santander (Spain) Santander OUTLINE 1) Lorenz '96 model. 2) Definition
More informationLocal Ensemble Transform Kalman Filter
Local Ensemble Transform Kalman Filter Brian Hunt 11 June 2013 Review of Notation Forecast model: a known function M on a vector space of model states. Truth: an unknown sequence {x n } of model states
More informationKarhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques
Institut für Numerische Mathematik und Optimierung Karhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques Oliver Ernst Computational Methods with Applications Harrachov, CR,
More informationNew Fast Kalman filter method
New Fast Kalman filter method Hojat Ghorbanidehno, Hee Sun Lee 1. Introduction Data assimilation methods combine dynamical models of a system with typically noisy observations to obtain estimates of the
More informationAn iterative ensemble Kalman filter in presence of additive model error
An iterative ensemble Kalman filter in presence of additive model error Pavel Sakov, Jean-Matthieu Haussaire, Marc Bocquet To cite this version: Pavel Sakov, Jean-Matthieu Haussaire, Marc Bocquet. An iterative
More informationBayesian Inference and the Symbolic Dynamics of Deterministic Chaos. Christopher C. Strelioff 1,2 Dr. James P. Crutchfield 2
How Random Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos Christopher C. Strelioff 1,2 Dr. James P. Crutchfield 2 1 Center for Complex Systems Research and Department of Physics University
More informationPractical Aspects of Ensemble-based Kalman Filters
Practical Aspects of Ensemble-based Kalman Filters Lars Nerger Alfred Wegener Institute for Polar and Marine Research Bremerhaven, Germany and Bremen Supercomputing Competence Center BremHLR Bremen, Germany
More informationDETERMINATION OF MODEL VALID PREDICTION PERIOD USING THE BACKWARD FOKKER-PLANCK EQUATION
.4 DETERMINATION OF MODEL VALID PREDICTION PERIOD USING THE BACKWARD FOKKER-PLANCK EQUATION Peter C. Chu, Leonid M. Ivanov, and C.W. Fan Department of Oceanography Naval Postgraduate School Monterey, California.
More informationA Spectral Approach to Linear Bayesian Updating
A Spectral Approach to Linear Bayesian Updating Oliver Pajonk 1,2, Bojana V. Rosic 1, Alexander Litvinenko 1, and Hermann G. Matthies 1 1 Institute of Scientific Computing, TU Braunschweig, Germany 2 SPT
More information(Extended) Kalman Filter
(Extended) Kalman Filter Brian Hunt 7 June 2013 Goals of Data Assimilation (DA) Estimate the state of a system based on both current and all past observations of the system, using a model for the system
More informationExpanding the validity of the ensemble Kalman filter without the intrinsic need for inflation
Nonlin. Processes Geophys.,, 645 66, 015 www.nonlin-processes-geophys.net//645/015/ doi:10.5194/npg--645-015 Authors) 015. CC Attribution 3.0 License. Expanding the validity of the ensemble Kalman filter
More informationFour-Dimensional Ensemble Kalman Filtering
Four-Dimensional Ensemble Kalman Filtering B.R. Hunt, E. Kalnay, E.J. Kostelich, E. Ott, D.J. Patil, T. Sauer, I. Szunyogh, J.A. Yorke, A.V. Zimin University of Maryland, College Park, MD 20742, USA Ensemble
More informationData assimilation with and without a model
Data assimilation with and without a model Tim Sauer George Mason University Parameter estimation and UQ U. Pittsburgh Mar. 5, 2017 Partially supported by NSF Most of this work is due to: Tyrus Berry,
More informationMethods of Data Assimilation and Comparisons for Lagrangian Data
Methods of Data Assimilation and Comparisons for Lagrangian Data Chris Jones, Warwick and UNC-CH Kayo Ide, UCLA Andrew Stuart, Jochen Voss, Warwick Guillaume Vernieres, UNC-CH Amarjit Budiraja, UNC-CH
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationThe Ensemble Kalman Filter:
p.1 The Ensemble Kalman Filter: Theoretical formulation and practical implementation Geir Evensen Norsk Hydro Research Centre, Bergen, Norway Based on Evensen, Ocean Dynamics, Vol 5, No p. The Ensemble
More informationAccounting for model error due to unresolved scales within ensemble Kalman filtering
Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. (24) DOI:.2/qj.245 Accounting for model error due to unresolved scales within ensemble Kalman filtering Lewis Mitchell a,b
More informationNonlinear error dynamics for cycled data assimilation methods
Nonlinear error dynamics for cycled data assimilation methods A J F Moodey 1, A S Lawless 1,2, P J van Leeuwen 2, R W E Potthast 1,3 1 Department of Mathematics and Statistics, University of Reading, UK.
More informationAn Iterative EnKF for Strongly Nonlinear Systems
1988 M O N T H L Y W E A T H E R R E V I E W VOLUME 140 An Iterative EnKF for Strongly Nonlinear Systems PAVEL SAKOV Nansen Environmental and Remote Sensing Center, Bergen, Norway DEAN S. OLIVER Uni Centre
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationP3.11 A COMPARISON OF AN ENSEMBLE OF POSITIVE/NEGATIVE PAIRS AND A CENTERED SPHERICAL SIMPLEX ENSEMBLE
P3.11 A COMPARISON OF AN ENSEMBLE OF POSITIVE/NEGATIVE PAIRS AND A CENTERED SPHERICAL SIMPLEX ENSEMBLE 1 INTRODUCTION Xuguang Wang* The Pennsylvania State University, University Park, PA Craig H. Bishop
More informationDATA ASSIMILATION FOR FLOOD FORECASTING
DATA ASSIMILATION FOR FLOOD FORECASTING Arnold Heemin Delft University of Technology 09/16/14 1 Data assimilation is the incorporation of measurement into a numerical model to improve the model results
More informationAccelerating the spin-up of Ensemble Kalman Filtering
Accelerating the spin-up of Ensemble Kalman Filtering Eugenia Kalnay * and Shu-Chih Yang University of Maryland Abstract A scheme is proposed to improve the performance of the ensemble-based Kalman Filters
More information+ + ( + ) = Linear recurrent networks. Simpler, much more amenable to analytic treatment E.g. by choosing
Linear recurrent networks Simpler, much more amenable to analytic treatment E.g. by choosing + ( + ) = Firing rates can be negative Approximates dynamics around fixed point Approximation often reasonable
More informationAn iterative ensemble Kalman smoother
Quarterly Journalof the RoyalMeteorologicalSociety Q. J. R. Meteorol. Soc. 140: 1521 1535, July 2014 A DOI:10.1002/qj.2236 An iterative ensemble Kalman smoother M. Bocquet a,b * and P. Saov c a Université
More informationA Cross-Associative Neural Network for SVD of Nonsquared Data Matrix in Signal Processing
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 12, NO. 5, SEPTEMBER 2001 1215 A Cross-Associative Neural Network for SVD of Nonsquared Data Matrix in Signal Processing Da-Zheng Feng, Zheng Bao, Xian-Da Zhang
More informationBred Vectors, Singular Vectors, and Lyapunov Vectors in Simple and Complex Models
Bred Vectors, Singular Vectors, and Lyapunov Vectors in Simple and Complex Models Adrienne Norwood Advisor: Eugenia Kalnay With special thanks to Drs. Kayo Ide, Brian Hunt, Shu-Chih Yang, and Christopher
More informationAsymptotic distribution of eigenvalues of Laplace operator
Asymptotic distribution of eigenvalues of Laplace operator 23.8.2013 Topics We will talk about: the number of eigenvalues of Laplace operator smaller than some λ as a function of λ asymptotic behaviour
More information4DEnVar. Four-Dimensional Ensemble-Variational Data Assimilation. Colloque National sur l'assimilation de données
Four-Dimensional Ensemble-Variational Data Assimilation 4DEnVar Colloque National sur l'assimilation de données Andrew Lorenc, Toulouse France. 1-3 décembre 2014 Crown copyright Met Office 4DEnVar: Topics
More informationEnsemble forecasting and flow-dependent estimates of initial uncertainty. Martin Leutbecher
Ensemble forecasting and flow-dependent estimates of initial uncertainty Martin Leutbecher acknowledgements: Roberto Buizza, Lars Isaksen Flow-dependent aspects of data assimilation, ECMWF 11 13 June 2007
More informationStability of Ensemble Kalman Filters
Stability of Ensemble Kalman Filters Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley, Heikki Haario and Tuomo Kauranne Lappeenranta University of Technology University of
More informationAbstract 2. ENSEMBLE KALMAN FILTERS 1. INTRODUCTION
J5.4 4D ENSEMBLE KALMAN FILTERING FOR ASSIMILATION OF ASYNCHRONOUS OBSERVATIONS T. Sauer George Mason University, Fairfax, VA 22030 B.R. Hunt, J.A. Yorke, A.V. Zimin, E. Ott, E.J. Kostelich, I. Szunyogh,
More informationAdaptive ensemble Kalman filtering of nonlinear systems
Adaptive ensemble Kalman filtering of nonlinear systems Tyrus Berry George Mason University June 12, 213 : Problem Setup We consider a system of the form: x k+1 = f (x k ) + ω k+1 ω N (, Q) y k+1 = h(x
More informationData Assimilation in the Geosciences
Data Assimilation in the Geosciences An overview on methods, issues and perspectives arxiv:1709.02798v3 [physics.ao-ph] 8 Jun 2018 Alberto Carrassi, Marc Bocquet, Laurent Bertino and Geir Evensen Article
More informationLecture 3: Review of Linear Algebra
ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak, scribe: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters,
More informationMatrix stabilization using differential equations.
Matrix stabilization using differential equations. Nicola Guglielmi Universitá dell Aquila and Gran Sasso Science Institute, Italia NUMOC-2017 Roma, 19 23 June, 2017 Inspired by a joint work with Christian
More informationPerformance of ensemble Kalman filters with small ensembles
Performance of ensemble Kalman filters with small ensembles Xin T Tong Joint work with Andrew J. Majda National University of Singapore Sunday 28 th May, 2017 X.Tong EnKF performance 1 / 31 Content Ensemble
More informationLearning gradients: prescriptive models
Department of Statistical Science Institute for Genome Sciences & Policy Department of Computer Science Duke University May 11, 2007 Relevant papers Learning Coordinate Covariances via Gradients. Sayan
More informationChapter 6: Ensemble Forecasting and Atmospheric Predictability. Introduction
Chapter 6: Ensemble Forecasting and Atmospheric Predictability Introduction Deterministic Chaos (what!?) In 1951 Charney indicated that forecast skill would break down, but he attributed it to model errors
More informationThe University of Texas at Austin Department of Electrical and Computer Engineering. EE381V: Large Scale Learning Spring 2013.
The University of Texas at Austin Department of Electrical and Computer Engineering EE381V: Large Scale Learning Spring 2013 Assignment Two Caramanis/Sanghavi Due: Tuesday, Feb. 19, 2013. Computational
More informationEfficient Data Assimilation for Spatiotemporal Chaos: a Local Ensemble Transform Kalman Filter
arxiv:physics/0511236v2 [physics.data-an] 29 Dec 2006 Efficient Data Assimilation for Spatiotemporal Chaos: a Local Ensemble Transform Kalman Filter Brian R. Hunt Institute for Physical Science and Technology
More informationLagrangian Data Assimilation and Its Application to Geophysical Fluid Flows
Lagrangian Data Assimilation and Its Application to Geophysical Fluid Flows Laura Slivinski June, 3 Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 / 3 Data Assimilation Setup:
More informationFull-State Feedback Design for a Multi-Input System
Full-State Feedback Design for a Multi-Input System A. Introduction The open-loop system is described by the following state space model. x(t) = Ax(t)+Bu(t), y(t) =Cx(t)+Du(t) () 4 8.5 A =, B =.5.5, C
More informationR. E. Petrie and R. N. Bannister. Department of Meteorology, Earley Gate, University of Reading, Reading, RG6 6BB, United Kingdom
A method for merging flow-dependent forecast error statistics from an ensemble with static statistics for use in high resolution variational data assimilation R. E. Petrie and R. N. Bannister Department
More informationLOCAL ENSEMBLE TRANSFORM KALMAN FILTER: A NON-STATIONARY CONTROL LAW FOR COMPLEX ADAPTIVE OPTICS SYSTEMS ON ELTS
Florence, Italy. May 2013 ISBN: 978-88-908876-0-4 DOI: 10.12839/AO4ELT3.13253 LOCAL ENSEMBLE TRANSFORM KALMAN FILTER: A NON-STATIONARY CONTROL LAW FOR COMPLEX ADAPTIVE OPTICS SYSTEMS ON ELTS GRAY Morgan
More informationSome ideas for Ensemble Kalman Filter
Some ideas for Ensemble Kalman Filter Former students and Eugenia Kalnay UMCP Acknowledgements: UMD Chaos-Weather Group: Brian Hunt, Istvan Szunyogh, Ed Ott and Jim Yorke, Kayo Ide, and students Former
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationMIT Final Exam Solutions, Spring 2017
MIT 8.6 Final Exam Solutions, Spring 7 Problem : For some real matrix A, the following vectors form a basis for its column space and null space: C(A) = span,, N(A) = span,,. (a) What is the size m n of
More informationPhase space, Tangent-Linear and Adjoint Models, Singular Vectors, Lyapunov Vectors and Normal Modes
Phase space, Tangent-Linear and Adjoint Models, Singular Vectors, Lyapunov Vectors and Normal Modes Assume a phase space of dimension N where Autonomous governing equations with initial state: = is a state
More informationComparing the SEKF with the DEnKF on a land surface model
Comparing the SEKF with the DEnKF on a land surface model David Fairbairn, Alina Barbu, Emiliano Gelati, Jean-Francois Mahfouf and Jean-Christophe Caret CNRM - Meteo France Partly funded by European Union
More informationA Stochastic Collocation based. for Data Assimilation
A Stochastic Collocation based Kalman Filter (SCKF) for Data Assimilation Lingzao Zeng and Dongxiao Zhang University of Southern California August 11, 2009 Los Angeles Outline Introduction SCKF Algorithm
More informationEnsemble square-root filters
Ensemble square-root filters MICHAEL K. TIPPETT International Research Institute for climate prediction, Palisades, New Yor JEFFREY L. ANDERSON GFDL, Princeton, New Jersy CRAIG H. BISHOP Naval Research
More informationComputer Intensive Methods in Mathematical Statistics
Computer Intensive Methods in Mathematical Statistics Department of mathematics johawes@kth.se Lecture 16 Advanced topics in computational statistics 18 May 2017 Computer Intensive Methods (1) Plan of
More information14 Singular Value Decomposition
14 Singular Value Decomposition For any high-dimensional data analysis, one s first thought should often be: can I use an SVD? The singular value decomposition is an invaluable analysis tool for dealing
More informationData assimilation in high dimensions
Data assimilation in high dimensions David Kelly Kody Law Andy Majda Andrew Stuart Xin Tong Courant Institute New York University New York NY www.dtbkelly.com February 3, 2016 DPMMS, University of Cambridge
More informationStochastic Collocation Methods for Polynomial Chaos: Analysis and Applications
Stochastic Collocation Methods for Polynomial Chaos: Analysis and Applications Dongbin Xiu Department of Mathematics, Purdue University Support: AFOSR FA955-8-1-353 (Computational Math) SF CAREER DMS-64535
More informationRandom matrices: Distribution of the least singular value (via Property Testing)
Random matrices: Distribution of the least singular value (via Property Testing) Van H. Vu Department of Mathematics Rutgers vanvu@math.rutgers.edu (joint work with T. Tao, UCLA) 1 Let ξ be a real or complex-valued
More informationCME 345: MODEL REDUCTION
CME 345: MODEL REDUCTION Proper Orthogonal Decomposition (POD) Charbel Farhat & David Amsallem Stanford University cfarhat@stanford.edu 1 / 43 Outline 1 Time-continuous Formulation 2 Method of Snapshots
More informationLinear Algebra in Computer Vision. Lecture2: Basic Linear Algebra & Probability. Vector. Vector Operations
Linear Algebra in Computer Vision CSED441:Introduction to Computer Vision (2017F Lecture2: Basic Linear Algebra & Probability Bohyung Han CSE, POSTECH bhhan@postech.ac.kr Mathematics in vector space Linear
More information