Accelerating minimizations in ensemble variational assimilation

Accelerating minimizations in ensemble variational assimilation G. Desroziers, L. Berre Météo-France/CNRS (CNRM/GAME) 1/20

Outline • Ensemble Variational assimilation • Accelerating minimizations • Conclusion and future work 2/20

Simulation of analysis errors: «hybrid EnKF/Var» or «consistent ensemble 4D-Var»? A consistent 4D-Var approach can be used in both ensemble and deterministic components : • Simple to implement (perturbed Var ~ unperturbed Var). • A full-rank hybrid B is consistently used in the 2 parts. • Non-linear aspects of 4D-Var can be represented (outer loop). 4/20

The operational Météo-France ensemble Var assimilation • Six perturbed global members, T399 L70 (50 km / 10 km), with global 4D-Var Arpege. • Spatial filtering of error variances, to further increase the sample size and robustness. • Inflation of ensemble B / model error contributions, soon replaced by inflation of perturbations. 5/20

Applications of the EnDA system at Météo-France • Flow-dependentbackground error variances in 4D-Var. • Flow-dependentbackground errorcorrelations experimented using wavelet filtering properties (Varella et al 2011 a,b) . • Initialisationof Météo-France ensemble predictionby EnDA. • Diagnosticson analysis consistency and observation impact (Desroziers et al 2009) . 6/20

Connexion between large sb’s and intense weather ( Klaus storm, 24/01/2009, 00/03 UTC ) Mean sea level pressure field Background error standard deviations 7/20

Background errorcorrelationsusing EnDA and wavelets Wavelet-implied horizontal length-scales (in km), for wind near 500 hPa, averaged over a 4-day period. (Varella et al 2011b, and also Fisher 2003, Deckmyn and Berre 2005, Pannekoucke et al 2007) 8/20

Ensemble variational assimilation • Ens. variational assimilation (similar to EnKF): for l = 1, L (size of ensemble) • dal= (I – KH) dbl+ K dol, • with H observation operator matrix, K implicit gain matrix, • dol = R1/2 hol and hol a vector of random numbers • db+l= M dal+ dml, • with dml = Q½ hml ,hml a vector of random numbers • and Q model error covariance matrix. • Minimize L cost-functions Jl, with perturbed innovations dl: Jl(dxl)=1/2dxlT B-1dxl+ 1/2(dl-Hldxl)T R-1(dl-Hldxl), with B and R bg and obs. error matrices, and dl = yo + R1/2 hol - H(M(xbl )), xal= xbl + dxl and xbl+ = M( xal) + Q1/2 hml . 10/20

Hessian matrix of the assimilation problem • Hessian of the cost-function: J’’ = B-1+ HTR-1 H. • Bad conditioning of J’’: very slow (or no) convergence. • Cost-function with B1/2 preconditioning ( dx = B1/2 c ): J(c) = 1/2cT c+ 1/2(d-H B1/2 c)T R-1(d-H B1/2 c) • Hessian of the cost-function: J’’ = I + BT/2 HTR-1 H B1/2 . • Far better conditioning and convergence! (Lorenc 1988, Haben et al 2011) 11/20

Lanczos algorithm • Generate iteratively a set of k orthonormal vectors q such as QkTJ’’ Qk = Tk, where Qk = (q1 q2 … qk ), and Tk is a tri-diagonal matrix. • The extremal eigenvalues of Tk quickly converge towards the extremal eigenvalues of J’’. • If Tk= YkLkYkT is the eigendecomposition of Tk, the Ritz vectors are obtained with Zk= QkYk and the Ritz pairs (zk,lk) approximate the eigenpairs of J’’. 12/20

Lanczos algorithm / Conjugate gradient • Use of the Lanczos vectors to get the solution of the variational problem: ck = c0 + Qk Wk . • Optimal coefficients Wk should make the gradient of J vanish at ck: J’(ck) = J’(c0)+ J’’ (ck-c0) = J’(c0)+ J’’ Qk Wk = 0, which gives Wk = - ( QkTJ’’ Qk )-1QkTJ’(c0) = - Tk-1J’(c0), and then ck= c0 - QkTk-1QkTJ’(c0). • Same solution as after k iterations of a Conjugate Gradient algorithm. (Paige and Saunders 1975, Fisher 1998) 13/20

Accelerating a « perturbed » minimization using « unperturbed » Lanczos vectors • Minimizations with - unperturbed innovations d and - perturbed innovations dl have basically the same Hessians: J’’(d) = I + BT/2 HTR-1 H B1/2 , J’’(dl) = I + BT/2 HlTR-1 HlB1/2 , • The solution obtained for the « unperturbed » problem ck = c0 - Qk ( QkTJ’’ Qk )-1QkTJ’(c0, d) can be transposed to the « perturbed » minimization ck,l = c0 - Qk ( QkTJ’’ Qk )-1QkTJ’(c0, dl) to improve its starting point. 14/20

Accelerating a « perturbed » minimization using « unperturbed » Lanczos vectors Perturbed analysis dashed line : starting point with 50 Lanczos vectors 15/20

Accelerating a « perturbed » minimization using « unperturbed » Lanczos vectors 1 stand-alone minim. 10 vectors 50 vectors 100 vectors Decrease of the cost function for a new « perturbed » minimization 16/20

Accelerating minimizations using « perturbed » Lanczos vectors • If L perturbed minim. with k iterations have already been performed, then the starting point of a perturbed (or unperturbed) minimization can be written under the form ck = c0 + Qk,L Wk,L, where Wk,L is a vector of k x L coefficients and Qk,L = (q1,1 … qk,1 … q1,L … qk,L ) is a matrix containing the k x L Lanczos vectors. • Following the same approach as above, the solution can be expressed and computed: ck,L= c0 – Qk,L ( Qk,LTJ’’ Qk,L )-1Qk,LTJ’(c0). • Matrix Qk,LTJ’’ Qk,L is no longer tri-diagonal, but can be easily inverted. 17/20

Accelerating minimizations using « perturbed » Lanczos vectors (k = 1) 1 stand-alone minim. 10x1 = 10 «synergetic» vectors 50x1 = 50 «synergetic» vectors 100x1 = 100 «synergetic» vectors Decrease of the cost function for a new « perturbed » minimization 18/20

Conclusion and future work • Ensemble Variational assimilation: error cycling can be simulated in a way consistent with 4D-Var. • Flow-dependent covariances can be estimated. • Positive impacts, in particular for intense/severe weather events, from both flow-dependent variances and correlations. • Accelerating minimizations seems possible (preliminary tests in the real size Ens. 4D-Var Arpege also encouraging). • Connection with Block Lanczos / CG algorithms (O’Leary 1980) . • Possible appl. in EnVar without TL/AD(Lorenc 2003, Buehner 2005) . 20/20

Accelerating minimizations in ensemble variational assimilation