POD/DEIM Strategies for reduced data assimilation systems

Transcription

1 POD/DEIM Strategies for reduced data assimilation systems Răzvan Ştefănescu 1 Adrian Sandu 1 Ionel M. Navon 2 1 Computational Science Laboratory, Department of Computer Science, Virginia Polytechnic Institute and State University 2 Department of Scientific Computing,The Florida State University POD/DEIM Strategies for reduced data assimilation systems 1/37

2 Outline 1 Data Assimilation 2 Reduced order data assimilation 3 Reduced Order Modelling 4 ROM 4D-Var DA systems - Choice of bases 5 4D-Var SWE DA reduced order systems 6 Numerical results 7 Conclusions and future research POD/DEIM Strategies for reduced data assimilation systems 2/37

3 Data Assimilation Data assimilation - process of combining information from models, measurements, and priors - all with associated uncertainties - to obtain the best estimate of the state of a physical system. Two families of methods: variational and statistical methods. Variational methods - rooted in control theory and require developments of tangent linear and adjoint models. Statistical methods: Ensemble based sequential data assimilation and Bayesian inversion POD/DEIM Strategies for reduced data assimilation systems 3/37

4 4D-Var Data Assimilation The objective function J to be optimized is defined based on model-data misfit penalty terms as: J(x 0 ) = 1 ( x b ) T ( x 0 B 1 0 x b ) x N ( y i H(x i ) ) T ( R 1 i y i H(x i ) ), 2 subject to i=1 (1) x i+1 = M i (x i ), i = 0,.., N 1, (2) The optimality conditions: Adjoint model: λ i = M T i λ i+1 + H T R 1 ( i y i H(x i ) ), i = N 1, 1; λ N =H T R 1 ( N y N H(x N ) ) and λ 0 = M T 0 λ 1. (3) Cost function gradient: x0 L = B 1 ( 0 x b ) x 0 λ0 = 0; (4) POD/DEIM Strategies for reduced data assimilation systems 4/37

5 4D-Var Data Assimilation POD/DEIM Strategies for reduced data assimilation systems 5/37

6 WRF model The Weather Research and Forecasting (WRF) Model is a next-generation mesoscale numerical weather prediction system designed to serve both atmospheric research and operational forecasting needs (Skamarock et al. [16], W. C. Skamarock and Powers [21]). It features a data assimilation system, and a software architecture facilitating parallel computation and system extensibility. The model serves: a wide range of meteorological applications across scales from tens of meters to thousands of kilometers; atmospheric simulations based on real data (observations, analyses) or idealized conditions. WRF is currently in operational use at NCEP, NCAR. POD/DEIM Strategies for reduced data assimilation systems 6/37

7 Lightning Data Assimilation The research uses convective available potential energy (CAPE) as a proxy between lightning data and model variables (pressure, temperature, water vapor mixing ration and geopotential height) Price and Rind [13], Barthe et al. [3]. Three strategies: 3D-Var, 1D+3D-Var, 1D+4D-Var (Marécal and Mahfouf [10, 11]). Development of tangent linear and adjoint lightning observation operators. R. Ştefănescu, I.M. Navon, Max Marchand, Henry Fuelberg. 1D+4D-VAR data assimilation of lightning with WRFDA system using nonlinear observation operators, Technical Report, ArXiv POD/DEIM Strategies for reduced data assimilation systems 7/37

8 Stage IV precipitation validation Figure: 1 h precipitation (mm) ending at 2000 UTC 15 June from the control run, various assimilation procedures, and stage IV precipitation.

9 Incremental 4D-Var Data Assimilation Courtier et al. [8] POD/DEIM Strategies for reduced data assimilation systems 9/37

10 Reduced order data assimilation POD/DEIM Strategies for reduced data assimilation systems 10/37

11 Why do we need reduced order data assimilation? Replace the current linearized cost function to be minimized in the inner loop Surrogate models that accurately represent sub-grid-scale processes instead of linearized low-resolution models Suitable reduced order models can be derived successively and give global convergence results - Arian et al. [2] - This approach can be interpreted in the context of trust region methods using general nonlinear model functions with inexact gradient information. Highly non-linear and non-smooth observation operators Increased space and time resolutions POD/DEIM Strategies for reduced data assimilation systems 11/37

12 Proper Orthogonal Decomposition For general nonlinear systems input-specified semi-empirical methods are usually employed Data analysis is conducted to extract basis functions, from experimental data or detailed simulations of high-dimensional systems Galerkin projections that yield low dimensional dynamical models. Standard POD models: high computational complexities for estimating the nonlinearities. Tensorial POD - Stefanescu et al. [18] - Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations We assume a Petrov-Galerkin projection for constructing the reduced order models. U denotes the POD basis and the test functions are stored in W. W T U = I k, I k being the identity matrix of order k. For simplicity we assume a POD expansion of x = U x.

13 ROM 4D-Var DA systems - Choice of bases What information the bases should contain? The reduced adjoint (RA) approach projects the first order optimality equations of the full system onto the POD reduced spaces Accurate low-order surrogate models; Its not clear what information should be included in the reduced basis used for full space gradient equation projection The adjoint of reduced (AR) model approach formulates the first order optimality conditions from the forward reduced order model Consistent KKT discrete optimality conditions; Reduced adjoint model approximates poorly its full counterpart and POD bases rely only on forward dynamics information. POD/DEIM Strategies for reduced data assimilation systems 13/37

14 ROM 4D-Var DA systems - Choice of bases To guide the POD bases snapshots selection process for Petrov-Galerkin based reduced data assimilation systems governed by non-linear models Consistent reduced Karush Kuhn Tucker (KKT) optimality conditions and accurate reduced POD adjoint model solutions with respect to the full adjoint model outputs Every type of reduced optimization involving adjoint models and projection based reduced order methods including reduced basis approach will benefit. POD/DEIM Strategies for reduced data assimilation systems 14/37

15 ROM 4D-Var DA systems - Choice of bases We derive the optimality conditions as in the AR approach Forward POD manifold U f is computed using snapshots of the full forward model solution only x U f x Petrov-Galerkin (PG) projection; the test functions POD basis W f is different than the trial functions POD manifold U f J POD ( x 0 ) = 1 2 ( x b ) T ( U f x 0 B 1 0 x b ) T U f x N i=1 ( y i H(U f x i ) ) T ( R 1 i y i H(U f x i ) ) T, (5) subject to x i+1 = W T f M i (U f x i ),, i = 0,.., N 1. (6) POD/DEIM Strategies for reduced data assimilation systems 15/37

16 ROM 4D-Var DA systems - Choice of bases Reduced adjoint model λ i = Uf T M T i W f λ i+1 + Uf T H T R 1 ( i y i H(U f x i ) ), i = N 1, 1; λ N = Uf T H T R 1 ( N y N H(U f x N ) ) and λ 0 = Uf T M T 0 W f λ 1 (7) Reduced Cost Function gradient x0 L POD = Uf T B 1 ( 0 x b ) U f x 0 λ 0 = 0; (8) POD/DEIM Strategies for reduced data assimilation systems 16/37

17 ROM 4D-Var DA systems - Choice of bases RA approach: the full forward and adjoint models are projected onto separate reduced manifolds U f and U a are the trial POD reduced subspaces and W f and W a are the test functions POD manifolds, x i U f x i, λ i U a λ i, i = 0,.., N. Reduced forward model: x i+1 = W T f M i (U f x i ), i = 0,.., N 1. (9) Reduced adjoint model: λ i = Wa T M T i U a λ i+1 + Wa T H T R 1 ( i y i H(U f x i ) ), i = N 1, 1 λ N = Wa T H T R 1 ( N y N H(U f x N ) ) and λ 0 = Wa T M T 0 U a λ 1, (10) POD/DEIM Strategies for reduced data assimilation systems 17/37

18 ROM 4D-Var DA systems - Choice of bases AR adjoint model: λ i = Uf T M T i W f λ i+1 + Uf T H T R 1 ( i y i H(U f x i ) ), i = N 1, 1; λ N = Uf T H T R 1 ( N y N H(U f x N ) ) and λ 0 = Uf T M T 0 W f λ 1 (11) RA adjoint model: λ i = Wa T M T i U a λ i+1 + Wa T H T R 1 ( i y i ) H(U f x i, i = N 1, 1 λ N = Wa T H T R 1 ( N y N H(U f x N ) ) and λ 0 = Wa T M T 0 U a λ 1, (12) For Petrov Galerkin and Galerkin projections - ARRA technique W f = U a and W a = U f, and U f = U a. (13) Gradient information must be also included inside the basis to obtain accurate reduced feasible KKT conditions.

19 4D-Var SWE DA reduced order systems Three POD based reduced order models will be considered for deriving reduced order SWE data assimilation systems: standard Proper Orthogonal Decomposition (SPOD), tensorial POD (TPOD) and standard POD/Discrete Empirical Interpolation Method (POD/DEIM) The methods differ in the way the nonlinear terms are treated - polynomial quadratic nonlinearity u 2. POD/DEIM Strategies for reduced data assimilation systems 19/37

20 4D-Var SWE DA reduced order systems Standard POD Ñ(ũ) = }{{} W T Uũ Uũ }{{}, Ñ(ũ) R k (14) k N state N state 1 where is the componentwise multiplication Matlab operator and n is usually the number of spatial mesh points. Tensorial POD Ñ(ũ) = [ Ñ i ]i=1,..,k Rk ; Ñ i = Ñ(ũ) = }{{} T }{{} Ũ k k 2 k 2 1 k j=1 l=1 T = ( T i,j,l )i,j,l=1,..,k Rk k k, T i,j,l = k T i,j,l ũ j ũ l. (15) n W i,r U j,r U l,r. POD/DEIM Strategies for reduced data assimilation systems 20/37 r=1

21 4D-Var SWE DA reduced order systems Standard POD/DEIM ( ) Ñ(ũ) W T V (P T V ) 1 (P T Uũ) (P T Uũ) }{{} k m }{{} m 1 (16) where m is the number of interpolation points, V R n m gathers the first m POD basis modes of the nonlinear term while P R n m is the DEIM interpolation selection matrix (Chaturantabut [5], Chaturantabut and Sorensen [7, 6], Stefanescu and Navon [17]). POD/DEIM Strategies for reduced data assimilation systems 21/37

22 4D-Var SWE DA reduced order systems w t = A(w) w x + B(w) w y 0 x L, 0 y D, t [0, t f ], + Cw, (17) where w = (u, v, φ) T, u, v are the velocity components in the x and y directions, respectively, h is the depth of the fluid, g is the acceleration due to gravity and φ = 2 gh. The matrices A, B and C are expressed A = u 0 φ/2 0 u 0 φ/2 0 u C =, B = 0 f 0 f v v φ/2 0 φ/2 v f = ˆf + β(y D/2) (Coriolis force), β = f y, with ˆf and β constants.,

23 4D-Var SWE DA reduced order systems F 11 (u, φ) = u A x u φ A xφ, F 12 (u, v) = v A y u F 21 (u, v) = u A x v, F 22 (v, φ) = v A y v φ A y φ, F 31 (u, φ) = 1 2 φ A x u + u A xφ, F 32 (v, φ) = 1 2 φ A y v + v A y φ. Full ADI SWE Standard POD Tensorial POD POD/DEIM m=180 POD/DEIM m=70 CPU time s u e e e e-5 v e e e e-5 φ e-5e 8.505e e e-5 Table: CPU time gains and the root mean square errors for each of the model variables at t f = 3h for a 3h time integration window. Number of POD modes was k = 50 and two tests with different number of DEIM points m = 180, 70 were simulated.103, 776 spatial points. POD/DEIM Strategies for reduced data assimilation systems 23/37

24 4D-Var SWE DA reduced order systems CPU time (seconds) SPOD TPOD POD/DEIM m=70 POD/DEIM m=180 FULL CPU time (seconds) SPOD TPOD POD/DEIM m=70 POD/DEIM m= No. of spatial discretization points (a) On-line stage No. of spatial discretization points (b) Off-line stage Figure: Cpu time vs. the number of spatial discretization points for t f = 3h ; number of POD modes = 50; two different numbers of DEIM points 70 and 180 have been employed. POD/DEIM Strategies for reduced data assimilation systems 24/37

25 4D-Var SWE DA reduced order systems Algorithm 1 Standard and Tensorial POD SWE DA systems Off-line stage 1: Generate background state u, v and φ. 2: Solve full forward ADI SWE model to generate state variables snapshots. 3: Solve full adjoint ADI SWE model to generate adjoint variables snapshots. 4: For each state variable compute a POD basis using snapshots describing dynamics of the forward and its corresponding adjoint trajectories plus gradient. 5: Compute tensors as T required for reduced Jacobian calculations. Calculate other POD coefficients corresponding to linear terms. POD/DEIM Strategies for reduced data assimilation systems 25/37

26 4D-Var SWE DA reduced order systems Algorithm 1 Standard and Tensorial POD SWE DA systems On-line stage - Minimize reduced cost functional J POD (5) 1: Solve forward reduced order model (6) 2: Solve adjoint reduced order model (7) 3: Compute reduced gradient (8) Decisional stage 4: Project the suboptimal reduced initial condition generated by the online stage and perform steps 1 and 2 of off-line stage. Using full forward information evaluate J in (1). If J > ε 3 then continue the off-line stage from step 3, otherwise STOP. POD/DEIM Strategies for reduced data assimilation systems 26/37

27 4D-Var SWE DA reduced order systems The on-line stage - minimization of the cost function J POD performed on a reduced POD manifold The stoping criteria are J POD ε 1, J POD (i+1) JPOD (i) ε 2, MXFUN iter Max (18) The off-line stage - outer iteration - general stopping criterion J ε 3 POD/DEIM Strategies for reduced data assimilation systems 27/37

28 Numerical Results ADI SWE model 0.5% uniform perturbations on the initial conditions of Grammeltvedt [9] and generate twin-experiment observations at every grid space point location and every time step Background state is computed using a 5% perturbations of the initial conditions The length of the assimilation window: 3h. BFGS optimization method (CONMIN) We use ε 1 = and ε 2 = We select mesh points 91 time steps and use 50 POD basis functions. MXFUN is set to 25 and ε 3 = Galerkin projection. The motivation was built for incremental 4D-Var - comparison against nonlinear 4D-Var

29 Numerical Results - Choice of POD basis 8 5 u v φ 8 5 u v φ logarithmic scale Number of eigenvalues (a) Forward model snapshots only logarithmic scale Number of eigenvalues (b) Forward and adjoint models snapshots Figure: The decay around the singular values of the snapshots solutions for u, v, φ for t = 960s and integration time window of 3h. POD/DEIM Strategies for reduced data assimilation systems 29/37

30 Numerical Results - Choice of POD basis Cost func red. of adj. + adj. of red. Cost func adj. of red. Cost func full J/J(1) Grad/Grad(1) Grad full Grad adj. of red. Grad red. of adj. + adj. of red Number of iterations Figure: Tensorial POD/4DVAR ADI 2D Shallow water equations Evolution of cost function and gradient norm as a function of the number of minimization iterations. The information from the adjoint equations and gradient has to be incorporated into POD basis.

31 POD based SWE 4D-Var DA systems n = space points, number of POD basis modes k = 50, MXFUN = 15 and ε 3 = hybrid POD/DEIM m=30 hybrid POD/DEIM m=50 standard POD tensorial POD Full configuration J/J(1) Number of iterations (a) Iteration performance J/J(1) hybrid POD/DEIM m=30 hybrid POD/DEIM m= standard POD tensorial POD Full configuration Cpu time (b) Time performance Figure: Number of iterations and CPU time comparisons for the reduced Order SWE DA systems vs. full SWE DA system. POD/DEIM Strategies for reduced data assimilation systems 31/37

32 Conclusions and future research New POD bases selection strategies for POD based reduced 4DVar data assimilation systems governed by nonlinear state models using both Petrov-Galerkin and Galerkin projections. Consistent reduced Karush Kuhn Tucker (KKT) optimality conditions + accurate feasible reduced POD KKT condtitions. Petrov-Galerkin projection - test functions POD bases of the forward and adjoint models have to match the trial functions POD bases of the adjoint and forward models Galerkin projection - one single POD basis is required and the correlation matrix must contain snapshots from both forward and adjoint full models. Include also the optimality condition information. POD/DEIM Strategies for reduced data assimilation systems 32/37

33 Conclusions and future research Every type of reduced optimization involving adjoint models and projected based reduced order methods including reduced basis approach benefit from the new strategies. For meshes of points or higher the hybrid POD/DEIM reduced data assimilation system is approximately 10 times faster then the full space data assimilation system POD/DEIM Strategies for reduced data assimilation systems 33/37

34 Conclusions and future research This rate increases directly proportional with the the mesh size Stabilization strategies proposed by Amsallem and Farhat [1], Bui-Thanh et al. [4] must be pursued in order to obtain feasible Petrov-Galerkin reduced order data assimilation systems Multifidelity techniques: Local in time adaptive ROMs. (Peherstorfer et al. [12], Rapún and Vega [15]); Exploit the structure of the weak constraints variational approach (Trémolet [20]), consistent reduced KKT conditions (Stefănescu et al. [19]) and formulate a piecewise-in-space-time approximation strategy that uses different ROMs on different subintervals, and constructs them concurrently; Develop reduced order parallel 4D-Var framework using Augmented Lagrangian (Rao and Sandu [14]). POD/DEIM Strategies for reduced data assimilation systems 34/37

35 Manuscripts related to the present research effort Ştefănescu, Răzvan, Adrian Sandu, and Ionel Michael Navon. POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation. Journal of Computational Physics 295 (2015): Ştefănescu, Răzvan, and Adrian Sandu. Efficient approximation of sparse Jacobians for time-implicit reduced order models. arxiv preprint arxiv: (2014). Ştefănescu, Răzvan, Adrian Sandu, and Ionel Michael Navon. Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations. International Journal for Numerical Methods in Fluids 76.8 (2014): Stefănescu, Răzvan and I.M. Navon, POD/DEIM Nonlinear model order reduction of an ADI implicit shallow water equations model, Journal of Computational Physics, Vol 237, pp

36 Manuscripts related to the present research effort Stefănescu, Răzvan, I.M. Navon, Max Marchand, Henry Fuelberg, 1D+ 4D-VAR data assimilation of lightning with WRFDA system using nonlinear observation operators. arxiv preprint arxiv: (2013). POD/DEIM Strategies for reduced data assimilation systems 36/37

37 POD/DEIM Strategies for reduced data assimilation systems POD/DEIM Strategies for reduced data assimilation systems 37/37

38 [1] D. Amsallem and C. Farhat. Stabilization of projection-based reduced-order models. Int. J. Numer. Meth. Engng., 91: , [2] E. Arian, M. Fahl, and E.W. Sachs. Trust-region proper orthogonal decomposition for flow control. ICASE: Technical Report , [3] C. Barthe, W. Deierling, and M. C. Barth. Estimation of total lightning from various storm parameters: A cloud-resolving model study. J. Geophys. Res., 110:D24202, [4] T. Bui-Thanh, K. Willcox, O. Ghattas, and B. van Bloemen Waanders. Goal-oriented, model-constrained optimization for reduction of large-scale systems. J. Comput. Phys., 224(2): , [5] S. Chaturantabut. Dimension Reduction for Unsteady Nonlinear Partial Differential Equations via Empirical Interpolation Methods. Technical Report TR09-38,CAAM, Rice University, [6] S. Chaturantabut and D.C. Sorensen. Nonlinear model reduction via POD/DEIM Strategies for reduced data assimilation systems 37/37

39 discrete empirical interpolation. SIAM Journal on Scientific Computing, 32(5): , [7] S. Chaturantabut and D.C. Sorensen. A state space error estimate for POD-DEIM nonlinear model reduction. SIAM Journal on Numerical Analysis, 50(1):46 63, [8] P. Courtier, J.-N. Thepaut, and A. Hollingsworth. A strategy for operational implementation of 4D Var using an incremental approach. Quarterly Journal of the Royal Meteorological Society, 120: , [9] A. Grammeltvedt. A survey of finite difference schemes for the primitive equations for a barotropic fluid. Monthly Weather Review, 97(5): , [10] V. Marécal and J.-F. Mahfouf. Four-dimensional variational assimilation of total column water vapor in rainy areas. Mon. Wea. Rev., 130(-):4358, [11] V. Marécal and J.-F. Mahfouf. Experiments on 4d-var assimilation of rainfal data using an incremental formulation. Q. J. R. Meteorol. Soc, 129(-): , POD/DEIM Strategies for reduced data assimilation systems 37/37

40 [12] B. Peherstorfer, D. Butnaru, K. Willcox, and H. Bungartz. Localized Discrete Empirical Interpolation Method. SIAM Journal on Scientific Computing, 36(1):A168 A192, [13] C. Price and D. Rind. A simple lightning parameterization for calculating global lightning distributions. J. Geophys. Res., 97(-): , [14] V. Rao and A. Sandu. Parallel 4D-Var framework using Augmented Lagrangian. In preparation, [15] M.L. Rapún and J.M. Vega. Reduced order models based on local POD plus Galerkin projection. Journal of Computational Physics, 229 (8): , [16] W. C. Skamarock, J. B. Klemp, J. Dudhia, D. O. Gill, D. M. Barker, W. Wang, and J. G. Powers. A description of the Advanced Research WRF Version 2. NCAR Tech Notes-468+STR, [17] R. Stefanescu and I.M. Navon. POD/DEIM Nonlinear model order reduction of an ADI implicit shallow water equations model. Journal of Computational Physics, 237:95 114, POD/DEIM Strategies for reduced data assimilation systems 37/37

41 [18] R. Stefanescu, A. Sandu, and I.M. Navon. Comparisons of POD reduced order strategies for the nonlinear 2D Shallow Water Equations Model. Technical Report TR 2, Virginia Polytechnic Institute and State University, February 2014, also submitted to International Journal for Numerical Methods in Fluids. [19] R. Stefănescu, A. Sandu, and I.M. Navon. POD/DEIM Reduced-Order Strategies for Efficient Four Dimensional Variational Data Assimilation. Technical Report TR 3, Virginia Polytechnic Institute and State University, March [20] Y. Trémolet. Accounting for an imperfect model in 4D-Var. Q.J.R. Meteorol. Soc., 132: , [21] J. Dudhia D. O. Gill D. M. Barker M. G. Duda X-Y Huang W. Wang W. C. Skamarock, J. B. Klemp and J. G. Powers. A description of the advanced research wrf version 3. NCAR/TN475+STR, POD/DEIM Strategies for reduced data assimilation systems 37/37