POD/DEIM 4DVAR Data Assimilation of the Shallow Water Equation Model

Transcription

1 nonlinear 4DVAR 4DVAR Data Assimilation of the Shallow Water Equation Model R. Ştefănescu and Ionel M. Department of Scientific Computing Florida State University Tallahassee, Florida May 23, 2013 (Florida State University) May 23, / 51

2 Contents nonlinear 4DVAR 1 2 nonlinear ADI FD SWE model 3 4DVAR scheme of the SWE (Florida State University) May 23, / 51

3 Contents nonlinear 4DVAR 1 2 nonlinear ADI FD SWE model 3 4DVAR scheme of the SWE (Florida State University) May 23, / 51

4 Contents nonlinear 4DVAR 1 2 nonlinear ADI FD SWE model 3 4DVAR scheme of the SWE (Florida State University) May 23, / 51

5 POD/EIM nonlinear 4DVAR Model order reduction : Reduce the computational complexity/time of large scale dynamical systems by approximations of much lower dimension with nearly the same input/output response characteristics. Goal : Construct reduced-order model for different types of discretization method (finite difference (FD), finite element (FEM), finite volume (FV)) of unsteady and/or parametrized nonlinear PDEs. E.g., PDE: y (x, t) = L(y(x, t)) + F(y(x, t)), t [0, T ] t where L is a linear function and F a nonlinear one. (Florida State University) May 23, / 51

6 applied to FD SCHEMES nonlinear 4DVAR The corresponding FD scheme is a n dimensional ordinary differential system d dt y(t) = Ay(t) + F(y(t)), A Rn n, where y(t) = [y 1 (t), y 2 (t),.., y n (t)] R n and y i (t) R are the spatial components y(x i, t), i = 1,.., n. F is a nonlinear function evaluated at y(t) componentwise, i.e. F = [F(y 1 (t)),.., F(y n (t))] T, F : I R R. (Florida State University) May 23, / 51

7 applied to FD SCHEMES nonlinear 4DVAR A common model order reduction method involves the Galerkin projection with basis V k R n k obtained from Proper Orthogonal Decomposition (POD), for k n, i.e. y V k ỹ(t), ỹ(t) R k. Applying an inner product to the ODE discrete system we get d dt ỹ(t) = V k T AV k ỹ(t) + Vk T F(V k ỹ(t)) }{{}}{{} k k Ñ(ỹ) (1) (Florida State University) May 23, / 51

8 applied to FD SCHEMES nonlinear 4DVAR The efficiency of POD - Galerkin technique is limited to the linear or bilinear terms. The projected nonlinear term still depends on the dimension of the original system Ñ(ỹ) = Vk T F(V k ỹ(t)). }{{}}{{} k n n 1 To mitigate this inefficiency we introduce (DEIM) for nonlinear approximation. DEIM is a discrete variation of the Empirical method proposed by Barrault et al. (2004). The application was suggested by Chaturantabut and Sorensen (2008, 2010, 2012). (Florida State University) May 23, / 51

9 applied to FD SCHEMES nonlinear 4DVAR M. Barrault, Y. Maday, N. Nguyen, A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris, Ser. I 339 (2004) It is an efficient reduced-basis discretization procedure for partial differential equations with non-affine parameter dependence. The method replaces non affine coefficient functions with a collateral reduced-basis expansion which then permits an (effectively affine) offline-online computational decomposition. (Florida State University) May 23, / 51

10 applied to FD SCHEMES nonlinear 4DVAR The efficiency of POD - Galerkin technique is limited to the linear or bilinear terms. The projected nonlinear term still depends on the dimension of the original system Ñ(ỹ) = Vk T F(V k ỹ(t)). }{{}}{{} k n n 1 For m n Ñ(ỹ) Vk T U(P T U) 1 F(P T V k ỹ(t)) }{{}}{{} precomputed k m m 1 (Florida State University) May 23, / 51

11 nonlinear nonlinear SWE nonlinear 4DVAR We applied DEIM to a POD alternating direction implicit (ADI) FD on a rectangular domain. We considered the alternating direction fully implicit finite-difference scheme (Gustafsson 1971, Fairweather and 1980, and De Villiers 1986, Kreiss and Widlund 1966) on a rectangular domain since the scheme remains stable at large Courant numbers (CFL). (Florida State University) May 23, / 51

12 nonlinear SWE model nonlinear 4DVAR w t = A(w) w x + B(w) w + C(y)w, (2) y 0 x L, 0 y D, t [0, t f ], 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 = 0 u 0 u 0 φ/2 φ/2 0 u C =, B = f 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., (Florida State University) May 23, / 51

13 nonlinear SWE model nonlinear 4DVAR We assume periodic solutions in the x-direction w(x, y, t) = w(x + L, y, t), while in the y direction we have v(x, 0, t) = v(x, D, t) = 0. The initial conditions are derived from the initial height-field condition No. 1 of Grammelvedt (1969), i.e. ( h(x, y) = H 0 +H 1 +tanh 9 D/2 y ( )+H 2 sech 2 9 D/2 y ) ( 2πx sin 2D 2D L ) (Florida State University) May 23, / 51

14 nonlinear The nonlinear Gustafsson ADI finite difference implicit scheme nonlinear 4DVAR First we introduce a network of N x N y equidistant points on [0, L] [0, D], with dx = L/(N x 1), dy = D/(N y 1). We also discretize the time interval [0, t f ] using NT equally distributed points and dt = t f /(NT 1). Next we define vectors of unknown variables of dimension n xy = N x N y containing approximate solutions such as u(t) u(x i, y j, t), v(t) v(x i, y j, t), φ φ(x i, y j, t) R nxy The idea behind the ADI method is to split the finite difference equations into two, one with the x-derivative taken implicitly and the next with the y-derivative taken implicitly, (Florida State University) May 23, / 51

15 nonlinear The POD version of SWE model nonlinear 4DVAR The POD reduced-order system is constructed by applying the Galerkin projection method to ADI FD discrete model by first replacing u, v, φ with their POD based approximation Uũ, V ṽ, Φ φ, respectively, and then premultiplying the corresponding equations by U T, V T and Φ T, the POD bases. (Florida State University) May 23, / 51

16 The resulting POD reduced system for the first step (t n+ 1 2 ) of the ADI FD scheme is ũ(t n+ 1 ) + t ( ) 2 2 UT F11 ũ(t n+ 1 ), φ(t 2 n+ 1 ) = ũ(t 2 n ) t ( ) 2 UT F12 ũ(t n ), ṽ(t n ) + t ( ) 2 UT [f, f,.., f ] T V ṽ(t }{{} n ), N x ṽ(t n+ 1 ) + t ( ) 2 2 V T F 21 ũ(t n+ 1 ), ṽ(t 2 n+ 1 ) + t ( ) 2 2 V T [f, f,.., f ] T Uũ(t }{{} n+ 1 ) 2 N x = ṽ(t n ) t ( ) 2 V T F22 ṽ(t n ), φ(t n ), φ(t n+ 1 ) + t ( ) 2 2 ΦT F31 ũ(t n+ 1 ), φ(t 2 n+ 1 ) = φ(t 2 n ) t ( ) 2 ΦT F32 ṽ(t n ), φ(t n ), where F 11, F 12, F 21, F 22, F 31, F 32 : R k R k R k are defined by (3)

17 nonlinear The POD version of SWE model nonlinear 4DVAR F 11 (ũ, φ) = (Uũ) (A x U }{{} ũ) (Φ φ) (A x Φ }{{} φ), F 12 (ũ, ṽ) = (V ṽ) (A y U }{{} ũ), F 21 (ũ, ṽ) = (Uũ) (A x V }{{} ṽ), F 22 (ṽ, φ) = (V ṽ) (A y V }{{} ṽ) (Φ φ) (A y Φ }{{} φ), F 31 (ũ, φ) = 1 2 (Φ φ) (A x U }{{} ũ) + (Uũ) (A xφ }{{} φ), F 32 (ṽ, φ) = 1 2 (Φ φ) (A y V }{{} ṽ) + (V ṽ) (A y Φ }{{} φ). (4) (Florida State University) May 23, / 51

18 nonlinear The POD version of SWE model nonlinear 4DVAR The coefficient matrices defined in the linear terms of the POD reduced system as well as the coefficient matrices in the nonlinear functions (i.e. A x U, A y U, A x V, A y V, A x Φ, A y Φ R n k grouped by the curly braces) can be precomputed, saved and re-used in all time steps. However, performing the componentwise multiplications in (4) and computing the projected nonlinear terms in (3) }{{} U T F11 (ũ, φ), U T F12 (ũ, ṽ), V T F21 (ũ, ṽ), }{{} k n xy n xy 1 V T F22 (ṽ, φ), Φ T F31 (ũ, φ), Φ T F32 (ṽ, φ), (5) still have computational complexities depending on the dimension n xy of the original system from both evaluating the nonlinear functions and performing matrix multiplications to project on POD bases. (Florida State University) May 23, / 51

19 nonlinear (DEIM) nonlinear 4DVAR Let f : D R n, D R n be a nonlinear function. If U = {u l } m l=1, u i R n, i = 1,.., m is a linearly independent set, for m n, then for τ D, the DEIM approximation of order m for f (τ) in the space spanned by {u l } m l=1 is given by f (τ) Uc(τ), U R n m, c(τ) R m. (6) The basis U can be constructed effectively by applying the POD method on the nonlinear snapshots f (τ t i ), i = 1,.., n s. (Florida State University) May 23, / 51

20 (DEIM) is used to determine the coefficient vector c(τ) by selecting m rows ρ 1,.., ρ m, ρ i N, of the overdetermined linear system (6) f 1 (τ)... } f n (τ) {{ } f (τ) R n u u 1m.. = } u n1... {{ u nm } U R n m to form a m-by-m linear system f ρ1 (τ) u ρ11... u ρ1m. =..... } f ρm (τ) {{ } f ρ (τ) R m u ρm1... u ρmm }{{} U ρ R m m c 1 (τ).. } c m (τ) {{ } c(τ) R m c 1 (τ).. } c m (τ) {{ } c(τ) R m

21 nonlinear (DEIM) nonlinear 4DVAR In the short notation form U ρ c(τ) = f ρ (τ). Lemma in Chaturantabut (2008) proves that U ρ is invertible, thus we can uniquely determine c(τ) c(τ) = U 1 ρ f ρ(τ). The DEIM approximation of F (τ) R n is f (τ) Uc(τ) = UU 1 ρ f ρ(τ). (Florida State University) May 23, / 51

22 nonlinear nonlinear 4DVAR (DEIM) U ρ and f ρ (τ) can be written in terms of U and f (τ) U ρ = P T U, f ρ (τ) = P T f (τ) where P = [e ρ1,.., e ρm ] R n m, e ρi = [0,..0, }{{} 1, 0,.., 0] T R n. ρ i The DEIM approximation of f R n becomes f (τ) U(P T U) 1 P T f (τ). Using the DEIM approximation, the complexity for computing the nonlinear term of the reduced system in each time step is now independent of the dimension n of the original full-order sytem. The only unknowns need to be specified are the indices ρ 1, ρ 2,..., ρ m or matrix P. The algorithm can be found in Chaturantabut (2008), Chaturantabut and Sorensen (2010). (Florida State University) May 23, / 51

23 DEIM: Algorithm for Indices INPUT: {u l } m l=1 Rn (linearly independent): OUTPUT: ρ = [ρ 1,.., ρ m ] R m 1 [ ψ ρ 1 ] = max u 1, ψ R and ρ 1 is the component position of the largest absolute value of u 1, with the smallest index taken in case of a tie. 2 U = [u 1 ], P = [e ρ1 ], ρ = [ρ 1 ]. 3 For l = 2,.., m do a Solve (P T U)c = P T u l for c b r = u l Uc c [ ψ ρ l ] = max{ r } [ ρ d U [U u l ], P [P e ρl ], ρ 4 end for. ρ l ]

24 nonlinear The DEIM version of SWE model nonlinear 4DVAR The projected nonlinear functions can be approximated by DEIM in a form that enables precomputation so that the computational cost is decreased and independent of the original system. Only a few entries of the nonlinear term corresponding to the specially selected interpolation indices from DEIM must be evaluated at each time step. DEIM approximation is applied to each of the nonlinear functions F 11, F 12, F 21, F 22, F 31, F 32 defined in (4). (Florida State University) May 23, / 51

25 The DEIM version of SWE model Let U F11 R nx y m, m n, be the POD basis matrix of rank m for snapshots of the nonlinear function F 11 (obtained from ADI FD scheme). The DEIM approximation of F 11 is F 11 U F11 (P T F 11 U F11 ) 1 F m 11, so the projected nonlinear term U T F11 (ũ, φ) in the POD reduced system (3) can be approximated as U T F11 (ũ, φ) U T U F11 (PF T 11 U F11 ) 1 }{{} E 1 R k m where F m 11 (ũ, φ) = P T F 11 F11 (ũ, φ). m F } 11(ũ, {{ φ), } m 1 Since F 11 is a pointwise function, F m 11 : Rk R k R m can be defined as F 11(ũ, m φ) = (PF T 11 Uũ) (PF T 11 A x U }{{} ũ) (PT F 11 Φ φ) (PF T 11 A x Φ φ) }{{} Similarly we obtain the DEIM approximation for the rest of the projected nonlinear terms in (5)

26 The domain was discretized using a mesh of points, with x = y = 20km. Thus the dimension of the full-order discretized model is The integration time window for ADI FD scheme was 24h and we used 91 time steps (NT = 91) with t = 960s. Contour of geopotential from to by Wind field y(km) y(km) x(km) x(km) Fig.1 Initial condition: Geopotential height field for the Grammeltvedt initial condition (left). Wind field (arrows are scaled by a factor of 1km) calculated from the geopotential field by using the geostrophic approximation (right).

27 Courant-Friedrichs-Levy (CFL) condition: gh( t x ) < The nonlinear algebraic systems of ADI FD SWE scheme were solved with the Quasi-Newton method and the LU decomposition was performed only once every 6 th time step. Contour of geopotential at time t f = 24h 4500 Wind field at time t f = 24h y(km) y(km) x(km) x(km) Fig.2 The geopotential field (left) and the wind field (the velocity unit is 1km/s) at t = t f = 24h obtained using the ADI FD SWE scheme for t = 960s.

28 10 5 Singular Values of Snapshots Solution u v φ 0 5 Singular Values of Nonlinear Snapshots F 11 F 12 F 21 0 F 22 F 31 log 10 scale 5 10 log 10 scale F Number of snapshots Number of snapshots Fig.3 The decay around the singular values of the snapshots solutions for u, v, φ and nonlinear functions for t = 960s. The POD basis functions were constructed using 91 snapshots.

29 The dimension of the POD bases for each variable was taken to be 35. DEIM POINTS for F 31 DEIM POINTS for F y(km) y(km) x(km) x(km) Fig.4 First 40 points selected by DEIM for the nonlinear functions F 31 (left) and F 32 (right)

30 We also propose an Euler explicit FD SWE scheme as the starting point for a POD, reduced model. The POD bases were constructed using the same 91 snapshots as in the POD ADI SWE case, only this time the Galerkin projection was applied to the Euler FD SWE model. The root mean square error was employed in order to compare the POD and techniques at time t = 24h. ADI SWE POD ADI SWE ADI SWE POD EE SWE EE SWE CPU time seconds RMSE φ e e e e-004 RMSE u e e e e-004 RMSE v e e e e-004 Table 2 CPU time gains and the root mean square errors for each of the model variables at t = t f. The POD bases dimensions were taken as 35 capturing more than 99.9% of the system energy. 90 DEIM points were chosen.

31 Nonlinear model order reduction of an ADI implicit shallow water equations model, R. Stefanescu and I.M., Journal of Computational Physics, Volume 237, 15 March 2013, Pages CPU time (seconds) CPU time vs. number of spatial discretization points 80 ADI SWE 70 POD ADI SWE ADI SWE POD EE SWE 60 EE SWE No. of spatial discretization points Fig.5 Cpu time vs. Spatial discretization points; POD DIM = 35, No. DEIM points = 90.

32 4DVAR 4DVAR scheme of SWE nonlinear 4DVAR We aim to incorporate data assimilation system into model reduction We applied DEIM to a POD Leapfrog on a rectangular domain. We build the 4D VAR minimization application to emphasize the DEIM impact in the context of reduced PDE constrained optimization. (Florida State University) May 23, / 51

33 4DVAR 4D VAR nonlinear 4DVAR The aim of 4-D VAR data assimilation is to reconcile observations with model predictions subject to the model serving as a strong constraint., In the full high-fidelity nonlinear 4-D VAR, this process is implemented by minimizing the following cost functional with respect to the control variables: Nt J(w 0 ) = (w 0 w b ) T B 1 (w 0 w b )+ γ w (H k w k wk o ) T O 1 (H k w k wk o ), k=1 (7) where w 0 = (u 0, v 0, φ 0 ) and γ w is a weight function. (Florida State University) May 23, / 51

34 4D VAR In the data assimilation process of the SWE model, we just consider the observation information and don t involve the background information. The reduced order cost functional in 4-D VAR assumes the form J (w 0 ) = Nt k=1 γ w (w k wk o)t O 1 (w k wk o) (8) where w 0 is the reduced order control vector w 0 = (u 0, v 0, φ 0 ) = (ũ 0, ṽ 0, φ 0 ) and w k is the model prediction obtained from the reduced order forward model.

35 4DVAR nonlinear 4DVAR 4D VAR In the process of minimizing the cost functional with respect to the control variables, the limited-memory BFGS (L-BFGS) quasi-newton method is applied. The gradient of the reduced cost functional with respect to the control variables can be expressed as ũ0 J POD = U POD T ( u0 J) u0=u POD ũ ṽ0 J POD = V POD T ( v0 J) v0=v POD ṽ φ0 J POD = Φ POD T ( φ0 J) φ0=φpod φ Since our work is in progress, for this presentation, the gradient of the cost functional was calculated using the adjoint of the forward POD model. (Florida State University) May 23, / 51

36 4DVAR 4D VAR nonlinear 4DVAR Consequently, computational savings are mainly achieved by a drastic reduction in the number of iterations due to the low dimension of the optimization problem. Since the validity of the reduced order model is limited to the vicinity of the control space (POD bases and bases), it might not be an appropriate model when the latest state is significantly different from the one on which the reduced order model is based. (Florida State University) May 23, / 51

37 4DVAR Ad-hoc 4D VAR nonlinear 4DVAR An ad-hoc adaptive 4-D VAR algorithm is proposed as follows: 1 Generate a set of snapshots (including the nonlinear ones corresponding to the quadratic non-linear terms) from the solution of the full forward model and construct the reduced order model 2 Perform iterations for the optimization problem using the reduced order model with the L-BFGS method and calculate the cost functional J n where n is the n th L-BFGS iteration. 3 Check the value of the cost functional. (Florida State University) May 23, / 51

38 4DVAR Ad-hoc 4D VAR nonlinear 4DVAR If J n < ɛ where ɛ is the tolerance for the optimization, then stop, the 4-D VAR data assimilation is completed; If J n > ɛ and J n 1 J n > η η > 0, then set n = n + 1 and go back to (2); If J n > ɛ and J n 1 J n < η, project back the reduced order control variables from the latest optimization iteration to the original space, then go to (1). (Florida State University) May 23, / 51

39 4DVAR nonlinear 4DVAR The domain was discretized using a mesh of points, x = 600km, y = 440km. The integration window was 5h and we used 60 time steps (Nt = 60) with t = 300s. Leapfrog scheme was employed to obtain the snapshots for ROMs models. (Florida State University) May 23, / 51

40 21500 Contour of PHI at time t Wind field at time t y(km) y(km) x(km) x(km) Fig.6 Initial condition: Geopotential height field for the Grammeltvedt initial condition (left). Wind field calculated from the geopotential field by using the geostrophic approximation (right).

41 Contour of PHI at time t f =5h Contour of PHI at time t f =5h y(km) y(km) x(km) x(km) Fig.7 Geopotential height field at t = t f obtained using Leapfrog full scheme (left) and Leapfrog scheme (right)

42 Wind field at time t f = 5h Wind field at time t f = 5h y(km) 2000 y(km) x(km) x(km) Fig.8 Wind field at t = t f obtained using Leapfrog full scheme (left) and Leapfrog scheme (right)

43 Contour of PHI at time t f =5h Contour of PHI at time t f =5h y(km) y(km) x(km) x(km) Fig.9 Geopotential height field at t = t f obtained using Leapfrog full scheme (left) and Leapfrog scheme (right)

44 Wind field at time t f = 5h Wind field at time t f = 5h y(km) 2000 y(km) x(km) x(km) Fig.10 Wind field at t = t f obtained using Leapfrog full scheme (left) and Leapfrog scheme (right)

45 We employed the root mean square error calculation in order to compare the POD and techniques at time t = 5h. Leapfrog SWE Leapfrog POD SWE Leapfrog SWE CPU time seconds E E-03 RMSE φ E E-04 RMSE u E E-07 RMSE v E E-07 Table 2 CPU time gains and the root mean square errors for each of the model variables at t = t f. The POD bases dimensions were taken as 15 capturing more than 99.9% of the system energy. 20 DEIM points were chosen.

46 18000 PHIOBS at time t f =5h First guess of PHI at time t f =5h y(km) y(km) x(km) x(km) Fig.11 Phi observations (left) and first guess (right) for t = t f

47 Wind field observations at time t f = 5h First guess wind field at time t f = 5h y(km) 2000 y(km) x(km) x(km) Fig.12 Wind field observations (left) and first guess (right) for t = t f

48 D Var Data Assimilation using SWE 4D Var full 4D Var POD 4D Var No of iterations Fig.13 The performance of minimization of the cost functional for the Ad Hoc POD 4-D VAR, 4-D VAR and full 4-D VAR.

49 Full 4D-VAR POD 4D-VAR 4D-VAR CPU time seconds CPU time for updating POD basis CPU time for updating basis Updating POD basis costs the same as one forward integration in time of the full model Updating basis costs the same as three forward integration in time of the full model

50 4DVAR Discussion and Future Work nonlinear 4DVAR For our test, we chose the number of POD basis to be equal with 10 and the number of DEIM interpolation points used was also 10. By increasing the control space dimension and the number of DEIM points we expect that and POD 4DVAR minimizations to perform better. The Leapfrog scheme should also include the Robert Asselin Williams (RAW) filter to dump the computational mode. 4DVAR gradient test was affected by the inconsistency between the forward model and the POD adjoint model. Using discretize and differentiate we ll obtain a faster adjoint model comparing with POD adjoint model. (Florida State University) May 23, / 51

51 4DVAR Discussion and Future Work nonlinear 4DVAR The discrete adjoint will reduce the computational complexity of POD adjoint model due to its depending on the nonlinear full dimension model and regain the full model reduction expected from the POD model. The trust region scheme will be combined with POD 4-D VAR data assimilation in order to solve the reduced order inverse problem more efficiently. (Florida State University) May 23, / 51

52 4DVAR Conclusion and future research nonlinear 4DVAR Nonlinear model order reduction of an ADI implicit shallow water equations model, R. Stefanescu and I.M., Journal of Computational Physics, Volume 237, 15 March 2013, Pages To obtain the approximate solution in case of both POD and reduced systems, one must store POD or solutions of order O(kNT ), k being the POD bases dimension and NT the number of time steps in the integration window. The coefficient matrices that must be retained while solving the POD reduced system are of order of O(k 2 ) for projected linear terms and O(n xy k) for the nonlinear term, where n xy is the space dimension. (Florida State University) May 23, / 51

53 4DVAR nonlinear 4DVAR Conclusion and future research In the case of solving reduced system the coefficient matrices that need to be stored are of order of O(k 2 ) for projected linear terms and O(mk) for the nonlinear terms, where m is the number of DEIM points determined by the DEIM indexes algorithm, m n xy. Therefore DEIM improves the efficiency of the POD approximation and achieves a complexity reduction of the nonlinear term with a complexity proportional to the number of reduced variables. We proved the efficiency of DEIM using three different schemes, the ADI FD SWE fully implicit model, the Euler explicit FD SWE scheme, the Leapfrog scheme. Trust Region 4DVAR algorithm shows great potential having the ability to decrease the CPU time of POD 4DVAR by the same rate as reduces the computational complexity of the POD forward model once the adjoint is constructed. (Florida State University) May 23, / 51