POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model

Transcription

1 .. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model North Carolina State University rstefan@ncsu.edu POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 1/64

2 Part1 - POD/DEIM nonlinear model reduction 1 POD/DEIM justification and methodology POD/DEIM as a discrete variant of EIM and their pseudo - algorithms 3 Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain 4 Numerical Results 5 Conclusion and future research

3 1.1. POD/DEIM justification and methodology POD/DEIM justification and methodology 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 [, T ] t where L is a linear function and F a nonlinear one. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 3/64

4 POD/DEIM justification and methodology 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 (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. 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 a discontinuous inner product to the ODE discrete system we get d dt ỹ(t) = V k T AV k ỹ(t) + Vk T }{{}} F(V {{ kỹ(t)) } k k Ñ(ỹ) (1)

5 1.1. POD/DEIM justification and methodology POD/DEIM justification and methodology POD is one of the most significant projection-based reduction methods for non-linear dynamical systems. It is also known as Karhunen - Loève expansion, principal component analysis in statistics, singular value decomposition (SVD) in matrix theory and empirical orthogonal functions (EOF) in meteorology and geophysical uid dynamics Introduced in the field of turbulence by Lumley (1967) It was Sirovich (1987 a,b,c) that introduced the method of snapshots obtained from either experiments or numerical simulation Error formula for the POD basis of rank l T y(t) l < y(t), v j > X v j X dt = j=1 j=l+1 λ j. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 5/64

6 POD/DEIM justification and methodology 1.1. POD/DEIM justification and methodology 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 Discrete Empirical Interpolation Method (DEIM) for nonlinear approximation. For m n Ñ(ỹ) Vk T U(PT U) 1 F(P T V }{{} k ỹ(t)). }{{} m 1 precomputed k m POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 6/64

7 1.. POD/DEIM as a discrete variant of EIM and their pseudo - algorithms Discrete Empirical Interpolation Method (DEIM) DEIM is a discrete variation of the Empirical Interpolation method proposed by Barrault et al. (4) - Comptes Rendus de l Acadèmie des Sciences. The application was suggested by Chaturantabut and Sorensen (8, 1, 1). 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. () The basis U can be constructed effectively by applying the POD method on the nonlinear snapshots f (τ t i ), i = 1,.., n s. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 7/64

8 Discrete Empirical Interpolation Method (DEIM) Interpolation 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 () f 1 (τ) u u 1m c. = (τ) } f n (τ) {{ } f (τ) R n u n1... u nm }{{} U R n m } c m (τ) {{ } c(τ) R m to form a m-by-m linear system f ρ1 (τ) u ρ u ρ1 m c 1 (τ). = f ρm (τ) u ρm1... u ρmm c m (τ) }{{}}{{}}{{} f ρ (τ) R m U ρ R m m c(τ) R m

9 1.. POD/DEIM as a discrete variant of EIM and their pseudo - algorithms Discrete Empirical Interpolation Method (DEIM) In the short notation form U ρ c(τ) = f ρ (τ). Lemma.3.1 in Chaturantabut (8) 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). POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 9/64

10 1.. POD/DEIM as a discrete variant of EIM and their pseudo - algorithms Discrete Empirical Interpolation Method (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 = [,.., }{{} 1,,.., ] T R n. ρ i The DEIM approximation of f R n becomes f (τ) U(P T U) 1 P T f (τ). POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 1/64

11 1.. POD/DEIM as a discrete variant of EIM and their pseudo - algorithms Discrete Empirical Interpolation Method (DEIM) By taking τ = y(t) R n, the DEIM approximation for the nonlinear function f (τ) = f (y(t)) = F(V k ỹ(t)) in the POD-Galerkin reduced system (1) is F(V k ỹ(t)) U(P T U) 1 P T F(V k ỹ(t)), but since F evaluates componentwise at its input we have F(V k ỹ(t)) U(P T U) 1 F(P T V k ỹ(t)). POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 11/64

12 Discrete Empirical Interpolation Method (DEIM) Thus, the DEIM approximation of the nonlinear POD-Galerkin term Ñ(ỹ) Ñ(ỹ) = Vk T F(V }{{} k ỹ(t)) }{{} k n n 1 is Ñ(ỹ) Vk T U(PT U) 1 F(P T V }{{} k ỹ(t)). }{{} m 1 precomputed k m 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,..., ρ m or matrix P.

13 DEIM: Algorithm for Interpolation 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. U = [u 1 ], P = [e ρ1 ], ρ = [ρ 1 ]. 3 For l =,.., 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 ρl ] 4 end for.

14 DEIM: Algorithm for Interpolation Indices The term r can be viewed as the residual or the error between the input basis u l and its approximation Uc from interpolating the basis {u 1, u,.., u l 1 } at the points x ρ1, x ρ,.., x ρl 1. The linear independence of the input basis {u l } m l=1 guarantees that, in each iteration, r is a nonzero vector and the output indices {ρ i } m i=1 are non - repeated. An error bound for the DEIM approximation is provided in Chaturantabut (8). A state space error analysis for POD-DEIM Nonlinear model reduction applied to ODE systems arising from spatial discretizations of parabolic PDEs can be found in Chaturantabut and Sorensen (1).

15 1.. POD/DEIM as a discrete variant of EIM and their pseudo - algorithms Discrete Empirical Interpolation Method (DEIM) The following example illustrates the efficiency of DEIM in approximating a highly nonlinear function defined on a discrete 1D spatial domain. Consider a nonlinear parameterized function s : Ω D R defined by ( ) s(x; µ) = (1 x)sin πµ(x + 1) e (1+x)µ, where x Ω = [ 1, 1] and µ D = [, π ] R. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 15/64

16 1.. POD/DEIM as a discrete variant of EIM and their pseudo - algorithms Discrete Empirical Interpolation Method (DEIM) Let [x 1, x,..., x n ] R n, x i R being equally distributed in Ω, for i = 1,,.., n, n = 11. We introduce f : D R n as follows f (µ) = [s(x 1 ; µ), s(x ; µ),.., s(x n ; µ) T ] R n, µ D We used 5 snapshots f (µ j ) 5 j=1 to construct POD basis {u l } m l=1 with µ j equidistantly points in [, π ]. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 16/64

17 1.. POD/DEIM as a discrete variant of EIM and their pseudo - algorithms Discrete Empirical Interpolation Method (DEIM) logarithmic scale Singular values of 5 Snapshots DEIM points and the first 6 POD basis functions PODbasis1 PODbasis PODbasis3 PODbasis4 PODbasis5 PODbasis6 DEIM points Exact function Figure 1: Singular eigenvalues using logarithmic scale and the corresponding first 6 POD basis functions with DEIM points of snapshots, µ = 1.38 POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 17/64

18 Figure : The selection process of DEIM interpolation points u 4 DEIM#1.15 DEIM#..4.6 u 1 current point.1.5 u r=u Uc current point previous point DEIM#3.15 DEIM# r=u 4 Uc current point previous points u 3 r=u 3 Uc current point previous points DEIM#5 u 5 r=u 5 Uc.3 DEIM#6.3 current point previous points u 6 r=u 6 Uc current point previous points

19 Figure 3: DEIM approximation for different values of m 1. 1 DEIM#1 exact DEIM approx 1. 1 DEIM# exact DEIM approx DEIM#3 exact 1. DEIM#4 exact 1 DEIM approx 1 DEIM approx DEIM#5 exact 1. DEIM#6 exact 1 DEIM approx. 1 DEIM approx

20 1.. POD/DEIM as a discrete variant of EIM and their pseudo - algorithms Discrete Empirical Interpolation Method (DEIM) The Exact function and its DEIM approximation for µ= Exact fuction DEIM solution Error in Euclidian Norm DEIM error POD error logarithmic scale m (Reduced dimension) Figure 4: The DEIM approximate function for m = compared with the exact function of dimension n = 11 at µ = 1.38 (left); Comparison of the spatial errors for POD and DEIM approximations (right) POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model /64

21 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain We consider a -D shallow-water(swe) equations model on a β-plane solved using an alternating direction fully implicit finite-difference scheme (Gustafsson 1971, Fairweather and Navon 198, Navon and De Villiers 1986, Kreiss and Widlund 1966) on a rectangular domain. The scheme was shown to be unconditionally stable for the linearized equations. The discretization yields a number of nonlinear systems of algebraic equations. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 1/64

22 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain Next we use a proper orthogonal decomposition to reduce the dimension of the S-W model. Due to nonlinearities, the computational complexity of the reduced model still depends on the number of variables of the nonlinear full shallow - water equations model. By employing the discrete empirical interpolation method (DEIM) we reduce the computational complexity and regain the full model reduction expected from the POD model. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model /64

23 SWE model w t = A(w) w x + B(w) w y x L, y D, t [, t f ], + C(y)w, (3) 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 = gh. The matrices A, B and C are expressed A = u φ/ u φ/ u C =, B = f f v v φ/ φ/ v f = ˆf +β(y D/) (Coriolis force), β = f y, with ˆf and β constants.,

24 SWE model 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain 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,, t) = v(x, D, t) =. The initial conditions are derived from the initial height-field condition No. 1 of Grammelvedt (1969), i.e. ( h(x, y) = H +H 1 +tanh 9 D/ y ( )+H sech 9 D/ y ) ( πx sin D D L ) POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 4/64

25 SWE model 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The initial velocity fields were derived from the initial height field using the geostrophic relationship ( ) ( ) g h g h u = f y, v = f x. The constants used were: L = 6km g = 1ms D = 44km H = m ˆf = 1 4 s 1 H 1 = mm β = s 1 m 1 H = 133m. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 5/64

26 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The nonlinear Gustafsson ADI finite difference implicit scheme First we introduce a network of N x N y equidistant points on [, L] [, D], with dx = L/(N x 1), dy = D/(N y 1). We also discretize the time interval [, 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 POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 6/64

27 For t n+1, the Gustafsson nonlinear ADI difference scheme is defined by I. First step - get solution at t(n + 1 ( ) ) u(t n+ 1 ) + t F 11 v(t n+ 1 ) + t ( F 1 φ(t n+ 1 ) + t ( F 31 u(t n+ 1 ), φ(t n+ 1 ) u(t n+ 1 u(t n+ 1 ) ), v(t n+ 1 ) = u(t n ) t F 1 ( ) u(t n ), v(t n ) + t [f, f,.., f ] T v(t }{{} n ), N x + t [f, f,.., f }{{} t F N x ( ) ] T u(t n+ v(t n ), φ(t n ), ) ), φ(t n+ 1 ) = φ(t n ) t F 3 1 ) = v(t n ) ( v(t n ), φ(t n ) with * denoting MATLAB componentwise multiplication and the nonlinear functions F 11, F 1, F 1, F, F 31, F 3 : R nxy R nxy R nxy are defined as follows ), (4)

28 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The nonlinear Gustafsson ADI finite difference implicit scheme F 11 (u, φ) = u A x u + 1 φ A xφ, F 1 (u, v) = v A y u, F 1 (u, v) = u A x v, F (v, φ) = v A y v + 1 φ A y φ, F 31 (u, φ) = 1 φ A x u + u A xφ, F 3 (v, φ) = 1 φ A y v + v A y φ, nxy nxy where A x, A y R are constant coefficient matrices for discrete first-order and second-order differential operators which take into account the boundary conditions. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 8/64

29 II. Second step - get solution at t(n + 1) ( ) u(t n+1 ), v(t n+1 ) u(t n+1 ) + t F 1 v(t n+1 ) + t ( ) F v(t n ), φ(t n ) φ(t n+1 ) + t ( ) F 3 v(t n+1 ), φ(t n+1 ) t [f, f,.., f }{{} t F 11 = v(t n+ 1 ) t t [f, f,.., f }{{} = φ(t n+ 1 ( t F 31 1 ) ] T v(t n+1 ) = u(t n+ N x ( ) u(t n+ 1 ), φ(t n+ 1 ), ( ) F 1 u(t n+ 1 ), v(t n+ 1 ) N x ] T u(t n+ ) u(t n+ 1 1 ), ) ), φ(t n+ 1 ). (5)

30 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The Quasi-Newton Method The nonlinear systems of algebraic equations (4) and (5) are written in the form g(α) =. where α is the vector of unknown. Due to the fact that no more than two variables are coupled to each other on the left-hand side of equations (4) and (5), we first solve system (4) for u = [u 1, u,..., u nxy ] and φ = [φ 1, φ,..., φ nxy ] and define α = (u 1, φ 1, u, φ,..., u nxy, φ nxy ) R nxy. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 3/64

31 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The Quasi-Newton Method The Newton method is given by α (m+1) = α (m) J 1 (α (m) )g(α (m) ), (6) nxy nxy where the superscript denotes the iteration and J R is the Jacobian J = g α. Owing to the structure of the Gustafsson algorithm for the SWE, the Jacobian matrix is either block cyclic tridiagonal or block tridiagonal. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 31/64

32 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The Quasi-Newton Method J 1 g in (6) is solved by first applying an LU decomposition to J. Then it is computed by backsubstitution in two stages. First z is solved from Lz = g, and then J 1 g is obtained from U(J 1 g) = z. In the quasi-newton method, the computationally expensive LU decomposition is performed only once every M th time-step, where M is a fixed integer. The quasi-newton formula is α (m+1) = α (m) ˆ J 1 (α (m) )g(α (m) ), where ˆ J = J(α () ) + O(dt). POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 3/64

33 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The Quasi-Newton Method The method works when M, the number of time-steps between successive updating of the LU decomposition of the Jacobian matrix J, is a relatively small number, in our case, M = 6 or M = 1. The second part of the system (4) is solved for v = [v 1, v,..., v nxy ] by employing the same quasi-newton method. Thus α is defined as α = (v 1, v, v nxy ) R nxy. In order to obtain the SWE numerical solution at t(n + 1) we applied the same quasi-newton technique for system (5). This time the variables coupled first were α = (v 1, φ 1, v, φ,..., v nxy, φ nxy ) R nxy, while u was solved from the remaining equations. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 33/64

34 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The POD version of SWE model Proper orthogonal decomposition provides a technique for deriving low order model of dynamical systems. It can be thought of as a Galerkin approximation in the spatial variable built from functions corresponding to the solution of the physical system at specified time instances. These are called snapshots. Let Y = [u 1, u,.., u NT ] R nxy NT, be a snapshot set, i.e. the numerical solution obtained with ADI implicit FD scheme at t 1, t,.., t NT, of the horizontal component of the velocity. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 34/64

35 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The POD version of SWE model The POD basis U R nxy k is determined by solving the eigen value problem Y T Y û i = λ i û i, i = 1,,.., NT, and retaining the set of right singular vectors of Y corresponding to the k largest singular values, i.e. U = {u i } k i=1, u i = 1 λ i Y û i. We determined the POD basis in this way by taking in consideration the relationship NT n xy. Similarly, let V, Φ R nxy k be the POD basis matrices of the vertical component of the velocity and geopotential, respectively. Now we can approximate u, v and φ as following u Uũ, v V ṽ, φ Φ φ, ũ, ṽ, φ R k. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 35/64

36 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The POD version of SWE model The POD reduced-order system is constructed by applying the Galerkin projection method to ADI FD discrete model (4) and (5) by first replacing u, v, φ with their approximation Uũ, V ṽ, Φ φ, respectively, and then premultiplying the corresponding equations by U T, V T and Φ T. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 36/64

37 The resulting POD reduced system for the first step (t n+ 1 ) of the ADI FD scheme is ũ(t n+ 1 ) + t UT F 11 ( ṽ(t n+ 1 ) + t ( V T F 1 φ(t n+ 1 ) + t ( ΦT F 31 ũ(t n+ 1 ), φ(t n+ 1 ) ũ(t n+ 1 ũ(t n+ 1 ) = ũ(t n ) t ( ) UT F 1 ũ(t n ), ṽ(t n ) + t ( ) UT [f, f,.., f, }{{} N x ] T V ṽ(t n ) ) ), ṽ(t n+ 1 ) + t ( V T [f, f,.., f }{{} ), φ(t n+ 1 ) ) ] T Uũ(t n+ 1 ) N x = ṽ(t n ) t ( ) V T F ṽ(t n ), φ(t n ), ) = φ(t n ) t ( ) ΦT F 3 ṽ(t n ), φ(t n ), where F 11, F 1, F 1, F, F 31, F 3 : R k R k R k are defined by (7)

38 The POD version of SWE model 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain F 11 (ũ, φ) = (Uũ) (A x U }{{} ũ) + 1 (Φ φ) (A x Φ }{{} φ), F 1 (ũ, ṽ) = (V ṽ) (A y U }{{} ũ), F 1 (ũ, ṽ) = (Uũ) (A x V }{{} ṽ), F (ṽ, φ) = (V ṽ) (A y V }{{} ṽ) + 1 (Φ φ) (A y Φ }{{} φ), (8) F 31 (ũ, φ) = 1 (Φ φ) (A x U }{{} ũ) + (Uũ) (A xφ }{{} φ), F 3 (ṽ, φ) = 1 (Φ φ) (A y V }{{} ṽ) + (V ṽ) (A y Φ }{{} φ). POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 38/64

39 The POD version of SWE model 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain By defining A 1, A R nxy k such as A 1 (:, i) = [f, f,.., f ] T V (:, i), A }{{} (:, i) = [f, f,.., f ] T U(:, i), i = 1,.., k, }{{} N x N x ) the linear terms in (7), t ([f UT, f,.., f ] T V ṽ(t }{{} n ) and ) N x t ([f V T, f,.., f ] T Uũ(t }{{} n+ 1 ) can be rewritten as N x t UT A }{{} 1 ṽ(t n) and t V T A }{{} ũ(t n+ 1 ) respectively. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 39/64

40 The POD version of SWE model The coefficient matrices U T A 1, V T A R k k 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 (8) and computing the projected nonlinear terms in (7) U T }{{} k n xy F 11 (ũ, φ), U T F }{{} 1 (ũ, ṽ), V T F 1 (ũ, ṽ), n xy 1 V T F (ṽ, φ), Φ T F31 (ũ, φ), Φ T F3 (ṽ, φ), (9) 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.

41 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain The DEIM version of SWE model DEIM is used to remove this dependency. 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 1, F 1, F, F 31, F 3 defined in (8). POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 41/64

42 The DEIM version of SWE model 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain Let U F 11 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). Using the DEIM algorithm we select a set of m DEIM indices corresponding to U F 11, denoting by [ρ F 11 1,.., ρf 11 m ] T R m. The DEIM approximation of F 11 is F 11 U F 11 (P T F 11 U F 11 ) 1 F m 11, so the projected nonlinear term U T F 11 (ũ, φ) in the POD reduced system (7) can be approximated as U T F 11 (ũ, φ) U T U F 11 (PF T 11 U F 11 ) 1 F 11(ũ, m φ), }{{}}{{} E 1 R k m m 1 where F m 11 (ũ, φ) = P T F 11 F 11 (ũ, φ). POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 4/64

43 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) + 1 (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 (9) U T F 1 (ũ, ṽ) U T U F 1 (PF T 1 U F 1 ) 1 F 1(ũ, m ṽ), }{{}}{{} E R k m m 1 V T F 1 (ũ, ṽ) V T U F 1 (PF T 1 U F 1 ) 1 F 1(ũ, m ṽ), }{{}}{{} E 3 R k m m 1 V T F (ṽ, φ) V T U F (PF T U F ) 1 F (ṽ, m φ), }{{}}{{} E 4 R k m m 1 Φ T F 31 (ũ, φ) Φ T U F 31 (PF T 31 U F 31 ) 1 F 31(ũ, m φ), }{{}}{{} E 5 R k m m 1 Φ T F 3 (ṽ, φ) Φ T U F 3 (PF T 3 U F 3 ) 1 F 3(ṽ, m φ), }{{}}{{} E 6 R k m m 1

44 The DEIM version of SWE model 1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain F 1(ũ, m ṽ) = (PF T 1 V ṽ) (PF T 1 A y U }{{} ũ), F 1(ũ, m ṽ) = (PF T 1 Uũ) (PF T 1 A x V }{{} ṽ), F (ṽ, m φ) = (PF T V ṽ) (PF T A y V }{{} ṽ) + 1 (PT F Φ φ) (PF T A y Φ φ), }{{} F 31(ũ, m φ) = (PF T 31 Φ φ) (PF T 31 A x U }{{} ũ) + (PT F 31 Uũ) (PF T 31 A x Φ φ), }{{} F 3(ṽ, m φ) = 1 (PT F 3 Φ φ) (PF T 3 A y V }{{} ṽ) + (PT F 3 V ṽ) (PF T 3 A y Φ φ). }{{} (1) POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 44/64

45 Each of the k m coefficient matrices grouped by the curly brackets in (1), as well as E i, i = 1,,.., 6 can be precomputed and re-used at all time steps, so that the computational complexity of the approximate nonlinear terms are independent of the full-order dimension n xy. Finally, the POD-DEIM reduced system for the first step of ADI FD SWE model is of the form ũ(t n+ 1 ) + t E 1 F 11 m ( ṽ(t n+ 1 ) + t ( E 3 F 1 m φ(t n+ 1 ) + t ( E 5 F 31 m ũ(t n+ 1 ), φ(t n+ 1 ) ũ(t n+ 1 ũ(t n+ 1 ), ṽ(t n+ 1 ) ), φ(t n+ 1 ) ) = ũ(t n ) t ( ) E F 1 m ũ(t n ), ṽ(t n ) + t UT A 1 ṽ(t n ), ) + t V T A ũ(t n+ 1 ) = ṽ(t n ) t ( ) E 4 F m ṽ(t n ), φ(t n ), ) = φ(t n ) t ( ) E 6 F 3 m ṽ(t n ), φ(t n ). (11)

46 1.4. Numerical Results Numerical Results The domain was discretized using a mesh of points. Thus the dimension of the full-order discretized model is The integration time window was 4h. ADI FD scheme proposed by Gustafsson (1971) was first employed in order to obtain the numerical solution of the SWE model. The initial condition were derived from the geopotential height formulation introduced by Grammelvedt (1969) using the geostrophic balance relationship. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 46/64

47 Numerical Results Contour of geopotential from to 18 by 5 45 Wind field y(km) y(km) x(km) x(km) Figure 5: 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).

48 1.4. Numerical Results Numerical Results Our first numerical experiment was done using a time step t = 48s. The Courant Friedrichs Levy criterion CFL= gh t x was determined CFL = The nonlinear algebraic systems obtained from ADI FD SWE were solved with the Quasi-Newton method, were the LU decomposition was performed only once every 6 th time step. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 48/64

49 Numerical Results 1.4. Numerical Results 4 Contour of geopotential at time t f = 4h 45 4 Wind field at time t f = 4h y(km) y(km) x(km) x(km) Figure 6: The geopotential field (left) and the wind field (arrows are scaled by a factor of 1km) at t = t f = 4h obtained using the ADI FD SWE scheme for t = 48s. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 49/64

50 1.4. Numerical Results Numerical Results The POD basis vectors were constructed using 181 snapshots obtained from the numerical solution of the full - order ADI FD SWE model at equally spaced time steps in the interval [ 4h]. The dimension of the POD bases for each variable was taken 4, capturing more than 99.9% of the system energy. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 5/64

51 Numerical Results 1.4. Numerical Results logarithmic scale Singular Values of Snapshots Solution u v φ logarithmic scale Singular Values of Nonlinear Snapshots FF11 FF1 FF1 FF FF31 FF Number of snapshots Number of snapshots Figure 7: The decay around the singular values of the snapshots solutions for u, v, φ and nonlinear functions ( t = 48s). POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 51/64

52 Numerical Results We applied the DEIM algorithm for interpolation indices to improve the efficiency of the POD approximation and to achieve a complexity reduction of the nonlinear terms with a complexity proportional to the number of reduced variables DEIM POINTS for FF DEIM POINTS for FF y(km) y(km) x(km) x(km) Figure 8: First 4 points selected by DEIM for the nonlinear functions FF31 (left) and FF3 (right)

53 1.4. Numerical Results Numerical Results Once the dimension of DEIM (no of points selected by DEIM algorithm) reaches 4 the approximation errors from the POD-DEIM and POD reduced systems are indistinguishable. We emphasize the performances of POD - DEIM method in comparison with the POD approach using the numerical solution of the ADI FD SWE model. Next three slides depict the space error behaviors between POD/POD - DEIM solution and ADI FD SWE solution. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 53/64

54 Numerical Results 1.4. Numerical Results Geopotential POD errors h POD h ADI FD x 1 4 Geopotential POD DEIM errors h DEIM h ADI FD x y(km) 5.5 y(km) x(km) x(km) Figure 9: Errors between the geopotential calculated with POD/POD -DEIM and geopotential determined with the ADI FD SWE model at t = 4h ( t = 48s). The number of DEIM points was taken 4. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 54/64

55 Numerical Results 1.4. Numerical Results u POD errors u POD u ADI FD x 1 4 u POD DEIM errors u DEIM u ADI FD x y(km) 5.5 y(km) x(km) x(km) 1.5 Figure 1: Errors between u calculated with POD/POD -DEIM and u determined with the ADI FD SWE model at t = 4h ( t = 48s). The number of DEIM points was taken 4. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 55/64

56 Numerical Results 1.4. Numerical Results v POD errors v POD v ADI FD x 1 5 v POD DEIM errors v DEIM v ADI FD x y(km) 5 4 y(km) x(km) x(km) 8 Figure 11: Errors between v calculated with POD/POD -DEIM and v determined with the ADI FD SWE model at t = 4h ( t = 48s). The number of DEIM points was taken 4. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 56/64

57 Numerical Results Using the following norms 1 t f 1 t f t f i=1 t f i=1 y ADI FD (:, i) y ADI POD (:, i) y ADI FD (:, i) y ADI FD (:, i) y ADI POD DEIM (:, i) y ADI FD (:, i) we obtained the following average errors. ADI FD ADI POD ADI POD-DEIM CPU time φ u v Table 1: CPU time gains and the average errors for each of the model variables. The POD bases dimensions were taken 4 capturing more than 99.9% of the system energy. 4 DEIM points were chosen.

58 Numerical Results If the POD bases dimensions are larger than a 1 1 CPU time reduction is gained when employing the POD -DEIM method in comparison with the POD approach. time(seconds) 1 3 CPU time Error DEIM4 1 4 DEIM5 DEIM1 POD FULL 1 Average Relative Error of Geopotential DEIM4 DEIM5 DEIM1 POD POD dimension POD dim Figure 1: CPU time of the full system, POD reduced sytem and POD - DEIM reduced system (left); Average relative error of φ from the POD-DEIM reduced system compared with the one from the POD reduced system (right)

59 1.5. Conclusion and future research Conclusion and future research The coefficient matrices that must be retained while solving the POD reduced system are of order of O(k ) for projected linear terms and O(n xy k) for the nonlinear term. In the case of solving the POD-DEIM reduced system the coefficient matrices that need to be stored are of order of O(k ) 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. DEIM therefore 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. POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 59/64

60 Conclusion and future research To prove the efficiency of DEIM we used the ADI FD SWE model. The CPU time was reduced by a factor of 1 for the ADI POD - DEIM SWE. Also we noticed that the approximation errors from the POD-DEIM and POD reduced systems are indistinguishable when the dimension of DEIM reached 4. Application to optimization and uncertainty quantification. Adaptive ROMs based on DEIM points. POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation, Journal of Computational Physics, Volume 95, 15 August 15, Pages

61 References G. Fairweather, I. M. Navon, A linear ADI method for the shallow water equations, Journal of Computational Physics, 37 (198) B. Gustafsson, An alternating direction implicit method for solving the shallow water equations, Journal of Computational Physics,7 (1971) I. M. Navon, R. De Villiers, GUSTAF: A Quasi-Newton Nonlinear ADI FORTRAN IV Program for Solving the Shallow-Water Equations with Augmented Lagrangians, Computers and Geosciences,1, No. (1986) S. Chaturantabut, D.C. Sorensen, A state space error estimate for POD-DEIM nonlinear model reduction,siam Journal on Numerical Analysis, 5, 1 (1) S. Chaturantabut, D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM Journal on Scientific Computing, 3, 5 (1)

62 References S. Chaturantabut, Master s Thesis: Dimension Reduction for Unsteady Nonlinear Partial Differential Equations via Empirical Interpolation Methods, November 8. S. Chaturantabut, D.C. Sorensen, Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media, Mathematical and Computer Modelling of Dynamical Systems, 17, 4 (11) A. R. Kellems, S. Chaturantabut, D. C. Sorensen, S. J. Cox, Morphologically accurate reduced order modeling of spiking neurons. Journal of Computational Neuroscience. 8, Number 3 (1), M. Barrault,Y. Maday, N.C. Nguyen; A.T. Patera An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, 339,9, (4)

63 References M. Hinze, M. Kunkel, Discrete Empirical Interpolation in POD Model Order Reduction of Drift-Diffusion Equations in Electrical Networks, Scientific Computing in Electrical Engineering SCEE 1 Mathematics in Industry, 16, Part 5,(1) O. Lass, S. Volkwein, POD Galerkin schemes for nonlinear elliptic-parabolic systems,konstanzer Schriften in Mathematik, Nr. 31, März 1, ISSN

64 Part - POD 4-D VAR of the limited area FEM SWE using Dual-Weighting and Trust Region method