POD/DEIM Nonlinear model order reduction of an ADI implicit shallow water equations model

Transcription

1 Manuscript Click here to view linked References POD/DEIM Nonlinear model order reduction of an ADI implicit shallow water equations model R. Ştefănescu, I.M. Navon The Florida State University, Department of Scientific Computing, Tallahassee, Florida 336, USA Corresponding author Preprint submitted to Elsevier June,

2 POD/DEIM Nonlinear model order reduction of an ADI implicit shallow water equations model R. Ştefănescu, I.M. Navon The Florida State University, Department of Scientific Computing, Tallahassee, Florida 336, USA Abstract In the present paper we consider a -D shallow-water equations (SWE model on a β- plane solved using an alternating direction fully implicit (ADI finite-difference scheme (Gustafsson [], Fairweather and Navon [], Navon and De Villiers [3], Kreiss and Widlund [4] 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. We then use a proper orthogonal decomposition (POD to reduce the dimension of the SWE model. Due to the model nonlinearities, the computational complexity of the reduced model still depends on the number of variables of the full shallow - water equations model. By employing the discrete empirical interpolation method (DEIM we reduce the computational complexity of the reduced order model due to its depending on the nonlinear full dimension model and regain the full model reduction expected from the POD model. To emphasize the CPU gain in performance due to use of POD/DEIM, we also propose testing an explicit Euler finite difference scheme (EE as an alternative to the ADI implicit scheme for solving the swallow water equations model. We then proceed to assess the efficiency of POD/DEIM as a function of number Corresponding author Preprint submitted to Elsevier June,

3 of spatial discretization points, time steps, and POD basis functions. The CPU time decreased by at least a factor of /5 in case of POD/DEIM implicit/explicit SWE scheme when the number of spatial discretization points exceeded than and the DEIM dimension was 9. Moreover, in most of our experiments, once the number of points selected by DEIM algorithm reached 5, the approximation errors due to POD/DEIM and POD reduced systems are almost indistinguishable supporting the theoretical results obtained by Chaturnabut and Sorensen in [5]. Keywords: shallow water equations; proper orthogonal decomposition; reduced-order models (ROMs; finite difference methods; discrete empirical interpolation method (DEIM;. Introduction The shallow water equations are the simplest form of the equations of motion that can be used to describe the horizontal (motion structure of the atmosphere. They describe the evolution of an incompressible and inviscid fluid in response to gravitational and rotational accelerations and their solutions represent East West propagating Rossby waves and inertia - gravity waves (David Randall [6]. To avoid the limitations imposed by the Courant Friedrichs - Lewy (CFL stability conditions restricting the time steps in explicit finite difference approximations, implicit scheme must be considered. We propose here the alternating direction implicit (ADI method (Gustafsson [], Fairweather and Navon [], Navon and De Villiers [3]. Such methods reduce multidimensional problem to systems of one dimensional problems (Douglas and Gunn [7], Yanenko [8], Marchuck [9]. The nonlinear algebraic systems corresponding to the discrete model were solved using the quasi Newton method proposed in Gustafsson []. This quasi - Newton method performs an LU decomposition done every M-th time step, where M is a fixed integer. Since back substitution is a fast operation the scheme will be efficient as long as the number of iterations is small. 3

4 In order to reduce the dimension of the above model we use a proper orthogonal decomposition (POD. However due to the nonlinearities of the implicit SWE model the computational complexity of the reduced shallow water equations model still depends on the number of variables of the full shallow - water equations model (Chaturantabut and Sorensen [], []. To mitigate this problem, we apply the discrete empirical interpolation method (DEIM to adress the reduction of the nonlinear components and thus reduce the computational complexity by implementing the POD/DEIM method. The paper is organized as follows. In Section we introduce the Gustafsson ADI fully implicit method applied to the shallow water equations model and briefly describe its algorithmic components since they are already available in archived literature. In Section 3 we describe in some detail the snapshot POD procedure and its implementation to the ADI method for the SWE model. Section 4 addresses the snapshot POD combined with DEIM methodology and provides the detailed algorithmic description of the DEIM implementation. In Section 5 we present the numerical experiments related to the POD/DEIM procedure for both explicit and implicit schemes applied to the SWE models. The POD/DEIM procedure amounts to replace orthogonal projection with an interpolation projection of the nonlinear terms that requires the evaluation of only a few selected components of the nonlinear terms. We evaluate the efficiency of DEIM as a function of number of spatial discretization points, time steps and basis functions for this quadratically nonlinear problem and more studies about the conservation of the integral invariants, RMSE and correlation coefficients between full model, POD and POD/DEIM systems were performed.. Brief description of the Gustafsson ADI method. Here we shortly describe the Gustafsson shallow water alternating direction implicit method (Gustafsson [], Fairweather and Navon [], Navon and De Villiers [3]. We are 4

5 solving the SWE model using the β-plane approximation on a rectangular domain. w t = A(w w x + B(w w + C(yw, ( y x L, y D, t (, t f ], where w = (u, v, φ T is a vector function, 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, B = v v φ/ φ/ v, C = f f, where f is the Coriolis term given by f = ˆf + β(y D/, β = f, y [, D], y with ˆf and β constants. We assume periodic solutions in the x-direction w(x, y, t = w(x + L, y, t, x =, y [, D], t (, t f ], while in the y direction v(x,, t = v(x, D, t =, x [, L], t (, t f ]. 5

6 Initially w(x, y, = ψ(x, y, ψ : R R R, (x, y [, L] [, D]. Note that no boundary conditions are necessary for u and φ at y =, D. Now we introduce a network of N x N y equidistant points on [, L] [, D], with x = L/(N x, y = D/(N y. We also discretize the time interval [, t f ] using NT equally distributed points and t = t f /(NT. Next we define vectors of unknown variables of dimension n xy = N x N y containing approximate solutions such as u(t n u(x i, y j, t n, v(t n v(x i, y j, t n, φ(t n φ(x i, y j, t n R nxy, i =,,.., N x, j =,,.., N y, n =,,..NT. Then Gustafsson s nonlinear ADI finite difference shallow water equations scheme (ADI FD SWE is defined by I. First step - get solution at t n+ u(t n+ + t ( F v(t n+ + t ( F φ(t n+ + t ( F 3 u(t n+ u(t n+ u(t n+, φ(t n+ = u(t n t ( F u(t n, v(t n +, v(t n+ t + t [f, f,.., f N x ] T v(t n, [f, f,.., f t F, φ(t n+ = φ(t n t F 3 ] T u(t n+ = v(t n N ( x v(t n, φ(t n, ( v(t n, φ(t n, ( with * denoting MATLAB componentwise multiplication, f is a N y -dimensional vector storing the Coriolis components f(y j, j =,,..N y and the nonlinear functions F, F, F, F, F 3, F 3 : R nxy R nxy R nxy are defined as follows F (u, φ = u A x u + φ A xφ, 6

7 F (u, v = v A y u, F (u, v = u A x v, F (v, φ = v A y v + φ A yφ, F 3 (u, φ = φ A xu + u A x φ, F 3 (v, φ = φ A yv + v A y φ, where A x, A y R nxy nxy are constant coefficient matrices for discrete first-order and second-order differential operators which take into account the boundary conditions. II. Second step - get solution at t n+ u(t n+ + t ( F u(t n+, v(t n+ v(t n+ + t ( F v(t n, φ(t n φ(t n+ + t ( F 3 v(t n+, φ(t n+ t t F = v(t n+ t t [f, f,.., f] T v(t n+ = u(t n+ N ( x u(t n+, φ(t n+, ( F u(t n+, v(t n+ [f, f,.., f N x ] T u(t n+ = φ(t n+ t F 3 (, u(t n+, φ(t n+. (3 The nonlinear systems of algebraic equations ( and (3 are solved using the quasi- Newton method. Thereby we rewrite ( and (3 in the form g(α = where α is the vector of unknowns. Due to the fact that no more than two variables are coupled to each other on the left-hand side of equations ( and (3, we first solve system ( for u = [u, u,..., u nxy ] and φ = [φ, φ,..., φ nxy ] i.e. the first and the third equations 7

8 in ( and define α = (u, φ, u, φ,..., u nxy, φ nxy R nxy. The iterative Newton method is given by α (m+ = α (m J (α (m g(α (m, where the superscript denotes the iteration and J R nxy nxy 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. J g 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 g is obtained from U(J 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. Because the backsubstitution is a fast operation, the quasi-newton method is computationally efficient especially when the number of nonlinear iterations at each time step is small. Gustafsson proved in [] that even one quasi-newton iteration is sufficient at each time step. 8

9 The quasi-newton formula is α (m+ = α (m Ĵ (α (m g(α (m, where Ĵ = J(α ( + O( t. 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. For our numerical experiments we took M = 6. The second part of the system (, the second equation in ( is solved for v = [v, v,..., v nxy ] by employing the same quasi-newton method. Thus α is defined as α = (v, v, v nxy R nxy. In order to obtain the SWE numerical solution at t n+ we applied the same quasi- Newton technique for system (3. This time the coupled variables were α = (v, φ, v, φ,..., v nxy, φ nxy R nxy for the second and third equation in (3, while u was solved from the remaining equation. 3. 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 us denote by Y = [u, u,..., u NT ] R nxy NT an ensemble of NT time instances of the numerical solution obtained from ADI FD SWE scheme at t,t,..,t NT for the horizontal component of the velocity. Due 9

10 to possible linear dependence, the snapshots themselves are not appropriate as a basis. Instead three methods can be employed, singular value decomposition (SVD for Y R nxy NT, eigenvalues decomposition for Y Y T R nxy nxy or eigenvalue decomposition for Y T NT NT Y R (see [] and [3] and the leading generalized eigenfunctions are chosen as a basis, referred to as POD basis. Error estimates for proper orthogonal decomposition models for nonlinear dynamical systems may be found in [4]. In our case, taking into consideration that NT n xy, we choose to construct the POD basis U R nxy k, k N by solving the eigenvalue problem Y T Y û i = λ i û i, i =,,.., 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=, u i = λ i Y û i. 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 follows u(t n Uũ(t n, v(t n V ṽ(t n, φ(t n Φ φ(t n, ũ(t n, ṽ(t n, φ(t n R k, n =,,.., NT. The POD reduced-order system is constructed by applying the Galerkin projection method to ADI FD SWE discrete model ( and (3 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. is The resulting POD reduced system for the first step t n+ of the ADI FD SWE scheme

11 ũ(t n+ + t ( U T F ṽ(t n+ + t ( V T F φ(t n+ + t ( ΦT F3 ũ(t n+ ũ(t n+ ũ(t n+, φ(t n+, ṽ(t n+ + t V T, φ(t n+ = ũ(t n t ( U T F ũ(t n, ṽ(t n + t ( U T [f, f,.., f] T V ṽ(t n, N ( x [f, f,.., f] T Uũ(t n+ N x = ṽ(t n t ( V T F ṽ(t n, φ(t n, = φ(t n t ( ΦT F3 ṽ(t n, φ(t n, (4 where F, F, F, F, F 3, F 3 : R k R k R k are defined by F (ũ, φ = (Uũ (A x U ũ + (Φ φ (A x Φ φ, F (ũ, ṽ = (V ṽ (A y U ũ, F (ũ, ṽ = (Uũ (A x V ṽ, F (ṽ, φ = (V ṽ (A y V ṽ + (Φ φ (A y Φ φ, (5 below F 3 (ũ, φ = (Φ φ (A x U ũ + (Uũ (A xφ φ, F 3 (ṽ, φ = (Φ φ (A y V ṽ + (V ṽ (A yφ φ. The second step of the POD reduced system for the ADI FD SWE scheme is depicted

12 ũ(t n+ + t ( U T F ũ(t n+, ṽ(t n+ t ( U T ṽ(t n+ + t ( V T F ṽ(t n+, φ(t n+ φ(t n+ + t ( ΦT F3 ṽ(t n+, φ(t n+ ( t U T F = ṽ(t n+ t V T ] T V ṽ(t n+ N x [f, f,.., f ũ(t n+, φ(t n+ ( t V T F ( [f, f,.., f ũ(t n+ ] T Uũ(t n+ N x ( = φ(t n+ t ΦT F3 ũ(t n+ = ũ(t n+, ṽ(t n+,, φ(t n+, (6 The initial conditions are obtain by multiplying the following three equations with U T, V T, Φ T u(t Uũ(t, v(t V ṽ(t, φ(t Φ φ(t. We get ũ(t U T u(t, ṽ(t V T v(t, φ(t Φ T φ(t. Next we define A, A R nxy k such as A (:, i = [f, f,.., f] T V (:, i, A (:, i = [f, f,.., f] T U(:, i, i =,.., k N x N x and the linear terms in (4 and (6, U ([f, T f,.., f N x can be rewritten as U T A ṽ and V T A ũ respectively. ] T V ṽ and V ([f, T f,.., f N x ] T Uũ The coefficient matrices U T A, 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 in (5 can be precomputed, saved and re-used in all time steps of the interval of integration [, t f ]. However, performing the componentwise multiplications in (5 and computing the projected nonlinear

13 terms in (4 and (6 U T k n xy F (ũ, }{{ φ, U T F (ũ, ṽ, V T F (ũ, ṽ, } n xy V T F (ṽ, φ, Φ T F3 (ũ, φ, Φ T F3 (ṽ, φ, (7 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. If we denote the complexity for evaluating the nonlinear function F by α(n xy, then the complexity for computing U T F (ũ, φ is approximately O(α(n xy + 4n xy k. By employing the discrete empirical interpolation method we aim to remove this dependency and regain the full model reduction expected from the POD model.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 algorithm described in the next section must be evaluated at each time step. DEIM approximation is applied to each of the nonlinear functions F, F, F, F, F 3, F 3 defined in (5. 4. The POD/DEIM method and its application to the ADI/SWE model 4.. Discrete Empirical Interpolation Method DEIM is a discrete variation of the Empirical Interpolation method (EIM proposed by Barrault et al. [5]. The application was suggested and analyzed by Chaturantabut and Sorensen in [5], [], []. In [5], authors present an error estimate of the POD/DEIM method. This discrete empirical interpolation method provides an efficient way to approximate nonlinear functions. It was also incorporated into the reduced-basis techniques to 3

14 provide a better reduced-basis treatment (in terms of CPU time of nonaffine and nonlinear parameterized PDEs. DEIM was succesfully applied in conjunction with POD for models governing the voltage dynamics of neurons in [6], the integrated circuits with semiconductors with modified nodal analysis and drift diffusion (see [7] and dynamics of the concentration of lithium ions in lithium ion batteries in [8]. In order to improve the stability of POD/DEIM reduced order schemes in case of a nonlinear transmission line, a micromachined switch and a nonlinear thermal model for a RF amplifier a few modifications to the DEIM based model reduction were proposed by Hochman et al. in [9]. Next we describe the DEIM approximation procedure applied to a nonlinear function. Let f : D R n, D R n be a nonlinear function. If U = {u l } m l=, u i R n, i =,.., 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= is given by f(τ Uc(τ, U R n m, c(τ R m. (8 The basis U can be constructed effectively by applying the POD method on the nonlinear snapshots f(τ t i, τ t i D ( τ may be a function defined from [, T ] D, and τ t i is the value of τ evaluated at t i, i =,.., n s, n s >. Next, interpolation is used to determine the coefficient vector c(τ by selecting m rows ρ,.., ρ m, ρ i N, of the overdetermined linear system (8 to form a m by m linear system P T Uc(τ = P T f(τ, where P = [e ρ,.., e ρm ] R n m, e ρi = [,..,,,.., ] T R n. The DEIM approximation of f R n ρ i becomes f(τ U(P T U P T f(τ. 4

15 Now the only unknowns that need to be specified are the indices ρ, ρ,..., ρ m or the matrix P whose dimensions are n m. These are determined by the following pseudo - algorithm DEIM: Algorithm for Interpolation Indices INPUT: {u l } m l= Rn (linearly independent: OUTPUT: ρ = [ρ,.., ρ m ] N m (A. [ ψ ρ ] = max u, ψ R and ρ is the component position of the largest absolute value of u, with the smallest index taken in case of a tie. (A. U = [u ] R n, P = [e ρ ] R n, ρ = [ρ ] N. (A3. For l =,.., m do a. Solve (P T Uc = P T u l for c R l ; U, P R n (l. b. r = u l Uc, r R n. c. [ ψ ρ l ] = max{ r }. d. U [U u l ], P [P e ρl ], ρ ρ ρ l. (A4. end for. The DEIM procedure inductively constructs a set of indices from a linearly independent set. An error analysis in [] shows that the POD basis is a suitable choice for this algorithm and the order of the input basis {u l } m l= Rn according to the dominant singular values must be utilized. Initially the algorithm searches for the largest value of the first POD basis u and the corresponding index represents the first DEIM interpolation index ρ {,,.., n}. The remaining interpolation indices ρ l, l =, 3.., m are selected so that each of them corresponds to the entry of the largest magnitude of r defined in step (A3 b. The vector r can be viewed as the residual or the error between the input basis u l, l =, 3.., m and its approximation Uc from interpolating the basis {u, u,.., u l } 5

16 at the indices ρ, ρ,.., ρ l. The linear independence of the input basis {u l } m l= guarantees that, in each iteration, r is a nonzero vector and the output indices {ρ i } m i= are not repeating. An error bound for the DEIM approximation is provided in Chaturantabut and Sorensen [5] and []. The following example illustrates the efficiency of DEIM in approximating a highly nonlinear function defined on a discrete D spatial domain. Consider a nonlinear parameterized function s : Ω D R defined by ( s(x; µ = ( xsin πµ(x + e (+xµ, where x Ω = [, ] and µ D = [, π] R. Let [x, x,..., x n ] R n, x i R be a set of equally distributed points in Ω, for i =,,.., n, n =. Then we introduce f : D R n as follows f(µ = [s(x ; µ, s(x ; µ,.., s(x n ; µ T ] R n, µ D We used 5 snapshots f(µ j 5 j= to construct POD basis {u l } m l= with µ j equidistant points in [, π ]. The energy captured by the leading POD modes as a function of the dimension of the POD reduced space and the first corresponding 6 POD basis vectors with the first 6 spatial points selected by the DEIM algorithm are depicted in Figure (. Figure ( illustrates the selection procedure performed by DEIM algorithm where the input set contains the first 6 POD modes. u l, r = u l Uc and U, l =,.., 6 are defined at iteration l in DEIM algorithm. 6

17 logarithmic scale Singular values of 5 Snapshots DEIM points and the first 6 POD basis functions PODbasis PODbasis PODbasis3 PODbasis4 PODbasis5 PODbasis6 DEIM points Exact function Figure : Singular eigenvalues using logarithmic scale and the corresponding first 6 POD basis functions with DEIM points of snapshots, µ =.38 DEIM#.5 DEIM#. DEIM# u current point..5 u r=u Uc current point previous point u 3.6 r=u 3 Uc.8.5. current point previous points DEIM#4 u 4.5 DEIM#5 u 5.3 DEIM#6..5 r=u 4 Uc current point previous points.4.3. r=u 5 Uc current point previous points u 6 r=u 6 Uc current point previous points Figure : The selection process of DEIM interpolation points A comparison between the exact function f and its DEIM approximation is displayed in Figure (3. Once the selected number of DEIM points is larger than 3 the euclidian norm of the error between f and its approximation is of order O( 4. The DEIM and POD approximations errors are similar when the number of DEIM points is (see Figure (4. 7

18 . DEIM# exact DEIM approx. DEIM# exact DEIM approx. DEIM#3 exact DEIM approx DEIM#4 exact DEIM approx.. DEIM#5 exact DEIM approx.. DEIM#6 exact DEIM approx Figure 3: DEIM approximation for different values of m The Exact function and its DEIM approximation for µ=.38. 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 = at µ =.38 (left; Comparison of the spatial errors for POD and DEIM approximations (right 4.. The DEIM SWE model The DEIM approximation presented earlier in this section is used to approximate the nonlinear terms in POD ADI SWE model described in (7 with the nonlinear approximations having computational complexity proportional to the numbered of reduced variables obtained with POD. Let U F R nxy m, m n xy, be the POD basis matrix of rank m for snapshots of the nonlinear function F (obtained from ADI FD SWE scheme. Using the DEIM algorithm we select a set of m DEIM indices corresponding to U F, denoting by [ρ F,.., ρ F m ] T N m, 8

19 and determine the matrix P F R nxy m. The DEIM approximation of F assumes the form F U F (PF T U F m F, so the projected nonlinear term U T F (ũ, φ in the POD reduced system can be approximated as U T F (ũ, φ U T U F (PF T U F E R k m where F m (ũ, φ = P T F F (ũ, φ and E R k m. m F } (ũ, {{ φ, } m Since F is a pointwise function (introduced in (5, F m : R k R k R m can be defined as F m (ũ, φ = (P T F Uũ (P T F A x U ũ + (P T F Φ φ (P T F A x Φ φ. If we denote by U F, U F, U F, U F 3, U F 3 R nxy m the POD bases matrices of rank m for the snapshots of the nonlinear functions F, F, F, F 3, F 3, we obtain in a similar manner the DEIM approximations for the rest of the projected nonlinear terms in (7 U T F (ũ, ṽ U T U F (PF T U F E R k m m F (ũ, ṽ, m V T F (ũ, ṽ V T U F (PF T U F m F (ũ, ṽ, E 3 R k m m V T F (ṽ, φ V T U F (PF T U F E 4 R k m m F } (ṽ, {{ φ, } m Φ T F3 (ũ, φ Φ T U F 3 (PF T 3 U F 3 m F 3(ũ, φ, E 5 R k m m Φ T F3 (ṽ, φ Φ T U F 3 (PF T 3 U F 3 E 6 R k m m F } 3(ṽ, {{ φ, } m 9

20 where E, E 3, E 4, E 5, E 6 R k m and F m (ũ, ṽ = (P T F V ṽ (P T F A y U ũ, F m (ũ, ṽ = (P T F Uũ (P T F A x V ṽ, F m (ṽ, φ = (P T F V ṽ (P T F A y V ṽ + (P T F Φ φ (P T F A y Φ φ, F m 3(ũ, φ = (P T F 3 Φ φ (P T F 3 A x U ũ + (P T F 3 Uũ (P T F 3 A x Φ φ, (9 F m 3(ṽ, φ = (P T F 3 Φ φ (P T F 3 A y V ṽ + (P T F 3 V ṽ (P T F 3 A y Φ φ. Each of the k m coefficient matrices grouped by the curly brackets in (9, as well as E i, i =,,.., 6 can be precomputed and reused at all time steps, so that the computational complexity for each of the approximate nonlinear terms is O(α(m + 4mk, thus not depending on 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+ + t E F m ṽ(t n+ + t E F 3 m φ(t n+ + t E F 5 3 m ( ( ( ũ(t n+ ũ(t n+ ũ(t n+, φ(t n+, ṽ(t n+, φ(t n+ = ũ(t n t ( E F m ũ(t n, ṽ(t n + t U T A ṽ(t n, + t V T A ũ(t n+ = ṽ(t n t ( E F 4 m ṽ(t n, φ(t n, = φ(t n t ( E F 6 3 m ṽ(t n, φ(t n, n =,.., NT, ( while the second step is introduced below ũ(t n+ + t ( E F m ũ(t n+, ṽ(t n+ t U T A ṽ(t n+ = ũ(t n+ t ( E F m ũ(t n+, φ(t n+, ṽ(t n+ + t ( E F 4 m ṽ(t n+, φ(t n+ = ṽ(t n+ t ( E F 3 m ũ(t n+, ṽ(t n+ t V T A ũ(t n+, φ(t n+ + t ( E F 6 3 m ṽ(t n+, φ(t n+ = φ(t n+ t ( E F 5 3 m ũ(t n+, φ(t n+, n =,.., NT. (

21 The initial conditions remain the same as in the case of POD reduced system. The nonlinear algebraic systems (9 and ( as well as (4 and (6 obtained by employing POD/DEIM and POD methods on ADI FD SWE were first splitted into subsystems according to the left-hand side of the equations where no more than two variables are coupled to each other and this was done in the same manner as in the case of Gustafsson ADI FD SWE nonlinear systems. The derived systems were solved using the Newton method. 5. Numerical experiments In this section, we present two main experiments for the two - dimensional shallow water equations model to validate the fesability and efficiency of the POD/DEIM method in comparison with POD technique. For all tests we derived the initial conditions from the initial height condition No. of Grammeltvedt [] i.e. ( h(x, y = H + H + tanh 9 D/ y ( + H sech 9 D/ y sin D D x L, y D. ( πx L, The initial velocity fields were derived from the initial height field using the geostrophic relationship ( g u = f h y, v = ( g h f x. Figure (5 depicts the initial geopotential isolines and the geostrophic wind field. The constants used were L = 6km, D = 44km, ˆf = 4 s, β =.5 s m, g = ms, H = m, H = m, H = 33m. For the first test, the domain was discretized using a mesh of 3 points, with x = y = km. Thus the dimension of the full-order discretized model is 665. The integration time window was 4h and we used 9 time steps (NT = 9 with t = 96s.

22 Contour of geopotential from to 8 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 km calculated from the geopotential field by using the geostrophic approximation (right. ADI FD SWE scheme proposed by Gustafsson in [] was first employed in order to obtain the numerical solution of the SWE model. The implicit scheme allowed us to integrate in time at the following Courant-Friedrichs-Levy (CFL condition gh( t x < The nonlinear algebraic systems of ADI FD SWE scheme were solved with the Quasi- Newton method, were the LU decomposition was performed only once every 6 th time step. The SWE solutions at t = 4h are illustrated in Figure (6. The POD basis functions were constructed using 9 snapshots obtained from the numerical solution of the full - order ADI FD SWE model at equally spaced time steps in the interval [, 4h]. Figure (7 shows the decay around the eigenvalues of the snapshot solutions for u, v, φ and the nonlinear snapshots F, F, F, F, F 3, F 3. The dimension of the POD bases for each variable was taken 35, capturing more than 99.9% of the system energy. 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.

23 Contour of geopotential at time t f = 4h 45 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 km at t = t f = 4h obtained using the ADI FD SWE scheme for t = 96s. Singular Values of Snapshots Solution u v φ Singular Values of Nonlinear Snapshots F F 5 5 F F F 3 F 3 logarithmic scale 5 logarithmic scale 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 for t = 96s. Figure (8 illustrates the distribution of the first 4 spatial points selected from the DEIM algorithm using the POD bases of F 3 and F 3 as inputs. We emphasize the performances of POD/DEIM method in comparison with the POD approach using the numerical solution of the ADI FD SWE model. Figure (9 depicts the grid point local error behaviors between POD, POD/DEIM ADI SWE solutions and ADI FD SWE solutions, where we used 9 DEIM points. 3

24 DEIM POINTS for F 3 DEIM POINTS for F y(km y(km x(km x(km Figure 8: First 4 points selected by DEIM for the nonlinear functions F 3 (left and F 3 (right Using the following norms NT t f i= w ADI F D (:, i w P OD ADI (:, i w ADI F D (:, i, NT t f i= w ADI F D (:, i w P OD/DEIM ADI (:, i w ADI F D (:, i, i =,,.., t f we calculated the global errors for all three variables of SWE model w = (u, v, φ. The results are presented in table (. POD ADI SWE POD/DEIM ADI SWE φ 7.7e-5.6e-4 u 4.95e e-3 v 6.356e e-3 Table : Global errors for each of the model variables. The POD bases dimensions were taken 35 capturing more than 99.9% of the system energy. 9 DEIM points were chosen. Additional to ADI FD SWE scheme we propose the Euler explicit FD SWE scheme as the starting point for a POD, POD/DEIM reduced model. The POD bases were constructed using the same 9 snapshots as in the POD ADI SWE case, only this time the Galerkin projection was applied to the Euler FD SWE model. The DEIM algorithm was used again and the numerical results are depicted in table (. This time we employed the root mean square error calculation in order to compare the POD and POD/DEIM 4

25 POD errors φ POD φ ADI FD x 4 POD/DEIM errors φ DEIM φ ADI FD x 4 POD errors u POD u ADI FD x y(km y(km y(km x(km x(km x(km POD/DEIM errors u DEIM u ADI FD x 4 POD errors v POD v ADI FD x 4 POD/DEIM errors v DEIM v ADI FD x y(km y(km y(km x(km x(km x(km Figure 9: Errors between POD, POD/DEIM ADI SWE solutions and the ADI FD SWE solutions at t = 4h ( t = 96s. The number of DEIM points was taken 9. techniques at time t = 4h. ADI SWE POD ADI SWE POD/DEIM ADI SWE POD EE SWE POD/DEIM EE SWE CPU time seconds φ e e-5.545e-4.79e-4 u -.65e-4.579e-4.98e-4 3.6e-4 v e-5.64e-4.667e-4.37e-4 Table : 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. 9 DEIM points were chosen. Applying DEIM method to POD ADI SWE model we reduced the computational time with a factor of In the case of the explicit scheme the DEIM algorithm decreased CPU time by The POD/DEIM EE SWE model was solved using the Runge-Kutta- Fehlberg method (RKF45. Due to the large number of spatial discretization points, small number of time steps and only one Newton iteration threshold imposed when solving the nonlinear algebraic systems of POD and POD/DEIM ADI SWE schemes made these two 5

26 implicit schemes faster than the POD and POD/DEIM EE SWE explicit schemes. The numerical results showed also that the implicit schemes are slightly more accurate than the explicit ones. Figure ( depicts the efficiency of POD/DEIM methods as a function of spatial discretization points. Once the number of spatial discrete points is larger than the POD/DEIM schemes are faster than the POD schemes by a factor of, for 9 points selected by DEIM algorithm. CPU time (seconds CPU time vs. number of spatial discretization points ADI SWE POD ADI SWE POD/DEIM ADI SWE POD EE SWE POD/DEIM EE SWE No. of spatial discretization points Figure : Cpu time vs. Spatial discretization points; POD DIM = 35, No. DEIM points = 9. Next we propose a second experiment to test the performances of POD/DEIM methods. We increased the number of time steps as well as the number of snapshots used to generate the POD bases. Thus, we took NT = 8 and the number of snapshots n s = 8. The memory burden was too high so we had to decrease the number of spatial discretization points. In consequence we choose N x = 5 and N y =, with x = y = 4km and t = 48s. We solved again the SWE model using the ADI FD SWE scheme in order to generate the 8 snapshots needed for POD and POD/DEIM reduced systems. This time the 6

27 Courant-Friedrichs-Levy (CFL condition was gh( t x <.797. The results are similar with the ones obtained for a CFL condition gh( t x < 7.88 underlying the performance of fully implicit Gustafsson scheme. The geopotential and wind field at final time t f = 4h are depicted in Figure (. Contour of geopotential at time t f = 4h 45 Wind field at time t f = 4h y(km y(km x(km x(km Figure : The geopotential field (left and the wind field (arrows are scaled by a factor of km at t = t f = 4h obtained using the ADI FD SWE scheme for t = 48s. Figure ( shows the decay for the singular values of the snapshot solutions for u, v, φ and the nonlinear snapshots F, F, F, F, F 3, F 3. We noticed that the singular values of F 3 and F 3 are decreasing slowly when comparing with the singular values of the other nonlinear functions. We can improve the accuracy of POD, POD/DEIM solutions and reduce the computational time by taking into account different POD bases dimensions according to the eigenvalues decay. The dimension of the POD bases for each variable was taken 35. Next we apply the DEIM algorithm using as input the POD bases corresponding to the nonlinear functions. The first 4 points selected by the discrete empirical interpolation method for F 3 and F 3 are illustrated in Figure (3. 7

28 5 Singular Values of Snapshots Solution u v φ 5 Singular Values of Nonlinear Snapshots FF FF FF FF FF3 FF3 logarithmic scale 5 logarithmic scale Number of snapshots Number of snapshots Figure : Singular values of the snapshots solutions for u, v, φ and nonlinear functions for t = 48s. 45 DEIM POINTS for FF DEIM POINTS for FF y(km y(km x(km x(km Figure 3: First 4 points selected by DEIM for the nonlinear functions FF3 (left and FF3 (right Next we determine solutions of the POD ADI SWE model and the POD/DEIM ADI SWE model using 8 DEIM points. The solutions of POD/DEIM implicit scheme are very accurate, local errors depicted in Figure (4, global errors in table (3 and RMSE results in table (4 confirm it and showing that POD/DEIM ADI SWE scheme is a much faster and almost as accurate as POD ADI SWE scheme. Comparing with the first experiment we reduced the number of spatial discretization points by a factor of 4. This does not affect the magnitude of the local errors even if they were decreased for both POD and POD/DEIM ADI methods with factors between 3 and 4, when comparing to the results obtained in the first case. The increasing number 8

29 of time steps and snapshots are responsible for improving the accuracy degree of the solutions. This can be observed in figure (5 where we illustrate the local errors for the POD and POD/DEIM ADI SWE solutions using the same configuration as in the second experiment but we decreased the number of time steps and snapshots at 9. POD errors φ POD φ ADI FD x 5 DEIM/POD errors φ DEIM φ ADI FD x 5 POD errors u POD u ADI FD x y(km y(km y(km x(km x(km x(km DEIM/POD errors u DEIM u ADI FD x 4 POD errors v POD v ADI FD x 4 DEIM/POD errors v DEIM v ADI FD x y(km y(km y(km x(km x(km x(km Figure 4: Errors between POD, POD/DEIM ADI SWE solutions and the ADI FD SWE solutions at t = 4h ( t = 48s. The number of DEIM points was taken 8. 9

30 POD errors φ POD φ ADI FD x 4 POD/DEIM errors φ DEIM φ ADI FD x 4 POD errors u POD u ADI FD x y(km y(km y(km x(km x(km x(km 6 POD/DEIM errors u DEIM u ADI FD x 4 POD errors v POD v ADI FD x 4 POD/DEIM errors v DEIM v ADI FD x y(km y(km y(km x(km x(km x(km Figure 5: Errors between POD, POD/DEIM ADI SWE solutions and the ADI FD SWE solutions at t = 4h ( t = 96s. The number of snapshots is 9 and the number of DEIM points was taken 8. Table (3 compares the accuracy of POD and POD/DEIM ADI SWE schemes measuring the global errors of the solutions with respect to the ADI FD SWE solutions. POD ADI SWE DEIM/POD ADI SWE φ.648e e-5 u.79e-3.9e-3 v.7e-3.47e-3 Table 3: Global errors for each of the model variables at t = t f, t = 48s. The POD bases dimensions were taken 35 capturing more than 99.9% of the system energy. 8 DEIM points were chosen. Once again we calculate the solution of SWE model using the POD and POD/DEIM EE SWE schemes. By employing the DEIM method on the POD ADI SWE model we reduce the CPU time with a factor of 3. In the case of explicit scheme the DEIM algorithm decreased the computational time by. Now, the adaptative Runge-Kutta- Fehlberg involved in the explicit reduced order models (ROMs was faster than the Newton method used to solve the nonlinear algebraic systems in implicit ROMs mostly because we doubled the number of time steps and thus the RKF45 didn t need to generate a large 3

31 amount of intermediary time steps as it did in the first experiment in order to generate an accurate solution. From table (4 we notice that the RMSE for both implicit and explicit POD/DEIM schemes are almost similar with the ones generated by the implicit and explicit POD systems with respect to the ADI FD SWE numerical solutions. Putting together the results obtained from experiment and we conclude that the POD and POD/DEIM ADI SWE schemes are more accurate than the POD and POD/DEIM EE SWE schemes. ADI SWE POD ADI SWE DEIM/POD ADI SWE POD EE SWE DEIM/POD EE SWE CPU time φ -.67e-5.743e e e-5 u e e e e-5 v -.397e-5.755e e e-5 Table 4: CPU time gains and the root mean square errors for each of the model variables at t = t f, t = 48s. The POD bases dimensions were taken 35 capturing more than 99.9% of the system energy. 8 DEIM points were chosen. Next we evaluate the efficiency of POD/DEIM method as a function of POD dimension. Figure (6 gives the root mean square errors of φ and the corresponding average CPU times for different dimensions of POD and DEIM approximations. time(seconds 5 5 CPU time FULL POD ADI SWE POD EE SWE DEIM/POD ADI SWE 5 DEIM/POD ADI SWE DEIM/POD EE SWE 5 DEIM/POD EE SWE. x 3 Root mean square error of φ POD ADI SWE POD EE SWE DEIM/POD ADI SWE 5 DEIM/POD ADI SWE DEIM/POD EE SWE 5 DEIM/POD EE SWE POD dimension POD dimension Figure 6: CPU time of the full system, POD reduced systems and POD - DEIM reduced systems (left; Root mean square error of φ calculated for POD/DEIM and POD reduced systems with respect to ADI FD SWE solutions The results in table (5 and (6 show the performances of POD/DEIM method. Once the POD dimension is larger than 5(5 in case of DEIM/POD ADI (EE SWE scheme 3

32 the CPU time is decreased at least by a factor of. When DEIM dimension reached 5, we notice that RMSE results between POD and POD/DEIM are almost identical. All the methods performed well when POD dimension 5 leading to RMSE results of order O( 5. PODDIM POD ADI SWE DEIM/POD ADI SWE5 DEIM/POD ADI SWE POD EE SWE DEIM/POD EE SWE5 DEIM/POD EE SWE Table 5: Comparison between CPU times of POD and DEIM/POD implicit and explicit schemes. The computational time of the full model (ADI FD SWE was PODDIM POD ADI SWE DEIM/POD ADI SWE5 DEIM/POD ADI SWE POD EE SWE DEIM/POD EE SWE5 DEIM/POD EE SWE.8e-3.8e-3.8e-3.8e-3.8e-3.8e e e e e-4 5.e-4 5.7e e-4 5.e e e e e e-4.68e-4.589e-4.584e-4.669e-4.595e-4.9e-4.e-4.e e-5.97e e e e e e e-5 6.8e e-5.898e-5.8e e e e e-5.846e-5.835e e e e-5 Table 6: Comparison between RMSE of POD and DEIM/POD implicit and explicit schemes. Numerical experiments carried out for a -day period showed that the POD/DEIM ADI SWE scheme as well as the POD/DEIM EE SWE discrete model conserve the average height of the free surface and the potential enstrophy while another integral invariant of the SWE model, the total energy, is not preserved largely due to the absence of a staggered C in the numerical discretization. Arakawa in [] showed that when the finite difference Jacobian expression for the advection term is restricted to a form which properly represents the interaction between grid points the computational instability are prevented thereby preserving all the integral invariants. Thus the POD and POD/DEIM systems behave similar as the ADI FD SWE full scheme in the matter of integral invariants and Figure (7 depicts their evolution in time. Tables (7 - (9 present integral invariants measures for all the schemes involved in this study using max min evaluation and Euclidian norm with respect to SWE invariants calculated with ADI FD SWE full scheme. 3

33 x Total Energy ADI SWE POD ADI SWE DEIM/POD ADI SWE POD EE SWE DEIM/POD EE SWE Potential Enstrophy ADI SWE POD ADI SWE DEIM/POD ADI SWE POD EE SWE DEIM/POD EE SWE.5..5 Average Height of the Free Surface ADI SWE POD ADI SWE DEIM/POD ADI SWE POD EE SWE DEIM/POD EE SWE Time Time Time Figure 7: Shallow Water Equations invariants, POD dimension =35, No of DEIM points = ADI SWE POD ADI SWE DEIM/POD ADI SWE POD EE SWE DEIM/POD EE SWE Max-Min Norm Table 7: Average Height of the free surface In Figure (8, the Pearson correlation coefficient defined bellow is used as an additional metric to evaluate the quality of POD/DEIM schemes where σ i = j=n xy j= ( Wi,j W j, σ = r i = covi, i =,.., NT, σσ i i j=n xy j= ( W scheme i,j W scheme j, i =,..., NT, cov = j=n xy j= ( Wi,j W j ( W scheme i,j W scheme j, i =,..., NT, where W = (u, v, φ represents the ADI FD SWE solution and W scheme = (u scheme, v scheme, φ scheme the solution calculated with one of the following schemes: POD ADI SWE, POD/DEIM ADI SWE, POD EE SWE and POD/DEIM EE SWE using 5 and DEIM points. W j and W scheme j are corresponding means over the simulation period [, t f ] at spatial node j. 33

34 ADI SWE POD ADI SWE DEIM/POD ADI SWE POD EE SWE DEIM/POD EE SWE Max-Min Norm Table 8: Potential Enstrophy ADI SWE POD ADI SWE DEIM/POD ADI SWE POD EE SWE DEIM/POD EE SWE Max-Min 5.644e e e e e+6 Norm.e+.489e e e e+6 Table 9: Total Energy.5 Correlation Coefficient of u. Correlation Coefficient of v. Correlation Coefficient of φ POD ADI SWE POD EE SWE DEIM/POD ADI SWE5 DEIM/POD ADI SWE DEIM/POD EE SWE5 DEIM/POD EE SWE Time POD ADI SWE POD EE SWE DEIM/POD ADI SWE5 DEIM/POD ADI SWE DEIM/POD EE SWE5 DEIM/POD EE SWE Time POD ADI SWE POD EE SWE DEIM/POD ADI SWE5 DEIM/POD ADI SWE DEIM/POD EE SWE5 DEIM/POD EE SWE Time Figure 8: Correlation coefficients for the SWE variables, POD dimension = Conclusions To obtain the approximate solution in case of both POD and POD/DEIM reduced systems, one must store POD or POD/DEIM 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 for projected linear terms and O(n xy k for the nonlinear term, where n xy is the space dimension. In the case of solving 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. 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 two different schemes, the ADI FD SWE fully 34

35 implicit model and the Euler explicit FD SWE scheme. We noticed, as we expected, that POD/DEIM CPU time is most sensitive to the number of spatial discretization points. The largest reduction of the CPU time was obtain in first experiment where it was reduced by a factor of 73.9 when using the POD/DEIM ADI SWE scheme while in the case of POD/DEIM EE SWE model we decreased the CPU time with a factor of Also we noticed that the approximation errors of POD/DEIM and POD reduced systems are almost identical once the dimension of DEIM reached 5, for any of the methods used, either explicit or implicit. In the second experiment, we increased the number of time steps and snapshots and consequently the solutions accuracy was higher in comparison with the results obtained in the first experiment. Future research work will address different POD bases dimensions for both the state variables as well the nonlinear functions appearing in the SWE model in order to increase the solution accuracy. Another future research task will be to apply the DEIM technique to different invers problems such as POD 4-D VAR of the limited area finite element shallow water equations and adaptive POD 4-D VAR applied to finite volume shallow water equations. We also plan to compare the Discrete Empirical Interpolation Method with the Best Points Interpolation (BPIM one in proper orthogonal decomposition framework applied to SWE equation. BPIM was proposed by Nguyen et al [3] where the interpolation points are defined as a solution of a least-squares minimization problem. Thus, BPIM replaces the greedy algorithm used in Empirical Interpolation Method(EIM by an optimization problem which provides higher accuracy at the cost of greater computational complexity. In [4], Galbally et al. made a comparison between gappy POD, EIM and BPIM techniques for a nonlinear combustion problem governed by an advection diffusion PDE. 35

36 Acknowledgments Prof. I.M. Navon acknowledges the support of NSF grant ATM References [] B. Gustafsson, An alternating direction implicit method for solving the shallow water equations, Journal of Computational Physics 7( [] G. Fairweather, I.M. Navon, A linear ADI method for the shallow water equations, Journal of Computational Physics 37(98 8. [3] 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 ( ( [4] H.O. Kreiss, O.B. Widlund, Difference approximations for initial-value problems for partial differential equations, Department of Computer Sciences, Report NR 7, Upsala University, 967. [5] S. Chaturantabut, D.C. Sorensen, A state space error estimate for POD-DEIM nonlinear model reduction, SIAM Journal on Numerical Analysis 5( ( [6] D.A. Randall, Geostrophic adjustment and the finite-difference shallow-water equations, Monthly Weather Review ( [7] J.Jr. Douglas, J.E. Gunn, A general formulation of alternating-direction methods, Numer. Math. 6( [8] N.N. Yanenko, The method of fractional steps, Sringer-Verlag, Berlin,

37 [9] G.I. Marchuk, Numerical solution of the problems of dynamics of atmosphere and oceans, Gidrometeoizdat, Leningrad, 974. [] S. Chaturantabut, D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM Journal on Scientific Computing, 3(5 ( [] 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, 7(4 ( [] G. Dimitriu, N. Apreutesei, Comparative Study with Data Assimilation Experiments Using Proper Orthogonal Decomposition Method, Lecture Notes in Computer Science 488( [3] G. Dimitriu, N. Apreutesei, R. Stefănescu, Numerical Simulations with Data Assimilation Using an Adaptive POD Procedure, Lecture Notes in Computer Science 59( [4] M. Hinze, S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control, in: P. Benner, V. Mehrmann, D. Sorensen (Eds., Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational and Applied Mathematics, 5, pp [5] 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 ( [6] A. R. Kellems, S. Chaturantabut, D.C. Sorensen, S.J. Cox, Morphologically accurate reduced order modeling of spiking neurons, Journal of Computational Neuroscience, 8(3 (,