Energy Stable Model Order Reduction for the Allen-Cahn Equation

Transcription

1 Energy Stable Model Order Reduction for the Allen-Cahn Equation Bülent Karasözen joint work with Murat Uzunca & Tugba Küçükseyhan Institute of Applied Mathematics & Department of Mathematics, METU, Ankara MODRED 2017, January

2 Outline 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

3 Conservative & Dissipative PDEs 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

4 Conservative & Dissipative PDEs Evolutionary PDEs u t = D δh, (x,t) Ω (0,T ], δu H = (H(x;u,u x,u xx,...)dx Ω Variational derivative: δh δu = H ( ) ( ) H H u x x 2 u x u xx

5 Conservative & Dissipative PDEs Conservative PDEs D = S constant skew-adjoint operator in B = L 2 (Ω). us udx = 0, B Ω H Hamiltonian (energy) conserving system dh δh = dt δu S δh δu dx = 0 Nonlinear Schrödinder (NLS) equation ( ) ( u 0 i u = i 0 t H = Ω Ω )( δh δu δh δu ( u 2 + γ 2 u 4) dx, S = ) ( 0 i i 0 Korteweg de Vries equation, Sine-Gordon equation, Maxwell equation )

6 Conservative & Dissipative PDEs Dissipative PDEs D = N constant negative (semi) definite operator un udx 0, B Ω H Lyapunov function, energy dissipating system H δh = Ω δu N δh δu dx 0 Allen-Cahn equation u = ε u + u u3 t ( ε H = Ω 2 u u2 + 1 ) 4 u4 dx, N = 1 Energy is monotonically decreasing H (u(t n )) < H (u(t m )), t n > t m Cahn-Hilliard Equation, Ginzburg-Landau equation

7 Conservative & Dissipative PDEs Fitzhugh-Nagumo equation τ 1 u t τ 2 v t = d 1 u + f (u,v) = d 2 v + g(u,v) Skew-gradient structure ( ) f (u,v) = v u τ 1 ( g(u,v) τ 2 ). ( d1 E(u,v) = Ω 2 u 2 d ) 2 2 v 2 + F(u,v) dx with the potential function F(u,v) = u4 4 + (1 + β)u3 3 βu2 2 uv + γv2 2 εv

8 Discontinuous Galerkin discretization 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

9 Discontinuous Galerkin discretization Symmetric interior penalty Galerkin (SIPG) ξ h = {K} is a partition of the computational domain Ω into a family of triangles. P k (K) is the space of polynomials of order k on the element K. The finite dimensional solution and test function space W h = W h (ξ h ) = { w L 2 (Ω) : w K P k (K), K ξ h }, where w W h is discontinuous along the inter-element boundaries. B. Rivière. Discontinuous Galerkin methods for solving elliptic and parabolic equations, Theory and implementation. SIAM, 2008.

10 Discontinuous Galerkin discretization The jump and average operators of a function w W h on an interior edge e K i K j : [[w]] := w Ki n Ki + w Kj n Kj, {{w}} := 1 2 (w K i + w Kj ), with [[w]] := w K n and {{w}} := w K on a boundary edge e Ω. K 2 Ω K 1 K 1 nk n e e The sets of inflow and outflow edges: Γ = {z Ω : b(v) n(v,x) < 0}, Γ + = Ω \ Γ, K = {z K : b(v) n K (v,x) < 0}, K + = K \ K, where n K is the outward unit vector on K.

11 Discontinuous Galerkin discretization Symmetric interior penalty Galerkin (SIPG) discretization Symmetric interior penalty Galerkin (SIPG) discretization ( t u h (µ),υ h ) Ω +a h (µ;u h (µ),υ h )+( f (u h (µ); µ),υ h ) Ω = 0, υ h V h, with the SIPG bilinear form a h (µ;u,υ) = ε u υ {ε u}[υ]ds K T K h E Eh 0 E σε {ε υ}[u] + [u][υ]ds, E Eh 0 E h E Eh 0 E E

12 Discontinuous Galerkin discretization Full order model u h (x,t) = n e n q i=1 j=1 u i j(t)ϕ i j(x) u ϕ := u(t) = (u 1 1 (t),u1 2 (t),...,un e n q (t)) T := (u 1 (t),u 2 (t),...,u N (t)) T := ϕ(x) = (ϕ 1 1 (x),ϕ1 2 (x),...,ϕn e n q (x)) T := (ϕ 1 (x),ϕ 2 (x),...,ϕ N (x)) T N = n e n q the dg degrees of freedom (DoFs) Mu t = Au f(u) M R N N is the mass matrix A R N N is the stiffness matrix

13 Energy preserving time discretization Average vector filed method Energy stable (conserving or dissipating) average vector field (AVF) method for ODEs ẏ = f(u) u n = u n 1 + t 1 0 f(τu n + (1 τ)u n 1 )dτ. Mu n+1 = Mu n t 1 2 A(un+1 +u n ) t f(τu n+1 +(1 τ)u n )dτ 0 The nonlinear function is approximated by symmetric Gauss-Legendre quadrature s i=1 b i = 1 2 f(τu n+1 + (1 τ)u n )dτ s i=1 b i f (c i u n+1 + (1 c i )u n ) AVF method coincides with the mid-point rule for quadratic f(u). Celledoni, E., Grimm, V., McLachlan, R., McLaren, D., ONeale, D., Owren, B., Quispel, G.: Preserving energy resp. dissipation in numerical PDEs using the average vector field method. Journal of Computational Physics 231,

14 Energy preserving time discretization Energy stability of the full order model The SIPG discretized energy function of the continuous energy E h (u n h ) = ε 2 un h 2 L 2 (Ω) + (F(un h ),1) Ω + E E 0 h ( ({ε u n h },[un h ]) E + σε 2h E ([u n h ],[un h ]) E E h (u n+1 h ) E h (u n h ) = 1 t un+1 h u n h 2 L 2 (Ω) 0, For Allen-Cahn equation: E h (u n+1 h ) E h (u n h ) ), For NLS equation: E h (u n h ) = E h(u 0 h )

15 POD & Discrete empirical interpolation method(deim) 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

16 POD & Discrete empirical interpolation method(deim) POD M-orthogonal reduced basis functions {ψ w,i }, i = 1,2,...,k J 1 min ψ w,1,...,ψ w,k J w j k 2 (w j,ψ w,i ) L 2 (Ω)ψ w,i j=1 i=1 L 2 (Ω) subject to (ψ w,i,ψ w, j ) L 2 (Ω) = Ψ T w,,imψ w,, j = δ i j, 1 i, j k, The minimization problem is equivalent to the eigenvalue problem U U T MΨ u,,i = σ 2 u,iψ u,,i i = 1,2,...,k for Ψ u,,i the coefficient vectors of the POD basis functions ψ u,i. Selection of POD modes: energy criteria E = l i=1 σ 2 i d i=1 σ 2 i

17 POD & Discrete empirical interpolation method(deim) Reduced order model The ROM solution u h,r (x,t) of dimension N N by POD u h (x,t) u h,r (x,t) = N i=1 SIPG weak formulation for ROM u i,r (t)ψ i (x), (ψ i (x),ψ j (x)) Ω = δ i j, ( t u h,r,υ h,r ) Ω + a h (u h,r,υ h,r ) + ( f (u h,r ),υ h,r ) Ω = 0, υ h,r V h,r ψ i (x) = N Ψ j,i ϕ j (x), Ψ T,iMΨ, j = δ i j. j=1 Ψ = [Ψ,1,...,Ψ,N ] R N N. u = Ψu r. The reduced semi-discrete ODE system: t u r + A r u r + f r (u r ) = 0, Reduced stiffness matrix A r = Ψ T AΨ Reduced non-linear vector f r (u r ) = Ψ T f(ψu r ).

18 POD & Discrete empirical interpolation method(deim) Discrete empirical interpolation (DEIM) Computation of f r (u r ) = Ψ T f(ψu r ) R N depends on the dimension N of the full system Use an approximation f r (u r ) = }{{} Ψ T N N f(ψu r (t)) }{{} N 1 f Ws, W R N m,c R m It is over determined. Find a projection P such that: P = [e 1,...,e m ] R N m with e i is the i -th column of the identity matrix I R N N

19 POD & Discrete empirical interpolation method(deim) DEIM Algorithm To find permutation matrix P and indices = [ 1,..., m ] T Algorithm 1 The DEIM Algorithm INPUT: {W i } m i=1 RN OUTPUT: = [ 1,..., m ] T R m, P R N m [ ρ, 1 ] = max{ W 1 } W = [W 1 ],P = [e 1 ], = [ 1 ] for i = 2 to m do Solve (P T W)c = P T W i for c r = W i Wc [ ρ, i ] = max{ r } [ ] W [W W i ],P [P e i ], end for S. Chaturantabut and D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM Journal on Scientific Computing 32 (5), , i

20 POD & Discrete empirical interpolation method(deim) Finding the basis W To find (POD) basis functions W = [W 1,...,W m ] R N m : Set the snapshot matrix F := [f 1,...,f J ] R N J, where f i := f(ψu r (t i )) Set {W i } m i=1 as the first m N left singular vectors {W i} m i=1 through the SVD of F F = W ΣZ T

21 POD & Discrete empirical interpolation method(deim) Approximating the nonlinearity After finding P and = [ 1,..., m ] T P T f = (P T W)s f Ws = W(P T W) 1 P T f f r (u r ) = Ψ T f Qf

22 POD & Discrete empirical interpolation method(deim) Approximating the nonlinearity After finding P and = [ 1,..., m ] T P T f = (P T W)s f Ws = W(P T W) 1 P T f f r (u r ) = Ψ T f Qf where Q = (Ψ T W (P T W) }{{} m m 1 ) }{{} N m (Precomputable), f = P T f(ψu r (t)) }{{} m 1

23 POD & Discrete empirical interpolation method(deim) A priori error bound DEIM approximation satisfies the a priori error bound f W(P T W) 1 f 2 (P T W) 1 2 (I WW T )f 2, where the term (P T W) 1 2 is of moderate size of order 100 or less. H. Antil, M. Heinkenschloss, and D. C. Sorensen, Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear and Parametric Systems, A. Quarteroni and G. Rozzas (eds.), Reduced Order Methods for Modeling and Computational Reduction, Model. Simul.& Appl. Vol. 9, 2014, pp ,

24 POD & Discrete empirical interpolation method(deim) Degrees of freedoms with linear dg basis Classical FEM DG H. Antil, M. Heinkenschloss, and D. C. Sorensen, Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear and Parametric Systems, A. Quarteroni and G. Rozzas (eds.), Reduced Order Methods for Modeling and Computational Reduction, Model. Simul.& Appl. Vol. 9, 2014, pp , Springer Italia, Milan.

25 POD & Discrete empirical interpolation method(deim) Reduced Jacobian evaluation The reduced Jacobian arising from DEIM u r f r (Ψu r ) Q(P T J f )Ψ. For m = 4, 2 = 6 and 3 = 2, and linear basis are used (N loc = 3): f f 1 P T f = = f 6 f 2, f N f 4 P T J f = f f 6 f 6 u 4 u 5 u f 2 f 2 f 2 u 1 u 2 u ,

26 Dynamic mode decomposition (DMD) 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

27 of Sciences, 17(5): , 1931 Dynamic mode decomposition (DMD) Dynamic mode decomposition Equation-free, snapshot based method. Arnoldi-like algorithm based on the Koopman analysis. Modes and eigenvalues represents the temporal dynamics inherent in the data with oscillatory components. The modes are not necessarily orthogonal. DMD modes have a single-frequency content contrary to the POD modes that possess multi temporal frequency. Optimality in POD Optimality in DMD Energy Criteria? B. O. Koopman. Hamiltonian systems and transformation in Hilbert space. Proceedings of the National Academy

28 Dynamic mode decomposition (DMD) For the dynamic system specified by x k+1 = g(x k ) on a manifold M, the Koopman operator maps any function f : M R into A f (x k ) = f (g(x k )). Koopman operator A linear, infinite dimensional operator The function f (x k ) can then be expressed as: f (x k ) = α j (x 0 )φ j e (ω j+iσ j )k, j=1 φ j :Koopman modes, w j : growth rates and σ j : frequencies of eigenvalues of A. Dynamic mode decomposition method approximates the Koopman operator with a finite dimensional linear operator.

29 Dynamic mode decomposition (DMD) Snapshot matrix: S = [u 1,...,u J ] in R N J with S 0 = [u 1,...,u J 1 ], S 1 = [u 2,...,u J ]. Assume that A R N N a linear operator with such that S 1 A S 0, A minimizes the Frobenious norm S 1 A S 0 F. Problem: When N is large, it is hard to compute the matrix A.

30 Dynamic mode decomposition (DMD) SVD of S 0 with rank(s 0 ) = k: S 0 = UΣV, where, U R N k and V R J 1 k are orthogonal matrices, Σ R k k is the diagonal matrix. POD projection of A : à = U AU = U S 1 (UΣV ) 1 U, = U S 1 V Σ 1. where à R k k with k < N. Eigenvectors of A: Φ DMD = S 1 Σ 1 VW, where W is the eigenvector of Ã. Φ DMD = DMD modes.

31 Dynamic mode decomposition (DMD) Exact DMD Algorithm 2 Exact DMD Algorithm Input: Snapshots S = [u 1,...,u J ] with u i = u ti. Output: DMD modes Φ DMD. 1: Define the matrices S 0 = [u 1,...,u J 1 ] and S 1 = [u 2,...,u J ]. 2: SVD of S 0 : S 0 = UΣV. 3: Ã = U S 1 Σ 1 V. 4: Eigenvalues and eigenvectors of ÃW = ΛW 5: Φ DMD = S 1 Σ 1 V W. J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, J. N. Kutz. On dynamic mode decomposition: theory and applications. J. Comput. Dyn., 1(2): , 2014

32 Dynamic mode decomposition (DMD) Equivalently, α 1 α 2 D α = S = [u 1,...,u J ] Φ DMD D α V and,... αk 1 α k where γ i = exp(ω i t), i = 1,2,...,k.,V and = 1 γ 1 γ1 J 1 1 γ 2 γ2 J γ k 1 γ J 1 k 1 1 γ k γ J 1 k

33 Dynamic mode decomposition (DMD) Optimal mode amplitudes Aim: To identify the optimal mode amplitudes α = [α 1 (0),...,α k (0)] with respect to the minimization problem: min α S ΦDMD D α V and 2 F. Let P = (W W) (V and V and ), q = diag(v andv Σ W). where denotes the conjugate transpose, denotes the elementwise multiplication. Then, α opt = P 1 q. M. R. Jovanovič, P. J. Schmid, J. W. Nichols, Physics of Fluids 26, , 2014

34 Dynamic mode decomposition (DMD) DMD approximation of the nonlinearity Reduced approximation of the nonlinearity in terms of DMD modes POD-DMD reduced system f(t,u) Φ DMD (diag(e ωdmdt )b M r u r = A r u r (Φ POD ) T Φ DMD diag(e ωdmdt )b u(0) = u r 0 b = Φ f 1 Φ : Moore-Penrose pseudo inverse of Φ DMD. Alessandro Alla and J. Nathan Kutz. Nonlinear model order reduction via dynamic mode decomposition. arxiv: , 2016.

35 Allen-Cahn equation 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

36 Allen-Cahn equation 2D Allen-Cahn equation with quartic potential u t = ε u + f (u) Cubic bi-stable nonlinearity f (u) = u 3 u Homogenous Neumann boundary conditions Ω = [0,0.1] 2 [0,0.025], t = , x = initial condition u(x,0) = tanh[(0.25 Parameter ε = 0.02 (x 1 0.5) 2 + (x 2 0.5) 2 )/( 2ε] Allen, M.S., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica 27(6), , 1979

37 Allen-Cahn equation 10 0 u f Figure 1: Decay of the singular values for u and the nonlinearity f.

38 Allen-Cahn equation Figure 2: Plots of errors between FOM and ROM solutions

39 Allen-Cahn equation Comparison POD-DEIM-DMD Energy FOM 3 POD 3 POD, 5 DEIM 3 POD, 5 DMD CPU (seconds) t 0 Full POD DEIM DMD Figure 3: Energy plots (left) and CPU times (right)

40 Allen-Cahn equation Table 1: Energy errors between FOM and ROMs, and Speed-Up Factors ROM # Modes Energy-Error Speed-Up Factor POD e DEIM e DMD e

41 Allen-Cahn equation L 2 Error u 10 0 POD POD-DEIM 10-5 POD-DMD Number of modes CPU-time POD POD-DEIM POD-DMD Full Number of modes Figure 4: L 2 L 2 FOM-ROM (left) and CPU times (right) with the same increasing number of POD, DEIM and DMD basis functions.

42 Diffusive FitzHugh-Nagumo Equation 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

43 Diffusive FitzHugh-Nagumo Equation Diffusive FHN u t = d 1 u u 3 + u v + κ, v t = d 2 v v + u space-time domain Q = [ 1,1] 2 [0,100]. Random initial conditions between -1 and 1 and zero flux boundary conditions x = y = and temporal step-size t = 0.1. Parameters d 1 = , d 2 = 0.005, κ = 0 TT Marquez-Lago, P. Padilla, A selection criterion for patterns in reaction diffusion systems. Theoretical Biology and Medical Modelling 11:7, 20167, 2014

44 Diffusive FitzHugh-Nagumo Equation Figure 5: Labyrinthic patterns at T = 100 for POD, DEIM, DMD

45 Diffusive FitzHugh-Nagumo Equation 10 5 u v 10 0 f Energy -0.4 FOM POD 20 POD, 30 DEIM 20 POD, 70 DMD t

46 Diffusive FitzHugh-Nagumo Equation ROM #POD #DEIM #DMD L 2 Error u L 2 Error v Speed-up POD e e DEIM e e DMD e e Table 2: FOM-ROM errors & speed-up

47 Diffusive FitzHugh-Nagumo Equation L 2 Error u POD POD-DEIM POD-DMD Number of modes Figure 7: L 2 L 2 FOM-ROM errors

48 Diffusive FitzHugh-Nagumo Equation CPU-time 10 1 POD POD-DEIM POD-DMD 10 0 Full Number of modes Figure 8: CPU times

49 Convective FitzHugh-Nagumo equation 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

50 Convective FitzHugh-Nagumo equation Convective FHN u t = d 1 u V u c 1 u(u c 2 )(u 1) v, v t = d 2 v V v ε(v c 3 u) On th space time cylinder Q = Ω (0,1), where Ω = (0,L) (0,H). For a given V max = 1 4 ah2, the velocity field V (y) = ay(h y) Space-time domain Q = Ω [0,T ] = [ 1,1] 2 [0,100]. Random initial conditions between -1 and 1 and zero flux boundary conditions. x = y = and temporal step-size t = 0.1. Parameters d 1 = , d 2 = 0.005, κ = 0 E. A. Ermakova, E. E. Shnol, M. A. Panteleev, A. A. Butylin, V. Volpert, F. I. Ataullakhanov. On propagation of excitation waves in moving media: The FitzHugh-Nagumo model. PLosOne, 4(2):E4454, 2009

51 Convective FitzHugh-Nagumo equation Figure 9: Standing waves at T = 100, POD, DEIM, DMD

52 Convective FitzHugh-Nagumo equation 10 5 u v 10 0 f Figure 10: Decay of the singular values

53 Convective FitzHugh-Nagumo equation ROM #POD #DEIM #DMD L 2 Error u L 2 Error v Speed-up POD e e DEIM e e DMD e e Table 3: FOM-ROM errors & speed-up

54 10 1 POD Convective FitzHugh-Nagumo equation 10 0 POD-DEIM POD-DMD 10-1 L 2 Error u Number of modes Figure 11: L 2 L 2 FOM-ROM errors

55 Convective FitzHugh-Nagumo equation CPU-time POD POD-DEIM POD-DMD 10-1 Full Number of modes Figure 12: CPU times

56 Nonlinear Schrödinger Equation (NLS) 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

57 Nonlinear Schrödinger Equation (NLS) Nonlinear Schrödinger iu t + u + 2 u 2 u = 0 in [0,2π] 2 (0,1] with periodic boundary conditions and with the exact solution u(x,y,t) = Aexp(i(c 1 x + c 2 y ωt)) ω = c c2 2 c A 2. A = c 1 = c 2 = 1. temporal and spatial mesh sizes t = 0.01, x = y = π/ Yan Xu and Chi-Wang Shu. Local discontinuous Galerkin methods for nonlinear Schrödinger equations. Journal of Computational Physics, 205(1):72 97, 2005

58 Nonlinear Schrödinger Equation (NLS) 10 5 Real p Imaginary q 10 0 (p 2 +q 2 ) 2 q (p 2 +q 2 ) 2 p Figure 13: Singular values for the solutions and nonlinearities

59 Nonlinear Schrödinger Equation (NLS) Figure 14: Solution profiles at the final time T =10

60 Nonlinear Schrödinger Equation (NLS) Figure 15: FOM-ROM errors at the final time

61 Nonlinear Schrödinger Equation (NLS) 1 FOM POD Energy Error Energy Error t 4 POD, 4 DEIM t 4 POD, 4 DMD 0.35 Energy Error Energy Error t t Figure 16: Energy errors Ω u 2 + u 4 dω

62 Nonlinear Schrödinger Equation (NLS) FOM x POD x Mass Error 0 1 Mass Error t 4 POD, 4 DEIM x t 7 x 4 POD, 4 DMD 10 7 Mass Error 1 2 Mass Error t t Figure 17: Mass errors Ω u 2 dω

63 Nonlinear Schrödinger Equation (NLS) 1 FOM POD Moment Error Moment Error t 4 POD, 4 DEIM t 4 POD, 4 DMD 0.3 Moment Error Moment Error t t Figure 18: Momentum errors 1 2 ((uu x u u x )) + (uu y u u y ))

64 Nonlinear Schrödinger Equation (NLS) Table 4: The computation time (in sec) and speed-up factors FOM POD DEIM DMD CPU Speed Up Table 5: Errors between FOM Solution Energy Mass Moment POD 2.16e e e e-03 POD-DEIM 2.17e e e e-03 POD-DMD 1.94e e e e-03

65 Extend & Kernel DMD 1 Conservative & Dissipative PDEs 2 Discontinuous Galerkin discretization 3 Energy preserving time discretization 4 POD & Discrete empirical interpolation method(deim) 5 Dynamic mode decomposition (DMD) 6 Allen-Cahn equation 7 Diffusive FitzHugh-Nagumo Equation 8 Convective FitzHugh-Nagumo equation 9 Nonlinear Schrödinger Equation (NLS) 10 Extend & Kernel DMD

66 Extend & Kernel DMD Koopman operator Discrete time dynamical system x k+1 = F f (x k ) Koopman operator on the observable function g K t g(x k ) = g(f t (x k )) = g(x k+1 ) System dynamics is characterized by K ϕ k = λϕ k g(x) = [ g 1 (x) g 2 (x) g p (x) ] T = k=1ϕ k (x)v k K g(x) = K k=1 ϕ k (x)v k = k=1 K ϕ k (x)v k = k=1 λ k ϕ k (x)v k

67 Extend & Kernel DMD Observables Observables g(x) mapping from the physical space to feature space System dynamics is characterized in the feature space by Koopman eigenvalues λ k, eigenfunctions ϕ k (x), and modes v k Construction of the feature space g(x) Polynomials g j (x) = {x,x 2,x 3,...,x n } Radial basis functions Hermite polynomials Discontinuous spectral elements

68 Extend & Kernel DMD Column size of the data matrix n, number of snapshots m Extended DMD when n >> m Examples for observables for NLS [ x g 1 (x) = x 2 x ], g 2 (x) = Kernel DMD m >> n, SVM based kernels polynomial kernel f (x,x ) = (a + x T x ) p [ x x 2 radial basis functions f (x,x ) = exp( a x x 2 ) sigmoid kernel f (x,x ) = tanh(x T x + a) Perform DMD on the data matrices of the observables ] J. Nathan Kutz and Steven L. Brunton, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, SIAM, 2016.

69 Extend & Kernel DMD From the foreword Oberwolfach Mini-Workshop: Applied Koopmanism, 7 13 February 2016 Poincaré suggested that complicated dynamics governed by non-linear partial differential equations can be reduced to and analyzed with the novel (at that time) linear infinite dimensional spectral methods advocated by Hilbert and Fredholm,...Poincaré s vision became reality a good two decades later by landmark contributions of Carleman, Koopman and von Neumann. Prediction from the data as fourth paradigm of scientific discovery, New Report on Research and Education in Computational Science and Engineering, September 2016 Data-Driven Methods for Reduced-Order Modeling and Stochastic Partial Differential Equations, Banff International Research Station University of British Columbia, January 29 - February 3, 2017.