Dynamical Low-Rank Approximation to the Solution of Wave Equations

Transcription

1 Dynamical Low-Rank Approximation to the Solution of Wave Equations Julia Schweitzer joint work with Marlis Hochbruck INSTITUT FÜR ANGEWANDTE UND NUMERISCHE MATHEMATIK 1 KIT Universität des Landes Baden-Württemberg und nationales Forschungszentrum in der Helmholtz-Gemeinschaft

2 Outline Motivation Model Adaptive Rank Numerical Results Conclusion 2

3 Example in 2d a tt a + g( a 2 )a = 0 2d simulation in cartesian coordinates (Gautschi type method including QEA...) Ch. Karle 3

4 10 2 t = 0 Properties of the solution spatial discretization on uniform grid large matrix t = 500 t = 1000 t =

5 Low-rank representation A(x, y, t) Y(x, y, t) = U(x, t)s(t)v(y, t) H C m,n n r r n r r m = m U C m,r, V C n,r, S C r,r, U and V with orthonormal columns S invertible A: m n complex entries (example: m = 2048, n = ) U, S and V: m r + n r + r 2 complex entries (example: r = ) 5

6 Best approximation solve full system in each time step after every time step compute singular value decomposition save first r singular values as well as right and left singular vectors Problem: required in each step full matrix singular value decomposition Goal: new system of equations for U, S and V Dynamical Low-Rank approximation 6

7 Physical motivation total energy of a system described by a(t, r) E = 1 2 energy conservation Ω ( at 2 + a 2 + G( a 2 ) G(0) ) dr G (x) = g(x) d dt Ω E = Re ( a t att a + g( a 2 )a ) dr =! 0 energy remains conserved if nonlinear wave equation is fulfilled a tt a + g( a 2 )a =: a tt F(a) = 0 7

8 Formulation on manifolds energy conservation Re a t, a tt F(a) = 0 since wave equation is fulfilled δa, a tt F(a) = 0 δa a weaker model let Y be in a manifold M and Y a let δy be an element of the tangential space T Y M of M Y tt F(Y) = 0 doesn t hold any more, but δy, Y tt F(Y) = 0 δy T Y M 8

9 Choice of the manifold shape functions in physics: Variation of Action Gauß-Hermite wave packets Hagedorn wave packets for first order systems: Lubich, Faou, Gradinaru low-rank matrices for first order systems: Lubich, Koch, Nonnenmacher extension to threedimensional tensors for first order systems Lubich, Koch application to chemical master equation Jahnke 9

10 Our choice of the manifold manifold of rank r matrices M = {Y C m,n : rank(y) = r} = {U C m,r, S C r,r inv., V C n,r : U H U = I r = V H V} S not necessarily diagonal decomposition not unique unique decomposition in tangent space choice of norm on the manifold δy = δusv H + UδSV H + USδV H δu H U = 0 = δv H V relation to physical back ground approximation of the integral over Ω for discretized function 10 A, B = a ij b ij, i,j A 2 = a ij 2 = A 2 F i,j

11 DLR equations δy, Y tt f(y) = 0 δy T Y M is equivalent to a system of equations for U, S and V: S = U H f(y)v + U H U S + SV H V U = (I UU H )f(y)vs 1 UU H U 2U S S 1 V = (I VV H )f(y) H US H VV H V 2V S H S H the properties U H U = I = V H V and U H U = 0 = V H V are conserved 11

12 Computational aspects numerical time integration scheme: standard scheme time step selection, error estimation orthogonality constraints inverse of S in case of singularity U = (I UU H )f(y)vs 1 UU H U 2U S S 1 evaluation of f(y): f(y) not required but only f(y)v or f(y) H U Lubich, Nonnenmacher 12

13 Example: Cubic Wave Equation a tt a = a 2 a initial condition: Gaußian pulse (slim in direction of transport, wide in perpendicular direction) periodic boundary conditions (compute spatial derivatives via fft) small example (n = m = 256) comparison with Störmer Verlet / leap frog method 13

14 Results rank storage error DLR error BA storage for Y: matrix entries 10 1 err vs. rank 10 2 err error of DLR error of best rank r approximation rank 14

15 15 Results

16 Rank control Why is this important? theory for first order equations: regularization does not affect the error considerably Lubich, Koch But: system is not very stable and time steps are chosen too small Solution: variable rank S too large: easy! S nearly singular remove small singular values and corresponding vectors S too small: not so easy indicator for rising rank initial condition for additional singular vectors 16

17 Embedded method integrate two sets of matrices: 1st set: U 1 C n,r 1, S 1 C r 1,r 1 and V 1 C m,r 1 S 1 invertible can be integrated in stable way contains relevant information 2nd set: U 2 C n,r 2, S 2 C r 2,r 2 and V 2 C m,r 2 r 2 = r S 2 not necessarily invertible regularization not too bad indicator for rising rank extract initial conditions for 1st set in case of rising rank 17

18 Rank change rank reduction Ũ 1, S1, Ṽ 1 r dr Ũ 2, S2, Ṽ 2 r + 2 dr U 1, S 1, V 1 r U 2, S 2, V 2 r + 2 rank extension Û 1, Ŝ 1, V1 r + dr Û 2, Ŝ 2, V2 r dr? 18

19 19 Rank extension project 2nd set onto orthogonal complement of 1st set keep only dominant parts (I U 1 U H 1 )U 2 S 2 V H 2 (I V 1 V H 1 ) = U S V H U = U(:, 1 : dr) S = S(1 : dr, 1 : dr) V = V(:, 1 : dr) combine with old data Ŝ 1 = [ ] S1 0 0 S V 1 = [V 1 V] Û 1 = [U 1 U] for derivatives use variation of 1st order system S = U H Y V U = (I ÛÛ H )Y V S 1 V = (I V V H )Y H US H Y = U 2 S 2V H 2 + U 2S 2 VH 2 + U 2S 2 V H 2 projection with respect to the complete new space Lubich, Koch

20 Initial conditions for particular applications Guess initial values for 2. set in case of the moving wave: S, S are set to zero transversal direction U: Ũ SṼ H = (I U 2 U H 2 )D 2 y U 2, Ũ = parallel direction V: V either random and orthogonalized or by the same scheme as U V such that it resembles the speed of the pulse 20

21 Results fixed rank = 7 = max rank of variable rank 21

22 Results 10 time step size rank 5 dlr vr dlr cr lf 3 t 10 0 dlr vr dlr cr t variable rank fixed rank 22

23 Conclusion Up to now: equations for DLR approximation for 2. order equations error estimator adaptive rank Goals: more sophisticated time integration scheme application to real laser-plasma problem extension to 3d problems 23