Model reduction of wave propagation

Transcription

1 Model reduction of wave propagation via phase-preconditioned rational Krylov subspaces Delft University of Technology and Schlumberger V. Druskin, R. Remis, M. Zaslavsky, Jörn Zimmerling November 8, 2017 Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

2 Motivation Second order wave equation with wave operator A Receiver Source PML 5500 Au s 2 u = δ(x x S ) Assume N grid steps in every spatial direction Scaling of surface seismic in 3D: # Grid points O(N 3 ) # Sources O(N 2 ) # Frequencies O(N) Overall O(N 6 )ψ(n 3 ) x-direction [m] y-direction [m] Wavespeed [m/s] (a) Section of wave speed profile of the Marmousi model. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

3 Goal of this work Simulate and compress large scale wave fields in modern high performance computing environment (parallel CPU and GPU environment) Use projection based model order reduction to Approximate transfer function reduce # of frequencies needed to solve reduce # of sources to be considered reduce # number of grid points needed Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

4 Introduction Simulating and compressing large scale wave fields Au [l] s 2 u [l] = δ(x x [l] S ), (1) With the wave operator given by A ν 2, Laplace frequency s We consider a Multiple-Input Multiple-Output problem Define fields U = [u [1], u [2],..., u [Ns] ] and sources B = [ δ(x x [1] S ), δ(x x[2] S ),..., δ(x x[ns] S )] We are interested in the transfer function (Receivers and Sources coincide) F(s) = B H U(s)dx (2) Open Domains Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

5 Problem Formulation After finite difference discretization with PML (A(s) s 2 I)U = ˆB Step sizes inside the PML h i = α i + β i s Frequency dependent A(s) caused by absorbing boundary Q(s)U = B with Q(s) C N N Q(s) propterties sparse complex symmetric (reciprocity) Schwarz reflection principle Q(s) = Q(s) passive (nonlinear NR 1 Re < 0) 1 NR:W {A(s)} = { s C : x H A(s)x = 0 x C k \0 } Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

6 Problem Formulation Transfer function from sources to receivers F(B, s) = B H Q(s) 1 B Reduced order modeling of transfer function over frequency range F m (B, s) = V m B H (VmQ(s)V H m ) 1 VmB H with V m C N m valid in a range of s, and easy to store Motivation FD grid overdiscretized w.r.t. Nyquist approximation F(B, s) to noise level PML introduces losses limited I/O map Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

7 Outline 1 Problem formulation 2 Model order reduction Rational Krylov subspaces 3 Phase-Preconditioning 4 Numerical Experiments 5 Conclusions Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

8 Structure preserving rational Krylov subspaces Preserve: symmetry (w.r.t. matrix, frequency), passivity Block rational Krylov subspace approach κ = s 1,..., s m K m (κ) = span { Q(s 1 ) 1 B,..., Q(s m ) 1 B } K 2m Re = span {Km (κ), K m (κ)} Let V m be a (real) basis for KRe m then with reduced order model (via Galerkin condition) we approximate R m (s) = V H mq(s)v m F m (s) = (V H mb) H R m (s) 1 V H mb, Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

9 Structure preserving rational Krylov subspaces Numerical Range: W {R m (s)} W {Q(s)} Proof: x H mr m (s)x m = (V m x m ) H Q(s)(V m x m ) R m (s) is passive F m (s) is Hermite interpolant of F(s) on κ κ Q(κ) 1 B KRe 2m + uniqueness of Galerkin Schwarz reflection and symmetry hold aswell Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

10 RKS for a Resonant cavity 6 5 FDFD Response RKS Response m=60 RKS Response m=20 Response [a.u] x-direction Receiver Source PML Normalized Frequency y-direction RKS has excellent convergence if singular Hankel values of system decay fast (few contributing eigenvectors) F m (s) is a [2m 1/2m] rational function Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

11 Problem of RKS in Geophysics - Nyquist Limit 0.2 Response [a.u.] Normalized Frequency Long travel times δ(t T) F exp( st) Nyquist sampling of ω = π T F(s) = B H Q(s) 1 Bdx is oscillatory Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

12 Filion Quadrature Filion quadrature deals with oscillatory integral F(s) = exp(st) f(t)dt quadrature requires s t 1 F(s) t n a n exp(s n t) f(n t) Filion quadrature makes a n function of s t F(s) t n a n (s t) exp(s n t) f(n t) Make projection basis s dependent Frequency dependence from asymptotic s i (WKB) Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

13 Phase-Preconditioning I - 1D We can overcome the Nyquist demand by splitting the wavefield into oscillatory and smooth part u(s j ) = exp( s j T eik )c out (s j ) + exp(s j T eik )c in (s j ). (3) Oscillatory phase term obtained from high frequency asymptotics Eikonal equation T [l] eik 2 = 1 ν 2 Amplitudes c out/in are smooth Motivated by Filon quadrature Handle oscillatory part analytically Quadrature with smooth amplitudes Note: Splitting not unique Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

14 Phase-Preconditioning II Projection on frequency dependent Reduced Order Basis K 2m EIK (κ, s) = span{ exp( st eik) c out (s 1 ),..., exp( st eik ) c out (s m ), exp( st eik ) c in (s s ),..., exp( st eik ) c in (s m )} Preserve Schwartz reflection principle K 4m EIK;R (κ, s) = span { KEIK 2m (κ, s), K2m EIK (κ, s)} (4) equivalent to changing Operator Coefficients from Galerkin condition m m u m (s) = α i (s) exp( st eik ) c out (s i )+ β i (s) exp(st eik ) c in (s i )+... i=1 i=1 Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

15 Phase-Preconditioning III Non-uniqueness of splitting resolved by one-way WEQ c out (s j ) = ν ( ) sj exp( s j T eik ) 2s j ν u(s j) x x S u(s j), (5) c in (s j ) = ν ( ) sj exp( s j T eik ) 2s j ν u(s j) + x x S u(s j). (6) Effects of Phase preconditioning on Number of Interpolation points Size of the computational Grid MIMO problems Computational Scheme Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

16 Projection on frequency dependent space Let V m;eik (s) be a real basis of KEIK;R 4m (κ, s) The reduced order model is given by large inner products on GPU This preserves symmetry Schwarz reflection principle passivity Interpolation R m;eik (s) = V H m;eik (s)q(s)v m;eik(s). Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

17 Number of interpolation points needed Double interpolation of transfer function still holds F m (s) = F m (s) and d ds F m(s) = d F(s) with s κ κ. (7) ds Number of interpolation point needed dependent on complexity medium, not latest arrival Proposition: Let a 1D problem have l homogenous layers. Then there exist m l + 1 non-coinciding interpolation points, such that the solution u m;eik (s) = u. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

18 Illustration of Proposition Wavespeed Source L1 L2 L3 L4 L5 L6 L7 Real part field Source L1 L2 L3 L4 L5 L6 L7 Imag(C) outgooing Source L1 L2 L3 L4 L5 L6 L7 Imag(C) incoming 2 0 Source L1 L2 L3 L4 L5 L6 L7 Amplitudes are constants in layers + left and right of source Basis is complete after l + 1 iterations Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

19 Phase-Preconditioning higher dimensions/mimo Split with dimension specific function u(s j ) [l] = g(s j T [l] eik )c out(s j ) [l] + g( s j T [l] eik )c in(s j ) [l], (8) g(x) obtained from WKB approximation One way wave equations along T eik used for decomposition In 2D is we use g(x) = K 0 (x) for outgoing Multiple T [l] direction c in (s j ) = eik s j T sign(im (s j ))iπ for multiple sources [l] account for multiple ] [K 1 (s j T) u(s j ) + K 0 (s j T) ν2 T u(s j ) s j Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

20 Size of the computational Grid Receiver Source PML x-direction [m] wavespeed [m/s] y-direction [m] (b) Section of the wave speed profile of the smoothed Marmousi model. (c) Real part of the wavefield u [4]. (d) Real part of the amplitude c [4] out. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

21 Numerical Experiments - I Configurations (Neumann boundary condition on top) Layered medium Travel time dominated 5 Sources and 5 Receivers x 4m Comp. Size 829x480 points Size 3160 m x 1920m range c m/s Range Quadrature 0-40 Hz x-direction [m] Receiver Source PML y-direction [m] wavespeed [m/s] (e) Section of the wave speed profile of the smoothed Marmousi model. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

22 Numerical Experiments response [a.u.] Full Response RKS m=20 PPRKS m=20 Interpolation points Source 1 to Receiver 5 PPRKS clearly outperforms RKS normalized frequency (f) Real part of the frequency-domain transfer function Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

23 Numerical Experiments I Time-domain convergence of the RKS and the PPRKS 1 0 RKS PPRKS 10 log of error number of interpolation frequencies m (g) Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

24 Computational Grid Amplitudes c in/out are much smoother than the wavefield ROM can extrapolate to high frequencies oscillatory part is handled analytically Two-grid approach: Amplitudes can be computed on coarse grid U c = Q course (s) 1 B c Interpolate amplitudes to fine grid Projection of operator and evaluation are performed on fine grid F c;m (s) = B H { [V c;m (s)] H Q fine (s)v c;m (s) } 1 B Solution gets gauged to the fine grid (no interpolation anymore) Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

25 Phase-Preconditioning SVD Amplitudes are smooth in space and can become redundant Reduce amplitudes via SVD of [c out c in ] c j SVD Amplitudes have no source information 10log normalized singular values RKS Contracted amplitude basis index singular value (h) Singular values of normalized (c out c in ) # singular values larger than 0.01 versus # sources with m = 40 N src [c out, c in ] u m N src Reduction of sources Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

26 Phase-Preconditioning SVD Assume s ir then we obtain [ u m [l] N src M SVD a [l] rj (s) = α [l] rj r=1 j=1 ] T [ ] g(st [r] eik )cj SVD g( st [r] eik )cj SVD (9) where M SVD 2mN src. Coefficients from Galerkin condition c j SVD is no longer source dependent Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

27 Computational Scheme - Coarse Grid - CPU ROM Construction phase Initialize Simulation Compute T [l] eik Solve coarse problem single shot/frequency Q coarse (κ i )u [l] (κ i ) = b [l] Embarrassingly Parallel Compute SVD of c [l] out and c [l] in Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

28 Computational Scheme - Fine Grid - GPU Compute SVD of c [l] out and c [l] in Evaluate ROM single Frequency s e R m;eik = V m;eik (s e ) H Q fine V m;eik (s e ) F c,m = B H s V m;eik (s e )R 1 m;eik V m;eik(s e ) H B s ROM Evaluation Phase Compute inverse Fourier Transform ˆF m;c (t) = F 1 F m;c (iω) Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

29 Numerical Experiments - Grid Coarsening Coarsen the computation mesh by factor 16 (4 in each direction) Interpolation points until 5.5 ppw (m = 40, N src = 12, M = 100) FD error averaged over all traces RKS m=40 stepsize=0.5 uniform shifts FDFD m=500 stepsize=0.6 PP-RKS m=40 stepsize=4.0 shifts used in PP-RKS normalized frequency (i) Relative error err average x-direction [m] y-direction [m] (j) Configuration Speed [m/s] Figure: Smooth Marmousi test configuration with grid coarsening. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

30 Numerical Experiments - Grid Coarsening Projection of fine operator gauges the ROM Direct evaluation of coarse operator not accurate FD error averaged over all traces Direct evaluation of 500 frequency points PPRKS with m=40 shifts points per wavelength (k) err average ROM;coarse versus erraverage FD;coarse Figure: Smooth Marmousi test configuration with grid coarsening. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

31 Numerical Experiments - Grid Coarsening Ricker Wavelet with cut-off frequency of 2.7 ppw on coarse grid 10-6 response [a.u.] Comparison ROM normalized time (l) Time domain trace from the left most source to the right most receiver after m = 40 interpolation points. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

32 Numerical Experiment III Reiceiver Source PML Resonant Borehole in smooth Geology Resonant behavior causes long runtimes 6 Surface- and 8 BH source-receiver pairs x-direction [m] wavespeed [m/s] y-direction [m] 1500 (m) Simulated configuration. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

33 Numerical Experiments III x-direction [m] Reiceiver Source PML Speed [m/s] x-direction [m] Reiceiver Source PML Speed [m/s] u [l] (s j ) = g(s j T [l] eik )c[l] out;eik (s j) + g( s j T [l] eik )c[l] in;eik (s j), u [l] (s j ) = g(s j T [l] eik;cm )c[l] out;cm (s j) + g( s j T [l] eik;cm )c[l] in;cm (s j) y-direction [m] (n) Isosurfaces T eik y-direction [m] (o) Isosurf. T eik;cm. m = 40, N src = 14, M SVD = 30 c in/out;eik/cm Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

34 Numerical Experiments III Ricker Wavelet with cut-off frequency of 2.7 ppw on coarse grid response [a.u.] Comparison ROM normalized time (p) Time-domain trace of the coinciding source receiver pair number 1 after m = 40 interpolation points, together with the comparison solution. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

35 Numerical Experiments III Ricker Wavelet with cut-off frequency of 2.7 ppw on coarse grid response [a.u.] normalized time (q) Time-domain trace from source number 7 inside the borehole to the rightmost surface receiver number 14 after m = 40 interpolation points. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

36 Conclusions All three challenges (Grid size, Nr of Sources, Nr of interpolation points) can be reduced with phase preconditioning Projection on frequency dependent basis allows ROM beyond the Nyquist limit Can be used for other oscillatory PDEs that have asymptotic solutions Work shifted from solvers to inner products Significantly compressed the ROM into coarse amplitudes Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

37 Paper V. Druskin, R. Remis, M. Zaslavsky and J. Zimmerling, Compressing Large-Scale Wave Propagation Models via Phase-Preconditioned Rational Krylov Subspaces, arxiv: Thanks 2 2 STW (project 14222, Good Vibrations) and Schlumberger Doll-Research Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 37

38 Numerical Experiments IIII Coarsen the computation mesh by factor 16 (4 in each direction) Interpolation points until 5.5 ppw (m = 40, N src = 12, M = 150) FD error averaged over all traces RKS m=40 step size=1.0 FDFD m=500 step size=4.0 FDFD m=500 step size=1.2 PP-RKS m=40 step size= normalized frequency (g) Relative error err average x-direction [m] Receiver Source PML y-direction [m] (h) Configuration Figure: Marmousi test configuration with grid coarsening Wavespeed [m/s] Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 4

39 Numerical Experiments IIII Ricker Wavelet with cut-off frequency of 2.7 ppw on coarse grid response [a.u.] Comparison ROM normalized time (a) Time domain trace from the left most source to the right most receiver after m = 40 interpolation points. Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 4

40 Computational Complexity Cost of the basis computation and evaluation of the ROM Basis Computation on CPU 3, Evaluation GPU 4 Basis Computation comparison Computation Time Block solve fine grid Q fine (s i ) 1 B 10.3s Single solve fine grid Q fine (s i ) 1 b 4.1s Block solve coarse grid Q coarse (s i ) 1 B 0.6s Single solve coarse grid Q coarse (s i ) 1 b 0.2s Evaluation Step Computation Time Scaling Computing phase functions exp(iωt eik ) s N src N f Hadamard Products exp(iωt eik ) c SVD s M SVD N src N f Galerkin inner product V m;eik (s e ) H Q fine V m;eik (s e ) 1.752s N f MSVD 2 N2 src 3 Solved using UMFPACK v on a 4-Core Intel i CPU@3.40 GHz with parallel BLAS level-3 routines 4 Double precision python implementation on an Nvidia GTX 1080 Ti Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 4

41 Dispersion Correction At 5.5 ppw with a second order scheme we dispersion analytical travel time does not correspond to numerical use ν [l] in decomposition to cancel highest s 2 term ( exp 2sT [l] eik ) k i=1 ( ) D xi exp st [l] eik 2 = s2 (10) ν [l]2 Jörn Zimmerling (TU Delft) Model reduction of wave propagation November 8, / 4