Post on 27-May-2018
transcript
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 1 / 37
Model reduction of wave propagationvia phase-preconditioned rational Krylov subspacesDelft University of Technology† and Schlumberger∗
V. Druskin∗, R. Remis†, M. Zaslavsky∗, Jorn Zimmerling†
November 8, 2017
Motivation
• Second order wave equation withwave operator A
Au− s2u = −δ(x− xS)
• Assume N grid steps in everyspatial direction
• Scaling of surface seismic in 3D:
• # Grid points O(N3)
• # Sources O(N2)
• # Frequencies O(N)
• Overall O(N6)ψ(N3)
500 1000 1500
y-direction [m]
500
1000
1500
2000
2500
3000
x-d
irection [m
]
Receiver
Source
PML
1500
2000
2500
3000
3500
4000
4500
5000
5500
Wavespeed [m
/s]
(a) Section of wave speed profileof the Marmousi model.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 2 / 37
Goal of this work
• Simulate and compress large scale wave fields in modern highperformance 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
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 3 / 37
Introduction
• Simulating and compressing large scale wave fields
Au[l] − s2u[l] = −δ(x− x[l]S ), (1)
• With the wave operator given by A ≡ ν2∆, 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 Sourcescoincide)
F(s) =
∫BHU(s)dx (2)
• Open Domains
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 4 / 37
Problem Formulation
• After finite difference discretization with PML
(A(s)− s2I)U = B
• Step sizes inside the PML hi = αi + βis
• Frequency dependent A(s) caused by absorbing boundary
Q(s)U = B with Q(s) ∈ CN×N
• Q(s) propterties• sparse• complex symmetric (reciprocity)• Schwarz reflection principle Q(s) = Q(s)• passive (nonlinear NR1 Re < 0)
1NR:W {A(s)} ={s ∈ C : xHA(s)x = 0 ∀x ∈ Ck\0
}
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 5 / 37
Problem Formulation
• Transfer function from sources to receivers
F(B, s) = BHQ(s)−1B
• Reduced order modeling of transfer function over frequency range
Fm(B, s) = VmBH(VHmQ(s)Vm)−1VH
mB with Vm ∈ CN×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
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 6 / 37
Outline
1 Problem formulation
2 Model order reduction – Rational Krylov subspaces
3 Phase-Preconditioning
4 Numerical Experiments
5 Conclusions
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 7 / 37
Structure preserving rational Krylov subspaces
• Preserve: symmetry (w.r.t. matrix, frequency), passivity
• Block rational Krylov subspace approach κ = s1, . . . , sm
Km(κ) = span{
Q(s1)−1B, . . . ,Q(sm)−1B}
K2mRe = span {Km(κ),Km(κ)}
• Let Vm be a (real) basis for KmRe then with reduced order model
(via Galerkin condition)
Rm(s) = VHmQ(s)Vm
we approximate
Fm(s) = (VHmB)HRm(s)−1VH
mB,
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 8 / 37
Structure preserving rational Krylov subspaces
• Numerical Range: W {Rm(s)} ⊆ W {Q(s)}Proof: xHmRm(s)xm = (Vmxm)HQ(s)(Vmxm)⇒ Rm(s) is passive
• Fm(s) is Hermite interpolant of F(s) on κ ∪ κQ(κ)−1B ∈ K2m
Re + uniqueness of Galerkin
• Schwarz reflection and symmetry hold aswell
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 9 / 37
RKS for a Resonant cavity
0 0.1 0.2 0.3 0.4 0.5
Normalized Frequency
0
1
2
3
4
5
6R
espo
nse
[a.u
]FDFD ResponseRKS Response m=60RKS Response m=20
20 40 60 80 100 120
y-direction
20
40
60
80
100
120
x-d
ire
ctio
n
Receiver
Source
PML
0
0.2
0.4
0.6
0.8
1
• RKS has excellent convergence if singular Hankel values ofsystem decay fast (few contributing eigenvectors)
• Fm(s) is a [2m − 1/2m] rational function
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 10 / 37
Problem of RKS in Geophysics - Nyquist Limit
0.05 0.1 0.15
Normalized Frequency
-0.3
-0.2
-0.1
0
0.1
0.2
Re
sp
on
se
[a
.u.]
• Long travel times ∗δ(t− T)F−→ exp(−sT)
• Nyquist sampling of ∆ω = πT
• F(s) =∫
BHQ(s)−1Bdx is oscillatory
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 11 / 37
Filion Quadrature
• Filion quadrature deals with oscillatory integral
F(s) =
∫exp(st) f(t)dt
quadrature requires s∆t � 1
F(s) ≈ ∆t∑n
an exp(s n∆t) f(n∆t)
• Filion quadrature makes an function of s∆t
F(s) ≈ ∆t∑n
an(s∆t) exp(s n∆t) f(n∆t)
• ⇒ Make projection basis s dependent
• ⇒ Frequency dependence from asymptotic s→ i∞ (WKB)
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 12 / 37
Phase-Preconditioning I - 1D
• We can overcome the Nyquist demand by splitting the wavefieldinto oscillatory and smooth part
u(sj) = exp(−sjTeik)cout(sj) + exp(sjTeik)cin(sj). (3)
• Oscillatory phase term obtained from high frequency asymptotics
• Eikonal equation |∇T[l ]eik|
2 = 1ν2
• Amplitudes cout/in are smooth
• Motivated by Filon quadrature• Handle oscillatory part analytically• Quadrature with smooth amplitudes
• Note: Splitting not unique
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 13 / 37
Phase-Preconditioning II
• Projection on frequency dependent Reduced Order Basis
K2mEIK(κ, s) = span{ exp(−sTeik) cout(s1), . . . , exp(−sTeik) cout(sm),
exp( sTeik) cin (ss), . . . , exp( sTeik) cin (sm)}
• Preserve Schwartz reflection principle
K4mEIK;R(κ, s) = span
{K2m
EIK(κ, s),K2mEIK(κ, s)
}(4)
• equivalent to changing Operator
• Coefficients from Galerkin condition
um(s) =m∑i=1
αi (s) exp(−sTeik) cout(si )+m∑i=1
βi (s) exp(sTeik) cin(si )+. . .
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 14 / 37
Phase-Preconditioning III
• Non-uniqueness of splitting resolved by one-way WEQ
cout(sj) =ν
2sjexp( sjTeik)
(sjνu(sj)−
∂
∂|x − xS|u(sj)
), (5)
cin (sj) =ν
2sjexp(−sjTeik)
(sjνu(sj) +
∂
∂|x − xS|u(sj)
). (6)
Effects of Phase preconditioning on
• Number of Interpolation points
• Size of the computational Grid
• MIMO problems
• Computational Scheme
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 15 / 37
Projection on frequency dependent space
• Let Vm;EIK(s) be a real basis of K4mEIK;R(κ, s)
• The reduced order model is given by
Rm;EIK(s) = VHm;EIK(s)Q(s)Vm;EIK(s).
• large inner products on GPU
• This preserves• symmetry• Schwarz reflection principle• passivity• Interpolation
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 16 / 37
Number of interpolation points needed
• Double interpolation of transfer function still holds
Fm(s) = Fm(s) andd
dsFm(s) =
d
dsF(s) with s ∈ κ ∪ κ. (7)
• Number of interpolation point needed dependent on complexitymedium, not latest arrival
• Proposition: Let a 1D problem have ` homogenous layers .Then there exist m ≤ `+ 1 non-coinciding interpolation points,such that the solution um;EIK(s) = u.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 17 / 37
Illustration of Proposition
Source L1 L2 L3 L4 L5 L6 L70
0.5
1
Wavespeed
Source L1 L2 L3 L4 L5 L6 L7-10
0
10Real part field
Source L1 L2 L3 L4 L5 L6 L70
5
10Imag(C) outgooing
Source L1 L2 L3 L4 L5 L6 L70
2
4Imag(C) incoming
• Amplitudes are constants in layers + left and right of source
• Basis is complete after `+ 1 iterations
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 18 / 37
Phase-Preconditioning higherdimensions/MIMO
• Split with dimension specific function
u(sj)[l] = g(sjT
[l]eik)cout(sj)
[l] + g(−sjT[l]eik)cin(sj)
[l], (8)
• g(x) obtained from WKB approximation
• One way wave equations along ∇Teik used for decomposition
• In 2D is we use g(x) = K0 (x) for outgoing
• Multiple T[l]eik for multiple sources [l] account for multiple
direction
cin(sj) =sjT
sign(Im (sj))iπ
[K1 (sjT) u(sj) +K0 (sjT)
ν2
sj∇T · ∇u(sj)
]
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 19 / 37
Size of the computational Grid
500 1000 1500
y-direction [m]
500
1000
1500
2000
2500
3000
x-d
irection [m
]
Receiver
Source
PML
1500
2000
2500
3000
3500
4000
4500
5000
wa
ve
sp
ee
d [
m/s
]
(b) Section of the wave speed profile ofthe smoothed Marmousi model.
(c) Real part of thewavefield u[4].
(d) Real part of the
amplitude c[4]out.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 20 / 37
Numerical Experiments - I
• Configurations (Neumann boundarycondition on top)
• Layered medium
• Travel time dominated
• 5 Sources and 5 Receivers∆x 4mComp. Size 829x480 pointsSize 3160 m x 1920mrange c 1500 - 5500 m/sRange Quadrature 0-40 Hz
500 1000 1500
y-direction [m]
500
1000
1500
2000
2500
3000
x-d
ire
ctio
n [
m]
Receiver
Source
PML
1500
2000
2500
3000
3500
4000
4500
5000
wavespeed [m
/s]
(e) Section of the wave speedprofile of the smoothed Marmousimodel.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 21 / 37
Numerical Experiments
0 0.05 0.1 0.15
normalized frequency
-0.6
-0.4
-0.2
0
0.2
0.4
response [a.u
.]
Full Response
RKS m=20
PPRKS m=20
Interpolation points
(f) Real part of the frequency-domain transfer function
• Source 1 toReceiver 5
• PPRKS clearlyoutperformsRKS
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 22 / 37
Numerical Experiments I• Time-domain convergence of the RKS and the PPRKS
0 50 100 150 200
number of interpolation frequencies m
-6
-5
-4
-3
-2
-1
0
1
10 log o
f err
or
RKS
PPRKS
(g)
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 23 / 37
Computational Grid
• Amplitudes cin/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 Uc = Qcourse(s)−1Bc
• Interpolate amplitudes to fine grid
• Projection of operator and evaluation are performed on fine grid
Fc;m(s) = BH{
[Vc;m(s)]HQfine(s)Vc;m(s)}−1
B
• Solution gets gauged to the fine grid(no interpolation anymore)
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 24 / 37
Phase-Preconditioning SVD• Amplitudes are smooth in space and can become redundant• Reduce amplitudes via SVD of [cout cin]⇒ c jSVD• Amplitudes have no source information
0 100 200 300 400 500index singular value
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
10
log
no
rma
lize
d s
ing
ula
r va
lue
s
RKSContracted amplitude basis
(h) Singular values of normalized (cout cin)
# singular values larger than 0.01
versus # sources with m = 40
Nsrc 12 24 48 96
[cout, cin] 69 72 73 73u 457 833 1369 1741m · Nsrc 480 960 1920 3840
⇒ Reduction of sources
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 25 / 37
Phase-Preconditioning SVD
• Assume s ∈ iR then we obtain
u[l ]m (s) =
Nsrc∑r=1
MSVD∑j=1
[a
[l ]rj
α[l ]rj
]T [g(sT
[r ]eik)c jSVD
g(−sT [r ]eik)c jSVD
](9)
where MSVD � 2mNsrc.
• Coefficients from Galerkin condition
• c jSVD is no longer source dependent
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 26 / 37
Computational Scheme - Coarse Grid - CPU
ROM Construction phase
EmbarrassinglyParallel
ROM Evaluation Phase
Initialize Simulation
Compute T[l ]eik
Solve coarse problemsingle shot/frequency
Qcoarse(κi )u[l ](κi ) = b[l ]
Compute SVD of
c[l ]out and c
[l ]in
Evaluate ROM single Frequency se
Hm;EIK = Vm;EIK(se)HQfineVm;EIK(se)
Fc,m = BHs Vm;EIK(se)H−1
m;EIKVm;EIK(se)HBs
Compute inverseFourier Transform
Fm;c(t) = F−1Fm;c(iω)
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 27 / 37
Computational Scheme - Fine Grid - GPU
ROM Evaluation Phase
Compute SVD of
c[l ]out and c
[l ]in
Evaluate ROM single Frequency se
Rm;EIK = Vm;EIK(se)HQfineVm;EIK(se)
Fc,m = BHs Vm;EIK(se)R−1
m;EIKVm;EIK(se)HBs
Compute inverseFourier Transform
Fm;c(t) = F−1Fm;c(iω)
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 28 / 37
Numerical Experiments - Grid Coarsening• Coarsen the computation mesh by factor 16 (4 in each direction)• Interpolation points until 5.5 ppw (m = 40,Nsrc = 12,M = 100)
0 0.02 0.04 0.06 0.08
normalized frequency
0
0.02
0.04
0.06
0.08
FD
err
or
ave
rag
ed
ove
r a
ll tr
ace
s
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
(i) Relative error erraverage
500 1000 1500
y-direction [m]
500
1000
1500
2000
2500
3000
x-d
irection [m
]
1500
2000
2500
3000
3500
4000
4500
5000
Speed [m
/s]
(j) Configuration
Figure: Smooth Marmousi test configuration with grid coarsening.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 29 / 37
Numerical Experiments - Grid Coarsening• Projection of fine operator gauges the ROM• Direct evaluation of coarse operator not accurate
10 20 30 40 50points per wavelength
10-3
10-2
10-1
100
FD
err
or
avera
ged
ove
r all
trace
s
Direct evaluation of 500 frequency points
PPRKS with m=40 shifts
(k) erraverageROM;coarse versus erraverageFD;coarse
Figure: Smooth Marmousi test configuration with grid coarsening.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 30 / 37
Numerical Experiments - Grid Coarsening• Ricker Wavelet with cut-off frequency of 2.7 ppw on coarse grid
0 1000 2000 3000 4000 5000 6000 7000
normalized time
-8
-6
-4
-2
0
2
4
6
response [a.u
.]
×10-6
Comparison
ROM
2000 2500 3000 3500 4000 4500 5000
(l) Time domain trace from the left most source to the right mostreceiver after m = 40 interpolation points.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 31 / 37
Numerical Experiment III
• Resonant Borehole in smoothGeology
• Resonant behavior causeslong runtimes
• 6 Surface- and 8 BHsource-receiver pairs
200 600 1000
y-direction [m]
500
1000
1500
2000
2500
x-d
ire
ctio
n [
m]
Reiceiver
Source
PML
1500
2000
2500
3000
3500
4000
4500
5000
wavespeed [m
/s]
1
2
3
4
5
6
7
8
9 14
(m) Simulated configuration.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 32 / 37
Numerical Experiments III
200 600 1000
y-direction [m]
500
1000
1500
2000
2500
x-di
rect
ion
[m]
ReiceiverSourcePML
1000
2000
3000
4000
5000
Spe
ed [m
/s]
(n) Isosurfaces Teik.
200 600 1000
y-direction [m]
500
1000
1500
2000
2500
x-di
rect
ion
[m]
ReiceiverSourcePML
1000
2000
3000
4000
5000
Spe
ed [m
/s]
(o) Isosurf. Teik;CM.
u[l ](sj) = g(sjT[l ]eik)c
[l ]out;eik(sj)
+ g(−sjT[l ]eik)c
[l ]in;eik(sj),
u[l ](sj) = g(sjT[l ]eik;CM)c
[l ]out;CM(sj)
+ g(−sjT[l ]eik;CM)c
[l ]in;CM(sj).
• m = 40, Nsrc = 14,MSVD = 30
• cin/out;eik/CM
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 33 / 37
Numerical Experiments III• Ricker Wavelet with cut-off frequency of 2.7 ppw on coarse grid
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
normalized time
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1re
sponse [a.u
.]×10
-4
Comparison
ROM
8000 8500 9000 9500
(p) Time-domain trace of the coinciding source receiver pair number 1 afterm = 40 interpolation points, together with the comparison solution.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 34 / 37
Numerical Experiments III• Ricker Wavelet with cut-off frequency of 2.7 ppw on coarse grid
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
normalized time
-1
-0.5
0
0.5
1
response [a.u
.]
×10-6
3500 4000 4500 5000 5500 6000
(q) Time-domain trace from source number 7 inside the borehole to therightmost surface receiver number 14 after m = 40 interpolation points.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 35 / 37
Conclusions
• All three challenges (Grid size, Nr of Sources, Nr of interpolationpoints) can be reduced with phase preconditioning
• Projection on frequency dependent basis allows ROM beyond theNyquist limit
• Can be used for other oscillatory PDEs that have asymptoticsolutions
• Work shifted from solvers to inner products
• Significantly compressed the ROM into coarse amplitudes
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 36 / 37
Paper
V. Druskin, R. Remis, M. Zaslavsky and J. Zimmerling,Compressing Large-Scale Wave Propagation Models viaPhase-Preconditioned Rational Krylov Subspaces, arXiv:1711.00942
Thanks2
2STW (project 14222, Good Vibrations) and Schlumberger Doll-Research
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 37 / 37
Numerical Experiments IIII• Coarsen the computation mesh by factor 16 (4 in each direction)• Interpolation points until 5.5 ppw (m = 40,Nsrc = 12,M = 150)
0 0.05 0.1 0.15
normalized frequency
0
0.2
0.4
0.6
0.8
1
FD
err
or
avera
ged o
ver
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=4.0
(g) Relative error erraverage
500 1000 1500
y-direction [m]
500
1000
1500
2000
2500
3000
x-d
ire
ctio
n [
m]
Receiver
Source
PML
1500
2000
2500
3000
3500
4000
4500
5000
5500
Wa
ve
sp
ee
d [
m/s
]
(h) Configuration
Figure: Marmousi test configuration with grid coarsening.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 1 / 4
Numerical Experiments IIII• Ricker Wavelet with cut-off frequency of 2.7 ppw on coarse grid
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
normalized time
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1re
sp
on
se
[a
.u.]
×10-5
Comparison
ROM
2000 2500 3000 3500 4000 4500 5000
(a) Time domain trace from the left most source to the right mostreceiver after m = 40 interpolation points.
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 2 / 4
Computational Complexity
• Cost of the basis computation and evaluation of the ROM
• Basis Computation on CPU3, Evaluation GPU4
Basis Computation comparison Computation Time
Block solve fine grid Qfine(si )−1B 10.3s
Single solve fine grid Qfine(si )−1b 4.1s
Block solve coarse grid Qcoarse(si )−1B 0.6s
Single solve coarse grid Qcoarse(si )−1b 0.2s
Evaluation Step Computation Time Scaling
Computing phase functions exp(iωTeik) 0.00546s NsrcNf
Hadamard Products exp(iωTeik) cSVD 0.01496s MSVDNsrcNf
Galerkin inner product Vm;EIK(se)H · QfineVm;EIK(se) 1.752s NfM2SVDN
2src
3Solved using UMFPACK v 5.4.0 on a 4-Core Intel i5-4670 CPU@3.40 GHzwith parallel BLAS level-3 routines
4Double precision python implementation on an Nvidia GTX 1080 Ti
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 3 / 4
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 s2 term
exp(
2sT[l ]eik
) k∑i=1
|Dxi exp(−sT
[l ]eik
)|2 =
s2
ν ′[l ]2
(10)
Jorn Zimmerling (TU Delft) Model reduction of wave propagation November 8, 2017 4 / 4