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Efficient Wavefield Simulators Based on Krylov Model-Order Reduction - - PowerPoint PPT Presentation

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Introduction Basic equations Lanczos algorithms PML Efficient Wavefield Simulators Based on Krylov Model-Order Reduction Techniques From Resonators to Open Domains Rob Remis Delft University of Technology November 3, 2017 ICERM Brown

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Introduction Basic equations Lanczos algorithms PML

Efficient Wavefield Simulators Based on Krylov Model-Order Reduction Techniques

From Resonators to Open Domains

Rob Remis Delft University of Technology

November 3, 2017 – ICERM Brown University 1

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Introduction Basic equations Lanczos algorithms PML

Thanks

A special thanks to Mikhail Zaslavsky, Schlumberger-Doll Research J¨

  • rn Zimmerling, Delft University of Technology

and Vladimir Druskin, Schlumberger-Doll Research

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Introduction Basic equations Lanczos algorithms PML

Happy birthday

Happy birthday Vladimir!

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Introduction Basic equations Lanczos algorithms PML

Introduction

Back in the day (late 80s, early 90s)

SLDM: Spectral Lanczos Decomposition Method Fast convergence for parabolic (diffusion) equations Applicable to lossless (hyperbolic) wave equation as well Not many advantages compared with explicit time-stepping (FDTD)

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Introduction Basic equations Lanczos algorithms PML

Main research question

What happens if we include losses? Lossy wavefield systems Perfectly Matched Layers (PML, after 1994)

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Introduction Basic equations Lanczos algorithms PML

Basic equations

First-order lossless wavefield system (D + M∂t) F = −w(t)Q Plus initial conditions Dirichlet boundary conditions (no PML) included Lossy wavefield system (D + S + M∂t) F = −w(t)Q

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Introduction Basic equations Lanczos algorithms PML

Maxwell’s equations

Field vector F = [Ex, Ey, Ez, Hx, Hy, Hz]T Source vector Q = [Jsp

x , Jsp y , Jsp z , K sp x , K sp y , K sp z ]T

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Introduction Basic equations Lanczos algorithms PML

Maxwell’s equations

Medium matrices M = ε µ

  • and

S = σ

  • 8
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Maxwell’s equations

Differentiation matrix D = −∇× ∇×

  • Signature matrix

δ− = diag(1, 1, 1, −1, −1, −1)

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Basic equations

Spatial discretization (D + S + M∂t) f = −w(t)q Order of this system can be very large especially in 3D Discretized counterpart of δ− is denoted by d−

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Introduction Basic equations Lanczos algorithms PML

Basic equations

Medium matrices (isotropic media)

S diagonal and semipositive definite M diagonal and positive definite

Differentiation matrix

W step size matrix = diagonal and positive definite Symmetry property DTW = −WD

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Introduction Basic equations Lanczos algorithms PML

Basic equations

System matrix for lossless media: A = M−1D System matrix for lossy media: A = M−1(D + S) Evolution operator = exp(−At)

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Introduction Basic equations Lanczos algorithms PML

Basic equations

Lossless media: A is skew-symmetric w.r.t. WM Evolution operator is orthogonal w.r.t. WM Inner product and norm x, y = yHWMx x = x, x1/2 Stored field energy in the computational domain 1 2f 2 Initial-value problem: norm of f is preserved

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Introduction Basic equations Lanczos algorithms PML

Lanczos algorithms

Lossless media: construct SLDM field approximations via Lanczos algorithm for skew-symmetric matrices FDTD can be written in a similar form as Lanczos algorithm recurrence relation for FDTD = recurrence relation for Fibonacci polynomials

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Introduction Basic equations Lanczos algorithms PML

Lanczos algorithms

Lanczos recurrence coefficients: βi Comparison with FDTD: 1/βi = time step of Lanczos Automatic time step adaptation – no Courant condition

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Introduction Basic equations Lanczos algorithms PML

Lanczos algorithms

Lossy media: system matrix A = M−1(D + S) is no longer skew-symmetric Introduce dp = 1 2(I + d−) and dm = 1 2(I − d−)

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Lanczos algorithms

Special case: S = ξdp σ(x) = ξε(x) for all x belonging to computational domain Exploit shift invariance of Lanczos decomposition Basis for lossless media can be used to describe wave propagation for lossy media (in this special case)

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Lanczos algorithms

Not possible for general lossy media Matrix D is symmetric with respect to Wd− DTWd− = Wd−D System matrix A is symmetric w.r.t. WMd−

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Introduction Basic equations Lanczos algorithms PML

Lanczos algorithms

System matrix A is symmetric w.r.t. bilinear form x, y = yHWMd−x Free-field Lagrangian 1 2f , f

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Introduction Basic equations Lanczos algorithms PML

Lanczos algorithms

Write f = f (q) to indicate that the field is generated by a source q Reciprocity:

Source vector: q = dpq, receiver vector r = dpr f (q), r = q, f (r) Source vector: q = dpq, receiver vector r = dmr f (q), r = −q, f (r)

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Introduction Basic equations Lanczos algorithms PML

Lanczos algorithms

SLDM field approximations for lossy media can be constructed via modified Lanczos algorithm Modified Lanczos algorithm = Lanczos algorithm for symmetric matrices with inner product replaced by bilinear form Modified Lanczos algorithm can also be obtained from two-sided Lanczos algorithm Can the modified Lanczos algorithm breakdown in exact arithmetic?

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Introduction Basic equations Lanczos algorithms PML

PML

No outward wave propagation has been included up to this point Implementation via Perfectly Matched Layers (PML) Coordinate stretching (Laplace domain) ∂k ← → χ−1

k ∂k

k = x, y, z Stretching function χk(k, s) = αk(k) + βk(k) s

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Introduction Basic equations Lanczos algorithms PML

PML

Stretched first-order system

  • D(s) + S + sM

ˆ F = − ˆ w(s)Q Direct spatial discretization

  • D(s) + S + sM

ˆ f = − ˆ w(s)q Leads to nonlinear eigenproblems for spatial dimensions > 1

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Introduction Basic equations Lanczos algorithms PML

PML

Linearization of the PML Spatial finite-difference discretization using complex PML step sizes (Dcs + S + sM) fcs = −w(s)q System matrix Acs = M−1(Dcs + S)

  • V. Druskin and R. F. Remis, “A Krylov stability-corrected coordinate stretching method to simulate wave

propagation in unbounded domains,” SIAM J. Sci. Comput., Vol. 35, 2013, pp. B376 – B400.

  • V. Druskin, S. G¨

uttel, and L. Knizhnerman, “Near-optimal perfectly matched layers for indefinite Helmholtz problems,” SIAM Rev. 58-1 (2016), pp. 90 – 116. 24

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PML

What about the spectrum of the system matrix?

Re(λ) Im(λ) Lossless resonator

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PML

Eigenvalues move into the complex plane

Re(λ) Im(λ) Complex scaling

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PML

Stable part of the spectrum

Re(λ) Im(λ) Stable part

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PML

Stability correction

Re(λ) Im(λ) Stable part

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Stability-Corrected Wave Function

Time-domain stability-corrected wave function f (t) = −w(t) ∗ 2η(t)Re

  • η(Acs) exp(−Acst)q
  • Complex Heaviside unit step function

η(z) =

  • 1

Re(z) > 0 Re(z) < 0

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Stability-Corrected Wave Function

Frequency-domain stability-corrected wave function ˆ f (s) = − ˆ w(s)

  • r(Acs, s) + r( ¯

Acs, s)

  • q

with r(z, s) = η(z) z + s Note that ˆ f (¯ s) = ¯ ˆ f (s) and the stability-corrected wave function is a nonentire function of the system matrix Acs

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Stability-Corrected Wave Function

Symmetry relations are preserved With a step size matrix W that has complex entries These entries correspond to PML locations

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Introduction Basic equations Lanczos algorithms PML

Stability-Corrected Wave Function

Stability-corrected wave function cannot be computed by FDTD SLDM field approximations via modified Lanczos algorithm Reduced-order model fm(t) = −w(t) ∗ 2M−1qη(t)Re [Vmη(Hm) exp(−Hmt)e1]

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Stability-Corrected Wave Function

m = 300

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10

−13

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 x 10

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Time [s] Electric Field Strength [V/m]

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Stability-Corrected Wave Function

m = 400

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10

−13

−8 −6 −4 −2 2 4 6 8 x 10

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Time [s] Electric Field Strength [V/m]

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Stability-Corrected Wave Function

m = 500

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10

−13

−8 −6 −4 −2 2 4 6 8 x 10

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Time [s] Electric Field Strength [V/m]

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Stability-Corrected Wave Function

Photonic crystal

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Stability-Corrected Wave Function

m = 1000 vs. 8200 FDTD iterations

0.5 1 1.5 2 2.5 3 3.5 4 x 10

−13

−5 −4 −3 −2 −1 1 2 3 4 5 x 10

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Time [s] Electric Field Strength [V/m]

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Introduction Basic equations Lanczos algorithms PML

Stability-Corrected Wave Function

m = 2000 vs. 8200 FDTD iterations

0.5 1 1.5 2 2.5 3 3.5 4 x 10

−13

−5 −4 −3 −2 −1 1 2 3 4 5 x 10

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Time [s] Electric Field Strength [V/m]

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Introduction Basic equations Lanczos algorithms PML

Stability-Corrected Wave Function

m = 3000 vs. 8200 FDTD iterations

0.5 1 1.5 2 2.5 3 3.5 4 x 10

−13

−5 −4 −3 −2 −1 1 2 3 4 5 x 10

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Time [s] Electric Field Strength [V/m]

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Introduction Basic equations Lanczos algorithms PML

Extensions

Approach has been extended for dispersive media in

  • J. Zimmerling, L. Wei, H. Urbach, and R. Remis, A Lanczos

model-order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media, Journal

  • f Computational Physics, Vol. 315, pp. 348 – 362, 2016.

Extended Krylov subspace implementations are discussed in

  • V. Druskin, R. Remis, and M. Zaslavsky, Journal of

Computational Physics, Vol. 272, pp. 608 – 618, 2014.

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Current and future work

Rational Krylov field approximations

No stability-correction required

Phase-preconditioned rational Krylov methods

Large travel times

More on this in the coming week!

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Thank you for your attention!

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