Krylov Model-Order Reduction Techniques for Time- and Frequency-Domain Wavefjeld Problems Rob Remis Delft University of Technology November 6 – 10, 2017 – ICERM Brown University 1
Acknowledgment This is joint work with Mikhail Zaslavsky, Schlumberger-Doll Research Jörn Zimmerling, Delft University of Technology Vladimir Druskin, Schlumberger-Doll Research 2
Outline Basic equations and symmetry properties Polynomial Krylov reduction 1 Perfectly matched layers Polynomial Krylov reduction 2 Laurent polynomial (extended) Krylov reduction Introducing rational and preconditioned rational Krylov reduction More on this in J. Zimmerling’s talk 3
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods Basic equations First-order wavefjeld system acoustics, seismics, electrodynamics Plus initial conditions Dirichlet boundary conditions (acoustics) 4 ( D + M ∂ t ) F = − w ( t ) Q Signature matrix δ − δ − = diag ( 1 , − 1 , − 1 , − 1 )
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods Basic equations Spatial discretization Order of this system can be very large especially in 3D 5 ( D + M ∂ t ) f = − w ( t ) q Discretized counterpart of δ − is denoted by d −
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods Basic equations Solution System matrix 6 f ( t ) = − w ( t ) ∗ η ( t ) exp( − At ) M − 1 q η ( t ) Heaviside unit step function A = M − 1 D
Basic equations & symmetry Polynomial Krylov reduction 1 Initial-value problem: norm of f is preserved 1 (sum of fjeld energies) Stored fjeld energy in the computational domain Inner product and norm System matrix A is skew-symmetric w.r.t. WM Symmetry (Preconditioned) rational Krylov methods Extended Krylov Reduction Polynomial Krylov Reduction 2 PML 7 Evolution operator exp( − At ) is orthogonal w.r.t. WM � x � = � x , x � 1 / 2 � x , y � = y H WMx 2 � f � 2
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods Symmetry Bilinear form Free fjeld Lagrangian (difgerence of fjeld energies) 1 Symmetry property related to reciprocity 8 System matrix A is symmetric w.r.t. WMd − � x , y � = y H WMd − x 2 � f , f �
Basic equations & symmetry Introduce Reciprocity source q Polynomial Krylov reduction 1 and Symmetry (Preconditioned) rational Krylov methods Extended Krylov Reduction Polynomial Krylov Reduction 2 PML 9 d p = 1 d m = 1 2 ( I + d − ) 2 ( I − d − ) Write f = f ( q ) to indicate that the fjeld is generated by a Source vector: q = d p q , receiver vector r = d p r � f ( q ) , r � = � q , f ( r ) � Source vector: q = d p q , receiver vector r = d m r � f ( q ) , r � = −� q , f ( r ) �
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods Polynomial Krylov reduction Exploit symmetry of system matrix in a Lanczos reduction algorithm For lossless media both symmetry properties lead to the same reduction algorithm First symmetry property is lost for lossy media with a system Second symmetry property still holds 10 matrix of the form A = M − 1 ( D + S )
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods Polynomial Krylov reduction For lossless media, FDTD can be written in a similar form as Lanczos algorithm recurrence relation for FDTD = recurrence relation for Fibonacci polynomials Stability of FDTD and numerical dispersion can be studied using this connection 11
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods Polynomial Krylov reduction Automatic time step adaptation – no Courant condition Lanczos reduction hardly provides any speedup compared with FDTD Both are polynomial fjeld approximations Lanczos: fjeld is approximated by a Lanczos polynomial in A FDTD: fjeld is approximated by a Fibonacci polynomial in A 12 Lanczos recurrence coeffjcients: β i Comparison with FDTD: 1 /β i = time step of Lanczos
Basic equations & symmetry No outward wave propagation has been included yet s Stretching function Polynomial Krylov reduction 1 Implementation via Perfectly Matched Layers (PML) Coordinate stretching (Laplace domain) PML (Preconditioned) rational Krylov methods Extended Krylov Reduction Polynomial Krylov Reduction 2 PML 13 ∂ k ← → χ − 1 k ∂ k k = x , y , z χ k ( k , s ) = α k ( k ) + β k ( k )
Basic equations & symmetry PML Direct spatial discretization Polynomial Krylov reduction 1 Stretched fjrst-order system 14 (Preconditioned) rational Krylov methods Extended Krylov Reduction Polynomial Krylov Reduction 2 PML � ˆ � D ( s ) + S + s M F = − ˆ w ( s ) Q � ˆ � D ( s ) + S + sM f = − ˆ w ( s ) q Leads to nonlinear eigenproblems for spatial dimensions > 1
Basic equations & symmetry Polynomial Krylov reduction 1 Helmholtz problems,” SIAM Rev. 58-1 (2016), pp. 90 – 116. V. Druskin, S. Güttel, and L. Knizhnerman, “Near-optimal perfectly matched layers for indefjnite propagation in unbounded domains,” SIAM J. Sci. Comput. , Vol. 35, 2013, pp. B376 – B400. V. Druskin and R. F. Remis, “A Krylov stability-corrected coordinate stretching method to simulate wave System matrix sizes Spatial fjnite-difgerence discretization using complex PML step Linearization of the PML PML (Preconditioned) rational Krylov methods Extended Krylov Reduction Polynomial Krylov Reduction 2 PML 15 ( D cs + S + sM ) f cs = − w ( s ) q A cs = M − 1 ( D cs + S )
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods PML Spectrum of the system matrix A cs 16 Im( λ ) Re( λ ) Lossless resonator
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods PML Eigenvalues move into the complex plane 17 Im( λ ) Re( λ ) Complex scaling
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods PML Stable part of the spectrum 18 Im( λ ) Re( λ ) Stable part
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods PML Stability correction 19 Im( λ ) Re( λ ) Stable part
Basic equations & symmetry Time-domain stability-corrected wave function 0 1 Complex Heaviside unit step function Polynomial Krylov reduction 1 20 PML (Preconditioned) rational Krylov methods Extended Krylov Reduction Polynomial Krylov Reduction 2 PML � � f ( t ) = − w ( t ) ∗ 2 η ( t ) Re η ( A cs ) exp( − A cs t ) q � Re ( z ) > 0 η ( z ) = Re ( z ) < 0
Basic equations & symmetry Frequency-domain stability-corrected wave function function is a nonentire function of the system matrix A cs with q Polynomial Krylov reduction 1 21 PML (Preconditioned) rational Krylov methods PML Extended Krylov Reduction Polynomial Krylov Reduction 2 ˆ r ( A cs , s ) + r (¯ � � f ( s ) = − ˆ w ( s ) A cs , s ) r ( z , s ) = η ( z ) z + s s ) = ¯ Note that ˆ ˆ f (¯ f ( s ) and the stability-corrected wave
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods PML Symmetry relations are preserved With a step size matrix W that has complex entries These entries correspond to PML locations 22
Basic equations & symmetry Polynomial Krylov reduction 1 PML Polynomial Krylov Reduction 2 Extended Krylov Reduction (Preconditioned) rational Krylov methods Polynomial Krylov Reduction Stability-corrected wave function cannot be computed by FDTD SLDM fjeld approximations via modifjed Lanczos algorithm Reduced-order model 23 f m ( t ) = − w ( t ) ∗ 2 � M − 1 q � η ( t ) Re [ V m η ( H m ) exp( − H m t ) e 1 ]
Basic equations & symmetry (Preconditioned) rational Krylov methods Polynomial Krylov reduction 1 Polynomial Krylov Reduction 24 Extended Krylov Reduction Polynomial Krylov Reduction 2 PML m = 300 8 1 x 10 0.8 0.6 0.4 Electric Field Strength [V/m] 0.2 0 − 0.2 − 0.4 − 0.6 − 0.8 − 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time [s] − 13 x 10
Basic equations & symmetry (Preconditioned) rational Krylov methods Polynomial Krylov reduction 1 Polynomial Krylov Reduction 25 Extended Krylov Reduction Polynomial Krylov Reduction 2 PML m = 400 7 8 x 10 6 4 Electric Field Strength [V/m] 2 0 − 2 − 4 − 6 − 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time [s] − 13 x 10
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