Fast 3D Helmholtz Solvers for Seismic Inversion in the Frequency - PowerPoint PPT Presentation

Fast 3D Helmholtz Solvers for Seismic Inversion in the Frequency Domain Russell J. Hewett Mathematics & CMDA, Virginia Tech Theory and Experience in Solving Inverse Problems in Geophysics Workshop Uppsala University April 10, 2019 RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 1 / 42

Collaborators ◮ Leonardo Zepeda-Nu˜ nez, Lawrence Berkeley National Lab ◮ Matthias Taus, TU Wien ◮ Laurent Demanet, MIT ◮ Adrien Scheuer, Universit` e Catholique de Louvain RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 2 / 42

FWI in the Frequency Domain PDE constrained optimization in frequency domain ◮ min J ( m ) = 1 2 || d −F ( m ) || 2 2 s.t. Lu = f Advantages: ◮ No need to invert source time series ˆ f ( ω ) = FFT ( f ( t )) ◮ Only need specific frequency components RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 3 / 42

FWI in the Frequency Domain PDE constrained optimization in frequency domain ◮ min J ( m ) = 1 2 || d −F ( m ) || 2 2 s.t. Lu = f Advantages: ◮ Reduced memory and disk requirements in inverse problem � T δm = − � q, ∂ tt u 0 � T = − q ( x, t ) ∂ tt u 0 ( x, t ) dt 0 becomes � q, − ω 2 u 0 q ( x, ω ) − ω 2 ˆ � � δm = − Ω = − ˆ u 0 ( x, ω ) ω ◮ Hybrid modeling: Use time-domain + DFT to achieve frequency domain update RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 3 / 42

FWI in the Frequency Domain PDE constrained optimization in frequency domain ◮ min J ( m ) = 1 2 || d −F ( m ) || 2 2 s.t. Lu = f Advantages: ◮ Multiple simultaneous right-hand sides ◮ With a factorization based method, only need to Helmholtz operator once per domain ◮ Compare to explicit time-stepping: matvec required for each time step for each source RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 3 / 42

FWI in the Frequency Domain PDE constrained optimization in frequency domain ◮ min J ( m ) = 1 2 || d −F ( m ) || 2 2 s.t. Lu = f Advantages: ◮ Heirarchichal frequency “sweeping” ⇒ Convergence guarantees (E. Beretta, M.V. de Hoop, F. Faucher, O. Scherzer (SIMA 2016)) RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 3 / 42

FWI in the Frequency Domain 0 20 40 60 80 100 120 140 0 100 200 300 400 500 0 20 40 60 80 100 120 140 0 100 200 300 400 500 0 20 40 60 80 100 120 140 0 100 200 300 400 500 Created with PySIT (www.pysit.org). RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 4 / 42

FWI in the Frequency Domain PDE constrained optimization in frequency domain ◮ min J ( m ) = 1 2 || d −F ( m ) || 2 2 s.t. Lu = f Challenges: ◮ Helmholtz in high frequency regime ◮ Helmholtz in 3D at high resolution ◮ Scalable Helmholtz in HPC environment RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 5 / 42

Motivation for Sweeping Solvers Helmholtz at high frequency is hard Hu = ( − ω 2 − △ ) u = f + ABCs ◮ Frequency ω grows with n ◮ Computational load N scales with n d RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 6 / 42

Motivation for Sweeping Solvers Helmholtz at high frequency is hard Hu = ( − ω 2 − △ ) u = f + ABCs ◮ Frequency ω grows with n ◮ Computational load N scales with n d Classical dense direct methods in 3D ◮ memory-intensive ◮ hard to parallelize RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 6 / 42

Motivation for Sweeping Solvers Helmholtz at high frequency is hard Hu = ( − ω 2 − △ ) u = f + ABCs ◮ Frequency ω grows with n ◮ Computational load N scales with n d Classical dense direct methods in 3D ◮ memory-intensive ◮ hard to parallelize Multigrid methods ◮ poor frequency scaling ◮ down-sampling oscillatory waves is hard RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 6 / 42

Motivation for Sweeping Solvers Helmholtz at high frequency is hard Hu = ( − ω 2 − △ ) u = f + ABCs ◮ Frequency ω grows with n ◮ Computational load N scales with n d Classical dense direct methods in 3D ◮ memory-intensive ◮ hard to parallelize Multigrid methods ◮ poor frequency scaling ◮ down-sampling oscillatory waves is hard Classical iterative schemes ◮ n iter grows with ω RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 6 / 42

Sweeping Solvers and Domain Decomposition Methods Sweeping Solvers/Preconditioners ◮ First O ( N ) claim (Engquist and Ying, 2010) ◮ First O ( N ) claim w/ domain decomposition (Stolk 2013) RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 7 / 42

Sweeping Solvers and Domain Decomposition Methods Sweeping Solvers/Preconditioners ◮ First O ( N ) claim (Engquist and Ying, 2010) ◮ First O ( N ) claim w/ domain decomposition (Stolk 2013) Other domain-decomposition methods (DDMs): ◮ Multifrontal w/ HSS compression (Xia, et al., 2013) ◮ Hierarchical Poincare-Steklov methods (Gillman, et al., 2014) ◮ Common challenges: ◮ Hazy scalability ◮ Issues with rough media RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 7 / 42

Sweeping Solvers and Domain Decomposition Methods Sweeping Solvers/Preconditioners ◮ First O ( N ) claim (Engquist and Ying, 2010) ◮ First O ( N ) claim w/ domain decomposition (Stolk 2013) Other domain-decomposition methods (DDMs): ◮ Multifrontal w/ HSS compression (Xia, et al., 2013) ◮ Hierarchical Poincare-Steklov methods (Gillman, et al., 2014) ◮ Common challenges: ◮ Hazy scalability ◮ Issues with rough media Our approach: DDMs + sweeping w/ polarized traces ◮ Use direct methods distributed over tractable subproblems ◮ Glue with boundary integral formulations ◮ Embedded within iterative scheme RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 7 / 42

Take-home Messages 1. Polarized traces: domain decomposition done right ◮ Maximizes leveraging legacy direct solvers in the subdomains ◮ Maximal flexibility in parallel computation RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 8 / 42

Take-home Messages 1. Polarized traces: domain decomposition done right ◮ Maximizes leveraging legacy direct solvers in the subdomains ◮ Maximal flexibility in parallel computation 2. The number of iterations is ◮ Weakly dependent on the frequency ◮ Weakly dependent on the number L of subdomains ◮ Robust to roughness in background model RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 8 / 42

Take-home Messages 1. Polarized traces: domain decomposition done right ◮ Maximizes leveraging legacy direct solvers in the subdomains ◮ Maximal flexibility in parallel computation 2. The number of iterations is ◮ Weakly dependent on the frequency ◮ Weakly dependent on the number L of subdomains ◮ Robust to roughness in background model 3. Precondition interdomain communication with polarizing integral conditions RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 8 / 42

Take-home Messages 1. Polarized traces: domain decomposition done right ◮ Maximizes leveraging legacy direct solvers in the subdomains ◮ Maximal flexibility in parallel computation 2. The number of iterations is ◮ Weakly dependent on the frequency ◮ Weakly dependent on the number L of subdomains ◮ Robust to roughness in background model 3. Precondition interdomain communication with polarizing integral conditions 4. Can achieve sublinear complexity: better than O ( RN ) RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 8 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 9 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 9 / 42

Half-space Problem Polarization condition: � G ( x, y ) ∂ n y u ↑ ( y ) ds y 0 = − u Γ � Γ i,i +1 ∂ n y G ( x, y ) u ↑ ( y ) ds y f + Γ RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 10 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 11 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 11 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 11 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 11 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 11 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 11 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 11 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 11 / 42

Method Of Polarized Traces RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 11 / 42

A System for Traces ◮ Assume local PDE is solved in the bulk v 1   n v 2   1  v 2  ◮ Traces can be found by solving M u = f =   n  .  .   .   v L 1 ◮ M is constructed from dense Green’s function blocks. . . ◮ . . . non-trivial to invert RJH (Virginia Tech) Polarized Traces Uppsala / April 10, 2019 12 / 42