Seismic waveform inversion aims at obtaining detailed estimates of subsurface medium parameters, such as the spatial distribution of soundspeed, from multiexperiment seismic data. A formulation of this inverse problem in the frequency domain leads to an optimization problem constrained by a Helmholtz equation with many right-hand sides. Application of this technique to industry-scale problems faces several challenges: First, we need to solve the Helmholtz equation for high wave numbers over large computational domains. Second, the data consist of many independent experiments, leading to a large number of PDE solves. This results in high computational complexity both in terms of memory and CPU time as well as input/output costs. Finally, the inverse problem is highly nonlinear and a lot of art goes into preprocessing and regularization. Ideally, an inversion needs to be run several times with different initial guesses and/or tuning parameters. In this paper, we discuss the requirements of the various components (PDE solver, optimization method, \dots) when applied to large-scale three-dimensional seismic waveform inversion and combine several existing approaches into a flexible inversion scheme for seismic waveform inversion. The scheme is based on the idea that in the early stages of the inversion we do not need all the data or very accurate PDE solves. We base our method on an existing preconditioned Krylov solver (CARP-CG) and use ideas from stochastic optimization to formulate a gradient-based (quasi-Newton) optimization algorithm that works with small subsets of the right-hand sides and uses inexact PDE solves for the gradient calculations. We propose novel heuristics to adaptively control both the accuracy and the number of right-hand sides. We illustrate the algorithms on synthetic benchmark models for which significant computational gains can be made without being sensitive to noise and without losing the accuracy of the inverted model.


  1. seismic inversion
  2. Helmholtz equation
  3. preconditioning
  4. Kaczmarz method
  5. inexact gradient
  6. block-cg

MSC codes

  1. 65N21
  2. 65K10

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: S192 - S217
ISSN (online): 1095-7197


Submitted: 26 April 2013
Accepted: 14 March 2014
Published online: 30 October 2014


  1. seismic inversion
  2. Helmholtz equation
  3. preconditioning
  4. Kaczmarz method
  5. inexact gradient
  6. block-cg

MSC codes

  1. 65N21
  2. 65K10



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