@article {louboutin2017fwip2,
	title = {Full-Waveform Inversion - Part 2: adjoint modeling},
	journal = {The Leading Edge},
	volume = {37},
	number = {1},
	year = {2018},
	note = {(The Leading Edge)},
	month = {1},
	pages = {69-72},
	abstract = {This tutorial is the second part of a three part tutorial series on full-waveform inversion (FWI), in which we provide a step by step walk through of setting up forward and adjoint wave equation solvers and an optimization framework for inversion. In part 1 (Louboutin et al., 2017), we demonstrated how to discretize the acoustic wave equation and how to set up a basic forward modeling scheme using Devito, a domain-specific language (DSL) in Python for automated finite-difference (FD) computations (Lange et al., 2016). Devito allows us to define wave equations as symbolic Python expressions (Meurer et al., 2017), from which optimized FD stencil code is automatically generated at run time. In part 1, we show how we can use Devito to set up and solve acoustic wave equations with (impulsive) seismic sources and sample wavefields at the receiver locations to model shot records.},
	keywords = {acoustic, devito, finite-difference, FWI, tutorial},
	doi = {10.1190/tle37010069.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/TheLeadingEdge/2018/louboutin2017fwip2/louboutin2017fwip2.html},
	author = {Mathias Louboutin and Philipp A. Witte and Michael Lange and Navjot Kukreja and Fabio Luporini and Gerard Gorman and Felix J. Herrmann}
}
@article {fang2017uqfip,
	title = {Uncertainty quantification for inverse problems with weak partial-differential-equation constraints},
	journal = {Geophysics},
	volume = {83},
	number = {6},
	year = {2018},
	note = {(Geophysics)},
	pages = {R629-R647},
	abstract = {In a statistical inverse problem, the objective is a complete statistical description of unknown parameters from noisy observations in order to quantify uncertainties of the parameters of interest. We consider inverse problems with partial-differential-equation-constraints, which are applicable to a variety of seismic problems. Bayesian inference is one of the most widely-used approaches to precisely quantify statistics through a posterior distribution, incorporating uncertainties in observed data, modeling kernel, and prior knowledge of the parameters. Typically when formulating the posterior distribution, the partial-differential-equation-constraints are required to be exactly satisfied, resulting in a highly nonlinear forward map and a posterior distribution with many local maxima. These drawbacks make it difficult to find an appropriate approximation for the posterior distribution. Another complicating factor is that traditional Markov chain Monte Carlo methods are known to converge slowly for realistically sized problems. In this work, we relax the partial-differential-equation-constraints by introducing an auxiliary variable, which allows for Gaussian deviations in the partial-differential-equations. Thus, we obtain a new bilinear posterior distribution consisting of both data and partial-differential-equation misfit terms. We illustrate that for a particular range of variance choices for the partial-differential-equation misfit term, the new posterior distribution has fewer modes and can be well-approximated by a Gaussian distribution, which can then be sampled in a straightforward manner. Since it is prohibitively expensive to explicitly construct the dense covariance matrix of the Gaussian approximation for intermediate to large-scale problems, we present a method to implicitly construct it, which enables efficient sampling. We apply this framework to two-dimensional seismic inverse problems with 1,800 and 92,455 unknown parameters. The results illustrate that our framework can produce comparable statistical quantities to those produced by conventional Markov chain Monte Carlo type methods while requiring far fewer partial-differential-equation solves, which are the main computational bottlenecks in these problems.},
	keywords = {acoustic, FWI, UQ, weak-constraint},
	doi = {10.1190/geo2017-0824.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2018/fang2017uqfip/fang2017uqfip.html},
	author = {Zhilong Fang and Curt Da Silva and Rachel Kuske and Felix J. Herrmann}
}
@conference {louboutin2015SEGtcs,
	title = {Time compressively sampled full-waveform inversion with stochastic optimization},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2015},
	note = {(SEG, New Orleans)},
	month = {10},
	pages = {5153-5157},
	abstract = {Time-domain Full-Waveform Inversion (FWI) aims to image the subsurface of the earth accurately from field recorded data and can be solved via the reduced adjoint-state method. However, this method requires access to the forward and adjoint wavefields that are meet when computing gradient updates. The challenge here is that the adjoint wavefield is computed in reverse order during time stepping and therefore requires storage or other type of mitigation because storing the full time history of the forward wavefield is too expensive in realistic 3D settings. To overcome this challenge, we propose an approximate adjoint-state method where the wavefields are subsampled randomly, which drastically the amount of storage needed. By using techniques from stochastic optimization, we control the errors induced by the subsampling. Examples of the proposed technique on a synthetic but realistic 2D model show that the subsampling-related artifacts can be reduced significantly by changing the sampling for each source after each model update. Combination of this gradient approximation with a quasi-Newton method shows virtually artifact free inversion results requiring only 5\% of storage compared to saving the history at Nyquist. In addition, we avoid having to recompute the wavefields as is required by checkpointing.},
	keywords = {acoustic, Full-waveform inversion, inversion, SEG, Stochastic optimization, Subsampling, Time-domain},
	doi = {10.1190/segam2015-5924937.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2015/louboutin2015SEGtcs/louboutin2015SEGtcs.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2015/louboutin2015SEGtcs/louboutin2015SEGtcs_poster.pdf},
	author = {Mathias Louboutin and Felix J. Herrmann}
}
@conference {miao2014SEGrhss,
	title = {Randomized HSS acceleration for full-wave-equation depth stepping migration},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2014},
	note = {(SEG)},
	month = {10},
	pages = {3752-3756},
	abstract = {In this work we propose to use the spectral projector (Kenney and Laub, 1995) and randomized HSS technique (Chandrasekaran et al., 2006) to achieve a stable and affordable two-way wave equation depth stepping migration algorithm.},
	keywords = {acoustic, depth extrapolation, full wave equation migration, randomized SVD, SEG, spectral projector},
	doi = {http://dx.doi.org/10.1190/segam2014-1417.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2014/miao2014SEGrhss/miao2014SEGrhss.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2014/miao2014SEGrhss/miao2014SEGrhss_pres.pdf},
	author = {Lina Miao and Polina Zheglova and Felix J. Herrmann}
}
