@conference {peters2019SEGans,
	title = {A numerical solver for least-squares sub-problems in 3D wavefield reconstruction inversion and related problem formulations},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2019},
	note = {(SEG, San Antonio)},
	month = {09},
	pages = {1536-1540},
	abstract = {Recent years saw a surge of interest in seismic waveform inversion approaches based on quadratic-penalty or augmented-Lagrangian methods, including Wavefield Reconstruction Inversion. These methods typically need to solve a least-squares sub-problem that contains a discretization of the Helmholtz equation. Memory requirements for direct solvers are often prohibitively large in three dimensions, and this limited the examples in the literature to two dimensions. We present an algorithm that uses iterative Helmholtz solvers as a black-box to solve the least-squares problem corresponding to 3D grids. This algorithm enables Wavefield Reconstruction Inversion and related formulations, in three dimensions. Our new algorithm also includes a root-finding method to convert a penalty into a constraint on the data-misfit without additional computational cost, by reusing precomputed quantities. Numerical experiments show that the cost of parallel communication and other computations are small compared to the main cost of solving one Helmholtz problem per source and one per receiver.},
	keywords = {Full-waveform inversion, least-squares, numerical linear algebra, penalty method, quadratic constraints, root finding, SEG},
	doi = {10.1190/segam2019-3216638.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2019/peters2019SEGans/peters2019SEGans.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2019/peters2019SEGans/peters2019SEGans_pres.pdf},
	author = {Bas Peters and Felix J. Herrmann}
}
@conference {peters2015AIPwri,
	title = {Wavefield-reconstruction inversion},
	booktitle = {Conference on Applied Inverse Problems},
	year = {2015},
	note = {(AIP, Helsinki)},
	month = {05},
	abstract = {Wavefield Reconstruction Inversion is a method for PDE-constrained optimization, which revolves around the estimation of fields using the PDE as well as the observed data in a least-squares sense. The method is quadratic penalty based, which offers some interesting possibilities for the construction of algorithms, compared to the Lagrangian form. One of the main benefits of the method is when the initial guess is far from the global minimizer. Reduced-space and full-space algorithms are discussed, including illustrative examples. The method was developed with seismic applications in mind, but applies to other PDE-constrained optimization problems as well.},
	keywords = {AIP, Full-waveform inversion, Optimization, penalty method, Waveform reconstruction inversion},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/AIP/2015/peters2015AIPwri/peters2015AIPwri.pdf},
	author = {Bas Peters and Felix J. Herrmann}
}
@conference {leeuwen2014EAGEntf,
	title = {A new take on FWI: wavefield reconstruction inversion},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2014},
	month = {06},
	abstract = {We discuss a recently proposed novel method for waveform inversion: Wavefield Reconstruction Inversion (WRI). As opposed to conventional FWI {\textendash} which attempts to minimize the error between observed and predicted data obtained by solving a wave equation {\textendash} WRI reconstructs a wave-field from the data and extracts a model-update from this wavefield by minimizing the wave-equation residual. The method does not require explicit computation of an adjoint wavefield as all the necessary information is contained in the reconstructed wavefield. We show how the corresponding model updates can be interpreted physically analogously to the conventional imaging-condition-based approach.},
	keywords = {EAGE, Full-waveform inversion, Optimization, penalty method, Wavefield Reconstruction Inversion},
	doi = {10.3997/2214-4609.20140703},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/leeuwen2014EAGEntf/leeuwen2014EAGEntf.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/leeuwen2014EAGEntf/leeuwen2014EAGEntf_pres.pdf},
	author = {Tristan van Leeuwen and Felix J. Herrmann and Bas Peters}
}
@conference {peters2014SEGsrh,
	title = {A sparse reduced Hessian approximation for multi-parameter wavefield reconstruction inversion},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2014},
	note = {(SEG)},
	month = {10},
	pages = {1206-1210},
	abstract = {Multi-Parameter full-waveform inversion is a challenging problem, because the unknown parameters appear in the same wave equation and the magnitude of the parameters can vary many orders of magnitude. This makes accurate estimation of multiple-parameters very difficult. To mitigate the problems, sequential strategies, regularization methods and scalings of gradients and quasi-Newton Hessians have been proposed. All of these require design, fine-tuning and adaptation to different waveform inversion problems. We propose to use a sparse approximation to the Hessian derived from a penalty-formulation of the objective function. Sparseness allows to have the Hessian in memory and compute update directions at very low cost. This results in decent reconstruction of the multiple parameters at very low additional memory and computational expense.},
	keywords = {Full-waveform inversion, Hessian, Optimization, penalty method, SEG},
	doi = {http://dx.doi.org/10.1190/segam2014-1667.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2014/peters2014SEGsrh/peters2014SEGsrh.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2014/peters2014SEGsrh/peters2014SEGsrh_pres.pdf},
	author = {Bas Peters and Felix J. Herrmann}
}
@conference {peters2014EAGEweb,
	title = {Wave-equation based inversion with the penalty method: adjoint-state versus wavefield-reconstruction inversion},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2014},
	month = {06},
	abstract = {In this paper we make a comparison between wave-equation based inversions based on the adjoint-state and penalty methods. While the adjoint-state method involves the minimization of a data-misfit and exact solutions of the wave-equation for the current velocity model, the penalty-method aims to first find a wavefield that jointly fits the data and honours the physics, in a least-squares sense. Given this reconstructed wavefield, which is a proxy for the true wavefield in the true model, we calculate updates for the velocity model. Aside from being less nonlinear{\textendash}the acoustic wave equation is linear in the wavefield and model parameters but not in both{\textendash}the inversion is carried out over a solution space that includes both the model and the wavefield. This larger search space allows the algortihm to circumnavigate local minima, very much in the same way as recently proposed model extentions try to acomplish. We include examples for low frequencies, where we compare full-waveform inversion results for both methods, for good and bad starting models, and for high frequencies where we compare reverse-time migration with linearized imaging based on wavefield-reconstruction inversion. The examples confirm the expected benefits of the proposed method.},
	keywords = {EAGE, Full-waveform inversion, Imaging, Optimization, penalty method},
	doi = {10.3997/2214-4609.20140704},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/peters2014EAGEweb/peters2014EAGEweb.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/peters2014EAGEweb/peters2014EAGEweb_pres.pdf},
	author = {Bas Peters and Felix J. Herrmann and Tristan van Leeuwen}
}
