@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 {fang2017AIPepm,
	title = {An efficient penalty method for PDE-constrained optimization problem with source estimation and stochastic optimization},
	booktitle = {Applied Inverse Problems Annual Conference Proceedings},
	year = {2017},
	note = {(AIP, Hangzhou)},
	month = {05-06},
	pages = {40},
	abstract = {Many inverse problems in applied science and engineering can be written as PDE-constrained optimization problems. Typically, one is interested in estimating the unknown medium parameters of the PDE describing the system, given multi-source measurements of the fields in the true medium. The objective function is often very oscillatory, which gives rise to problematic local minima, and merely evaluating one or more of its derivatives requires solving multiple PDEs, which are expensive. In many applications such as seismic exploration, only the source location is available and the source signature itself is unknown, which results in erroneous estimated parameters. In this work, we propose a penalty method to solve the PDE-constrained optimization problem, whereby we replace the PDE constraints by a two-norm penalty term, which leads to a bi-linear optimization problem with a much less oscillatory objective function. By applying the variable projection method, we develop an efficient method to invert the unknown parameters and source signatures simultaneously. To reduce the computational cost, we randomly select a subset of all data to update the parameters at each iteration. Numerical examples demonstrate that this method is less prone to becoming stuck in harmful local minima and is able to successfully invert the unknown parameters and source signature with less computational cost compared to using all the data.},
	keywords = {Optimization, penalty method, source estimation, Stochastic optimization},
	url = {http://aip2017.csp.escience.cn/dct/page/1},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/AIP/2017/fang2017AIPepm/fang2017AIPepm_pres.pdf},
	author = {Zhilong Fang and Curt Da Silva and Felix J. Herrmann}
}
@article {vanleeuwen2015IPpmp,
	title = {A penalty method for PDE-constrained optimization in inverse problems},
	journal = {Inverse Problems},
	volume = {32},
	number = {1},
	year = {2015},
	note = {(Inverse Problems)},
	month = {12},
	pages = {015007},
	abstract = {Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the parameters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the nonlinearity of the problem and is less sensitive to the initial iterate.},
	keywords = {Inverse problems, Optimization, PDE, penalty method},
	url = {https://slim.gatech.edu/Publications/Public/Journals/InverseProblems/2015/vanleeuwen2015IPpmp/vanleeuwen2015IPpmp.pdf},
	url2 = {http://stacks.iop.org/0266-5611/32/i=1/a=015007},
	author = {Tristan van Leeuwen 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}
}
@techreport {herrmann2014pmpde,
	title = {A penalty method for PDE-constrained optimization},
	number = {WO 2014/172787},
	year = {2014},
	note = {(International publication date 30 October 2014. International publication number WO 2014/172787.)},
	month = {10},
	institution = {UBC},
	type = {Patent},
	abstract = {The invention is directed to a computer-implemented method for obtaining a physical model having physical model parameters wherein solutions to one or more partial-differential-equations (PDE{\textquoteright}s) are calculated ans wherein (i) an appropriate initial model is selected, (ii) setup a system of equations referred to as the data-augmented PDE for the field, comprising of the discretized PDE, the sampling operator, the source function and the observed data, and solve the data-augmented PDE in a suitable manner to obtain a field that both satisfies the PDE and fits the data to some degree, (iii) setup a system of equations by using the PDE, the source function and the field obtained in step (ii) and solve this system of equations in a suitable manner to obtain an update of the physical model parameters and repeat steps (ii)-(iii) until a predetermined stopping criterion is met.},
	keywords = {Optimization, patent, penalty method},
	url = {https://slim.gatech.edu/Publications/Public/Patents/2014/herrmann2014pmpde/herrmann2014pmpde_WO2014172787.pdf},
	url2 = {http://patentscope.wipo.int/search/en/WO2014172787},
	author = {Felix J. Herrmann and Tristan van Leeuwen}
}
@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}
}
@conference {herrmann2013SEGpmc,
	title = {A penalty method for PDE-constrained optimization with applications to wave-equation based seismic inversion},
	booktitle = {SEG Workshop on Computational Mathematics for Geophysics; Houston},
	year = {2013},
	month = {09},
	keywords = {Optimization, penalty method, SEG, workshop},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2013/herrmann2013SEGpmc/herrmann2013SEGpmc_pres.pdf},
	author = {Tristan van Leeuwen and Felix J. Herrmann}
}
