@article {peters2018pmf,
	title = {Projection methods and applications for seismic nonlinear inverse problems with multiple constraints},
	journal = {Geophysics},
	volume = {84},
	number = {2},
	year = {2018},
	note = {(Geophysics)},
	month = {02},
	pages = {R251-R269},
	abstract = {Nonlinear inverse problems are often hampered by non-uniqueness and local minima because of missing low frequencies and far offsets in the data, lack of access to good starting models, noise, and modeling errors. A well-known approach to counter these deficiencies is to include prior information on the unknown model, which regularizes the inverse problem. While conventional regularization methods have resulted in enormous progress in ill-posed (geophysical) inverse problems, challenges remain when the prior information consists of multiple pieces. To handle this situation, we propose an optimization framework that allows us to add multiple pieces of prior information in the form of constraints. Compared to additive regularization penalties, constraints have a number of advantages making them more suitable for inverse problems such as full-waveform inversion. The proposed framework is rigorous because it offers assurances that multiple constraints are imposed uniquely at each iteration, irrespective of the order in which they are invoked. To project onto the intersection of multiple sets uniquely, we employ Dykstra{\textquoteright}s algorithm that scales to large problems and does not rely on trade-off parameters. In that sense, our approach differs substantially from approaches such as Tikhonov regularization, penalty methods, and gradient filtering. None of these offer assurances, which makes them less suitable to full-waveform inversion where unrealistic intermediate results effectively derail the iterative inversion process. By working with intersections of sets, we keep expensive objective and gradient calculations unaltered, separate from projections, and we also avoid trade-off parameters. These features allow for easy integration into existing code bases. In addition to more predictable behavior, working with constraints also allows for heuristics where we built up the complexity of the model gradually by relaxing the constraints. This strategy helps to avoid convergence to local minima that represent unrealistic models. We illustrate this unique feature with examples of varying complexity.},
	keywords = {constraints, Full-waveform inversion, intersection, Optimization, projection, regularization},
	doi = {10.1190/geo2018-0192.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2018/peters2018pmf/peters2018pmf.html},
	author = {Bas Peters and Brendan R. Smithyman and Felix J. Herrmann}
}
@conference {peters2016SEGprs,
	title = {Parallel reformulation of the sequential adjoint-state method},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {1411-1415},
	abstract = {This abstract is about reducing the total computation time to solve full-waveform inversion problems. A common problem formulation is a nonlinear least-squares problem where the gradient is computed using the adjoint-state algorithm. This formulation offers parallelism for the computation of gradients for different sources and frequencies. The adjoint-state algorithm itself is sequential however and this is a limiting factor when a lot of compute nodes are available and only a few wavefields need to be computed. This situation occurs when stochastic optimization strategies are used to minimize the objective function. We present a parallel reformulation of the sequential adjoint-state algorithm, which allows the forward- and adjoint wavefields to be computed in parallel. Both algorithms are mathematically equivalent but the parallel version is twice as fast in run time. An important characteristic of the proposed algorithm is that one wavefield needs to be computed per source and one per receiver. These fields can be used to apply the (inverse) Gauss-Newton Hessian to a vector without recomputing wavefields. A 2D example shows that good full-waveform inversion results are obtained, even when a small number of sources and receivers is used.},
	keywords = {FWI, numerical linear algebra, parallel computing, PDE-constrained optimization, SEG},
	doi = {10.1190/segam2016-13966771.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/peters2016SEGprs/peters2016SEGprs.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/peters2016SEGprs/peters2016SEGprs_pres.pdf},
	author = {Bas Peters and Tristan van Leeuwen and Felix J. Herrmann}
}
@conference {herrmann2015SIAMpcf,
	title = {Pros and cons of full- and reduced-space methods for Wavefield Reconstruction Inversion},
	booktitle = {SIAM Conference on Mathematical and Computational Issues in the Geosciences},
	year = {2015},
	note = {(SIAM Conference on Mathematical and Computational Issues in the Geosciences, Stanford University, California)},
	month = {06-07},
	abstract = {By insisting on fitting observed data, Wavefield Reconstruction Inversion (WRI) is no longer cycle skipped and therefore less reliant on the accuracy of starting models. While extending the search space mitigates local minima, there are challenges scaling to 3D seismic when using reduced-space methods that require accurate solves. Conversely, full-space methods allow for inaccurate solves but require storage of all wavefields. We weigh pros and cons of these two approaches in the seismic context.},
	keywords = {FWI, SIAM, WRI},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SIAM/2015/herrmann2015SIAMpcf/herrmann2015SIAMpcf_pres.pdf},
	author = {Felix J. Herrmann and Bas Peters}
}
