@article {li2015GEOPmgn,
	title = {Modified Gauss-Newton full-waveform inversion explained{\textendash}-why sparsity-promoting updates do matter},
	journal = {Geophysics},
	volume = {81},
	number = {3},
	year = {2016},
	note = {(Geophysics)},
	month = {05},
	pages = {R125-R138},
	abstract = {Full-waveform inversion can be formulated as a nonlinear least-squares optimization problem. This non-convex problem can be extremely computationally expensive because it requires repeatedly solving large linear systems that correspond to discretized partial differential equations. Randomized subsampling techniques allow us to work with small subsets of (monochromatic) source experiments, reducing the computational cost. However, this subsampling weakens subsurface illumination and introduces subsampling related incoherent artifacts. These subsampling-related artifacts{\textendash}-in conjunction with local minima that are known to plague full-waveform inversion{\textendash}-motivate us to come up with a technique to "regularize" this problem. Following earlier work, we take advantage of the fact that curvelets represent subsurface models and perturbations parsimoniously. At first impulse promoting sparsity on the model directly seems the most natural way to proceed, but we will demonstrate that in certain cases it can be advantageous to promote sparsity on the Gauss-Newton updates instead. While constraining the one-norm of the descent directions does not change not change the underlying full-waveform inversion objective, the constrained model updates remain descent directions, remove subsampling-related artifacts and improve the overall inversion result. We empirically observe this phenomenon in situations where the different model updates occur at roughly the same locations in the curvelet domain. We further investigate and analyze this phenomenon, where nonlinear inversions benefit from sparsity-promoting constraints on the updates, by means of a set of carefully selected examples including phase retrieval and full-waveform inversion. In all cases, we observe a faster decay of the residual and model error as a function of the number of iterations.},
	keywords = {Full-waveform inversion, Gauss-Newton method, sparsity promotion},
	doi = {10.1190/geo2015-0266.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2016/li2015GEOPmgn/li2015GEOPmgn.pdf},
	author = {Xiang Li and Ernie Esser and Felix J. Herrmann}
}
@mastersthesis {li2015THsps,
	title = {Sparsity promoting seismic imaging and full-waveform inversion},
	year = {2015},
	note = {(PhD)},
	month = {07},
	school = {The University of British Columbia},
	type = {phd},
	address = {Vancouver},
	abstract = {This thesis will address the large computational costs of solving least-squares migration and full-waveform inversion problems. Least-squares seismic imaging and full-waveform inversion are seismic inversion techniques that require iterative minimizations of large least-squares misfit functions. Each iteration requires an evaluation of the Jacobian operator and its adjoint, both of which require two wave-equation solves for all sources, creating prohibitive computational costs. In order to reduce costs, we utilize randomized dimensionality reduction techniques, reducing the number of sources used during inversion. The randomized dimensionality reduction techniques create subsampling related artifacts, which we mitigate by using curvelet-domain sparsity-promoting inversion techniques. Our method conducts least-squares imaging at the approximate cost of one reverse-time migration with all sources, and computes the Gauss-Newton full-waveform inversion update at roughly the cost of one gradient update with all sources. Finally, during our research of the full-waveform inversion problem, we discovered that we can utilize our method as an alternative approach to add sparse constraints on the entire velocity model by imposing sparsity constraints on each model update separately, rather than regularizing the total velocity model as typically practiced. We also observed this alternative approach yields a faster decay of the residual and model error as a function of iterations. We provided empirical arguments why and when imposing sparsity on the updates can lead to improved full-waveform inversion results.},
	keywords = {Full-waveform inversion, Gauss-Newton method, least-squares imaging, seismic imaging, sparsity promoting},
	url = {https://slim.gatech.edu/Publications/Public/Thesis/2015/li2015THsps/li2015THsps.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Thesis/2015/li2015THsps/li2015THsps_pres.pdf},
	author = {Xiang Li}
}
