@article {liu2018ssi,
	title = {Simultaneous-shot inversion for PDE-constrained optimization problems with missing data},
	journal = {Inverse Problems},
	volume = {35},
	number = {2},
	year = {2018},
	note = {(Inverse Problems)},
	pages = {025003},
	abstract = {Stochastic optimization is key to efficient inversion in PDE-constrained optimization. Using {\textquoteleft}simultaneous shots{\textquoteright}, or random superposition of source terms, works very well in simple acquisition geometries where all sources see all receivers, but this rarely occurs in practice. We develop an approach that interpolates data to an ideal acquisition geometry while solving the inverse problem using simultaneous shots. The approach is formulated as a joint inverse problem, combining ideas from low-rank interpolation with full-waveform inversion. Results using synthetic experiments illustrate the flexibility and efficiency of the approach.},
	keywords = {Full-waveform inversion, low-rank interpolation, Optimization},
	doi = {10.1088/1361-6420/aaf317},
	url = {https://slim.gatech.edu/Publications/Public/Journals/InverseProblems/2018/liu2018ssi/liu2018ssi.pdf},
	author = {Michelle Liu and Rajiv Kumar and Eldad Haber and Aleksandr Y. Aravkin}
}
@article {esser2016tvr,
	title = {Total-variation regularization strategies in full-waveform inversion},
	journal = {SIAM Journal on Imaging Sciences},
	volume = {11},
	number = {1},
	year = {2018},
	note = {(SIAM Journal on Imaging Sciences)},
	pages = {376-406},
	abstract = {We propose an extended full-waveform inversion formulation that includes general convex constraints on the model. Though the full problem is highly nonconvex, the overarching optimization scheme arrives at geologically plausible results by solving a sequence of relaxed and warm-started constrained convex subproblems. The combination of box, total-variation, and successively relaxed asymmetric total-variation constraints allows us to steer free from parasitic local minima while keeping the estimated physical parameters laterally continuous and in a physically realistic range. For accurate starting models, numerical experiments carried out on the challenging 2004 BP velocity benchmark demonstrate that bound and total-variation constraints improve the inversion result significantly by removing inversion artifacts, related to source encoding, and by clearly improved delineation of top, bottom, and flanks of a high-velocity high-contrast salt inclusion. The experiments also show that for poor starting models these two constraints by themselves are insufficient to detect the bottom of high-velocity inclusions such as salt. Inclusion of the one-sided asymmetric total-variation constraint overcomes this issue by discouraging velocity lows to buildup during the early stages of the inversion. To the author{\textquoteright}s knowledge the presented algorithm is the first to successfully remove the imprint of local minima caused by poor starting models and band-width limited finite aperture data.},
	keywords = {constrained optimization, Full-waveform inversion, hinge loss, salt, total variation},
	doi = {10.1137/17M111328X},
	url = {https://slim.gatech.edu/Publications/Public/Journals/SIAMJournalOnImagingSciences/2018/esser2016tvr/esser2016tvr.pdf},
	url2 = {https://doi.org/10.1137/17M111328X},
	author = {Ernie Esser and Llu{\'\i}s Guasch and Tristan van Leeuwen and Aleksandr Y. Aravkin and Felix J. Herrmann}
}
@article {kumar2016bls,
	title = {Beating level-set methods for 5D seismic data interpolation: a primal-dual alternating approach},
	journal = {IEEE Transactions on Computational Imaging},
	year = {2017},
	note = {(published online in IEEE Transactions on Computational Imaging)},
	month = {04},
	abstract = {Acquisition cost is a crucial bottleneck for seismic workflows, and low-rank formulations for data interpolation allow practitioners to {\textquoteright}fill in{\textquoteright} data volumes from critically subsampled data acquired in the field. Tremendous size of seismic data volumes required for seismic processing remains a major challenge for these techniques. Residual-constrained formulations require less parameter tuning when the target noise floor is known. We propose a new approach to solve residual constrained formulations for interpolation. We represent the data volume in a compressed manner using low-rank matrix factors, and build a block-coordinate algorithm with constrained convex subproblems that are solved with a primal-dual splitting scheme. The develop optimization framework works on the whole seismic temporal frequency slices and does not require windowing or non-trivial sorting of seismic data. The new approach is competitive with state of the art level-set algorithms that interchange the role of objectives with constraints. We use the new algorithm to successfully interpolate a large scale 5D seismic data volume (upto 1010 data-points), generated from the geologically complex synthetic 3D Compass velocity model, where 80\% of the data has been removed. We also develop a robust extension of the primal-dual approach to deal with the outliers (or noise) in the data.},
	keywords = {alternating minimization, matrix completion, primal-dual splitting, seismic data, seismic trace interpolation},
	doi = {10.1109/TCI.2017.2693966},
	url = {https://slim.gatech.edu/Publications/Public/Journals/IEEETransComputationalImaging/2017/kumar2016bls/kumar2016bls.pdf},
	author = {Rajiv Kumar and Oscar Lopez and Damek Davis and Aleksandr Y. Aravkin and Felix J. Herrmann}
}
@article {tu2015GJIsem,
	title = {Source estimation with surface-related multiples{\textendash}-fast ambiguity-resolved seismic imaging},
	journal = {Geophysical Journal International},
	volume = {205},
	year = {2016},
	note = {(published online in Geophysical Journal International)},
	month = {03},
	pages = {1492-1511},
	abstract = {We address the problem of obtaining a reliable seismic image without prior knowledge of the source wavelet, especially from data that contain strong surface-related multiples. Conventional reverse-time migration requires prior knowledge of the source wavelet, which is either technically or computationally challenging to accurately determine; inaccurate estimates of the source wavelet can result in seriously degraded reverse-time migrated images, and therefore wrong geological interpretations. To solve this problem, we present a "wavelet-free" imaging procedure that simultaneously inverts for the source wavelet and the seismic image, by tightly integrating source estimation into a fast least-squares imaging framework, namely compressive imaging, given a reasonably accurate background velocity model. However, this joint inversion problem is difficult to solve as it is plagued with local minima and the ambiguity with respect to amplitude scalings because of the multiplicative, and therefore nonlinear, appearance of the source wavelet in the otherwise linear formalism. We have found a way to solve this nonlinear joint-inversion problem using a technique called variable projection, and a way to overcome the scaling ambiguity by including surface-related multiples in our imaging procedure following recent developments in surface-related multiple prediction by sparse inversion. As a result, we obtain without prior knowledge of the source wavelet high-resolution seismic images, comparable in quality to images obtained assuming the true source wavelet is known. By leveraging the computationally efficient compressive-imaging methodology, these results are obtained at affordable computational costs compared with conventional processing work flows that include surface-related multiple removal and reverse-time migration.},
	keywords = {computational seismology, free surface, Inverse theory, multiples, seismic imaging, time series analysis, wave propagation},
	url = {https://slim.gatech.edu/Publications/Public/Journals/GeophysicalJournalInternational/2016/tu2015GJIsem/tu2015GJIsem.pdf},
	url2 = {http://gji.oxfordjournals.org/content/205/3/1492.full.pdf?keytype=ref\&ijkey=zFydRGQpiJzeCS9},
	author = {Ning Tu and Aleksandr Y. Aravkin and Tristan van Leeuwen and Tim T.Y. Lin and Felix J. Herrmann}
}
@article {kumar2014GEOPemc,
	title = {Efficient matrix completion for seismic data reconstruction},
	journal = {Geophysics},
	volume = {80},
	number = {5},
	year = {2015},
	note = {(Geophysics)},
	month = {09},
	pages = {V97-V114},
	abstract = {Despite recent developments in improved acquisition, seismic data often remain undersampled along source and receiver coordinates, resulting in incomplete data for key applications such as migration and multiple prediction. We have interpreted the missing-trace interpolation problem in the context of matrix completion (MC), and developed three practical principles for using low-rank optimization techniques to recover seismic data. Specifically, we strive for recovery scenarios wherein the original signal is low rank and the subsampling scheme increases the singular values of the matrix. We use an optimization program that restores this low-rank structure to recover the full volume. Omitting one or more of these principles can lead to poor interpolation results, as we found experimentally. In light of this theory, we compensate for the high-rank behavior of data in the source-receiver domain by using the midpoint-offset transformation for 2D data and a source-receiver permutation for 3D data to reduce the overall singular values. Simultaneously, to work with computationally feasible algorithms for large-scale data, we use a factorization-based approach to MC, which significantly speeds up the computations compared with repeated singular value decompositions without reducing the recovery quality. In the context of our theory and experiments, we also find that windowing the data too aggressively can have adverse effects on the recovery quality. To overcome this problem, we carried out our interpolations for each frequency independently while working with the entire frequency slice. The result is a computationally efficient, theoretically motivated framework for interpolating missing-trace data. Our tests on realistic 2D and 3D seismic data sets show that our method compares favorably in terms of computational speed and recovery quality with existing curvelet- and tensor-based techniques.},
	keywords = {2D, 3D, algorithm, interpolation, low-rank, signal processing},
	doi = {10.1190/geo2014-0369.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2015/kumar2014GEOPemc/kumar2014GEOPemc.pdf},
	url2 = {http://library.seg.org/doi/abs/10.1190/geo2014-0369.1},
	author = {Rajiv Kumar and Curt Da Silva and Okan Akalin and Aleksandr Y. Aravkin and Hassan Mansour and Ben Recht and Felix J. Herrmann}
}
@article {vanLeeuwen2014GEOPcav,
	title = {Comment on: {\textquotedblleft}Application of the variable projection scheme for frequency-domain full-waveform inversion{\textquotedblright} (M. Li, J. Rickett, and A. Abubakar, Geophysics, 78, no. 6, R249{\textendash}R257)},
	journal = {Geophysics},
	volume = {79},
	number = {3},
	year = {2014},
	note = {(discussion by Tristan van Leeuwen, Aleksandr Y. Aravkin, and Felix J. Herrmann)},
	month = {05},
	pages = {X11-X17},
	keywords = {variable projection, waveform inversion},
	doi = {10.1190/geo2013-0466.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2014/vanLeeuwen2014GEOPcav/vanLeeuwen2014GEOPcav.pdf},
	author = {Tristan van Leeuwen and Aleksandr Y. Aravkin and Felix J. Herrmann}
}
@article {aravkin2014SISCfmd,
	title = {Fast methods for denoising matrix completion formulations, with applications to robust seismic data interpolation},
	journal = {SIAM Journal on Scientific Computing},
	volume = {36},
	number = {5},
	year = {2014},
	note = {(SISC)},
	month = {10},
	pages = {S237-S266},
	abstract = {Recent SVD-free matrix factorization formulations have enabled rank minimization for systems with millions of rows and columns, paving the way for matrix completion in extremely large-scale applications, such as seismic data interpolation. In this paper, we consider matrix completion formulations designed to hit a target data-fitting error level provided by the user, and we propose an algorithm called LR-BPDN that is able to exploit factorized formulations to solve the corresponding optimization problem. Since practitioners typically have strong prior knowledge about target error level, this innovation makes it easy to apply the algorithm in practice, leaving only the factor rank to be determined. Within the established framework, we propose two extensions that are highly relevant to solving practical challenges of data interpolation. First, we propose a weighted extension that allows known subspace information to improve the results of matrix completion formulations. We show how this weighting can be used in the context of frequency continuation, an essential aspect to seismic data interpolation. Second, we propose matrix completion formulations that are robust to large measurement errors in the available data. We illustrate the advantages of LR-BPDN on collaborative filtering problems using the MovieLens 1M and 10M and Netflix 100M datasets. Then we use the new method, along with its robust and subspace reweighted extensions, to obtain high-quality reconstructions for large-scale seismic interpolation problems with real data, even in the presence of data contamination.},
	keywords = {factorized formulations, matrix completion, robust statistics, seismic trace interpolation, variational optimization},
	doi = {10.1137/130919210},
	url = {https://slim.gatech.edu/Publications/Public/Journals/SIAMJournalOnScientificComputing/2014/aravkin2014SISCfmd/aravkin2014SISCfmd.pdf},
	url2 = {http://epubs.siam.org/doi/abs/10.1137/130919210},
	author = {Aleksandr Y. Aravkin and Rajiv Kumar and Hassan Mansour and Ben Recht and Felix J. Herrmann}
}
@article {aravkin2012IPNuisance,
	title = {Estimating nuisance parameters in inverse problems},
	journal = {Inverse Problems},
	volume = {28},
	number = {11},
	year = {2012},
	month = {10},
	abstract = {Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is variable projection, where nonlinear least squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with nuisance parameters, such as variance or degrees of freedom. As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, degree of freedom (d.o.f.) parameter estimation in the context of robust inverse problems, automatic calibration, and optimal experimental design. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several large- scale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms.},
	keywords = {full waveform inversion, students t, variance},
	doi = {10.1088/0266-5611/28/11/115016},
	url = {http://arxiv.org/abs/1206.6532},
	url2 = {https://slim.gatech.edu/Publications/Public/Journals/InverseProblems/2012/aravkin2012IPNuisance/aravkin2012IPNuisance.pdf},
	author = {Aleksandr Y. Aravkin and Tristan van Leeuwen}
}
@article {Li11TRfrfwi,
	title = {Fast randomized full-waveform inversion with compressive sensing},
	journal = {Geophysics},
	volume = {77},
	number = {3},
	year = {2012},
	month = {05},
	pages = {A13-A17},
	abstract = {Wave-equation based seismic inversion can be formulated as a nonlinear inverse problem where the medium properties are obtained via minimization of a least- squares misfit functional. The demand for higher resolution models in more geologically complex areas drives the need to develop techniques that explore the special structure of full-waveform inversion to reduce the computational burden and to regularize the inverse problem. We meet these goals by using ideas from compressive sensing and stochastic optimization to design a novel Gauss-Newton method, where the updates are computed from random subsets of the data via curvelet-domain sparsity promotion. Application of this idea to a realistic synthetic shows improved results compared to quasi-Newton methods, which require passes through all data. Two different subset sampling strategies are considered: randomized source encoding, and drawing sequential shots firing at random source locations from marine data with missing near and far offsets. In both cases, we obtain excellent inversion results compared to conventional methods at reduced computational costs.},
	keywords = {Compressive Sensing, FWI, Optimization, SLIM},
	doi = {10.1190/geo2011-0410.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2012/Li11TRfrfwi/Li11TRfrfwi.pdf},
	author = {Xiang Li and Aleksandr Y. Aravkin and Tristan van Leeuwen and Felix J. Herrmann}
}
@article {Aravkin11TRridr,
	title = {Robust inversion, dimensionality reduction, and randomized sampling},
	journal = {Mathematical Programming},
	volume = {134},
	number = {1},
	year = {2012},
	month = {08},
	pages = {101-125},
	abstract = {We consider a class of inverse problems in which the forward model is the solution operator to linear ODEs or PDEs. This class admits several dimensionality-reduction techniques based on data averaging or sampling, which are especially useful for large-scale problems. We survey these approaches and their connection to stochastic optimization. The data-averaging approach is only viable, however, for a least-squares misfit, which is sensitive to outliers in the data and artifacts unexplained by the forward model. This motivates us to propose a robust formulation based on the Student{\textquoteright}s t-distribution of the error. We demonstrate how the corresponding penalty function, together with the sampling approach, can obtain good results for a large-scale seismic inverse problem with 50 \% corrupted data.},
	keywords = {FWI, Inverse problems, Optimization, Robust estimation, Seismic inversion, Stochastic optimization},
	doi = {10.1007/s10107-012-0571-6},
	url = {http://www.springerlink.com/content/35rwr101h5736340/},
	url2 = {https://slim.gatech.edu/Publications/Public/Journals/MathematicalProgramming/aravkin2012MPrid/aravkin2012MPrid.pdf},
	author = {Aleksandr Y. Aravkin and Michael P. Friedlander and Felix J. Herrmann and Tristan van Leeuwen}
}
@article {vanLeeuwen2010IJGswi,
	title = {Seismic waveform inversion by stochastic optimization},
	journal = {International Journal of Geophysics},
	volume = {2011},
	year = {2011},
	note = {Article ID: 689041, 18pages},
	month = {12},
	abstract = {We explore the use of stochastic optimization methods for seismic waveform inversion. The basic principle of such methods is to randomly draw a batch of realizations of a given misfit function and goes back to the 1950s. The ultimate goal of such an approach is to dramatically reduce the computational cost involved in evaluating the misfit. Following earlier work, we introduce the stochasticity in waveform inversion problem in a rigorous way via a technique called randomized trace estimation. We then review theoretical results that underlie recent developments in the use of stochastic methods for waveform inversion. We present numerical experiments to illustrate the behavior of different types of stochastic optimization methods and investigate the sensitivity to the batch size and the noise level in the data. We find that it is possible to reproduce results that are qualitatively similar to the solution of the full problem with modest batch sizes, even on noisy data. Each iteration of the corresponding stochastic methods requires an order of magnitude fewer PDE solves than a comparable deterministic method applied to the full problem, which may lead to an order of magnitude speedup for waveform inversion in practice.},
	keywords = {FWI, Optimization, SLIM},
	doi = {10.1155/2011/689041},
	url = {https://slim.gatech.edu/Publications/Public/Journals/InternationJournalOfGeophysics/2011/vanLeeuwen10IJGswi/vanLeeuwen10IJGswi.pdf},
	author = {Tristan van Leeuwen and Aleksandr Y. Aravkin and Felix J. Herrmann}
}
