@article {yang2020tdsp,
	title = {Time-domain sparsity promoting least-squares reverse time migration with source estimation},
	journal = {Geophysical Prospecting},
	volume = {68},
	number = {9},
	year = {2020},
	note = {(Geophysical Prospecting)},
	month = {08},
	pages = {2697-2711},
	abstract = {Least-squares reverse time migration is well-known for its capability to generate artifact-free true-amplitude subsurface images through fitting observed data in the least-squares sense. However, when applied to realistic imaging problems, this approach is faced with issues related to overfitting and excessive computational costs induced by many wave-equation solves. The fact that the source function is unknown complicates this situation even further. Motivated by recent results in stochastic optimization and transform-domain sparsity-promotion, we demonstrate that the computational costs of inversion can be reduced significantly while avoiding imaging artifacts and restoring amplitudes. While powerfull, these new approaches do require accurate information on the source-time function, which is often lacking. Without this information, the imaging quality deteriorates rapidly. We address this issue by presenting an approach where the source-time function is estimated on the fly through a technique known as variable projection. Aside from introducing negligible computational overhead, the proposed method is shown to perform well on imaging problems with noisy data and problems that involve complex settings such as salt. In either case, the presented method produces high resolution high-amplitude fidelity images including an estimates for the source-time function. In addition, due to its use of stochastic optimization, we arrive at these images at roughly one to two times the cost of conventional reverse time migration involving all data.},
	keywords = {penalty, source estimation, sparsity inversion},
	doi = {10.1111/1365-2478.13021},
	url = {https://slim.gatech.edu/Publications/Public/Journals/GeophysicalProspecting/2020/yang2020tdsp/yang2020tdsp.html},
	author = {Mengmeng Yang and Zhilong Fang and Philipp A. Witte and Felix J. Herrmann}
}
@article {fang2017sewri,
	title = {Source estimation for wavefield-reconstruction inversion},
	journal = {Geophysics},
	volume = {83},
	number = {4},
	year = {2018},
	note = {(Geophysics)},
	pages = {R345-R359},
	abstract = {Source estimation is essential to all the wave-equation-based seismic inversions, including full-waveform inversion and the recently proposed wavefield-reconstruction inversion. When the source estimation is inaccurate, errors will propagate into the predicted data and introduce additional data misfit. As a consequence, inversion results that minimize this data misfit may become erroneous. To mitigate the errors introduced by the incorrect and pre-estimated sources, an embedded procedure that updates sources along with medium parameters is necessary for the inversion. So far, such a procedure is still missing in the context of wavefield-reconstruction inversion, a method that is, in many situations, less prone to local minima related to the so-called cycle skipping, compared to full-waveform inversion through exact data-fitting. While wavefield-reconstruction inversion indeed helps to mitigate issues related to cycle skipping by extending the search space with wavefields as auxiliary variables, it relies on having access to the correct source functions. In this paper, we remove the requirement of having the accurate source functions by proposing a source estimation technique specifically designed for wavefield-reconstruction inversion. To achieve this task, we consider the source functions as unknown variables and arrive at an objective function that depends on the medium parameters, wavefields, and source functions. During each iteration, we apply the so-called variable projection method to simultaneously project out the source functions and wavefields. After the projection, we obtain a reduced objective function that only depends on the medium parameters and invert for the unknown medium parameters by minimizing this reduced objective. Numerical experiments illustrate that this approach can produce accurate estimates of the unknown medium parameters without any prior information of the source functions.},
	keywords = {estimation, FWI, source, variable projection, WRI},
	doi = {10.1190/geo2017-0700.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2018/fang2017sewri/fang2017sewri.html},
	author = {Zhilong Fang and Rongrong Wang and Felix J. Herrmann}
}
@article {fang2017uqfip,
	title = {Uncertainty quantification for inverse problems with weak partial-differential-equation constraints},
	journal = {Geophysics},
	volume = {83},
	number = {6},
	year = {2018},
	note = {(Geophysics)},
	pages = {R629-R647},
	abstract = {In a statistical inverse problem, the objective is a complete statistical description of unknown parameters from noisy observations in order to quantify uncertainties of the parameters of interest. We consider inverse problems with partial-differential-equation-constraints, which are applicable to a variety of seismic problems. Bayesian inference is one of the most widely-used approaches to precisely quantify statistics through a posterior distribution, incorporating uncertainties in observed data, modeling kernel, and prior knowledge of the parameters. Typically when formulating the posterior distribution, the partial-differential-equation-constraints are required to be exactly satisfied, resulting in a highly nonlinear forward map and a posterior distribution with many local maxima. These drawbacks make it difficult to find an appropriate approximation for the posterior distribution. Another complicating factor is that traditional Markov chain Monte Carlo methods are known to converge slowly for realistically sized problems. In this work, we relax the partial-differential-equation-constraints by introducing an auxiliary variable, which allows for Gaussian deviations in the partial-differential-equations. Thus, we obtain a new bilinear posterior distribution consisting of both data and partial-differential-equation misfit terms. We illustrate that for a particular range of variance choices for the partial-differential-equation misfit term, the new posterior distribution has fewer modes and can be well-approximated by a Gaussian distribution, which can then be sampled in a straightforward manner. Since it is prohibitively expensive to explicitly construct the dense covariance matrix of the Gaussian approximation for intermediate to large-scale problems, we present a method to implicitly construct it, which enables efficient sampling. We apply this framework to two-dimensional seismic inverse problems with 1,800 and 92,455 unknown parameters. The results illustrate that our framework can produce comparable statistical quantities to those produced by conventional Markov chain Monte Carlo type methods while requiring far fewer partial-differential-equation solves, which are the main computational bottlenecks in these problems.},
	keywords = {acoustic, FWI, UQ, weak-constraint},
	doi = {10.1190/geo2017-0824.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2018/fang2017uqfip/fang2017uqfip.html},
	author = {Zhilong Fang and Curt Da Silva and Rachel Kuske and Felix J. Herrmann}
}
@article {Herrmann11TRfcd,
	title = {Fighting the curse of dimensionality: compressive sensing in exploration seismology},
	journal = {Signal Processing Magazine, IEEE},
	volume = {29},
	number = {3},
	year = {2012},
	month = {05},
	pages = {88-100},
	abstract = {Many seismic exploration techniques rely on the collection of massive data volumes that are mined for information during processing. This approach has been extremely successful, but current efforts toward higher resolution images in increasingly complicated regions of Earth continue to reveal fundamental shortcomings in our typical workflows. The "curse" of dimensionality is the main roadblock and is exemplified by Nyquist{\textquoteright}s sampling criterion, which disproportionately strains current acquisition and processing systems as the size and desired resolution of our survey areas continues to increase.},
	keywords = {dimensionality curse, Earth, geographic information systems, higher-resolution images, massive data volumes, Nyquist sampling criterion, seismic exploration techniques, seismology, strains current acquisition system, strains current processing system},
	issn = {1053-5888},
	doi = {10.1109/MSP.2012.2185859},
	url = {https://slim.gatech.edu/Publications/Public/Journals/IEEESignalProcessingMagazine/2012/Herrmann11TRfcd/Herrmann11TRfcd.pdf},
	author = {Felix J. Herrmann and Michael P. Friedlander and Ozgur Yilmaz}
}
@article {Friedlander11TRhdm,
	title = {Hybrid deterministic-stochastic methods for data fitting},
	journal = {SIAM Journal on Scientific Computing},
	volume = {34},
	number = {3},
	year = {2012},
	month = {01},
	pages = {A1380-A1405},
	abstract = {Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms (both deterministic and randomized) offer inexpensive iterations by sampling only subsets of the terms in the sum. These methods can make great progress initially, but often slow as they approach a solution. In contrast, full gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate of convergence analysis and numerical experiments illustrate the potential for the approach.},
	keywords = {Optimization},
	doi = {10.1137/110830629},
	author = {Michael P. Friedlander and Mark Schmidt}
}
@article {mansour2012IEEETITrcs,
	title = {Recovering compressively sampled signals using partial support information},
	journal = {IEEE Transactions on Information Theory},
	volume = {58},
	number = {2},
	year = {2012},
	month = {02},
	pages = {1122-1134},
	publisher = {Department of Computer Science},
	address = {University of British Columbia, Vancouver},
	abstract = {We study recovery conditions of weighted $\ell_1$ minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that if at least 50\% of the (partial) support information is accurate, then weighted $\ell_1$ minimization is stable and robust under weaker sufficient conditions than the analogous conditions for standard $\ell_1$ minimization. Moreover, weighted $\ell_1$ minimization provides better upper bounds on the reconstruction error in terms of the measurement noise and the compressibility of the signal to be recovered. We illustrate our results with extensive numerical experiments on synthetic data and real audio and video signals.},
	keywords = {Compressive Sensing},
	doi = {10.1109/TIT.2011.2167214},
	url = {https://slim.gatech.edu/Publications/Public/Journals/IEEETransInformationTheory/2012/mansour2012IEEETITrcs/mansour2012IEEETITrcs.pdf},
	author = {Michael P. Friedlander and Hassan Mansour and Rayan Saab and Ozgur Yilmaz}
}
@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 {vandenberg2011SIAMsol,
	title = {Sparse optimization with least-squares constraints},
	journal = {SIAM Journal on Optimization},
	volume = {21},
	number = {4},
	year = {2011},
	month = {11},
	pages = {1201{\textendash}1229},
	abstract = {The use of convex optimization for the recovery of sparse signals from incomplete or compressed data is now common practice. Motivated by the success of basis pursuit in recovering sparse vectors, new formulations have been proposed that take advantage of different types of sparsity. In this paper we propose an efficient algorithm for solving a general class of sparsifying formulations. For several common types of sparsity we provide applications, along with details on how to apply the algorithm, and experimental results.},
	keywords = {basis pursuit, compressed sensing, convex program, duality, group sparsity, matrix completion, Newton{\textquoteright}s method, root-finding, sparse solutions},
	doi = {10.1137/100785028},
	url = {http://www.math.ucdavis.edu/\%7Empf/2010-sparse-optimization-with-least-squares.html},
	author = {Ewout van den Berg and Michael P. Friedlander}
}
@article {hennenfent2010GEOPnct,
	title = {Nonequispaced curvelet transform for seismic data reconstruction: a sparsity-promoting approach},
	journal = {Geophysics},
	volume = {75},
	number = {6},
	year = {2010},
	month = {12},
	pages = {WB203-WB210},
	publisher = {SEG},
	abstract = {We extend our earlier work on the nonequispaced fast discrete curvelet transform (NFDCT) and introduce a second generation of the transform. This new generation differs from the previous one by the approach taken to compute accurate curvelet coefficients from irregularly sampled data. The first generation relies on accurate Fourier coefficients obtained by an l2-regularized inversion of the nonequispaced fast Fourier transform (FFT) whereas the second is based on a direct l1-regularized inversion of the operator that links curvelet coefficients to irregular data. Also, by construction the second generation NFDCT is lossless unlike the first generation NFDCT. This property is particularly attractive for processing irregularly sampled seismic data in the curvelet domain and bringing them back to their irregular record-ing locations with high fidelity. Secondly, we combine the second generation NFDCT with the standard fast discrete curvelet transform (FDCT) to form a new curvelet-based method, coined nonequispaced curvelet reconstruction with sparsity-promoting inversion (NCRSI) for the regularization and interpolation of irregularly sampled data. We demonstrate that for a pure regularization problem the reconstruction is very accurate. The signal-to-reconstruction error ratio in our example is above 40 dB. We also conduct combined interpolation and regularization experiments. The reconstructions for synthetic data are accurate, particularly when the recording locations are optimally jittered. The reconstruction in our real data example shows amplitudes along the main wavefronts smoothly varying with limited acquisition imprint.},
	keywords = {curvelet transforms, data acquisition, geophysical techniques, Processing, seismology, SLIM},
	doi = {10.1190/1.3494032},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2010/hennenfent2010GEOPnct/hennenfent2010GEOPnct.pdf},
	author = {Gilles Hennenfent and Lloyd Fenelon and Felix J. Herrmann}
}
@article {vandenberg2010IEEEter,
	title = {Theoretical and empirical results for recovery from multiple measurements},
	journal = {IEEE Transactions on Information Theory},
	volume = {56},
	number = {5},
	year = {2010},
	month = {05},
	pages = {2516-2527},
	abstract = {The joint-sparse recovery problem aims to recover, from sets of compressed measurements, unknown sparse matrices with nonzero entries restricted to a subset of rows. This is an extension of the single-measurement-vector (SMV) problem widely studied in compressed sensing. We analyze the recovery properties for two types of recovery algorithms. First, we show that recovery using sum-of-norm minimization cannot exceed the uniform recovery rate of sequential SMV using L1 minimization, and that there are problems that can be solved with one approach but not with the other. Second, we analyze the performance of the ReMBo algorithm [M. Mishali and Y. Eldar, IEEE Trans. Sig. Proc., 56 (2008)] in combination with L1 minimization, and show how recovery improves as more measurements are taken. From this analysis it follows that having more measurements than number of nonzero rows does not improve the potential theoretical recovery rate.},
	keywords = {convex optimization, joint sparsity, multiple channels, Sparse recovery},
	doi = {10.1109/TIT.2010.2043876},
	url = {http://www.math.ucdavis.edu/\%7Empf/2010-joint-sparsity.html},
	author = {Ewout van den Berg and Michael P. Friedlander}
}
@article {vandenberg2009ACMstf,
	title = {Sparco: a testing framework for sparse reconstruction},
	journal = {ACM Transactions on Mathematical Software},
	volume = {35},
	number = {4},
	year = {2009},
	month = {02},
	pages = {1-16},
	abstract = {Sparco is a framework for testing and benchmarking algorithms for sparse reconstruction. It includes a large collection of sparse reconstruction problems drawn from the imaging, compressed sensing, and geophysics literature. Sparco is also a framework for implementing new test problems and can be used as a tool for reproducible research. Sparco is implemented entirely in Matlab, and is released as open-source software under the GNU Public License.},
	keywords = {compressed sensing, linear operators, Sparse recovery},
	url = {http://doi.acm.org/10.1145/1462173.1462178},
	author = {Ewout van den Berg and Michael P. Friedlander and Gilles Hennenfent and Felix J. Herrmann and Rayan Saab and Ozgur Yilmaz}
}
@article {hennenfent2008GEOPnii,
	title = {New insights into one-norm solvers from the Pareto curve},
	journal = {Geophysics},
	volume = {73},
	number = {4},
	year = {2008},
	month = {07},
	pages = {A23-A26},
	abstract = {Geophysical inverse problems typically involve a trade off between data misfit and some prior. Pareto curves trace the optimal trade off between these two competing aims. These curves are commonly used in problems with two-norm priors where they are plotted on a log-log scale and are known as L-curves. For other priors, such as the sparsity-promoting one norm, Pareto curves remain relatively unexplored. We show how these curves lead to new insights in one-norm regularization. First, we confirm the theoretical properties of smoothness and convexity of these curves from a stylized and a geophysical example. Second, we exploit these crucial properties to approximate the Pareto curve for a large-scale problem. Third, we show how Pareto curves provide an objective criterion to gauge how different one-norm solvers advance towards the solution.},
	keywords = {Acquisition, Geophysics, Optimization, Pareto, Processing, SLIM},
	doi = {10.1190/1.2944169},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2008/hennenfent08GEOnii/hennenfent08GEOnii.pdf},
	author = {Gilles Hennenfent and Ewout van den Berg and Michael P. Friedlander and Felix J. Herrmann}
}
@article {berg2008SJSCpareto,
	title = {Probing the Pareto frontier for basis pursuit solutions},
	journal = {SIAM Journal on Scientific Computing},
	volume = {31},
	number = {2},
	year = {2008},
	month = {01},
	pages = {890-912},
	publisher = {SIAM},
	abstract = {The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously differentiable over all points of interest, and show that it gives an explicit relationship to two other optimization problems closely related to BPDN. We describe a root-finding algorithm for finding arbitrary points on this curve; the algorithm is suitable for problems that are large scale and for those that are in the complex domain. At each iteration, a spectral gradient-projection method approximately minimizes a least-squares problem with an explicit one-norm constraint. Only matrix-vector operations are required. The primal-dual solution of this problem gives function and derivative information needed for the root-finding method. Numerical experiments on a comprehensive set of test problems demonstrate that the method scales well to large problems.},
	keywords = {basis pursuit, convex program, duality, Newton{\textquoteright}s method, one-norm regularization, Optimization, projected gradient, root-finding, sparse solutions},
	doi = {10.1137/080714488},
	url = {https://slim.gatech.edu/Publications/Public/Journals/SIAMJournalOnScientificComputing/2008/vanderberg08SIAMptp/vanderberg08SIAMptp.pdf},
	author = {Ewout van den Berg and Michael P. Friedlander}
}
@article {friedlander2007TASdtd,
	title = {Discussion: the Dantzig selector: statistical estimation when p is much larger than n},
	journal = {The Annals of Statistics},
	volume = {35},
	number = {6},
	year = {2007},
	month = {03},
	pages = {2385-2391},
	keywords = {dantzig, SLIM, statistics},
	doi = {10.1214/009053607000000479},
	author = {Michael P. Friedlander and M. A. Saunders}
}
@article {friedlander2007SJOero,
	title = {Exact regularization of convex programs},
	journal = {SIAM Journal on Optimization},
	volume = {18},
	number = {4},
	year = {2007},
	month = {05},
	pages = {1326-1350},
	abstract = {The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a Lagrange multiplier. Moreover, the regularization parameter threshold is inversely related to the Lagrange multiplier. We use this result to generalize an exact regularization result of Ferris and Mangasarian [Appl. Math. Optim., 23(1991), pp. 266{\textendash}273] involving a linearized selection problem. We also use it to derive necessary and sufficient conditions for exact penalization, similar to those obtained by Bertsekas [Math. Programming, 9(1975), pp. 87{\textendash}99] and by Bertsekas, Nedi , Ozdaglar [Convex Analysis and Optimization, Athena Scientific, Belmont, MA, 2003]. When the regularization is not exact, we derive error bounds on the distance from the regularized solution to the original solution set. We also show that existence of a {\textquoteleft}{\textquoteleft}weak sharp minimum{\textquoteright}{\textquoteright} is in some sense close to being necessary for exact regularization. We illustrate the main result with numerical experiments on the l1 regularization of benchmark (degenerate) linear programs and semidefinite/second-order cone programs. The experiments demonstrate the usefulness of l1 regularization in finding sparse solutions.},
	keywords = {Optimization, SLIM},
	doi = {10.1137/060675320},
	author = {Michael P. Friedlander and P. Tseng}
}
