@article {silva2018alr,
	title = {Applications of low-rank compressed seismic data to full waveform inversion and extended image volumes},
	journal = {Geophysics},
	year = {2019},
	note = {Accepted on January 22, 2019.},
	abstract = {Conventional oil and gas fields are increasingly difficult to explore and image, resulting in the call for more complex wave-equation based inversion algorithms that require dense long-offset samplings. Consequently, there is an exponential growth in the size of data volumes and prohibitive demands on computational resources. In this work, we propose a method to compress and process seismic data directly in a low-rank tensor format, which drastically reduces the amount of storage required to represent the data. We first outline how seismic data exhibits low-rank structure in a particular transform-domain, which can be exploited to compress the dense data in one extremely storage-efficient tensor format when the data is fully sampled. In the more realistic case of missing data, we can use efficient interpolation techniques to approximate the fully sampled volume in compressed form. In either case, once we have our data represented in its compressed tensor form, we design an algorithm to extract source or receiver gathers directly from the compressed parameters. This extraction process can be done on-the-fly directly on the compressed data, in a full waveform inversion context, and does not require scanning through the entire dataset in order to form shot gathers. To the best of our knowledge, this work is one of the first major contributions to working with seismic data applications directly in the compressed domain without reconstructing the entire data volume. We use a stochastic inversion approach, which works with small subsets of source experiments at each iteration, further reducing the computational and memory costs of full waveform inversion. We also demonstrate how this data compression and extraction technique can be applied to forming full subsurface image gathers through probing techniques.},
	keywords = {common shot/receiver gather, extended image volume, full waveform inversion, low rank, tensor},
	doi = {10.1190/geo2018-0116.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2019/silva2018alr/silva2018alr.pdf},
	author = {Curt Da Silva and Yiming Zhang and Rajiv Kumar and Felix J. Herrmann}
}
@article {dasilva2017uls,
	title = {A unified 2D/3D large scale software environment for nonlinear inverse problems},
	journal = {ACM Transactions on Mathematical Software},
	year = {2019},
	note = {Accepted on January 27, 2019.},
	abstract = {Large scale parameter estimation problems are some of the most computationally demanding problems. An academic researcher{\textquoteright}s domain-specific knowledge often precludes that of software design, which results in software frameworks for inversion that are technically correct, but not scalable to realistically-sized problems. On the other hand, the computational demands of the problem for realistic problems result in industrial codebases that are geared solely for performance, rather than comprehensibility or flexibility. We propose a new software design that bridges the gap between these two seemingly disparate worlds. A hierarchical and modular design allows a user to delve into as much detail as she desires, while using high performance primitives at the lower levels. Our code has the added benefit of actually reflecting the underlying mathematics of the problem, which lowers the cognitive load on user using it and reduces the initial startup period before a researcher can be fully productive. We also introduce a new preconditioner for the Helmholtz equation that is suitable for fault-tolerant distributed systems. Numerical experiments on a variety of 2D and 3D test problems demonstrate the effectiveness of this approach on scaling algorithms from small to large scale problems with minimal code changes.},
	keywords = {large scale, Matlab, Optimization, PDE-constrained inversion},
	url = {https://slim.gatech.edu/Publications/Public/Journals/ACMTOMS/2019/dasilva2017uls/dasilva2017uls.html},
	author = {Curt Da Silva 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}
}
@conference {herrmann2017SIAMdsa,
	title = {Domain-specific abstractions for full-waveform inversion},
	booktitle = {SIAM Conference on Computational Science and Engineering},
	year = {2017},
	note = {(SIAM Conference on Computational Science and Engineering, Atlanta)},
	month = {02-03},
	keywords = {FWI, Inverse problems, SIAM, software},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SIAM/2017/herrmann2017SIAMdsa/herrmann2017SIAMdsa_pres.pdf},
	author = {Felix J. Herrmann and Curt Da Silva}
}
@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}
}
@mastersthesis {dasilva2017THlso,
	title = {Large-scale optimization algorithms for missing data completion and inverse problems},
	year = {2017},
	note = {(PhD)},
	month = {09},
	school = {The University of British Columbia},
	type = {phd},
	address = {Vancouver},
	abstract = {Inverse problems are an important class of problems found in many areas of science and engineering. In these problems, one aims to estimate unknown parameters of a physical system through indirect multi-experiment measurements. Inverse problems arise in a number of fields including seismology, medical imaging, and astronomy, among others. An important aspect of inverse problems is the quality of the acquired data itself. Real-world data acquisition restrictions, such as time and budget constraints, often results in measured data with missing entries. Many inversion algorithms assume that the input data is fully sampled and relatively noise free and produce poor results when these assumptions are violated. Given the multidimensional nature of real-world data, we propose a new low-rank optimization method on the smooth manifold of Hierarchical Tucker tensors. Tensors that exhibit this low-rank structure can be recovered from solving this non-convex program in an efficient manner. We successfully interpolate realistically sized seismic data volumes using this approach. If our low-rank tensor is corrupted with non-Gaussian noise, the resulting optimization program can be formulated as a convex-composite problem. This class of problems involves minimizing a non-smooth but convex objective composed with a nonlinear smooth mapping. In this thesis, we develop a level set method for solving composite-convex problems and prove that the resulting subproblems converge linearly. We demonstrate that this method is competitive when applied to examples in noisy tensor completion, analysis-based compressed sensing, audio declipping, total-variation deblurring and denoising, and one-bit compressed sensing. With respect to solving the inverse problem itself, we introduce a new software design framework that manages the cognitive complexity of the various components involved. Our framework is modular by design, which enables us to easily integrate and replace components such as linear solvers, finite difference stencils, preconditioners, and parallelization schemes. As a result, a researcher using this framework can formulate her algorithms with respect to high-level components such as objective functions and hessian operators. We showcase the ease with which one can prototype such algorithms in a 2D test problem and, with little code modification, apply the same method to large-scale 3D problems.},
	keywords = {convex composite, interpolation, low-rank, Optimization, PhD, software design, tensor, tensor completion},
	url = {https://slim.gatech.edu/Publications/Public/Thesis/2017/dasilva2017THlso/dasilva2017THlso.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Thesis/2017/dasilva2017THlso/dasilva2017THlso_pres.pdf},
	author = {Curt Da Silva}
}
@conference {zhang2017SEGmsd,
	title = {Massive 3D seismic data compression and inversion with hierarchical Tucker},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2017},
	note = {(SEG, Houston)},
	month = {09},
	pages = {1347-1352},
	abstract = {Modern-day oil and gas exploration, especially in areas of complex geology such as fault belts and sub-salt areas, is an increasingly expensive and risky endeavour. Typically long-offset and dense sampling seismic data are required for subsequent shot based processing procedures, e.g. wave-equation based inversion (WEI) and surface-related multiple elimination (SRME). However, these strict requirements result in an exponential growth in data volume size and prohibitive demands on computational resources, given the multidimensional nature of the data volumes. Moreover the physical constraints and cost limitations impose restrictions on acquiring fully sampled data. In this work, we propose to invert our large-scale data from a set of subsampled measurements, resulting in an estimate of the true volume in a compressed low-rank tensor format. Rather than expanding the data to its fully-sampled form for later downstream processes, we demonstrate how to use this compressed data directly via on-the-fly common shot or receiver gathers extraction. The combination of massive compression and fast on demand data reconstruction of 3D shot or receiver gathers leads to a substantial reduction in memory costs but with minimal effects on results in the subsequent processing procedures. We demonstrate the effective implementation of our proposed framework on full-waveform inversion on a 3D seismic synthetic data set generated from a Overthrust model.},
	keywords = {3D, FWI, SEG, tensor algebra},
	doi = {10.1190/segam2017-17742951.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2017/zhang2017SEGmsd/zhang2017SEGmsd.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2017/zhang2017SEGmsd/zhang2017SEGmsd_pres.pdf},
	author = {Yiming Zhang and Curt Da Silva and Rajiv Kumar and Felix J. Herrmann}
}
@presentation {zhang2017SINBADFmsd,
	title = {Massive seismic data compression \& recovery w/ on-the-fly data extraction},
	year = {2017},
	publisher = {SINBAD},
	abstract = {Industrial seismic exploration has moved towards complex geological areas, which requires typically long-offset and dense sampling data in order to avoid aliasing and inaccuracy in wave-equation based inversion algorithms. These strict requirements lead to massive data volume size and prohibitive demands on computational resources. In this work, we propose to compress our dense data in hierarchical Tucker tensor format by exploiting the low-rank structure of the data in a transformed domain. Then, we devise on-the-fly common shot or receiver gather extraction directly via the highly compressed factors. In subsampling scenarios, by interpolating this novel tensor format, we can also reconstruct the shot or receiver gather on a per-query basis rather than expanding the data to its fully-sampled form. We demonstrate the effective performance of our proposed technique on 3D stochastic full-waveform inversion, which allows the stochastic algorithm to extract shot gathers as it requires them throughout the inversion process. Moreover, we finally show how to computational effectively generate the CIGs from this compressed low-rank tensor representation of the data with the help of fast simultaneous shot or receiver gather generation.},
	keywords = {Presentation, SINBAD, SINBADFALL2017, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2017/Fall/zhang2017SINBADFmsd/zhang2017SINBADFmsd.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2017/Fall/zhang2017SINBADFmsd/zhang2017SINBADFmsd.mov},
	author = {Yiming Zhang and Curt Da Silva and Rajiv Kumar and Felix J. Herrmann}
}
@conference {fang2017WAVESuqf,
	title = {Uncertainty quantification for inverse problems with a weak wave-equation constraint},
	booktitle = {WAVES 2017 {\textendash}- 13th International Conference on Mathematical and Numerical Aspects of Wave Propagation},
	year = {2017},
	note = {(WAVES, Minneapolis)},
	month = {05},
	pages = {127-128},
	abstract = {In this work, we present a new posterior distribution to quantify uncertainties in solutions of wave-equation based inverse problems. By introducing an auxiliary variable for the wavefields, we weaken the strict wave-equation constraint used by conventional Bayesian approaches. With this weak constraint, the new posterior distribution is a bi-Gaussian distribution with respect to both model parameters and wavefields, which can be directly sampled by the Gibbs sampling method.},
	keywords = {constraint, Gibbs sampling, uncertainty, wave equation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/WAVES/2017/fang2017WAVESuqf/fang2017WAVESuqf.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/WAVES/2017/fang2017WAVESuqf/fang2017WAVESuqf_pres.pdf},
	author = {Zhilong Fang and Curt Da Silva and Rachel Kuske and Felix J. Herrmann}
}
@presentation {dasilva2016SINBADFccs,
	title = {Composite convex smooth optimization with seismic data processing applications},
	year = {2016},
	publisher = {SINBAD},
	abstract = {In this work, we show a general technique for solving optimization problems that are comprised of minimizing a composition of a convex, non-smooth function with a smooth function. We demonstrate this technique in the seismic data processing context applied to robust missing trace interpolation in the low-rank Hierarchical Tucker tensor format as well as cosparsity-based missing trace interpolation.},
	keywords = {Presentation, SINBAD, SINBADFALL2016, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/dasilva2016SINBADFccs/dasilva2016SINBADFccs.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/dasilva2016SINBADFccs/dasilva2016SINBADFccs.mov},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {fang2016SINBADFeaq,
	title = {Efficient approach for quantifying uncertainty of wavefield reconstruction inversion},
	year = {2016},
	publisher = {SINBAD},
	abstract = {Due to the noisy measurements, sparse observations, uncertain forward models, and uncertain prior parameter information, significant uncertainty arises in the inverted velocity model of full-waveform inversion. The uncertain velocity model leads to the uncertainty in the correct positions of events in the migrated images, which subsequently influences the important decision-makings in the oil and gas exploration and production (E\&P) business such as the drilling decisions and economic evaluations. Therefore, understanding the uncertainty in the inverted velocity model is essential for the mitigation of the E\&P risks. A common approach to accounting for uncertainty in seismic inverse problems like full-waveform inversion is to describe the unknown parameters in probabilistic by the Bayesian inference. The Bayesian inference aims at incorporating the information from the data, physics, and one{\textquoteright}s prior knowledge to formulate a posterior distribution function to characterize the statistical properties of the unknown parameters. However, characterizing uncertainties for the large-scale full-waveform inversion in seismic exploration is prohibitively expensive since it requires thousands of function evaluations to sample the parameter space (e.g. via Markov chain Monte Carlo sampling). Apart from this, the existing methods of uncertainty quantification for full-waveform inversion is through applying the Bayesian inference to the standard method such as the adjoint-state method with the assumption that no uncertainty arises from the wave-equation. However, the wave-equation may not be able to reflect all the physics, hence, incorporating uncertainty from wave-equation might benefit the quantification of uncertainty. In this work, we propose to perform uncertainty quantification using the recently proposed wavefield reconstruction inversion technique, which gives an appropriate framework to tackle the uncertainty in wave-equation solve by adding the misfit of the wave-equation to the likelihood distribution and relaxing the wave-equation constraint. Moreover, we propose a computationally feasible approach to analyzing the posterior distribution of the wavefield reconstruction inversion method, which incorporates a Gaussian distribution that approximates the posterior distribution with an optimization-driven Gaussian simulator to sample the Gaussian distribution efficiently. Numerical examples show that comparing to the McMC method, our approach is able to produce comparable results with less computational cost. This is joint work with Curt da Silva and Rachel Kuske.},
	keywords = {Presentation, SINBAD, SINBADFALL2016, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/fang2016SINBADFeaq/fang2016SINBADFeaq.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/fang2016SINBADFeaq/fang2016SINBADFeaq.mov},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Tristan van Leeuwen and Felix J. Herrmann and Rachel Kuske}
}
@presentation {kumar2016SINBADFlrm,
	title = {Low-rank methods for on-the-fly slicing \& dicing of seismic data \& image volumes},
	year = {2016},
	publisher = {SINBAD},
	abstract = {Conventional oil and gas fields are increasingly difficult to explore and produce, which calls for more complex wave-equation based inversion (WEI) algorithms that require dense long-offset samplings. These requirements result in an exponential growth in data volumes and prohibitive demands on computational resources. In this work, we propose a fast, resilient, and scalable wave-equation based inversion methodology for both the data and image-domain, which can handle complicated wave physics. First we show that both the data and image domains exhibit low-rank structure in a transform-domain, which can be exploited to compress the dense data or image volumes. Then, by accessing information from the compressed volumes on-the fly, we devise a scalable computational inversion framework driven by gradient calculations, which works with small subsets of source experiments. In the full-waveform inversion context, we demonstrate the efficacy of low-rank interpolation to improve downstream inversion results compared to merely inverting the velocity model using the subsampled data volume directly. We demonstrate the effectiveness of the proposed framework on two different waveform-inversion formulations, specifically performing full-waveform inversion on 5D data set generated using the overthrust model, and wave-equation based migration velocity analysis on the Marmousi model. This is joint work with Curt Da Silva and Yiming Zhang.},
	keywords = {Presentation, SINBAD, SINBADFALL2016, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/kumar2016SINBADFlrm/kumar2016SINBADFlrm.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/kumar2016SINBADFlrm/kumar2016SINBADFlrm.mov},
	author = {Rajiv Kumar and Curt Da Silva and Yiming Zhang and Felix J. Herrmann}
}
@conference {fang2016SEGuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion using a PDE free semidefinite Hessian and randomize-then-optimize method},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {1390-1394},
	abstract = {The data of full-waveform inversion often contains noise, which induces uncertainties in the inversion results. Ideally, one would like to run a number of independent inversions with different realizations of the noise and assess model-side uncertainties from the resulting models, however this is not feasible because we collect the data only once. To circumvent this restriction, various sampling schemes have been devised to generate an ensemble of models that fit the data to within the noise level. Such sampling schemes typically involve running multiple inversions or evaluating the Hessian of the cost function, both of which are computationally expensive. In this work, we propose a new method to quantify uncertainties based on a novel formulation of the full-waveform inversion problem {\textendash} wavefield reconstruction inversion. Based on this formulation, we formulate a semidefinite approximation of the corresponding Hessian matrix. By precomputing certain quantities, we are able to apply this Hessian to given input vectors without additional solutions of the underlying partial differential equations. To generate a sample, we solve an auxiliary stochastic optimization problem involving this Hessian. The result is a computationally feasible method that, with little overhead, can generate as many samples as required at small additional cost. We test our method on the synthetic BG Compass model and compare the results to a direct-sampling approach. The results show the feasibility of applying our method to computing statistical quantities such as the mean and standard deviation in the context of wavefield reconstruction inversion.},
	keywords = {SEG, Uncertainty quantification, WRI},
	doi = {10.1190/segam2016-13879108.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/fang2016SEGuqw/fang2016SEGuqw.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/fang2016SEGuqw/fang2016SEGuqw_pres.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Tristan van Leeuwen and Felix J. Herrmann}
}
@presentation {dasilva2016SINBADFuse,
	title = {A unified 2D/3D software environment for large scale time-harmonic full waveform inversion},
	year = {2016},
	publisher = {SINBAD},
	abstract = {Full Waveform Inversion is a costly and complex procedure, resulting in industrial codebases that are often written entirely in low-level languages, making them hard to understand, maintain, improve, and extend. In this work, we propose a software framework for organizing a 3D FWI environment that helps mitigate design complexities inherent in the problem in a straightforward way while delegating the performance-critical portions to low level languages to maintain efficiency. The result is a software framework that has a unified interface for 2D and 3D and is flexible, efficient, scalable, and demonstrably correct. We demonstrate the effectiveness of this approach on a large scale 3D FWI example.},
	keywords = {Presentation, SINBAD, SINBADFALL2016, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/dasilva2016SINBADFuse/dasilva2016SINBADFuse.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/dasilva2016SINBADFuse/dasilva2016SINBADFuse.mov},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@conference {dasilva2016SEGuse,
	title = {A unified 2D/3D software environment for large scale time-harmonic full waveform inversion},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {1169-1173},
	abstract = {Full-Waveform Inversion is a costly and complex procedure for realistically sized 3D seismic data. The performance-critical nature of this problem often results in software environments that are written entirely in low-level languages, making them hard to understand, maintain, improve, and extend. The unreadability of such codes can stymie research developments, where the translation from higher level mathematical ideas to high performance codes can be lost, inhibiting the uptake of new ideas in production-level codebases. We propose a new software organization paradigm for Full-Waveform Inversion and other PDE-constrained optimization problems that is flexible, efficient, scalable, and demonstrably correct. We decompose the various structural components of FWI in to its constituent components, from parallel computation to assembling objective functions and gradients to lower level matrix-vector multiplications. This decomposition allows us to create a framework where individual components can be easily swapped out to suit a particular user{\textquoteright}s existing software environment. The ease of applying high-level algorithms to the FWI problem allows us to easily implement stochastic FWI and demonstrate its effectiveness on a large scale 3D problem.},
	keywords = {Full-waveform inversion, large scale, numerical analysis, SEG, software},
	doi = {10.1190/segam2016-13869051.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/dasilva2016SEGuse/dasilva2016SEGuse.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/dasilva2016SEGuse/dasilva2016SEGuse_pres.pdf},
	author = {Curt Da Silva 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}
}
@conference {dasilva2015WSLRigt,
	title = {Irregular grid tensor completion},
	booktitle = {Workshop on Low-rank Optimization and Applications},
	year = {2015},
	note = {(Workshop on Low-rank Optimization and Applications, University of Bonn, Germany)},
	month = {06},
	abstract = {Low rank tensor completion has recently garnered the attention of researchers owing to the ubiquity of tensors in the sciences and the theoretical and numerical challenges compared to matrix completion. Here we consider a tensor completion scheme where the data arises from sampling a continuous function. When the sampling grid is not periodic, the resulting tensor may not be low rank in the Hierarchical Tucker sense, which can adversely affect reconstruction quality when there are missing samples. In order to compensate for this off-grid sampling, we introduce a resampling operator (here, the non-uniform Fourier transform) that accounts for this non-uniformity moreso than merely treating the sampling grid as periodic. Numerical experiments demonstrate that this approach can improve reconstruction quality when there is relatively little data.},
	keywords = {hierarchical tucker, irregular sampling, off the grid, structured tensor, tensor interpolation},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/WS_LR/2015/dasilva2015WSLRigt/dasilva2015WSLRigt_poster.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {dasilva2015SINBADFnso,
	title = {A new software organization for 3D FWI},
	year = {2015},
	publisher = {SINBAD},
	abstract = {In this work, I have implemented a scalable and extendable framework for 3D Full Waveform Inversion in Matlab. This approach uses modern software design principles to create a codebase that is easy to understand, maintain, and extend and that also allows for rapid prototyping of new algorithmic ideas that can be easily transferred to large scale problems. The code itself is modularized in a proper way, which allows straightforward testing of each component (e.g., Taylor error test, adjoint tests, etc.). Improvements to the computational kernel (i.e., Helmholtz solves with more efficient matrix-vector products, new preconditioners, etc.) propagate to the entire framework.},
	keywords = {Presentation, SINBAD, SINBADFALL2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/dasilva2015SINBADFnso/dasilva2015SINBADFans.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/dasilva2015SINBADFnso/dasilva2015SINBADFsdd.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {dasilva2015SINBADogt,
	title = {Off the grid tensor completion for seismic data interpolation},
	year = {2015},
	publisher = {SINBAD},
	abstract = {In this work, we extend our previous low-rank tensor techniques for seismic data interpolation to the {\textquotedblleft}off-the-grid{\textquotedblright} case, wherein the data is not acquired on a regularly-spaced grid. This irregular sampling scheme causes the underlying tensor to become high-rank, thus preventing our previous Hierarchical Tucker tensor algorithms from successfully recovering the volume. Instead of applying these techniques naively, we introduce a fictitious, regular grid on which the volume exhibits low-rank behaviour and a resampling operator that interpolates the data from the regular, low-rank domain to the irregular sampling domain. By incorporating these changes in to our previous framework, we can interpolate irregularly sampled data volumes with costs determined primarily by the choice of resampling transform. In our case, we use a non-uniform Fast Fourier Transform, but other choices are possible.},
	keywords = {Presentation, SINBAD, SINBADSPRING2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Spring/dasilva2015SINBADogt/dasilva2015SINBADogt.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@conference {dasilva2015EAGEogt,
	title = {Off the grid tensor completion for seismic data interpolation},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2015},
	note = {(EAGE, Madrid)},
	month = {06},
	abstract = {The practical realities of acquiring seismic data in a realistic survey are often at odds with the stringent requirements of Shannon-Nyquist-based sampling theory. The unpredictable movement of the ocean{\textquoteleft}s currents can be detrimental in acquiring exactly equally-spaced samples while sampling at Nyquist-rates are expensive, given the huge dimensionality and size of the data volume. Recent work in matrix and tensor completion for seismic data interpolation aim to alleviate such stringent Nyquist-based sampling requirements but are fundamentally posed on a regularly-spaced grid. In this work, we extend our previous results in using the so-called Hierarchical Tucker (HT) tensor format for recovering seismic data to the irregularly sampled case. We introduce an interpolation operator that resamples our tensor from a regular grid (in which we impose our low-rank constraints) to our irregular sampling grid. Our framework is very flexible and efficient, depending primarily on the computational costs of this operator. We demonstrate the superiority of this approach on a realistic BG data set compared to using low-rank tensor methods that merely use binning.},
	keywords = {EAGE, hierarchical tucker, irregular sampling, off the grid, structured tensor, tensor interpolation},
	doi = {10.3997/2214-4609.201412978},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/dasilva2015EAGEogt/dasilva2015EAGEogt.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/dasilva2015EAGEogt/dasilva2015EAGEogt_pres.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@article {dasilva2014htuck,
	title = {Optimization on the Hierarchical Tucker manifold - applications to tensor completion},
	journal = {Linear Algebra and its Applications},
	volume = {481},
	year = {2015},
	note = {(Linear Algebra and its Applications)},
	month = {09},
	pages = {131-173},
	abstract = {In this work, we develop an optimization framework for problems whose solutions are well-approximated by Hierarchical Tucker (HT) tensors, an efficient structured tensor format based on recursive subspace factorizations. By exploiting the smooth manifold structure of these tensors, we construct standard optimization algorithms such as Steepest Descent and Conjugate Gradient for completing tensors from missing entries. Our algorithmic framework is fast and scalable to large problem sizes as we do not require SVDs on the ambient tensor space, as required by other methods. Moreover, we exploit the structure of the Gramian matrices associated with the HT format to regularize our problem, reducing overfitting for high subsampling ratios. We also find that the organization of the tensor can have a major impact on completion from realistic seismic acquisition geometries. These samplings are far from idealized randomized samplings that are usually considered in the literature but are realizable in practical scenarios. Using these algorithms, we successfully interpolate large-scale seismic data sets and demonstrate the competitive computational scaling of our algorithms as the problem sizes grow.},
	keywords = {differential geometry, Gauss{\textendash}Newton, hierarchical tucker tensors, low-rank tensor, Riemannian manifold optimization, tensor completion},
	doi = {10.1016/j.laa.2015.04.015},
	url = {https://slim.gatech.edu/Publications/Public/Journals/LinearAlgebraAndItsApplications/2015/dasilva2014htuck/dasilva2014htuck.pdf},
	url2 = {http://www.sciencedirect.com/science/article/pii/S0024379515002530},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@conference {kumar2015IIPFWIrmb,
	title = {Rank minimization based seismic data processing and inversion},
	booktitle = {Inaugural Full-Waveform Inversion Workshop},
	year = {2015},
	note = {(Natal, Brazil)},
	month = {08-09},
	abstract = {We present the low-rank extensions of full-waveform inversion to mitigate the local-minima issues. The purpose formulation is computationally tractable and does not involve any extra computational cost compared to the adjoint-state method.},
	keywords = {FWI, inversion, Processing, rank minimization},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/IIPFWI/kumar2015IIPFWIrmb/kumar2015IIPFWIrmb_pres.pdf},
	author = {Rajiv Kumar and Curt Da Silva and Oscar Lopez and Aleksandr Y. Aravkin and Hassan Mansour and Haneet Wason and Ernie Esser and Felix J. Herrmann}
}
@presentation {dasilva2015SINBADrii,
	title = {Recent improvements to the 2/3{\textemdash}D imaging \& inversion algorithms in the SLIM software release},
	year = {2015},
	publisher = {SINBAD},
	abstract = {The SLIM software release contains algorithms for frequency domain 2D imaging and inversion based on direct LU and QR factorizations. Recent improvments include significant speedups, by utilizing the latest parallel direct factorization algorithms and inversion strategies which solve for each frequency on its corresponding grid. The iterative Helmholtz solvers available in the software release are now also applicable to 2D problems, extending the 2D algorithms to very large \& high frequency 2D imaging and inversion.},
	keywords = {Presentation, SINBAD, SINBADSPRING2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Spring/dasilva2015SINBADrii/dasilva2015SINBADrii.pdf},
	author = {Curt Da Silva and Bas Peters and Felix J. Herrmann}
}
@presentation {dasilva2015SINBADFsss,
	title = {Scaling SINBAD software to 3-D on Yemoja},
	year = {2015},
	publisher = {SINBAD},
	abstract = {We present early results on the scalability of SINBAD{\textquoteright}s wavefield reconstruction and wave-equation based inversion technologies on Yemoja, a 17k core cluster made available to us by BG Group at SENAI CIMATEC Supercomputing Centre in Brazil.},
	keywords = {Presentation, SINBAD, SINBADFALL2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/dasilva2015SINBADFsss/dasilva2015SINBADFsss.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/dasilva2015SINBADFsss/dasilva2015SINBADFsss.mov},
	author = {Curt Da Silva and Haneet Wason and Mathias Louboutin and Bas Peters and Shashin Sharan and Zhilong Fang and Felix J. Herrmann}
}
@conference {lin2015IIPFWIsws,
	title = {Software and workflows at SLIM/SINBAD},
	booktitle = {Inaugural Full-Waveform Inversion Workshop},
	year = {2015},
	note = {(Natal, Brazil)},
	month = {08-09},
	keywords = {FWI, SLIM, software},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/IIPFWI/lin2015IIPFWIsws/lin2015IIPFWIsws_pres.pdf},
	author = {Tim T.Y. Lin and Curt Da Silva and Felix J. Herrmann}
}
@conference {lopez2015SEGWSsfm,
	title = {SVD-free matrix completion in randomized marine acquisition},
	booktitle = {SEG Workshop on Rank-Reduction and Other Sparse Transform Methods with Application to Data Reconstruction, De-Noising, De-Blending and Imaging; New Orleans},
	year = {2015},
	note = {(SEG, New Orleans)},
	month = {10},
	abstract = {During this presentation, we will illustrate how insights from matrix-free matrix completion can be used to solve problems in (on- and off-the-grid) missing trace interpolation and time-jittered (simultaneous) marine acquisition. In particular, we will demonstrate the importance of data organization to favor recovery by rank minimization and the use of deterministic spectral properties to predict the recovery. We also make make comparisons between wavefield reconstructions based on transform-domain sparsity promotion and matrix completion.},
	keywords = {Acquisition, interpolation, matrix completion, rank reduction, SEG, workshop},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2015/lopez2015SEGWSsfm/lopez2015SEGWSsfm_pres.pdf},
	author = {Oscar Lopez and Haneet Wason and Curt Da Silva and Rajiv Kumar and Felix J. Herrmann}
}
@conference {fang2015EAGEuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2015},
	note = {(EAGE, Madrid)},
	month = {06},
	abstract = {In this work, we propose a method to quantify the uncertainty of wavefield reconstruction inversion under the framework of Bayesian inference. Unlike the conventional method using the wave equation as the forward mapping, we involve the wave equation misfit in the posterior distribution and propose a new posterior distribution. The negative log-likelihood of the new distribution is less oscillatory than that of the conventional posterior distribution, and its Gauss-Newton Hessian is a diagonal matrix that can be generated without any additional computational cost. We use the diagonal Gauss-Newton Hessian to derive an approximate Gaussian distribution at the maximum likelihood point to quantify the uncertainty. This method makes the uncertainty quantification for WRI computationally tractable and is able to provide reasonable uncertainty analysis based on our numerical results.},
	keywords = {EAGE, FWI, UQ, WRI},
	doi = {10.3997/2214-4609.201413198},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/fang2015EAGEuqw/fang2015EAGEuqw.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/fang2015EAGEuqw/fang2015EAGEuqw_pres.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@conference {fang2015IIPFWIuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	booktitle = {Inaugural Full-Waveform Inversion Workshop},
	year = {2015},
	note = {(Natal, Brazil)},
	month = {08-09},
	abstract = {In this work, we propose a method to quantify the uncertainty of wavefield reconstruction inversion under the framework of Bayesian inference. Unlike the conventional method using the wave equation as the forward mapping, we involve the wave equation misfit in the posterior distribution and propose a new posterior distribution. The negative log-likelihood of the new distribution is less oscillatory than that of the conventional posterior distribution, and its Gauss-Newton Hessian is a diagonal matrix that can be generated without any additional computational cost. We use the diagonal Gauss-Newton Hessian to derive an approximate Gaussian distribution at the maximum likelihood point to quantify the uncertainty. This method makes the uncertainty quantification for WRI computationally tractable and is able to provide reasonable uncertainty analysis based on our numerical results.},
	keywords = {FWI, uncertainty, WRI},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/IIPFWI/fang2015IIPFWIuqw/fang2015IIPFWIuqw_pres.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@presentation {fang2015SINBADFuqw,
	title = {Uncertainty quantification for Wavefield-Reconstruction Inversion using a positive-definite approximated Hessian},
	year = {2015},
	publisher = {SINBAD},
	abstract = {We analyze the Hessian of wavefield reconstruction inversion (WRI) and propose a new approximated Hessian. Instead of requiring PDE solves, the matrix-vector multiplication action of the approximate Hessian can be achieved with several matrix-vector multiplications. The diagonal part of the approximated Hessian can be also calculated without additional PDE solves. We apply this approximated Hessian to uncertainty quantification and obtain statistical parameters such as standard deviation and confidence interval. Numerical example illustrate the accuracy of the estimated Hessian and the feasibility of the method to quantify uncertainties.},
	keywords = {Presentation, SINBAD, SINBADFALL2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/fang2015SINBADFuqw/fang2015SINBADFuqw.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/fang2015SINBADFuqw/fang2015SINBADFuqw.mov},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@conference {fang2014EAGEfuq,
	title = {Fast uncertainty quantification for 2D full-waveform inversion with randomized source subsampling},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2014},
	month = {06},
	abstract = {Uncertainties arise in every area of seismic exploration, especially in full-waveform inversion, which is highly non-linear. In the framework of Bayesian inference, uncertainties can be analyzed by sampling the posterior probability density distribution with a Markov chain Monte-Carlo (McMC) method. We reduce the cost of computing the posterior distribution by working with randomized subsets of sources. These approximations, together with the Gaussian assumption and approximation of the Hessian, leads to a computational tractable uncertainty quantification. Application of this approach to a synthetic leads to standard deviations and confidence intervals that are qualitatively consistent with our expectations.},
	keywords = {EAGE, FWI, Markov chain Monte Carlo, randomized sources subsampling, Uncertainty quantification},
	doi = {10.3997/2214-4609.20140715},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/fang2014EAGEfuq/fang2014EAGEfuq.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/fang2014EAGEfuq/fang2014EAGEfuq_pres.pdf},
	author = {Zhilong Fang and Curt Da Silva and Felix J. Herrmann}
}
@presentation {dasilva2014SINBADlss,
	title = {Large-scale seismic data interpolation in a parallel computing environment},
	year = {2014},
	publisher = {SINBAD},
	abstract = {This talk will outline the details for a parallel implementation of the SPG-LR algorithm for matrix completion and its applications to large-scale seismic data interpolation. Previous work done in our group on LR-based matrix factorization has been shown to be extremely efficient for seismic data interpolation that fits in memory. We extend the SPGL1 framework that this method is built on to instances where the full seismic data volume cannot entirely fit in the memory of one computational node and must be run in a distributed environment. We will also look at off-the-grid regularization for tensor completion using a non-uniform FFT operator and examine its effects on interpolating an irregularly sampled, 3D seismic frequency slice.},
	keywords = {Presentation, SINBAD, SINBADFALL2014, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2014/Fall/dasilva2014SINBADlss/dasilva2014SINBADlss.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {dasilva2014SINBADlrp,
	title = {Low-rank promoting transformations and tensor interpolation - applications to seismic data denoising},
	year = {2014},
	publisher = {SINBAD},
	abstract = {In this presentation, we extend our previous work in Hierarchical Tucker (HT) tensor completion, which uses an extremely efficient representation for representing high-dimensional tensors exhibiting low-rank structure, to handle subsampled tensors with noisy entries. We consider a {\textquoteleft}low-noise{\textquoteright} case, so that the energies of the noise and the signal are nearly indistinguishable, and a {\textquoteleft}high-noise{\textquoteright} case, in which the noise energy is now scaled to the amplitude of the entire data volume. For the low-noise case in particular, standard trace-by-trace energy comparisons cannot distinguish noise from signal. We examine the behaviour of noise in terms of the singular values along different matricizations of the data, i.e. reshaping of the tensor along different dimensions. By interpreting this effect in the context of tensor completion, we demonstrate the inefficacy of denoising by this method in the source-receiver domain. In light of this observation, we transform the decimated, noisy data in to the midpoint-offset domain, which promotes low-rank behaviour in the signal and high-rank behaviour in the noise. This distinction between signal and noise allows low-rank interpolation to effectively denoise the signal, without knowledge of the noise location, with only a marginal increase in computational cost. We demonstrate the effectiveness of this approach on a 4D frequency slice.},
	keywords = {Presentation, SINBAD, SINBADSPRING2014, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2014/Spring/dasilva2014SINBADlrp/dasilva2014SINBADlrp.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@conference {dasilva2014EAGEhtucknoisy,
	title = {Low-rank promoting transformations and tensor interpolation - applications to seismic data denoising},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2014},
	month = {06},
	abstract = {In this abstract, we extend our previous work in Hierarchical Tucker (HT) tensor completion, which uses an extremely efficient representation for representing high-dimensional tensors exhibit- ing low-rank structure, to handle subsampled tensors with noisy entries. We consider a "low-noise" case, so that the energies of the noise and the signal are nearly indistinguishable, and a {\textquoteright}high-noise{\textquoteright} case, in which the noise energy is now scaled to the amplitude of the entire data volume. We examine the effect of the noise in terms of the singular values along different matricizations of the data, i.e., reshaping of the tensor along different modes. By interpreting this effect in the context of tensor completion, we demonstrate the inefficacy of denoising by this method in the source-receiver do- main. In light of this observation, we transform the decimated, noisy data in to the midpoint-offset domain, which promotes low-rank behaviour in the signal and high-rank behaviour in the noise. This distinction between signal and noise allows low-rank interpolation to effectively denoise the signal with only a marginal increase in computational cost. We demonstrate the effectiveness of this approach on a 4D frequency slice.},
	keywords = {EAGE, hierarchical tucker, low-rank transform, riemannian optimization, seismic denoising, structured tensor, tensor interpolation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/dasilva2014EAGEhtucknoisy/dasilva2014EAGEhtucknoisy.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/dasilva2014EAGEhtucknoisy/dasilva2014EAGEhtucknoisy_pres.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {fang2014SINBADuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	year = {2014},
	publisher = {SINBAD},
	abstract = {This talk discusses the uncertainty quantification for the wavefield reconstruction inversion. The Hessian and gradient of the WRI can be easily obtained once the wave field is generated. As a result, we use a Newton type McMC method to sample the posterior distribution. Statistical property such as mean, standard deviation, confidence interval are calculated based on the sampled distribution.},
	keywords = {Presentation, SINBAD, SINBADFALL2014, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2014/Fall/fang2014SINBADuqw/fang2014SINBADuqw.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@conference {dasilva2013EAGEhtucktensor,
	title = {Hierarchical Tucker tensor optimization - applications to 4D seismic data interpolation},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2013},
	month = {06},
	abstract = {In this work, we develop optimization algorithms on the manifold of Hierarchical Tucker (HT) tensors, an extremely efficient format for representing high-dimensional tensors exhibiting particular low-rank structure. With some minor alterations to existing theoretical developments, we develop an optimization framework based on the geometric understanding of HT tensors as a smooth manifold, a generalization of smooth curves/surfaces. Building on the existing research of solving optimization problems on smooth manifolds, we develop Steepest Descent and Conjugate Gradient methods for HT tensors. The resulting algorithms converge quickly, are immediately parallelizable, and do not require the computation of SVDs. We also extend ideas about favourable sampling conditions for missing-data recovery from the field of Matrix Completion to Tensor Completion and demonstrate how the organization of data can affect the success of recovery. As a result, if one has data with randomly missing source pairs, using these ideas, coupled with an efficient solver, one can interpolate large-scale seismic data volumes with missing sources and/or receivers by exploiting the multidimensional dependencies in the data. We are able to recover data volumes amidst extremely high subsampling ratios (in some cases, > 75\%) using this approach.},
	keywords = {3D data interpolation, EAGE, riemannian optimization, structured tensor},
	doi = {10.3997/2214-4609.20130390},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2013/dasilva2013EAGEhtucktensor/dasilva2013EAGEhtucktensor.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2013/dasilva2013EAGEhtucktensor/dasilva2013EAGEhtucktensor_pres.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {dasilva2013SINBADSPRhtuck,
	title = {Hierarchical Tucker tensor optimization - applications to 4D seismic data interpolation},
	year = {2013},
	publisher = {SINBAD},
	abstract = {In this work, we develop optimization algorithms on the manifold of Hierarchical Tucker (HT) tensors, an extremely efficient format for representing high-dimensional tensors exhibiting particular low-rank structure. With some minor alterations to existing theoretical developments, we develop an optimization framework based on the geometric understanding of HT tensors as a smooth manifold, a generalization of smooth curves/surfaces. Building on the existing research of solving optimization problems on smooth manifolds, we develop Steepest Descent and Conjugate Gradient methods for HT tensors. The resulting algorithms converge quickly, are immediately parallelizable, and do not require the computation of SVDs. We also derive efficient Gauss-Newton based algorithms which converge much faster than standard, first-order methods. We also extend ideas about favourable sampling conditions for missing-data recovery from the field of Matrix Completion to Tensor Completion and demonstrate how the organization of data can affect the success of recovery. As a result, if one has data with randomly missing source pairs, using these ideas, coupled with an efficient solver, one can interpolate large-scale seismic data volumes with missing sources and/or receivers by exploiting the multidimensional dependencies in the data. We are able to recover data volumes amidst extremely high subsampling ratios (in some cases, > 75\%) using this approach and we demonstrate our recovery on a synthetic 5D data set provided to us by BG.},
	keywords = {3D data interpolation, hierarchical tucker, Presentation, SINBAD, SINBADSPRING2013, SLIM, structured tensor},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Spring/dasilva2013SINBADSPRhtuck/dasilva2013SINBADSPRhtuck.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@conference {dasilva2013SAMPTAhtuck,
	title = {Hierarchical Tucker tensor optimization - applications to tensor completion},
	year = {2013},
	month = {07},
	abstract = {In this work, we develop an optimization framework for problems whose solutions are well-approximated by Hierarchical Tucker tensors, an efficient structured tensor format based on recursive subspace factorizations. Using the differential geometric tools presented here, we construct standard optimization algorithms such as Steepest Descent and Conjugate Gradient, for interpolating tensors in HT format. We also empirically examine the importance of one{\textquoteright}s choice of data organization in the success of tensor recovery by drawing upon insights from the Matrix Completion literature. Using these algorithms, we recover various seismic data sets with randomly missing source pairs.},
	keywords = {differential geometry, hierarchical tucker, riemannian optimization, SAMPTA, structured tensor, tensor interpolation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SAMPTA/2013/dasilva2013SAMPTAhtuck/dasilva2013SAMPTAhtuck.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SAMPTA/2013/dasilva2013SAMPTAhtuck/dasilva2013SAMPTAhtuck_pres.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {akalin2013SINBADmtc,
	title = {Matrix and tensor completion for large-scale seismic interpolation: a comparative study},
	year = {2013},
	publisher = {SINBAD},
	abstract = {Owing to their high dimensionality, interpolating 3D seismic data volumes remains a computationally daunting task. In this work, we outline a comprehensive framework for sampling and interpolating such volumes based on the well-understood theory of Matrix and Tensor completion. This interpolation theory consists of three components major signal structure, structure-destroying sampling, and structure-restoring optimization. By viewing interpolation in the context of this theory, we are able to specify exactly when these approaches are expected to perform well. We also introduce structure-revealing transformations that promote the inherent low-rank structure in seismic data as well as a factorization approach that scales to large problem sizes. Our methods are able to handle large-scale data volumes more accurately and more quickly compared to other more ad-hoc approaches, as we will demonstrate. This is joint work with Curt Da Silva, Rajiv Kumar, Ben Recht, and Felix J. Herrmann.},
	keywords = {Presentation, SINBAD, SINBADFALL2013, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Fall/akalin2013SINBADmtc/akalin2013SINBADmtc.pdf},
	author = {Okan Akalin and Curt Da Silva and Rajiv Kumar and Ben Recht and Felix J. Herrmann}
}
@presentation {dasilva2013SINBADstf,
	title = {Structured tensor formats for missing-trace interpolation and beyond},
	year = {2013},
	publisher = {SINBAD},
	abstract = {High-dimensional data, alternatively known as tensors, occurs in a variety of seismic problems. By exploiting the fact that seismic data can be well represented as a structured tensor, we design algorithms that operate on much lower dimensional parameters. In this talk, we will review some recent developments in interpolating seismic data volumes in the so-called Hierarchical Tucker format as well as demonstrate the need for such formats when tackling high-dimensional problems such as Uncertainty Quantification.},
	keywords = {Presentation, SINBAD, SINBADFALL2013, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Fall/dasilva2013SINBADstf/dasilva2013SINBADstf.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@conference {dasilva2013SEGhtuck,
	title = {Structured tensor missing-trace interpolation in the Hierarchical Tucker format},
	booktitle = {SEG Technical Program Expanded Abstracts},
	volume = {32},
	year = {2013},
	month = {09},
	pages = {3623-3627},
	publisher = {SEG},
	organization = {SEG},
	abstract = {Owing to the large scale and dimensionality of a 3D seismic experiment, acquiring fully-sampled data according to the Nyquist criterion is an exceedingly arduous and cost-prohibitive task. In this paper, we develop tools to interpolate 5D seismic volumes with randomly missing sources or receivers using a relatively novel tensor format known as the Hierarchical Tucker (HT) format. By exploiting the underlying smooth structure of HT tensors, specifically its smooth manifold structure, we develop solvers which are fast, immediately parallelizable, and SVD-free, making these solvers amenable to large-scale problems where SVD-based projection methods are far too costly. We also build on intuition of multidimensional sampling from the perspective of matrix-completion and demonstrate the ability of our algorithms to recover frequency slices even amidst very high levels of source subsampling on a synthetic large-scale 3D North Sea dataset.},
	keywords = {differential geometry, gauss newton, hierarchical tucker, riemannian optimization, SEG, structured tensor, tensor interpolation},
	doi = {10.1190/segam2013-0709.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2013/dasilva2013SEGhtuck/dasilva2013SEGhtuck.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2013/dasilva2013SEGhtuck/dasilva2013SEGhtuck_pres.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {dasilva2012SINBADhtt,
	title = {Hierarchical Tucker tensor optimization - applications to 4D seismic data interpolation},
	year = {2012},
	publisher = {SINBAD},
	abstract = {There has been a swell of research in the scientific computing community in the last couple of years which tries to extend notions of linear algebra (rank, the SVD, linear systems, etc.) to higher dimensional arrays, or tensors. Much work has been proposed to try to overcome the so called "curse of dimensionality", the O(N^d) storage required for a d-dimensional array, where N is the size of each dimension. The hierarchical Tucker format is one such tensor representation which manages to decompose a hierarchy of dimensions into parameter matrices of very manageable size, requiring at most dNK + (d - 2)K^3 + K^2 parameters, where K is an internal rank parameter. In this work, we extend ideas of matrix completion to the tensor case, where we only know a small number of randomly distributed entries from various 4D frequency slices, and try to recover the fully sampled tensor based on the knowledge that it has low hierarchical tucker rank in a particular arrangement of dimensions. Using this approach, we exploit the multi-dimensional dependencies within the full data in order to achieve very promising interpolation results even from heavily subsampled data.},
	keywords = {Presentation, SINBAD, SINBADFALL2012, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2012/Fall/dasilva2012SINBADhtt/dasilva2012SINBADhtt_pres.pdf},
	author = {Curt Da Silva}
}
@conference {dasilva2012EAGEprobingprecond,
	title = {Matrix probing and simultaneous sources: a new approach for preconditioning the Hessian},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2012},
	month = {06},
	abstract = {Recent advances based on the mathematical understanding of the Hessian as, under certain conditions, a pseudo-differential operator have resulted in a new preconditioner by L. Demanet et al. Basing their approach on a suitable basis expansion for the Hessian, by suitably {\textquoteright}probing{\textquoteright} the Hessian, i.e. applying the Hessian to a small number of randomized model perturbations, one can obtain an approximation to the inverse Hessian in an efficient manner. Building upon this approach, we consider this preconditioner in the context of least-squares migration and Full Waveform Inversion and specifically dimensionality reduction techniques in these domains. By utilizing previous work in simultaneous sources, we are able to develop an efficient least-squares migration scheme which recovers higher quality images and hence higher quality search directions in the context of a Gauss-Newton method for Full Waveform Inversion while simultaneously avoiding inordinate amounts of additional work.},
	keywords = {EAGE, matrix probing, pseudo-differential operator},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2012/dasilva2012EAGEprobingprecond/dasilva2012EAGEprobingprecond.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2012/dasilva2012EAGEprobingprecond/dasilva2012EAGEprobingprecond_pres.pdf},
	url2 = {http://earthdoc.eage.org/publication/publicationdetails/?publication=59193},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {dasilva2012SINBADrdp,
	title = {Recent developments in preconditioning the wave-equation Hessian},
	year = {2012},
	publisher = {SINBAD},
	keywords = {Presentation, SINBAD, SINBADSPRING2012, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2012/Spring/dasilva2012SINBADrdp/dasilva2012SINBADrdp_pres.pdf},
	author = {Curt Da Silva}
}
@presentation {dasilva2011SINBADrdp,
	title = {Recent developments in preconditioning the FWI Hessian},
	year = {2011},
	publisher = {SINBAD},
	keywords = {Presentation, SINBAD, SINBADFALL2011, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2011/Fall/dasilva2011SINBADrdp/dasilva2011SINBADrdp_pres.pdf},
	author = {Curt Da Silva}
}
