@unpublished{fang2015EAGEsew,
	title = {Source estimation for {Wavefield} {Reconstruction} {Inversion}},
	year = {2015},
	note = {(to be presented at the EAGE Conference)},
	publisher = {UBC},
	organization = {UBC},
	abstract = {Wavefield reconstruction inversion is a new approach to waveform based inversion that helps overcome the {\textquoteleft}cycle skipping{\textquoteright} problem. However, like most waveform based inversion methods, wavefield reconstruction inversion also requires good source wavelets. Without correct source wavelets, wavefields cannot be reconstructed correctly and the velocity model cannot be updated correctly neither. In this work, we propose a source estimation method for wavefield reconstruction inversion based on the variable projection method. In this method, we reconstruct wavefields and estimate source wavelets simultaneously by solving an extended least-squares problem, which contains source wavelets. This approach does not increase the computational cost compared to conventional wavefield reconstruction inversion. Numerical results illustrates with our source estimation method we are able to recover source wavelets and obtain inversion results that are comparable to results obtained with true source wavelets.},
	keywords = {EAGE, source estimation, WRI},
	url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2015/fang2015EAGEsew/fang2015EAGEsew.html},
	author = {Zhilong Fang and Felix J. Herrmann}
}

@article{GPR:GPR279,
author = {Shin, Changsoo and Jang, Seonghyung and Min, Dong-Joo},
title = {Improved amplitude preservation for prestack depth migration by inverse scattering theory},
journal = {Geophysical Prospecting},
volume = {49},
number = {5},
publisher = {Blackwell Science Ltd},
issn = {1365-2478},
url = {http://dx.doi.org/10.1046/j.1365-2478.2001.00279.x},
doi = {10.1046/j.1365-2478.2001.00279.x},
pages = {592--606},
year = {2001},
}

@article{eckstein2012augmented,
  title={Augmented Lagrangian and alternating direction methods for convex optimization: A tutorial and some illustrative computational results},
  author={Eckstein, J and Yao, W},
  journal={RUTCOR Research Reports},
  volume={32},
  year={2012}
}

@conference {baumstein,
	title = {POCS-based geophysical constraints in multi-parameter Full Wavefield Inversion},
	year = {2013},
	month = {06},
	publisher = {EAGE},
	author = {A. Baumstein }
}

@article{doi:10.1190/1.1635056,
author = {Sen, M. and Roy, I.},
title = {Computation of differential seismograms and iteration adaptive regularization in prestack waveform inversion},
journal = {GEOPHYSICS},
volume = {68},
number = {6},
pages = {2026-2039},
year = {2003},
doi = {10.1190/1.1635056},
URL = { 
        http://dx.doi.org/10.1190/1.1635056   
},
eprint = { 
        http://dx.doi.org/10.1190/1.1635056    
}
}

@article{Prieux01092013,
author = {Prieux, Vincent and Brossier, Romain and Operto, Stéphane and Virieux, Jean}, 
title = {Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation},
volume = {194}, 
number = {3}, 
pages = {1640-1664}, 
year = {2013}, 
doi = {10.1093/gji/ggt177}, 
abstract ={Multiparameter full waveform inversion (FWI) is a challenging quantitative seismic imaging method for lithological characterization and reservoir monitoring. The difficulties in multiparameter FWI arise from the variable influence of the different parameter classes on the phase and amplitude of the data, and the trade-off between these. In this framework, choosing a suitable parametrization of the subsurface and designing the suitable FWI workflow are two key methodological issues in non-linear waveform inversion. We assess frequency-domain visco-acoustic FWI to reconstruct the compressive velocity (VP), the density (ρ) or the impedance (IP) and the quality factor (QP), from the hydrophone component, using a synthetic data set that is representative of the Valhall oil field in the North Sea. We first assess which of the (VP, ρ) and (VP, IP) parametrizations provides the most reliable FWI results when dealing with wide-aperture data. Contrary to widely accepted ideas, we show that the (VP, ρ) parametrization allows a better reconstruction of both the VP, ρ and IP parameters, first because it favours the broad-band reconstruction of the dominant VP parameter, and secondly because the trade-off effects between velocity and density at short-to-intermediate scattering angles can be removed by multiplication, to build an impedance model. This allows for the matching of the reflection amplitudes, while the broad-band velocity model accurately describes the kinematic attributes of both the diving waves and reflections. Then, we assess different inversion strategies to recover the quality factor QP, in addition to parameters VP and ρ. A difficulty related to attenuation estimation arises because, on the one hand the values of QP are on average one order of magnitude smaller than those of VP and ρ, and on the other hands model perturbations relative to the starting models can be much higher for QP than for VP and ρ during FWI. In this framework, we show that an empirical tuning of the FWI regularization, which is adapted to each parameter class, is a key issue to correctly account for the attenuation in the inversion. We promote a hierarchical approach where the dominant parameter VP is reconstructed first from the full data set (i.e. without any data preconditioning) to build a velocity model as kinematically accurate as possible before performing the joint update of the three parameter classes during a second step. This hierarchical imaging of compressive wave speed, density and attenuation is applied to a real wide-aperture ocean-bottom-cable data set from the Valhall oil field. Several geological features, such as accumulation of gas below barriers of claystone and soft quaternary sediment are interpreted in the FWI models of density and attenuation. The models of VP, ρ and QP that have been developed by visco-acoustic FWI of the hydrophone data can be used as initial models to perform visco-elastic FWI of the geophone data for the joint update of the compressive and shear wave speeds.}, 
URL = {http://gji.oxfordjournals.org/content/194/3/1640.abstract}, 
eprint = {http://gji.oxfordjournals.org/content/194/3/1640.full.pdf+html}, 
journal = {Geophysical Journal International} 
}


@article{shin09,
author = {Shin, C and Cha, Y H},
journal = {Geophysical Journal International},
pages = {1067--1079},
title = {{Waveform inversion in the Laplace-Fourier domain}},
url = {http://dx.doi.org/10.1111/j.1365-246X.2009.04102.x},
volume = {177},
year = {2009}
}

@article{Bauschke1994418,
title = "Dykstras Alternating Projection Algorithm for Two Sets ",
journal = "Journal of Approximation Theory ",
volume = "79",
number = "3",
pages = "418 - 443",
year = "1994",
note = "",
issn = "0021-9045",
doi = "http://dx.doi.org/10.1006/jath.1994.1136",
url = "http://www.sciencedirect.com/science/article/pii/S0021904584711361",
author = "H.H. Bauschke and J.M. Borwein"
}

@book{Nocedal:2000,
author = {Nocedal, J and Wright, S J},
title = {{Numerical optimization}},
publisher = {Springer},
year = {2000}
}

@book{Boyd:2004:CO:993483,
 author = {Boyd, Stephen and Vandenberghe, Lieven},
 title = {Convex Optimization},
 year = {2004},
 isbn = {0521833787},
 publisher = {Cambridge University Press},
 address = {New York, NY, USA}
} 

@inbook{pnmethods,
author = {Mark Schmidt and Dongmin Kim and Suvrit Sra},
booktitle = {Optimization for Machine Learning},
title = {Projected Newton-type Methods in Machine Learning},
year = {2012},
publisher = {MIT Press},
chapter = {11},
pages = {305-327},
month = {04},
volume = {35}
}

@conference {costlyfsimplec,
author = {Mark Schmidt and Ewout van den Berg and Michael Friedlander and Kevin Murphy},
title = {Optimizing Costly Functions with Simple Constraints: A Limited-Memory Projected Quasi-Newton Algorithm},
year = {2009},
month = {01},
volume = {5},
pages = {456-463},
organization = {JMLR},
}

@unpublished {peters2014SEGsrh,
	title = {A sparse reduced Hessian approximation for multi-parameter Wavefield Reconstruction Inversion},
	year = {2014},
	month = {04},
	publisher = {UBC},
	organization = {UBC},
	abstract = {Multi-Parameter full-waveform inversion is a challenging problem, because the unknown parameters appear in the same wave equation and the magnitude of the parameters can vary many orders of magnitude. This makes accurate estimation of multiple-parameters very difficult. To mitigate the problems, sequential strategies, regularization methods and scalings of gradients and quasi-Newton Hessians have been proposed. All of these require design, fine-tuning and adaptation to different waveform inversion problems. We propose to use a sparse approximation to the Hessian derived from a penalty-formulation of the objective function. Sparseness allows to have the Hessian in memory and compute update directions at very low cost. This results in decent reconstruction of the multiple parameters at very low additional memory and computational expense.},
	keywords = {Full-waveform inversion, Hessian, Optimization, penalty method, private},
	url = {https://www.slim.eos.ubc.ca/Publications/Private/Conferences/SEG/2014/peters2014SEGsrh.pdf},
	author = {Bas Peters and Felix J. Herrmann}
}

@conference {leeuwen2014EAGEntf,
	title = {A new take on FWI: Wavefield Reconstruction Inversion},
	booktitle = {EAGE},
	year = {2014},
	month = {01},
	publisher = {UBC},
	organization = {UBC},
	abstract = {We discuss a recently proposed novel method for waveform inversion: Wavefield Reconstruction Inversion (WRI). As opposed to conventional FWI {\textendash} which attempts to minimize the error between observed and predicted data obtained by solving a wave equation {\textendash} WRI reconstructs a wave-field from the data and extracts a model-update from this wavefield by minimizing the wave-equation residual. The method does not require explicit computation of an adjoint wavefield as all the necessary information is contained in the reconstructed wavefield. We show how the corresponding model updates can be interpreted physically analogously to the conventional imaging-condition-based approach.},
	keywords = {EAGE, Full-waveform inversion, Optimization, penalty method, Wavefield Reconstruction Inversion},
	url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/leeuwen2014EAGEntf.pdf},
	author = {Tristan van Leeuwen and Felix J. Herrmann and Bas Peters}
}

@conference {peters2014EAGEweb,
	title = {Wave-equation based inversion with the penalty method: adjoint-state versus wavefield-reconstruction inversion},
	booktitle = {EAGE},
	year = {2014},
	month = {01},
	publisher = {UBC},
	organization = {UBC},
	abstract = {In this paper we make a comparison between wave-equation based inversions based on the adjoint-state and penalty methods. While the adjoint-state method involves the minimization of a data-misfit and exact solutions of the wave-equation for the current velocity model, the penalty- method aims to first find a wavefield that jointly fits the data and honours the physics, in a least-squares sense. Given this reconstructed wavefield, which is a proxy for the true wavefield in the true model, we calculate updates for the velocity model. Aside from being less nonlinear{\textendash}the acoustic wave equation is linear in the wavefield and model parameters but not in both{\textendash}the inversion is carried out over a solution space that includes both the model and the wavefield. This larger search space allows the algortihm to circumnavigate local minima, very much in the same way as recently proposed model extentions try to acomplish. We include examples for low frequencies, where we compare full-waveform inversion results for both methods, for good and bad starting models, and for high frequencies where we compare reverse-time migration with linearized imaging based on wavefield-reconstruction inversion. The examples confirm the expected benefits of the proposed method.},
	keywords = {EAGE, Full-waveform inversion, Imaging, Optimization, penalty method},
	url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/peters2014EAGEweb.pdf},
	author = {Bas Peters and Felix J. Herrmann and Tristan van Leeuwen}
}

@article{Lavoue01042014,
author = {Lavoué, F. and Brossier, R. and Métivier, L. and Garambois, S. and Virieux, J.}, 
title = {Two-dimensional permittivity and conductivity imaging by full waveform inversion of multioffset GPR data: a frequency-domain quasi-Newton approach},
volume = {197}, 
number = {1}, 
pages = {248-268}, 
year = {2014}, 
doi = {10.1093/gji/ggt528}, 
abstract ={Full waveform inversion of ground-penetrating radar data is an emerging technique for the quantitative, high-resolution imaging of the near subsurface. Here, we present a 2-D frequency-domain full waveform inversion for the simultaneous reconstruction of the dielectric permittivity and of the electrical conductivity. The inverse problem is solved with a quasi-Newton optimization scheme, where the influence of the Hessian is approximated by the L-BFGS-B algorithm. This formulation can be considered to be fully multiparameter since it enables to update permittivity and conductivity values within the same descent step, provided we define scales of measurement through a reference permittivity, a reference conductivity, and an additional scaling factor. Numerical experiments on a benchmark from the literature demonstrate that the inversion is very sensitive to the parameter scaling, despite the consideration of the approximated Hessian that should correct for parameter dimensionalities. A proper scaling should respect the natural sensitivity of the misfit function and give priority to the parameter that has the most impact on the data (the permittivity, in our case). We also investigate the behaviour of the inversion with respect to frequency sampling, considering the selected frequencies either simultaneously or sequentially. As the relative imprint of permittivity and conductivity in the data varies with frequency, the simultaneous reconstruction of both parameters takes a significant benefit from broad frequency bandwidth data, so that simultaneous or cumulative strategies should be favoured. We illustrate our scaling approach with a realistic synthetic example for the imaging of a complex subsurface from on-ground multioffset data. Considering data acquired only from the ground surface increases the ill-posedness of the inverse problem and leads to a strong indetermination of the less-constrained conductivity parameters. A Tikhonov regularization can prevent the creation of high-wavenumber artifacts in the conductivity model that compensate for erroneous low-wavenumber structures, thus enabling to select model solutions. We propose a workflow for multiparameter imaging involving both parameter scaling and regularization. Optimal combinations of scaling factors and regularization weights can be identified by seeking regularization levels that exhibit a clear minimum of final data misfit with respect to parameter scaling. We confirm this workflow by inverting noise-contaminated synthetic data. In a surface-to-surface acquisition configuration, we have been able to reconstruct an accurate permittivity structure and a smooth version of the conductivity distribution, based entirely on the analysis of the data misfit with respect to parameter scaling, for different regularization levels.}, 
URL = {http://gji.oxfordjournals.org/content/197/1/248.abstract}, 
eprint = {http://gji.oxfordjournals.org/content/197/1/248.full.pdf+html}, 
journal = {Geophysical Journal International} 
}

@article{Prieux01092013,
author = {Prieux, Vincent and Brossier, Romain and Operto, Stéphane and Virieux, Jean}, 
title = {Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation},
volume = {194}, 
number = {3}, 
pages = {1640-1664}, 
year = {2013}, 
doi = {10.1093/gji/ggt177}, 
abstract ={Multiparameter full waveform inversion (FWI) is a challenging quantitative seismic imaging method for lithological characterization and reservoir monitoring. The difficulties in multiparameter FWI arise from the variable influence of the different parameter classes on the phase and amplitude of the data, and the trade-off between these. In this framework, choosing a suitable parametrization of the subsurface and designing the suitable FWI workflow are two key methodological issues in non-linear waveform inversion. We assess frequency-domain visco-acoustic FWI to reconstruct the compressive velocity (VP), the density (ρ) or the impedance (IP) and the quality factor (QP), from the hydrophone component, using a synthetic data set that is representative of the Valhall oil field in the North Sea. We first assess which of the (VP, ρ) and (VP, IP) parametrizations provides the most reliable FWI results when dealing with wide-aperture data. Contrary to widely accepted ideas, we show that the (VP, ρ) parametrization allows a better reconstruction of both the VP, ρ and IP parameters, first because it favours the broad-band reconstruction of the dominant VP parameter, and secondly because the trade-off effects between velocity and density at short-to-intermediate scattering angles can be removed by multiplication, to build an impedance model. This allows for the matching of the reflection amplitudes, while the broad-band velocity model accurately describes the kinematic attributes of both the diving waves and reflections. Then, we assess different inversion strategies to recover the quality factor QP, in addition to parameters VP and ρ. A difficulty related to attenuation estimation arises because, on the one hand the values of QP are on average one order of magnitude smaller than those of VP and ρ, and on the other hands model perturbations relative to the starting models can be much higher for QP than for VP and ρ during FWI. In this framework, we show that an empirical tuning of the FWI regularization, which is adapted to each parameter class, is a key issue to correctly account for the attenuation in the inversion. We promote a hierarchical approach where the dominant parameter VP is reconstructed first from the full data set (i.e. without any data preconditioning) to build a velocity model as kinematically accurate as possible before performing the joint update of the three parameter classes during a second step. This hierarchical imaging of compressive wave speed, density and attenuation is applied to a real wide-aperture ocean-bottom-cable data set from the Valhall oil field. Several geological features, such as accumulation of gas below barriers of claystone and soft quaternary sediment are interpreted in the FWI models of density and attenuation. The models of VP, ρ and QP that have been developed by visco-acoustic FWI of the hydrophone data can be used as initial models to perform visco-elastic FWI of the geophone data for the joint update of the compressive and shear wave speeds.}, 
URL = {http://gji.oxfordjournals.org/content/194/3/1640.abstract}, 
eprint = {http://gji.oxfordjournals.org/content/194/3/1640.full.pdf+html}, 
journal = {Geophysical Journal International} 
}

@article{truncNewtonoperto,
author = {Métivier, L. and Brossier, R. and Virieux, J. and Operto, S.},
title = {Full Waveform Inversion and the Truncated Newton Method},
journal = {SIAM Journal on Scientific Computing},
volume = {35},
number = {2},
pages = {B401-B437},
year = {2013},
doi = {10.1137/120877854},

URL = {http://epubs.siam.org/doi/abs/10.1137/120877854},
eprint = {http://epubs.siam.org/doi/pdf/10.1137/120877854}
}

@article {JGRB:JGRB13943,
author = {Gallardo, Luis A. and Meju, Max A.},
title = {Joint two-dimensional DC resistivity and seismic travel time inversion with cross-gradients constraints},
journal = {Journal of Geophysical Research: Solid Earth},
volume = {109},
number = {B3},
issn = {2156-2202},
url = {http://dx.doi.org/10.1029/2003JB002716},
doi = {10.1029/2003JB002716},
pages = {n/a--n/a},
keywords = {joint inversion, DC resistivity, seismic refraction},
year = {2004},
}

@article{TarantolaA,
author = {Tarantola, A.},
title = {A strategy for nonlinear elastic inversion of seismic reflection data},
journal = {GEOPHYSICS},
volume = {51},
number = {10},
pages = {1893-1903},
year = {1986},
doi = {10.1190/1.1442046},
URL = {http://library.seg.org/doi/abs/10.1190/1.1442046},
eprint = {http://library.seg.org/doi/pdf/10.1190/1.1442046}
}

@article {GJI:GJI2978,
author = {Plessix, R.-E.},
title = {A review of the adjoint-state method for computing the gradient of a functional with geophysical applications},
journal = {Geophysical Journal International},
volume = {167},
number = {2},
publisher = {Blackwell Publishing Ltd},
issn = {1365-246X},
url = {http://dx.doi.org/10.1111/j.1365-246X.2006.02978.x},
doi = {10.1111/j.1365-246X.2006.02978.x},
pages = {495--503},
keywords = {adjoint state, gradient, migration, tomography},
year = {2006},
}

@ARTICLE{vanLeeuwen2013GJImlm,
  author = {Tristan van Leeuwen and Felix J. Herrmann}, 
  title = {Mitigating local minima in full-waveform inversion by expanding the search space},
  year = {2013}, 
  month = {10},
  volume = {195}, 
  pages = {661-667},
  journal = {Geophysical Journal International}, 
  abstract = {Wave equation based inversions, such as full-waveform
                  inversion and reverse-time migration, are
                  challenging because of their computational costs,
                  memory requirements and reliance on accurate initial
                  models. To confront these issues, we propose a novel
                  formulation of wave equation based inversion based
                  on a penalty method. In this formulation, the
                  objective function consists of a data-misfit term
                  and a penalty term, which measures how accurately
                  the wavefields satisfy the wave equation. This new
                  approach is a major departure from current
                  formulations where forward and adjoint wavefields,
                  which both satisfy the wave equation, are correlated
                  to compute updates for the unknown model
                  parameters. Instead, we carry out the inversions
                  over two alternating steps during which we first
                  estimate the wavefield everywhere, given the current
                  model parameters, source and observed data, followed
                  by a second step during which we update the model
                  parameters, given the estimate for the wavefield
                  everywhere and the source. Because the inversion
                  involves both the synthetic wavefields and the
                  medium parameters, its search space is enlarged so
                  that it suffers less from local minima. Compared to
                  other formulations that extend the search space of
                  wave equation based inversion, our method differs in
                  several aspects, namely (i) it avoids storage and
                  updates of the synthetic wavefields because we
                  calculate these explicitly by finding solutions that
                  obey the wave equation and fit the observed data and
                  (ii) no adjoint wavefields are required to update
                  the model, instead our updates are calculated from
                  these solutions directly, which leads to significant
                  computational savings. We demonstrate the validity
                  of our approach by carefully selected examples and
                  discuss possible extensions and future research.}, 
  url1 = {http://gji.oxfordjournals.org/content/195/1/661}, 
  url2 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/GeophysicalJournalInternational/2013/vanLeeuwen2013GJImlm/vanLeeuwen2013mlm.pdf},
  doi = {10.1093/gji/ggt258}, 
  eprint = {http://gji.oxfordjournals.org/content/early/2013/07/30/gji.ggt258.full.pdf+html} 
}

@UNPUBLISHED{vanLeeuwen2013Penalty2,
  author   = {Tristan van Leeuwen and Felix J. Herrmann},
  title    = {A penalty method for PDE-constrained optimization},
  year     = {2013},
  institution = {UBC},
  abstract = {We present a method for solving PDE constrained
                  optimization problems based on a penalty
                  formulation. This method aims to combine advantages
                  of both full-space and reduced methods by exploiting
                  a large search-space (consisting of both control and
                  state variables) while allowing for an efficient
                  implementation that avoids storing and updating the
                  state-variables. This leads to a method that has
                  roughly the same per-iteration complexity as
                  conventional reduced approaches while dening an
                  objective that is less non-linear in the control
                  variable by implicitly relaxing the constraint. We
                  apply the method to a seismic inverse problem where
                  it leads to a particularly ecient implementation
                  when compared to a conventional reduced approach as
                  it avoids the use of adjoint
                  state-variables. Numerical examples illustrate the
                  approach and suggest that the proposed formulation
                  can indeed mitigate some of the well-known problems
                  with local minima in the seismic inverse problem.},
  keywords = {waveform inversion, optimization, private},
  month    = {Apr},
  url = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2013/vanLeeuwen2013Penalty2/vanLeeuwen2013Penalty2.pdf}
}

@article{Pratt98,
author = {Pratt, G.R. and Shin, Changsoo and Hicks, G.J.},
doi = {10.1046/j.1365-246X.1998.00498.x},
file = {:Users/tristanvanleeuwen/Documents/Mendeley Desktop/Pratt, Shin, Hicks - 1998 - Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion.pdf:pdf},
issn = {0956540X},
journal = {Geophysical Journal International},
keywords = {di,erence methods,inversion,nite-di,numerical techniques,raction,seismic velocities,wave equation},
month = may,
number = {2},
pages = {341--362},
title = {{Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion}},
url = {http://doi.wiley.com/10.1046/j.1365-246X.1998.00498.x},
volume = {133},
year = {1998}
}
@article{Haber2012,
author = {Haber, Eldad and Chung, Matthias and Herrmann, Felix},
doi = {10.1137/11081126X},
file = {:Volumes/Users/tristan/Documents/Mendeley Desktop//Haber, Chung, Herrmann - 2012 - An Effective Method for Parameter Estimation with PDE Constraints with Multiple Right-Hand Sides.pdf:pdf},
issn = {1052-6234},
journal = {SIAM Journal on Optimization},
month = jul,
number = {3},
pages = {739--757},
title = {{An Effective Method for Parameter Estimation with PDE Constraints with Multiple Right-Hand Sides}},
url = {http://epubs.siam.org/doi/abs/10.1137/11081126X},
volume = {22},
year = {2012}
}
@article{Li2012,
author = {Li, X. and Aravkin, A.Y. and van Leeuwen, T. and Herrmann, F.J.},
doi = {10.1190/geo2011-0410.1},
issn = {00168033},
journal = {Geophysics},
keywords = {journal},
mendeley-tags = {journal},
number = {3},
pages = {A13},
title = {{Fast randomized full-waveform inversion with compressive sensing}},
volume = {77},
year = {2012}
}
@article{VanLeeuwen2012b,
author = {van Leeuwen, Tristan and Herrmann, Felix J.},
doi = {10.1111/j.1365-2478.2012.01096.x},
issn = {00168025},
journal = {Geophysical Prospecting},
keywords = {journal},
mendeley-tags = {journal},
month = jul,
pages = {no--no},
title = {{Fast waveform inversion without source-encoding}},
url = {http://doi.wiley.com/10.1111/j.1365-2478.2012.01096.x},
year = {2012}
}
@article{Friedlander2012,
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 offer inexpensive iterations by sampling a subset 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 shows that by controlling the sample size in an incremental-gradient algorithm, it is possible to maintain the steady convergence rates of full-gradient methods. We detail a practical quasi-Newton implementation based on this approach. Numerical experiments illustrate its potential benefits.},
author = {Friedlander, Michael P. and Schmidt, Mark},
doi = {10.1137/110830629},
issn = {1064-8275},
journal = {SIAM Journal on Scientific Computing},
keywords = {47N10,90C06,90C25,94A20,data fitting,gradient descent,incremental gradient,optimization},
language = {en},
month = may,
number = {3},
pages = {A1380--A1405},
publisher = {Society for Industrial and Applied Mathematics},
title = {{Hybrid Deterministic-Stochastic Methods for Data Fitting}},
url = {http://epubs.siam.org/doi/abs/10.1137/110830629},
volume = {34},
year = {2012}
}

@book{Bertsekas1996,
author = {Bertsekas, D.P.},
isbn = {1-886529-04-3},
title = {{Constrained Optimization and Lagrange Multiplier Methods}},
year = {1996},
publisher = {Athena Scientific}
}
@article{Eckstein2012,
author = {Eckstein, J.},
journal = {Rutcor Research Report},
title = {{Augmented Lagrangian and Alternating Direction Methods for Convex Optimization: A Tutorial and Some Illustrative Computational Results}},
volume = {32-2012},
year = {2012}
}
@article{Haber2000,
author = {Haber, Eldad and Ascher, Uri M and Oldenburg, Doug},
doi = {10.1088/0266-5611/16/5/309},
file = {:Volumes/Users/tristan/Documents/Mendeley Desktop/Haber, Ascher, Oldenburg - 2000 - On optimization techniques for solving nonlinear inverse problems.pdf:pdf},
issn = {0266-5611},
journal = {Inverse Problems},
month = oct,
number = {5},
pages = {1263--1280},
title = {{On optimization techniques for solving nonlinear inverse problems}},
url = {http://stacks.iop.org/0266-5611/16/i=5/a=309?key=crossref.98f435f9ee66231b63da02b10f82a60b},
volume = {16},
year = {2000}
}
@techreport{chavent96,
    hal_id = {inria-00073799},
    url = {http://hal.inria.fr/inria-00073799},
    title = {{Primal-Dual Formulations for Parameter Estimation Problems}},
    author = {Chavent, Guy and Kunisch, Karl and Roberts, Jean},
    abstract = {{A new method for formulating and solving parameter estimation problems based on Fenchel duality is presented. The partial differential equation is considered as a contraint in a least squares type formulation and is realized as a penalty term involving the primal and dual energy functionals associated with the differential equation. Splitting algorithms and mixed finite element discretizations are discussed and some numerical examples are given.}},
    keywords = {PARAMETER ESTIMATION / FENCHEL DUALITY / INVERSE PROBLEMS / REGULARIZATION},
    language = {Anglais},
    affiliation = {ESTIME - INRIA Paris-Rocquencourt , ONDES - INRIA Rocquencourt},
    type = {Rapport de recherche},
    institution = {INRIA},
    number = {RR-2891},
    year = {1996},
    pdf = {http://hal.inria.fr/inria-00073799/PDF/RR-2891.pdf},
}
@article{Richter1981,
author = {Richter, R.G.},
file = {:Volumes/Users/tristan/Documents/Mendeley Desktop/Richter - 1981 - Numerical Identification of a Spatially Varying Diffusion Coefficient.html:html},
journal = {Mathematics of Computation},
number = {154},
pages = {375--386},
title = {{Numerical Identification of a Spatially Varying Diffusion Coefficient}},
url = {http://www.jstor.org/stable/2007648},
volume = {36},
year = {1981}
}
@article{Hanke1998,
author = {Hanke, Martin and Scherzer, Otmar},
doi = {10.1137/S0036139997331628},
issn = {0036-1399},
journal = {SIAM Journal on Applied Mathematics},
month = jan,
number = {3},
pages = {1012--1027},
title = {{Error Analysis of an Equation Error Method for the Identification of the Diffusion Coefficient in a Quasi-linear Parabolic Differential Equation}},
url = {http://epubs.siam.org/doi/abs/10.1137/S0036139997331628},
volume = {59},
year = {1998}
}
@article{Censor1983,
author = {Censor, Yair and Eggermont, Paul P. B. and Gordon, Dan},
doi = {10.1007/BF01396307},
issn = {0029-599X},
journal = {Numerische Mathematik},
month = feb,
number = {1},
pages = {83--92},
title = {{Strong underrelaxation in Kaczmarz's method for inconsistent systems}},
url = {http://www.springerlink.com/index/10.1007/BF01396307},
volume = {41},
year = {1983}
}
@book{biegler2007real,
  title={Real-time PDE-constrained Optimization},
  author={Biegler, L.T.},
  isbn={9780898718935},
  series={Computational science and engineering},
  url={http://books.google.ca/books?id=tflUtbrn7NQC},
  year={2007},
  publisher={Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104)}
}
@article{Biros2005,
author = {Biros, George and Ghattas, Omar},
doi = {10.1137/S1064827502415661},
issn = {1064-8275},
journal = {SIAM Journal on Scientific Computing},
month = jan,
number = {2},
pages = {714--739},
title = {{Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part II: The Lagrange--Newton Solver and Its Application to Optimal Control of Steady Viscous Flows}},
url = {http://epubs.siam.org/doi/abs/10.1137/S1064827502415661},
volume = {27},
year = {2005}
}
@article{Abdoulaev2005,
author = {Abdoulaev, Gassan S and Ren, Kui and Hielscher, Andreas H},
doi = {10.1088/0266-5611/21/5/002},
file = {:Volumes/Users/tristan/Documents/Mendeley Desktop/Abdoulaev, Ren, Hielscher - 2005 - Optical tomography as a PDE-constrained optimization problem.pdf:pdf},
issn = {0266-5611},
journal = {Inverse Problems},
month = oct,
number = {5},
pages = {1507--1530},
title = {{Optical tomography as a PDE-constrained optimization problem}},
url = {http://stacks.iop.org/0266-5611/21/i=5/a=002?key=crossref.49523d7f9f3f814ad391621358a1c759},
volume = {21},
year = {2005}
}
@article{Haber2004,
author = {Haber, Eldad and Ascher, Uri M. and Oldenburg, Douglas W.},
doi = {10.1190/1.1801938},
file = {:Volumes/Users/tristan/Documents/Mendeley Desktop/Haber, Ascher, Oldenburg - 2004 - Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach.pdf:pdf},
issn = {00168033},
journal = {Geophysics},
number = {5},
pages = {1216},
title = {{Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach}},
url = {http://link.aip.org/link/GPYSA7/v69/i5/p1216/s1\&Agg=doi},
volume = {69},
year = {2004}
}

@article{Bjorck1979,
author = {Bj\"{o}rck, {\AA}. and Elfving, T.},
doi = {10.1007/BF01930845},
issn = {0006-3835},
journal = {BIT},
month = jun,
number = {2},
pages = {145--163},
title = {{Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations}},
url = {http://www.springerlink.com/content/r6414ut31765364j/},
volume = {19},
year = {1979}
}
@article{VanLeeuwen2012e,
abstract = {The Helmholtz equation arises in many applications, such as seismic and medical imaging. These application are characterized by the need to propagate many wavelengths through an inhomogeneous medium. The typical size of the problems in 3D applications precludes the use of direct factorization to solve the equation and hence iterative methods are used in practice. For higher wavenumbers, the system becomes increasingly indefinite and thus good preconditioners need to be constructed. In this note we consider an accelerated Kazcmarz method (CGMN) and present an expression for the resulting iteration matrix. This iteration matrix can be used to analyze the convergence of the CGMN method. In particular, we present a Fourier analysis for the method applied to the 1D Helmholtz equation. This analysis suggests an optimal choice of the relaxation parameter. Finally, we present some numerical experiments.},
journal={arXiv},
archivePrefix = {arXiv},
arxivId = {1210.2644},
author = {van Leeuwen, Tristan},
eprint = {1210.2644},
keywords = {preprint},
mendeley-tags = {preprint},
month = oct,
title = {{Fourier analysis of the CGMN method for solving the Helmholtz equation}},
year = {2012}
}
@article{Gordon2013,
author = {Gordon, Dan and Gordon, Rachel},
doi = {10.1016/j.cam.2012.07.024},
issn = {03770427},
journal = {Journal of Computational and Applied Mathematics},
month = jan,
number = {1},
pages = {182--196},
title = {{Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers}},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0377042712003147},
volume = {237},
year = {2013}
}
@article{Aravkin2012c,
abstract = {Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is variable projection, where nonlinear least squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with nuisance parameters, such as variance or degrees of freedom. As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, degree of freedom (d.o.f.) parameter estimation in the context of robust inverse problems, automatic calibration, and optimal experimental design. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several large- scale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms.},
author = {Aravkin, Aleksandr Y and van Leeuwen, Tristan},
issn = {0266-5611},
journal = {Inverse Problems},
keywords = {journal},
mendeley-tags = {journal},
number = {11},
pages = {115016},
title = {{Estimating nuisance parameters in inverse problems}},
volume = {28},
year = {2012}
}
@book{Saad,
author = {Saad, Yousef},
edition = {2$^\mathrm{nd}$},
isbn = {9780898715347},
publisher = {SIAM},
title = {{Iterative methods for sparse linear systems}},
year = {2003}
}
@article{Epanomeritakis08,
author = {Epanomeritakis, I and Ak\c{c}elik, V and Ghattas, O and Bielak, J},
doi = {10.1088/0266-5611/24/3/034015},
issn = {0266-5611},
journal = {Inverse Problems},
month = jun,
number = {3},
pages = {034015},
title = {{A Newton-CG method for large-scale three-dimensional elastic full-waveform seismic inversion}},
url = {http://stacks.iop.org/0266-5611/24/i=3/a=034015?key=crossref.59250bd240c7ffa6a3c05544b0ffa82c},
volume = {24},
year = {2008}
}
@article{virieux09,
author = {Virieux, J and Operto, S},
journal = {Geophysics},
keywords = {geophysical prospecting; seismic waves; seismology},
number = {6},
pages = {WCC1--WCC26},
publisher = {SEG},
title = {{An overview of full-waveform inversion in exploration geophysics}},
url = {http://dx.doi.org/10.1190/1.3238367},
volume = {74},
year = {2009}
}
@article{Bunks1995,
author = {Bunks, Carey},
doi = {10.1190/1.1443880},
file = {:Users/tristanvanleeuwen/Documents/Mendeley Desktop/Bunks - 1995 - Multiscale seismic waveform inversion.pdf:pdf},
issn = {1070485X},
journal = {Geophysics},
month = sep,
number = {5},
pages = {1457},
title = {{Multiscale seismic waveform inversion}},
url = {http://link.aip.org/link/?GPY/60/1457/1\&Agg=doi},
volume = {60},
year = {1995}
}
@article{Tarantola82,
author = {Tarantola, A and Valette, A},
journal = {Reviews of Geophysics and Space Physics},
number = {2},
pages = {129--232},
title = {{Generalized nonlinear inverse problems solved using the least squares criterion}},
volume = {20},
year = {1982}
}
@article{Krebs09,
author = {Krebs, Jerome R. and Anderson, John E. and Hinkley, David and Baumstein, Anatoly and Lee, Sunwoong and Neelamani, Ramesh and Lacasse, Martin-Daniel},
doi = {10.1190/1.3255314},
journal = {Geophysics},
number = {1},
pages = {2273--2277},
publisher = {SEG},
title = {{Fast full wave seismic inversion using source encoding}},
url = {http://link.aip.org/link/SEGEAB/v28/i1/p2273/s1\&Agg=doi http://library.seg.org/vsearch/servlet/VerityServlet?KEY=SEGLIB\&smode=strresults\&sort=chron\&maxdisp=25\&threshold=0\&pjournals=SAGEEP,JEEGXX\&pjournals=JEEGXX\&pjournals=SAGEEP\&pjournals=GPYSA7,LEEDFF,SEGEAB,SEGBKS\&pjournals=GPYSA7\&pjournals=LEEDFF\&pjournals=SEGEAB\&pjournals=SEGBKS\&possible1zone=article\&possible4=krebs\&possible4zone=author\&bool4=and\&OUTLOG=NO\&viewabs=SEGEAB\&key=DISPLAY\&docID=5\&page=1\&chapter=0},
volume = {28},
year = {2009}
}
@article{Haber12,
author = {Haber, Eldad and Chung, Matthias and Herrmann, Felix},
doi = {10.1137/11081126X},
file = {:Users/tristanvanleeuwen/Documents/Mendeley Desktop//Haber, Chung, Herrmann - 2012 - An Effective Method for Parameter Estimation with PDE Constraints with Multiple Right-Hand Sides.pdf:pdf},
issn = {1052-6234},
journal = {SIAM Journal on Optimization},
month = jul,
number = {3},
pages = {739--757},
title = {{An Effective Method for Parameter Estimation with PDE Constraints with Multiple Right-Hand Sides}},
url = {http://epubs.siam.org/doi/abs/10.1137/11081126X},
volume = {22},
year = {2012}
}

@inbook{doi:10.1190/segam2012-1228.1,
author = {Shah, N. and Warner, M. and Nangoo, T. and Umpleby, A. and Stekl, I. and Morgan, J. and Guasch, L.},
title = {Quality assured full-waveform inversion: Ensuring starting model adequacy},
booktitle = {SEG Technical Program Expanded Abstracts 2012},
chapter = {499},
pages = {1-5},
doi = {10.1190/segam2012-1228.1},
URL = {http://library.seg.org/doi/abs/10.1190/segam2012-1228.1},
eprint = {http://library.seg.org/doi/pdf/10.1190/segam2012-1228.1},
year = {2012}
}

@book{machinelearning,
author = {Murphy, K},
title = {{Machine Learning: A Probabilistic Perspective}},
publisher = {The MIT Press},
year = {2012}
}

@article{Brenders01012007,
author = {Brenders, A. J. and Pratt, R. G.}, 
title = {Full waveform tomography for lithospheric imaging: results from a blind test in a realistic crustal model},
volume = {168}, 
number = {1}, 
pages = {133-151}, 
year = {2007}, 
doi = {10.1111/j.1365-246X.2006.03156.x}, 
URL = {http://gji.oxfordjournals.org/content/168/1/133.abstract}, 
eprint = {http://gji.oxfordjournals.org/content/168/1/133.full.pdf+html}, 
journal = {Geophysical Journal International} 
}