@article {Friedlander11TRhdm,
	title = {Hybrid deterministic-stochastic methods for data fitting},
	journal = {SIAM Journal on Scientific Computing},
	volume = {34},
	number = {3},
	year = {2012},
	month = {01},
	pages = {A1380-A1405},
	abstract = {Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms (both deterministic and randomized) offer inexpensive iterations by sampling only subsets of the terms in the sum. These methods can make great progress initially, but often slow as they approach a solution. In contrast, full gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate of convergence analysis and numerical experiments illustrate the potential for the approach.},
	keywords = {Optimization},
	doi = {10.1137/110830629},
	author = {Michael P. Friedlander and Mark Schmidt}
}
@conference {vanleeuwen2011EAGEhsdomwi,
	title = {A hybrid stochastic-deterministic optimization method for waveform inversion},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2011},
	month = {05},
	abstract = {Present-day high quality 3D acquisition can give us lower frequencies and longer offsets with which to invert. However, the computational costs involved in handling this data explosion are tremendous. Therefore, recent developments in full-waveform inversion have been geared towards reducing the computational costs involved. A key aspect of several approaches that have been proposed is a dramatic reduction in the number of sources used in each iteration. A reduction in the number of sources directly translates to less PDE-solves and hence a lower computational cost. Recent attention has been drawn towards reducing the sources by randomly combining the sources in to a few supershots, but other strategies are also possible. In all cases, the full data misfit, which involves all the sequential sources, is replaced by a reduced misfit that is much cheaper to evaluate because it involves only a small number of sources (batchsize). The batchsize controls the accuracy with which the reduced misfit approximates the full misfit. The optimization of such an inaccurate, or noisy, misfit is the topic of stochastic optimization. In this paper, we propose an optimization strategy that borrows ideas from the field of stochastic optimization. The main idea is that in the early stage of the optimization, far from the true model, we do not need a very accurate misfit. The strategy consists of gradually increasing the batchsize as the iterations proceed. We test the proposed strategy on a synthetic dataset. We achieve a very reasonable inversion result at the cost of roughly 13 evaluations of the full misfit. We observe a speed-up of roughly a factor 20.},
	keywords = {EAGE, Full-waveform inversion, Optimization, Presentation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2011/vanleeuwen11EAGEhsdomwi/vanleeuwen11EAGEhsdomwi.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2011/vanleeuwen11EAGEhsdomwi/vanleeuwen11EAGEhsdomwi_pres.pdf},
	url2 = {http://earthdoc.eage.org/publication/publicationdetails/?publication=50341},
	author = {Tristan van Leeuwen and Felix J. Herrmann and Mark Schmidt and Michael P. Friedlander}
}
@techreport {vandenberg08gsv,
	title = {Group sparsity via linear-time projection},
	number = {TR-2008-09},
	year = {2008},
	month = {06},
	institution = {UBC - Department of Computer Science},
	abstract = {We present an efficient spectral projected-gradient algorithm for optimization subject to a group one-norm constraint. Our approach is based on a novel linear-time algorithm for Euclidean projection onto the one- and group one-norm constraints. Numerical experiments on large data sets suggest that the proposed method is substantially more efficient and scalable than existing methods.},
	keywords = {Optimization, SLIM},
	url = {http://www.optimization-online.org/DB_FILE/2008/07/2056.pdf},
	author = {Ewout van den Berg and Mark Schmidt and Michael P. Friedlander and K. Murphy}
}
