@article {peters2018pmf,
	title = {Projection methods and applications for seismic nonlinear inverse problems with multiple constraints},
	journal = {Geophysics},
	volume = {84},
	number = {2},
	year = {2018},
	note = {(Geophysics)},
	month = {02},
	pages = {R251-R269},
	abstract = {Nonlinear inverse problems are often hampered by non-uniqueness and local minima because of missing low frequencies and far offsets in the data, lack of access to good starting models, noise, and modeling errors. A well-known approach to counter these deficiencies is to include prior information on the unknown model, which regularizes the inverse problem. While conventional regularization methods have resulted in enormous progress in ill-posed (geophysical) inverse problems, challenges remain when the prior information consists of multiple pieces. To handle this situation, we propose an optimization framework that allows us to add multiple pieces of prior information in the form of constraints. Compared to additive regularization penalties, constraints have a number of advantages making them more suitable for inverse problems such as full-waveform inversion. The proposed framework is rigorous because it offers assurances that multiple constraints are imposed uniquely at each iteration, irrespective of the order in which they are invoked. To project onto the intersection of multiple sets uniquely, we employ Dykstra{\textquoteright}s algorithm that scales to large problems and does not rely on trade-off parameters. In that sense, our approach differs substantially from approaches such as Tikhonov regularization, penalty methods, and gradient filtering. None of these offer assurances, which makes them less suitable to full-waveform inversion where unrealistic intermediate results effectively derail the iterative inversion process. By working with intersections of sets, we keep expensive objective and gradient calculations unaltered, separate from projections, and we also avoid trade-off parameters. These features allow for easy integration into existing code bases. In addition to more predictable behavior, working with constraints also allows for heuristics where we built up the complexity of the model gradually by relaxing the constraints. This strategy helps to avoid convergence to local minima that represent unrealistic models. We illustrate this unique feature with examples of varying complexity.},
	keywords = {constraints, Full-waveform inversion, intersection, Optimization, projection, regularization},
	doi = {10.1190/geo2018-0192.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2018/peters2018pmf/peters2018pmf.html},
	author = {Bas Peters and Brendan R. Smithyman and Felix J. Herrmann}
}
@article {louboutin2016ppf,
	title = {Performance prediction of finite-difference solvers for different computer architectures},
	journal = {Computers \& Geosciences},
	volume = {105},
	year = {2017},
	note = {(Computers \& Geosciences)},
	month = {08},
	pages = {148-157},
	abstract = {The life-cycle of a partial differential equation (PDE) solver is often characterized by three development phases: the development of a stable numerical discretization; development of a correct (verified) implementation; and the optimization of the implementation for different computer architectures. Often it is only after significant time and effort has been invested that the performance bottlenecks of a PDE solver are fully understood, and the precise details varies between different computer architectures. One way to mitigate this issue is to establish a reliable performance model that allows a numerical analyst to make reliable predictions of how well a numerical method would perform on a given computer architecture, before embarking upon potentially long and expensive implementation and optimization phases. The availability of a reliable performance model also saves developer effort as it both informs the developer on what kind of optimisations are beneficial, and when the maximum expected performance has been reached and optimisation work should stop. We show how discretization of a wave-equation can be theoretically studied to understand the performance limitations of the method on modern computer architectures. We focus on the roofline model, now broadly used in the high-performance computing community, which considers the achievable performance in terms of the peak memory bandwidth and peak floating point performance of a computer with respect to algorithmic choices. A first principles analysis of operational intensity for key time-stepping finite-difference algorithms is presented. With this information available at the time of algorithm design, the expected performance on target computer systems can be used as a driver for algorithm design.},
	keywords = {finite differences, HPC, Modeling, performance},
	doi = {https://doi.org/10.1016/j.cageo.2017.04.014},
	url = {https://slim.gatech.edu/Publications/Public/Journals/ComputersAndGeosciences/2016/louboutin2016ppf/louboutin2016ppf.pdf},
	author = {Mathias Louboutin and Michael Lange and Felix J. Herrmann and Navjot Kukreja and Gerard Gorman}
}
@article {vanleeuwen2015IPpmp,
	title = {A penalty method for PDE-constrained optimization in inverse problems},
	journal = {Inverse Problems},
	volume = {32},
	number = {1},
	year = {2015},
	note = {(Inverse Problems)},
	month = {12},
	pages = {015007},
	abstract = {Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the parameters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the nonlinearity of the problem and is less sensitive to the initial iterate.},
	keywords = {Inverse problems, Optimization, PDE, penalty method},
	url = {https://slim.gatech.edu/Publications/Public/Journals/InverseProblems/2015/vanleeuwen2015IPpmp/vanleeuwen2015IPpmp.pdf},
	url2 = {http://stacks.iop.org/0266-5611/32/i=1/a=015007},
	author = {Tristan van Leeuwen and Felix J. Herrmann}
}
@article {berg2008SJSCpareto,
	title = {Probing the Pareto frontier for basis pursuit solutions},
	journal = {SIAM Journal on Scientific Computing},
	volume = {31},
	number = {2},
	year = {2008},
	month = {01},
	pages = {890-912},
	publisher = {SIAM},
	abstract = {The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously differentiable over all points of interest, and show that it gives an explicit relationship to two other optimization problems closely related to BPDN. We describe a root-finding algorithm for finding arbitrary points on this curve; the algorithm is suitable for problems that are large scale and for those that are in the complex domain. At each iteration, a spectral gradient-projection method approximately minimizes a least-squares problem with an explicit one-norm constraint. Only matrix-vector operations are required. The primal-dual solution of this problem gives function and derivative information needed for the root-finding method. Numerical experiments on a comprehensive set of test problems demonstrate that the method scales well to large problems.},
	keywords = {basis pursuit, convex program, duality, Newton{\textquoteright}s method, one-norm regularization, Optimization, projected gradient, root-finding, sparse solutions},
	doi = {10.1137/080714488},
	url = {https://slim.gatech.edu/Publications/Public/Journals/SIAMJournalOnScientificComputing/2008/vanderberg08SIAMptp/vanderberg08SIAMptp.pdf},
	author = {Ewout van den Berg and Michael P. Friedlander}
}
@article {bernabe2004JGRpas,
	title = {Permeability and storativity of binary mixtures of high-and low-porosity materials},
	journal = {Journal of Geophysical Research: Solid Earth},
	volume = {109},
	year = {2004},
	month = {12},
	pages = {B12207},
	abstract = {As a first step toward determining the mixing laws for the transport properties of rocks, we prepared binary mixtures of high- and low-permeability materials by isostatically hot-pressing mixtures of fine powders of calcite and quartz. The resulting rocks were marbles containing varying concentrations of dispersed quartz grains. Pores were present throughout the rock, but the largest ones were preferentially associated with the quartz particles, leading us to characterize the material as being composed of two phases, one with high permeability and the second with low permeability. We measured the permeability and storativity of these materials using the oscillating flow technique, while systematically varying the effective pressure and the period and amplitude of the input fluid oscillation. Control measurements performed using the steady state flow and pulse decay techniques agreed well with the oscillating flow tests. The hydraulic properties of the marbles were highly sensitive to the volume fraction of the high-permeability phase (directly related to the quartz content). Below a critical quartz content, slightly less than 20 wt \%, the high-permeability volume elements were disconnected, and the overall permeability was low. Above the critical quartz content the high-permeability volume elements formed throughgoing paths, and permeability increased sharply. We numerically simulated fluid flow through binary materials and found that permeability approximately obeys a percolation-based mixing law, consistent with the measured permeability of the calcite-quartz aggregates.},
	keywords = {Modeling, permeability, porosity, SLIM},
	doi = {10.1029/2004JB003111},
	url = {https://slim.gatech.edu/Publications/Public/Journals/JournalOfGeophysicalResearch/2004/bernabe04JGRpas/bernabe04JGRpas.pdf},
	url2 = {http://onlinelibrary.wiley.com/doi/10.1029/2004JB003111/abstract},
	author = {Y. Bernab{\'e} and U. Mok and B. Evans and Felix J. Herrmann}
}
