@conference {peters2019SEGans,
	title = {A numerical solver for least-squares sub-problems in 3D wavefield reconstruction inversion and related problem formulations},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2019},
	note = {(SEG, San Antonio)},
	month = {09},
	pages = {1536-1540},
	abstract = {Recent years saw a surge of interest in seismic waveform inversion approaches based on quadratic-penalty or augmented-Lagrangian methods, including Wavefield Reconstruction Inversion. These methods typically need to solve a least-squares sub-problem that contains a discretization of the Helmholtz equation. Memory requirements for direct solvers are often prohibitively large in three dimensions, and this limited the examples in the literature to two dimensions. We present an algorithm that uses iterative Helmholtz solvers as a black-box to solve the least-squares problem corresponding to 3D grids. This algorithm enables Wavefield Reconstruction Inversion and related formulations, in three dimensions. Our new algorithm also includes a root-finding method to convert a penalty into a constraint on the data-misfit without additional computational cost, by reusing precomputed quantities. Numerical experiments show that the cost of parallel communication and other computations are small compared to the main cost of solving one Helmholtz problem per source and one per receiver.},
	keywords = {Full-waveform inversion, least-squares, numerical linear algebra, penalty method, quadratic constraints, root finding, SEG},
	doi = {10.1190/segam2019-3216638.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2019/peters2019SEGans/peters2019SEGans.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2019/peters2019SEGans/peters2019SEGans_pres.pdf},
	author = {Bas Peters and Felix J. Herrmann}
}
@conference {peters2016SEGprs,
	title = {Parallel reformulation of the sequential adjoint-state method},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {1411-1415},
	abstract = {This abstract is about reducing the total computation time to solve full-waveform inversion problems. A common problem formulation is a nonlinear least-squares problem where the gradient is computed using the adjoint-state algorithm. This formulation offers parallelism for the computation of gradients for different sources and frequencies. The adjoint-state algorithm itself is sequential however and this is a limiting factor when a lot of compute nodes are available and only a few wavefields need to be computed. This situation occurs when stochastic optimization strategies are used to minimize the objective function. We present a parallel reformulation of the sequential adjoint-state algorithm, which allows the forward- and adjoint wavefields to be computed in parallel. Both algorithms are mathematically equivalent but the parallel version is twice as fast in run time. An important characteristic of the proposed algorithm is that one wavefield needs to be computed per source and one per receiver. These fields can be used to apply the (inverse) Gauss-Newton Hessian to a vector without recomputing wavefields. A 2D example shows that good full-waveform inversion results are obtained, even when a small number of sources and receivers is used.},
	keywords = {FWI, numerical linear algebra, parallel computing, PDE-constrained optimization, SEG},
	doi = {10.1190/segam2016-13966771.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/peters2016SEGprs/peters2016SEGprs.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/peters2016SEGprs/peters2016SEGprs_pres.pdf},
	author = {Bas Peters and Tristan van Leeuwen and Felix J. Herrmann}
}
