A short note on rank-2 relaxation for waveform inversion

Demanet, Laurent; Shank, Stephen D; Cosse, Augustin

doi:10.1190/segam2015-5925071.1

Cited by 5 publications

(3 citation statements)

References 22 publications

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“…However, a lifting approach similar to that used in [2] to expand the search space of the model could be applied. We replace the outer product of the two identical vectors with two identical low-rank matrices to expand the space of search directions (note that we cannot get a rank one solution unless an extra low-rank penalty is added.)…”

Section: A New Algorithm For Blind Deconvolutionmentioning

confidence: 99%

Resolving scaling ambiguities with the ℓ1/ℓ2 norm in a blind deconvolution problem with feedback

Esser

Lin

Herrmann

et al. 2015

2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

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Compared to more mundane blind deconvolution problems, blind deconvolution in seismic applications involves a feedback mechanism related to the free surface. The presence of this feedback mechanism gives us an unique opportunity to remove ambiguities that have plagued blind deconvolution for a long time. While beneficial, this feedback by itself is insufficient to remove the ambiguities even with`1 constraints. However, when paired with an`1/`2 constraint the feedback allows us to resolve the scaling ambiguity under relatively mild assumptions. Inspired by lifting approaches, we propose to split the sparse signal into positive and negative components and apply an`1/`2 constraint to the difference, thereby obtaining a constraint that is easy to implement. Numerical experiments demonstrate robustness to the initialization as well as to noise in the data.

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Section: A New Algorithm For Blind Deconvolutionmentioning

confidence: 99%

Resolving scaling ambiguities with the ℓ1/ℓ2 norm in a blind deconvolution problem with feedback

Esser

Lin

Herrmann

et al. 2015

2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

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“…Since X ∈ C (nu+ng+1) 2 , we are not able to optimize over X directly for large-scale realistic applications. Nonetheless, as stated by Cosse et al [2015], it is possible for us to obtain a computationally feasible formulation with a reasonable storage requirement by introducing a rank-r factorization RR for the matrix X:…”

Section: Wri With a Rank-r Relaxationmentioning

confidence: 99%

“…Through tuning the penalty parameter, the resulting approach does not enforce the PDE constraints at each iteration and arguably yields a less non-linear problem in the model parameter. The "Lift" strategy follows the early work in Cosse et al [2015] that borrows ideas from recent developments in the semidefinite relaxation for polynomial equations to mitigate non-convexity [Lasserre, 2001, Laurent, 2009. We lift both unknown wavefields and model parameters from 1D vectors to rank-2 matrices, and reformulate the WRI problem as a set of constraints on a rank-2 moment matrix in a higher dimensional space.…”

Section: Introductionmentioning

confidence: 99%

Lift and Relax for PDE-constrained inverse problems in seismic imaging

Fang,

Demanet

2020

Preprint

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We present Lift and Relax for Waveform Inversion (LRWI), an approach that mitigates the local minima issue in seismic full waveform inversion (FWI) via a combination of two convexification techniques. The first technique (Lift) extends the set of variables in the optimization problem to products of those variables, arranged as a moment matrix. This algebraic idea is a celebrated way to replace a hard polynomial optimization problem by a semidefinite programming approximation. Concretely, both the model and the wavefield are lifted from vectors to rank-2 matrices. The second technique (Relax) invites to consider the wave equation, not as a hard constraint, but as a soft constraint to be satisfied only approximately -a technique known as wavefield reconstruction inversion (WRI). WRI weakens wave-equation constraints by introducing wave-equation misfits as a weighted penalty term in the objective function. The relaxed penalty formulation enables balancing the data and wave-equation misfits by tuning a penalty parameter. Together, "Lift" and "Relax" help reformulate the inverse problem as a set of constraints on a rank-2 moment matrix in a higher dimensional space. Such a lifting strategy permits a good data and wave-equation fit throughout the inversion process, while leaving the numerical rank of the rank-2 moment matrix to be minimized down to one. Numerical examples indicate that compared to FWI and WRI, LRWI can conduct successful inversions using an initial model that would be considered too poor, and data with a starting frequency that would be considered too high, for either method in isolation. Specifically, LRWI increases the acceptable starting frequency from 1.0 Hz and 0.5 Hz to 2.0 Hz and 2.5 for the Marmousi model and the Overthrust model, respectively, in the cases of a linear gradient starting model.

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