@article{jordan1999introduction, title={{An Introduction to Variational Methods for Graphical Models}}, author={Jordan, Michael I and Ghahramani, Zoubin and Jaakkola, Tommi S and Saul, Lawrence K}, journal={{Machine Learning}}, volume={37}, number={2}, pages={183--233}, year={1999}, publisher={{Springer}}, doi={10.1023/A:1007665907178} } @book{wainwright2008graphical, title={Graphical models, exponential families, and variational inference}, author={Wainwright, Martin J and Jordan, Michael Irwin}, year={2008}, publisher={Now Publishers Inc} } @book{robert2004monte, title={Monte {C}arlo statistical methods}, author={Robert, C. and Casella, G.}, publisher={Springer-Verlag}, year={2004} } @article{blei2017variational, title={{Variational inference: A review for statisticians}}, author={Blei, David M and Kucukelbir, Alp and McAuliffe, Jon D}, journal={{Journal of the American statistical Association}}, volume={112}, number={518}, pages={859--877}, year={2017}, publisher={Taylor \& Francis} } @InProceedings{rezende2015variational, title = {{Variational Inference with Normalizing Flows}}, author = {Danilo Rezende and Shakir Mohamed}, booktitle = {{Proceedings of the 32nd International Conference on Machine Learning}}, pages = {1530--1538}, year = {2015}, volume = {37}, series = {{Proceedings of Machine Learning Research}}, month = {07--09 Jul}, publisher = {{PMLR}}, url = { http://proceedings.mlr.press/v37/rezende15.html } } @misc{lensink2019fully, title={Fully {H}yperbolic {C}onvolutional {N}eural {N}etworks}, author={Lensink, K. and Haber, E. and Peters, B.}, year={2019}, eprint={1905.10484}, archivePrefix={arXiv}, primaryClass={cs.CV} } @inproceedings{Putzky2019, author = {Putzky, Patrick and Welling, Max}, booktitle = {Advances in Neural Information Processing Systems}, editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett}, pages = {}, publisher = {Curran Associates, Inc.}, title = {Invert to Learn to Invert}, url = {https://proceedings.neurips.cc/paper/2019/file/ac1dd209cbcc5e5d1c6e28598e8cbbe8-Paper.pdf}, volume = {32}, year = {2019} } @techreport{WesternGeco2012, title = {{Parihaka 3D PSTM Final Processing Report}}, number = {New Zealand Petroleum Report 4582}, year = {2012}, institution = {New Zealand Petroleum \& Minerals, Wellington}, author={WesternGeco.} } @article{dinh2016density, title={Density estimation using {R}eal {NVP}}, author={Dinh, Laurent and Sohl-Dickstein, Jascha and Bengio, Samy}, journal={arXiv preprint arXiv:1605.08803}, year={2016} } @article{kruse2019hint, title={{HINT}: {H}ierarchical {I}nvertible {N}eural {T}ransport for {D}ensity {E}stimation and {B}ayesian {I}nference}, author={Kruse, Jakob and Detommaso, Gianluca and Scheichl, Robert and K{\"o}the, Ullrich}, journal={arXiv preprint arXiv:1905.10687}, year={2019} } @book{villani2009optimal, title={{Optimal transport: old and new}}, author={Villani, C{\'e}dric}, year={2009}, doi={10.1007/978-3-540-71050-9}, publisher={{Springer-Verlag Berlin Heidelberg}} } @article{marzouk2016sampling, title={Sampling via measure transport: An introduction}, author={Marzouk, Youssef and Moselhy, Tarek and Parno, Matthew and Spantini, Alessio}, journal={Handbook of uncertainty quantification}, pages={1--41}, year={2016} } @article{marzouk2018multifidelity, title={A transport-based multifidelity preconditioner for {M}arkov chain {M}onte {C}arlo}, author={Benjamin Peherstorfer and Youssef Marzouk}, journal={Advances in Computational Mathematics}, volume={45}, number={5-6}, pages={2321--2348}, year={2019}, publisher={Springer} } @article{marzuk2018, title = {Transport {M}ap {A}ccelerated {M}arkov {C}hain {M}onte {C}arlo}, author={Parno, Matthew D and Marzouk, Youssef M}, journal={SIAM/ASA Journal on Uncertainty Quantification}, volume={6}, number={2}, pages={645--682}, year={2018}, publisher={SIAM}, doi = {10.1137/17M1134640} } @article{adler2018deep, title={{Deep Bayesian Inversion}}, author={Adler, Jonas and {\"O}ktem, Ozan}, journal={arXiv preprint arXiv:1811.05910}, year={2018} } @article{devito-compiler, author = { {Luporini}, F. and {Lange}, M. and {Louboutin}, M. and {Kukreja}, N. and {H{\"u}ckelheim}, J. and {Yount}, C. and {Witte}, P. and {Kelly}, P.~H.~J. and {Herrmann}, F.~J. and {Gorman}, G.~J.}, title = { Architecture and performance of Devito, a system for automated stencil computation }, journal = { CoRR }, volume = { abs/1807.03032 }, month = { jul }, year = { 2018 }, url = { http://arxiv.org/abs/1807.03032 }, archivePrefix = { arXiv }, eprint = { 1807.03032 } } @article{devito-api, author = {Louboutin, M. and Lange, M. and Luporini, F. and Kukreja, N. and Witte, P. A. and Herrmann, F. J. and Velesko, P. and Gorman, G. J.}, title = {Devito (v3.1.0): an embedded domain-specific language for finite differences and geophysical exploration}, journal = {Geoscientific Model Development}, volume = {12}, year = {2019}, number = {3}, pages = {1165--1187}, url = {https://www.geosci-model-dev.net/12/1165/2019/}, doi = {10.5194/gmd-12-1165-2019} } @software{invnet, author = {Philipp Witte and Gabrio Rizzuti and Mathias Louboutin and Ali Siahkoohi and Felix Herrmann and Bas Peters}, title = {{InvertibleNetworks.jl: A Julia framework for invertible neural networks}}, month = mar, year = 2021, publisher = {Zenodo}, version = {v2.0.0}, doi = {10.5281/zenodo.4610118}, url = {https://doi.org/10.5281/zenodo.4610118} } @article{kingma2015adam, title={{A}dam: {A} {M}ethod for {S}tochastic {O}ptimization}, author={Diederik P. Kingma and Jimmy Ba}, journal={CoRR}, year={2014}, archivePrefix = {arXiv}, arxivId = {1412.6980}, eprint = {1412.6980}, volume={abs/1412.6980} } @article{Zhang2020, author = {Zhang, Xin and Curtis, Andrew}, title = "{Variational full-waveform inversion}", journal = {Geophysical Journal International}, volume = {222}, number = {1}, pages = {406-411}, year = {2020}, month = {04}, abstract = "{Seismic full-waveform inversion (FWI) can produce high-resolution images of the Earth’s subsurface. Since full-waveform modelling is significantly nonlinear with respect to velocities, Monte Carlo methods have been used to assess image uncertainties. However, because of the high computational cost of Monte Carlo sampling methods, uncertainty assessment remains intractable for larger data sets and 3-D applications. In this study, we propose a new method called variational FWI, which uses Stein variational gradient descent to solve FWI problems. We apply the method to a 2-D synthetic example and demonstrate that the method produces accurate approximations to those obtained by Hamiltonian Monte Carlo. Since variational inference solves the problem using optimization, the method can be applied to larger data sets and 3-D applications by using stochastic optimization and distributed optimization.}", issn = {0956-540X}, doi = {10.1093/gji/ggaa170}, url = {https://doi.org/10.1093/gji/ggaa170}, eprint = {https://academic.oup.com/gji/article-pdf/222/1/406/33181729/ggaa170.pdf}, } @article{zhang2020seismic, title={Seismic tomography using variational inference methods}, author={Zhang, Xin and Curtis, Andrew}, journal={Journal of Geophysical Research: Solid Earth}, volume={125}, number={4}, pages={e2019JB018589}, year={2020}, publisher={Wiley Online Library} } @article{curtis2001prior, title={Prior information, sampling distributions, and the curse of dimensionality}, author={Curtis, Andrew and Lomax, Anthony}, journal={Geophysics}, volume={66}, number={2}, pages={372--378}, year={2001}, publisher={Society of Exploration Geophysicists} } @article{kotsi2020, author = {Kotsi, M and Malcolm, A and Ely, G}, title = {{Uncertainty quantification in time-lapse seismic imaging: a full-waveform approach}}, journal = {{Geophysical Journal International}}, volume = {222}, number = {2}, pages = {1245--1263}, year = {2020}, month = {05}, issn = {0956-540X}, doi = {10.1093/gji/ggaa245}, } @article {fang2018uqfip, title = {Uncertainty quantification for inverse problems with weak partial-differential-equation constraints}, journal={{GEOPHYSICS}}, volume = {83}, number = {6}, year = {2018}, pages = {R629--R647}, doi = {10.1190/geo2017-0824.1}, author = {Zhilong Fang and Curt Da Silva and Rachel Kuske and Felix J. Herrmann} } @article{malinverno2006two, title={Two ways to quantify uncertainty in geophysical inverse problems}, author={Malinverno, Alberto and Parker, Robert L}, journal={{GEOPHYSICS}}, volume={71}, number={3}, pages={W15--W27}, year={2006}, publisher={Society of Exploration Geophysicists}, doi={10.1190/1.2194516} } @article{malinverno2004expanded, title={Expanded uncertainty quantification in inverse problems: Hierarchical Bayes and empirical Bayes}, author={Malinverno, Alberto and Briggs, Victoria A}, journal={{GEOPHYSICS}}, volume={69}, number={4}, pages={1005--1016}, year={2004}, publisher={Society of Exploration Geophysicists}, doi={10.1190/1.1778243} } @article{MartinMcMC2012, Author = {Martin, James and Wilcox, Lucas C. and Burstedde, Carsten and Ghattas, OMAR}, Date-Modified = {2016-06-24 20:16:07 +0000}, Eprint = {http://epubs.siam.org/doi/pdf/10.1137/110845598}, Journal = {SIAM Journal on Scientific Computing}, Number = {3}, Pages = {A1460-A1487}, Title = {A {S}tochastic {N}ewton {MCMC} {M}ethod for {L}arge-Scale {S}tatistical {I}nverse {P}roblems with {A}pplication to {S}eismic {I}nversion}, Url = {http://epubs.siam.org/doi/abs/10.1137/110845598}, Volume = {34}, Year = {2012}, } @article{stuart2019two, title={A two-stage {Markov chain Monte Carlo} method for seismic inversion and uncertainty quantification}, author={Stuart, Georgia K and Minkoff, Susan E and Pereira, Felipe}, journal = {{GEOPHYSICS}}, volume={84}, number={6}, pages={R1003--R1020}, year={2019}, month={11}, publisher={{Society of Exploration Geophysicists}}, doi = {10.1190/geo2018-0893.1} } @conference{zhao2019gradient, title={A gradient based {MCMC} method for {FWI} and uncertainty analysis}, author={Zhao, Zeyu and Sen, Mrinal K}, booktitle = {{SEG Technical Program Expanded Abstracts 2019}}, pages={1465--1469}, year = {2019}, month = {8}, doi={10.1190/segam2019-3216560.1} } @conference {siahkoohi2020SEGhorizonUQ, title = {{Uncertainty quantification in imaging and automatic horizon tracking---a Bayesian deep-prior based approach}}, author = {Ali Siahkoohi and Gabrio Rizzuti and Felix J. Herrmann}, booktitle = {SEG Technical Program Expanded Abstracts 2020}, year = {2020}, month = {9}, pages = {1636--1640}, doi={10.1190/segam2020-3417560.1}, } @article {siahkoohi2020EAGEdlb, title = {A deep-learning based Bayesian approach to seismic imaging and uncertainty quantification}, year = {2020}, month = {6}, journal={{82nd EAGE Conference and Exhibition 2020}}, keywords = {deep learning, EAGE, seismic imaging, stochastic gradient Langevin dynamics, Uncertainty quantification}, url = {https://slim.gatech.edu/Publications/Public/Submitted/2020/siahkoohi2020EAGEdlb/siahkoohi2020EAGEdlb.html}, author = {Siahkoohi, A. and Rizzuti, G. and Herrmann, F. J.} } @article{chevron2017, author = {Ray, Anandaroop and Kaplan, Sam and Washbourne, John and Albertin, Uwe}, title = "{Low frequency full waveform seismic inversion within a tree based Bayesian framework}", journal = {Geophysical Journal International}, volume = {212}, number = {1}, pages = {522--542}, year = {2017}, month = {10}, abstract = "{Limited illumination, insufficient offset, noisy data and poor starting models can pose challenges for seismic full waveform inversion. We present an application of a tree based Bayesian inversion scheme which attempts to mitigate these problems by accounting for data uncertainty while using a mildly informative prior about subsurface structure. We sample the resulting posterior model distribution of compressional velocity using a trans-dimensional (trans-D) or Reversible Jump Markov chain Monte Carlo method in the wavelet transform domain of velocity. This allows us to attain rapid convergence to a stationary distribution of posterior models while requiring a limited number of wavelet coefficients to define a sampled model. Two synthetic, low frequency, noisy data examples are provided. The first example is a simple reflection + transmission inverse problem, and the second uses a scaled version of the Marmousi velocity model, dominated by reflections. Both examples are initially started from a semi-infinite half-space with incorrect background velocity. We find that the trans-D tree based approach together with parallel tempering for navigating rugged likelihood (i.e. misfit) topography provides a promising, easily generalized method for solving large-scale geophysical inverse problems which are difficult to optimize, but where the true model contains a hierarchy of features at multiple scales.}", issn = {0956-540X}, doi = {10.1093/gji/ggx428}, url = {https://doi.org/10.1093/gji/ggx428}, eprint = {https://academic.oup.com/gji/article-pdf/212/1/522/21782947/ggx428.pdf}, } @conference {herrmann2019NIPSliwcuc, title = {Learned imaging with constraints and uncertainty quantification}, booktitle = {{Neural Information Processing Systems (NeurIPS) 2019 Deep Inverse Workshop}}, year = {2019}, month = {12}, abstract = {We outline new approaches to incorporate ideas from convolutional networks into wave-based least-squares imaging. The aim is to combine hand-crafted constraints with deep convolutional networks allowing us to directly train a network capable of generating samples from the posterior. The main contributions include combination of weak deep priors with hard handcrafted constraints and a possible new way to sample the posterior.}, keywords = {constraint, deep learning, Imaging, Uncertainty quantification}, url = {https://arxiv.org/pdf/1909.06473.pdf}, author = {Felix J. Herrmann and Ali Siahkoohi and Gabrio Rizzuti} } @article{kovachki2020conditional, title={{Conditional Sampling With Monotone GANs}}, author={Kovachki, Nikola and Baptista, Ricardo and Hosseini, Bamdad and Marzouk, Youssef}, journal={arXiv preprint arXiv:2006.06755}, year={2020} } @inproceedings{Goodfellow2014, title = {{G}enerative {A}dversarial {N}ets}, author={Goodfellow, Ian and Pouget-Abadie, Jean and Mirza, Mehdi and Xu, Bing and Warde-Farley, David and Ozair, Sherjil and Courville, Aaron and Bengio, Yoshua}, booktitle = {Proceedings of the 27th International Conference on Neural Information Processing Systems}, series = {NIPS'14}, pages={2672--2680}, location = {Montreal, Canada}, year={2014}, url = {http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf}, eprint = {http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf} } @article{vandeLeemput2019MemCNN, journal = {{Journal of Open Source Software}}, doi = {10.21105/joss.01576}, issn = {2475-9066}, number = {39}, publisher = {The Open Journal}, title = {{MemCNN: A Python/PyTorch package for creating memory-efficient invertible neural networks}}, url = {http://dx.doi.org/10.21105/joss.01576}, volume = {4}, author = {Sil C. {van de} Leemput and Jonas Teuwen and Bram {van} Ginneken and Rashindra Manniesing}, pages = {1576}, date = {2019-07-30}, year = {2019}, month = {7}, day = {30}, } @inproceedings{peters2020fully, title={Fully reversible neural networks for large-scale 3D seismic horizon tracking}, author={Peters, Bas and Haber, Eldad}, booktitle={82nd EAGE Annual Conference \& Exhibition}, pages={1--5}, year={2020}, organization={European Association of Geoscientists \& Engineers} } @article{zhao2020bayesian, title={Bayesian Seismic Tomography using Normalizing Flows}, author={Zhao, Xuebin and Curtis, Andrew and Zhang, Xin}, year={2020}, publisher={EarthArXiv}, doi={10.31223/X53K6G}, url={https://eartharxiv.org/repository/view/1940/} } @inproceedings{NIPS2017_17ed8abe, author = {Liu, Qiang}, booktitle = {Advances in Neural Information Processing Systems}, editor = {I. Guyon and U. V. Luxburg and S. Bengio and H. Wallach and R. Fergus and S. Vishwanathan and R. Garnett}, pages = {}, publisher = {Curran Associates, Inc.}, title = {Stein Variational Gradient Descent as Gradient Flow}, url = {https://proceedings.neurips.cc/paper/2017/file/17ed8abedc255908be746d245e50263a-Paper.pdf}, volume = {30}, year = {2017} } @conference {siahkoohi2020ABIpto, title = {Preconditioned training of normalizing flows for variational inference in inverse problems}, booktitle = {{3rd Symposium on Advances in Approximate Bayesian Inference}}, year = {2021}, month = {1}, abstract = {In the context of inverse problems with computationally expensive forward operators, specially for domains with limited access to high-fidelity training unknown and observed data pairs, we propose a preconditioned scheme for training a conditional normalizing flow (NF) capable of directly sampling the posterior distribution. Our training objective consists of the Kullback-Leibler divergence between the predicted and the desired posterior density. To minimize the costs associated with the forward operator, we initialize the NF via the weights of another pretrained low-fidelity NF, which is trained beforehand on available low-fidelity model and data pairs. Our numerical experiments, including a 2D toy and a seismic image compressed sensing example, demonstrate the improved performance and speed-up of the proposed method compared to training a NF from scratch.}, author = {Ali Siahkoohi and Gabrio Rizzuti and Mathias Louboutin and Philipp Witte and Felix J. Herrmann}, url={https://openreview.net/pdf?id=P9m1sMaNQ8T} } @techreport {siahkoohi2020TRfuqf, title = {Faster Uncertainty Quantification for Inverse Problems with Conditional Normalizing Flows}, number = {TR-CSE-2020-2}, year = {2020}, month = {07}, institution = {Georgia Institute of Technology}, url = {https://arxiv.org/abs/2007.07985}, author = {Ali Siahkoohi and Gabrio Rizzuti and Philipp A. Witte and Felix J. Herrmann} } @book{tarantola2005inverse, title={Inverse problem theory and methods for model parameter estimation}, author={Tarantola, Albert}, year={2005}, publisher={SIAM}, doi={10.1137/1.9780898717921}, ISBN={978-0-89871-572-9} } @InProceedings{wgan, title = {{W}asserstein Generative Adversarial Networks}, author = {Martin Arjovsky and Soumith Chintala and L{\'e}on Bottou}, booktitle = {Proceedings of the 34th International Conference on Machine Learning}, pages = {214--223}, year = {2017}, volume = {70}, series = {Proceedings of Machine Learning Research}, month = {06--11 Aug}, publisher = {PMLR}, url = {http://proceedings.mlr.press/v70/arjovsky17a.html}, } @inproceedings{welling2011bayesian, author = {Welling, Max and Teh, Yee Whye}, title = {{Bayesian Learning via Stochastic Gradient Langevin Dynamics}}, year = {2011}, isbn = {9781450306195}, publisher = {Omnipress}, address = {Madison, WI, USA}, booktitle = {{Proceedings of the 28th International Conference on Machine Learning}}, pages = {681--688}, numpages = {8}, location = {{Bellevue, Washington, USA}}, series = {{ICML’11}}, url={https://dl.acm.org/doi/abs/10.5555/3104482.3104568}, doi={10.5555/3104482.3104568} } @book{bishop2006pattern, title={{Pattern Recognition and Machine Learning}}, author={Bishop, Christopher M}, year={2006}, publisher={{Springer-Verlag New York}} } @article{Mosser2020, abstract = {We present an application of deep generative models in the context of partial differential equation constrained inverse problems. We combine a generative adversarial network representing an a priori model that generates geological heterogeneities and their petrophysical properties, with the numerical solution of the partial-differential equation governing the propagation of acoustic waves within the earth's interior. We perform Bayesian inversion using an approximate Metropolis-adjusted Langevin algorithm to sample from the posterior distribution of earth models given seismic observations. Gradients with respect to the model parameters governing the forward problem are obtained by solving the adjoint of the acoustic wave equation. Gradients of the mismatch with respect to the latent variables are obtained by leveraging the differentiable nature of the deep neural network used to represent the generative model. We show that approximate Metropolis-adjusted Langevin sampling allows an efficient Bayesian inversion of model parameters obtained from a prior represented by a deep generative model, obtaining a diverse set of realizations that reflect the observed seismic response.}, author = {Mosser, Lukas and Dubrule, Olivier and Blunt, Martin J.}, da = {2020/01/01}, date-added = {2021-04-01 11:53:59 -0400}, date-modified = {2021-04-01 11:53:59 -0400}, doi = {10.1007/s11004-019-09832-6}, id = {Mosser2020}, isbn = {1874-8953}, journal = {Mathematical Geosciences}, number = {1}, pages = {53--79}, title = {Stochastic Seismic Waveform Inversion Using Generative Adversarial Networks as a Geological Prior}, ty = {JOUR}, url = {https://doi.org/10.1007/s11004-019-09832-6}, volume = {52}, year = {2020}, Bdsk-Url-1 = {https://doi.org/10.1007/s11004-019-09832-6}} @book{anderson1979, title={{Optimal Filtering}}, author={Anderson, Brian D.O. and Moore, John B.}, year = {1979}, publisher = {Prentice-Hall, Englewood Cliffs, NJ.}, } @conference {rizzuti2020SEGuqavp, title = {Parameterizing uncertainty by deep invertible networks, an application to reservoir characterization}, booktitle = {{SEG Technical Program Expanded Abstracts}}, year = {2020}, month = {09}, pages = {1541--1545}, doi = {10.1190/segam2020-3428150.1}, software = {https://github.com/slimgroup/Software.SEG2020}, author = {Gabrio Rizzuti and Ali Siahkoohi and Philipp A. Witte and Felix J. Herrmann} }