@article {fang2017uqfip,
	title = {Uncertainty quantification for inverse problems with weak partial-differential-equation constraints},
	journal = {Geophysics},
	volume = {83},
	number = {6},
	year = {2018},
	note = {(Geophysics)},
	pages = {R629-R647},
	abstract = {In a statistical inverse problem, the objective is a complete statistical description of unknown parameters from noisy observations in order to quantify uncertainties of the parameters of interest. We consider inverse problems with partial-differential-equation-constraints, which are applicable to a variety of seismic problems. Bayesian inference is one of the most widely-used approaches to precisely quantify statistics through a posterior distribution, incorporating uncertainties in observed data, modeling kernel, and prior knowledge of the parameters. Typically when formulating the posterior distribution, the partial-differential-equation-constraints are required to be exactly satisfied, resulting in a highly nonlinear forward map and a posterior distribution with many local maxima. These drawbacks make it difficult to find an appropriate approximation for the posterior distribution. Another complicating factor is that traditional Markov chain Monte Carlo methods are known to converge slowly for realistically sized problems. In this work, we relax the partial-differential-equation-constraints by introducing an auxiliary variable, which allows for Gaussian deviations in the partial-differential-equations. Thus, we obtain a new bilinear posterior distribution consisting of both data and partial-differential-equation misfit terms. We illustrate that for a particular range of variance choices for the partial-differential-equation misfit term, the new posterior distribution has fewer modes and can be well-approximated by a Gaussian distribution, which can then be sampled in a straightforward manner. Since it is prohibitively expensive to explicitly construct the dense covariance matrix of the Gaussian approximation for intermediate to large-scale problems, we present a method to implicitly construct it, which enables efficient sampling. We apply this framework to two-dimensional seismic inverse problems with 1,800 and 92,455 unknown parameters. The results illustrate that our framework can produce comparable statistical quantities to those produced by conventional Markov chain Monte Carlo type methods while requiring far fewer partial-differential-equation solves, which are the main computational bottlenecks in these problems.},
	keywords = {acoustic, FWI, UQ, weak-constraint},
	doi = {10.1190/geo2017-0824.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2018/fang2017uqfip/fang2017uqfip.html},
	author = {Zhilong Fang and Curt Da Silva and Rachel Kuske and Felix J. Herrmann}
}
@conference {fang2017WAVESuqf,
	title = {Uncertainty quantification for inverse problems with a weak wave-equation constraint},
	booktitle = {WAVES 2017 {\textendash}- 13th International Conference on Mathematical and Numerical Aspects of Wave Propagation},
	year = {2017},
	note = {(WAVES, Minneapolis)},
	month = {05},
	pages = {127-128},
	abstract = {In this work, we present a new posterior distribution to quantify uncertainties in solutions of wave-equation based inverse problems. By introducing an auxiliary variable for the wavefields, we weaken the strict wave-equation constraint used by conventional Bayesian approaches. With this weak constraint, the new posterior distribution is a bi-Gaussian distribution with respect to both model parameters and wavefields, which can be directly sampled by the Gibbs sampling method.},
	keywords = {constraint, Gibbs sampling, uncertainty, wave equation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/WAVES/2017/fang2017WAVESuqf/fang2017WAVESuqf.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/WAVES/2017/fang2017WAVESuqf/fang2017WAVESuqf_pres.pdf},
	author = {Zhilong Fang and Curt Da Silva and Rachel Kuske and Felix J. Herrmann}
}
@conference {fang2016SEGuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion using a PDE free semidefinite Hessian and randomize-then-optimize method},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {1390-1394},
	abstract = {The data of full-waveform inversion often contains noise, which induces uncertainties in the inversion results. Ideally, one would like to run a number of independent inversions with different realizations of the noise and assess model-side uncertainties from the resulting models, however this is not feasible because we collect the data only once. To circumvent this restriction, various sampling schemes have been devised to generate an ensemble of models that fit the data to within the noise level. Such sampling schemes typically involve running multiple inversions or evaluating the Hessian of the cost function, both of which are computationally expensive. In this work, we propose a new method to quantify uncertainties based on a novel formulation of the full-waveform inversion problem {\textendash} wavefield reconstruction inversion. Based on this formulation, we formulate a semidefinite approximation of the corresponding Hessian matrix. By precomputing certain quantities, we are able to apply this Hessian to given input vectors without additional solutions of the underlying partial differential equations. To generate a sample, we solve an auxiliary stochastic optimization problem involving this Hessian. The result is a computationally feasible method that, with little overhead, can generate as many samples as required at small additional cost. We test our method on the synthetic BG Compass model and compare the results to a direct-sampling approach. The results show the feasibility of applying our method to computing statistical quantities such as the mean and standard deviation in the context of wavefield reconstruction inversion.},
	keywords = {SEG, Uncertainty quantification, WRI},
	doi = {10.1190/segam2016-13879108.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/fang2016SEGuqw/fang2016SEGuqw.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/fang2016SEGuqw/fang2016SEGuqw_pres.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Tristan van Leeuwen and Felix J. Herrmann}
}
@conference {fang2015EAGEuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2015},
	note = {(EAGE, Madrid)},
	month = {06},
	abstract = {In this work, we propose a method to quantify the uncertainty of wavefield reconstruction inversion under the framework of Bayesian inference. Unlike the conventional method using the wave equation as the forward mapping, we involve the wave equation misfit in the posterior distribution and propose a new posterior distribution. The negative log-likelihood of the new distribution is less oscillatory than that of the conventional posterior distribution, and its Gauss-Newton Hessian is a diagonal matrix that can be generated without any additional computational cost. We use the diagonal Gauss-Newton Hessian to derive an approximate Gaussian distribution at the maximum likelihood point to quantify the uncertainty. This method makes the uncertainty quantification for WRI computationally tractable and is able to provide reasonable uncertainty analysis based on our numerical results.},
	keywords = {EAGE, FWI, UQ, WRI},
	doi = {10.3997/2214-4609.201413198},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/fang2015EAGEuqw/fang2015EAGEuqw.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/fang2015EAGEuqw/fang2015EAGEuqw_pres.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@conference {fang2015IIPFWIuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	booktitle = {Inaugural Full-Waveform Inversion Workshop},
	year = {2015},
	note = {(Natal, Brazil)},
	month = {08-09},
	abstract = {In this work, we propose a method to quantify the uncertainty of wavefield reconstruction inversion under the framework of Bayesian inference. Unlike the conventional method using the wave equation as the forward mapping, we involve the wave equation misfit in the posterior distribution and propose a new posterior distribution. The negative log-likelihood of the new distribution is less oscillatory than that of the conventional posterior distribution, and its Gauss-Newton Hessian is a diagonal matrix that can be generated without any additional computational cost. We use the diagonal Gauss-Newton Hessian to derive an approximate Gaussian distribution at the maximum likelihood point to quantify the uncertainty. This method makes the uncertainty quantification for WRI computationally tractable and is able to provide reasonable uncertainty analysis based on our numerical results.},
	keywords = {FWI, uncertainty, WRI},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/IIPFWI/fang2015IIPFWIuqw/fang2015IIPFWIuqw_pres.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@presentation {fang2015SINBADFuqw,
	title = {Uncertainty quantification for Wavefield-Reconstruction Inversion using a positive-definite approximated Hessian},
	year = {2015},
	publisher = {SINBAD},
	abstract = {We analyze the Hessian of wavefield reconstruction inversion (WRI) and propose a new approximated Hessian. Instead of requiring PDE solves, the matrix-vector multiplication action of the approximate Hessian can be achieved with several matrix-vector multiplications. The diagonal part of the approximated Hessian can be also calculated without additional PDE solves. We apply this approximated Hessian to uncertainty quantification and obtain statistical parameters such as standard deviation and confidence interval. Numerical example illustrate the accuracy of the estimated Hessian and the feasibility of the method to quantify uncertainties.},
	keywords = {Presentation, SINBAD, SINBADFALL2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/fang2015SINBADFuqw/fang2015SINBADFuqw.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/fang2015SINBADFuqw/fang2015SINBADFuqw.mov},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@presentation {fang2014SINBADuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	year = {2014},
	publisher = {SINBAD},
	abstract = {This talk discusses the uncertainty quantification for the wavefield reconstruction inversion. The Hessian and gradient of the WRI can be easily obtained once the wave field is generated. As a result, we use a Newton type McMC method to sample the posterior distribution. Statistical property such as mean, standard deviation, confidence interval are calculated based on the sampled distribution.},
	keywords = {Presentation, SINBAD, SINBADFALL2014, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2014/Fall/fang2014SINBADuqw/fang2014SINBADuqw.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@presentation {fang2013SINBADuafwi,
	title = {Uncertainty analysis for FWI},
	year = {2013},
	publisher = {SINBAD},
	abstract = {Uncertainty analysis is important for seismic interpretation. Based on the framework of Bayesian, we can analyse different statistic parameters of our FWI result. However, directly sampling the posterior probability density function (pdf) is computationally intractable. In order to make this problem computationally tractable, in this work, we use Gaussian distribution approximation and low rank approximation to generate the posterior pdf. Simultaneous shots are also used to reduce the computational costs.},
	keywords = {Presentation, SINBAD, SINBADFALL2013, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Fall/fang2013SINBADuafwi/fang2013SINBADuafwi.pdf},
	author = {Zhilong Fang and Felix J. Herrmann}
}
