@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 {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}
}
@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}
}
