@article {dasilva2017uls,
	title = {A unified 2D/3D large scale software environment for nonlinear inverse problems},
	journal = {ACM Transactions on Mathematical Software},
	year = {2019},
	note = {Accepted on January 27, 2019.},
	abstract = {Large scale parameter estimation problems are some of the most computationally demanding problems. An academic researcher{\textquoteright}s domain-specific knowledge often precludes that of software design, which results in software frameworks for inversion that are technically correct, but not scalable to realistically-sized problems. On the other hand, the computational demands of the problem for realistic problems result in industrial codebases that are geared solely for performance, rather than comprehensibility or flexibility. We propose a new software design that bridges the gap between these two seemingly disparate worlds. A hierarchical and modular design allows a user to delve into as much detail as she desires, while using high performance primitives at the lower levels. Our code has the added benefit of actually reflecting the underlying mathematics of the problem, which lowers the cognitive load on user using it and reduces the initial startup period before a researcher can be fully productive. We also introduce a new preconditioner for the Helmholtz equation that is suitable for fault-tolerant distributed systems. Numerical experiments on a variety of 2D and 3D test problems demonstrate the effectiveness of this approach on scaling algorithms from small to large scale problems with minimal code changes.},
	keywords = {large scale, Matlab, Optimization, PDE-constrained inversion},
	url = {https://slim.gatech.edu/Publications/Public/Journals/ACMTOMS/2019/dasilva2017uls/dasilva2017uls.html},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@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 {dasilva2016SEGuse,
	title = {A unified 2D/3D software environment for large scale time-harmonic full waveform inversion},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {1169-1173},
	abstract = {Full-Waveform Inversion is a costly and complex procedure for realistically sized 3D seismic data. The performance-critical nature of this problem often results in software environments that are written entirely in low-level languages, making them hard to understand, maintain, improve, and extend. The unreadability of such codes can stymie research developments, where the translation from higher level mathematical ideas to high performance codes can be lost, inhibiting the uptake of new ideas in production-level codebases. We propose a new software organization paradigm for Full-Waveform Inversion and other PDE-constrained optimization problems that is flexible, efficient, scalable, and demonstrably correct. We decompose the various structural components of FWI in to its constituent components, from parallel computation to assembling objective functions and gradients to lower level matrix-vector multiplications. This decomposition allows us to create a framework where individual components can be easily swapped out to suit a particular user{\textquoteright}s existing software environment. The ease of applying high-level algorithms to the FWI problem allows us to easily implement stochastic FWI and demonstrate its effectiveness on a large scale 3D problem.},
	keywords = {Full-waveform inversion, large scale, numerical analysis, SEG, software},
	doi = {10.1190/segam2016-13869051.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/dasilva2016SEGuse/dasilva2016SEGuse.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/dasilva2016SEGuse/dasilva2016SEGuse_pres.pdf},
	author = {Curt Da Silva and Felix J. Herrmann}
}
@presentation {dasilva2016SINBADFuse,
	title = {A unified 2D/3D software environment for large scale time-harmonic full waveform inversion},
	year = {2016},
	publisher = {SINBAD},
	abstract = {Full Waveform Inversion is a costly and complex procedure, resulting in industrial codebases that are often written entirely in low-level languages, making them hard to understand, maintain, improve, and extend. In this work, we propose a software framework for organizing a 3D FWI environment that helps mitigate design complexities inherent in the problem in a straightforward way while delegating the performance-critical portions to low level languages to maintain efficiency. The result is a software framework that has a unified interface for 2D and 3D and is flexible, efficient, scalable, and demonstrably correct. We demonstrate the effectiveness of this approach on a large scale 3D FWI example.},
	keywords = {Presentation, SINBAD, SINBADFALL2016, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/dasilva2016SINBADFuse/dasilva2016SINBADFuse.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/dasilva2016SINBADFuse/dasilva2016SINBADFuse.mov},
	author = {Curt Da Silva 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}
}
@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}
}
