@conference {fang2017AIPepm,
	title = {An efficient penalty method for PDE-constrained optimization problem with source estimation and stochastic optimization},
	booktitle = {Applied Inverse Problems Annual Conference Proceedings},
	year = {2017},
	note = {(AIP, Hangzhou)},
	month = {05-06},
	pages = {40},
	abstract = {Many inverse problems in applied science and engineering can be written as PDE-constrained optimization problems. Typically, one is interested in estimating the unknown medium parameters of the PDE describing the system, given multi-source measurements of the fields in the true medium. The objective function is often very oscillatory, which gives rise to problematic local minima, and merely evaluating one or more of its derivatives requires solving multiple PDEs, which are expensive. In many applications such as seismic exploration, only the source location is available and the source signature itself is unknown, which results in erroneous estimated parameters. In this work, we propose a penalty method to solve the PDE-constrained optimization problem, whereby we replace the PDE constraints by a two-norm penalty term, which leads to a bi-linear optimization problem with a much less oscillatory objective function. By applying the variable projection method, we develop an efficient method to invert the unknown parameters and source signatures simultaneously. To reduce the computational cost, we randomly select a subset of all data to update the parameters at each iteration. Numerical examples demonstrate that this method is less prone to becoming stuck in harmful local minima and is able to successfully invert the unknown parameters and source signature with less computational cost compared to using all the data.},
	keywords = {Optimization, penalty method, source estimation, Stochastic optimization},
	url = {http://aip2017.csp.escience.cn/dct/page/1},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/AIP/2017/fang2017AIPepm/fang2017AIPepm_pres.pdf},
	author = {Zhilong Fang and Curt Da Silva and Felix J. Herrmann}
}
@presentation {fang2016SINBADFeaq,
	title = {Efficient approach for quantifying uncertainty of wavefield reconstruction inversion},
	year = {2016},
	publisher = {SINBAD},
	abstract = {Due to the noisy measurements, sparse observations, uncertain forward models, and uncertain prior parameter information, significant uncertainty arises in the inverted velocity model of full-waveform inversion. The uncertain velocity model leads to the uncertainty in the correct positions of events in the migrated images, which subsequently influences the important decision-makings in the oil and gas exploration and production (E\&P) business such as the drilling decisions and economic evaluations. Therefore, understanding the uncertainty in the inverted velocity model is essential for the mitigation of the E\&P risks. A common approach to accounting for uncertainty in seismic inverse problems like full-waveform inversion is to describe the unknown parameters in probabilistic by the Bayesian inference. The Bayesian inference aims at incorporating the information from the data, physics, and one{\textquoteright}s prior knowledge to formulate a posterior distribution function to characterize the statistical properties of the unknown parameters. However, characterizing uncertainties for the large-scale full-waveform inversion in seismic exploration is prohibitively expensive since it requires thousands of function evaluations to sample the parameter space (e.g. via Markov chain Monte Carlo sampling). Apart from this, the existing methods of uncertainty quantification for full-waveform inversion is through applying the Bayesian inference to the standard method such as the adjoint-state method with the assumption that no uncertainty arises from the wave-equation. However, the wave-equation may not be able to reflect all the physics, hence, incorporating uncertainty from wave-equation might benefit the quantification of uncertainty. In this work, we propose to perform uncertainty quantification using the recently proposed wavefield reconstruction inversion technique, which gives an appropriate framework to tackle the uncertainty in wave-equation solve by adding the misfit of the wave-equation to the likelihood distribution and relaxing the wave-equation constraint. Moreover, we propose a computationally feasible approach to analyzing the posterior distribution of the wavefield reconstruction inversion method, which incorporates a Gaussian distribution that approximates the posterior distribution with an optimization-driven Gaussian simulator to sample the Gaussian distribution efficiently. Numerical examples show that comparing to the McMC method, our approach is able to produce comparable results with less computational cost. This is joint work with Curt da Silva and Rachel Kuske.},
	keywords = {Presentation, SINBAD, SINBADFALL2016, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/fang2016SINBADFeaq/fang2016SINBADFeaq.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/fang2016SINBADFeaq/fang2016SINBADFeaq.mov},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Tristan van Leeuwen and Felix J. Herrmann and Rachel Kuske}
}
