Uncertainty Quantification for Ill-Posed Inverse Problems in the Physical Sciences: Confidence Intervals via Optimization and Sampling

 Ill-posed inverse problems frequently arise in the physical sciences through the modeling of natural processes and phenomena. Given an observation, there is often a vast space of model parameters such that every parameter setting in that space is consistent with that observation. Since scientific inference is impossible with such loose constraints on the truth, inference and uncertainty quantification (UQ) typically require some additional way to bound this space and report uncertainty within these new bounds. In particular, we are motivated by two applications where such a scenario arises; particle unfolding from high-energy physics and carbon flux inversion from atmospheric science, and how to compute confidence intervals for UQ on key scientific quantities in these settings. More generally, this thesis is motivated by inverse problem settings where an observation is the sum of a deterministic model on a finite-dimensional parameter space and Gaussian noise, where there are known physical constraints on the model parameters. 

Typically, the aforementioned ill-posedness is addressed using a prior distribution on the parameters to regularize the problem and from which UQ follows by reporting posterior uncertainty or credible intervals on a quantities of interest. Instead, we propose to handle the ill-posedness by implicitly regularizing the problem by including known physical constraints and doing inference directly on one-dimensional functionals of the model parameters to avoid over-optimism and under-coverage that can arise when using a probabilistic prior. Statistically, the inclusion of parameter constraints and the potential presence of a null space break the assumptions of standard statistical models, leaving unclear the task of providing rigorous UQ in these regimes and motivating the development of a statistical framework to accommodate this setting. Scientifically and computationally, computing confidence intervals in high dimensional scenarios, like carbon flux inversion, involves computing in a “matrix-free” setting, motivating the development of new computational approaches. 

This thesis develops a statistical framework and theory for confidence intervals in constrained parameter settings to implement the above implicit regularization idea for ill-posed inverse problem UQ. Computational tools are then developed both via optimization for high-dimensional problems and sampling for low- and moderate-dimensional problems to compute such intervals. Finally, these techniques are applied to the aforementioned applications, addressing the wide-bin bias problem in high-energy physics and over-optimistic UQ in carbon flux inversion. In total, these innovations have led to tighter confidence interval constructions maintaining nominal coverage and the first prior-free UQ for a high-dimensional 4D-Var data assimilation application.