Uncertainty quantification and efficient design of experiments for data- and simulation-driven inverse problem solving

As scientists and engineers increasingly complement or surrogate observational data with numerical simulations, principled approaches are needed to enable discoveries in contexts where acquiring information from one or both sources is limited and comes with uncertainties. In this project we will investigate novel spatial statistical models, uncertainty quantification approaches and sequential design algorithms for parsimonious uncertainty reduction with main test cases in geosciences and cosmology. Two types of inverse problems will be considered: i) predicting (probability) distribution-valued fields and exploiting resulting approaches to accelerate algorithms for approximate posterior sampling and global optimization under uncertainty and, ii) estimating sets that are implicitly defined in terms of expensive-to-evaluate functions, under indirect function evaluations. Regarding i), our main focus will be on modelling complex systems in cases where the output of interest arises as a probability distribution. We will investigate spatial prediction of probability distributions from several viewpoints ranging from variations of Kriging to Bayesian nonparametric approaches. The developments will be tested first for predicting distributions of distances between observed and simulated statistics of cosmological images as a function of cosmological parameters, towards accelerated and enhanced Approximate Bayesian Computation. Second, to predict the distribution of misfits between observed and simulated concentration curves in the case of a contaminant localization hydrology problem under uncertain geology, developing in turn an original class of non-convex stochastic optimization algorithms. The main contributions on ii) will consist in theoretically and practically developing set estimators and associated confidence statements under indirect measurements. A volcano gravimetry application will serve as baseline linear problem, and will be used to develop a new approach to design targeted geophysical measurements under campaigning constraints. We will also tackle the non-linear problem of building confidence regions on filamentary structures from density information, notably by relying on approximate conditional simulations of derivative random fields and integral curves. Both mathematical and applied results of this project, supported by test cases from different horizons, will highlight the potential impact of its methodological contributions beyond disciplinary boundaries.
University of California at Davis
Swiss National Science Foundation
Nov 01, 2018
Aug 31, 2022