Forward and backward uncertainty quantification: methods, analysis, and applications

Abstract

The field of uncertainty quantification (UQ) deals with physical systems described by an input-output mapping. Randomness is present either in the parameters of the input or in the quantities of interest which are functionals of the output. This dissertation studies problems that arise in both forward and backward UQ and proposes methods to address them. Forward UQ quantifies the probability distribution on the output when the uncertainty on the input is propagated through the mapping. A probabilistically accurate and computationally efficient surrogate model is necessary to avoid numerous solutions of the input-output map. In contrast, backward UQ infers the distribution on the input assuming that the law of the output is known. Interest is on the well-posedness of this stochastic inverse problem and in particular, the uniqueness of the resulting solution. We survey existing methods developed to tackle problems in these settings and examine challenges associated with them. With respect to forward UQ, commonly used surrogate models may not possess requisite convergence properties or may be constructed without regard to the distribution on the input. We therefore develop an adaptive method based on Voronoi cells that constructs the surrogate accurately in high probability regions of the input and in regions where the input-output mapping exhibits substantial variation. For backward UQ, recently proposed approaches make assumptions on the law of the input which do not necessarily guarantee recovery of its true distribution. As such, we investigate what additional information must be specified on the input to solve this inverse problem and outline how approaches based on optimization, the principle of maximum entropy, and Bayes' theorem can incorporate such information. We conclude by considering a different type of inverse problem where the objective is on identifying samples of the input which yield large quantities of interest. This enables one to study the law of the input conditioned on extreme events and predict which inputs cause such events. We achieve this through a general framework which leverages on multifidelity surrogate models for input-output maps and machine learning classifiers.