Sliced inverse approach and domain recovery for stochastic inverse problems - Chi, Jiarui, author; Wang, Haonan, advisor; Estep, Don, advisor; Breidt, F. Jay, committee member; Tavener, Simon, committee member; Zhou, Wen, committee member (Colorado State University. Libraries, 2021)
This dissertation tackles several critical challenges related to the Stochastic Inverse Problem (SIP) to perform scientific inference and prediction for complex physical systems which are characterized by mathematical models, e.g. differential equations. We treat both discrete and continuous cases. The SIP concerns inferring the values and quantifying the uncertainty of the inputs of a model, which are considered as random and unobservable quantities governing system behavior, by using observational data on the model outputs. Uncertainty of the inputs is quantified through probability distributions on the input domain which induce the probability distribution on the outputs realized by the observational data. The formulation of the SIP is based on rigorous measure-theoretic probability theory that uses all the information encapsulated in both the model and data. We introduce a problem in which a portion of the inputs can be observed and varied to study the hidden inputs, and we employ a formulation of the problem that uses all the knowledge in multiple experiments by varying the observable inputs. Since the map that the model induces is typically not 1-1, an ansatz, i.e. an assumption of some prior information, is necessary to be imposed in order to determine a specific solution of the SIP. The resulting solution is heavily conditioned on the observable inputs and we seek to combine solutions from different values of the observable inputs in order to reduce that dependence. We propose an approach of combining the individual solutions based on the framework of the Dempster-Shafer theory, which removes the dependency on the experiments as well as the ansatz and provides useful distributional information about the unobservable inputs, more specifically, about the ansatz. We develop an iterative algorithm that updates the ansatz information in order to obtain a best form of the solution for all experiments. The philosophy of Bayesian approaches is similar to that of the SIP in the sense that they both consider random variables as the model inputs and they seek to update the unobservable solution using information obtained from observations. We extend the classical Bayesian in the context of the SIP by incorporating the knowledge of the model. The input domain is a pre-specified condition for the SIP given by the knowledge from scientists and is often assumed to be a compact metric space. The supports of the probability distributions computed in the SIP are restricted to the domain, and thus an inappropriate choice of domain might cause a massive loss of information in the solutions. Similarly, we combine the individual solutions from multiple experiments to recover a unique domain among many choices of domain induced by the distribution of the inputs in general cases. In particular, results on the convergence of the domain recovery in linear models are investigated.