| Inverse problems in continuum transport processes (governed by partial differential equations (PDEs)) have major applications in a variety of scientific and engineering areas. The ill-posedness, high computational cost, and other complications of these problems pose significant intellectual challenges. In this thesis, a computational framework is developed that integrates computational mathematics, Bayesian statistics, statistical computation, and reduced-order modeling to address data-driven inverse heat and mass transfer problems. The Bayesian computational approach is advantageous in many aspects. In particular, it is able to quantify system uncertainty and random data error, to derive a probabilistic description of the inverse solution, to provide flexible spatial/temporal regularization to the ill-posedness of the inverse problem, and to allow adaptive sequential estimation. The components of this framework include hierarchical Bayesian formulation, prior modeling of distributed parameters via spatial statistics, exploration of implicit posterior distributions using Markov chain Monte Carlo (MCMC) simulation, proper orthogonal decomposition (POD)-based reduced-order modeling of the PDE system, and sequential Bayesian estimation. These methodologies are applied to the solution of a number of inverse problems in transport processes including inverse heat conduction, inverse heat radiation, contaminant detection in porous media flows, control of directional solidification, and multiscale permeability estimation in heterogeneous media. These problems are selected due to their technological significance as well as their ability to demonstrate the attributes of the Bayesian computational approach. The developed methodologies are general and applicable to many other inverse continuum problems. A summary of achievements and suggestions for future research are given at the end of the thesis. |
