Parameter and State Model Reduction for Bayesian Statistical Inverse Problems

Decisions based on single-point estimates of uncertain parameters neglect regions of significant probability. We consider a paradigm based on decision-making under uncertainty including three steps: identification of parametric probability by solution of the statistical inverse problem, propagation of that uncertainty through complex models, and solution of the resulting stochastic or robust mathematical programs. In this thesis we consider the first of these steps, solution of the statistical inverse problem, for partial differential equations (PDEs) parameterized by field quantities. When these field variables and forward models are discretized, the resulting system is high-dimensional in both parameter and state space. The system is therefore ex-pensive to solve. The statistical inverse problem is one of Bayesian inference. With assumption on prior belief about the form of the parameter and an assignment of normal error in sensor measurements, we derive the solution to the statistical inverse problem analytically, up to a constant of proportionality. The parametric probabilit