Quantifying uncertainties in a quantity of interest (QoI), arising from space-time evolution of a non-deterministic physical process, is important for several applications. The stochastic physical process is typically modeled by a linear or nonlinear partial differential equation (PDE) in a stationary or uncertain time-dependent domain, with several uncertain input parameters governing the PDE and an initial state that induces the evolutionary process. Simulations of statistical moments of the QoI play a crucial role in uncertainty quantification (UQ) of the stochastic model. The UQ process requires efficient approximations, fully-discrete algebraic system modeling of the PDE, and computer implementation of the physical process in space, time, and high-dimensional stochastic variables.

Developing algorithms for efficient simulations of deterministic moving-domain linear PDE models and three-dimensional nonlinear space-time models is still an active area of research. Further augmenting such algorithmic complexities, with high-stochastic-dimensional approximations, leads to large scale scientific computing models. Practical realizations of these models may become computationally prohibitive using standard low-order methods, such as Monte Carlo (MC). In general, large scale space-time stochastic models require developing efficient high-order algorithms, and high-level language implementation using high performance computing (HPC) techniques. In practical terms, simulations of such parallel scientific computing models on HPC environments require substantial power consumption. Reducing the power use for simulating large scale models also requires future research on the application of HPC environment machine learning and hence power adaptive techniques.

The main focus of this thesis is on the development and HPC implementation of algorithms to address such stochastic-space-time computational challenges, for a class of deterministic and stochastic models on time-dependent and stationary domains, governed by the Schrodinger and nonlinear Allen-Cahn PDEs with high-dimensional uncertainties. Novel contributions in this thesis include: (1) A moving-domain finite element method (MD-FEM), applying the MD-FEM in conjunction with an adaptive multilevel MC (MLMC) algorithm, and parallel implementation. We developed the MD-FEM-MLMC algorithm and simulation for a diffraction-in-time stochastic model, governed by the Schrodinger equation on a time-dependent domain, with nondeterministic domain size and an initial state modeled by a random field. (2) High-order quasi Monte Carlo (QMC) stochastic approximations and adaptive MLQMC algorithms, applied in conjunction with FEM approximations, for simulating statistical moments of a QoI induced by a stochastic order parameter phase separation field. The field is modeled by a nonlinear two-/three-space dimensional Allen-Cahn PDE with random gradient energy and an uncertain initial state induced by a random field. (3) Preliminary investigation and application of a class of machine learning techniques for reduction of HPC power consumption, applied in conjunction with several order QMC approximations.

A large number of numerical experiments in the thesis, for the above stochastic-space-time dimensional models, demonstrate the efficiency of our adaptive hybrid FEM and stochastic sampling approximations, parallel performance, and practical realizations of such large stochastic dimensional moving domain and nonlinear models.