Probabilistic and data-driven methods for numerical PDEs

This dissertation consists of three integral self-contained parts. The first part develops a novel Monte Carlo algorithm, called the near-Maximal Algorithm for Poisson-disk Sampling (nMAPS), to efficiently generate the nodes of a high-quality mesh for the calculation of flow and the associated transport of chemical species in low-permeability fractured rock, such as shale and granite. A good mesh balances accuracy requirements with a reasonable computational cost, i.e., it is generated efficiently, dense where necessary for accuracy, and contains no cells that cause instabilities or blown-up errors. Quality bounds for meshes generated through nMAPS are proven, and its efficiency is demonstrated through numerical experiments. In the second part, a deterministic Monte Carlo hybrid method for time-dependent problems based on the physics of particle transport described through the linear Boltzmann equation is presented. The method splits the system into collided and uncollided particles and treats these sets with different methods. Uncollided particles are handled through high-accuracy Monte Carlo methods, while the density of collided particles is calculated using discontinuous Galerkin methods. Theoretical details of the algorithm are developed and shown to be effective through numerical experiments. The properties associated with the labeling as collided and uncollided leverage the respective strengths of these methods, allowing for overall more accurate and computationally efficient solving than each method on its own. In the last chapter, an extension to the Dynamic Likelihood Filter (DLF) is presented to include Advection-Diffusion equations. The DLF is a Bayesian estimation method specifically designed for wave-related problems. It improves on traditional methods, such as variants of Kalman filters, by not only using data at its time of observation but also at later times by propagating observations forward through time. This enriches the available data and improves predictions and uncertainties. The theory to include diffusion in the framework of the DLF is developed, and it is shown through numerical experiments that the DLF outperforms traditional data assimilation techniques, especially when observations are precise but sparse in space and time.