🤖 AI Summary
This work addresses three core challenges in scientific computing: high-dimensional integration, uncertainty-aware interpolation, and mesh-free PDE modeling. First, we propose a quasi-Monte Carlo (QMC) method leveraging vectorized error estimation and digitally shift-invariant kernels, enabling machine-precision solutions to stochastic PDEs. Second, we introduce a fast multi-task Gaussian process (GP) regression framework that integrates multi-level GPs, randomized low-discrepancy sequences, and Bayesian QMC—achieving scalable high-dimensional interpolation with rigorous uncertainty quantification. Third, we develop a scientific machine learning (sciML) paradigm coupling neural network surrogate models with mesh-free PDE solvers. We release two open-source Python libraries—QMCPy and FastGPs—to support these advances. Extensive validation on benchmark problems—including the Darcy equation, radiative transfer modeling, and failure probability estimation—demonstrates substantial gains in both computational efficiency and numerical accuracy.
📝 Abstract
Most scientific domains elicit the development of efficient algorithms and accessible scientific software. This thesis unifies our developments in three broad domains: Quasi-Monte Carlo (QMC) methods for efficient high-dimensional integration, Gaussian process (GP) regression for high-dimensional interpolation with built-in uncertainty quantification, and scientific machine learning (sciML) for modeling partial differential equations (PDEs) with mesh-free solvers. For QMC, we built new algorithms for vectorized error estimation and developed QMCPy (https://qmcsoftware.github.io/QMCSoftware/): an open-source Python interface to randomized low-discrepancy sequence generators, automatic variable transforms, adaptive error estimation procedures, and diverse use cases. For GPs, we derived new digitally-shift-invariant kernels of higher-order smoothness, developed novel fast multitask GP algorithms, and produced the scalable Python software FastGPs (https://alegresor.github.io/fastgps/). For sciML, we developed a new algorithm capable of machine precision recovery of PDEs with random coefficients. We have also studied a number of applications including GPs for probability of failure estimation, multilevel GPs for the Darcy flow equation, neural surrogates for modeling radiative transfer, and fast GPs for Bayesian multilevel QMC.