Algorithms and Scientific Software for Quasi-Monte Carlo, Fast Gaussian Process Regression, and Scientific Machine Learning
It addresses computational challenges in scientific domains like uncertainty quantification and PDE solving, but is incremental as it builds on existing QMC, GP, and sciML frameworks.
This thesis tackled the development of efficient algorithms and software for high-dimensional integration, interpolation, and PDE modeling, resulting in new methods like vectorized error estimation for QMC, fast multitask GP algorithms, and machine precision recovery for PDEs with random coefficients.
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.