Algorithms and Scientific Software for Quasi-Monte Carlo, Fast Gaussian Process Regression, and Scientific Machine Learning

TL;DR

This paper presents new algorithms and open-source software across three scientific computing domains: Quasi-Monte Carlo methods, Gaussian process regression, and scientific machine learning for PDEs, advancing efficiency and scalability.

Contribution

It introduces novel algorithms and software tools for high-dimensional integration, Gaussian process modeling, and PDE-based machine learning, unifying these developments in a comprehensive framework.

Findings

Developed QMCPy for efficient QMC error estimation and variable transforms.

Created FastGPs for scalable Gaussian process regression with higher-order kernels.

Designed a PDE solver capable of machine precision recovery with random coefficients.

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.