三类科学计算方法与开源工具，提升高维积分、插值与偏微分方程求解效率。
Algorithms and Scientific Software for Quasi-Monte Carlo, Fast Gaussian Process Regression, and Scientific Machine Learning
- 提出高阶光滑核函数与快速多任务高斯过程算法
- 实现机器精度的随机系数偏微分方程求解，误差可自适应估计
- 开源工具支持高维积分、不确定性量化与科学建模，适合科研开发者
本论文统一了三类科学计算方向的算法与软件开发：用于高效高维积分的准蒙特卡洛（QMC）方法、具备不确定性量化能力的高维插值高斯过程（GP）回归，以及基于无网格求解器建模偏微分方程（PDE）的科学机器学习（sciML）。针对QMC，构建了向量化误差估计算法，并开发了开源工具QMCPy，支持随机低差异序列生成、变量变换、自适应误差估计及多种应用场景。针对GP，推导出更高阶光滑性的数字平移不变核，提出新型快速多任务GP算法，发布可扩展的FastGPs软件。针对sciML，提出新算法可在机器精度下恢复含随机系数的PDE解。应用包括概率失效估计、达西流方程的多层级GP、辐射传输的神经代理模型，以及贝叶斯多层级QMC中的快速GP。
原文摘要 · Abstract (English)
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.
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