Gaussian processes (GPs) are widely used in non-parametric Bayesian modeling, and play an important role in various statistical and machine learning applications. In a variety tasks of uncertainty quantification, generating random sample paths of GPs is of interest. As GP sampling requires generating high-dimensional Gaussian random vectors, it is computationally challenging if a direct method, such as the Cholesky decomposition, is used. In this paper, we propose a scalable algorithm for sampling random realizations of the prior and posterior of GP models. The proposed algorithm leverages inducing points approximation with sparse grids, as well as additive Schwarz preconditioners, which reduce computational complexity, and ensure fast convergence. We demonstrate the efficacy and accuracy of the proposed method through a series of experiments and comparisons with other recent works.

翻译：高斯过程（GPs）广泛应用于非参数贝叶斯建模，并在各种统计与机器学习应用中扮演重要角色。在不确定性量化的诸多任务中，生成高斯过程的随机样本路径具有重要意义。由于高斯过程采样需要生成高维高斯随机向量，若采用直接法（如Cholesky分解），计算上将面临挑战。本文提出一种可扩展的算法，用于生成高斯过程模型先验与后验的随机实现。该算法结合了稀疏网格的诱导点近似与加性施瓦茨预条件子，既降低了计算复杂度，又保证了快速收敛。通过一系列实验及与近期其他工作的对比，我们验证了所提方法的有效性与准确性。