Physics-based covariance models provide a systematic way to construct covariance models that are consistent with the underlying physical laws in Gaussian process analysis. The unknown parameters in the covariance models can be estimated using maximum likelihood estimation, but direct construction of the covariance matrix and classical strategies of computing with it requires $n$ physical model runs, $n^2$ storage complexity, and $n^3$ computational complexity. To address such challenges, we propose to approximate the discretized covariance function using hierarchical matrices. By utilizing randomized range sketching for individual off-diagonal blocks, the construction process of the hierarchical covariance approximation requires $O(\log{n})$ physical model applications and the maximum likelihood computations require $O(n\log^2{n})$ effort per iteration. We propose a new approach to compute exactly the trace of products of hierarchical matrices which results in the expected Fischer information matrix being computable in $O(n\log^2{n})$ as well. The construction is totally matrix-free and the derivatives of the covariance matrix can then be approximated in the same hierarchical structure by differentiating the whole process. Numerical results are provided to demonstrate the effectiveness, accuracy, and efficiency of the proposed method for parameter estimations and uncertainty quantification.

翻译：基于物理的协方差模型提供了一种系统化的方法，可以在高斯过程分析中构建与底层物理定律一致的协方差模型。协方差模型中的未知参数可通过最大似然估计进行求解，但直接构造协方差矩阵并使用经典计算策略需要$n$次物理模型运行、$n^2$的存储复杂度以及$n^3$的计算复杂度。为应对这些挑战，我们提出使用分层矩阵来近似离散化的协方差函数。通过利用随机化范围草图对非对角块进行近似，分层协方差近似的构建过程仅需$O(\log{n})$次物理模型应用，而每次迭代中的最大似然计算只需$O(n\log^2{n})$的计算量。我们提出了一种新方法，能够精确计算分层矩阵乘积的迹，从而使期望的Fisher信息矩阵也能在$O(n\log^2{n})$内完成计算。该构建过程完全无需显式矩阵存储，且协方差矩阵的导数可通过差异化整个流程在相同分层结构中进行近似。数值结果展示了所提方法在参数估计和不确定性量化中的有效性、准确性和高效性。