标题： 基于层次矩阵的可扩展物理最大似然估计 显示英文标题

标题： Scalable Physics-based Maximum Likelihood Estimation using Hierarchical Matrices

摘要： 基于物理的协方差模型为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提供了一种系统的方法。 协方差模型中的未知参数可以使用最大似然估计进行估计，但直接构造协方差矩阵以及使用它的经典策略需要 $n$次物理模型运行，$n^2$存储复杂度，和$n^3$计算复杂度。 为了解决这些挑战，我们提出使用分层矩阵来近似离散化的协方差函数。 通过利用随机范围压缩对每个非对角块进行处理，分层协方差近似的构造过程需要$O(\log{n})$次物理模型应用，而最大似然计算每次迭代需要$O(n\log^2{n})$的工作量。 我们提出一种新方法来精确计算分层矩阵乘积的迹，这使得期望的费舍尔信息矩阵可以在$O(n\log^2{n})$内计算出来。 该构造完全是无矩阵的，然后可以通过对整个过程求导，在相同的分层结构中近似协方差矩阵的导数。 数值结果用于展示所提出方法在参数估计和不确定性量化方面的有效性、准确性和效率。 显示英文摘要

摘要： 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.