基于层次矩阵的可扩展物理的最大似然估计

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Yian Chen, M. Anitescu
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引用次数: 6

摘要

在高斯过程分析中,基于物理的协方差模型为构建符合基本物理规律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极大似然估计进行估计,但直接构建协方差矩阵和使用协方差矩阵计算的经典策略需要$n$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,我们建议使用层次矩阵来近似离散协方差函数。通过对单个非对角线块使用随机范围草图,分层协方差近似的构建过程需要$O(\log{n})$物理模型应用,最大似然计算需要$O(n\log^2{n})$每次迭代的努力。我们提出了一种精确计算层次矩阵乘积轨迹的新方法,使得期望的Fischer信息矩阵在$O(n\log^2{n})$中也是可计算的。这种构造是完全无矩阵的,通过微分整个过程可以在同一层次结构中逼近协方差矩阵的导数。数值结果证明了该方法在参数估计和不确定度量化方面的有效性、准确性和高效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Scalable Physics-based Maximum Likelihood Estimation using Hierarchical Matrices
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.
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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