Scalable Physics-Based Maximum Likelihood Estimation Using Hierarchical Matrices
Yian Chen, Mihai Anitescu · SIAM/ASA Journal on Uncertainty Quantification · 2023
Abstract. 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 require [Formula: see text] physical model runs, [Formula: see text] storage complexity, and [Formula: see text] 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 [Formula: see text] physical model applications and the maximum likelihood computations require [Formula: see text] effort per iteration. We propose a new approach to compute exactly the trace of products of hierarchical matrices which results in the expected Fisher information matrix being computable in [Formula: see text] 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.